Éléments de géométrie algébrique

A. Grothendieck and J. DieudonnéPublications mathématiques de l’I.H.É.S

ContributorsTim Hosgood, Ryan Keleti

autobuild :: 2020-02-12 17:00 UTC, git hash :: 1e16140

Contents

What this is 7In defense of a translation 7Notes from the translators 8Mathematical warnings 8

Introduction 9

Chapter 0. Preliminaries (EGA 0) 131. Rings of fractions 13

1.0. Rings and Algebras 131.1. Radical of an ideal. Nilradical and radical of a ring 141.2. Modules and rings of fractions 141.3. Functorial properties 151.4. Change of multiplicative subset 161.5. Change of ring 171.6. Identification of the module M f as an inductive limit 181.7. Support of a module 19

2. Irreducible spaces. Noetherian spaces 192.1. Irreducible spaces 192.2. Noetherian spaces 20

3. Supplement on sheaves 213.1. Sheaves with values in a category 213.2. Presheaves on an open basis 223.3. Gluing sheaves 233.4. Direct images of presheaves 243.5. Inverse images of presheaves 253.6. Simple and locally simple sheaves 273.7. Inverse images of presheaves of groups or rings 273.8. Sheaves on pseudo-discrete spaces 27

4. Ringed spaces 284.1. Ringed spaces, sheaves of A -modules, A -algebras 284.2. Direct image of an A -module 304.3. Inverse image of an A -module 314.4. Relation between direct and inverse images 32

5. Quasi-coherent and coherent sheaves 335.1. Quasi-coherent sheaves 335.2. Sheaves of finite type 345.3. Coherent sheaves 355.4. Locally free sheaves 365.5. Sheaves on a locally ringed space 39

6. Flatness 406.1. Flat modules 406.2. Change of ring 406.3. Local nature of flatness 416.4. Faithfully flat modules 416.5. Restriction of scalars 426.6. Faithfully flat rings 426.7. Flat morphisms of ringed spaces 43

2

CONTENTS 3

7. Adic rings 437.1. Admissible rings 437.2. Adic rings and projective limits 457.3. Preadic Noetherian rings 477.4. Quasi-finite modules over local rings 487.5. Rings of restricted formal series 497.6. Completed rings of fractions 517.7. Completed tensor products 537.8. Topologies on modules of homomorphisms 54

8. Representable functors 558.1. Representable functors 558.2. Algebraic structures in categories 57

9. Constructible sets 599.1. Constructible sets 599.2. Constructible subsets of Noetherian spaces 619.3. Constructible functions 62

10. Supplement on flat modules 6310.1. Relations between flat modules and free modules 6310.2. Local flatness criteria 6310.3. Existence of flat extensions of local rings 65

11. Supplement on homological algebra 6611.1. Review of spectral sequences 6611.2. The spectral sequence of a filtered complex 6911.3. The spectral sequences of a bicomplex 70

12. Supplement on sheaf cohomology 7113. Projective limits in homological algebra 7114. Combinatorial dimension of a topological space 71

14.1. Combinatorial dimension of a topological space 7114.2. Codimension of a closed subset 7314.3. The chain condition 75

Chapter I. The language of schemes (EGA I) 77Summary 771. Ane schemes 77

1.1. The prime spectrum of a ring 771.2. Functorial properties of prime spectra of rings 801.3. Sheaf associated to a module 811.4. Quasi-coherent sheaves over a prime spectrum 851.5. Coherent sheaves over a prime spectrum 861.6. Functorial properties of quasi-coherent sheaves over a prime spectrum 871.7. Characterization of morphisms of ane schemes 881.8. Morphisms from locally ringed spaces to ane schemes 89

2. Preschemes and morphisms of preschemes 922.1. Definition of preschemes 922.2. Morphisms of preschemes 922.3. Gluing preschemes 942.4. Local schemes 942.5. Preschemes over a prescheme 95

3. Products of preschemes 963.1. Sums of preschemes 963.2. Products of preschemes 963.3. Formal properties of the product; change of the base prescheme 993.4. Points of a prescheme with values in a prescheme; geometric points 1013.5. Surjections and injections 1033.6. Fibres 1053.7. Application: reduction of a prescheme mod. J 106

4. Subpreschemes and immersion morphisms 106

CONTENTS 4

4.1. Subpreschemes 1064.2. Immersion morphisms 1084.3. Products of immersions 1104.4. Inverse images of a subprescheme 1104.5. Local immersions and local isomorphisms 111

5. Reduced preschemes; separation condition 1125.1. Reduced preschemes 1125.2. Existence of a subprescheme with a given underlying space 1145.3. Diagonal; graph of a morphism 1155.4. Separated morphisms and separated preschemes 1175.5. Separation criteria 118

6. Finiteness conditions 1216.1. Noetherian and locally Noetherian preschemes 1216.2. Artinian preschemes 1236.3. Morphisms of finite type 1236.4. Algebraic preschemes 1266.5. Local determination of a morphism 1276.6. Quasi-compact morphisms and morphisms locally of finite type 129

7. Rational maps 1317.1. Rational maps and rational functions 1317.2. Domain of definition of a rational map 1337.3. Sheaf of rational functions 1357.4. Torsion sheaves and torsion-free sheaves 136

8. Chevalley schemes 1378.1. Allied local rings 1378.2. Local rings of an integral scheme 1388.3. Chevalley schemes 139

9. Supplement on quasi-coherent sheaves 1409.1. Tensor product of quasi-coherent sheaves 1409.2. Direct image of a quasi-coherent sheaf 1429.3. Extension of sections of quasi-coherent sheaves 1429.4. Extension of quasi-coherent sheaves 1439.5. Closed image of a prescheme; closure of a subprescheme 1459.6. Quasi-coherent sheaves of algebras; change of structure sheaf 147

10. Formal schemes 14810.1. Formal ane schemes 14810.2. Morphisms of formal ane schemes 14910.3. Ideals of definition for a formal ane scheme 15010.4. Formal preschemes and morphisms of formal preschemes 15110.5. Sheaves of ideals of definition for formal preschemes 15210.6. Formal preschemes as inductive limits of preschemes 15310.7. Products of formal preschemes 15610.8. Formal completion of a prescheme along a closed subset 15710.9. Extension of morphisms to completions 16010.10. Application to coherent sheaves on formal ane schemes 16110.11. Coherent sheaves on formal preschemes 16310.12. Adic morphisms of formal preschemes 16510.13. Morphisms of finite type 16610.14. Closed subpreschemes of formal preschemes 16710.15. Separated formal preschemes 169

Chapter II. Elementary global study of some classes of morphisms (EGA II) 171Summary 1711. Ane morphisms 171

1.1. S-preschemes and OS -algebras 1711.2. Ane preschemes over a prescheme 1721.3. Ane preschemes over S associated to an OS -algebra 173

CONTENTS 5

1.4. Quasi-coherent sheaves over a prescheme ane over S 1741.5. Change of base prescheme 1751.6. Ane morphisms 1771.7. Vector bundle associated to a sheaf of modules 177

2. Homogeneous prime spectra 1802.1. Generalities on graded rings and modules 1802.2. Rings of fractions of a graded ring 1832.3. Homogeneous prime spectrum of a graded ring 184

3. Homogeneous spectrum of a sheaf of graded algebras 1843.1. Homogeneous spectrum of a quasi-coherent graded OY -algebra 184

4. Projective bundles; ample sheaves 1844.1. Definition of projective bundles 184

5. Quasi-ane morphisms; quasi-projective morphisms; proper morphisms; projective morphisms 1855.1. Quasi-ane morphisms 1855.2. Serre’s criterion 1875.3. Quasi-projective morphisms 1885.4. Proper morphisms and universally closed morphisms 1895.5. Projective morphisms 1915.6. Chow’s lemma 193

6. Integral morphisms and finite morphisms 1956.1. Preschemes integral over another prescheme 195

7. Valuative criteria 1957.1. Reminder on valuation rings 195

8. Blowup schemes; based cones; projective closure 1968.1. Blowup preschemes 1968.2. Preliminary results on the localisation of graded rings 1988.3. Based cones 2028.4. Projective closure of a vector bundle 2068.5. Functorial behaviour 2078.6. A canonical isomorphism for pointed cones 2088.7. Blowing up based cones 2108.8. Ample sheaves and contractions 2128.9. Grauert’s ampleness criterion: statement 2158.10. Grauert’s ampleness criterion: proof 2178.11. Uniqueness of contractions 2178.12. Quasi-coherent sheaves on based cones 217

Chapter III. Cohomological study of coherent sheaves (EGA III) 218Summary 2181. Cohomology of ane schemes 218

1.1. Review of the exterior algebra complex 2181.2. Čech cohomology of an open cover 2211.3. Cohomology of an ane scheme 2221.4. Application to the cohomology of arbitrary preschemes 223

2. Cohomological study of projective morphisms 2272.1. Explicit calculations of certain cohomology groups 227

3. Finiteness theorem for proper morphisms 2273.1. The dévissage lemma 2273.2. The finiteness theorem: the case of usual schemes 2283.3. Generalization of the finiteness theorem (usual schemes) 2293.4. Finiteness theorem: the case of formal schemes 229

4. The fundamental theorem of proper morphisms. Applications 2314.1. The fundamental theorem 231

5. An existence theorem for coherent algebraic sheaves 2325.1. Statement of the theorem 232

6. Local and global Tor functors; Künneth formula 2326.1. Introduction 232

CONTENTS 6

7. Base change for homological functors of sheaves of modules 2327.1. Functors of A-modules 232

Chapter IV. Local study of schemes and their morphisms (EGA IV) 233Summary 2331. Relative finiteness conditions. Constructible sets of preschemes 234

1.1. Quasi-compact morphisms 234

Bibliography 235

What this is

This whole chapter is written by the translators.

This is a community translation of Alexander Grothendieck’s and Jean Dieudonné’s Éléments de géométriealgébrique (EGA). As it is a work in progress by multiple people, there will probably be a few mistakes—if youspot any then please do let us know1.To contribute, please visit

https://github.com/ryankeleti/ega.On est désolés, Grothendieck.

In defense of a translation

From Wikipedia2:

In January 2010, Grothendieck wrote the letter “Déclaration d’intention de non-publication”to Luc Illusie, claiming that all materials published in his absence have been publishedwithout his permission. He asks that none of his work be reproduced in whole or in part andthat copies of this work be removed from libraries.3 [...] This order may have been reversedlater in 2010.4

It is a matter of often heated contention as to whether or not any translation of Grothendieck’s work shouldtake place, given his extremely explicit views on the matter. By no means do we mean to argue that somehowGrothendieck’s wishes should be invalidated or ignored, nor do we wish to somehow twist his earlier wordsaround in order to justify what we have done: we fully accept that he himself would probably have branded thisproject “an abomination”. With this in mind, it remains to explain why we have gone ahead anyway.

First, and possibly foremost, it does not make sense (to us) for an individual to own the rights to knowledge.Arguments can be made about how the EGA is the product of years and years of intense work by Grothendieck,and so this is something that he ‘owns’ and has full control over. Indeed, it is true that there are almostinnumerably many sentences in these works that only Grothendieck himself could have engineered, but, intranslation, we have never improved anything, but only (regrettably, but almost certainly) worsened. The work inthese pages is that of Grothendieck; we have been not much more than typesetters and eager readers. However,there is some important point to be made about the fact that Grothendieck collaborated and worked withmany other incredibly proficient mathematicians during the writing of this treatise; although it is impossible topinpoint which parts exactly others may have contributed (and by no means do we wish to imply that any ofthis work is derivative or fraudulent in any way whatsoever—EGA was written by Grothendieck) it seems fair that,in some amount, there are bits of the EGAs that ‘belong’ to a broader collection of minds.

It is a very good idea here to repeat the oft-quoted aphorism: “the work here is not ours, but any mistakesare”—it is very understandable for an author to not want their name on something that they have not themselveswritten, or, at the very least, read. This may be, in part, a reason for Grothendieck’s wishes, but that is purespeculation. Even so, we include this above disclaimer.

Secondly, then, we note that the French version of EGA is still entirely readily accessible. Anybody readingthese copies who is not a native French speaker, will probably be translating at least some part of EGA intoEnglish in their head, or into their notebooks, as they read. This document is just the product of a few peopledoing exactly that, but then passing on their eorts to make things just that little bit easier for anyone else whofollows.

1https://github.com/ryankeleti/ega/issues2https://en.wikipedia.org/wiki/Alexander_Grothendieck#Retirement_into_reclusion_and_death3Grothendieck’s letter. Secret Blogging Seminar. 9 February 2010. Retrieved 3 September 2019.4Réédition des SGA. Archived from the original on 29 June 2016. Retrieved 12 November 2013.

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MATHEMATICAL WARNINGS 8

Lastly, to quote another adage, “the guilty person is often the loudest”. If it seems like we are over-eager todefend ourselves because we know that we are somehow in the wrong, it is because we are, at least partially.Working on this translation has meant going against Grothendieck’s explicit requests, and for that we are sorry.We only hope that the freedom of knowledge is an excusable defense.

Notes from the translators

Grothendieck’s writing style in EGA is quite particular, most notably for its long sentence structure. Astranslators, we have tried to give the best possible approximation of this style in English, resisting the temptationto “streamline” things in places where the language is more dense than usual.

***Any translations about which we are not entirely sure will be marked with a (?).

***Whenever a note is made by the translators, it will be prefaced by “[Trans.]”.

***Along the margins we have provided the page numbers corresponding to the original text, as published by

Publications mathématiques de l’I.H.É.S., where the EGA were published as the following volumes:5

– EGA I (tome 4, 1960)– EGA II (tome 8, 1961)– EGA III, part 1 (tome 11, 1961)– EGA III, part 2 (tome 17, 1963)– EGA IV, part 1 (tome 20, 1964)– EGA IV, part 2 (tome 24, 1965 )– EGA IV, part 3 (tome 28, 1966 )– EGA IV, part 4 (tome 32, 1967 ).

Due to EGA being a collection of volumes (one non-preliminary chapter, or part of a chapter, per volume),the page numbers reset at every new chapter. In addition, the preliminary section is stretched out over multiplevolumes. To combat this, we label the pages as

X | p,referring to Chapter X, page p. For EGA III and IV, which are split across multiple chapters, we label the pagesas

X-n | p,referring to Chapter X, part n, page p. In the case of the preliminaries (which are often collectively referred toas EGA 0), the preliminaries from volume Y are denoted as 0Y.

***Later volumes (EGA II, III, and IV) include errata for earlier chapters. Where possible, we have used these

to ‘update’ our translation, and entirely replace whatever mistakes might have been in the original copies ofEGA I and II. If the change is minor (e.g. ‘intersection’ replacing ‘inter-section’) then we will not mention it; if itis anything more fundamental (e.g. X ′ replacing X ) then we will include some margin note on the relevant linedetailing the location of the erratum (e.g. ErrII to denote that the correction is listed in the Errata section ofEGA II).

Mathematical warnings

EGA uses prescheme for what is now usually called a scheme, and scheme for what is now usually called aseparated scheme.

In some cases, we (the translators) have changed “→” to “ 7→” where appropriate.

5PDFs of which can be found online, hosted by the Grothendieck circle.

Introduction

To Oscar Zariski and André Weil.

I | 5This memoir, and the many others will undoubtedly follow, are intended to form a treatise on the foundations

of algebraic geometry. They do not, in principle, presume any particular knowledge of the subject, and it haseven been recognised that such knowledge, despite its obvious advantages, could sometimes (because of themuch-too-narrow interpretation—through the birational point of view—that it usually implies) be a hindranceto the one who wants to become familiar with the point of view and techniques presented here. However, weassume that the reader has a good knowledge of the following topics:

(a) Commutative algebra, as it is laid out, for example, in the volumes (in progress of being written) of theÉléments of N. Bourbaki (and, pending the publication of these volumes, in Samuel–Zariski [SZ60]and Samuel [Sam53b, Sam53a]).

(b) Homological algebra, for which we refer to Cartan–Eilenberg [CE56] (cited as (M)) and Godement [God58](cited as (G)), as well as the recent article by A. Grothendieck [Gro57] (cited as (T)).

(c) Sheaf theory, where our main references will be (G) and (T); this theory provides the essential languagefor interpreting, in “geometric” terms, the essential notions of commutative algebra, and for “globalizing”them.

(d) Finally, it will be useful for the reader to have some familiarity with functorial language, which will beconstantly used in this treatise, and for which the reader may consult (M), (G), and especially (T); theprinciples of this language and the main results of the general theory of functors will be described inmore detail in a book currently in preparation by the authors of this treatise.

***

It is not the place, in this introduction, to give a more or less summary description from the “schemes”point of view in algebraic geometry, nor the long list of reasons which made its adoption necessary, and inparticular the systematic acceptance of nilpotent elements in the local rings of “manifolds” that we consider(which necessarily shifts the idea of rational maps into the background, in favor of those of regular maps or“morphisms”). To be precise, this treatise aims to systematically develop the language of schemes, and willdemonstrate, we hope, its necessity. Although it would be easy to do so, we will not try to give here an “intuitive” I | 6introduction to the notions developed in Chapter I. For the reader who would like to have a glimpse of thepreliminary study of the subject matter of this treatise, we refer them to the conference by A. Grothendieck atthe International Congress of Mathematicians in Edinburgh in 1958 [Gro58], and the exposé [Gro] of the sameauthor. The work [Ser55a] (cited as (FAC)) of J.-P. Serre can also be considered as an intermediary expositionbetween the classical point of view and the schemes point of view in algebraic geometry, and, as such, its readingmay be an excellent preparation for the reading of our Éléments.

***

We give below the general outline planned for this treatise, subject to later modifications, especiallyconcerning the later chapters.

9

INTRODUCTION 10

Chapter I. — The language of schemes.— II. — Elementary global study of some classes of morphisms.— III. — Cohomology of algebraic coherent sheaves. Applications.— IV. — Local study of morphisms.— V. — Elementary procedures of constructing schemes.— VI. — Descent. General method of constructing schemes.— VII. — Group schemes, principal fibre bundles.— VIII. — Dierential study of fibre bundles.— IX. — The fundamental group.— X. — Residues and duality.— XI. — Theories of intersection, Chern classes, Riemann–Roch theorem.— XII. — Abelian schemes and Picard schemes.— XIII. — Weil cohomology.

In principle, all chapters are considered open to changes, and supplementary sections could always be addedlater; such sections would appear in separate fascicles in order to minimize the inconveniences accompanyingwhatever mode of publication adopted. When the writing of such a section is foreseen or in progress duringthe publication of a chapter, it will be mentioned in the summary of the chapter in question, even if, owing tocertain orders of urgency, its actual publication clearly ought to have been later. For the convenience of thereader, we give in “Chapter 0” the necessary tools in commutative algebra, homological algebra, and sheaftheory, that will be used throughout this treatise, that are more or less well known but for which it was notpossible to give convenient references. It is recommended for the reader to not read Chapter 0 except whilstreading the actual treatise, when the results to which we refer seem unfamiliar. Besides, we think that in this way, I | 7the reading of this treatise could be a good method for the beginner to familiarize themselves with commutativealgebra and homological algebra, whose study, when not accompanied with tangible applications, is consideredtedious, or even depressing, by many.

***

It is outside of our capabilities to give a historic overview, or even a summary thereof, of the ideas and resultsdescribed herein. The text will contain only those references considered particularly useful for comprehension,and we indicate the origin of only the most important results. Formally, at least, the subjects discussed in ourwork are reasonably new, which explains the scarcity of references made to the fathers of algebraic geometryfrom the 19th to the beginning of the 20th century, whose works we know only by hear-say. It is suitable, however,to say some words here about the works which have most directly influenced the authors and contributed tothe development of scheme-theoretic point of view. We absolutely must mention the fundamental work (FAC)of J.-P. Serre first, which has served as an introduction to algebraic geometry for more that one young student(the author of this treatise being one), deterred by the dryness of the classic Foundations of A. Weil [Wei46].It is there that it is shown, for the first time, that the “Zariski topology” of an “abstract” algebraic variety isperfectly suited to applying certain techniques from algebraic topology, and notably to be able to define acohomology theory. Further, the definition of an algebraic variety given therein is that which translates mostnaturally to the idea that we develop here6. Serre himself had incidentally noted that the cohomology theory ofane algebraic varieties could be translated without diculty by replacing the ane algebras over a field byarbitrary commutative rings. Chapters I and II of this treatise, and the first two paragraphs of Chapter III, canthus be considered, for the most part, as easy translations, to this bigger framework, of the principal results of(FAC) and a later article of the same author [Ser57]. We have also vastly profited from the Séminaire de géométriealgébrique de C. Chevalley [CC]; in particular, the systematic usage of “constructible sets” introduced by him hasturned out to be extremely useful in the theory of schemes (cf. Chapter IV). We have also borrowed from himthe study of morphisms from the point of view of dimension (Chapter IV), that translates with negligible change I | 8to the framework of schemes. It also merits noting that the idea of “schemes of local rings”, introduced byChevalley, naturally lends itself to being extended to algebraic geometry (not having, however, all the flexibility

6Just as J.-P. Serre informed us, it is right to note that the idea of defining the structure of a manifold by the data of a sheaf of rings isdue to H. Cartan, who took this idea as the starting point of his theory of analytic spaces. Of course, just as in algebraic geometry, it wouldbe important in “analytic geometry” to give the allow the use of nilpotent elements in local rings of analytic spaces. This extension of thedefinition of H. Cartan and J.-P. Serre has recently been broached by H. Grauert [Gra60], and there is room to hope that a systematicreport of analytic geometry in this setting will soon see the light of day. It is also evident that the ideas and techniques developed in thistreatise retain a sense of analytic geometry, even though one must expect more considerable technical diculties in this latter theory. Wecan foresee that algebraic geometry, by the simplicity of its methods, will be able to serve as a sort of formal model for future developmentsin the theory of analytic spaces.

INTRODUCTION 11

and generality that we intend to give it here); for the connections between this idea and our theory, see Chapter I,§8. One such extension has been developed by M. Nagata in a series of memoirs [Nag58a], which contain manyspecial results concerning algebraic geometry over Dedekind rings7.

***

It goes without saying that a book on algebraic geometry, and especially a book dealing with the fundamentals,is of course influenced, if only by proxy, by mathematicians such as O. Zariski and A. Weil. In particular, theThéorie des fonctions holomorphes by Zariski [Zar51], reasonably flexible thanks to the cohomological methods andan existence theorem (Chapter III, §§4 and 5), is (along with the method of descent described in Chapter VI)one of the principal tools used in this treatise, and it seems to us one of the most powerful at our disposal inalgebraic geometry.

The general technique in which it is employed can be sketched as follows (a typical example of which willbe given in Chapter XI, in the study of the fundamental group). We have a proper morphism (Chapter II)f : X → Y of algebraic varieties (or, more generally, of schemes) that we wish to study on the neighborhood of apoint y ∈ Y , with the aim of resolving a problem P relative to a neighborhood of y . We proceed step by step:

1st We can suppose that Y is ane, so that X becomes a scheme defined on the ane ring A of Y , and wecan even replace A by the local ring of y . This reduction is always easy in practice (Chapter V) andbrings us to the case where A is a local ring.

2nd We study the problem in question when A is a local Artinian ring. So that the problem P still makessense when A is not assumed to be integral, we sometimes have to reformulate P, and it appears thatwe often obtain a better understanding of the problem in doing so, in an “infinitesimal” way.

3rd The theory of formal schemes (Chapter III, §§3, 4, and 5) lets us pass from the case of an Artinianring to a complete local ring.

4th Finally, if A is an arbitrary local ring, considering “multiform (?) sections” of suitable schemes over X ,approximating a given “formal” section (Chapter IV), will let us pass, by extension of scalars, to the I | 9completion of A, from a known result (about the schemed induced by X by extension of scalars to thecompletion of A) to an analogous result for a finite simple (e.g. unramified) extension of A.

This sketch shows the importance of the systematic study of schemes defined over an Artinian ring A. Thepoint of view of Serre in his formulation of the theory of local class fields, and the recent works of Greenberg,seem to suggest that such a study could be undertaken by functorially attaching, to some such scheme X , ascheme X ′ over the residue field k of A (assumed perfect) of dimension equal (in nice cases) to n dim X , where nis the height of A.

As for the influence of A. Weil, it suces to say that it is the need to develop the tools necessary to formulate,with full generality, the definition of “Weil cohomology”, and to tackle the proof8 of all the formal propertiesnecessary to establish the famous conjectures in Diophantine geometry [Wei49], that has been one of theprincipal motivations for the writing of this treatise, as well as the desire to find the natural setting of the usualideas and methods of algebraic geometry, and to give the authors the chance to understand said ideas andmethods.

***

Finally, we believe it useful to warn the reader that they, as did all the authors themselves, will almostcertainly have diculty before becoming accustomed to the language of schemes, and to convince themselvesthat the usual constructions that suggest geometric intuition can be translated, in essentially only one sensibleway, to this language. As in many parts of modern mathematics, the first intuition seems further and furtheraway, in appearance, from the correct language needed to express the mathematics in question with completeprecision and the desired level of generality. In practice, the psychological diculty comes from the need toreplicate some familiar set-theoretic constructions to a category that is already quite dierent from that of sets(the category of preschemes, or the category of preschemes over a given prescheme): Cartesian products, grouplaws, ring laws, module laws, fibre bundles, principal homogeneous fibre bundles, etc. It will most likely bedicult for the mathematician, in the future, to shy away from this new eort of abstraction (maybe rathernegligible, on the whole, in comparison with that supplied by our fathers) to familiarize themselves with thetheory of sets.

***7Among the works that come close to our point of view of algebraic geometry, we pick out the work of E. Kähler [Käh58] and a recent

note of Chow and Igusa [CI58], which go back over certain results of (FAC) in the context of Nagata–Chevalley theory, as well as giving aKünneth formula.

8To avoid any misunderstanding, we point out that this task has barely been undertaken at the moment of writing this introduction,and still hasn’t led to the proof of the Weil conjectures.

INTRODUCTION 12

The references are given following the numerical system; for example, in III, 4.9.3, the III indicates thevolume, the 4 the chapter, the 9 the section, and the 3 the paragraph.9 If we reference a volume from withinitself then we omit the volume number.

I | 10[Trans] Page 10 in the original is left blank.

9[Trans] This is not a direct translation of the original, but instead uses the language more familiar to modern book (and LATEX document) layouts.

CHAPTER 0

Preliminaries (EGA 0)

§1. Rings of fractions0I | 11

1.0. Rings and Algebras.

(1.0.1). All the rings considered in this treatise will have a unit element; all the modules over such a ring willbe assumed to be unitary; the ring homomorphisms will always be assumed to send the unit element to the unitelement; unless otherwise stated, a subring of a ring A will be assumed to contain the unit element of A. We willfocus in particular on commutative rings, and when we speak of a ring without specifying any details, it will beimplied that it is commutative. If A is a not-necessarily-commutative ring, by A-module we will we mean a leftmodule unless stated otherwise.

(1.0.2). Let A and B be not-necessarily-commutative rings and φ : A→ B a homomorphism. Any left (resp. right)B-module M can be provided with a left (resp. right) A-module structure by a ·m=φ (a)·m (resp. m·a = m·φ (a));when it will be necessary to distinguish M as an A-module or a B-module, we will denote by M[φ ] the left (resp.right) A-module defined as such. If L is an A-module, then a homomorphism u : L→ M[φ ] is a homomorphism ofabelian groups such that u(a · x) =φ (a) ·u(x) for a ∈ A, x ∈ L; we will also say that it is a φ -homomorphism L→ M ,and that the pair (φ , u) (or, by abuse of language, u) is a di-homomorphism from (A, L) to (B, M). The pairs (A, L)consisting of a ring A and an A-module L thus form a category whose morphisms are di-homomorphisms.

(1.0.3). Under the hypotheses of (1.0.2), if J is a left (resp. right) ideal of A, we denote by BJ (resp. JB)the left (resp. right) ideal Bφ (J) (resp. φ (J)B) of B generated by φ (J); it is also the image of the canonicalhomomorphism B ⊗A J→ B (resp. J⊗A B→ B) of left (resp. right) B-modules.

(1.0.4). If A is a (commutative) ring, and B a not-necessarily-commutative ring, then the data of a structure ofan A-algebra on B is equivalent to the data of a ring homomorphism φ : A→ B such that φ (A) is contained in thecenter of B. For all ideals J of A, JB = BJ is then a two-sided ideal of B, and for every B-module M , JM is thena B-module equal to (BJ)M .

(1.0.5). We will not dwell much on the notions of modules of nite type and (commutative) algebras of nite type;to say that an A-module M is of finite type means that there exists an exact sequence Ap → M → 0. We say that 0I | 12an A-module M admits a nite presentation if it is isomorphic to the cokernel of a homomorphism Ap → Aq, or, inother words, if there exists an exact sequence Ap → Aq → M → 0. We note that for a Noetherian ring A, everyA-module of finite type admits a finite presentation.

Let us recall that an A-algebra B is said to be integral over A if every element in B is a root in B of a monicpolynomial with coecients in A; equivalently, if every element of B is contained in a subalgebra of B which isan A-module of nite type. When this is so, and if B is commutative, the subalgebra of B generated by a finitesubset of B is an A-module of finite type; for a commutative algebra B to be integral and of finite type over A, itis necessary and sucient that B be an A-module of finite type; we also say that B is an integral A-algebra of nitetype (or simply nite, if there is no chance of confusion). It should be noted that in these definitions, it is notassumed that the homomorphism A→ B defining the A-algebra structure is injective.

(1.0.6). An integral ring (or an integral domain) is a ring in which the product of a finite family of elements 6= 0is 6= 0; equivalently, in such a ring, we have 0 6= 1, and the product of two elements 6= 0 is 6= 0. A prime ideal ofa ring A is an ideal p such that A/p is integral; this implies that p 6= A. For a ring A to have at least one primeideal, it is necessary and sucient that A 6= 0.

(1.0.7). A local ring is a ring A in which there exists a unique maximal ideal, which is thus the complement ofthe invertible elements, and contains all the ideals 6= A. If A and B are local rings, and m and n their respectivemaximal ideals, then we say that a homomorphism φ : A→ B is local if φ (m) ⊂ n (or, equivalently, if φ−1(n) = m).By passing to quotients, such a homomorphism then defines a monomorphism from the residue field A/m to theresidue field B/n. The composition of any two local homomorphisms is a local homomorphism.

13

1. RINGS OF FRACTIONS 14

1.1. Radical of an ideal. Nilradical and radical of a ring.

(1.1.1). Let a be an ideal of a ring A; the radical of a, denoted by r(a), is the set of x ∈ A such that xn ∈ a foran integer n > 0; it is an ideal containing a. We have r(r(a)) = r(a); the relation a ⊂ b implies r(a) ⊂ r(b); theradical of a finite intersection of ideals is the intersection of their radicals. If φ is a homomorphism from anotherring A′ to A, then we have r(φ−1(a)) =φ−1(r(a)) for any ideal a ⊂ A. For an ideal to be the radical of an ideal, itis necessary and sucient that it be an intersection of prime ideals. The radical of an ideal a is the intersectionof the minimal prime ideals which contain a; if A is Noetherian, there are finitely many of these minimal primeideals.

The radical of the ideal (0) is also called the nilradical of A; it is the set N of the nilpotent elements of A. Wesay that the ring A is reduced if N= (0); for every ring A, the quotient A/N of A by its nilradical is a reduced ring.

(1.1.2). Recall that the nilradical N(A) of a (not-necessarily-commutative) ring A is the intersection of themaximal left ideals of A (and also the intersection of maximal right ideals). The nilradical of A/N(A) is (0).

0I | 131.2. Modules and rings of fractions.

(1.2.1). We say that a subset S of a ring A is multiplicative if 1 ∈ S and the product of two elements of S is in S.The examples which will be the most important in what follows are: 1st, the set S f of powers f n (n¾ 0) of anelement f ∈ A; and 2nd, the complement A− p of a prime ideal p of A.

(1.2.2). Let S be a multiplicative subset of a ring A, and M an A-module; on the set M × S, the relation betweenpairs (m1, s1) and (m2, s2):

“there exists an s ∈ S such that s(s1m2 − s2m1) = 0”

is an equivalence relation. We denote by S−1M the quotient set of M × S by this relation, and by m/s thecanonical image of the pair (m, s) in S−1M ; we call iS

M : m 7→ m/1 (also denoted iS) the canonical map from Mto S−1M . This map is, in general, neither injective nor surjective; its kernel is the set of m ∈ M such that thereexists an s ∈ S for which sm= 0.

On S−1M we define an additive group law by setting

(m1/s1) + (m2/s2) = (s2m1 + s1m2)/(s1s2)

(one can check that it is independent of the choice of representative of the elements of S−1M , which areequivalence classes). On S−1A we further define a multiplicative law by setting (a1/s1)(a2/s2) = (a1a2)/(s1s2),and finally an exterior law on S−1M , acted on by the set of elements of S−1A, by setting (a/s)(m/s′) = (am)/(ss′).It can then be shown that S−1A is endowed with a ring structure (called the ring of fractions of A with denominatorsin S) and S−1M with the structure of an S−1A-module (called the module of fractions of M with denominators inS); for all s ∈ S, s/1 is invertible in S−1A, its inverse being 1/s. The canonical map iS

A (resp. iSM ) is a ring

homomorphism (resp. a homomorphism of A-modules, S−1M being considered as an A-module by means of thehomomorphism iS

A : A→ S−1A).

(1.2.3). If S f = f nn¾0 for a f ∈ A, we write A f and M f instead of S−1f A and S−1

f M ; when A f is considered asalgebra over A, we can write A f = A[1/ f ]. A f is isomorphic to the quotient algebra A[T]/( f T − 1)A[T]. Whenf = 1, A f and M f are canonically identified with A and M ; if f is nilpotent, then A f and M f are 0.

When S = A− p, with p a prime ideal of A, we write Ap and Mp instead of S−1A and S−1M ; Ap is a localring whose maximal ideal q is generated by iS

A(p), and we have (iSA)−1(q) = p; by passing to quotients, iS

A gives amonomorphism from the integral ring A/p to the field Ap/q, which can be identified with the field of fractions ofA/p.

(1.2.4). The ring of fractions S−1A and the canonical homomorphism iSA are a solution to a universal mapping

problem: any homomorphism u from A to a ring B such that u(S) is composed of invertible elements in B factorsuniquely as

u : AiSA−→ S−1A

u∗−→ B

where u∗ is a ring homomorphism. Under the same hypotheses, let M be an A-module, N a B-module, and 0I | 14v : M → N a homomorphism of A-modules (for the B-module structure on N defined by u : A→ B); then vfactors uniquely as

v : MiSM−→ S−1M

v∗−→ N

where v∗ is a homomorphism of S−1A-modules (for the S−1A-module structure on N defined by u∗).

1. RINGS OF FRACTIONS 15

(1.2.5). We define a canonical isomorphism S−1A⊗A M ' S−1M of S−1A-modules, sending the element (a/s)⊗mto the element (am)/s, with the inverse isomorphism sending m/s to (1/s)⊗m.

(1.2.6). For every ideal a′ of S−1A, a= (iSA)−1(a′) is an ideal of A, and a′ is the ideal of S−1A generated by iS

A(a),which can be identified with S−1a (1.3.2). The map p′ 7→ (iS

A)−1(p′) is an isomorphism, for the structure given by

ordering, from the set of prime ideals of S−1A to the set of prime ideals p of A such that p∩ S =∅. In addition,the local rings Ap and (S−1A)S−1p are canonically isomorphic (1.5.1).

(1.2.7). When A is an integral ring, for which K denotes its field of fractions, the canonical map iSA : A→ S−1A is

injective for any multiplicative subset S not containing 0, and S−1A is then canonically identified with a subringof K containing A. In particular, for every prime ideal p of A, Ap is a local ring containing A, with maximal idealpAp, and we have pAp ∩ A= p.

(1.2.8). If A is a reduced ring (1.1.1), so is S−1A: indeed, if (x/s)n = 0 for x ∈ A, s ∈ S, then this means that thereexists an s′ ∈ S such that s′xn = 0, hence (s′x)n = 0, which, by hypothesis, implies s′x = 0, so x/s = 0.

1.3. Functorial properties.

(1.3.1). Let M and N be A-modules, and u an A-homomorphism M → N . If S is a multiplicative subset of A, wedefine a S−1A-homomorphism S−1M → S−1N , denoted by S−1u, by setting S−1u(m/s) = u(m)/s; if S−1M andS−1N are canonically identified with S−1A⊗A M and S−1A⊗A N (1.2.5), then S−1u is considered as 1⊗u. If P is athird A-module, and v an A-homomorphism N → P, we have S−1(v u) = (S−1v) (S−1u); in other words, S−1Mis a covariant functor in M , from the category of A-modules to that of S−1A-modules (A and S being fixed).

(1.3.2). The functor S−1M is exact; in other words, if the sequence

Mu−→ N

v−→ P

is exact, then so is the sequence

S−1MS−1u−−→ S−1N

S−1 v−−→ S−1P.

In particular, if u : M → N is injective (resp. surjective), the same is true for S−1u; if N and P are submodules of 0I | 15M , S−1N and S−1P are canonically identified with submodules of S−1M , and we have

S−1(N + P) = S−1N + S−1P and S−1(N ∩ P) = (S−1N)∩ (S−1P).

(1.3.3). Let (Mα,φβα) be an inductive system of A-modules; then (S−1Mα, S−1φβα) is an inductive system ofS−1A-modules. Expressing the S−1Mα and S−1φβα as tensor products ((1.2.5) and (1.3.1)), it follows from thepermutability of the tensor product and inductive limit operations that we have a canonical isomorphism

S−1 lim−→Mα ' lim−→S−1Mα

which is we can further express by saying that the functor S−1M (in M) commutes with inductive limits.

(1.3.4). Let M and N be A-modules; there is a canonical functorial (in M and N) isomorphism

(S−1M)⊗S−1A (S−1N)' S−1(M ⊗A N)

which sends (m/s)⊗ (n/t) to (m⊗ n)/st.

(1.3.5). We also have a functorial (in M and N) homomorphism

S−1 HomA(M , N) −→ HomS−1A(S−1M , S−1N)

which sends u/s to the homomorphism m/t 7→ u(m)/st . When M has a finite presentation, the above homomor-phism is an isomorphism: it is immediate when M is of the form Ar , and we pass to the general case by startingwith the exact sequence Ap → Aq → M → 0 and using the exactness of the functor S−1M and the left-exactnessof the functor HomA(M , N) in M . Note that this is always the case when A is Noetherian and the A-module M isof nite type.

1. RINGS OF FRACTIONS 16

1.4. Change of multiplicative subset.

(1.4.1). Let S and T be multiplicative subsets of a ring A such that S ⊂ T ; there exists a canonical homomorphismρT,S

A (or simply ρT,S) from S−1A to T−1A, sending the element denoted a/s of S−1A to the element denoted a/s inT−1A; we have iT

A = ρT,SA iS

A . For every A-module M , there exists, in the same way, an S−1A-linear map from S−1Mto T−1M (the latter considered as an S−1A-module by the homomorphism ρT,S

A ), which sends the element m/s ofS−1M to the element m/s of T−1M ; we denote this map by ρT,S

M , or simply ρT,S , and we still have iTM = ρ

T,SM iS

M ;by the canonical identification (1.2.5), ρT,S

M is identified with ρT,SA ⊗ 1. The homomorphism ρT,S

M is a functorialmorphism (or natural transformation) from the functor S−1M to the functor T−1M , in other words, the diagram

S−1MS−1u //

ρT,SM

S−1N

ρT,SN

T−1M T−1u // T−1N

is commutative, for every homomorphism u : M → N ; T−1u is entirely determined by S−1u, since, for m ∈ M 0I | 16and t ∈ T , we have

(T−1u)(m/t) = (t/1)−1ρT,S((S−1u)(m/1)).

(1.4.2). With the same notation, for A-modules M and N , the diagrams (cf. (1.3.4) and (1.3.5))

(S−1M)⊗S−1A (S−1N) ∼ //

S−1(M ⊗A N)

(T−1M)⊗T−1A (T−1N) ∼ // T−1(M ⊗A N)

S−1 HomA(M , N) //

HomS−1A(S−1M , S−1N)

T−1 HomA(M , N) // HomT−1A(T−1M , T−1N)

are commutative.

(1.4.3). There is an important case, in which the homomorphism ρT,S is bijective, when we then know that everyelement of T is a divisor of an element of S; we then identify the modules S−1M and T−1M via ρT,S . We saythat S is saturated if every divisor in A of an element of S is in S; by replacing S with the set T of all the divisorsof the elements of S (a set which is multiplicative and saturated), we see that we can always, if we wish, consideronly modules of fractions S−1M , where S is saturated.

(1.4.4). If S, T , and U are three multiplicative subsets of A such that S ⊂ T ⊂ U , then we have

ρU ,S = ρU ,T ρT,S .

(1.4.5). Consider an increasing ltered family (Sα) of multiplicative subsets of A (we write α ¶ β for Sα ⊂ Sβ),

and let S be the multiplicative subset⋃

α Sα; let us put ρβα = ρSβ ,SαA for α ¶ β; according to (1.4.4), the

homomorphisms ρβα define a ring A′ as the inductive limit of the inductive system of rings (S−1α A,ρβα). Let ρα

be the canonical map S−1α A→ A′, and let φα = ρ

S,SαA ; as φα = φβ ρβα for α ¶ β according to (1.4.4), we can

uniquely define a homomorphism φ : A′→ S−1A such that the diagram

S−1α A

ρα

ρβα

φα

S−1β A

ρβ~~

φβ ""

(α ¶ β)

A′φ // S−1A

is commutative. In fact, φ is an isomorphism; it is indeed immediate by construction that φ is surjective. Onthe other hand, if ρα(a/sα) ∈ A′ is such that φ (ρα(a/sα)) = 0, then this means that a/sα = 0 in S−1A, that isto say that there exists an s ∈ S such that sa = 0; but there is a β ¾ α such that s ∈ Sβ , and consequently, as

1. RINGS OF FRACTIONS 17

ρα(a/sα) = ρβ(sa/ssα) = 0, we find that φ is injective. The case for an A-module M is treated likewise, and wehave thus defined canonical isomorphisms

lim−→S−1α A' (lim−→Sα)

−1A, lim−→S−1α M ' (lim−→Sα)

−1M ,

the second being functorial in M .0I | 17

(1.4.6). Let S1 and S2 be multiplicative subsets of A; then S1S2 is also a multiplicative subset of A. Let us denoteby S′2 the canonical image of S2 in the ring S−1

1 A, which is a multiplicative subset of this ring. For every A-moduleM there is then a functorial isomorphism

S′2−1(S−1

1 M)' (S1S2)−1M

which sends (m/s1)/(s2/1) to the element m/(s1s2).

1.5. Change of ring.

(1.5.1). Let A and A′ be rings, φ a homomorphism A′→ A, and S (resp. S′) a multiplicative subset of A (resp.A′), such that φ (S′) ⊂ S; the composition homomorphism A′

φ−→ A→ S−1A factors as

A′ −→ S′−1A′φ S′

−→ S−1A,

by (1.2.4); where φ S′(a′/s′) = φ (a′)/φ (s′). If A= φ (A′) and S = φ (S′), then φ S′ is surjective. If A′ = A and φ isthe identity, then φ S′ is exactly the homomorphism ρS,S′

A defined in (1.4.1).

(1.5.2). Under the hypotheses of (1.5.1), let M be an A-module. There exists a canonical functorial morphism

σ : S′−1(M[φ ]) −→ (S−1M)[φ S′ ]

of S′−1A′-modules, sending each element m/s′ of S′−1(M[φ ]) to the element m/φ (s′) of (S−1M)[φ S′ ]; indeed, weimmediately see that this definition does not depend on the representative m/s′ of the element in question.When S =φ (S′), the homomorphism σ is bijective. When A′ = A and φ is the identity, σ is none other than thehomomorphism ρS,S′

M defined in (1.4.1).When, in particular, we take M = A the homomorphism φ defines an A′-algebra structure on A; S′−1(A[φ ]) is

then endowed with a ring structure, with which it can be identified with (φ (S′))−1A, and the homomorphismσ : S′−1(A[φ ])→ S−1A is a homomorphism of S′−1A′-algebras.

(1.5.3). Let M and N be A-modules; by composing the homomorphisms defined in (1.3.4) and (1.5.2), we obtaina homomorphism

(S−1M ⊗S−1A S−1N)[φ S′ ]←− S′−1((M ⊗ A)[φ ])

which is an isomorphism when φ (S′) = S. Similarly, by composing the homomorphisms in (1.3.5) and (1.5.2),we obtain a homomorphism

S′−1((HomA(M , N))[φ ]) −→ (HomS−1A(S−1M , S−1N))[φ S′ ]

which is an isomorphism when φ (S′) = S and M admits a finite presentation.

(1.5.4). Let us now consider an A′-module N ′, and form the tensor product N ′ ⊗A′ A[φ ], which can be consideredas an A-module by setting a · (n′ ⊗ b) = n′ ⊗ (ab). There is a functorial isomorphism of S−1A-modules

τ : (S′−1N ′)⊗S′−1A′ (S−1A)[φ S′ ] ' S−1(N ′ ⊗A′ A[φ ])

which sends the element (n′/s′)⊗(a/s) to the element (n′⊗a)/(φ (s′)s); indeed, we can show that when we replace 0I | 18n′/s′ (resp. a/s) by another expression of the same element, (n′ ⊗ a)/(φ (s′)s) does not change; on the otherhand, we can define a homomorphism inverse to τ by sending (n′ ⊗ a)/s to the element (n′/1)⊗ (a/s): we usethe fact that S−1(N ′⊗A′ A[φ ]) is canonically isomorphic to (N ′⊗A′ A[φ ])⊗A S−1A (1.2.5), so also to N ′⊗A′ (S−1A)[ψ],where we denote by ψ the composite homomorphism a′ 7→φ (a′)/1 from A′ to S−1A.

(1.5.5). If M ′ and N ′ are A′-modules, then by composing the isomorphisms (1.3.4) and (1.5.4), we obtain anisomorphism

S′−1M ⊗S′−1A′ S′−1N ′ ⊗S′−1A′ S

−1A' S−1(M ′ ⊗A′ N ′ ⊗A′ A).Likewise, if M ′ admits a finite presentation, we have by (1.3.5) and (1.5.4) an isomorphism

HomS′−1A′(S′−1M ′, S′−1N ′)⊗S′−1A′ S

−1A' S−1(HomA′(M′, N ′)⊗A′ A).

1. RINGS OF FRACTIONS 18

(1.5.6). Under the hypotheses of (1.5.1), let T (resp. T ′) be another multiplicative subset of A (resp. A′) suchthat S ⊂ T (resp. S′ ⊂ T ′) and φ (T ′) ⊂ T . Then the diagram

S′−1A′φ S′

//

ρT ′ ,S′

S−1A

ρT,S

T ′−1A′

φ T ′

// T−1A

is commutative. If M is an A-module, then the diagram

S′−1(M[φ ])σ //

ρT ′ ,S′

(S−1M)[φ S′ ]

ρT,S

T ′−1(M[φ ])

σ // (T−1M)[φ T ′ ]

is commutative. Finally, if N ′ is an A′-module, then the diagram

(S′−1N ′)⊗S′−1A′ (S−1A)[φ S′ ]

∼τ//

S−1(N ′ ⊗A′ A[φ ])

ρT,S

(T ′−1N ′)⊗T ′−1A′ (T

−1A)[φ T ′ ]∼τ// T−1(N ′ ⊗A′ A[φ ])

is commutative, the left vertical arrow obtained by applying ρT ′,S′

N ′ to S′−1N ′ and ρT,SA to S−1A.

0I | 19

(1.5.7). Let A′′ be a third ring, φ ′ : A′′→ A′ a ring homomorphism, and S′′ a multiplicative subset of A′′ suchthat φ ′(S′′) ⊂ S′. Let φ ′′ =φ φ ′; then we have

φ ′′S′′=φ S′ φ ′S

′′.

Let M be an A-module; evidently we have M[φ ′′] = (M[φ ])[φ ′]; if σ′ and σ′′ are the homomorphisms defined by φ ′and φ ′′ in the same way as how σ is defined in (1.5.2) by φ , then we have the transitivity formula

σ′′ = σ σ′.

Finally, let N ′′ be an A′′-module; the A-module N ′′⊗A′′A[φ ′′] is canonically identified with (N ′′⊗A′′A′[φ ′])⊗A′A[φ ],

and likewise the S−1A-module (S′′−1N ′′) ⊗S′′−1A′′ (S−1A)[φ ′′S′′ ] is canonically identified with ((S′′−1N ′′) ⊗S′′−1A′′

(S′−1A′)[φ ′S′′ ])⊗S′−1A′ (S−1A)[φ S′ ]. With these identifications, if τ′ and τ′′ are the isomorphisms defined by φ ′ and

φ ′′ in the same way as how τ is defined in (1.5.4) by φ , then we have the transitivity formula

τ′′ = τ (τ′ ⊗ 1).

(1.5.8). Let A be a subring of a ring B; for every minimal prime ideal p of A, there exists a minimal prime idealq of B such that p= A∩ q. Indeed, Ap is a subring of Bp (1.3.2) and has a single prime ideal p′ (1.2.6); since Bp isnot 0, it has at least one prime ideal q′ and we necessarily have q′ ∩ Ap = p′; the prime ideal q1 of B, the inverseimage of q′, is thus such that q1 ∩ A= p, and a fortiori we have q∩ A= p for every minimal prime ideal q of Bcontained in q1.

1.6. Identication of the module M f as an inductive limit.

(1.6.1). Let M be an A-module and f an element of A. Consider a sequence (Mn) of A-modules, all identical to M ,and for each pair of integers m¶ n, let φnm be the homomorphism z 7→ f n−mz from Mm to Mn; it is immediatethat ((Mn), (φnm)) is an inductive system of A-modules; let N = lim−→Mn be the inductive limit of this system. Wedefine a canonical functorial A-isomorphism from N to M f . For this, let us note that, for all n, θn : z 7→ z/ f n is anA-homomorphism from M = Mn to M f , and it follows from the definitions that we have θn φnm = θm for m¶ n.As a result, there exists an A-homomorphism θ : N → M f such that, if φn denotes the canonical homomorphismMn→ N , then we have θn = θ φn for all n. Since, by hypothesis, every element of M f is of the form z/ f n for atleast one n, it is clear that θ is surjective. On the other hand, if θ(φn(z)) = 0, or, in other words, if z/ f n = 0,then there exists an integer k > 0 such that f kz = 0, so φn+k,n(z) = 0, which gives φn(z) = 0. We can thereforeidentify M f with lim−→Mn via θ.

2. IRREDUCIBLE SPACES. NOETHERIAN SPACES 19

(1.6.2). Now write M f ,n, φ fnm, and φ

fn instead of Mn, φnm, and φn. Let g be another element of A. Since f n

divides f n gn, we have a functorial homomorphism

ρ f g, f : M f −→ M f g ((1.4.1) and (1.4.3));

if we identify M f and M f g with lim−→M f ,n and lim−→M f g,n respectively, then ρ f g, f identifies with the inductive 0I | 20limit of the maps ρn

f g, f : M f ,n → M f g,n, defined by ρnf g, f (z) = gnz. Indeed, this follows immediately from the

commutativity of the diagram

M f ,n

ρnf g, f //

φ fn

M f g,n

φ f gn

M f

ρ f g, f // M f g .

1.7. Support of a module.

(1.7.1). Given an A-module M , we define the support of M , denoted by Supp(M), to be the set of prime ideals pof A such that Mp 6= 0. For it to be the case that M = 0, it is necessary and sucient that Supp(M) =∅, becauseif Mp = 0 for all p, then the annihilator of an element x ∈ M cannot be contained in any prime ideal of A, andso is the whole of A.

(1.7.2). If 0→ N → M → P → 0 is an exact sequence of A-modules, then we have

Supp(M) = Supp(N)∪ Supp(P)

because, for every prime ideal p of A, the sequence 0→ Np→ Mp→ Pp→ 0 is exact (1.3.2) and in order thatMp = 0, it is necessary and sucient that Np = Pp = 0.

(1.7.3). If M is the sum of a family (Mλ) of submodules, then Mp is the sum of the (Mλ)p for every prime idealp of A ((1.3.3) and (1.3.2)), so Supp(M) =

⋃

λ Supp(Mλ).

(1.7.4). If M is an A-module of nite type, then Supp(M) is the set of prime ideals containing the annihilator of M .Indeed, if M is cyclic and generated by x , then to say that Mp = 0 is to say that that there exists an s 6∈ p suchthat s · x = 0, and thus that p does not contain the annihilator of x . Now if M admits a finite system (x i)1¶i¶nof generators, and if ai is the annihilator of x i , then it follows from (1.7.3) that Supp(M) is the set of the pcontaining one of the ai , or equivalently, the set of the p containing a=

⋂

i ai , which is the annihilator of M .

(1.7.5). If M and N are two A-modules of nite type, then we have

Supp(M ⊗A N) = Supp(M)∩ Supp(N).

It is a question of seeing that, if p is a prime ideal of A, then the condition Mp⊗ApNp 6= 0 is equivalent to “Mp 6= 0

and Np 6= 0” (taking (1.3.4) into account). In other words, it is a question of seeing that, if P and Q are modulesof finite type over a local ring B 6= 0, then P ⊗B Q 6= 0. Let m be the maximal ideal of B. By Nakayama’s Lemma,the vector spaces P/mP and Q/mQ are not 0, and so it is the same for the tensor product (P/mP)⊗B/m (Q/mQ)= (P ⊗B Q)⊗B (B/m), whence the conclusion.

In particular, if M is an A-module of finite type, and a an ideal of A, then Supp(M/aM) is the set of primeideals containing both a and the annihilator n of M (1.7.4), that is, the set of prime ideals containing a+ n.

§2. Irreducible spaces. Noetherian spaces0I | 21

2.1. Irreducible spaces.

(2.1.1). We say that a topological space X is irreducible if it is nonempty and if it is not a union of two distinctclosed subspaces of X . It is equivalent to say that X 6=∅ and the intersection of two nonempty open sets (andconsequently of a finite number of open sets) of X is nonempty, or that every nonempty open set is everywheredense, or that any closed set is rare1, or, lastly, that all open sets of X are connected.

(2.1.2). For a subspace Y of a topological space X to be irreducible, it is necessary and sucient that its closureY be irreducible. In particular, any subspace which is the closure x of a singleton is irreducible; we will expressthe relation y ∈ x (equivalent to y ⊂ x) by saying that y is a specialization of x or that x is a generalizationof y . When there exists, in an irreducible space X , a point x such that X = x, we will say that x is a genericpoint of X . Any nonempty open subset of X then contains x , and any subspace containing x has x as a genericpoint.

1[Trans] also known as nowhere dense.

2. IRREDUCIBLE SPACES. NOETHERIAN SPACES 20

(2.1.3). Recall that a Kolmogoro space is a topological space X satisfying the axiom of separation:

(T0) If x 6= y are any two points of X , there is an open set containing one of the points x and y, but notthe other.

If an irreducible Kolmogoro space admits a generic point, it admits exactly one, since a nonempty open setcontains any generic point.

Recall that a topological space X is said to be quasi-compact if, from any collection of open sets of X , onecan extract a finite cover of X (or, equivalently, if any decreasing filtered family of nonempty closed sets hasa nonempty intersection). If X is a quasi-compact space, then any nonempty closed subset A of X contains aminimal nonempty closed set M , because the set of nonempty closed subsets of A is inductive under the relation⊃; if, in addition, X is a Kolmogoro space, M is necessarily a single point (or, as we say by abuse of language,is a closed point).

(2.1.4). In an irreducible space X , every nonempty open subspace U is irreducible, and if X admits a genericpoint x , x is also a generic point of U .

To prove this, let (Uα) be a cover (whose set of indices is nonempty) of a topological space X , consisting ofnonempty open sets; if X is irreducible, it is necessary and sucient that Uα is irreducible for all α, and thatUα ∩Uβ 6=∅ for any α, β. The condition is clearly necessary; to see that it is sucient, it suces to prove that ifV is a nonempty open subset of X , then V ∩Uα is nonempty for all α, since then V ∩Uα is dense in Uα for all α,and consequently V is dense in X . Now there is at least one index γ such that V ∩Uγ 6=∅, so V ∩Uγ is dense inUγ , and as for all α, Uα ∩ Vα 6=∅, we also have V ∩ Uα ∩ Uγ 6=∅.

0I | 22

(2.1.5). Let X be an irreducible space, and f a continuous map from X into a topological space Y . Then f (X )is irreducible, and if x is a generic point of X , then f (x) is a generic point of f (X ) and hence also of f (X ). Inparticular, if, in addition, Y is irreducible and with a single generic point y, then for f (X ) to be everywheredense, it is necessary and sucient that f (x) = y .

(2.1.6). Any irreducible subspace of a topological space X is contained in a maximal irreducible subspace, whichis necessarily closed. Maximal irreducible subspaces of X are called the irreducible components of X . If Z1 andZ2 are two irreducible components distinct from the space X , then Z1 ∩ Z2 is a closed rare set in each of thesubspaces Z1, Z2; in particular, if an irreducible component of X admits a generic point (2.1.2), such a pointcannot belong to any other irreducible component. If X has only a nite number of irreducible components Zi(1¶ i ¶ n), and if, for each i, we put Ui =

⋃

j 6=i Z j , then the Ui are open, irreducible, disjoint, and their union is ErrIIdense in X . Let U be an open subset of a topological space X . If Z is an irreducible subset of X that intersectsU , then Z ∩ U is open and dense in Z , thus irreducible; conversely, for any irreducible closed subset Y of U , theclosure Y of Y in X is irreducible and Y ∩ U = Y . We conclude that there is a bijective correspondence between theirreducible components of U and the irreducible components of X which intersect U .

(2.1.7). If a topological space X is a union of a nite number of irreducible closed subspaces Yi , then theirreducible components of X are the maximal elements of the set of the Yi , because if Z is an irreducible closedsubset of X , then Z is the union of the Z ∩ Yi , from which one sees that Z must be contained in one of theYi . Let Y be a subspace of a topological space X , and suppose that Y has only a finite number of irreduciblecomponents Yi , (1¶ i ¶ n); then the closures Yi in X are the irreducible components of Y .

(2.1.8). Let Y be an irreducible space admitting a single generic point y . Let X be a topological space, andf a continuous map from X to Y . Then, for any irreducible component Z of X intersecting f −1(y), f (Z) isdense in Y . The converse is not necessarily true; however, if Z has a generic point z, and if f (Z) is dense in Y ,then we must have f (z) = y (2.1.5); in addition, Z ∩ f −1(y) is then the closure of z in f −1(y) and is thereforeirreducible, and as an irreducible subset of f −1(y) containing z is necessarily contained in Z (2.1.6), z is ageneric point of Z ∩ f −1(y). As any irreducible component of f −1(y) is contained in an irreducible componentof X , we see that, if any irreducible component Z of X intersecting f −1(y) admits a generic point, then there isa bijective correspondence between all these components and all the irreducible components Z ∩ f −1(y) of f −1(y),the generic points of Z being identical to those of Z ∩ f −1(y).

0I | 232.2. Noetherian spaces.

(2.2.1). We say that a topological space X is Noetherian if the set of open subsets of X satisfies the maximalcondition, or, equivalently, if the set of closed subsets of X satisfies the minimal condition. We say that X islocally Noetherian if each x ∈ X admits a neighborhood which is a Noetherian subspace.

3. SUPPLEMENT ON SHEAVES 21

(2.2.2). Let E be an ordered set satisfying the minimal condition, and let P be a property of the elements ofE subject to the following condition: if a ∈ E is such that for any x < a, P(x) is true, then P(a) is true. Underthese conditions, P(x) is true for all x ∈ E (“principle of Noetherian recurrence”). Indeed, let F be the set ofx ∈ E for which P(x) is false; if F were not empty, it would have a minimal element a, and as then P(x) is truefor all x < a, P(a) would be true, which is a contradiction.

We will apply this principle in particular when E is a set of closed subsets of a Noetherian space.

(2.2.3). Any subspace of a Noetherian space is Noetherian. Conversely, any topological space that is a finiteunion of Noetherian subspaces is Noetherian.

(2.2.4). Any Noetherian space is quasi-compact; conversely, any topological space in which all open sets arequasi-compact is Noetherian.

(2.2.5). A Noetherian space has only a nite number of irreducible components, as we see by Noetherianrecurrence.

§3. Supplement on sheaves

3.1. Sheaves with values in a category.

(3.1.1). Let C be a category, (Aα)α∈I , (Aαβ)(α,β)∈I×I two families of objects of C such that Aβα = Aαβ , and(ραβ)(α,β)∈I×I a family of morphisms ραβ : Aα → Aαβ . We say that a pair consisting of an object A of C and afamily of morphisms ρα : A→ Aα is a solution to the universal problem defined by the data of the families (Aα), (Aαβ),and (ραβ) if, for every object B of C, the map which sends f ∈ Hom(B, A) to the family (ρα f ) ∈ Πα Hom(B, Aα)is a bijection of Hom(B, A) to the set of all ( fα) such that ραβ fα = ρβα fβ for any pair of indices (α,β). If sucha solution exists, it is unique up to an isomorphism.

(3.1.2). We will not recall the definition of a presheaf U 7→ F (U) on a topological space X with values in acategory C (G, I, 1.9); we say that such a presheaf is a sheaf with values in C if it satisfies the following axiom:

(F) For any covering (Uα) of an open set U of X by open sets Uα contained in U , if we denote by ρα (resp. ραβ) therestriction morphism

F (U) −→F (Uα) (resp. F (Uα) −→F (Uα ∩ Uβ)),

the pair formed byF (U) and the family (ρα) are a solution to the universal problem for (F (Uα)), (F (Uα∩Uβ)), 0I | 24and (ραβ) in (3.1.1)2.

Equivalently, we can say that, for each object T of C, that the family U 7→ Hom(T,F (U)) is a sheaf of sets.

(3.1.3). Assume that C is the category defined by a “type of structure with morphisms” Σ, the objects of C beingthe sets with structures of type Σ and morphisms those of Σ. Suppose that the category C also satisfies thefollowing condition:

(E) If (A, (ρα)) is a solution of a universal mapping problem in the category C for families (Aα), (Aαβ), (ραβ),then it is also a solution of the universal mapping problem for the same families in the category of sets(that is, when we consider A, Aα, and Aαβ as sets, ρα and ραβ as functions)3.

Under these conditions, the condition (F) gives that, when considered as a presheaf of sets, U 7→ F (U) is asheaf. In addition, for a map u : T →F (U) to be a morphism of C, it is necessary and sucient, according to(F), that each map ρα u is a morphism T →F (Uα), which means that the structure of type Σ on F (U) is theinitial structure for the morphisms ρα. Conversely, suppose a presheaf U 7→ F (U) on X , with values in C, is asheaf of sets and satisfies the previous condition; it is then clear that it satisfies (F), so it is a sheaf with values in C.

(3.1.4). When Σ is a type of a group or ring structure, the fact that the presheaf U 7→ F (U) with values in C is asheaf of sets implies ipso facto that it is a sheaf with values in C (in other words, a sheaf of groups or rings within themeaning of (G))4. But it is not the same when, for example, C is the category of topological rings (with morphismsas continuous homomorphisms): a sheaf with values in C is a sheaf of rings U 7→ F (U) such that for any open Uand any covering of U by open sets Uα ⊂ U , the topology of the ring F (U) is to be the least ne making thehomomorpisms F (U)→F (Uα) continuous. We will say in this case that U 7→ F (U), considered as a sheaf of

2This is a special case of the more general notion of a (non-filtered) projective limit (see (T, I, 1.8) and the book in preparation announcedin the introduction).

3It can be proved that it also means that the canonical functor C→ Set commutes with projective limits (not necessarily filtered).4This is because in the category C, any morphism that is a bijection (as a map of sets) is an isomorphism. This is no longer true when C

is the category of topological spaces, for example.

3. SUPPLEMENT ON SHEAVES 22

rings (without a topology), is underlying the sheaf of topological rings U 7→ F (U). Morphisms uV :F (V )→G (V )(V an arbitrary open subset of X ) of sheaves of topological rings are therefore homomorphisms of the underlyingsheaves of rings, such that uV is continuous for all open V ⊂ X ; to distinguish them from any homomorphisms ofthe sheaves of the underlying rings, we will call them continuous homomorphisms of sheaves of topologicalrings. We have similar definitions and conventions for sheaves of topological spaces or topological groups.

0I | 25

(3.1.5). It is clear that for any category C, if there is a presheaf (respectively a sheaf) F on X with values in Cand U is an open set of X , the F (V ) for open V ⊂ U constitute a presheaf (or a sheaf) with values in C, whichwe call the presheaf (or sheaf) induced by F on U and denote it by F|U .

For any morphism u : F → G of presheaves on X with values in C, we denote by u|U the morphismF|U →G|U consisting of the uV for V ⊂ U .

(3.1.6). Suppose now that the category C admits inductive limits (T, 1.8); then, for any presheaf (and in particularany sheaf) F on X with values in C and each x ∈ X , we can define the stalk Fx as the object of C defined by theinductive limit of the F (U) with respect to the filtered set (for ⊃) of the open neighborhoods U of x in X , andthe morphisms ρV

U :F (V )→F (U). If u :F →G is a morphism of presheaves with values in C, we define foreach x ∈ X the morphism ux : Fx → Gx as the inductive limit of uU : F (U)→ G (U) with respect to all openneighborhoods of x ; we thus define Fx as a covariant functor in F , with values in C, for all x ∈ X .

When C is further defined by a kind of structure with morphisms Σ, we call sections over U of a sheaf Fwith values in C the elements of F (U), and we write Γ(U ,F ) instead of F (U); for s ∈ Γ(U ,F ), V an open setcontained in U , we write s|V instead of ρU

V (s); for all x ∈ U , the canonical image of s in Fx is the germ of s atthe point x , denoted by sx (we will never replace the notation s(x) in this sense, this notation being reserved foranother notion relating to sheaves which will be considered in this treatise (5.5.1)).

If then u : F → G is a morphism of sheaves with values in C, we will write u(s) instead of uV (s) for alls ∈ Γ(V,F ). ErrII

If F is a sheaf of commutative groups, or rings, or modules, we say that the set of x ∈ X such that Fx 6= 0is the support of F , denoted Supp(F ); this set is not necessarily closed in X .

When C is defined by a type of structure with morphisms, we systematically refrain from using the point of viewof “étalé spaces” in terms of relating to sheaves with values in C; in other words, we will never consider a sheafas a topological space (nor even as the whole union of its stalks), and we will not consider also a morphismu :F →G of such sheaves on X as a continuous map of topological spaces.

3.2. Presheaves on an open basis.

(3.2.1). We will restrict to the following categories C admitting projective limits (generalized, that is, correspondingto not necessarily filtered preordered sets, cf. (T, 1.8)). Let X be a topological space, B an open basis for thetopology of X . We will call a presheaf on B, with values in C, a family of objects F (U) ∈ C, corresponding to eachU ∈B, and a family of morphisms ρV

U :F (V )→F (U) defined for any pair (U , V ) of elements of B such thatU ⊂ V , with the conditions ρU

U = identity and ρWU = ρ

VU ρ

WV if U , V , W in B are such that U ⊂ V ⊂W . We can 0I | 26

associate a presheaf with values in C: U 7→ F (U) in the ordinary sense, taking for all open U , F ′(U) = lim←−F (V ),where V runs through the ordered set (for ⊂, not ltered in general) of V ∈B sets such that V ⊂ U , since the(V ) form a projective system for the ρW

V (V ⊂W ⊂ U , V ∈B, W ∈B). Indeed, if U , U ′ are two open sets of X

such that U ⊂ U ′, we define ρ′U′

U as the projective limit (for V ⊂ U) of the canonical morphisms F ′(U ′)→F (V ),in other words the unique morphism F ′(U ′)→F ′(U), which, when composed with the canonical morphismsF ′(U)→F (V ), gives the canonical morphisms F ′(U ′)→F (V ); the verification of the transitivity of ρ′U

′

U isthen immediate. Moreover, if U ∈B, the canonical morphism F ′(U)→F (U) is an isomorphism, allowing usto identify these two objects5.

(3.2.2). For the presheaf F ′ thus defined to be a sheaf, it is necessary and sucient that the presheaf F on Bsatisfies the condition:

5If X is a Noetherian space, we can still define F ′(U) and show that it is a presheaf (in the ordinary sense) when one supposes only thatC admits projective limits for nite projective systems. Indeed, if U is any open set of X , there is a nite covering (Vi) of U consisting of setsof B; for every couple (i, j) of indices, let (Vi jk) be a finite covering of Vi ∩ Vj formed by sets of B. Let I be the set of i and triples (i, j, k),ordered only by the relations i > (i, j, k), j > (i, j, k); we then take F ′(U) to be the projective limit of the system of F (Vi) and F (Vi jk); it iseasy to verify that this does not depend on the coverings (Vi) and (Vi jk) and that U 7→ F ′(U) is a presheaf.

3. SUPPLEMENT ON SHEAVES 23

(F0) For any covering (Uα) of U ∈B by sets Uα ∈B contained in U , and for any object T ∈ C, the map which sendsf ∈ Hom(T,F (U)) to the family (ρU

Uα f ) ∈ Πα Hom(T,F (Uα)) is a bijection from Hom(T,F (U)) to the set

of all ( fα) such that ρUαV fα = ρ

UβV fβ for any pair of indices (α,β) and any V ∈B such that V ⊂ Uα ∩Uβ

6.

The condition is obviously necessary. To show that it is sucient, consider first a second basis B′

of the topology of X , contained in B, and show that if F ′′ denotes the presheaf induced by the subfamily(F (V ))V∈B′ , F ′′ is canonically isomorphic to F ′. Indeed, first the projective limit (for V ∈ B′, V ⊂ U) ofthe canonical morphisms F ′(U) → F (V ) is a morphism F ′(U) → F ′′(U) for all open U . If U ∈ B, thismorphism is an isomorphism, because by hypothesis the canonical morphisms F ′′(U) → F (V ) for V ∈ B′,V ⊂ U , factorize as F ′′(U)→F (U)→F (V ), and it is immediate to see that the composition of morphismsF (U)→F ′′(U) andF ′′(U)→F (U) thus defined are the identities. This being so, for all open U , the morphismsF ′′(U)→F ′′(W ) =F (W ) for W ∈B and W ⊂ U satisfy the conditions characterizing the projective limit ofF (W ) (W ∈B, W ⊂ U), which proves our assertion given the uniqueness of a projective limit up to isomorphism.

This being so, let U be any open set of X , (Uα) a covering of U by the open sets contained in U , and B′ thesubfamily of B formed by the sets of B contained in at least one Uα; it is clear that B

′ is still a basis of the 0I | 27topology of U , so F ′(U) (resp. F ′′(Uα)) is the projective limit of F (V ) for V ∈B′ and V ⊂ U (resp., V ⊂ Uα),the axiom (F) is then immediately verified by virtue of the definition of the projective limit.

When (F0) is satisfied, we will say by abuse of language that the presheaf F on the basis B is a sheaf.

(3.2.3). Let F , G be two presheaves on a basis B, with values in C; we define a morphism u :F →G as a family(uV )V∈B of morphisms uV : F (V ) → G (V ) satisfying the usual compatibility conditions with the restrictionmorphisms ρW

V . With the notation of (3.2.1), we have a morphism u′ : F ′ → G ′ of (ordinary) presheaves bytaking for u′U the projective limit of the uV for V ∈B and V ⊂ U ; the verification of the compatibility conditions

with the ρ′U′

U follows from the functorial properties of the projective limit.

(3.2.4). If the category C admits inductive limits, and if F is a presheaf on the basis B, with values in C, for eachx ∈ X the neighborhoods of x belonging toB form a cofinal set (for ⊃) in the set of neighborhoods of x , therefore,if F ′ is the (ordinary) presheaf corresponding to F , the stalk F ′x is equal to lim−→B

F (V ) over the set of V ∈B

containing x . If u :F →G is morphism of presheaves on B with values in C, u′ :F ′→G ′ the correspondingmorphism of ordinary presheaves, u′x is likewise the inductive limit of the morphisms uV :F (V )→G (V ) forV ∈B, x ∈ V .

(3.2.5). We return to the general conditions of (3.2.1). If F is an ordinary sheaf with values in C, F1 the sheafon B obtained by the restriction of F to B, then the ordinary sheaf F ′1 obtained from F1 by the procedureof (3.2.1) is canonically isomorphic to F , by virtue of the condition (F) and the uniqueness properties of theprojective limit. We identify the ordinary sheaf F with F ′1.

If G is a second (ordinary) sheaf on X with values in C, and u :F →G a morphism, the preceding remarkshows that the data of the uV : F (V ) → G (V ) for only the V ∈ B completely determines u; conversely, it issucient, the uV being given for V ∈B, to verify the commutative diagram with the restriction morphisms ρW

Vfor V ∈B, W ∈B, and V ⊂W , for there to exist a morphism u′ and a unique F in G such that u′V = uV foreach V ∈B (3.2.3).

(3.2.6). Suppose that C admits projective limits. Then the category of sheaves on X with values in C admitsprojective limits; if (Fλ) is a projective system of sheaves on X with values in C, the F (U) = lim←−λFλ(U) indeeddefine a presheaf with values in C, and the verification of the axiom (F) follows from the transitivity of projectivelimits; the fact that F is then the projective limit of the Fλ is immediate.

When C is the category of sets, for each projective system (Hλ) such that Hλ is a subsheaf of Fλ for each λ, 0I | 28lim←−λHλ canonically identifies with a subsheaf of lim←−λFλ . If C is the category of abelian groups, the covariantfunctor lim←−λFλ is additive and left exact.

3.3. Gluing sheaves.

(3.3.1). Suppose still that the category C admits (generalized) projective limits. Let X be a topological space,U= (Uλ)λ∈L an open cover of X , and for each λ ∈ L, let Fλ be a sheaf on Uλ , with values in C; for each pair ofindices (λ,µ), suppose that we are given an isomorphism θλµ :Fµ|(Uλ ∩ Uµ)'F|(Uλ ∩ Uµ); in addition, supposethat for each triple (λ,µ,ν), if we denote by θ ′λµ, θ

′µν , θ

′λν the restrictions of θλµ, θµν , θλν to Uλ ∩Uµ ∩Uν , then we

6It also means that the pair formed by F (U) and the ρα = ρUUα

is a solution to the universal problem defined in (3.1.1) by the data of

Aα =F (Uα), Aαβ = ΠF (V ) (for V ∈B such that V ⊂ Uα ∩Uβ) and ραβ = (ρ′′V ) :F (Uα)→ ΠF (V ) defined by the condition that for V ∈B,

V ′ ∈B, W ∈B, V ∪ V ′ ⊂ Uα ∩ Uβ , W ⊂ V ∩ V ′, ρVW ρ

′′V = ρ

V ′W ρ

′′V ′ .

3. SUPPLEMENT ON SHEAVES 24

have θ ′λν = θ′λµ θ

′µν (gluing condition for the θλµ). Then there exists a sheaf F on X , with values in C, and for

each λ an isomorphism ηλ :F|Uλ 'Fλ such that, for each pair (λ,µ), if we denote by η′λ and η′µ the restrictions

of ηλ and ηµ to Uλ ∩ Uµ, then we have θλµ = η′λ η′−1µ ; in addition, F and the ηλ are determined up to unique

isomorphism by these conditions. The uniqueness indeed follows immediately from (3.2.5). To establish theexistence of F , denote by B the open basis consisting of the open sets contained in at least one Uλ , and for eachU ∈B, choose (by the Hilbert function τ) one of the Fλ(U) for one of the λ such that U ⊂ Uλ ; if we denote thisobject by F (U), the ρV

U for U ⊂ V , U ∈B, V ∈B are defined in an evident way (by means of the θλµ), and thetransitivity conditions is a consequence of the gluing condition; in addition, the verification of (F0) is immediate,so the presheaf on B thus clearly defines a sheaf, and we deduce by the general procedure (3.2.1) an (ordinary)sheaf still denoted F and which answers the question. We say that F is obtained by gluing the Fλ by means of theθλµ and we usually identify the Fλ and F|Uλ by means of the ηλ .

It is clear that each sheaf F on X with values in C can be considered as being obtained by the gluing of thesheaves Fλ =F|Uλ (where (Uλ) is an arbitrary open cover of X ), by means of the isomorphisms θλµ reduced tothe identity.

(3.3.2). With the same notation, let Gλ be a second sheaf on Uλ (for each λ ∈ L) with values in C, and for eachpair (λ,µ) let us be given an isomorphism ωλµ : Gµ|(Uλ ∩ Uµ)' Gλ |(Uλ ∩ Uµ), these isomorphisms satisfying thegluing condition. Finally, suppose that we are given for each λ a morphism uλ :Fλ →Gλ , and that the diagrams

(3.3.2.1) Fµ|(Uλ ∩ Uµ)uµ //

Gµ|(Uλ ∩ Uµ)

Fλ |(Uλ ∩ Uµ)

uλ // Gλ |(Uλ ∩ Uµ)

are commutative. Then, if G is obtained by gluing the Gλ by means of the ωλµ, there exists a unique morphismu :F →G such that the diagrams 0I | 29

F|Uλu|Uλ //

G|Uλ

Fλ

uλ // Gλare commutative; this follows immediately from (3.2.3). The correspondence between the family (uλ) and u is ina functorial bijection with the subset of Πλ Hom(Fλ ,Gλ) satisfying the conditions (3.3.2.1) on Hom(F ,G ).

(3.3.3). With the notation of (3.3.1), let V be an open set of X ; it is immediate that the restrictions to V ∩Uλ ∩Uµof the θλµ satisfy the gluing condition for the induced sheaves Fλ |(V ∩ Uλ) and that the sheaves on V obtainedby gluing the latter identifies canonically with F|V .

3.4. Direct images of presheaves.

(3.4.1). Let X , Y be two topological spaces, ψ : X → Y a continuous map. Let F be a presheaf on X with valuesin a category C; for each open U ⊂ Y , let G (U) =F (ψ−1(U)), and if U , V are two open subsets of Y such thatU ⊂ V , let ρV

U be the morphism F (ψ−1(V ))→F (ψ−1(U)); it is immediate that the G (U) and the ρVU define a

presheaf on Y with values in C, that we call the direct image of F by ψ and we denote it by ψ∗(F ). If F is a sheaf,we immediately verify the axiom (F) for the presheaf G , so ψ∗(F ) is a sheaf.

(3.4.2). Let F1, F2 be two presheaves of X with values in C, and let u : F1 → F2 be a morphism. WhenU varies over the set of open subsets of Y , the family of morphisms uψ−1(U) : F1(ψ−1(U)) → F2(ψ−1(U))satisfies the compatibility conditions with the restriction morphisms, and as a result defines a morphismψ∗(u) : ψ∗(F1)→ ψ∗(F2). If v :F2→F3 is a morphism from F2 to a third presheaf on X with values in C, wehave ψ∗(v u) = ψ∗(v)ψ∗(u); in other words, ψ∗(F ) is a covariant functor in F , from the category of presheaves(resp. sheaves) on X with values in C, to that of presheaves (resp. sheaves) on Y with values in C.

(3.4.3). Let Z be a third topological space, ψ′ : Y → Z a continuous map, and let ψ′′ = ψ′ ψ. It is clear that wehave ψ′′∗ (F ) = ψ

′∗(ψ∗(F )) for each presheaf F on X with values in C; in addition, for each morphism u :F →G

of such presheaves, we have ψ′′∗ (u) = ψ′∗(ψ∗(u)). In other words, ψ′′∗ is the composition of the functors ψ′∗ and ψ∗,

and this can be written as(ψ′ ψ)∗ = ψ′∗ ψ∗.

In addition, for each open set U of Y , the image under the restriction ψ|ψ−1(U) of the induced presheafF|ψ−1(U) is none other than the induced presheaf ψ∗(F )|U .

3. SUPPLEMENT ON SHEAVES 25

(3.4.4). Suppose that the category C admits inductive limits, and let F be a presheaf on X with values in C;for all x ∈ X , the morphisms Γ(ψ−1(U),F )→Fx (U an open neighborhood of ψ(x) in Y ) form an inductivelimit, which gives by passing to the limit a morphism ψx : (ψ∗(F ))ψ(x) → Fx of the stalks; in general, these 0I | 30morphisms are neither injective or surjective. It is functorial; indeed, if u :F1→F2 is a morphism of presheaveson X with values in C, the diagram

(ψ∗(F1))ψ(x)ψx //

(ψ∗(u))ψ(x)

(F1)x

ux

(ψ∗(F2))ψ(x)

ψx // (F2)x

is commutative. If Z is a third topological space, ψ′ : Y → Z a continuous map, and ψ′′ = ψ′ ψ, then we haveψ′′x = ψx ψ′ψ(x) for x ∈ X .

(3.4.5). Under the hypotheses of (3.4.4), suppose in addition that ψ is a homeomorphism from X to the subspaceψ(X ) of Y . Then, for each x ∈ X , ψx is an isomorphism. This applies in particular to the canonical injection j ofa subset X of Y into Y .

(3.4.6). Suppose that C be the category of groups, or of rings, etc. If F is a sheaf on X with values in C, ofsupport S, and if y 6∈ ψ(S), then it follows from the definition of ψ∗(F ) that (ψ∗(F ))y = 0, or in other words,

that the support of ψ∗(F ) is contained in ψ(S); but it is not necessarily contained in ψ(S). Under the samehypotheses, if j is the canonical injection of a subset X of Y into Y , the sheaf j∗(F ) induces F on X ; if moreoverX is closed in Y , j∗(F ) is the sheaf on Y which induces F on X and 0 on Y − X (G, II, 2.9.2), but it is in generaldistinct from the latter when we suppose that X is locally closed but not closed.

3.5. Inverse images of presheaves.

(3.5.1). Under the hypotheses of (3.4.1), if F (resp. G ) is a presheaf on X (resp. Y ) with values in C, theneach morphism u : G → ψ∗(F ) of presheaves on Y is called a φ -morphism from G to F , and we denote italso by G → F . We denote also by Homφ (G ,F ) the set of HomY (G ,ψ∗(F )) the ψ-morphisms from G toF . For each pair (U , V ), where U is an open set of X , V an open set of Y such that ψ(U) ⊂ V , we have amorphism uU ,V : G (U)→F (U) by composing the restriction morphism F (ψ−1(V ))→F (U) and the morphismuV : G (V )→ ψ∗(F )(V ) =F (ψ−1(V )); it is immediate that these morphisms render commutative the diagrams

(3.5.1.1) G (V )uU ,V //

F (U)

G (V ′)

uU′ ,V ′ // F (U ′)

for U ′ ⊂ U , V ′ ⊂ V , ψ(U ′) ⊂ V ′. Conversely, the data of a family (uU ,V ) of morphisms rendering commutativethe diagrams (3.5.1.1) define a ψ-morphism u, since it suces to take uV = uψ−1(V ),V . 0I | 31

If the category C admits (generalized) projective limits, and if B, B′ are bases for the topologies of X andY respectively, to define a ψ-morphism u of sheaves, we can restrict to giving the uU ,V for U ∈B, V ∈B′, andψ(U) ⊂ V , satisfying the compatibility conditions of (3.5.1.1) for U , U ′ in B and V , V ′ in B′; it indeed suces todefine uW , for each open W ⊂ Y , as the projective limit of the uU ,V for V ∈B′ and V ⊂W , U ∈B and ψ(U) ⊂ V .

When the category C admits inductive limits, we have, for each x ∈ X , a morphism G (V )→F (ψ−1(V ))→Fx ,for each open neighborhood V of ψ(x) in Y , and these morphisms form an inductive system which gives bypassing to the limit a morphism Gψ(x)→Fx .

(3.5.2). Under the hypotheses of (3.4.3), let F , G ,H be presheaves with values in C on X , Y , Z respectively, andlet u : G → ψ∗(F ), v :H → ψ′∗(G ) be a ψ-morphism and a ψ′-morphism respectively. We obtain a ψ′′-morphism

w : Hv−→ ψ′∗(G )

ψ′∗(u)−−−→ ψ′∗(ψ∗(F )) = ψ′′∗ (F ), that we call, by definition, the composition of u and v. We cantherefore consider the pairs (X ,F ) consisting of a topological space X and a presheaf F on X (with values in C)as forming a category, the morphisms being the pairs (ψ,θ) : (X ,F )→ (Y,G ) consisting of a continuous mapψ : X → Y and of a ψ-morphism θ : G →F .

(3.5.3). Let ψ : X → Y be a continuous map, G a presheaf on Y with values in C. We call the inverse image ofG under ψ the pair (G ′,ρ), where G ′ is a sheaf on X with values in C, and ρ : G →G ′ a ψ-morphism (in otherwords a homomorphism G → ψ∗(G ′)) such that, for each sheaf F on X with values in C, the map

(3.5.3.1) HomX (G ′,F ) −→ Homψ(G ,F ) −→ HomY (G ,ψ∗(F ))

3. SUPPLEMENT ON SHEAVES 26

sending v to ψ∗(v) ρ, is a bijection; this map, being functorial in F , then defines an isomorphism of functors inF . The pair (G ′,ρ) is the solution of a universal problem, and we say it is determined up to unique isomorphismwhen it exists. We then write G ′ = ψ∗(G ), ρ = ρG , and by abuse of language, we say that ψ∗(G ) is the inverseimage sheaf of G under ψ, and we agree that ψ∗(G ) is considered as equipped with a canonical ψ-morphismρG : G → ψ∗(G ), that is to say the canonical homomorphism of presheaves on Y :

(3.5.3.2) ρG : G −→ ψ∗(ψ∗(G )).For each homomorphism v : ψ∗(G )→F (where F is a sheaf on X with values in C), we put v[ = ψ∗(v)ρG :

G → ψ∗(F ). By definition, each morphism of presheaves u : G → ψ∗(F ) is of the form v[ for a unique v, whichwe will denote u]. In other words, each morphism u : G → ψ∗(F ) of presheaves factorizes in a unique way as

(3.5.3.3) u : GρG−→ ψ∗(ψ∗(G ))

ψ∗(u])−−−→ ψ∗(F ). 0I | 32

(3.5.4). Suppose now that the category C be such7 that each presheaf F on Y with values in C admits an inverseimage under ψ, and we denote it by ψ∗(G ).

We will see that we can define ψ∗(G ) as a covariant functor in G , from the category of presheaves on Y withvalues in C, to that of sheaves on X with values in C, in such a way that the isomorphism v 7→ v[ is an isomorphismof bifunctors

(3.5.4.1) HomX (ψ∗(G ),F )' HomY (G ,ψ∗(F ))in G and F .

Indeed, for each morphism w : G1 → G2 of presheaves on Y with values in C, consider the composite

morphism G1w−→G2

ρG2−→ ψ∗(ψ∗(G2)); to it corresponds a morphism (ρG2w)] : ψ∗(G1)→ ψ∗(G2), that we denote

by ψ∗(w). We therefore have, according to (3.5.3.3),

(3.5.4.2) ψ∗(ψ∗(w)) ρG1= ρG2

w.

For each morphism u : G2→ ψ∗(F ), where F is a sheaf on X with values in C, we have, according to (3.5.3.3),(3.5.4.2), and the definition of u[, that

(u] ψ∗(w))[ = ψ∗(u]) ψ∗(ψ∗(w)) ρG1= ψ∗(u]) ρG2

w= u w

where again

(3.5.4.3) (u w)] = u] ψ∗(w).

If we take in particular for u a morphism G2w′−→G3

ρG3−→ ψ∗(ψ∗(G3)), it becomes ψ∗(w′ w) = (ρG3w′ w)] =

(ρG3w′)] ψ∗(w) = ψ∗(w′) ψ∗(w), hence our assertion.Finally, for each sheaf F on X with values in C, let iF be the identity morphism of ψ∗(F ) and denote by

σF : ψ∗(ψ∗(F ) −→Fthe morphism (iF )]; the formula (3.5.4.3) gives in particular the factorization

(3.5.4.4) u] : ψ∗(G )ψ∗(u)−−−→ ψ∗(ψ∗(F ))

σF−→F

for each morphism u : G → ψ∗(F ). We say that the morphism σF is canonical.

(3.5.5). Let ψ′ : Y → Z be a continuous map, and suppose that each presheaf H on Z with values in C admitsan inverse image ψ′∗(H ) under ψ′. Then (with the hypotheses of (3.5.4)) each presheaf H on Z with values inC admits an inverse image under ψ′′ = ψ′ ψ and we have a canonical functorial isomorphism

(3.5.5.1) ψ′′∗(H )' ψ∗(ψ′∗(H )).This indeed follows immediately from the definitions, taking into account that ψ′′∗ = ψ

′∗ ψ∗. In addition, if 0I | 33

u : G → ψ∗(F ) is a ψ-morphism, v :H → ψ′∗(G ) a ψ′-morphism, and w = ψ′∗(u) v their composition (3.5.2),

then we have immediately that w] is the composite morphism

w] : ψ∗(ψ′∗(H ))ψ∗(v])−−−→ ψ∗(G ) u]

−→F .

(3.5.6). We take in particular for ψ the identity map 1X : X → X . Then if the inverse image under ψ of a presheafF on X with values in C exists, we say that this inverse image is the sheaf associated to the presheaf F . Each

morphism u :F →F ′ from F to a sheaf F ′ with values in C factorizes in a unique way as FρF−→ 1∗X (F )

u]−→F ′.

7In the book mentioned in the introduction, we will give very general conditions on the category C ensuring the existence of inverseimages of presheaves with values in C.

3. SUPPLEMENT ON SHEAVES 27

3.6. Simple and locally simple sheaves.

(3.6.1). We say that a presheaf F on X , with values in C, is constant if the canonical morphisms F (X )→F (U)are isomorphisms for each nonempty open U ⊂ X ; we note that F is not necessarily a sheaf. We say that a sheafis simple if it is the associated sheaf (3.5.6) of a constant presheaf. We say that a sheaf F is locally simple if eachx ∈ X admits an open neighborhood U such that F|U is simple.

(3.6.2). Suppose that X is irreducible (2.1.1); then the following properties are equivalent:(a) F is a constant presheaf on X ;(b) F is a simple sheaf on X ;(c) F is a locally simple sheaf on X .

Indeed, let F be a constant presheaf on X ; if U , V are two nonempty open sets in X , then U ∩V is nonempty,so F (X )→F (U)→F (U ∩ V ) and F (X )→F (U) are isomorphisms, and similarly both F (U)→F (U ∩ V )and F (V )→F (U ∩ V ) are isomorphisms. We therefore conclude immediately that the axiom (F) of (3.1.2) isclearly satisfied, F is isomorphic to its associated sheaf, and as a result (a) implies (b).

Now let (Uα) be an open cover of X by nonempty open sets and let F be a sheaf on X such that F|Uαis simple for each α; as Uα is irreducible, F|Uα is a constant presheaf according to the above. As Uα ∩ Uβis not empty, F (Uα)→F (Uα ∩ Uβ) and F (Uβ)→F (Uα ∩ Uβ) are isomorphisms, hence we have a canonicalisomorphism θαβ :F (Uα)→F (Uβ) for each pair of indices. But then if we apply the condition (F) for U = X ,we see that for each index α0, F (Uα0

) and the θα0α are solutions to the universal problem, which (according tothe uniqueness) implies that F (X )→F (Uα0

) is an isomorphism, and hence proves that (c) implies (a).0I | 34

3.7. Inverse images of presheaves of groups or rings.

(3.7.1). We will show that when we take C to be the category of sets, the inverse image under ψ for each presheafG with values in C always exists (the notation and hypotheses on X , Y , ψ being that of (3.5.3)). Indeed, for eachopen U ⊂ X , define G ′(U) as follows: an element s′ of G ′(U) is a family (s′x)x∈U , where s′x ∈ Gψ(x) for each x ∈ U ,and where, for each x ∈ U , the following condition is satisfied: there exists an open neighborhood V of ψ(x) inY , a neighborhood W ⊂ ψ−1(V )∩ U of x , and an element s ∈ G (V ) such that s′z = sψ(x) for all z ∈W . We verifyimmediately that U 7→ G ′(U) clearly satisfies the axioms of a sheaf.

Now let F be a sheaf of sets on X , and let u : G → ψ∗(F ), v : G ′→F be morphisms. We define u] and v[ inthe following manner: if s′ is a section of G ′ over a neighborhood U of x ∈ X and if V is an open neighborhoodof ψ(x) and s ∈ G (V ) such that we have s′z = sψ(x) for z in a neighborhood of x contained in ψ−1(V )∩ U , wetake u]x(s

′x) = uψ(x)(sψ(x)). Similarly, if s ∈ G (V ) (V open in Y ), v[(s) is the section of F over ψ−1(V ), the image

under v of the section s′ of G ′ such that s′x = sψ(x) for all x ∈ ψ−1(V ). In addition, the canonical homomorphism(3.5.3) ρ : G → ψ∗(ψ∗(G )) is defined in the following manner: for each open V ⊂ Y and each section s ∈ Γ(V,G ),ρ(s) is the section (sψ(x))x∈ψ−1(V ) of ψ∗(G ) over ψ−1(V ). The verification of the relations (u])[ = u, (v[)] = v, andv[ = ψ∗(v) ρ is immediate, and proves our assertion.

We check that, if w : G1 → G2 is a homomorphism of sheaves of sets on Y , ψ∗(w) is expressed in thefollowing manner: if s′ = (s′x)x∈U is a section of ψ∗(G1) over an open set U of X , then (ψ∗(w))(s′) is the family(wψ(x)(s′x))x∈U . Finally, it is immediate that for each open set V of Y , the inverse image of G|V under therestriction of ψ to ψ−1(V ) is identical to the induced sheaf ψ∗(G )|ψ−1(V ).

When ψ is the identity 1X , we recover the definition of a sheaf of sets associated to a presheaf (G, II, 1.2).The above considerations apply without change when C is the category of groups or of rings (not necessarilycommutative).

When X is any subset of a topological space Y , and j the canonical injection X → Y , for each sheaf G on Ywith values in a category C, we call the induced sheaf of X by G the inverse image j∗(G ) (whenever it exists); forthe sheaves of sets (or of groups, or of rings) we recover the usual definition (G, II, 1.5).

(3.7.2). Keeping the notation and hypotheses of (3.5.3), suppose that G is a sheaf of groups (resp. of rings) on Y .The definition of sections of ψ∗(G ) (3.7.1) shows (taking into account (3.4.4)) that the homomorphism of stalksψx ρψ(x) : Gψ(x)→ (ψ∗(G ))x is a functorial isomorphism in G , that identifies the two stalks; with this identification,u]x is identical to the homomorphism defined in (3.5.1), and in particular, we have Supp(ψ∗(G )) = ψ−1(Supp(G )).

An immediate consequence of this result is that the functor ψ∗(G ) is exact in G on the abelian category ofsheaves of abelian groups.

0I | 353.8. Sheaves on pseudo-discrete spaces.

(3.8.1). Let X be a topological space whose topology admits a basis B consisting of open quasi-compact subsets.Let F be a sheaf of sets on X ; if we equip each of the F (U) with the discrete topology, U 7→ F (U) is a presheaf

4. RINGED SPACES 28

of topological spaces. We will see that there exists a sheaf of topological spaces F ′ associated to F (3.5.6) such thatΓ(U ,F ′) is the discrete space F (U) for each open quasi-compact subsets U . It will suce to show that thepresheaf U 7→ F (U) of discrete topological spaces on B satisfy the condition (F0) of (3.2.2), and more generallythat if U is an open quasi-compact subset and if (Uα) is a cover of U by sets of B, then the least fine topologyT on Γ(U ,F ) renders continuous the maps Γ(U ,F )→ Γ(Uα,F ) is the discrete topology. There exists a finitenumber of indices αi such that U =

⋃

i Uαi. Let s ∈ Γ(U ,F ) and let si be its image in Γ(Uαi

,F ); the intersectionof the inverse images of the sets si is by definition a neighborhood of s for T ; but since F is a sheaf of setsand the Uαi

cover U , this intersection is reduced to s, hence our assertion.We note that if U is an open non quasi-compact subset of X , the topological space Γ(U ,F ′) still has Γ(U ,F )

as the underlying set, but the topology is not discrete in general: it is the least fine rendering commutative themaps Γ(U ,F )→ Γ(V,F ), for V ∈B and V ⊂ U (the Γ(V,F ) being discrete).

The above considerations apply without modification to sheaves of groups or of rings (not necessarilycommutative), and associate to them sheaves of topological groups or topological rings, respectively. To summarize,we say that the sheaf F ′ is the pseudo-discrete sheaf of spaces (resp. groups, rings) associated to a sheaf of sets (resp.groups, rings) F .

(3.8.2). Let F , G be two sheaves of sets (resp. groups, rings) on X , u :F → G a homomorphism. Then u isthus a continuous homomorphism F ′→G ′, if we denote by F ′ and G ′ the pseudo-discrete sheaves associated toF and G ; this follows in eect from (3.2.5).

(3.8.3). Let F be a sheaf of sets, H a subsheaf of F , F ′ and H ′ the pseudo-discrete sheaves associated to Fand H respectively. Then, for each open U ⊂ X , Γ(U ,H ′) is closed in Γ(U ,F ′): indeed, it is the intersectionof the inverse images of the Γ(V,H ) (for V ∈B, V ⊂ U) under the continuous maps Γ(U ,F )→ Γ(V,F ), andΓ(V,H ) is closed in the discrete space Γ(V,F ).

§4. Ringed spaces

4.1. Ringed spaces, sheaves of A -modules, A -algebras.

(4.1.1). A ringed space (resp. topologically ringed space) is a pair (X ,A ) consisting of a topological space X anda sheaf of (not necessarily commutative) rings (resp. of a sheaf of topological rings) A on X ; we say that X isthe underlying topological space of the ringed space (X ,A ), and A the structure sheaf. The latter is denoted OX ,and its stalk at a point x ∈ X is denoted OX ,x or simply Ox when there is no chance of confusion.

We denote by 1 or e the unit section of OX over X (the unit element of Γ(X ,OX )).As in this treatise we will have to consider in particular sheaves of commutative rings, it will be understood,

when we speak of a ringed space (X ,A ) without specification, that A is a sheaf of commutative rings.The ringed spaces with not-necessarily-commutative structure sheaves (resp. the topologically ringed spaces)

form a category, where we define a morphism (X ,A )→ (Y,B) as a couple (ψ,θ) =Ψ consisting of a continuousmap ψ : X → Y and a ψ-morphism θ :B →A (3.5.1) of sheaves of rings (resp. of sheaves of topological rings); ErrIIthe composition of a second morphism Ψ′ = (ψ′,θ ′) : (Y,B)→ (Z ,C ) and of Ψ, denoted Ψ′′ =Ψ′ Ψ, is themorphism (ψ′′,θ ′′) where ψ′′ = ψ′ ψ, and θ ′′ is the composition of θ and θ ′ (equal to ψ′∗(θ) θ

′, cf. (3.5.2)).For ringed spaces, remember that we then have θ ′′] = θ] ψ∗(θ ′]) (3.5.5); therefore if θ ′] and θ] are injective(resp. surjective), then the same is true of θ ′′], taking into account that ψx ρψ(x) is an isomorphism for all x ∈ X(3.7.2). We verify immediately, thanks to the above, that when ψ is an injective continuous map and when θ] is asurjective homomophism of sheaves of rings, the morphism (ψ,θ) is a monomorphism (T, 1.1) in the category ofringed spaces.

By abuse of language, we will often replace ψ by Ψ in notation, for example in writing Ψ−1(U) in place ofψ−1(U) for a subset U of Y , when the is no risk of confusion.

(4.1.2). For each subset M of X , the pair (M ,A|M) is evidently a ringed space, said to be induced on M by theringed space (X ,A ) (and is still called the restriction of (X ,A ) to M). If j is the canonical injection M → Xand ω is the identity map of A|M , ( j,ω[) is a monomorphism (M ,A|M)→ (X ,A ) of ringed spaces, calledthe canonical injection. The composition of a morphism Ψ : (X ,A )→ (Y,B) and this injection is called therestriction of Ψ to M .

(4.1.3). We will not revisit the definitions of A -modules or algebraic sheaves on a ringed space (X ,A ) (G, II, 2.2);when A is a sheaf of not necessarily commutative rings, by A -module we will always mean “left A -module”unless expressly stated otherwise. TheA -submodules ofA will be called sheaves of ideals (left, right, or two-sided)in A or A -ideals.

4. RINGED SPACES 29

When A is a sheaf of commutative rings, and in the definition of A -modules, we replace everywhere themodule structure by that of an algebra, we obtain the definition of an A -algebra on X . It is the same to say thatan A -algebra (not necessarily commutative) is a A -module C , given with a homomorphism of A -modulesφ :C ⊗A C →C and a section e over X , such that: 1st the diagram

C ⊗A C ⊗A Cφ⊗1 //

1⊗φ

C ⊗A C

φ

C ⊗A Cφ // C

is commutative; 2nd for each open U ⊂ X and each section s ∈ Γ(U ,C ), we have φ ((e|U)⊗ s) =φ (s⊗ (e|U)) = s.We say that C is a commutative A -algebra if the diagram

C ⊗A Cσ //

φ$$

C ⊗A C

φzz

C

is commutative, σ denoting the canonical symmetry (twist) map of the tensor product C ⊗A C .The homomorphisms of A -algebras are also defined as the homomorphisms of A -modules in (G, II, 2.2),

but naturally no longer form an abelian group.IfM is an A -submodule of an A -algebra C , the A -subalgebra of C generated byM is the sum of the images

of the homomorphisms⊗nM →C (for each n¾ 0). This is also the sheaf associated to the presheaf U 7→ B(U)

of algebras, B(U) being the subalgebra of Γ(U ,C ) generated by the submodule Γ(U ,M ).

(4.1.4). We say that a sheaf of rings A on a topological space X is reduced at a point x in X if the stalk Ax is areduced ring (1.1.1); we say that A is reduced if it is reduced at all points of X . Recall that a ring A is calledregular if each of the local rings Ap (where p varies over the set of prime ideals of A) is a regular local ring; wewill say that a sheaf of rings A on X is regular at a point x (resp. regular) if the stalk Ax is a regular ring (resp.if A is regular at each point). Finally, we will say that a sheaf of rings A on X is normal at a point x (resp.normal) if the stalk Ax is an integral and integrally closed ring (resp. if A is normal at each point). We will saythat a ringed space (X ,A ) has any of these preceeding properties if the sheaf of rings A has that property.

A graded sheaf of rings A is by definition a sheaf of rings that is the direct sum (G, II, 2,7) of a family(An)n∈Z of sheaves of abelian groups with the conditions AmAn ⊂Am+n; a graded A -module is an A -module Fthat is the direct sum of a family (Fn)n∈Z of sheaves of abelian groups, satisfying the conditions AmFn ⊂Fm+n.It is equivalent to say that (Am)x(An)x ⊂ (Am+n)x (resp. (Am)x(Fn)x ⊂ (Fm+n)x) for each point x .

(4.1.5). Given a ringed space (X ,A ) (not necessarily commutative), we will not recall here the definitions ofthe bifunctors F ⊗A G , HomA (F ,F ), and HomA (F ,G ) (G, II, 2.8 and 2.2) in the categories of left or right(depending on the case) A -modules, with values in the category of sheaves of abelian groups (or more generallyof C -modules, if C is the center of A ). The stalk (F ⊗A G )x for each point x ∈ X canonically identifies withFx ⊗Ax

Gx and we define a canonical and functorial homomorphism (HomA (F ,G ))x → HomAx(Fx ,Gx) which

is in general neither injective nor surjective. The bifunctors considered above are additive and in particular,commute with finite direct limits; F ⊗A G is right exact in F and in G , commutes with inductive limits, andA ⊗A G (resp. F ⊗A A ) canonically identifies with G (resp. F ). The functorsHomA (F ,G ) and HomA (F ,G )are left exact in F and G ; more precisely, if we have an exact sequence of the form 0→ G ′ → G → G ′′, thesequence

0 −→HomA (F ,G ′) −→HomA (F ,G ) −→HomA (F ,G ′′)is exact, and if we have an exact sequence of the form F ′→F →F ′′→ 0, the sequence

0 −→HomA (F ′′,G ) −→HomA (F ,G ) −→HomA (F ′,G )is exact, with the analogous properties for the functor Hom. In addiiton, HomA (A ,G ) canonically identifieswith G ; finally, for each open U ⊂ X , we have

Γ(U ,HomA (F ,G ) = HomA|U(F|U ,G|U).

For each left (resp. right) A -module, we define the dual of F and denote it by F∨ the right (resp. left)A -module HomA (F ,A ).

Finally, if A is a sheaf of commutative rings, F an A -module, U 7→ ∧pΓ(U ,F ) is a presheaf whoseassociated sheaf is an A -module denoted ∧pF and is called the p-th exterior power of F ; we verify easily that

4. RINGED SPACES 30

the canonical map of the presheaf U 7→ ∧pΓ(U ,F ) to the associated sheaf ∧pF is injective, and for each x ∈ X ,(∧pF )x = ∧p(Fx). It is clear that ∧pF is a covariant functor in F .

(4.1.6). Suppose that A is a sheaf of not-necessarily-commutative rings, J a left sheaf of ideals of A , F anleft A -module; we then denote by JF the A -submodule of F , the image of J ⊗Z F (where Z is the sheafassociated to the constant presheaf U 7→ Z) under the canonical map J ⊗Z F → F ; it is clear that for eachx ∈ X , we have (JF )x = JxFx . When A is commutative, JF is also the canonical image of J ⊗A F →F .It is immediate that JF is also the A -module associated to the presheaf U 7→ Γ(U ,J )Γ(U ,F ). If J1, J2 aretwo left sheaves of ideals of A , we have J1(J2F ) = (J1J2)F .

(4.1.7). Let (Xλ ,Aλ)λ∈L be a family of ringed spaces; for each couple (λ,µ), suppose we are given an opensubset Vλµ of Xλ , and an isomorphism of ringed spaces φλµ : (Vµλ ,Aµ|Vλµ) ' (Vλµ,Aλ |Vλµ), with Vλλ = Xλ , φλλbeing the identity. Furthermore, suppose that, for each triple (λ,µ,ν), if we denote by φ ′µλ the restriction of φµλto Vλµ ∩ Vλν , φ ′µλ is an isomorphism from (Vλµ ∩ Vλν ,Aλ |(Vλµ ∩ Vλν)) to (Vµν ∩ Vµλ ,Aµ|(Vµν ∩ Vµλ)) and that wehave φ ′λν =φ

′λµ φ

′µν (gluing condition for the φλµ). We can first consider the topological space obtained by gluing

(by means of the φλµ) of the Xλ along the Vλµ; if we identify Xλ with the corresponding open subset X ′λ in X , 0I | 39the hypotheses imply that the three sets Vλµ ∩ Vλν , Vµν ∩ Vµλ , Vνλ ∩ Vνµ identify with X ′λ ∩ X ′µ ∩ X ′ν . We can alsotransport to X ′λ the ringed space structure of Xλ , and if A ′λ are the transported sheaves of rings correspondingto the Aλ , the A ′λ satisfy the gluing condition (3.3.1) and therefore define a sheaf of rings A on X ; we say that(X ,A ) is the ringed space obtained by gluing the (Xλ ,Aλ) along the Vλµ, by means of the φλµ.

4.2. Direct image of an A -module.

(4.2.1). Let (X ,A ), (Y,B) be two ringed spaces, Ψ = (ψ,θ) a morphism (X ,A )→ (Y,B); ψ∗(A ) is then asheaf of rings on Y , and θ a homomorphism B → ψ∗(A ) of sheaves of rings. Then let F be an A -module; thedirect image ψ∗(F ) is a sheaf of abelian groups on Y . In addition, for each open U ⊂ Y ,

Γ(U ,ψ∗(F )) = Γ(ψ−1(U),F )

is equipped with the structure of a module over the ring Γ(U ,ψ∗(A )) = Γ(ψ−1(U),A ); the bilinear maps whichdefine these structures are compatible with the restriction operations, defining on ψ∗(F ) the structure of aψ∗(A )-module. The homomorphism θ :B → ψ∗(A ) then defines also on ψ∗(F ) a B -module structure; we saythat this B -module is the direct image of F under the morphism Ψ, and we denote it by Ψ∗(F ). If F1, F2 are twoA -modules over X and u an A -homomorphism F1→F2, it is immediate (by considering the sections over theopen subsets of Y ) that ψ∗(u) is a ψ∗(A )-homomorphism ψ∗(F1)→ ψ∗(F2), and a fortiori a B -homomorphismΨ∗(F1)→Ψ∗(F2); as a B -homomorphism, we denote it by Ψ∗(u). So we see that Ψ∗ is a covariant functor fromthe category of A -modules to that of B -modules. In addition, it is immediate that this functor is left exact (G,II, 2.12).

On ψ∗(A ), the structure of a B -module and the structure of a sheaf of rings define a B -algebra structure;we denote by Ψ∗(A ) this B -algebra.

(4.2.2). LetM , N be two A -modules. For each open set U of Y , we have a canonical map

Γ(ψ−1(U),M )×Γ(ψ−1(U),N ) −→ Γ(ψ−1(U),M ⊗A N )

which is bilinear over the ring Γ(ψ−1(U),A ) = Γ(U ,ψ∗(A )), and a fortiori over Γ(U ,B); it therefore defines ahomomorphism

Γ(U ,Ψ∗(M ))⊗Γ(U ,B) Γ(U ,Ψ∗(N )) −→ Γ(U ,Ψ∗(M ⊗A N ))and as we check immediately that these homomorphisms are compatible with the restriction operations, theygive a canonical functorial homomorphism of B -modules

(4.2.2.1) Ψ∗(M )⊗B Ψ∗(N ) −→Ψ∗(M ⊗A N )

which is in general neither injective nor surjective. If P is a third A -module, we check immediately that the 0I | 40diagram

(4.2.2.2) Ψ∗(M )⊗B Ψ∗(N )⊗B Ψ∗(P ) //

Ψ∗(M ⊗A N )⊗B Ψ∗(P )

Ψ∗(M )⊗B Ψ∗(N ⊗A P ) // Ψ∗(M ⊗A N ⊗A P )

is commutative.

4. RINGED SPACES 31

(4.2.3). LetM ,N be twoA -modules. For each open U ⊂ Y , we have by definition that Γ(ψ−1(U),HomA (M ,N )) =HomA|V (M|V,N |V ), where we put V = ψ−1(U); the map u 7→Ψ∗(u) is a homomorphism

HomA|V (M|V,N |V ) −→ HomB|U(Ψ∗(M )|U ,Ψ∗(N )|U)

on the Γ(U ,B)-module structures; these homomorphisms are compatible with the restriction operations, hencethey define a canonical functorial homomorphism of B -modules

(4.2.3.1) Ψ∗(HomA (M ,N )) −→HomB (Ψ∗(M ),Ψ∗(N )).(4.2.4). If C is an A -algebra, the composite homomorphism

Ψ∗(C )⊗B Ψ∗(C ) −→Ψ∗(C ⊗A C ) −→Ψ∗(C )

defines on Ψ∗(C ) the structure of a B -algebra, as a result of (4.2.2.2). We see similarly that ifM is a C -module,Ψ∗(M ) is canonically equipped with the structure of a Ψ∗(C )-module.

(4.2.5). Consider in particular the case where X is a closed subspace of Y and where ψ is the canonical injectionj : X → Y . If B ′ = B|X = j∗(B) is the restriction of the sheaf of rings B to X , an A -module M can beconsidered as a B ′-module by means of the homomorphism θ] :B ′→A ; then Ψ∗(M ) is the B -module whichinducesM on X and 0 elsewhere. If N is a second A -module, Ψ∗(M )⊗B Ψ∗(N ) canonically identifies withΨ∗(M ⊗B ′ N ) and HomB (Ψ∗(M ),Ψ∗(N )) with Ψ∗(HomB ′(M ,N )).

(4.2.6). Let (Z ,C ) be a third ringed space, Ψ′ = (ψ′,θ ′) a morphism (Y,B)→ (Z ,C ); if Ψ′′ is the compositemorphism Ψ′ Ψ, it is clear that we have Ψ′′∗ =Ψ′∗ Ψ∗.

4.3. Inverse image of an A -module.

(4.3.1). The hypotheses and notation being the same as (4.2.1), let G be a B -module and ψ∗(G ) the inverseimage (3.7.1) which is therefore a sheaf of abelian groups on X . The definition of sections of ψ∗(G ) and ofψ∗(B) (3.7.1) shows that ψ∗(G ) is canonically equipped with a ψ∗(B)-module structure. On the other hand,the homomorphism θ] : ψ∗(B)→A endows A with the a ψ∗(B)-module structure, which we denote by A[θ]when necessary to avoid confusion; the tensor product ψ∗(G )⊗ψ∗(B)A[θ] is then equipped with an A -modulestructure. We say that this A -module is the inverse image of G under the morphism Ψ and we denote it by Ψ∗(G ). 0I | 41If G1, G2 are two B -modules over Y , v a B -homomorphism G1→G2, then ψ∗(v), as we check immediately, is aψ∗(B)-homomorphism from ψ∗(G1) to ψ∗(G2); as a result ψ∗(v)⊗1 is an A -homomorphism Ψ∗(G1)→Ψ∗(G2),which we denote by Ψ∗(v). So we define Ψ∗ as a covariant functor from the category of B -modules to thatof A -modules. Here, this functor (contrary to ψ∗) is no longer exact in general, but only right exact, thetensorization by A being a right exact functor to the category of ψ∗(B)-modules.

For each x ∈ X , we have (Ψ∗(G ))x = Gψ(x) ⊗Bψ(x)Ax , according to (3.7.2). The support of Ψ∗(G ) is thus

contained in ψ−1(Supp(G )).

(4.3.2). Let (Gλ) be an inductive system of B -modules, and let G = lim−→Gλ be its inductive limit. The canonicalhomomorphisms Gλ →G define the ψ∗(B)-homomorphisms ψ∗(Gλ)→ ψ∗(G ), which give a canonical homo-morphism lim−→ψ∗(Gλ)→ ψ∗(G ). As the stalk at a point of an inductive limit of sheaves is the inductive limit ofthe stalks at the same point (G, II, 1.11), the preceding canonical homomorphism is bijective (3.7.2). In addition,the tensor product commutes with inductive limits of sheaves, and we thus have a canonical functorial isomorphismlim−→Ψ∗(Gλ)'Ψ∗(lim−→Gλ) of A -modules.

On the other hand, for a finite direct sum⊕

i Gi of B -modules, it is clear that ψ∗(⊕

i Gi) =⊕

i ψ∗(Gi),therefore, by tensoring with A[θ],

(4.3.2.1) Ψ∗⊕

i

Gi

=⊕

i

Ψ∗(Gi).

By passing to the inductive limit, we deduce, in light of the above, that the above equality is still true for anydirect sum.

(4.3.3). Let G1, G2 be two B -modules; from the definition of the inverse images of sheaves of abelian groups(3.7.1), we obtain immediately a canonical homomorphism ψ∗(G1)⊗ψ∗(B) ψ∗(G2)→ ψ∗(G1 ⊗B G2) of ψ∗(B)-modules, and the stalk at a point of a tensor product of sheaves being the tensor product of the stalks at thispoint (G, II, 2.8), we deduce from (3.7.2) that the above homomorphism is in fact a isomorphism. By tensoringwith A , we obtain a canonical functorial isomorphism

(4.3.3.1) Ψ∗(G1)⊗A Ψ∗(G2)'Ψ∗(G1 ⊗B G2).

4. RINGED SPACES 32

(4.3.4). Let C be a B -algebra; the data of the algebra structure on C is the same as the data of a B -homomorphism C ⊗B C →C satisfying the associativity and commutativity conditions (conditions which arechecked stalk-wise); the above isomorphism allows us to consider this homomorphism as a homomorphism ofA -modules Ψ∗(C )⊗A Ψ∗(C )→ Ψ∗(C ) satisfying the same conditions, so Ψ∗(C ) is thus equipped with anA -algebra structure. In particular, it follows immediately from the definitions that the A -algebra Ψ∗(B) isequal to A (up to a canonical isomorphism).

Similarly, ifM is a C -module, the data of this module structure is the same as that of a B -homomorphism 0I | 42C ⊗BM →M satisfying the associativity condition; hence we give a Ψ∗(C )-module structure on Ψ∗(M ).

(4.3.5). Let J be a sheaf of ideals of B ; as the functor ψ∗ is exact, the ψ∗(B)-module ψ∗(J ) canonicallyidentifies with a sheaf of ideals of ψ∗(B); the canonical injection ψ∗(J )→ ψ∗(B) then gives a homomorphismof A -modules Ψ∗(J ) = ψ∗(J ) ⊗ψ∗(B) A[θ] → A ; we denote by Ψ∗(J )A , or JA if there is no fear ofconfusion, the image of Ψ∗(J ) under this homomorphism. So we have by definition JA = θ](ψ∗(J ))Aand in particular, for each x ∈ X , (JA )x = θ]x(Jψ(x))Ax , taking into account the canonical identificationbetween the stalks of ψ∗(J ) and those of J (3.7.2). If J1, J2 are two sheaves of ideals of B , then we have(J1J2)A = J1(J2A ) = (J1A )(J2A ).

If F is an A -module, we set JF = (JA )F .

(4.3.6). Let (Z ,C ) be a third ringed space, Ψ′ = (ψ′,θ ′) a morphism (Y,B)→ (Z ,C ); if Ψ′′ is the compositemorphism Ψ′ Ψ, it follows from the definition (4.3.1) and from (4.3.3.1) that we have Ψ′′∗ =Ψ∗ Ψ′∗.

4.4. Relation between direct and inverse images.

(4.4.1). The hypotheses and notation being the same as in (4.2.1), let G be a B -module. By definition, ahomomorphism u : G →Ψ∗(F ) ofB -modules is still called a Ψ-morphisms from G toF , or simply a homomorphismfrom G to F and we write it as u : G →F when no confusion will occur. To give such a homomorphism is thesame as giving, for each pair (U , V ) where U is an open set of X , V an open set of Y such that ψ(U) ⊂ V , ahomomorphism uU ,V : Γ(V,G )→ Γ(U ,F ) of Γ(V,B)-modules, Γ(U ,F ) being considered as a Γ(V,B)-module bymeans of the ring homomorphism θU ,V : Γ(V,B)→ Γ(U ,A ); the uU ,V must in addition render commutative thediagrams (3.5.1.1). It suces, moreover, to define u by the data of the uU ,V when U (resp. V ) varies over a basisB (resp. B′) for the topology of X (resp. Y ) and to check the commutativity of (3.5.1.1) for these restrictions.

(4.4.2). Under the hypotheses of (4.2.1) and (4.2.6), let H be a C -module, v :H → Ψ′∗(G ) a Ψ′-morphism;

then w :Hv−→Ψ′∗(G )

Ψ′∗(u)−−−→Ψ′∗(Ψ∗(F )) is a Ψ′′-morphism which we call the composition of u and v.

(4.4.3). We will now see that we can define a canonical isomorphism of bifunctors in F and G

(4.4.3.1) HomA (Ψ∗(G ),F )' HomB (G ,Ψ∗(F ))

which we denote by v 7→ v[θ (or simply v 7→ v[ if there is no chance of confusion); we denote by u 7→ u]θ , oru 7→ u], the inverse isomorphism. This definition is the following: by composing v : Ψ∗(G ) → F with thecanonical map ψ∗(G )→Ψ∗(G ), we obtain a homomorphism of sheaves of groups v′ : ψ∗(G )→F , which is alsoa homomorphism of ψ∗(B)-modules. We obtain (3.7.1) a homomorphism v′[ : G → ψ∗(F ) =Ψ∗(F ), which isalso a homomorphism of B -modules as we check easily; we take v[θ = v′[. Similarly, for u : G →Ψ∗(F ), which 0I | 43is a homomorphism of B -modules, we obtain (3.7.1) a homomorphism u] : ψ∗(G )→ F of ψ∗(B)-modules,hence by tensoring with A we have a homomorphism of A -modules Ψ∗(G )→F , which we denote by u]θ . It

is immediate to check that (u]θ)[θ = u and (v[θ)

]

θ = v, so we have established the functorial nature in F of the

isomorphism v 7→ v[θ . The functorial nature in G of u 7→ u]θ is then formally shown as in (3.5.4) (reasoning thatwould also prove the functorial nature of Ψ∗ established in (4.3.1) directly).

If we take for v the identity homomorphism of Ψ∗(B), v[θ is a homomorphism

(4.4.3.2) ρG : G −→Ψ∗(Ψ∗(G ));

if we take for u the identity homomorphism of Ψ∗(F ), u]θ is a homomorphism

(4.4.3.3) σF : Ψ∗(Ψ∗(F )) −→F ;

these homomorphisms will be called canonical. They are in general neither injective or surjective. We havecanonical factorizations analogous to (3.5.3.3) and (3.5.4.4).

We note that if s is a section of G over an open set V of Y , ρG (s) is the section s′⊗1 of Ψ∗)(G ) over ψ−1(V ),s′ being such that s′x = sψ(x) for all x ∈ ψ−1(V ). We also note that if u : G → ψ∗(F ) is a homomorphism, it definesfor all x ∈ X a homomorphism ux : Gψ(x)→Fx on the stalks, obtained by composing (u])x : (Ψ∗(G ))x →Fx

5. QUASI-COHERENT AND COHERENT SHEAVES 33

and the canonical homomorphism sx 7→ sx⊗1 from Gψ(x) to (Ψ∗(G ))x = Gψ(x)⊗Bψ(x)Ax . The homomorphism ux

is obtained also by passing to the inductive limit relative to the homomorphisms Γ(V,G )u−→ Γ(ψ−1(V ),F )→Fx ,

where V varies over the neighborhoods of ψ(x).

(4.4.4). Let F1, F2 be A -modules, G1, G2 be B -modules, ui (i = 1,2) a homomorphism from Gi to Fi . Wedenote by u1⊗u2 the homomorphism u : G1⊗B G2→F1⊗A F2 such that u] = (u1)]⊗ (u2)] (taking into account(4.3.3.1)); we check that u is also the composition G1 ⊗B G2→Ψ∗(F1)⊗B Ψ∗(F2)→Ψ∗(F1 ⊗A F2), where thefirst arrow is the ordinary tensor product u1 ⊗B u2 and the second is the canonical homomorphism (4.2.2.1).

(4.4.5). Let (Gλ)λ∈L be an inductive system of B -modules, and, for each λ ∈ L, let uλ be a homomorphismGλ →Ψ∗(F ), form an inductive limit; we put G = lim−→Gλ and u= lim−→uλ ; then the (uλ)] form an inductive system

of homomorphisms Ψ∗(Gλ)→F , and the inductive limit of this system is none other than u].

(4.4.6). LetM , N be twoB -modules, V an open set of Y , U = ψ−1(V ); the map v 7→Ψ∗(v) is a homomorphism

HomB|V (M|V,N |V ) −→ HomA|U(Ψ∗(M )|U ,Ψ∗(N )|U)for the Γ(V,B)-module structures (HomA|U(Ψ∗(M )|U ,Ψ∗(N )|U) is normaly equipped with the a Γ(U ,ψ∗(B))-module structure, and thanks to the canonical homomorphism (3.7.2) Γ(V,B) → Γ(U ,ψ∗(B)), it is also a 0I | 44Γ(V,B)-module). We see immediately that these homomorphisms are compatible with the restriction morphisms,and as a result define a canonical functorial homomorphism

γ :HomB (M ,N ) −→Ψ∗(HomA (Ψ∗(M ),Ψ∗(N ));it also corresponds to this homomorphism the homomorphism

γ ] : Ψ∗(HomB (M ,N )) −→HomA (Ψ∗(M ),Ψ∗(N ))and these canonical morphisms are functorial inM and N .

(4.4.7). Suppose that F (resp. G ) is an A -algebra (resp. a B -algebra). If u : G →Ψ∗(F ) is a homomorphismof B -algebras, u] is a homomorphism Ψ∗(G )→F of A -algebras; this follows from the commutativity of thediagram

G ⊗B G //

G

u

Ψ∗(F ⊗A F ) // Ψ∗(F )

and from (4.4.4). Similarly, if v : Ψ∗(G ) → F is a homomorphism of A -algebras, v[ : G → Ψ∗(F ) is ahomomorphism of B -algebras.

(4.4.8). Let (Z ,C ) be a third ringed space, Ψ′ = (ψ′,θ ′) a morphism (Y,B)→ (Z ,C ), and Ψ′′ : (X ,A )→ (Z ,C )the composite morphism Ψ′ Ψ. Let H be a C -module, u′ a homomorphsim from H to G ; the composition

v′′ = v v′ is by definition the homomorphism fromH to F defined byHv′−→Ψ′∗(G )

Ψ′∗(v)−−−→Ψ′∗(Ψ∗(F )); we checkthat v′′] is the homomorphism

Ψ∗(Ψ′∗(H ))Ψ∗(v′])−−−−→Ψ∗(G ) v]

−→F .

§5. Quasi-coherent and coherent sheaves

5.1. Quasi-coherent sheaves.

(5.1.1). Let (X ,OX ) be a ringed space,F an OX -module. The data of a homomorphism u : OX →F of OX -modulesis equivalent to that of the section s = u(1) ∈ Γ(X ,F ). Indeed, when s is given, for each section t ∈ Γ(U ,OX ),we necessarily have u(t) = t · (s|U); we say that u is dened by the section s. If now I is any set of indices, considerthe direct sum sheaf O (I)X , and for each i ∈ I , let hi be the canonical injection of the i-th factor into O (I)X ; weknow that u 7→ (u hi) is an isomorphism from HomOX

(O (I)X ,F ) to the product (HomOX(OX ,F ))I . So there is a

canonical one-to-one correspondence between the homomorphisms u : O (I)X →F and the families of sections (si)i∈I

of F over X . The homomorphism u corresponding to (si) sends an element (ai) ∈ (Γ(U ,OX ))(I) to∑

i∈I ai · (si |U).We say that F is generated by the family (si) if the homomorphism O (I)X → F defined for each family is 0I | 45

surjective (in other words, if, for each x ∈ X , Fx is an Ox -module generated by the (si)x). We say that F isgenerated by its sections over X if it is generated by the family of all these sections (or by a subfamily), in otherwords, if there exists a surjective homomorphism O (I)X →F for a suitable I .

5. QUASI-COHERENT AND COHERENT SHEAVES 34

We note that a OX -module F can be such that there exists a point x0 ∈ X for which F|U is not generated byits sections over U , regardless of the choice of neighborhood U of x0: it suces to take X = R, for OX the simple sheafZ, for F the algebraic subsheaf of OX such that F0 = 0, Fx = Z for x 6= 0, and finally x0 = 0: the only sectionof F|U over U is 0 for a neighborhood U of 0.

(5.1.2). Let f : X → Y be a morphism of ringed spaces. If F is a OX -module generated by its sections over X ,then the canonical homomorphism f ∗( f∗(F ))→F (4.4.3.3) is surjective; indeed, with the notation of (5.1.1),si ⊗ 1 is a section of f ∗( f∗(F )) over X , and its image in F is si . The example in (5.1.1) where f is the identityshows that the inverse of this proposition is false in general.

If G is an OY -module generated by its sections over Y , then f ∗(G ) is generated by its sections over X , sincef ∗ is a right exact functor. ErrII

(5.1.3). We say that an OX -module F is quasi-coherent if for each x ∈ X there is an open neighborhood U of xsuch that F|U is isomorphic to the cokernel of a homomorphism of the form O (I)X |U →O

(J)X |U , where I and J

are sets of arbitrary indices. It is clear that OX is itself a quasi-coherent OX -module, and that any direct sum ofquasi-coherent OX -modules is again a quasi-coherent OX -module. We say that an OX -algebra A is quasi-coherent ifit is quasi-coherent as an OX -module.

(5.1.4). Let f : X → Y be a morphism of ringed spaces. If G is a quasi-coherent OY -module, then f ∗(G ) is aquasi-coherent OX -module. Indeed, for each x ∈ X , there is an open neighborhood V of f (x) in Y such that G|Vis the cokernel of a homomorphism O (I)Y |V →O

(J)Y |V . If U = f −1(V ), and if fU is the restriction of f to U , then

we have f ∗(G )|U = f ∗U (G|V ); as f ∗U is right exact and commutes with direct sums, f ∗U (G|V ) is the cokernel of ahomomorphism O (I)X |U →O

(J)X |U .

5.2. Sheaves of nite type.

(5.2.1). We say that an OX -module F is of nite type if for each x ∈ X there exists an open neighborhood U of xsuch that F|U is generated by a nite family of sections over U , or if it is isomorphic to a sheaf quotient of asheaf of the form (OX |U)p where p is finite. Each sheaf quotient of a sheaf of finite type is again a sheaf of finitetype, as well as each finite direct sum and each finite tensor product of sheaves of finite type. An OX -module offinite type is not necessarily quasi-coherent, as we can see for the OX -module OX/F , where F is the example in(5.1.1). If F is of finite type, then Fx is a Ox -module of finite type for each x ∈ X , but the example in (5.1.1)shows that this condition is necessary but not sucient in general.

(5.2.2). Let F be an OX -module of nite type. If si (1¶ i ¶ n) are the sections of F over an open neighborhoodU of a point x ∈ X and the (si)x generate Fx , then there exists an open neighborhood V ⊂ U of x such that the(si)y generate Fy for all y ∈ Y (FAC, I, 2, 12, prop. 1). In particular, we conclude that the support of F is closed. 0I | 46

Similarly, if u :F →G is a homomorphism such that ux = 0, then there exists a neighborhood U of x suchthat uy = 0 for all y ∈ U .

(5.2.3). Suppose that X is quasi-compact, and let F and G be two OX -modules such that G is of nite type,u :F →G a surjective homomorphism. In addition, suppose that F is the inductive limit of an inductive system(Fλ) of OX -modules. Then there exists an index µ such that the homomorphism Fµ→G is surjective. Indeed,for each x ∈ X , there exists a finite system of sections si of G over an open neighborhood U(x) of x such thatthe (si)y generate Gy for all y ∈ U(x); there is then an open neighborhood V (x) ⊂ U(x) of x and n sections t iof F over V (x) such that si |V (s) = u(t i) for all i; we can also suppose that the t i are the canonical images ofsections of a similar sheaf Fλ(x) over V (x). We then cover X with a finite number of neighborhoods V (xk), andlet µ be the maximal index of the λ(xk); it is clear that this index gives the answer.

Suppose still that X is quasi-compact, and let F be an OX -module of finite type generated by its sectionsover X (5.1.1); then F is generated by a nite subfamily of these sections: indeed, it suces to cover X by afinite number of open neighborhoods Uk such that, for each k, there is a finite number of sections sik of F overX whose restrictions to Uk generate F|Uk; it is clear that the sik then generate F .

(5.2.4). Let f : X → Y be a morphism of ringed spaces. If G is an OY -module of finite type, then f ∗(G ) isan OX -module of finite type. Indeed, for each x ∈ X , there is an open neighborhood V of f (x) in Y and asurjective homomorphism v : O p

Y |V →G|V . If U = f −1(V ) and if fU is the restriction of f to U , then we havef ∗(G )|U = f ∗U (G|V ); since f ∗U is right exact (4.3.1) and commutes with direct sums (4.3.2), f ∗U (v) is a surjective ErrIIhomomorphism O p

X |U → f ∗(G )|U .

(5.2.5). We say that an OX -moduleF admits a nite presentation if for each x ∈ X there exists an open neighborhoodU of x such that F|U is isomorphic to a cokernel of a (OX |U)-homomorphism O

pX |U → O

qX |U , p and q being two

5. QUASI-COHERENT AND COHERENT SHEAVES 35

integers > 0. Such an OX -module is therefore of finite type and quasi-coherent. If f : X → Y is a morphism ofringed spaces, and if G is an OY -module admitting a finite presentation, then f ∗(G ) admits a finite presentation,as shown in the argument of (5.1.4).

(5.2.6). Let F be an OX -module admitting a finite presentation (5.2.5); then, for each OX -module H , thecanonical functorial homomorphism

(HomOX(F ,H ))x −→ HomOx

(Fx ,Hx)

is bijective (T, 4.1.1).

(5.2.7). Let F and G be two OX -modules admitting a finite presentation. If for some x ∈ X , Fx and Gxare isomorphic as Ox -modules, then there exists an open neighborhood U of x such that F|U and G|U areisomorphic. Indeed, if φ : Fx → Gx and ψ : Gx → Fx are an isomorphism and its inverse isomorphism, thenthere exists, according to (5.2.6), an open neighborhood V of x and a section u (resp. v) of HomOX

(F ,G ) (resp.HomOX

(G ,F )) over V such that ux =φ (resp. vx = ψ). As (u v)x and (v u)x are the identity automorphisms, 0I | 47there exists an open neighborhood U ⊂ V of x such that (u v)|U and (v u)|U are the identity automorphisms,hence the proposition.

5.3. Coherent sheaves.

(5.3.1). We say that an OX -module F is coherent if it satisfies the two following conditions:

(a) F is of finite type.(b) for each open U ⊂ X , integer n> 0, and homomorphism u : O n

X |U →F|U , the kernel of u is of finitetype.

We note that these two conditions are of a local nature.For most of the proofs of the properties of coherent sheaves in what follows, cf. (FAC, I, 2).

(5.3.2). Each coherent OX -module admits a finite presentation (5.2.5); the inverse is not necessarily true, sinceOX itself is not necessarily a coherent OX -module.

Each OX -submodule of nite type of a coherent OX -module is coherent; each nite direct sum of coherentOX -modules is a coherent OX -module.

(5.3.3). If 0 → F → G → H → 0 is an exact sequence of OX -modules and if two of these OX -modules arecoherent, then so is the third.

(5.3.4). If F and G are two coherent OX -modules, u : F → G a homomorphism, then Im(u), Ker(u), andCoker(u) are coherent OX -modules. In particular, if F and G are OX -submodules of a coherent OX -module, thenF +G and F ∩G are coherent.

If A →B →C →D → E is an exact sequence of OX -modules, and if A , B , D, E are coherent, then C iscoherent. ErrII

(5.3.5). If F and G are two coherent OX -modules, then so are F ⊗OXG are HomOX

(F ,G ).

(5.3.6). Let F be a coherent OX -module, J a coherent sheaf of ideals of OX . Then the OX -module JF iscoherent, as the image of J ⊗OX

F under the canonical homomorphism J ⊗OXF →F ((5.3.4) and (5.3.5)).

(5.3.7). We say that an OX -algebra A is coherent if it is coherent as an OX -module. In particular, OX is a coherentsheaf of rings if and only if for each open U ⊂ X and each homomorphism of the form u : O p

X |U → OX |U , thekernel of u is an (OX |U)-module of finite type.

If OX is a coherent sheaf of rings, then each OX -module F admitting a finite presentation (5.2.5) is coherent,according to (5.3.4).

The annihilator of an OX -module F is the kernel J of the canonical homomorphism OX →HomOX(F ,F )

which sends each section s ∈ Γ(U ,OX ) to the multiplication by s map in Hom(F|U ,F|U); if OX is coherent andif F is a coherent OX -module, then J is coherent ((5.3.4) and (5.3.5)) and for each x ∈ X , Jx is the annihilatorof Fx (5.2.6).

0I | 48(5.3.8). Suppose that OX is coherent; let F be a coherent OX -module, x a point of X , M a submodule of finitetype of Fx ; then there exists an open neighborhood U of x and a coherent (OX |U)-submodule G of F|U suchthat Gx = M (T, 4.1, Lemma 1).

This result, along with the properties of the OX -submodules of a coherent OX -module, impose the necessaryconditions on the rings Ox such that OX is coherent. For example (5.3.4), the intersection of two ideals of finitetype of Ox must still be an ideal of finite type.

5. QUASI-COHERENT AND COHERENT SHEAVES 36

(5.3.9). Suppose that OX is coherent, and let M be an Ox -module admitting a finite presentation, thereforeisomorphic to a cokernel of a homomorphism φ : O p

x → Oqx ; then there exists an open neighborhood U of X

and a coherent (OX |U)-module F such that Fx is isomorphic to M . Indeed, according to (5.2.6), there existsa section u of HomOX

(O pX ,O q

X ) over an open neighborhood U of x such that ux = φ ; the cokernel F of thehomomorphism u : O p

X |U →OqX |U gives the answer (5.3.4).

(5.3.10). Suppose that OX is coherent, and let J be a coherent sheaf of ideals of OX . For a (OX/J )-moduleF to be coherent, it is necessary and sucient for it to be coherent as a OX -module. In particular, OX/J is acoherent sheaf of rings.

(5.3.11). Let f : X → Y be a morphism of ringed spaces, and suppose that OX is coherent; then, for eachcoherent OY -module G , f ∗(G ) is a coherent OX -module. Indeed, with the notation of (5.2.4), we can assumethat G|V is the cokernel of a homomorphism v : O q

Y |V →Op

Y |V ; as f ∗U is right exact, f ∗(G )|U = f ∗U (G|V ) is thecokernel of the homomorphism f ∗U (v) : O q

X |U →Op

X |U , hence our assertion.

(5.3.12). Let Y be a closed subset of X , j : Y → X the canonical injection, OY a sheaf of rings on Y , and setOX = j∗(OY ). For a OY -module G to be of finite type (resp. quasi-coherent, coherent), it is necessary and sucientfor j∗(G ) to be an OX -module of finite type (resp. quasi-coherent, coherent).

5.4. Locally free sheaves.

(5.4.1). Let X be a ringed space. We say that an OX -module F is locally free if for each x ∈ X there exists an openneighborhood U of x such that F|U is isomorphic to a (OX |U)-module of the form O (I)X |U , where I can dependon U . If for each U , I is finite, then we say that F is of nite rank; if for each U , I has the same finite number ofelements n, we say that F is of rank n. A locally free OX -module of rank 1 is called invertible (cf. (5.4.3)). If Fis a locally free OX -module of finite rank, then for each x ∈ X , Fx is a free Ox -module of finite rank n(x), andthere exists a neighborhood U of x such that F|U , is of rank n(x); if X is connected, then n(x) is constant.

It is clear that each locally free sheaf is quasi-coherent, and if OX is a coherent sheaf of rings, then eachlocally free OX -module of finite rank is coherent.

If L is locally free, then L ⊗OXF is an exact functor in F to the category of OX -modules.

We will mostly consider locally free OX -modules of finite rank, and when we speak of locally free sheaves 0I | 49without specifying, it will be understood that they are of nite rank.

Suppose that OX is coherent, and let F be a coherent OX -module. Then, if at a point x ∈ X , Fx is an Ox -modulefree of rank n, there exists a neighborhood U of x such that F|U is locally free of rank n; in fact, Fx is thenisomorphic to O n

x , and the proposition follows from (5.2.7). ErrII

(5.4.2). If L , F are two OX -modules, we have a canonical functorial homomorphism

(5.4.2.1) L ∨ ⊗OXF =HomOX

(L ,OX )⊗OXF −→HomOX

(L ,F )

defined in the following way: for each open set U , send any pair (u, t), where u ∈ Γ(U ,HomOX(L ,OX )) =

Hom(L|U ,OX |U) and t ∈ Γ(U ,F ), to the element of Hom(L|U ,F|U) which, for each x ∈ U , sends sx ∈ Lx tothe element ux(sx)t x of Fx . If L is locally free of nite rank, then this homomorphism is bijective; the propertybeing local, we can in fact reduce to the case where L = O n

X ; as for each OX -module G , HomOX(O n

X ,G ) iscanonically isomorphic to G n, we have reduced to the case L = OX , which is immediate.

(5.4.3). If L is invertible, then so is its dual L ∨ =HomOX(L ,OX ), since we can immediately reduce (as the

question is local) to the case L = OX . In addition, we have a canonical isomorphism

(5.4.3.1) HomOX(L ,OX )⊗OX

L ' OX

as, according to (5.3.2), it suces to define a canonical isomorphism HomOX(L ,L )' OX . For each OX -module

F , we have a canonical homomorphism OX ' HomOX(F ,F ) (5.3.7). It remains to prove that if F = L is

invertible, then this homomorphism is bijective, and as the question is local, it reduces to the case L = OX ,which is immediate.

Due to the above, we put L −1 =HomOX(L ,OX ), and we say that L −1 is the inverse of L . The terminology

“invertible sheaf” can be justified in the following way when X is a point and OX is a local ring A with maximalideal m; if M and M ′ are two A-modules (M being of finite type) such that M ⊗A M ′ is isomorphic to A, then as(A/m)⊗A (M ⊗A M ′) identifies with (M/mM)⊗A/m (M ′/mM ′), this latter tensor product of vector spaces over thefield A/m is isomorphic to A/m, which requires M/mM and M ′/mM ′ to be of dimension 1. For each elementz ∈ M not in mM , we have M = Az+mM , which implies that M = Az according to Nakayama’s Lemma, M beingof finite type. Moreover, as the annihilator of z kills M ⊗A M ′, which is isomorphic to A, this annihilator is 0,and as a result M is isomorphic to A. In the general case, this shows that L is an OX -module of finite type, such

5. QUASI-COHERENT AND COHERENT SHEAVES 37

that there exists an OX -module F for which L ⊗OXF is isomorphic to OX , and if in addition the rings Ox are

local rings, then Lx is an Ox -module isomorphic to Ox for each x ∈ X . If OX and L are assumed to be coherent,then we conclude that L is invertible according to (5.2.7).

(5.4.4). If L and L ′ are two invertible OX -modules, then so is L ⊗OXL ′, since the question is local, we can

assume that L = OX , and the result is then trivial. For each integer n¾ 1, we denote by L ⊗n the tensor productof n copies of the sheaf L ; we set by convention L ⊗0 = OX , and for n ¾ 1, L ⊗(−n) = (L −1)⊗n. With these 0I | 50notation, there is then a canonical functorial isomorphism

(5.4.4.1) L ⊗m ⊗OXL ⊗n 'L ⊗(n+m)

for any rational integers m and n: indeed, by definition, we immediately reduce to the case where m= −1, n= 1,and the isomorphism in question is then that defined in (5.4.3).

(5.4.5). Let f : Y → X be a morphism of ringed spaces. If L is a locally free (resp. invertible) OX -module,then f ∗(L ) is a locally free (resp. invertible) OY -module: this follows immediately from that the inverseimages of two locally isomorphic OX -modules are locally isomorphic, that f ∗ commutes with finite direct sums,and that f ∗(OX ) = OY (4.3.4). In addition, we know that we have a canonical functorial homomorphismf ∗(L ∨) → ( f ∗(L ))∨ (4.4.6), and when L is locally free, this homomorphism is bijective: indeed, we againreduce to the case where L = OX which is trivial. We conclude that if L is invertible, then f ∗(L ⊗n) canonicallyidentifies with ( f ∗(L ))⊗n for each rational integer n.

(5.4.6). Let L be an invertible OX -module; we denote by Γ∗(X ,L ) or simply Γ∗(L ) the abelian group directsum

⊕

n∈Z Γ(X ,L ⊗n); we equip it with the structure of a graded ring, by corresponding to a pair (sn, sm), wheresn ∈ Γ(X ,L ⊗n), sm ∈ Γ(X ,L ⊗m), the section of L ⊗(n+m) over X which corresponds canonically (5.4.4.1) to thesection sn ⊗ sm of L ⊗n ⊗OX

L ⊗m; the associativity of this multiplication is verified in an immediate way. It isclear that Γ∗(X ,L ) is a covariant functor in L , with values in the category of graded rings.

If now F is any OX -module, then we set

Γ∗(L ,F ) =⊕

n∈Z

Γ(X ,F ⊗OXL ⊗n).

We equip this abelian group with the structure of a graded module over the graded ring Γ∗(L ) in the following way:to a pair (sn, um), where sn ∈ Γ(X ,L ⊗n) and um ∈ Γ(X ,F ⊗OX

L ⊗m), we associate the section of F ⊗OXL ⊗(m+n)

which canonically corresponds (5.4.4.1) to sn ⊗ um; the verification of the module axioms are immediate. For Xand L fixed, Γ∗(L ,F ) is a covariant functor in F with values in the category of graded Γ∗(L )-modules; for Xand F fixed, it is a covariant functor in L with values in the category of abelian groups.

If f : Y → X is a morphism of ringed spaces, the canonical homomorphism (4.4.3.2) ρ :L ⊗n→ f∗( f ∗(L ⊗n))defines a homomorphism of abelian groups Γ(X ,L ⊗n)→ Γ(Y, f ∗(L ⊗n)), and as f ∗(L ⊗n) = ( f ∗(L ))⊗n), it fol-lows from the definitions of the canonical homomorphisms (4.4.3.2) and (5.4.4.1) that the above homomorphismsdefine a functorial homomorphism of graded rings Γ∗(L )→ Γ∗( f ∗(L )). The same canonical homomorphism (4.4.3)similarly defines a homomorphism of abelian groups Γ(X ,F ⊗OX

L ⊗n)→ Γ(Y, f ∗(F ⊗OXL ⊗n)), and as

f ∗(F ⊗OXL ⊗n) = f ∗(F )⊗OY

( f ∗(L ))⊗n (4.3.3.1),

these homomorphism (for n variable) define a di-homomorphism of graded modules Γ∗(L ,F )→ Γ∗( f ∗(L ), f ∗(F )). 0I | 51

(5.4.7). One can show that there exists a set M (also denoted M(X )) of invertible OX -modules such that eachinvertible OX -module is isomorphic to a unique element of M 8; we define on M a composition law by sendingtwo elements L and L ′ of M to the unique element of M isomorphic to L ⊗OX

L ′. With this composition law,M is a group isomorphic to the cohomology group H1(X ,O ∗X ), where O

∗X is the subsheaf of OX such that Γ(U ,O ∗X ) is the

group of invertible elements of the ring Γ(U ,OX ) for each open U ⊂ X (O ∗X is therefore a sheaf of multiplicativeabelian groups).

We will note that for all open U ⊂ X , the group of sections Γ(U ,O ∗X ) canonically identifies with theautomorphism group of the (OX |U)-module OX |U , the identification sending a section ε of O ∗X over U to theautomorphism u of OX |U such that ux(sx) = εx sx for all x ∈ X and all sx ∈ Ox . Then let U = (Uλ) be an opencover of X ; the data, for each pair of indices (λ,µ), of an automorphism θλµ of OX |(Uλ ∩ Uµ) is the same asgiving a 1-cochain of the cover U, with values in O ∗X , and say that the θλµ satisfy the gluing condition (3.3.1),meaning that the corresponding cochain is a cocycle. Similarly, the data, for each λ, of an automorphism ωλ ofOX |Uλ is the same as the data of a 0-cochain of the cover U, with values in O ∗X , and its coboundary corresponds tothe family of automorphisms (ωλ |Uλ ∩ Uµ) (ωµ|Uλ ∩ Uµ)−1. We can send each 1-cocycle of U with values in O ∗X

8See the book in preparation cited in the introduction.

5. QUASI-COHERENT AND COHERENT SHEAVES 38

to the element of M isomorphic to an invertible OX -module obtained by gluing with respect to the family ofautomorphisms (θλµ) corresponding to this cocycle, and to two cohomologous coycles correspond two equalelements of M (3.3.2); in other words, we thus define a map φU : H1(U,O ∗X )→M. In addition, if B is a secondopen cover of X , finer than U, then the diagram

H1(U,O ∗X )φU

$$

M

H1(B,O ∗X )φB

::

where the vertical arrow is the canonical homomorphism (G, II, 5.7), is commutative, as a result of (3.3.3). Bypassing to the inductive limit, we therefore obtain a map H1(X ,O ∗X )→M, the Čech cohomology group H1(X ,O ∗X )identifying as we know with the first cohomology group H1(X ,O ∗X ) (G, II, 5.9, Cor. of Thm. 5.9.1). This map issurjective: indeed, by definition, for each invertible OX -module L , there is an open cover (Uλ) of X such thatL is obtained by gluing the sheaves OX |Uλ (3.3.1). It is also injective, since it suces to prove for the mapsH1(U,OX ) →M, and this follows from (3.3.2). It remains to show that the bijection thus defined is a group 0I | 52homomorphism. Given two invertible OX -modules L and L ′, there is an open cover (Uλ) such that L|Uλ andL ′|Uλ are isomorphic to OX |Uλ for each λ; so there is for each index λ an element aλ (resp. a′λ) of Γ(Uλ ,L )(resp. Γ(Uλ ,L ′)) such that the elements of Γ(Uλ ,L ) (resp. Γ(Uλ ,L ′)) are the sλ · aλ (resp. sλ · a′λ), where sλvaries over Γ(Uλ ,OX ). The corresponding cocycles (ελµ), (ε ′λµ) are such that sλ · aλ = sµ · aµ (resp. sλ · a′λ = sµ · a′µ)over Uλ ∩ Uµ is equivalent to sλ = ελµsµ (resp. sλ = ε ′λµsµ) over Uλ ∩ Uµ. As the sections of L ⊗OX

L ′ over Uλ arethe finite sums of the sλs

′λ · (aλ ⊗ a′λ) where sλ and s′λ vary over Γ(Uλ ,OX ), it is clear that the cocycle (ελµ,ε ′λµ)

corresponds to L ⊗OXL ′, which finishes the proof. 9

(5.4.8). Let f = (ψ,ω) be a morphism Y → X of ringed spaces. The functor f ∗(L ) to the category of freeOX -modules defines a map (which we still denote f ∗ by abuse of language) from the set M(X ) to the set M(Y ).Second, we have a canonical homomorphism (T, 3.2.2)

(5.4.8.1) H1(X ,O ∗X ) −→ H1(Y,O ∗Y ).

When we canonically identify (5.4.7) M(X ) and H1(X ,O ∗X ) (resp. M(Y ) and H1(Y,O ∗Y )), the homomorphism(5.4.8.1) identies with the map f ∗. Indeed, if L comes from a cocycle (ελµ) corresponding to an open cover(Uλ) of X , then it suces to show that f ∗(L ) comes from a cocycle whose cohomology class is the image under(5.4.8.1) of (ελµ). If θλµ is the automorphism of OX |(Uλ ∩ Uµ) which corresponds to ελµ, then it is clear thatf ∗(L ) is obtained by gluing the OY |ψ−1(Uλ) by means of the automorphisms f ∗(θλµ), and it then suces tocheck that these latter automorphisms corresponds to the cocycle (ω](ελµ)), which follows immediately from thedefinitions (we can identify ελµ with its canonical image under ρ (3.7.2), a section of ψ∗(O ∗X ) over ψ

−1(Uλ ∩Uµ)).

(5.4.9). Let E and F be two OX -modules, F assumed to be locally free, and let G be an OX -module extension of

F by E , in other words there exists an exact sequence 0 → Ei−→ G

p−→ F → 0. Then, for each x ∈ X , there

exists an open neighborhood U of x such that G|U is isomorphic to the direct sum E|U ⊕F|U . We can reduceto the case where F = O n

X ; let ei (1¶ i ¶ n) be the canonical sections (5.5.5) of O nX ; there then exists an open

neighborhood U of x and n sections si of G over U such that p(si |U) = ei |U for 1 ¶ i ¶ n. That being so, letf be the homomorphism F|U →G|U defined by the sections si |U (5.1.1). It is immediate that for each openV ⊂ U , and each section s ∈ Γ(V,G ) we have s− f (p(s)) ∈ Γ(V,E ), hence our assertion.

(5.4.10). Let f : X → Y be a morphism of ringed spaces, F an OX -module, and L a locally free OY -module offinite rank. Then there exists a canonical isomorphism

(5.4.10.1) f∗(F )⊗OYL ' f∗(F ⊗OX

f ∗(L ))0I | 53

Indeed, for each OY -module L , we have a canonical homomorphism

f∗(F )⊗OYL

1⊗ρ−−→ f∗(F )⊗OY

f∗( f∗(L ))

α−→ f∗(F ⊗OX

f ∗(L )),

9For a general form of this result, see the book cited in the note on p. 51.

5. QUASI-COHERENT AND COHERENT SHEAVES 39

ρ the homomorphism (4.4.3.2) and α the homomorphism (4.2.2.1). To show that when L is locally free, thishomomorphism is bijective, it suces, the question being local, to consider the case where L = O n

X ; in addition,f∗ and f ∗ commute with finite direct sums, so we can assume n = 1, and in this case the proposition followsimmediately from the definitions and from the relation f ∗(OY ) = OX .

5.5. Sheaves on a locally ringed space.

(5.5.1). We say that a ringed space (X ,OX ) is a locally ringed space if for each x ∈ X , Ox is a local ring; theseringed spaces will be by far the most frequent ringed spaces that we will consider in this work. We then denoteby mx the maximal ideal of Ox , by k(x) the residue eld Ox/mx ; for each OX -module F , each open set U of X ,each point x ∈ U , and each section f ∈ Γ(U ,F ), we denote by f (x) the class of the germ fx ∈ Fx mod. mxFx ,and we say that this is the value of f at the point x . The relation f (x) = 0 then means that fx ∈ mxFx ; when thisis so, we say (by abuse of language) that f is zero at x . We will take care not to confuse this relation with fx = 0.

(5.5.2). Let X be a locally ringed space, L an invertible OX -module, and f a section of L over X . There is thenan equivalence between the three following properties for a point x ∈ X :

(a) fx is a generator of Lx ;(b) fx 6∈ mxLx (in other words, f (x) 6= 0);(c) there exists a section g of L −1 over an open neighborhood V of x such that the canonical image of f ⊗ g in

Γ(V,OX ) (5.4.3) is the unit section.Indeed, the question being local, we can reduce to the case where L = OX ; the equivalence of (a) and

(b) are then evident, and it is clear that (c) implies (b). Conversely, if fx 6∈ mx , then fx is invertible in Ox , sayfx gx = 1x . By definition of germs of sections, this means that there exists a neighborhood V of x and a sectiong of OX over V such that f g = 1 in V , hence (c).

It follows immediately from the condition (c) that the set X f of x satisfying the equivalent conditions (a),(b), (c) is open in X ; following the terminology introduced in (5.5.1), this is the set of the x for which f does notvanish.

(5.5.3). Under the hypotheses of (5.5.2), let L ′ be a second invertible OX -module; then, if f ∈ Γ(X ,L ),g ∈ Γ(X ,L ′), we have

X f ∩ X g = X f ⊗g .

We can in fact reduce immediately to the case where L =L ′ = OX (the question being local); as f ⊗ g thencanonically identifies with the product f g, the proposition is evident.

0I | 54

(5.5.4). Let F be a locally free OX of rank n; it is immediate that ∧pF is a locally free OX -module of rankn

p

ifp ¶ n and 0 if p > n, since the question is local and we can reduce to the case where F = O n

X ; in addition, foreach x ∈ X , (∧pF )x/mx(∧pF )x is a vector space of dimension

np

over k(x), which canonically identifies with∧p(Fx/mxFx). Let s1, . . . , sp be the sections of F over an open subset U of X , and let s = s1∧ · · ·∧ sp, which is asection of ∧pF over U (4.1.5); we have s(x) = s1(x)∧ · · · ∧ sp(x), and as a result, we say that the s1(x), . . . , sp(x)are linearly dependent means s(x) = 0. We conclude that the set of the x ∈ X such that s1(x), . . . , sp(x) are linearlyindependent is open in X : it suces in fact, by reducing to the case where F = O n

X , to apply (5.5.2) to the section

image of s under one of the projections of ∧pF = O (np)

X to then

p

factors.In particular, if s1, . . . , sn are n sections of F over U such that s1(x), . . . , sn(x) are linearly independent for

each point x ∈ U , then the homomorphism u : O nX |U →F|U defined by the si (5.1.1) is an isomorphism: indeed,

we can restrict to the case where F = O nX and where we canonically identify ∧nF and OX ; s = s1 ∧ · · · ∧ sn is

then an invertible section of OX over U , and we define an inverse homomorphism for u by means of the Cramerformulas.

(5.5.5). Let E and F be two locally free OX -modules (of finite rank), and let u : E →F be a homomorphism. Forthere to exist a neighborhood U of x ∈ X such that u|U is injective and that F|U is the direct sum of the u(E )|U andof a locally free (OX |U)-submodule G , it is necessary and sucient that ux : Ex →Fx gives, by passing to quotients,an injective homomorphism of vector spaces Ex/mxEx →Fx/mxFx . The condition is indeed necessary, since Fxis then the direct sum of the free Ox -modules ux(Ex) and Gx , so Fx/mxFx is the direct sum of ux(Ex)/mxux(Ex)and of Gx/mxGx . The condition is sucient, since we can reduce to the case where E = O m

X ; let s1, . . . , sm be theimages under u of the sections ei of O m

X such that (ei)y is equal to the i-th element of the canonical basis of O my

for each y ∈ Y (canonical sections of O mX ); by hypothesis, the s1(x), . . . , sm(x) are linearly independent, so if F

is of rank n, then there exist n−m sections sm+1, . . . , sn of F over a neighborhood V of x such that the si(x)(1¶ i ¶ n) form a basis for Fx/mxFx . There then exists (5.5.4) a neighborhood U ⊂ V of x such that the si(y)

6. FLATNESS 40

(1 ¶ i ¶ n) form a basis for Fy/myFy for each y ∈ V , and we conclude (5.5.4) that there is an isomorphismfrom F|U to O n

X |U , sending the si |U (1¶ i ¶ m) to the ei |U , which finishes the proof.

§6. Flatness

(6.0). The notion of flatness is due to J.-P. Serre [Ser56]; in the following, we omit the proofs of the results whichare presented in the Algèbre commutative of N. Bourbaki, to which we refer the reader. We assume that all ringsare commutative.10 0I | 55

If M , N are two A-modules, M ′ (resp. N ′) a submodule of M (resp. N), we denote by Im(M ′ ⊗A N ′) thesubmodule of M ⊗A N , the image under the canonical map M ′ ⊗A N ′→ M ⊗A N .

6.1. Flat modules.

(6.1.1). Let M be an A-module. The following conditions are equivalent:

(a) The functor M ⊗A N is exact in N on the category of A-modules;(b) TorA

i (M , N) = 0 for each i > 0 and for each A-module N ;(c) TorA

1(M , N) = 0 for each A-module N .

When M satisfies these conditions, we say that M is a at A-module. It is clear that each free A-module is flat.For M to be a flat A-module, it suces that for each ideal J of A, of nite type, the canonical map M ⊗A J→

M ⊗A A= M is injective.

(6.1.2). Each inductive limit of flat A-modules is a flat A-module. For a direct sum⊕

λ∈L Mλ of A-modules to be aflat A-modules, it is necessary and sucient that each of the A-modules Mλ is flat. In particular, every projectiveA-module is flat.

Let 0 → M ′ → M → M ′′ → 0 be an exact sequence of A-modules, such that M ′′ is at. Then, for eachA-module N , the sequence

0 −→ M ′ ⊗A N −→ M ⊗A N −→ M ′′ ⊗A N −→ 0

is exact. In addition, for M to be flat, is it necessary and sucient that M ′ is (but it can be that M and M ′ areflat without M ′′ = M/M ′ being so).

(6.1.3). Let M be a flat A-module, N any A-module; for two submodules N ′ N ′′ of N , we then have

Im(M ⊗A (N′ + N ′′)) = Im(M ⊗A N ′) + Im(M ⊗A N ′′),

Im(M ⊗A (N′ ∩ N ′′)) = Im(M ⊗A N ′)∩ Im(M ⊗A N ′′)

(images taken in M ⊗A N).

(6.1.4). Let M and N be two A-modules, M ′ (resp. N ′) a submodule of M (resp. N), and suppose that one ofthe modules M/M ′, N/N ′ is flat. Then we have Im(M ′ ⊗A N ′) = Im(M ′ ⊗A N)∩ (M ⊗A N ′) (images in M ⊗A N).In particular, if J is an ideal of A and if M/M ′ is flat, then we have JM ′ = M ′ ∩ JM .

6.2. Change of ring. When an additive group M is equipped with multiple modules structures relative tothe rings A, B, ..., we say that M is flat as an A-module, B-module, ..., we sometimes also say that M is A-at,B-at, ....

(6.2.1). Let A and B be two rings, M an A-module, N an (A, B)-bimodule. If M is flat and if N is B-flat, thenM ⊗A N is B-flat. In particular, if M and N are two flat A-modules, then M ⊗A N is a flat A-module. If B is anA-algebra and if M is a flat A-module, then the B-module M(B) = M ⊗A B is flat. Finally, if B is an A-algebra which 0I | 56is flat as an A-module, and if N is a flat B-module, then N is also A-flat.

(6.2.2). Let A be a ring, B an A-algebra which is flat as an A-module. Let M , N be two A-modules, such that Madmits a finite presentation; then the canonical homomorphism

(6.2.2.1) HomA(M , N)⊗A B −→ HomB(M ⊗A B, N ⊗A B)

(sending u⊗ b to the homomorphism m⊗ b′ 7→ u(m)⊗ b′b) is an isomorphism.

(6.2.3). Let (Aλ ,φµλ) be a filtered inductive system of rings; let A= lim−→Aλ . On the other hand, for each λ, let Mλbe an Aλ -module, and for λ ¶ µ let θµλ : Mλ → Mµ be a φµλ -homomorphism, such that (Mλ ,θµλ) is an inductivesystem; M = lim−→Mλ is then an A-module. This being so, if for each λ, Mλ is a at Aλ -module, then M is a atA-module. Indeed, let J be an ideal of nite type of A; by definition of the inductive limit, there exists an index λ

10See the exposé cited of N. Bourbaki for the generalization from most of the results to the noncommutative case.

6. FLATNESS 41

and an ideal Jλ of Aλ such that J= JλA. If we put J′µ = JλAµ for µ¾ λ, we also have J= lim−→J′µ (where µ varies

over the indices ¾ λ), hence (the functor lim−→ being exact and commuting with tensor products)

M ⊗A J= lim−→(Mµ ⊗Aµ J′µ) = lim−→J′µMµ = JM .

6.3. Local nature of atness.

(6.3.1). If A is a ring, S a multiplicative subset of A, S−1A is a at A-module. Indeed, for each A-module N ,N ⊗A S−1A identifies with S−1N (1.2.5) and we know (1.3.2) that S−1N is an exact functor in N .

If now M is a flat A-module, S−1M = M ⊗A S−1A is a flat S−1A-module (6.2.1), so it is also A-flat according tothe above and from (6.2.1). In particular, if P is an S−1A-module, we can consider it as an A-module isomorphicto S−1P; for P to be A-flat, it is necessary and sucient that it is S−1A-flat.

(6.3.2). Let A be a ring, B an A-algebra, and T a multiplicative subset of B. If P is a B-module which is A-at,T−1P is A-at. Indeed, for each A-module N , we have (T−1P)⊗A N = (T−1B ⊗B P)⊗A N = T−1B ⊗B (P ⊗A N)= T−1(P ⊗A N); T−1(P ⊗A N) is an exact functor in N , being the composition of the two exact functors P ⊗A N(in N) and T−1Q (in Q). If S is a multiplicative subset of A such that its image in B is contained in T , then T−1Pis equal to S−1(T−1P), so it is also S−1A-flat according to (6.3.1).

(6.3.3). Let φ : A→ B be a ring homomorphism, M a B-module. The following properties are equivalent:

(a) M is a flat A-module.(b) For each maximal ideal n of B, Mn is a flat A-module.(c) For each maximal ideal n of B, by setting m=φ−1(n), Mn is a flat Am-module.

Indeed, as Mn = (Mn)m, the equivalence of (b) and (c) follows from (6.3.1), and the fact that (a) implies(b) is a particular case of (6.3.2). It remains to see that (b) implies (a), that is to say, that for each injective 0I | 57homomorhism u : N ′→ N of A-modules, the homomorphism v = 1⊗u : M ⊗A N ′→ M ⊗A N is injective. We havethat v is also a homomorphism of B-modules, and we know that for it to be injective, it suces that for eachmaximal ideal n of B, vn : (M ⊗A N ′)n→ (M ⊗A N)n is injective. But as

(M ⊗A N)n = Bn ⊗B (M ⊗A N) = Mn ⊗A N ,

vn is none other that the homomorphism 1⊗ u : Mn ⊗A N ′→ Mn ⊗A N , which is injective since Mn is A-flat.In particular (by taking B = A), for an A-module M to be flat, it is necessary and sucient that Mm is Am-flat

for each maximal ideal m of A.

(6.3.4). Let M be an A-module; if M is flat, and if f ∈ A does not divide 0 in A, f does not kill any element 6= 0in M , since the homomorphism m 7→ f ·m is expressed as 1⊗u, where u is the multiplication a 7→ f · a on A andM is identified with M ⊗A A; if u is injective, it is the same for 1⊗ u since M is flat. In particular, if A is integral,M is torsion-free.

Conversely, suppose that A is integral, M is torsion-free, and suppose that for each maximal ideal m of A,Am is a discrete valuation ring ; then M is A-at. Indeed, it suces (6.3.3) to prove that Mm is Am-flat, and we cantherefore suppose that A is already a discrete valuation ring. But as M is the inductive limit of its submodules offinite type, and these latter submodules are torsion-free, we can in addition reduce to the case where M is offinite type (6.1.2). The proposition follows in this case from that M is a free A-module.

In particular, if A is an integral ring, φ : A→ B a ring homomorphism making B a at A-module and 6= 0,φ is necessarily injective. Conversely, if B is integral, A a subring of B, and if for each maximal ideal m of A, Am

is a discrete valuation ring, then B is A-at.

6.4. Faithfully at modules.

(6.4.1). For an A-module M , the following four properties are equivalent:

(a) For a sequence N ′→ N → N ′′ of A-modules to be exact, it is necessary and sucient that the sequenceM ⊗A N ′→ M ⊗A N → M ⊗A N ′′ is exact;

(b) M is flat for each A-module N , the relation M ⊗A N = 0 implies N = 0;(c) M is flat for each homomorphism v : N → N ′ of A-modules, the relation 1M ⊗ v = 0, 1M being the

identity automorphism of M ;(d) M is flat for each maximal ideal m of A, mM 6= M .

When M satisfies these conditions, we say that M is a faithfully at A-module; M is then necessarily a faithfulmodule. In addition, if u : N → N ′ is a homomorphism of A-modules, then for u to be injective (resp. surjective,bijective), it is necessary and sucient that 1⊗ u : M ⊗A N → M ⊗A N ′ is so.

0I | 58

6. FLATNESS 42

(6.4.2). A free module 6= 0 is faithfully flat; it is the same for the direct sum of a flat module and a faithfullyflat module. If S is a multiplicative subset of A, then S−1A is a faithfully flat A-module if S consists of invertibleelements (so S−1A= A).

(6.4.3). Let 0→ M ′→ M → M ′′→ 0 be an exact sequence of A-modules; if M ′ and M ′′ are flat, and if one ofthe two is faithfully flat, then M is also faithfully flat.

(6.4.4). Let A and B be two rings, M an A-module, N an (A, B)-bimodule. If M is faithfully flat and if N is afaithfully flat B-module, then M ⊗A N is a faithfully flat B-module. In particular, if M and N are two faithfullyflat A-modules, then so is M ⊗A N . If B is an A-algebra and if M is a faithfully flat A-module, the B-module M(B) isfaithfully flat.

(6.4.5). If M is a faithfully flat A-modules and if S is a multiplicative subset of A, S−1M is a faithfully flatS−1A-module, since S−1M = M ⊗A (S−1A) (6.4.4). Conversely, if for each maximal ideal m of A, Mm is a faithfullyflat Am-module, then M is a faithfully flat A-module, since M is A-flat (6.3.3), and we have

Mm/mMm = (M ⊗A Am)⊗Am(Am/mAm) = M ⊗A (A/m) = M/mM ,

so the hypotheses imply that M/mM 6= 0 for each maximal ideal m of A, which proves our assertion (6.4.1).

6.5. Restriction of scalars.

(6.5.1). Let A be a ring, φ : A→ B a ring homomorphism making B an A-algebra. Suppose that there exists aB-module N which is a faithfully at A-module. Then, for each A-module M , the homomorphism x 7→ 1⊗ x fromM to B ⊗A M = M(B) is injective. In particular, φ is injective; for each ideal a of A, we have φ−1(aB) = a; for eachmaximal (resp. prime) ideal m of A, there exists a maximal (resp. prime) ideal n of B such that φ−1(n) = m.

(6.5.2). When the conditions of (6.5.1) are satisfied, we identify A with the subring of B by φ and more generally,for each A-module M , we identify M with an A-submodule of M(B). We note that if B is also Noetherian, thenso is A, since the map a 7→ aB is an increasing injection from the set of ideals of A to the set of ideals of B; theexistence of an infinite strictly increasing sequence of ideals of A thus implies the existence of an analogoussequence of ideals of B.

6.6. Faithfully at rings.

(6.6.1). Let φ : A→ B be a ring homomorphism making B an A-algebra. The following five properties areequivalent:

(a) B is a faithfully flat A-module (in other words, M(B) is an exact and faithful functor in M).(b) The homomorphism φ is injective and the A-module B/φ (A) is flat. 0I | 59(c) The A-module B is flat (in other words, the functor M(B) is exact), and for each A-module M . the

homomorphism x 7→ 1⊗ x from M to M(B) is injective.(d) The A-module B is flat and for each ideal a of A, we have φ−1(aB) = a.(e) The A-module B is flat and for each maximal ideal m of A, there exists a maximal ideal n of B such

that φ−1(n) = m.

When these conditions are satisfied, we identify A with a subring of B.

(6.6.2). Let A be a local ring, m its maximal ideal, and B an A-algebra such that mB 6= B (which is so whenfor example B is a local ring and A→ B is a local homomorphism). If B is a at A-module, B is a faithfullyat A-module. Indeed, this follows from (6.4.1, (d)). Under the indicated conditions, we thus see that if B is ErrIINoetherian, then so too is A (6.5.2).

(6.6.3). Let B be an A-algebra which is a faithfully flat A-module. For each A-module M and each A-submoduleM ′ of M , we have (by identifying M with an A-submodule of M(B)) M ′ = M ∩ M ′(B). For M to be a flat (resp.faithfully flat) A-module, it is necessary and sucient that M(B) is a flat (resp. faithfully flat) B-module.

(6.6.4). Let B be an A-algebra, N a faithfully flat B-module. For B to be a flat (resp. faithfully flat) A-module, it isnecessary and sucient that N is.

In particular, let C be a B-algebra; if the ring C is faithfully flat over B and B is faithfully flat over A, then Cis faithfully flat over A; if C is faithfully flat over B and over A, then B is faithfully flat over A.

7. ADIC RINGS 43

6.7. Flat morphisms of ringed spaces.

(6.7.1). Let f : X → Y be a morphism of ringed spaces, and let F be a OX -module. We say that F is f -at (orY -at when there is no chance of confusion with f ) at a point x ∈ X if Fx is a flat O f (x)-module; we say thatF is f -at over y ∈ Y if F is f -flat for all the points x ∈ f −1(y); we say that F is f -at if F is f -flat at all thepoints of X . We say that the morphism f is at at x ∈ X (resp. at over y ∈ Y , resp. at) if OX is f -flat at x(resp. f -flat over y , resp. f -flat). If f is a flat morphism, we then say that X is at over Y , or Y -at. ErrII

(6.7.2). With the notation of (6.7.1), if F is f -flat at x , for each open neighborhood U of y = f (x), the functor( f ∗(G )⊗OX

F )x in G is exact on the category of (OY |U)-modules; indeed, this stalk canonically identifies withGy ⊗Oy

Fx , and our assertion follows from the definition. In particular, if f is a at morphism, the functor f ∗ isexact on the category of OY -modules.

(6.7.3). Conversely, suppose the sheaf of rings OY is coherent, and suppose that for each open neighborhood U ofy , the functor ( f ∗(G )⊗OX

F )x is exact in G on the category of coherent (OY |U)-modules. Then F is f -at at x .In fact, it suces to prove that for each ideal of finite type J of Oy , the canonical homomorphism J⊗Oy

Fx →Fx

is injective (6.1.1). We know (5.3.8) that there then exists an open neighborhood U of y and a coherent sheaf of 0I | 60ideals J of OY |U such that Jy = J, hence the conclusion.

(6.7.4). The results of (6.1) for flat modules are immediately translated into propositions for sheaves with aref -flat at a point:

If 0→F ′→F →F ′′→ 0 is an exact sequence of OX -modules and if F ′′ is f -flat at a point x ∈ X , then,for each open neighborhood U of y = f (x) and each (OY |U)-module G , the sequence

0 −→ ( f ∗(G )⊗OXF ′)x −→ ( f ∗(G )⊗OX

F )x −→ ( f ∗(G )⊗OXF ′′)x −→ 0

is exact. For F to be f -flat at x , it is necessary and sucient that F ′ is. We have similar statements for thecorresponding notions of a f -flat OX -modules over y ∈ Y , or of a f -flat OX -module.

(6.7.5). Let f : X → Y , g : Y → Z be two morphisms of ringed spaces; let x ∈ X , y = f (x), and F be anOX -module. If F is f -flat at the point x and if the morphism g is flat at the point y , then F is (g f )-flat at x(6.2.1). In particular, if f and g are flat morphisms, then g f is flat.

(6.7.6). Let X , Y be two ringed spaces, f : X → Y a at morphism. Then the canonical homomorphism ofbifunctors (4.4.6)

(6.7.6.1) f ∗(HomOY(F ,G )) −→HomOX

( f ∗(F ), f ∗(G ))

is an isomorphism when F admits a nite presentation (5.2.5).Indeed, the question being local, we can assume that there exists an exact sequence O m

Y →OnY →F → 0.

The two sides of (6.7.6.1) are right exact functors in F according to the hypotheses on f ; we then have reducedto proving the proposition in the case where F = OY , in which the result is trivial.

(6.7.8). We say that a morphism f : X → Y of ringed spaces is faithfully at if f is surjective and if, for eachx ∈ X , Ox is a faithfully at O f (x)-module. When X and Y are locally ringed spaces (5.5.1), it is equivalent to saythat the morphism f is surjective and at (6.6.2). When f is faithfully flat, f ∗ is an exact and faithful functor onthe category of OY -modules (6.6.1, a), and for an OY -module G to be Y -flat, it is necessary and sucient thatf ∗(G ) is (6.6.3).

§7. Adic rings

7.1. Admissible rings.

(7.1.1). Recall that in a topological ring A (not necessarily separated), we say that an element x is topologicallynilpotent if 0 is a limit of the sequence (xn)n¾0. We say that a topological ring A is linearly topologized if thereexists a fundamental system of neighborhoods of 0 in A of (necessarily open) ideals.

Denition (7.1.2). — In a linearly topologized ring A, we say that an ideal J is an ideal of denition if J is openand if, for each neighborhood V of 0, there exists a integer n> 0 such that Jn ⊂ V (which we express, by abuse of 0I | 61language, by saying that the sequence (Jn) tends to 0). We say that a linearly topologized ring A is preadmissible ifthere exists in A an ideal of definition; we say that A is admissible if it is preadmissible and if in addition it isseparated and complete.

It is clear that if J is an ideal of definition, L an open ideal of A, then J∩L is also an ideal of definition; theideals of definition of a preadmissible ring A thus form a fundamental system of neighborhoods of 0.

7. ADIC RINGS 44

Lemma (7.1.3). — Let A be a linearly topologized ring.

(i) For x ∈ A to be topologically nilpotent, it is necessary and sucient that for each open ideal J of A, the canonicalimage of x in A/J is nilpotent. The set T of topologically nilpotent elements of A is an ideal.

(ii) Suppose that in addition A is preadmissible, and let J be an ideal of denition for A. For x ∈ A to be topologicallynilpotent, it is necessary and sucient that its canonical image in A/J is nilpotent; the ideal T is the inverseimage of the nilradical of A/J and is thus open.

Proof. (i) follows immediately from the definitions. To prove (ii), it suces to note that for each neighborhoodV of 0 in A, there exists an n> 0 such that Jn ⊂ V ; if x ∈ A is such that xm ∈ J, we have xmq ∈ V for q ¾ n, so xis topologically nilpotent.

Proposition (7.1.4). — Let A be a preadmissible ring, J an ideal of denition for A.

(i) For an ideal J′ of A to be contained in an ideal of denition, it is necessary and sucient that there exists aninteger n> 0 such that J′n ⊂ J.

(ii) For an x ∈ A to be contained in an ideal of denition, it is necessary and sucient that it is topologicallynilpotent.

Proof.

(i) If J′n ⊂ J, then for each open neighborhood V of 0 in A, there exists an m such that Jm ⊂ V , thusJ′

mn ⊂ V .(ii) The condition is evidently necessary; it is sucient, since if it satisfied, then there exists an n such

that xn ∈ J, so J′ = J+ Ax is an ideal of definition, because it is open, and J′n ⊂ J.

Corollary (7.1.5). — In a preadmissible ring A, an open prime ideal contains all the ideals of denition.

Corollary (7.1.6). — The notation and hypotheses being that of (7.1.4), the following properties of an ideal J0 of A areequivalent:

(a) J0 is the largest ideal of denition of A;(b) J0 is a maximal ideal of denition;(c) J0 is an ideal of denition such that the ring A/J0 is reduced.

For there to exist an ideal J0 to have these properties, it is necessary and sucient that the nilradical of A/J to be nilpotent;J0 is then equal to the ideal T of topologically nilpotent elements of A.

Proof. It is clear that (a) implies (b), and (b) implies (c) according to (7.1.4, ii), and (7.1.3, ii); for the samereason, (c) implies (a). The latter assertion follows from (7.1.4, i) and (7.1.3, ii).

When T/J, the nilradical of A/J, is nilpotent, and we denote by Ared the (reduced) quotient ring A/T. 0I | 62

Corollary (7.1.7). — A preadmissible Noetherian ring admits a largest ideal of denition.

Corollary (7.1.8). — If a preadmissible ring A is such that, for an ideal of denition J, the powers Jn (n> 0) form afundamental system of neighborhoods of 0, it is the same for the powers J′n for each ideal of denition J′ of A.

Denition (7.1.9). — We say that a preadmissible ring A is preadic if there exists an ideal of definition J for Asuch that the Jn form a fundamental system of neighborhoods of 0 in A (or equivalently, such that the Jn are open).We call a ring adic if it is a separated and complete preadic ring.

If J is an ideal of definition for a preadic (resp. adic) ring A, we say that A is a J-preadic (resp. J-adic) ring,and that its topology is the J-preadic (resp. J-adic) topology. More generally, if M is an A-module, the topologyon M having for a fundamental system of neighborhoods of 0 the submodules JnM is called the J-preadic (resp.J-adic) topology. According to (7.1.8), these topologies are independent of the choice of the ideal of definition J.

Proposition (7.1.10). — Let A be an admissible ring, J an ideal of denition for A. Then J is contained in the radicalof A.

This statement is equivalent to any of the following corollaries:

Corollary (7.1.11). — For each x ∈ J, 1+ x is invertible in A.

Corollary (7.1.12). — For f ∈ A to be invertible in A, it is necessary and sucient that its canonical image in A/J isinvertible in A/J.

7. ADIC RINGS 45

Corollary (7.1.13). — For each A-module M of nite type, the relation M = JM (equivalent to M ⊗A (A/J) = 0)implies that M = 0.

Corollary (7.1.14). — Let u : M → N be a homomorphism of A-modules, N being of nite type; for u to be surjective, itis necessary and sucient that u⊗ 1 : M ⊗A (A/J)→ N ⊗A (A/J) is.

Proof. The equivalence of (7.1.10) and (7.1.11) follows from Bourbaki, Alg., chap. VIII, §6, no. 3, th. 1, andthe equivalence of (7.1.10) and (7.1.10) and (7.1.13) follows from loc. cit., th. 2; the fact that (7.1.10) implies(7.1.14) follows from loc. cit., cor. 4 of the prop. 6; on the other hand, (7.1.14) implies (7.1.13) by applyingthe zero homomorphism. Finally, (7.1.10) implies that if f is invertible in A/J, then f is not contained in anymaximal ideal of A, thus f is invertible in A, in other words, (7.1.10) implies (7.1.12); conversely, (7.1.12) implies(7.1.11).

It therefore remains to prove (7.1.11). Now as A is separated and complete, and the sequence (Jn) tendsto 0, it is immediate that the series

∑∞n=0(−1)n xn is convergent in A, and that if y is its sum, then we have

y(1+ x) = 1.

7.2. Adic rings and projective limits.

(7.2.1). Each projective limit of discrete rings is evidently a linearly topologized ring, separated and compact.Conversely, let A be a linearly topologized ring, and let (Jλ) be a fundamental system of open neighborhoodsof 0 in A consisting of ideals. The canonical maps φλ : A → A/Jλ form a projective system of continuous 0I | 63representations and therefore define a continuous representation φ : A→ lim←−A/Jλ ; if A is separated, then φ is atopological isomorphism from A to an everywhere dense subring of lim←−A/Jλ ; if in addition A is complete, then φis a topological isomorphism from A to lim←−A/Jλ .

Lemma (7.2.2). — For a linearly topologized ring to be admissible, it is necessary and sucient that it is isomorphic toa projective limit A= lim←−Aλ , where (Aλ ,µλµ) is a projective limit of discrete rings having for the set of indices a lteredordered (by ¶) L which admits a smallest element denoted 0 and satises the following conditions: 1st. the uλ : A→ Aλ aresujective; 2nd. the kernel Jλ of u0λ : Aλ → A0 is nilpotent. When this is so, the kernel J of u0 : A→ A0 is equal to lim←−Jλ .

Proof. The necessity of the condition follows from (7.2.1), by choosing (Jλ) to be a fundamental system ofneighborhoods of 0 consisting of ideals of definitions contained in an ideal of definition J0 and by applying(7.1.4, i). The converse follows from the definition of the projective limit and from (7.2.1), and the latter assertionis immediate.

(7.2.3). Let A be an admissible topological ring, J an ideal of A contained in an ideal of definition (in other words(7.1.4) such that (Jn) tends to 0); we can consider on A the ring topology having for a fundamental system ofneighborhoods of 0 the powers Jn (n > 0); we call again this the J-preadic topology. The hypothesis that A isadmissible implies that

⋃

n Jn = 0, therefore the J-preadic topology on A is separated ; let bA= lim←−A/Jn be the

completion of A for this topology (where the A/Jn are equipped with the discrete topology), and denote by u the(not necessarily continuous) ring homomorphism A→ bA, the projective limit of the sequence of homomorphismsun : A→ A/Jn. On the other hand, the J-preadic topology on A is finer than the given topology T on A; as A isseparated and complete for T , we can extend by continuity the identity map of A (equipped with the J-preadictopology) to A equipped with T ; this gives a continuous representation v : bA→ A.

Proposition (7.2.4). — If A is an admissible ring and J is contained in an ideal of denition of A, then A is separatedand complete for the J-preadic topology.

Proof. With the notation of (7.2.3), it is immediate that v u is the identity map of A. On the other hand,un v : bA→ A/Jn is the extension by continuity (for the J-preadic topology on A and the discrete topology onA/Jn) of the canonical map un; in other words, it is the canonical map from bA = lim←−k

A/Jk to A/Jn; u v is

therefore the projective limit of this sequence of maps, which is by definition the identity map of bA; this provesthe proposition.

Corollary (7.2.5). — Under the hypotheses of (7.2.3), the following conditions are equivalent:(a) the homomorphism u is continuous; 0I | 64(b) the homomorphism v is bicontinous;(c) A is a J-adic ring.

Corollary (7.2.6). — Let A be an admissible ring A, J an ideal of denition for A. For A to be Noetherian, it is necessaryand sucient for A/J to be Noetherian and for J/J2 to be an A/J-module of nite type.

7. ADIC RINGS 46

These conditions are evidently necessary. Conversely, suppose the conditions are satisfied; as according to(7.2.4) A is complete for the J-preadic topology, for it to be Noetherian, it is necessary and sucient that theassociated graded ring grad(A) (for the filtration on the Jn) is ([CC, p .18–07, th. 4]). Now, let a1, . . . , an be theelements of J whose classes mod. J2 are the generators of J/J2 as a A/J-module. It is immediate by inductionthat the classes mod. Jm+1 of the monomials of total degree m in the ai (1¶ i ¶ n) form a system of generatorsfor the A/J-module Jm/Jm+1. We conclude that grad(A) is a ring isomorphic to a quotient of (A/J)[T1, . . . , Tn](Ti indeterminates), which finishes the proof.

Proposition (7.2.7). — Let (Ai , ui j) be a projective system (i ∈ N) of discrete rings, and for each integer i, let Ji be thekernel in Ai of the homomorphism u0i : Ai → A0. We suppose that:

(a) For i ¶ j, ui j is surjective and its kernel is Ji+1j (therefore Ai is isomorphic to A j/J

i+1j ).

(b) J1/J21 (= J1) is a module of nite type over A0 = A1/J1.

Let A = lim←−iAi , and for each integer n ¾ 0, let un be the canonical homomorphism A→ An, and let J

(n+1) ⊂ A be itskernel. Then we have these conditions:

(i) A is an adic ring, having J= J(1) for an ideal of denition.(ii) We have J(n) = Jn for each n¾ 1.(iii) J/J2 is isomorphic to J1 = J1/J

21, and as a result is a module of nite type over A0 = A/J.

Proof. The hypothesis of surjectivity on the ui j implies that un is surjective; in addition, the hypothesis

(a) implies that J j+1j = 0, therefore A is an admissible ring (7.2.2); by definition, the J(n) form a fundamental

system of neighborhoods of 0 in A, so (ii) implies (i). In addition, we have J= lim←−iJi and the maps J→ Ji are

surjective, so (ii) implies (iii), and it remains to prove (ii). By definition, J(n) consists of the elements (xk)k¾0 ofA such that xk = 0 for k < n, therefore J(n)J(m) ⊂ J(n+m), in other words the J(n) form a ltration of A. On theother hand, J(n)/J(n+1) is isomorphic to the projection from J(n) to An; as J

(n) = lim←−i>nJn

i , this projection is none

other than Jnn, which is a module over A0 = An/Jn. Now let a j = (a jk)k¾0 be r elements of J = J(1) such that

a11, . . . , ar1 form a system of generators for J1 over A0; we will see that the set Sn of monomials of total degree nand the a j generate the ideal J

(n) of A. As Ji+1i = 0, it is clear first of all that Sn ⊂ J(n); since A is complete for the

filtration (J(m)), it suces to prove that the set Sn of classes mod. J(n+1) of elements of Sn generate the gradedmodule grad(J(n)) over the graded ring grad(A) for the above filtration ([CC, p. 18–06, lemme]); according tothe definition of the multiplication on grad(A), it suces to prove that for each m, Sm is a system of generators 0I | 65for the A0-module J(m)/J(m+1), or that Jm

m is generated by the monomials of degree m in the a jm (1 ¶ j ¶ r).For this, it remains to show that Jm is generated (as an Am-module) by the monomials of degree ¶ m relativeto a jm; the proposition being evident by definition for m = 1, we argue by induction on m, and let J′m be theAm-submodule of Jm generated by these monomials. The relation Jm−1 = Jm/J

mm and the induction hypothesis

prove that Jm = J′m + Jmm, hence, since Jm+1

m = 0, we have Jmm = J′m

m, and finally Jm = J′m.

Corollary (7.2.8). — Under the conditions of Proposition (7.2.7), for A to be Noetherian, it is necessary and sucientthat A0 is.

Proof. This follows immediately from Corollary (7.2.6).

Proposition (7.2.9). — Suppose the hypotheses of Proposition (7.2.7): for each integer i, let Mi be an Ai -module, andfor i ¶ j, let vi j : M j → Mi be a ui j -homomorphism, such that (Mi , vi j) is a projective system. In addition, suppose thatM0 is an A0-module of nite type and that the vi j are surjective with kernel J

i+1j M j . Then M = lim←−Mi is an A-module

of nite type, and the kernel of the surjective un-homomorphism vn : M → Mn is Jn+1M (such that Mn identifies with

M/Jn+1M = M ⊗A (A/Jn+1)).

Proof. Let zh = (zhk)k¾0 be a system of s elements of M such that the zh0 (1 ¶ h ¶ s) forms a system ofgenerators for M0; we will show that the zh generate the A-module M . The A-module M is separated and completefor the filtration by the M (n), where M (n) is the set of y = (yk)k¾0 in M such that yk = 0 for k < n; it is clearthat we have J(n)M ⊂ M (n) and that M (n)/M (n+1) = Jn

nMn. We therefore have reduced to showing that the classesof the zh modulo M (0) generate the graded module grad(M) (by the above filtration) over the graded ringgrad(A) [CC, p. 18–06, lemme]; for this, we observe that it suces to prove that the zhn (1¶ h¶ s) generate theAn-module Mn. We argue by induction on n, the proposition being evident by definition for n= 0; the relationMn−1 = Mn/J

nnMn and the induction hypothesis show that if M ′n is the submodule of Mn generated by the zhn, we

have that Mn = M ′n+JnnMn, and as Jn is nilpotent, this implies that Mn = M ′n. Similarly, passing to the associated

7. ADIC RINGS 47

graded modules shows that the canonical map from J(n) to M (n) is surjective (thus bijection), in other wordsthat J(n)M = JnM is the kernel of M → Mn−1. ErrII

Corollary (7.2.10). — Let (Ni , wi j) be a second projective system of Ai -modules satisfying the conditions of Proposition(7.2.9), and let N = lim←−Ni . There is a bijective correspondence between the projective systems (hi) of Ai -homomorphismshi : Mi → Ni and the homomorphisms of A-modules h : M → N (which is necessarily continuous for the J-adic topologies).

Proof. It is clear that if h : M → N is an A-homomorphism, then we have h(JnM) ⊂ JnN , hence the continuityof h; by passing to quotients, there corresponds to h a projective system of Ai -homomorphisms hi : Mi → Ni ,whose projective limit is h, hence the corollary.

Remark (7.2.11). — Let A be an adic ring with an ideal of definition J such that J/J2 is an A/J-module offinite type; it is clear that the Ai = A/Ji+1 satisfy the conditions of Proposition (7.2.7); as A is the projective limit 0I | 66of the Ai , we see that Proposition (7.2.7) gives the description of all the adic rings of the type considered (andin particular of all the adic Noetherian rings).

Example (7.2.12). — Let B be a ring, J an ideal of B such that J/J2 is a module of finite type over B/J(or over B, equivalently); set A= lim←−n

B/Jn+1; A is the separated completion of B equipped with the J-preadic

topology. If An = B/Jn+1, then it is immediate that the An satisfy the conditions of Proposition (7.2.7); thereforeA is an adic ring and if J is the closure in A of the canonical image of J, then J is an ideal of definition for A,Jn is the closure of the canonical image of Jn, A/Jn identifies with B/Jn and J/J2 is isomorphic to J/J2 as anA/J-module. Similarly, if N is such that N/JN is a B-module of finite type, and if we set Mi = N/Ji+1N , thenM = lim←−Mi is an A-module of finite type, isomorphic to the separated completion of N for the J-preadic topology,

and JnM identifies with the closure of the canonical image of JnN , and M/JnM identifies with N/JnN .

7.3. Preadic Noetherian rings.

(7.3.1). Let A be a ring, J an ideal of A, and M an A-module; we denote by bA= lim←−nA/Jn (resp. ÒM = lim←−n

M/JnM)

the separated completion of A (resp. M) for the J-preadic topology. Let M ′u−→ M

v−→ M ′′ → 0 be an exact

sequence of A-modules; as M/JnM = M ⊗A (A/Jn), the sequence

M ′/JnM ′un−→ M/JnM

vn−→ M ′′/JnM ′′ −→ 0

is exact for each n. In addition, as v(JnM) = Jnv(M) = JnM ′′, bv = lim←− vn is surjective (Bourbaki, Top. gén.,Chap. IX, 2nd ed., p. 60, Cor. 2). On the other hand, if z = (zk) is an element of the kernel of bv, then for eachinteger k, there exists a z′k ∈ M ′/Jk M ′ such that uk(z′k) = zk; we conclude that there exists a z′ = (z′n) ∈ÓM ′ suchthat the first k components of bu(z′) coincide with the z; in other words, the image of ÓM ′ under bu is dense in thekernel of bv.

If we suppose that A is Noetherian, then so is bA, according to (7.2.12), J/J2 is then an A-module of finite type.In addition, we have the following theorem.

Theorem (7.3.2). — (Krull’s Theorem). Let A be a Noetherian ring, J an ideal of A, M an A-module of finite type,and M ′ a submodule of M ; then the induced topology on M ′ by the J-preadic topology of M is identical to the J-preadictopology of M ′.

This follows immediately from

Lemma (7.3.2.1). — (Artin–Rees Lemma). Under the hypotheses of (7.3.2), there exists an integer p such that, forn¾ p, we have

M ′ ∩ JnM = Jn−p(M ′ ∩ Jp M).

For the proof, see [CC, p. 2–04]. 0I | 67

Corollary (7.3.3). — Under the hypotheses of (7.3.2), the canonical map M ⊗A bA→ ÒM is bijective, and the functorM ⊗A bA is exact in M on the category of A-modules of nite type; as a result, the separated J-adic completion bA is a atA-module (6.1.1).

Proof. We first note that ÒM is an exact functor in M on the category of A-modules of finite type. Indeed, let0→ M ′

u−→ M

v−→ M ′′→ 0 be an exact sequence; we have seen that bv : ÒM →dM ′′ is surjective (7.3.1); on the other

hand, if i is the canonical homomorphism M → ÒM , it follows from Krull’s Theorem (7.3.2) that the closure ofi(u(M ′)) in ÒM identifies with the separated completion of M ′ for the J-preadic topology; thus bu is injective, andaccording to (7.3.1), the image of bu is equal to the kernel of bv.

7. ADIC RINGS 48

This being so, the canonical map M ⊗A bA→ ÒM is obtained by passing to the projective limit of the mapsM ⊗A bA→ M ⊗A (A/Jn) = M/JnM . It is clear that this map is bijective when M is of the form Ap. If M is anA-module of finite type, then we have an exact sequence Ap → Aq → M → 0, hence, by virtue of the right exactnessof the functors M ⊗A bA and ÒM (in M) on the category of A-modules of finite type, we have the commutativediagram

Ap ⊗A bA //

Aq ⊗A bA //

M ⊗A bA //

0

cAp //ÒAq //

ÒM // 0,

where the two rows are exact and the first two vertical arrows are isomorphisms; this immediately finishes theproof.

Corollary (7.3.4). — Let A be a Noetherian ring, J an ideal of A, M and N two A-modules of nite type; we have thecanonical functorial isomorphisms

(M ⊗A N)∧ ' ÒM ⊗bAÒN , (HomA(M , N))∧ ' Hom

bA(ÒM ,ÒN).

Proof. This follows from Corollary (7.3.3), (6.2.1), and (6.2.2).

Corollary (7.3.5). — Let A be a Noetherian ring, J an ideal of A. The following conditions are equivalent:(a) J is contained in the radical of A.(b) bA is a faithfully at A-module (6.4.1).(c) Each A-module of nite type is separated for the J-preadic topology.(d) Each submodule of an A-module of nite type is closed for the J-preadic topology.

Proof. As bA is a flat A-module, the conditions (b) and (c) are equivalent, since (b) is equivalent to sayingthat if M is an A-module of finite type, then the canonical map M → ÒM = M ⊗A bA is injective (6.6.1, c). It isimmediate that (c) implies (d), since if N is a submodule of an A-module M of finite type, then M/N is separatedfor the J-preadic topology, so N is closed in M . We show that (d) implies (a): if m is a maximal ideal of A, thenm is closed in A for the J-preadic topology, so m=

⋂

p¾0(m+ Jp), and as m+ Jp is necessarily equal to A or tom, we have that m+ Jp = m for large enough p, hence Jp ⊂ m, and J ⊂ m when m is prime. Finally, (a) implies 0I | 68(b): indeed, let P be the closure of 0 in an A-module M of finite type, for the J-preadic topology; according toKrull’s Theorem (7.3.2), the topology induced on P by the J-preadic topology of M is the J-preadic topology ofP, so JP = P; as P is of finite type, it follows from Nakayama’s Lemma that P = 0 (J being contained in theradical of A).

We note that the conditions of Corollary (7.3.5) are satisfied when A is a local Noetherian ring and J 6= A isany ideal of A.

Corollary (7.3.6). — If A is a J-preadic Noetherian ring, then each A-module of nite type is separated and completefor the J-preadic topology.

Proof. As we then have bA= A, this follows immediately from Corollary (7.3.3).

We conclude that Proposition (7.2.9) gives the description of all the modules of finite type over an adicNoetherian ring.

Corollary (7.3.7). — Under the hypotheses of (7.3.2), the kernel of the canonical map M → ÒM = M ⊗A bA is the set ofthe x ∈ M killed by an element of 1+ J.

Proof. For each x ∈ M in this kernel, it is necessary and sucient that the separated completion of thesubmodule Ax is 0 (by Krull’s Theorem (7.3.2)), in other words, that x ∈ Jx .

7.4. Quasi-nite modules over local rings.

Denition (7.4.1). — Given a local ring A, with maximal ideal m, we say that an A-module M is quasi-finite(over A) if M/mM is of finite rank over the residue field k = A/m.

When A is Noetherian, the separated completion ÒM of M for the m-preadic topology is then an bA-module ofnite type; indeed, as m/m2 is then an A-module of finite type, this follows from Example (7.2.12) and from thehypothesis on M/mM .

In particular, if we suppose that in addition A is complete and M is separated for the m-preadic topology (inother words,

⋂

n mnM = 0), then M is also an A-module of nite type: indeed, ÒM is then an A-module of finite type,

7. ADIC RINGS 49

and as M identifies with a submodule of ÒM , M is also of finite type (and is indeed identical to its completionaccording to Corollary (7.3.6)).

Proposition (7.4.2). — Let A, B be two local rings, m, n their maximal ideals, and suppose that B is Noetherian. Letφ : A→ B be a local homomorphism, M a B-module of nite type. If M is a quasi-nite A-module, then the m-preadic andn-preadic topologies on M are identical, thus separated.

Proof. We note that by hypothesis M/mM is of nite length as an A-module, thus also a fortiori as a B-module.We conclude that n is the unique prime ideal of B containing the annihilator of M/mM : indeed, we immediatelyreduce (according to (1.7.4) and (1.7.2)) to the case where M/mM is simple, thus necessarily isomorphic to B/n,and our assertion is evident in this case. On the other hand, as M is a B-module of finite type, the prime idealswhich contain the annihilator of M/mM are those which contain mB+b, where we denote by b the annihilator ofthe B-module M (1.7.5). As B is Noetherian, we conclude ([Sam53b, p. 127, Cor. 4]) that mB + b is an ideal of 0I | 69definition for B, in other words there exists a k > 0 such that nk ⊂ mB+b ⊂ n; as a result, for each h> 0, we have

nhk ⊂ (mB + b)hM = mhM ⊂ nhM ,

which proves that the m-preadic and n-preadic topologies on M are the same; the second is separated accordingto Corollary (7.3.5).

Corollary (7.4.3). — Under the hypotheses of Proposition (7.2.4), if in addition A is Noetherian and complete for them-preadic topology, then M is an A-module of nite type.

Proof. Indeed, M is then separated for the m-preadic topology, and our assertion follows from the remarkafter Definition (7.4.1).

(7.4.4). The most important case of Proposition (7.4.2) is when B is a quasi-finite A-module, i.e., B/mB is analgebra of nite rank over k = A/m; furthermore, this condition can be broken down into the combination of thefollowing:

(i) mB is an ideal of denition for B;(ii) B/n is an extension of nite rank of the eld A/m.

When this is so, every B-module of finite type is evidently a quasi-finite A-module.

Corollary (7.4.5). — Under the hypotheses of Proposition (7.4.2), if b is the annihilator of the B-module M , then B/bis a quasi-nite A-module.

Proof. Suppose M 6= 0 (otherwise the corollary is evident). We can consider M as a module over the localNoetherian ring B/b; its annihilator then being 0, the proof of Proposition (7.4.2) shows that m(B/b) is an idealof definition for B/b. On the other hand, M/nM is a vector space of finite rank over A/m, being a quotient ofM/mM , which is by hypothesis of finite rank over A/m; as M 6= 0, we have M 6= nM by virtue of Nakayama’sLemma; as M/nM is a vector space 6= 0 over B/n, the fact that it is of finite rank over A/m implies that B/n isalso of finite rank over A/m; the conclusion follows from (7.4.4) applied to the ring B/b.

7.5. Rings of restricted formal series.

(7.5.1). Let A be a topological ring, linearly topologized, separated and complete; let (Jλ) be a fundamentalsystem of neighborhoods of 0 in A consisting of (open) ideals, such that A canonically identifies with lim←−A/Jλ(7.2.1). For each λ, let Bλ = (A/Jλ)[T1, . . . , Tr], where the Ti are indeterminates; it is clear that the Bλ forma projective system of discrete rings. We set lim←−Bλ = AT1, . . . , Tr, and we will see that this topological ringis independent of the fundamental system of ideals (Jλ) considered. More precisely, let A′ be the subring ofthe ring of formal series A[[T1, . . . , Tr]] consisting of formal series

∑

α cαTα (with α = (α1, . . . ,αr) ∈ Nr) suchthat lim cα = 0 (according to the filter by compliments of finite subsets of Nr); we say that these series are therestricted formal series in the Ti , with coecients in A. For each neighborhood V of 0 in A, let V ′ be the set of 0I | 70x =

∑

α cαTα ∈ A′ such that cα ∈ V for all α. We verify immediately that the V ′ form a fundamental system ofneighborhoods of 0 defining on A′ a separated ring topology; we will canonically define a topological isomorphismfrom the ring AT1, . . . , Tr to A′. For each α ∈ Nr and each λ, let φλ,α be the map from (A/Jλ)[T1, . . . , Tr] to A/Jλwhich sends each polynomial in the first ring to coecient of Tα in that polynomial. It is clear that the φλ,α forma projective system of homomorphisms of A/Jλ -modules, so their projective limit is a continuous homomorphismφα : AT1, . . . , Tr → A; we will see that, for each y ∈ AT1, . . . , Tr, the formal series

∑

αφα(y)Tα is restricted.

Indeed, if yλ is the component of y in Bλ , and if we denote by Hλ the finite set of the α ∈ Nr for which thecoecients of the polynomial yλ are nonzero, then we have φλ,α(yµ) ∈ Jλ for Jµ ⊂ Jλ and α 6∈ Hλ , and by passingto the limit, φα(y) ∈ Jλ for α 6∈ Hλ . We thus define a ring homomorphism φ : AT1, . . . , Tr → A′ by setting

7. ADIC RINGS 50

φ (y) =∑

αφα(y)Tα, and it is immediate that φ is continuous. Conversely, if θλ is the canonical homomorphism

A→ A/Jλ , then for each element z =∑

α cαTα ∈ A′ and each λ, there are only a finite number of indices α suchthat θλ(cα) 6= 0, and as a result ψλ(z) =

∑

α θλ(cα)Tα is in Bλ ; the ψλ are continuous and form a projective system

of homomorphisms whose projective limit is a continuous homomorphism ψ : A′→ AT1, . . . , Tr; is remains toverify that φ ψ and ψ φ are the identity automorphisms, which is immediate.

(7.5.2). We identify AT1, . . . , Tr with the ring A′ of restricted formal series by means of the isomorphismsdefined in (7.5.1). The canonical isomorphisms

((A/Jλ)[T1, . . . , Tr])[Tr+1, . . . , Ts]' (A/Jλ)[T1, . . . , Ts]

define, by passing to the projective limit, a canonical isomorphism

(AT1, . . . , Tr)Tr+1, . . . , Ts ' AT1, . . . , Ts.

(7.5.3). For every continuous homomorphism u : A→ B from A to a linearly topologized ring B, separatedand complete, and each system (b1, . . . , br) of r elements of B, there exists a unique continuous homomorphismu : AT1, . . . , Tr → B, such that u(a) = u(a) for all a ∈ A and u(T j) = b j for 1¶ j ¶ r. It suces to set

u

∑

αcαTα

=∑

αu(cα)b

α11 · · · b

αrr ;

the verification of the fact that the family (u(cα)bα11 · · · b

αrr ) is summable in B and that u is continuous are

immediate and left to the reader. We note that this property (for arbitrary B and b j) characterize the topologicalring AT1, . . . , Tr up to unique isomorphism.

Proposition (7.5.4). —

(i) If A is an admissible ring, then so is A′ = AT1, . . . , Tr.(ii) Let A be an adic ring, J an ideal of denition for A such that J/J2 is of nite type over A/J. If we set J′ = JA′, 0I | 71

then A′ is also a J′-adic ring, and J′/J′2 is of nite type over A′/J′. If in addition A is Noetherian, then so is A′.

Proof.

(i) If J is an ideal of A, J′ the ideal of A′ consisting of the∑

α cαTα such that cα ∈ J for all α, then(J′)n ⊂ (Jn)′; if J is an ideal of definition for A, then J′ is also an ideal of definition for A′.

(ii) Set Ai = A/Ji+1, and for i ¶ j, let ui j be the canonical homomorphism A/J j+1 → A/Ji+1; set A′i =Ai[T1, . . . , Tr], and let u′i j be the homomorphism A′j → A′i (i ¶ j) obtained by applying ui j to thecoecients of the polynomials in A′j . We will show that the projective system (A′i , u′i j) satisfies theconditions of Proposition (7.2.7); as J′ is the kernel of A′→ A′0, this proves the first assertion of (ii). Itis clear that the u′i j are surjective; the kernel J

′i of u0i is the set of polynomials in Ai[T1, . . . , Tr] whose

coecients are in J/Ji+1; in particular, J′1 is the set of polynomials in A1[T1, . . . , Tr] whose coecientsare in J/J2. As J/J2 is of finite type over A1 = A/J2, we see that J′1/J

′1

2 is a module of finite typeover A′1 (or equivalently, over A′0 = A′1/J

′1). We will now show that the kernel of ui j is J′j

i+1. It is

evident that J′ji+1 is contained in this kernel. On the other hand, let a1, . . . , am be the elements of J

whose classes mod J2 generate J/J2; we verify immediately that the classes mod J j+1 of monomialsof degree ¶ j in the ak (1 ¶ k ¶ m) generate J/J j+1, and the classes of monomials of degree > iand ¶ j generate Ji+1/J j+1; a monomial in the Tk having such an element for a coecient is thus aproduct of i + 1 elements of J′i , which establishes our assertion. Finally, if A is Noetherian, then so isA′/J′ = (A/J)[T1, . . . , Tr], hence A′ is Notherian (7.2.8).

Proposition (7.5.5). — Let A be a Notherian J-adic ring, B an admissible topological ring, φ : A→ B a continuoushomomorphism, making B and A-algebra. The following conditions are equivalent:

(a) B is Noetherian and JB-adic, and B/JB is an algebra of nite type over A/J.(b) B is topologically A-isomorphic to lim←−Bn, where Bn = Bm/J

n+1Bm for m¾ n, and B1 is an algebra of nite type

over A1 = A/J2.(c) B is topologically A-isomorphic to a quotient of an algebra of the form AT1, . . . , Tr by an ideal (necessarily

closed according to Corollary (7.3.6) and Proposition (7.5.4, ii)).

7. ADIC RINGS 51

Proof. As A is Noetherian, so is A′ = AT1, . . . , Tr (7.5.4), so (c) implies that B is Noetherian; as J′ = JA′ isan open neighborhood of 0 in A′ such that the J′

n form a fundamental system of neighborhoods of 0, the imagesJnB of the J′

n form a fundamental system of neighborhoods of 0 in B, and as B is separated and complete, B isa JB-adic ring. Finally, B/JB is an algebra (over (A/J) quotient of A′/JA′ = (A/J)[T1, . . . , Tr], so it is of finitetype, which proves that (c) implies (a).

If B is JB-adic and Noetherian, then B is isomorphic to lim←−Bn, where Bn = B/Jn+1B (7.2.11), and JB/J2B is

a module of finite type over B/JB. Let (a j)1¶ j¶s be a system of generators for the B/JB-module JB/J2B, and let(ci)1¶i¶r be a system of elements of B/J2B such that the classes mod JB/J2B form a system of generators for 0I | 72the A/J-algebra B/JB; we see immediately that the cia j form a system of generators for the A/J2-algebra B/J2B,hence (a) implies (b).

It remains to prove that (b) implies (c). The hypotheses imply that B1 is a Noetherian ring, and asB1 = B2/J

2B2, we have J2B1 = 0, hence JB1 = JB1/J2B1 is a B0-module of finite type. The conditions of

Proposition (7.2.7) are thus satisfied by the projective system (Bn) and B is a JB-adic ring. Let (ci)1¶i¶r be a finitesystem of elements of B whose classes mod JB generate the A/J-algebra B/JB, and whose linear combinationswith coecients in J are such that their classes mod J2B generate the B0-module JB/J2B. There exists acontinuous A-homomorphism u from A′ = AT1, . . . , Tr to B which reduces to φ on A and is such that u(Ti) = cifor 1¶ i ¶ r (7.5.3); if we prove that u is surjective, then (c) will be established, since from u(A′) = B we deducethat u(JnA′) = JnB, in other words that u is a strict morphism of topological rings and B is this isomorphic toa quotient of A′ by a closed ideal. As B is complete for the JB-adic topology, it suces ([CC, p. 18–07]) toshow that the homomorphism grad(A′)→ grad(B), induced canonically by u for the J-adic filtrations on A′ andB, is surjective. But by definition, the homomorphisms A′/JA′ → B/JB and JA′/J2A′ → JB/J2B induced by uare surjective; by induction on n, we immediately deduce that so is JA′/JnA′ → JB/JnB, and a fortiori so isJnA′/Jn+1A′→ JnB/Jn+1B, which finishes the proof.

7.6. Completed rings of fractions.

(7.6.1). Let A be a linearly topologized ring, (Jλ) a fundamental system of neighborhoods of 0 in A consisting ofideals, S a multiplicative subset of A. Let uλ be the canonical homomorphism A→ Aλ = A/Jλ , and for Jµ ⊂ Jλ ,let uλµ be the canonical homomorphism Aµ → Aλ . Set Sλ = uλ(S), so that uλµ(Sµ) = Sλ . The uλµ canonicallyinduce surjective homomorphisms S−1

µ Aµ→ S−1λ Aλ , for which these rings form a projective system; we denote

by AS−1 the projective limit of this system. This definition does not depend on the fundamental system ofneighborhoods (Jλ) chosen; indeed:

Proposition (7.6.2). — The ring AS−1 is topologically isomorphic to the separated completion of the ring S−1A forthe topology which has a fundamental system of neighborhoods of 0 consisting of the S−1Jλ .

Proof. If vλ is the canonical homomorphism S−1A→ S−1λ Aλ induced by uλ , then the kernel of vλ is surjective,

hence the proposition (7.2.1).

Corollary (7.6.3). — If S′ is the canonical image of S in the separated completion bA of A, then AS−1 canonicallyidenties with bAS′−1.

We note that if A is separated and complete, then it is not necessarily the same for S−1A with the topologydefined by the S−1Jλ , as we see for example by taking S to be the set of the f n (n¾ 0), where f is topologicallynilpotent but not nilpotent: indeed, S−1A is not 0 and on the other hand, for each λ there exists an n such thatf n ∈ Jλ , so 1= f n/ f n ∈ S−1Jλ and S−1Jλ = S−1A. 0I | 73

Corollary (7.6.4). — If, in A, 0 does not belong to S, then the ring AS−1 is not 0.

Proof. Indeed, 0 does not belong to 1 in the ring S−1A; otherwise, we would have that 1 ∈ S−1Jλ for eachopen ideal Jλ of A, and it follows that Jλ ∩ S 6=∅ for all λ, contradicting the hypothesis.

(7.6.5). We say that AS−1 is the completed ring of fractions of A with denominators in S. With the above notation,it is clear that the inverse image of S−1Jλ in A contains Jλ , hence the canonical map A→ S−1A is continuous,and if we compose it with the canonical map S−1A→ AS−1, we obtain a canonical continuous homomorphismA→ AS−1, the projective limit of the homomorphisms A→ S−1

λ Aλ .

(7.6.6). The couple consisting of AS−1 and the canonical homomorphism A→ AS−1 are characterized by thefollowing universal property: every continuous homomorphism u from A to a linearly topologized ring B, separated

and complete, such that u(S) consists of the invertible elements of B, uniquely factorizes as A→ AS−1u′−→ B,

where u′ is continuous. Indeed, u uniquely factorizes as A→ S−1Av′−→ B; as for each open ideal K of B we have

7. ADIC RINGS 52

that u−1(K) contains a Jλ , v′−1(K) necessarily contains S−1Jλ , so v′ is continuous; since B is separated and

complete, v′ uniquely factorizes as S−1A→ AS−1u′−→ B, where u′ is continuous; hence our assertion.

(7.6.7). Let B be a second linearly topologized ring, T a multiplicative subset of B, φ : A→ B a continuous

homomorphism such that φ (S) ⊂ T . According to the above, the continuous homomorphism Aφ−→ B→ BT−1

uniquely factorizes as A→ AS−1φ ′−→ BT−1, where φ ′ is continuous. In particular, if B = A and if φ is the

identity, we see that for S ⊂ T we have a continuous homomorphism ρT,S : AS−1 → AT−1 obtained by passingto the separated completion from S−1A→ T−1A; if U is a third multiplicative subset of A such that S ⊂ T ⊂ U ,then we have ρU ,S = ρU ,T ρT,S .

(7.6.8). Let S1, S2 be two multiplicative subsets of A, and let S′2 be the canonical image of S2 in AS−11 ; we

then have a canonical topological isomorphism A(S1S2)−1 ' AS−11 S

′2−1, as we see from the canonical

isomorphism (S1S2)−1A' S′′2−1(S−1

1 A) (where S′′2 is the canonical image of S2 in S−11 A), which is bicontinuous.

(7.6.9). Let a be an open ideal of A; we can assume that Jλ ⊂ a for all λ, and as a result S−1Jλ ⊂ S−1a in thering S−1A, in other words, S−1a is an open ideal of S−1A; we denote by aS−1 its separated completion, equalto lim←−(S

−1a/S−1Jλ), which is an open ideal of AS−1, isomorphic to the closure of the canonical image of S−1a.

In addition, the discrete ring AS−1/aS−1 is canonically isomorphic to S−1A/S−1a= S−1(A/a). Conversely, if a′ isan open ideal of AS−1, then a′ contains an ideal of the form JλS−1 which is the inverse image of an idealof S−1A/S−1Jλ , which is necessarily (1.2.6) of the form S−1a, where a ⊃ Jλ . We conclude that a′ = aS−1. Inparticular (1.2.6):

Proposition (7.6.10). — The map p 7→ pS−1 is an increasing bijection from the set of open prime ideals p of Asuch that p∩ S = ∅ to the set of open prime ideals of AS−1; in addition, the eld of fractions of AS−1/pS−1 is 0I | 74canonically isomorphic to that of A/p.

Proposition (7.6.11). —(i) If A is an admissible ring, then so is A′ = AS−1, and for every ideal of denition J for A, J′ = JS−1 is an

ideal of denition for A′.(ii) Let A be an adic ring, J an ideal of denition for A such that J/J2 is of nite type over A/J; then A′ is a J′-adic

ring and J′/J′2 is of nite type over A′/J′. If in addition A is Noetherian, then so is A′.

Proof.(i) If J is an ideal of definition for A, then it is clear that S−1J is an ideal of definition for the topological

ring S−1A, since we have (S−1J)n = S−1Jn. Let A′′ be the separated ring associated to S−1A, J′′ theimage of S−1J in A′′; the image of S−1Jn is J′′n, so J′′

n tends to 0 in A′′; as J′ is the closure of J′′ in A′,J′

n is contained in the closure of J′′n, hence tends to 0 in A′.(ii) Set Ai = A/Ji+1, and for i ¶ j, let ui j be the canonical homomorphism A/J j+1 → A/Ji+1; let Si be

the canonical image of S in Ai , and set A′i = S−1i Ai ; finally, let u′i j : A′j → A′i be the homomorphism

canonically induced by ui j . We show that the projective system (A′i , u′i j) satisfies the conditions ofProposition (7.2.7): it is clear that the u′i j are surjective; on the other hand, the kernel of u′i j is

S−1j (J

i+1/J j+1) (1.3.2), equal to J′ji+1, where J′j = S−1

j (J/Jj+1); finally, J′1/J

′1

2 = S−11 (J/J

2), and as

J/J2 is of finite type over A/J2, J′1/J′1

2 is of finite type over A′1. Finally, if A is Noetherian, then so isA′0 = S−1

0 (A/J), which finishes the proof (7.2.8).

Corollary (7.6.12). — Under the hypotheses of Proposition (7.6.11, ii), we have (JS−1)n = JnS−1.

Proof. This follows from Proposition (7.2.7) and the proof of Proposition (7.6.11).

Proposition (7.6.13). — Let A be an adic Noetherian ring, S a multiplicative subset of A; then AS−1 is a atA-module.

Proof. If J is an ideal of definition for A, then AS−1 is the separated completion of the Noetherian ringS−1A equipped with the S−1J-preadic topology; as a result (7.3.3) AS−1 is a flat S−1A-module; as S−1A is a flatA-module (6.3.1), the proposition follows from the transitivity of flatness (6.2.1).

Corollary (7.6.14). — Under the hypotheses of Proposition (7.6.13), let S′ ⊂ S be a second multiplicative subset of A;then AS−1 is a at AS′−1-module.

7. ADIC RINGS 53

Proof. By (7.6.8), AS−1 canonically identifies with AS′−1S−10 , where S0 is the canonical image of S in

AS′−1, and AS′−1 is Noetherian (7.6.11).

(7.6.15). For each element f of a linearly topologized ring A, we denote by A f the completed ring of fractionsAS−1

f , where S f is the multiplicative set of the f n (n ¾ 0); for each open ideal a of A, we write a f foraS−1

f . If g is a second element of A, then we have a canonical continuous homomorphism A f → A f g (7.6.7).When f varies over a multiplicative subset S of A, the A f form a filtered inductive system with the abovehomomorphisms; we set AS = lim−→ f ∈S

A f . For every f ∈ S, we have a homomorphism A f → AS−1 (7.6.7),and these homomorphisms form an inductive system; by passing to the inductive limit, they thus define a 0I | 75canonical homomorphism AS→ AS−1.

Proposition (7.6.16). — If A is a Noetherian ring, then AS−1 is a at module over AS.

Proof. By (7.6.14), AS−1 is flat for each of the rings A f for f ∈ S, and the conclusion follows from(6.2.3).

Proposition (7.6.17). — Let p be an open prime ideal in an admissible ring A, and let S = A− p. Then the ringsAS−1 and AS are local rings, the canonical homomorphism AS→ AS−1 is local, and the residue elds of AS andAS−1 are canonically isomorphic to the eld of fractions of A/p.

Proof. Let J ⊂ p be an ideal of definition for A; we have S−1J ⊂ S−1p = pAp, so Ap/S−1J is a local ring;

we conclude from Corollary (7.1.12), (7.6.9), and Proposition (7.6.11, i) that AS−1 is a local ring. Setm = lim−→ f ∈S

p f , which is an ideal in AS; we will see that each element in AS not in m is invertible. Indeed,

such an element is the image in AS of an element z ∈ A f not in p f , for an f ∈ S; its canonical image z0 in

A f /J f = S−1f (A/J) therefore is not in S−1

f (p/J) (7.6.9), which means that z0 = x/ fk, where x 6∈ p and x , f

are the classes of x , f mod J. As x ∈ S, we have g = x f ∈ S, and in S−1g A, the canonical image y0 = x k+1/gk

of x/ f k ∈ S−1f A admits an inverse x k−1 f 2k/gk. This implies a fortiori that the image of y0 in S−1

g A/S−1g J is

invertible, so ((7.6.9) and Corollary (7.1.12)) the canonical image y of z in Ag is invertible; the image of z inAS (equal to that of y) is as a result invertible. We thus see that AS a local ring with maximal ideal m; inaddition, the image of p f in AS−1 is contained in the maximal ideal pS−1 of this ring; a fortiori, the imageof m in AS−1 is contained in pS−1, so the canonical homomorphism AS→ AS−1 is local. Finally, as eachelement of AS−1/pS−1 is the image of an element in the ring S−1

f A for a suitable f ∈ S, the homomorphismAS→ AS−1/pS−1 is surjective, and gives an isomorphism of the residue fields by passing to quotients.

Corollary (7.6.18). — Under the hypotheses of Proposition (7.6.17), if we suppose also that A is an adic Noetherianring, then the local rings AS−1 and AS are Noetherian, and AS−1 is a faithfully at AS-module.

Proof. We know from before (7.6.11, ii) that AS−1 is Noetherian and AS-flat (7.6.16); as the homomorphismAS→ AS−1 is local, we conclude that AS−1 is a faithfully flat AS-module (6.6.2), and as a result that AS isNoetherian (6.5.2).

7.7. Completed tensor products.

(7.7.1). Let A be a linearly topologized ring, M , N two linearly topologized A-modules. Let J, V , W be openneighborhoods of 0 in A, M , N respectively, which are A-modules, and such that JM ⊂ V , JN ⊂W , so that M/Vand N/W can be considered as A/J-modules. When J, V , W vary over the systems of open neighborhoodssatisfying these properties, it is immediate that the modules (M/V )⊗A/J (N/W ) form a projective system ofmodules over the projective system of rings A/J; by passing to the projective limit, we thus obtain a module over 0I | 76the separated completion bA of A, which we call the completed tensor product of M and N and denote by (M ⊗A N)∧.If we have that M/V is canonically isomorphic to ÒM/V , where ÒM is the separated completion of M and V theclosure in ÒM of the image of V , then we see that the completed tensor product (M ⊗A N)∧ canonically identifieswith (ÒM ⊗

bAÒN)∧, which we denote by ÒM b⊗

bAÒN .

(7.7.2). With the above notation, the tensor products (M/V )⊗A (N/W ) and (M/V )⊗A/J (N/W ) identify canon-ically; they identify with (M ⊗A N)/(Im(V ⊗A N) + Im(M ⊗A W )). We conclude that (M ⊗A N)∧ is the separatedcompletion of the A-module M ⊗A N , equipped with the topology for which the submodules

Im(V ⊗A N) + Im(M ⊗A W )

form a fundamental system of neighborhoods of 0 (V and W varying over the set of open submodules of M and Nrespectively); we say for brevity that this topology is the tensor product of the given topologies on M and N .

7. ADIC RINGS 54

(7.7.3). Let M ′, N ′ be two linearly topologized A-modules, u : M → M ′, v : N → N ′ two continuous ho-momorphisms; it is immediate that u ⊗ v is continuous for the tensor product topologies on M ⊗A N andM ′ ⊗A N ′ respectively; by passing to the separated completions, we obtain a continuous homomorphism(M ⊗A N)∧→ (M ′ ⊗A N ′)∧, which we denote by ub⊗v; (M ⊗A N)∧ is thus a bifunctor in M and N on the categoryof linearly topologized A-modules.

(7.7.4). We similarly define the completed tensor product of any finite number of linearly topologized A-modules;it is immediate that this product has the usual properties of associativity and commutativity.

(7.7.5). If B, C are two linearly topologized A-algebras, then the tensor product topology on B ⊗A C has for afundamental system of neighborhoods of 0 the ideals Im(K⊗A C) + Im(B⊗A L) of the algebra B⊗A C , K (resp. L)varying over the set of open ideals of B (resp. C). As a result, (B ⊗A C)∧ is equipped with the structure of atopological bA-algebra, the projective limit of the projective system of A/J-algebras (B/K)⊗A/J (C/L) (J the openideal of A such that JB ⊂ K, JC ⊂ L; it always exists). We say that this algebra is the completed tensor product ofthe algebras B and C .

(7.7.6). The A-algebra homomorphisms b 7→ b⊗ 1, c 7→ 1⊗ c from B and C to B ⊗A C are continuous when weequip the latter algebra with the tensor product topology; by composing with the canonical homomorphismfrom B⊗A C to its separated completion, they give canonical homomorphisms ρ : B→ (B⊗A C)∧, σ : C → (B⊗A C)∧.The algebra (B ⊗A C)∧ and the homomorphisms ρ and σ have in addition the following universal property: forevery linearly topologized A-algebra D, separated and complete, and each pair of continuous A-homomorphismsu : B→ D, v : C → D, there exists a unique continuous A-homomorphism w : (B ⊗A C)∧→ D such that u= w ρand v = wσ. Indeed, there already exists a unique A-homomorphism w0 : B⊗A C → D such that u(b) = w0(b⊗1)and v(c) = w0(1⊗c), and it remains to prove that w0 is continuous, since it then gives a continuous homomorphism 0I | 77w by passing to the separated completion. If M is an open ideal of D, then there exists by hypothesis openideals K ⊂ B, L ⊂ C such that u(K) ⊂M, v(L) ⊂M; the image under w0 of Im(K⊗A C) + Im(B ⊗A L) is againcontained in M, hence our assertion.

Proposition (7.7.7). — If B and C are two preadmissible A-algebras, then (B⊗A C)∧ is admissible, and if K (resp. L) isan ideal of denition for B (resp. C ), then the closure in (B⊗A C)∧ of the canonical image of H= Im(K⊗A C)+ Im(B⊗AL)is an ideal of denition.

Proof. It suces to show that Hn tends to 0 for the tensor product topology, which follows immediately fromthe inclusion

H2n ⊂ Im(Kn ⊗A C) + Im(B ⊗A Ln).

Proposition (7.7.8). — Let A be a preadic ring, J an ideal of dention for A, M an A-module of finite type, equippedwith the J-preadic topology. For every topological adic Noetherian A-algebra B, B ⊗A M identies with the completed tensorproduct (B ⊗A M)∧.

Proof. If K is an ideal of definition for B, there exists by hypothesis an integer m such that JmB ⊂ K, soIm(B ⊗A J

nmM) = Im(JnmB ⊗A M) ⊂ Im(KnB ⊗A M) = Kn(B ⊗A M); we conclude that over B ⊗A M , the tensorproducts of the topologies of B and M is the K-preadic topology. As B ⊗A M is a B-module of finite type, theproposition follows from Corollary (7.3.6).

7.8. Topologies on modules of homomorphisms.

(7.8.1). Let A be a Noetherian J-adic ring, M and N two A-modules of finite type, equipped with the J-preadictopology; we know (7.3.6) that they are separated and complete; in addition, every A-homomorphism M → Nis automatically continuous, and the A-module HomA(M , N) is of finite type. For every integer i ¾ 0, setAi = A/Ji+1, Mi = M/Ji+1M , Ni = N/Ji+1N ; for i ¶ j, every homomorphism u j : M j → N j maps Ji+1M j to Ji+1N j ,thus giving by passage to quotients a homomorphism ui : Mi → Ni , which defines a canonial homomorphismHomA j

(M j , N j)→ HomAi(Mi , Ni); in addition, the HomAi

(Mi , Ni) form a projective system for these homomorphisms,and it follows from Corollary (7.2.10) that there is a canonical isomorphismφ : HomA(M , N)→ lim←−i

HomAi(Mi , Ni).

In addition:

Proposition (7.8.2). — If M and N are modules of nite type over a J-adic Noetherian ring A, then the submodulesHomA(M ,Ji+1N) form a fundamental system of neighborhoods of 0 in HomA(M , N) for the J-adic topology, and thecanonical isomorphism φ : HomA(M , N)→ lim←−i

HomAi(Mi , Ni) is a topological isomorphism.

8. REPRESENTABLE FUNCTORS 55

Proof. We can consider M as the quotient of a free A-module L of finite type, and as a result iden-tify HomA(M , N) as a submodule of HomA(L, N); in this identification, HomA(M ,Ji+1N) is the intersection ofHomA(M , N) and HomA(L,Ji+1N); as the induced topology on HomA(M , N) by the J-adic topology of HomA(L, N)is the J-adic (7.3.2), we have reduced to proving the first assertion for M = L = Am; but then HomA(L, N) = N m, 0I | 78HomA(L,Ji+1N) = (Ji+1N)m = Ji+1N m and the result is evident. To establish the second assertion, we note thatthe image of HomA(M ,Ji+1N) in HomA j

(M j , N j) is zero for j ¶ i, hence φ is continuous; conversely, the inverseimage in HomA(M , N) of 0 of HomAi

(Mi , Ni) is HomA(M ,Ji+1N), so φ is bicontinuous.

If we only suppose that A is a Noetherian J-preadic ring, M and N two A-modules of finite type, separated forthe J-preadic topology, then the following proof shows that the first assertion of Propositon (7.8.2) remains valid,and that φ is a topological isomorphism from HomA(M , N) to a submodule of lim←−i

HomAi(Mi , Ni).

Proposition (7.8.3). — Under the hypotheses of Proposition (7.8.2), the set of injective (resp. surjective, bijective)homomorphisms from M to N is an open subset of HomA(M , N).

Proof. According to Corollaries (7.3.5) and (7.1.14), for u to be injective, it is necessary and sucient thatthe corresponding homomorphism u0 : M/JM → N/JN is, and the set of surjective homomorphisms from M toN is thus the inverse image under the continuous map HomA(M , N)→ HomA0

(M0, N0) of a subset of a discretespace. We now show that the set of injective homomorphisms is open; let v be such a homomorphism and setM ′ = v(M); by the Artin–Rees Lemma (7.3.2.1), there exists an integer k ¾ 0 such that M ′ ∩ Jm+kN ⊂ JmM ′ forall m> 0; we will see that for all w ∈ Jk+1 HomA(M , N), u= v+w is injective, which will finish the proof. Indeed,let x ∈ M be such that u(x) = 0; we prove that for every i ¾ 0 the relation x ∈ Ji M implies that x ∈ Ji+1M ; thisfollows from x ∈

⋂

i¾0 Ji M = (0). Indeed, we then have w(x) ∈ Ji+k+1N , and as a result w(x) = −v(x) ∈ M ′, so

v(x) ∈ M ′ ∩ Ji+k+1N ⊂ Ji+1M ′, and as v is an isomorphism from M to M ′, x ∈ Ji+1M ; q.e.d.

§8. Representable functors0III | 5

8.1. Representable functors.

(8.1.1). We denote by Set the category of sets. Let C be a category; for two objects X , Y of C, we sethX (Y ) = Hom(Y, X ); for each morphism u : Y → Y ′ in C, we denote by hX (u) the map v 7→ vu from Hom(Y ′, X )to Hom(Y, X ). It is immediate that with these definitions, hX : C→ Set is a contravariant functor, i.e., an objectof the category Hom(Cop,Set), of covariant functors from the category Cop (the dual of the category C) to thecategory Set (T, 1.7, (d) and [Car]).

(8.1.2). Now let w : X → X ′ be a morphism in C; for each Y ∈ C and each v ∈ Hom(Y, X ) = hX (Y ), we havewv ∈ Hom(Y, X ′) = hX ′(Y ); we denote by hw(Y ) the map v 7→ wv from hX (Y ) to hX ′(Y ). It is immediate that foreach morphism u : Y → Y ′ in C, the diagram

hX (Y ′)hX (u) //

hw(Y ′)

hX (Y )

hw(Y )

hX ′(Y ′)

hX ′ (u) // hX ′(Y )

is commutative; in other words, hw is a natural transformation (or functorial morphism) hX → hX ′ (T, 1.2), also amorphism in the category Hom(Cop,Set) (T, 1.7, (d)). The definitions of hX and of hw therefore constitute thedefinition of a canonical covariant functor

(8.1.2.1) h : C −→ Hom(Cop,Set), X 7−→ hX .

(8.1.3). Let X be an object in C, F a contravariant functor from C to Set (an object of Hom(Cop,Set)). Letg : hX → F be a natural transformation: for all Y ∈ C, g(Y ) is thus a map hX (Y ) → F(Y ) such that for each 0III | 6morphism u : Y → Y ′ in C, the diagram

(8.1.3.1) hX (Y ′)hX (u) //

g(Y ′)

hX (Y )

g(Y )

F(Y ′)F(u) // F(Y )

is commutative. In particular, we have a map g(X ) : hX (X ) = Hom(X , X )→ F(X ), hence an element

(8.1.3.2) α(g) = (g(X ))(1X ) ∈ F(X )

8. REPRESENTABLE FUNCTORS 56

and as a result a canonical map

(8.1.3.3) α : Hom(hX , F) −→ F(X ).

Conversely, consider an element ξ ∈ F(X ); for each morphism v : Y → X in C, F(v) is a map F(X )→ F(Y );consider the map

(8.1.3.4) v 7−→ (F(v))(ξ )from hX (Y ) to F(Y ); if we denote by (β(ξ ))(Y ) this map,

(8.1.3.5) β(ξ ) : hX −→ F

is a natural transformation, since for each morphism u : Y → Y ′ in C we have (F(vu))(ξ ) = (F(v) F(u))(ξ ),which makes (8.1.3.1) commutative for g = β(ξ ). We have thus defined a canonical map

(8.1.3.6) β : F(X ) −→ Hom(hX , F).

Proposition (8.1.4). — The maps α and β are the inverse bijections of each other.

Proof. We calculate α(β(ξ )) for ξ ∈ F(X ); for each Y ∈ C, (β(ξ ))(Y ) is a map g1(Y ) : v 7→ (F(v))(ξ ) fromhX (Y ) to F(Y ). We thus have

α(β(ξ )) = (g1(X ))(1X ) = (F(1X ))(ξ ) = 1F(X )(ξ ) = ξ .

We now calculate β(α(g)) for g ∈ Hom(hX , F); for each Y ∈ C, (β(α(g)))(Y ) is the map v 7→ (F(v))((g(X ))(1X ));according to the commutativity of (8.1.3.1), this map is none other than v 7→ (g(Y ))((hX (v))(1X )) = (g(Y ))(v)by definition of hX (v), in other words, it is equal to g(Y ), which finishes the proof.

(8.1.5). Recall that a subcategory C′ of a category C is defined by the condition that its objects are objects of C,and that if X ′, Y ′ are two objects of C′, then the set HomC′(X ′, Y ′) of morphisms X ′→ Y ′ in C′ is a subset of theset HomC(X ′, Y ′) of morphisms X ′→ Y ′ in C, the canonical map of “composition of morphisms”

HomC′(X′, Y ′)×HomC′(Y

′, Z ′) −→ HomC′(X′, Z ′)

being the restriction of the canonical map 0III | 7

HomC(X′, Y ′)×HomC(Y

′, Z ′) −→ HomC(X′, Z ′).

We say that C′ is a full subcategory of C if HomC′(X ′, Y ′) = HomC(X ′, Y ′) for every pair of objects in C′. Thesubcategory C′′ of C consisting of the objects of C isomorphic to objects of C′ is then again a full subcategory ofC, equivalent (T, 1.2) to C′ as we verify easily.

A covariant functor F : C1 → C2 is called fully faithful if for every pair of objects X1, Y1 of C1, the mapu 7→ F(u) from Hom(X1, Y1) to Hom(F(X1), F(Y1)) is bijective; this implies that the subcategory F(C1) of C2 is full.In addition, if two objects X1, X ′1 have the same image X2, then there exists a unique isomorphism u : X1→ X ′1such that F(u) = 1X1

. For each object X2 of F(C1), let G(X2) be one of the objects X1 of C1 such that F(X1) = X2(G is defined by means of the axiom of choice); for each morphism v : X2→ Y2 in F(C1), G(v) will be the uniquemorphism u : G(X2)→ G(Y2) such that F(u) = v; G is then a functor from F(C1) to C1; FG is the identity functoron F(C1), and the above shows that there exists an isomorphism of functors φ : 1C1

→ GF such that F , G, φ , andthe identity 1F(C1)→ FG defines an equivalence between the category C1 and the full subcategory F(C1) of C2 (T,1.2).

(8.1.6). We apply Proposition (8.1.4) to the case where F is hX ′ , X ′ being any object of C; the map β :Hom(X , X ′)→ Hom(hX , hX ′) is none other than the map w 7→ hw defined in (8.1.2); this map being bijective, wesee with the terminology of (8.1.5) that:

Proposition (8.1.7). — The canonical functor h : C→ Hom(Cop,Set) is fully faithful.

(8.1.8). Let F be a contravariant functor from C to Set; we say that F is representable if there exists an objectX ∈ C such that F is isomorphic to hX ; it follows from Proposition (8.1.7) that the data of an X ∈ C and anisomorphism of functors g : hX → F determines X up to unique isomorphism. Proposition (8.1.7) then impliesthat h defines an equivalence between C and the full subcategory of Hom(Cop,Set) consisting of the contravariantrepresentable functors. It follows from Proposition (8.1.4) that the data of a natural transformation g : hX → F isequivalent to that of an element ξ ∈ F(X ); to say that g is an isomorphism is equivalent to the following conditionon ξ : for every object Y of C the map v 7→ (F(v))(ξ ) from Hom(Y, X ) to F(Y ) is bijective. When ξ satsifies thiscondition, we say that the pair (X ,ξ ) represents the representable functor F , By abuse of language, we also saythat the object X ∈ C represents F if there exists a ξ ∈ F(X ) such that (X ,ξ ) represents F , in other words if hX isisomorphic to F .

8. REPRESENTABLE FUNCTORS 57

Let F , F ′ be two contravariant representable functors from C to Set, hX → F and hX ′ → F ′ two isomorphismsof functors. Then it follows from (8.1.6) that there is a canonical bijective correspondence between Hom(X , X ′)and the set Hom(F, F ′) of natural transformations F → F ′.

(8.1.9). Example I. Projective limits. The notion of a contravariant representable functor covers in particular the“dual” notion of the usual notion of a “solution to a universal problem”. More generally, we will see that the 0III | 8notion of the projective limit is a special case of the notion of a representable functor. Recall that in a category C,we define a projective system by the data of a preordered set I , a family (Aα)α∈I of objects of C, and for every pairof indices (α,β) such that α ¶ β, a morphism uαβ : Aβ → Aα. A projective limit of this system in C consists ofan object B of C (denoted lim←−Aα), and for each α ∈ I , a morphism uα : B→ Aα such that: 1st. uα = uαβuβ for

α ¶ β; 2nd. for every object X of C and every family (vα)α∈I of morphisms vα : X → Aα such that vα = uαβ vβ forα ¶ β, there exists a unique morphism v : X → B (denoted lim←− vα) such that vα = uαv for all α ∈ I (T, 1.8). Thiscan be interpreted in the following way: the uαβ canonically define maps

uαβ : Hom(X , Aβ) −→ Hom(X , Aα)

which define a projective system of sets (Hom(X , Aα), uαβ), and (vα) is by definition an element of the setlim←−Hom(X , Aα); it is clear that X 7→ lim←−Hom(X , Aα) is a contravariant functor from C to Set, and the exis-tence of the projective limit B is equivalent to saying that (vα) 7→ lim←− vα is an isomorphism of functors in X

(8.1.9.1) lim←−Hom(X , Aα)' Hom(X , B),

in other words, that the functor X 7→ lim←−Hom(X , Aα) is representable.

(8.1.10). Example II. Final objects. Let C be a category, a a singleton set. Consider the contravariant functorF : C→ Set which sends every object X of C to the set a, and every morphism X → X ′ in C to the unique mapa → a. To say that this functor is representable means that there exists an object e ∈ C such that for everyY ∈ C, Hom(Y, e) = he(Y ) is a singleton set; we say that e is an nal object of C, and it is clear that two final objectsof C are isomorphic (which allows us to define, in general with the axiom of choice, one final object of C whichwe then denote eC). For example, in the category Set, the final objects are the singleton sets; in the category ofaugmented algebras over a field K (where the morphisms are the algebra homomorphisms compatible with theaugmentation), K is a final object; in the category of S-preschemes (I, 2.5.1), S is a final object.

(8.1.11). For two objects X and Y of a category C, set h′X (Y ) = Hom(X , Y ), and for every morphism u : Y → Y ′,let hX (u) be the map v 7→ vu from Hom(X , Y ) to Hom(X , Y ′); h′X is then a covariant functor C → Set, so wededuce as in (8.1.2) the definition of a canonical covariant functor h′ : Cop→ Hom(C,Set); a covariant functorF from C to Set, in other words an object of Hom(C,Set), is then representable if there exists an object X ∈ C(necessarily unique up to unique isomorphism) such that F is isomorphic to h′X ; we leave it to the reader to developthe “dual” notions of the above, which this time cover the notion of an inductive limit, and in particular the usualnotion of a “solution to a universal problem”.

8.2. Algebraic structures in categories.0III | 9

(8.2.1). Given two contravariant functors F and F ′ from C to Set, recall that for every object Y ∈ C, we set(F × F ′)(Y ) = F(Y )× F ′(Y ), and for every morphism u : Y → Y ′ in C, we set (F × F ′)(u) = F(u)× F ′(u), which isthe map (t, t ′) 7→ (F(u)(t), F ′(u)(t ′)) from F(Y ′)×F ′(Y ′) to F(Y )×F ′(Y ); F×F ′ : C→ Set is thus a contravariantfunctor (which is none other than the product of the objects F and F ′ in the category Hom(Cop,Set)). Given anobject X ∈ C, we call an internal composition law on X a natural transformation

(8.2.1.1) γX : hX × hX −→ hX .

In other words (T, 1.2), for every object Y ∈ C, γX (Y ) is a map hX (Y )× hX (Y )→ hX (Y ) (thus by definitionan internal composition law on the set hX (Y )) with the condition that for every morphism u : Y → Y ′ in C, thediagram

hX (Y ′)× hX (Y ′)hX (u)×hX (u) //

γX (Y ′)

hX (Y )× hX (Y )

γX (Y )

hX (Y ′)

hX (u) // hX (Y )

8. REPRESENTABLE FUNCTORS 58

is commutative; this implies that for the composition laws γX (Y ) and γX (Y ′), hX (u) is a homomorphism fromhX (Y ′) to hX (Y ).

In a similar way, given two objects Z and X of C, we call an external composition law on X , with Z as its domainof operators a natural transformation

(8.2.1.2) ωX ,Z : hZ × hX −→ hX .

We see as above that for every Y ∈ C, ωX ,Z(Y ) is an external composition law on hX (Y ), with hZ(Y ) as itsdomain of operators and such that for every morphism u : Y → Y ′, hX (u) and hZ(u) form a di-homomorphismfrom (hZ(Y ′), hX (Y ′)) to (hZ(Y ), hX (Y )).

(8.2.2). Let X ′ be a second object of C, and suppose we are given an internal composition law γX ′ on X ′;we say that a morphism w : X → X ′ in C is a homomorphism for the composition laws if for every Y ∈ C,hw(Y ) : hX (Y )→ hX ′(Y ) is a homomorphism for the composition laws γX (Y ) and γX ′(Y ). If X ′′ is a third object of C 0III | 10equipped with an internal composition law γX ′′ and w′ : X ′→ X ′′ is a morphism in C which is a homomorphismfor γX ′ and γX ′′ , then it is clear that the morphism w′w : X → X ′′ is a homomorphism for the composition lawsγX and γX ′′ . An isomorphism w : X ' X ′ in C is called an isomorphism for the composition laws γX and γX ′ if wis a homomorphism for these composition laws, and if its inverse morphism w−1 is a homomorphism for thecomposition laws γX ′ and γX .

We define in a similar way the di-homomorphisms for pairs of objects of C equipped with external compositionlaws.

(8.2.3). When an internal composition law γX on an object X ∈ C is such that γX (Y ) is a group law on hX (Y ) forevery Y ∈ C, we say that X , equipped with this law, is a C-group or a group object in C. We similarly define C-rings,C-modules, etc.

(8.2.4). Suppose that the product X × X of an object X ∈ C by itself exists in C; by definition, we then havehX×X = hX × hX up to canonical isomorphism, since it is a particular case of the projective limit (8.1.9); aninternal composition law on X can thus be considered as a functorial morphism γX : hX×X → hX , and thuscanonically determine (8.1.6) an element cX ∈ Hom(X × X , X ) such that hcX

= γX ; in this case, the data of aninternal composition law on X is equivalent to the data of a morphism X × X → X ; when C is the category Set,we recover the classical notion of an internal composition law on a set. We have an analogous result for anexternal composition law when the product Z × X exists in C.

(8.2.5). With the above notation, suppose that in addition X × X × X exists in C; the characterization of theproduct as an object representing a functor (8.1.9) implies the existence of canonical isomorphisms

(X × X )× X ' X × X × X ' X × (X × X );

if we canonically identity X × X × X with (X × X )× X , then the map γX (Y )× 1hX (Y ) identifies with hcX×1X(Y ) for

all Y ∈ C. As a result, it is equivalent to say that for every Y ∈ C, the internal law γX (Y ) is associative, or thatthe diagram of maps

hX (Y )× hX (Y )× hX (Y )γX (Y )×1 //

1×γX (Y )

hX (Y )× hX (Y )

γX (Y )

hX (Y )× hX (Y )

γX (Y ) // hX (Y )

is commutative, or that the diagram of morphisms 0III | 11

X × X × XcX×1X //

1X×cX

X × X

cX

X × X

cX // X

is commutative.

9. CONSTRUCTIBLE SETS 59

(8.2.6). Under the hypotheses of (8.2.5), if we want to express, for every Y ∈ C, the internal law γX (Y ) as a grouplaw, then it is first necessary that it is associative, and second that there exists a map αX (Y ) : hX (Y )→ hX (Y )having the properties of the inverse operation of a group; as for every morphism u : Y → Y ′ in C, we have seenthat hX (u) must be a group homomorphism hX (Y ′)→ hX (Y ), we first see that αX : hX → hX must be a naturaltransformation. On the other hand, one can express the chacteristic properties of the inverse s 7→ s−1 of a groupG without involving the identity element: it suces to check that the two composite maps

(s, t) 7−→ (s, s−1, t) 7−→ (s, s−1 t) 7−→ s(s−1 t),

(s, t) 7−→ (s, s−1, t) 7−→ (s, ts−1) 7−→ (ts−1)s

are equal to the second projection (s, t) 7→ t from G×G to G. By (8.1.3), we have αX = haX, where aX ∈ Hom(X , X );

the first condition above then expresses that the composite morphism

X × X(1X ,aX )×1X−−−−−−→ X × X × X

1X×cX−−−→ X × XcX−→ X

is the second projection X × X → X in C, and the second condition is similar.

(8.2.7). Now suppose that there exists a nal object e (8.1.10) in C. Let us always assume that γX (Y ) is a grouplaw on hX (Y ) for every Y ∈ C, and denote by ηX (Y ) the identity element of γX (Y ). As, for every morphismu : Y → Y ′ in C, hX (u) is a group homomorphism, we have ηX (Y ) = (hX (u))(ηX (Y ′)); taking in particular Y ′ = e,in which case u is the unique element ε of Hom(Y, e), we see that the element ηX (e) completely determinesηX (Y ) for every Y ∈ C. Set eX = ηX (X ), the identity element of the group hX (X ) = Hom(X , X ); the commutativityof the diagram

hX (e)hX (ε) //

heX (e)

hX (Y )

heX (Y )

hX (e)

hX (ε) // hX (Y )

(cf. (8.1.2)) shows that, on the set hX (Y ), the map heX(Y ) is none other than s 7→ ηX (Y ) sending every element 0III | 12

to the identity element. We then verify that the fact that ηX (Y ) is the identity element of γX (Y ) for every Y ∈ Cis equivalent to saying that the composite morphism

X(1X ,1X )−−−−→ X × X

1X×eX−−−→ X × XcX−→ X ,

and the analog in which we swap 1X and eX , are both equal to 1X .

(8.2.8). One could of course easily extend the examples of algebraic structures in categories. The example ofgroups was treated with enough detail, but latter on we will usually leave it to the reader to develop analogousnotions for the examples of algebraic structures we will encounter.

§9. Constructible sets

9.1. Constructible sets.

Denition (9.1.1). — We say that a continuous map f : X → Y is quasi-compact if for every quasi-compactopen subset U of Y , f −1(U) is quasi-compact. We say that a subset Z of a topological space X is retrocompact inX if the canonical injection Z → X is quasi-compact, in other words, if for every quasi-compact open subset U ofX , U ∩ Z is quasi-compact.

A closed subset of X is retrocompact in X , but a quasi-compact subset of X is not necessarily retrocompact inX . If X is quasi-compact, every retrocompact open subset of X is quasi-compact. It is clear that every nite unionof retrocompact sets in X is retrocompact in X , as every finite union of quasi-compact sets is quasi-compact.Every finite intersection of retrocompact open sets in X is a retrocompact open set in X . In a locally Noetherianspace X , every quasi-compact set is a Noetherian subspace, and as a result every subset of X is retrocompact in X .

Denition (9.1.2). — Given a topological space X , we say that a subset of X is constructible if it belongs tothe smallest set of subsets F of X containing all the retrocompact open subsets of F and is stable under finiteintersections and complements (which implies that F is also stable under finite unions).

Proposition (9.1.3). — For a subset of X to be constructible, it is necessary and sucient for it to be a nite union ofsets of the form U ∩ ûV , where U and V are retrocompact open sets in X .

9. CONSTRUCTIBLE SETS 60

Proof. It is clear that the condition is sucient. To see that it is necessary, consider the set G of finiteunions of sets of the form U ∩ ûV , where U and V are retrocompact open sets in X ; it suces to see that everycomplement of a set in G is in G. Let Z =

⋃

i∈I (Ui ∩ ûVi), where I is finite, Ui and Vi retrocompact open sets inX ; we have ûZ =

⋂

i∈I (Vi ∪ ûUi), so Z is a finite union of sets which are intersections of a certain number of theVi and of a certain number of the ûUi , thus of the form V ∩ ûU , where U is the union of a certain number of 0III | 13the Ui and V is the intersection of a certain number of the Vi ; but we have noted above that finite unions andintersections of retrocompact open sets in X are retrocompact open sets in X , hence the conclusion.

Corollary (9.1.4). — Every constructible subset of X is retrocompact in X .

Proof. It suces to show that if U and V are retrocompact open sets in X , then U ∩ ûV is retrocompact inX ; if W is a quasi-compact open set in X , then W ∩ U ∩ ûV is closed in the quasi-compact space W ∩ U , hence itis quasi-compact.

In particular:

Corollary (9.1.5). — For an open subset U of X to be constructible, it is necessary and sucient for it to be retrocompactin X . For a closed subset F of X to be constructible, it is necessary and sucient for the open set ûF to be retrocompact.

(9.1.6). An important case is when every quasi-compact open subset of X is retrocompact, in other words,when the intersection of two quasi-compact open subsets of X is quasi-compact (cf. (I, 5.5.6)). When X is alsoquasi-compact, this implies that the retrocompact open subsets of X are identical to the quasi-compact opensubsets of X , and the constructible subsets of X are finite unions of sets of the form U ∩ ûV , where U and V arequasi-compact open sets.

Corollary (9.1.7). — For a subset of a Noetherian space to be constructible, it is necessary and sucient for it to be anite union of locally closed subsets of X .

Proposition (9.1.8). — Let X be a topological space, U an open subset of X .(i) If T is a constructible subset of X , then T ∩ U is a constructible subset of U .(ii) In addition, suppose that U is retrocompact in X . For a subset Z of U to be constructible in X , it is necessary and

sucient for it to be constructible in U .

Proof.(i) Using Proposition (9.1.3), we reduce to showing that if T is a retrocompact open set in X , then T ∩U is

a retrocompact open set in U , in other words, for every quasi-compact open W ⊂ U , T ∩U∩W = T ∩Wis quasi-compact, which immediately follows from the hypothesis.

(ii) The condition is necessary by (i), so it remains to show that it is sucient. By Proposition (9.1.3), itsuces to consider the case where Z is a retrocompact open set in U , because it will then follow thatU − Z is constructible in X , and if Z and Z ′ are two retrocompact opens in U , then Z ∩ (U − Z ′) will beconstructible in X . If W is a quasi-compact open set in X and Z a retrocompact open set in U , then wehave Z ∩W = Z ∩ (W ∩ U), and by hypothesis W ∩ U is a quasi-compact open set in U ; so W ∩ Z isquasi-compact, and as a result Z is a retrocompact open set in X , and a fortiori constructible in X .

Corollary (9.1.9). — Let X be a topological space, (Ui)i∈I a nite cover of X by retrocompact open sets in X . For asubset Z of X to be constructible in X , it is necessary and sucient for Z ∩ Ui to be constructible in Ui for all i ∈ I .

(9.1.10). In particular, suppose that X is quasi-compact and every point of X admits a fundamental system of 0III | 14retrocompact open neighborhoods in X (and a fortiori quasi-compact); then the condition for a subset Z of X tobe constructible in X is of a local nature, in other words, it is necessary and sucient that for evert x ∈ X , thereexists an open neighborhood V of x such that V ∩ Z is constructible in V . Indeed, if this condition is satisfied,then there exists for every x ∈ X an open neighborhood V of x which is retrocompact in X and such that V ∩ Z isconstructible in V , by the hypotheses on X and by Proposition (9.1.8, i); it then suces to cover X by a finitenumber of these neighborhoods and to apply Corollary (9.1.9).

Denition (9.1.11). — Let X be a topological space. We say that a subset T of X is locally constructible in X iffor every x ∈ X there exists an open neighborhood V of x such that T ∩ V is constructible in V .

It follows from Proposition (9.1.8, i) that if V is such that V ∩ T is constructible in V , then for every openW ⊂ V , W ∩ T is constructible in W . If T is locally constructible in X , then for every open set U in X , T ∩ Uis locally constructible in U , as a result of the above remark. The same remark shows that the set of locally

9. CONSTRUCTIBLE SETS 61

constructible subsets of X is stable under finite unions and finite intersections; on ther other hand, it is clearthat it is also stabble under taking complements.

Proposition (9.1.12). — Let X be a topological space. Every constructible set in X is locally constructible in X . Theconverse is true if X is quasi-compact and if its topology admits a basis formed by the retrocompact sets in X .

Proof. The first assertion follows from Definition (9.1.11) and the second from (9.1.10).

Corollary (9.1.13). — Let X be a topological space whose topology admits a basis formed by the retrocompact sets in X .Then every locally constructible subset T of X is retrocompact in X .

Proof. Let U be a quasi-compact open set in X ; T ∩U is locally constructible in U , hence constructible in Uby Proposition (9.1.12), and as a result quasi-compact by Corollary (9.1.4).

9.2. Constructible subsets of Noetherian spaces.

(9.2.1). We have seen (9.1.7) that in a Noetherian space X , the constructible subsets of X are the nite unions oflocally closed subsets of X .

The inverse image of a constructible set in X by a continuous map from a Noetherian space X ′ to X isconstructible in X ′. If Y is a constructible subset of a Noetherian space X , then the subsets of Y are constructibleas subspaces of Y and are identical to those which are constructible as subspaces of X .

Proposition (9.2.2). — Let X be an irreducible Noetherian space, E a constructible subset of X . For E to be everywheredense in X , it is necessary and sucient for E to contain a nonempty open subset of X .

Proof. The condition is evidently sucient, as every nonempty open set is dense in X . Conversely, letE =

⋃ni=1(Ui ∩ Fi) be a constructible subset of X , the Ui being nonempty open sets and the Fi closed in X ; we

then have E ⊂⋃

i Fi . As a result, if E = X , then X is equal to one of the Fi , hence E ⊃ Ui , which finishes theproof.

When X admits a generic point x (0, 2.1.2), the condition of Proposition (9.2.2) is equivalent to the relationx ∈ E.

Proposition (9.2.3). — Let X be a Noetherian space. For a subset E of X to be constructible, it is necessary and sucientthat, for every irreducible closed subset Y of X , E ∩ Y is rare in Y or contains a nonempty open subset of Y .

Proof. The necessity of the condition follows from the fact that E ∩ Y must be a constructible subset ofY and from Proposition (9.2.2), since a nondense subset of Y is necessarily rare in the irreducible space Y](0, )0.2.1.1. To prove that the condition is sucient, apply the principle of Noetherian induction (0, 2.2.2)to the set F of closed subsets Y of X such that Y ∩ E is constructible (relative to Y or relative to X , which areequivalent): we can thus assume that for every closed subset Y 6= X of X , E ∩ Y is constructible. First supposethat X is not irreducible, and let X i (1 ¶ i ¶ m) are its irreducible components, necessarily of finite number(0, 2.2.5); by hypothesis the E ∩ X i are constructible, hence their union E is as well. Suppose now that X isirreducible; then by hypothesis, if E is rare, then E 6= X and E = E ∩ E is constructible; if E contains a nonemptyopen subset U of X , then it is the union of U and E ∩ (X − U); but X − U is a closed set distinct from X , soE ∩ (X − U) is constructible; as a result, E is itself constructible, which finishes the proof.

Corollary (9.2.4). — Let X be a Noetherian space, (Eα) an increasing ltered family of constructible subsets of X , suchthat

(1st) X is the union of the family (Eα).(2nd) Every irreducible closed subset of X is contained in the closure of one of the Eα.Then there exists an index α such that X = Eα.When every irreducible closed subset of X admits a generic point, the hypothesis (1st) can be omitted.

Proof. We apply the principle of Noetherian induction (0, 2.2.2) to the setM of closed subsets of X containedin at least one of the Eα; we can thus suppose that every closed subset Y 6= X of X is contained in one of the Eα.The proposition is evident if X is not irreducible, because each of the irreducible components X i of X (1¶ i ¶ m)is contained in an Eαi

, and there exists an Eα containing all of the Eαi. Now suppose that X is irreducible. By

hypothesis, there exists a β such that X = Eβ , so (9.2.2) Eβ contains a nonempty open subset U of X . But thenthe closed set X − U is contained in an Eγ , and it suces to take an Eα containing Eβ and Eγ . When every

irreducible closed subset Y of X admits a generic point y , there exists α such that y ∈ Eα, so Y = y ⊂ Eα, and 0III | 16condition (2nd) is therefore a consequence of (1st).

9. CONSTRUCTIBLE SETS 62

Proposition (9.2.5). — Let X be a Noetherian space, x a point of X , and E a constructible subset of X . For E to be aneighborhood of x , it is necessary and sucient that for every irreducible closed subset Y of X containing x , E ∩ Y is densein Y (if there exists a generic point y of Y , this also implies (9.2.2) that y ∈ E).

Proof. The condition is evidently necessary; we will prove that it is sucient. Applying the principle ofNoetherian induction to the set M of closed subsets Y of X containing x and such that E ∩ Y is a neighborhoodof x in Y , we can assume that every closed subset Y 6= X of X containing x belongs to M. If X is not irreducible,then each of the irreducible components X i of X containing x are distinct from X , hence E∩X i is a neighborhoodof x with respect to X i ; as a result, E is a neighborhood of x in the union of the irreducible components ofX containing x , and as this union is a neighborhood of x in X , so is E. If X is irreducible, then E is densein X by hypothesis, so it contains a nonempty open subset U of X (9.2.2); the proposition is then evident ifx ∈ U ; otherwise, x is by hypothesis inside E ∩ (X − U) with respect to X − U , so the closure of X − E in X doesnot contain x , and the complement of this closure is a neighborhood of x contained in E, which finishes theproof.

Corollary (9.2.6). — Let X be a Noetherian space, E a subset of X . For E to be an open set in X , it is necessary andsucient that for every irreducible closed subset Y of X intersecting E, E ∩ Y contains a nonempty open subset of Y .

Proof. The condition is evidently necessary; conversely, if it is satisfied, then it implies that E is constructibleby Proposition (9.2.3). In addition, Proposition (9.2.5) shows that E is then a neighborhood of each of its points,hence the conclusion.

9.3. Constructible functions.

Denition (9.3.1). — Let h be a map from a topological spacee X to a set T . We say that h is constructible ifh−1(t) is constructible for every t ∈ T , and empty except for finitely many values of t; then for every subset S ofT , h−1(S) is constructible. We say that h is locally constructible if every x ∈ X has an open neighborhood V suchthat h|V is constructible.

Every constructible function is locally constructible; the converse is true when X is quasi-compact andadmits a basis formed by the retrocompact open sets in X (in particular, when X is Noetherian).

Proposition (9.3.2). — Let h be a map from a Noetherian space X to a set T . For h to be constructible, it is necessaryand sucient that for every irreducible closed subset Y of X , there exists a nonempty subset U of Y , open relative to Y , inwhich h is constant.

Proof. The condition is necessary: indeed, by hypothesis, h does not take finitely many values t i on Y , andeach of the sets h−1(t i)∩ Y is constructible in Y (9.2.1); as they can not all be rare subsets of the space Y , atleast one of them contains a nonempty open set (9.2.3). 0III | 17

To see that the condition is sucient, we apply the principle of Noetherian induction on the set M of closedsubsets Y of X such that the restriction h|Y is constructible; we can thus assume that for every closed subsetY 6= X of X , h|Y is constructible. If X is not irreducible, then the restriction of h to each of the (finitely many)irreducible components X i of X is constructible, and it then follows immediately from Definition (9.3.1) that his constructible. If X is irreducible, then there exists by hypothesis a nonempty open subset U of X on whichh is constant; on the other hand, the restriction of h to X − U is constructible by hypothesis, and it followsimmediately that h is constructible.

Corollary (9.3.3). — Let X be a Noetherian space in which every irreducible closed subset admits a generic point. If his a map from X to a set T such that, for every t ∈ T , h−1(t) is constructible, then h is constructible.

Proof. If Y is an irreducible closed subset of X and y its generic point, then Y ∩ h−1(h(y)) is constructibleand contains y , hence (9.2.2) this set contains a nonempty open subset of Y , and it suces to apply Proposition(9.3.2).

Proposition (9.3.4). — Let X be a Noetherian space in which every irreducible closed subset admits a generic point, ha constructible map from X to an ordered set. For h to be upper semi-continuous on X , it is necessary and sucient that forevery x ∈ X and every specialization (0, 2.1.2) x ′ of x , we have h(x ′)¶ h(x).

Proof. The function h does not take a finite number of values; therefore, to say that it is upper semi-continuous means that for every x ∈ X , the set E of the y ∈ X such that h(y)¶ h(x) is a neighborhood of x . Byhypothesis, E is a constructible subset of X ; on the other hand,to say that an irreducible closed subset Y of Xcontains x means that its generic point y is a specialization of x ; the conclusion then follows from Proposition(9.2.5).

10. SUPPLEMENT ON FLAT MODULES 63

§10. Supplement on at modules

For any proofs missing in (10.1) and (10.2), we refer the reader to Bourbaki, Alg. comm., chap. II and III.

10.1. Relations between at modules and free modules.

(10.1.1). Let A be a ring, J an ideal of A, and M an A-module; for every integer p ¾ 0, we have a canonicalhomomorphism of (A/J)-modules

(10.1.1.1) φp : (M/JM)⊗A/J (Jp/Jp+1) −→ Jp M/Jp+1M ,

which is evidently surjective. We denote by gr(A) = ⊕p¾0Jp/Jp+1 the graded ring associated to A filtered by the Jp,

and by gr(M) = ⊕p¾0Jp M/Jp+1M the graded gr(A)-module associated to M filtered by the Jp M ; we then have

grp(A) = Jp/Jp+1, and grp(M) = Jp M/Jp+1M ; the φ define a surjective homomorphism of graded gr(A)-modules

(10.1.1.2) φ : gr0(M)⊗gr0(A) gr(A) −→ gr(M).0III-1 | 18

(10.1.2). Suppose that one of the following hypotheses is satisfied:

(i) J is nilpotent;(ii) A is Noetherian, J is contained in the radical of A, and M is of finite type.

Then the following properties are equivalent.

(a) M is a free A-module.(b) M/JM = N ⊗A (A/J) is a free (A/J)-module, and TorA

1(M , A/J) = 0.(c) M/JM is a free (A/J)-module, and the canonical homomorphism (10.1.1.2) is injective (and thus

bijective).

(10.1.3). Suppose that A/J is a eld (in other words, that J is maximal), and that one of the hypotheses, (i) and(ii), of (10.1.2) is satisfied. Then the following properties are equivalent.

(a) M is a free A-module.(b) M is a projective A-module.(c) M is a flat A-module.(d) TorA

1(M , A/J) = 0.(e) The canonical homomorphism (10.1.1.2) is bijective.

This result can be applied, in particular, to the following two cases:

(i) M is an arbitrary module, over a local ring A whose maximal ideal J is nilpotent (for example, a localArtinian ring);

(ii) M is a module of nite type over a local Noetherian ring.

10.2. Local atness criteria.

(10.2.1). With the hypotheses and notation of (10.1.1), consider the following conditions.

(a) M is a flat A-module.(b) M/JM is a flat (A/J)-module, and TorA

1(M , A/J) = 0.(c) M/JM is a flat (A/J)-module, and the canonical homomorphism (10.1.1.2) is bijective.(d) For all n¾ 1, M/JnM is a flat (A/Jn)-module.

Then we have the implications(a) =⇒ (b) =⇒ (c) =⇒ (d),

and, if J is nilpotent, then the four conditions are equivalent. This is also the case if A is Noetherian and M isideally separated, that is to say, for every ideal a of A, the A-module a⊗A M is separated for the J-preadic topology.

(10.2.2). Let A be a Noetherian ring, B a commutative Noetherian A-algebra, J an ideal of A such that JB iscontained in the radical of B, and M a B-module of finite type. Then, when M is considered as an A-module, thefour conditions of (10.2.1) are equivalent. This result applies first and foremost in the case where A and B are 0III-1 | 10local Noetherian rings, with the homomorphism A→ B being a local homomorphism. More specifically, if J isthen the maximal ideal of A, we can, in conditions (b) and (c), remove the hypothesis that M/JM is flat, since itis automatically satisfied, and condition (d) implies that the modules M/JnM are free over the A/Jn.

(10.2.3). With the hypotheses on A, B, J, and M from the start of (10.2.2), let bA be the separated completion ofA for the J-preadic topology, and ÒM the separated completion of M for the JB-preadic topology. Then, for M tobe a flat A-module, it is necessary and sucient for ÒM to be a flat bA-module.

10. SUPPLEMENT ON FLAT MODULES 64

(10.2.4). Let ρ : A→ B be a local homomorphism of local Noetherian rings, k the residue field of A, and M andN both B-modules of finite type, with N assumed to be A-at. Let u : M → N be a B-homomorphism. Then thefollowing conditions are equivalent.

(a) u is injective, and Coker(u) is a flat A-module.(b) u⊗ 1 : M ⊗A k→ N ⊗A k is injective.

(10.2.5). Let ρ : A→ B and σ : B→ C be local homomorphisms of local Noetherian rings, k the residue fieldof A, and M a C -module of finite type. Suppose that B is a at A-module. Then the following conditions areequivalent.

(a) M is a flat B-module.(b) M is a flat A-module, and M ⊗A k is a flat (B ⊗A k)-module.

Proposition (10.2.6). — Let A and B be local Noetherian rings, ρ : A→ B a local homomorphism, J an ideal of Bcontained in the maximal ideal, and M a B-module of nite type. Suppose that, for all n¾ 0, Mn = M/Jn+1M is a atA-module. Then M is a at A-module.

Proof. We have to prove that, for every injective homomorphism u : N ′ → N of A-modules of finite type,v = 1⊗u : M⊗AN ′→ M⊗AN is injective. But M⊗AN ′ and M⊗AN are B-modules of finite type, and thus separated

for the J-preadic topology (0I, 7.3.5); it thus suces to prove that the homomorphism bv : ÛM ⊗A N ′→ÛM ⊗A N ofthe separated completions is injective. But bv = lim←− vn, where vn is the homomorphism 1⊗u : Mn⊗A N ′→ Mn⊗A N ;since, by hypothesis, Mn is A-flat, vn is injective for all n, and thus so too is v, because the functor lim←− is leftexact.

Corollary (10.2.7). — Let A be a Noetherian ring, B a local Noetherian ring, ρ : A→ B a homomorphism, f anelement of the maximal ideal of B, and M a B-module of nite type. Suppose that the homothety fM : x → f x on M isinjective, and that M/ f M is a at A-module. Then M is a at A-module.

Proof. Let Mi = f i M for i ¾ 0; since fM is injective, Mi/Mi+1 is isomorphic to M/ f M , and thus A-flat forall i ¾ 0; the exact sequence

0 −→ Mi/Mi+1 −→ M/Mi+1 −→ M/Mi −→ 0gives us, by induction on i, that M/Mi is A-at for all i ¾ 0 (0I, 6.1.2); we can thus apply (10.2.6). We canalso argue directly as follows: for every A-module N of finite type, M ⊗A N is a B-module of finite type; since fbelongs to the radical n of B, the ( f )-adic topology on M ⊗A N is finer than the u-adic topology, and we know 0III-1 | 20that the latter is separated (0I,0.7.3.5, ) Now, since M/Mi is A-flat, we have that

f i(M ⊗A N) = Im(Mi ⊗A N −→ M ⊗A N) = Ker(M ⊗A N −→ (M/Mi)⊗A N)

by (0I, 6.1.2). So let N be an A-module of finite type, and N ′ a submodule of N , with canonical injectionj : N ′→ N ; in the commutative diagram

M ⊗A N ′ //

1M⊗ j

(M/Mi)⊗A N ′

1M/Mi⊗ j

M ⊗A N // (M/Mi)⊗A N

1M/Mi⊗ j is injective, because M/Mi is A-flat; we thus conclude that

Ker(M ⊗A N ′ −→ M ⊗A N) ⊂ Ker(M ⊗A N ′ −→ (M/Mi)⊗A N ′)

for any i; since the intersection (over i) of the latter kernel is 0, as we saw above, so too is the intersection (overi) of the former, and so M is A-flat.

Proposition (10.2.8). — Let A be a reduced Noetherian ring, and M an A-module of nite type. Suppose that, for everyA-algebra B (which is then a discrete valuation ring), M ⊗A B is a at B-module (and thus free (10.1.3)). Then M is a atA-module.

Proof. We know that, for M to be flat, it is necessary and suces for Mm to be a flat Am-module for everymaximal ideal m of A (0I, 6.3.3); we can thus restrict to the case where A is local (0I, 1.2.8). So let m be themaximal ideal of A, pi (1 ¶ i ¶ r) the minimal prime ideals of A, and k the residue field A/ f km. We know(II, 7.1.7) that there exists, for each i, a discrete valuation ring Bi that has the same field of fractions Ki as theintegral ring A/pi , and that, further, dominates A/pi . Let Mi = M ⊗A Bi . By hypothesis, Mi is free over Bi , andso, denoting by ki the residue field of Bi , we have

(10.2.8.1) rgki(Mi ⊗Bi

ki) = rgKi(Mi ⊗Bi

Ki).

10. SUPPLEMENT ON FLAT MODULES 65

But it is clear that the composite homomorphism A→ A/pi → Bi is local, and so k is an extension of ki , andthat we have Mi ⊗Bi

ki = M ⊗A ki = (M ⊗A k)⊗k ki , and also that Mi ⊗BiKi = M ⊗A Ki . Equation (10.2.8.1) thus

implies thatrgk(M ⊗A k) = rgKi

(M ⊗A Ki) for 1¶ i ¶ r

and since A is reduced, we know that this condition implies that M is a free A-module (Bourbaki, Alg. comm.,chap. II, § 3, no 2, prop. 7).

10.3. Existence of at extensions of local rings.

Proposition (10.3.1). — Let A be a local Noetherian ring, with maximal ideal J, and residue eld k = A/J. Let K bea eld extension of k. Then there exists a local homomorphism from A to a local Noetherian ring B, such that B/JB isk-isomorphic to K , and such that B is a at A-module.

The rest of this section is devoted to proving this proposition, step-by-step.

(10.3.1.1). First suppose that K = k(T ), where T is an indeterminate. In the ring of polynomials A′ = A[T],consider the prime ideal J′ = JA, consisting of the polynomials that have coecients in the ideal J; it is clear 0III-1 | 21that A′/J′ is canonically isomorphic to k[T]. We will show that the ring of fractions B = A′J is that for which weare searching (that is, a ring which satisfies the conditions of the conclusion of the proposition); it is clearly alocal Noetherian ring, with maximal ideal L = JB. Further, B/L = (A′/J′)J′ = (k[T])J′ is exactly the field offractions K of k[T]. Finally, B is a flat A′-module, and A′ is a free A-module, so B is a flat A-module (0I, 6.2.1).

(10.3.1.2). Now suppose that K = k(t) = k[t], where t is algebraic over k; let f ∈ k[T] be the minimalpolynomial of t; there exists a monic polynomial F ∈ A[T] whose canonical image in k[T] is f . So let A′ = A[T],and let J′ be the ideal JA′ + (F) in A′. We will see that the quotient ring B = A′/(F) is that for which we aresearching. First of all, it is clear that B is a free A-module, and thus flat. The ring A′/J′ is isomorphic to

(A′/JA′)/

(JA′ + (F))/JA′

= k[T]/( f ) = K;

the image L of J′ in B is thus maximal, and we evidently have that L= JB. Finally, B is a semi-local ring, becauseit is an A-module of finite type (Bourbaki, Alg. comm., chap. IV, § 2, no 5, cor. 3 of prop. 9), and its maximalideals are in bijective correspondence with those of B/JB ([SZ60, vol. I, p. 259]); the previous arguments thenprove that B is a local ring.

Lemma (10.3.1.3). — Let (Aλ , fµλ) be a ltered inductive system of local rings, such that the fµλ are local homomor-phisms; let mλ be the maximal ideal of Aλ , and let Kλ = Aλ/mλ . Then A′ = lim−→Aλ is a local ring, with maximal idealm = lim−→mλ , and residue eld K = lim−→Kλ . Further, if mµ = mλAµ with λ < µ, then we have m′ = mλA

′ for all λ. If,further, for λ < µ, Aµ is a at Aλ -module, and if all the Aλ are Noetherian, then A′ is a at Noetherian Aλ -modules for allλ.

Proof. Since, by hypothesis, ( fµλ)(mλ) ⊂ mµ for λ < µ, the mλ form an inductive system, and its limit m′

is evidently an ideal of A′. Further, if x ′ 6∈ m′, there exists a λ such that x ′ = fλ(xλ) for some xλ ∈ Aλ (wherefλ : Aλ → A′ denotes the canonical homomorphism); because x ′ 6∈ m′, we necessarily have that xλ 6∈ mλ , and soxλ admits an inverse yλ in Aλ , and y ′ = fλ(yλ) is the inverse of x ′ in A′, which proves that A′ is a local ringwith maximal ideal m′; the claim about K follows immediately from the fact that lim−→ is an exact functor. Thehypothesis that mµ = mλAµ implies that the canonical map mλ ⊗Aλ Aµ→ mµ is surjective; the equality m′ = mλA

′

then follows from, again, the fact that the functor lim−→ is exact and commutes with the tensor product.Now suppose that, for λ < µ, we have mµ = mλAµ, and that Aµ is a flat Aλ -module. Then A′ is a flat

Aλ -module for all λ, by (0I, 6.2.3); since A′ and Aλ are local rings, and since m′ = mλA′, A′ is even a faithfully

at Aλ -module (0I, 6.6.2). Finally, suppose further that the Aλ are Noetherian; the mλ -adic topologies are thenseparated (0I, 7.3.5); we now show that, from this, it follows that the m′-adic topology on A′ is separated. Indeed,if x ′ ∈ A′ belongs to all the m

′n (n¾ 0), then it is the image of some xµ ∈ Aµ for a specific index µ, and since theinverse image in Aµ of m

′n = mnµA′ is mn

µ (0I, 6.6.1), xµ belongs to all the mnµ, so xµ = 0, by hypothesis, and so

x ′ = 0. Denote by bA′ the completion of A′ for the m′-adic topology; the above shows that we have A′ ⊂ bA′. Wewill now show that bA′ is Noetherian and Aλ -at for all λ; from this, it will follow that bA′ is A′-flat (0I, 6.2.3), and 0III-1 | 22since m′bA′ 6= bA′, that bA′ is a faithfully flat A′-module (0I, 6.6.2), whence the final conclusion that A′ is Noetherian(0I, 6.5.2), which will finish the proof of the lemma.

We have bA′ = lim←−nA′/m′n; by the fact that A′ is Aλ -flat, we have that

m′n/m′n+1 = (mnλ/m

n+1λ )⊗Aλ A′ = (mn

λ/mn+1λ )⊗Kλ (Kλ ⊗Aλ A′) = (mn

λ/mn+1λ )⊗Kλ K;

11. SUPPLEMENT ON HOMOLOGICAL ALGEBRA 66

since mnλ/m

n+1λ is a Kλ -vector space of finite dimension, m′nλ /m

′n+1λ is a K -vector space of finite dimension for all

n¾ 0. It thus follows from (0I, 7.2.12) and (0I, 7.2.8) that bA′ is Noetherian. We further know that the maximal

ideal of bA′ is m′A′, and that bA′/m′nbA′ is isomorphic to A′/m′n; since A′/m′n = (Aλ/mnλ)⊗Aλ A′, we see that A′/m′n is

a flat (Aλ/mnλ)-module (0I, 6.2.1); criterion (10.2.2) is thus applicable to the Noetherian Aλ -algebra bA

′, and showsthat bA′ is Aλ -at.

(10.3.1.4). We now treat the general case. There exists an ordinal γ and, for every ordinal λ ¶ γ , a subfield kλof K that contains k, such that (i) for all λ < γ , kλ+1 is an extension of kλ generated by a single element; (ii)for every limit ordinal µ, kµ =

⋃

λ<µ kλ ; and (iii) K = kγ . In fact, it suces to consider a bijection ξ 7→ tξ fromthe set of ordinals ξ ¶ β (for some suitable β) to K , and to define kλ by transfinite induction (for λ ¶ β) as theunion of the kµ for µ< λ if λ is a limit ordinal, and as kν(tξ ) if λ = ν + 1, where ξ is the smallest ordinal suchthat tξ 6∈ kν ; γ is then, by definition, the smallest ordinal ¶ β such that kγ = K .

With this in mind, we will define, by transfinite induction, a family of local Noetherian rings Aλ for λ ¶ γ ,and local homomorphisms fµλ : Aλ → Aµ for λ ¶ µ, satisfying the following conditions:

(i) (Aλ , fµλ) is an inductive system, and A0 = A;(ii) for all λ, we have a k-isomorphism Aλ/JAλ ' kλ ;(iii) for λ ¶ µ, Aµ is a flat Aλ -module.

So suppose that the Aλ and the fµλ are defined for λ < µ < ξ , and suppose, first of all, that ξ = ζ + 1, sothat kξ = kζ (t). If t is transcendental over kζ , we define Aζ , following the procedure of (10.3.1.1), to be equal to(Aζ [t])JAζ [t] ; the canonical map is fζ ξ , and, for λ < ζ , we take fξλ = fξ ζ fζλ ; the verification of conditions (i) to(iii) is then immediate, given that what we have shown in (10.3.1.1). So now suppose that t is algebraic, and leth be its minimal polynomial in kζ [T], and H a monic polynomial in Aζ [T] whose image in kζ [T] is h; we thentake Aξ to be equal to Aζ [T](H), with the fξλ being defined as berofe; the verification of conditions (i) to (iii)then follows from what we have shown in (10.3.1.2).

Now suppose that ξ has no predecessor; we then take Aξ to be the inductive limit of the inductive systemof local rings (Aλ , fµλ) for λ < ξ ; we define fξλ as the canonical map for λ < ξ . The fact that Aξ is local andNoetherian, that the fξλ are local homomorphism, and that conditions (i) to (iii) are satisfied for λ ¶ ξ then 0III-1 | 23follows from the induction hypothesis, and from Lemma (10.3.1.3). With this construction, it is clear that thering B = Aγ satisfies the conditions of (10.3.1).

We note that, by (10.2.1, c), we have a canonical isomorphism

(10.3.1.5) gr(A)⊗k K∼−→ gr(B).

We can also replace B by its JB-adic completion bB without changing the conclusions of (10.3.1), because bBis a flat B-module (0I, 7.3.3), and thus a flat A-module (0I, 6.2.1).

We have also shown the following:

Corollary (10.3.2). — If K is an extension of nite degree, then we can assume that B is a nite A-algebra.

§11. Supplement on homological algebra

11.1. Review of spectral sequences.

(11.1.1). In the following, we use a more general notion of a spectral sequence than that defined in (T, 2.4);keeping the notations of (T, 2.4), we call a spectral sequence in an abelian category C a system E consisting of thefollowing parts:

(a) A family (Epqr ) of objects of C defined for p, q ∈ Z and r ¾ 2.

(b) A family of morphisms d pqr : Epq

r → Ep+r,q−r+1r such that d p+r,q−r+1

r d pqr = 0. We set Zr+1(Epq

r ) = Ker(d pqr )

and Br+1(Epqr ) = Im(d p+r,q−r+1

r ), so that

Br+1(Epqr ) ⊂ Zr+1(E

pqr ) ⊂ Epq

r .

(c) A family of isomorphisms αpqr : Zr+1(Epq

r )/Br+1(Epqr )' Epq

r+1.We then define for k ¾ r + 1, by induction on k, the subobjects Bk(Epq

r ) and Zk(Epqr ) as the

inverse images, under the canonical morphism Epqr → Epq

r /Br+1(Epqr ) of the subobjects of this quotuent

identified via αpqr with the subobjects Bk(E

pqr+1) and Zk(E

pqr+1) respectively. It is clear that we then have,

up to isomorphism,

(11.1.1.1) Zk(Epqr )/Bk(E

pqr ) = Epq

k for k ¾ r + 1,

11. SUPPLEMENT ON HOMOLOGICAL ALGEBRA 67

and, if we set Br(Epqr ) = 0 and Zr(Epq

r ) = Epqr , then we have the inclusion relations

(11.1.1.2) 0= Br(Epqr ) ⊂ Br+1(E

pqr ) ⊂ Br+2(E

pqr ) ⊂ · · · ⊂ Zr+2(E

pqr ) ⊂ Zr+1(E

pqr ) ⊂ Zr(E

pqr ) = Epq

r .

The other parts of the data of E are then:(d) Two subobjects B∞(E

pq2 ) and Z∞(E

pq2 ) of Epq

2 such that we have B∞(Epq2 ) ⊂ Z∞(E

pq2 ) and, for every

k ¾ 2,Bk(E

pq2 ) ⊂ B∞(E

pq2 ) and Z∞(E

pq2 ) ⊂ Zk(E

pq2 ).

We set

(11.1.1.3) Epq∞ = Z∞(E

pq2 )/B∞(E

pq2 ).

0III | 24(e) A family (En) of objects of C, each equipped with a decreasing ltration (F p(En))p∈Z. As usual, we

denote by gr(En) the graded object associated to the filtered object En, the direct sum of the grp(En) =

F p(En)/F p+1(En).(f) For every pair (p, q) ∈ Z× Z, an isomorphism βpq : Epq

∞ ' grp(Ep+q).

The family (En), without the filtrations, is called the abutment (or limit) of the spectral sequence E.Suppose that the category C admits infinite direct sums, or that for every r ¾ 2 and every n ∈ Z, there

are finitely many pairs (p, q) such that p + q = n and Epqr 6= 0 (it suces for it to hold for r = 2). Then the

E(n)r =∑

p+q=n Epqr are defined, and we if denote by d(n)r the morphism E(n)r → E(n+1)

r whose restriction to Epqr is

d pqr for every pair (p, q) such that p+ q = n, then d(n+1)

r d(n)r = 0, in other words, (E(n)r )n∈Z is a complex E(•)r in C,with dierentials of degree +1, and it follows from (c) that Hn(E(•)n ) is isomorphic to E(n)r+1 for every r ¾ 2.

(11.1.2). A morphism u : E → E′ from a spectral sequence E to a spectral sequence E′ = (E′pqr , E′n) consists of

systems of morphisms upqr : Epq

r → E′pqr and un : En→ E′n, the un compatible with the filtrations on En and E′n,

and the diagrams

Epqr

d pqr //

upqr

Ep+r,q−r+1r

up+r,q−r+1r

E′pq

r

d ′pqr // E′′p+r,q−r+1

r

being commutative; in addition, by passing to quotients, upqr gives a morphism upq

r : Zr+1(Z pqr )/Br+1(Epq

r )→Zr+1(E′pq

r )/Br+1(E′pqr ) and we must have α′pq

r upqr = upq

r+1 αpqr ; finally, we must have upq

2 (B∞(Epq2 )) ⊂ B∞(E

′pq2 )

and upq2 (Z∞(E

pq2 )) ⊂ Z∞(E

′pq2 ); by passing to quotients, upq

2 then gives a morphism u′pq∞ : Epq

∞ → E′pq∞ , and the

diagram

Epq∞

u′pq∞ //

βpq

E′pq∞

β′pq

grp(E

p+q)grp(u

p+q)// grp(E

′p+q)

must be commutative.The above definitions show, by induction on r, that if the upq

2 are isomorphisms, then so are the upqr for r ¾ 2;

if in addition we know that upq2 (B∞(E

pq2 )) = B∞(E

′pq2 ) and upq

2 (Z∞(Epq2 )) = Z∞(E

′pq2 ) and the un are isomorphisms,

then we can conclude that u is an isomorphism.

(11.1.3). Recall that if (F p(X ))p∈Z is a (decreasing) ltration of an object X ∈ C, then we say that this filtrationis separated if inf(F p(X )) = 0, discrete if there exists a p such that F p(X ) = 0, exhaustive (or coseparated) ifsup(F p(X )) = X , codiscrete if there exists a p such that F p(X ) = X .

We say that a spectral sequence E = (Epqr , En) is weakly convergent if we have B∞(E

pq2 ) = supk(Bk(E

pq2 )) and

Z∞(Epq2 ) = infk(Zk(E

pq2 )) (in other words, the objects of B∞(E

pq2 ) and Z∞(E

pq2 ) are determined from the data of

(a) and (c) of the spectral sequence E). We say that the spectral sequence E is regular if it is weakly convergentand if in addition:

(1st) For every pair (p, q), the decreasing sequence (Zk(Epq2 ))k¾2 is stable; the hypothesis that E is weakly

convergent then implies that Z∞(Epq2 ) = Zk(E

pq2 ) for k large enough (depending on p and q).

(2nd) For every n, the filtration (F p(En))p∈Z of En is discrete and exhaustive.

We say that the spectral sequence E is coregular if it is weakly convergent and if in addition:

11. SUPPLEMENT ON HOMOLOGICAL ALGEBRA 68

(3rd) For every pair (p, q), the increasing sequence (Bk(Epq2 ))k¾2 is stable, which implies that B∞(E

pq2 ) =

Bk(Epq2 ), and as a result, Epq

∞ = inf Epqk .

(4th) For every n, the filtration of En is codiscrete.Finally, we say that E is biregular if it is both regular and coregular, in other words if we have the following

conditions:(a) For every pair (p, q), the sequences (Bk(E

pq2 ))k¾2 and (Zk(E

pq2 ))k¾2 are stable and we have B∞(E

pq2 ) =

Bk(Epq2 ) and Z∞(E

pq2 ) = Zk(E

pq2 ) for k large enough (which implies that Epq

∞ = Epqk ).

(b) For every n, the filatration (F p(En))p∈Z is discrete and codiscrete (which we also call nite).The spectral sequences defined in (T, 2.4) are thus biregular spectral sequences.

(11.1.4). Suppose that in the category C, filtered inductive limits exist and the functor lim−→ is exact (which isequivalent to saying that the axiom (AB 5) of (T, 1.5) is satisfied (cf. T, 1.8)). The condition that the filtration(F p(X ))p∈Z of an object X ∈ C is exhaustive is then expressed as lim−→p→−∞

F p(X ) = X . If a spectral sequence E is

weakly convergent, then we have B∞(Epq2 ) = lim−→k→∞

Bk(Epq2 ); if in addition u : E→ E′ is a morphism from E to

a weakly convergent spectral sequence E′ in C, then we have upq2 (B∞(E

pq2 )) = B∞(E

′pq2 ), by the exactness of lim−→.

In addition:

Proposition (11.1.5). — Let C be an abelian category in which ltered inductive limits are exact, E and E′ two regularspectral sequences in C, u : E→ E′ a morphism of spectral sequences. If the upq

2 are isomorphisms, then so is u.

Proof. We already know (11.1.2) that the upqr are isomorphisms and that

upq2 (B∞(E

pq2 )) = B∞(E

′pq2 );

the hypothesis that E and E′ are regular also implies that upq2 (Z∞(E

pq2 )) = Z∞(E

′pq2 ), and as upq

2 is an isomorphism, 0III | 26so is u′pq

∞ ; we thus conclude that grp(up+q) is also an isomorphism. But as the filtrations of the En and the E′n

are discrete and exhaustive, this implies that the un are also isomorphisms (Bourbaki, Alg. comm., chap. III, §2,no8, th. 1).

(11.1.6). It follows from (11.1.1.2) and the definition (11.1.1.3) that if, for a spectral sequence E, we have Epqr = 0,

then we have Epqk = 0 for k ¾ r and Epq

∞ = 0. We say that a spectral sequence degenerates if there exists an integerr ¾ 2 and, for every integer n ∈ Z, an integer q(n) such that En−q,q

r = 0 for every q 6= q(n). We first deducefrom the previous remark that we also have En−q,q

k = 0 for k ¾ r (including k =∞) and q 6= q(n). In addition,

the definition of Epqr+1 shows that we have En−q(n),q(n)

r+1 = En−q(n),q(n)r ; if E is weakly convergent, then we also have

En−q(n),q(n)∞ = En−q(n),q(n)

r ; in other words, for every n ∈ Z, grp(En) = 0 for p 6= q(n) and grq(n)(E

n) = En−q(n),q(n)r . If

in addition the filtration of En is discrete and exhaustive, then the spectral sequence E is regular, and we haveEn = En−q(n),q(n)

r up to unique isomorphism.

(11.1.7). Suppose that filtered inductive limits exist and are exact in the category C, and let (Eλ , uµλ) be aninductive system (over a filtered set of indices) of spectral sequeneces in C. Then the inductive limit of thisinductive system exists in the additive category of spectral sequences of objects of C: to see this, it suces todefine Epq

r , d pqr , αpq

r , B∞(Epq2 ), Z∞(E

pq2 ), En, F p(En), and βpq as the respective inductive limits of the Epq

r,λ , d pqr,λ ,

αpqr,λ , B∞(E

pq2,λ), Z∞(E

pq2,λ), En

λ , F p(Enλ ), and β

pqλ ; the verification of the conditions of (11.1.1) follows from the

exactness of the functor lim−→ on C.

Remark (11.1.8). — Suppose that the category C is the category of A-modules over a Noetherian ring A (resp. aring A). Then the definitions of (11.1.1) show that if, for a given r, the Epq

r are A-modules of nite type (resp. ofnite length), then so are each of the modules Epq

s for s ¾ r, hence so is Epq∞. If in addition the filtration of the

abutment/limit (En) is discrete or codiscrete for all n, then we conclude that each of the En is also an A-module ofnite type (resp. of nite length).

(11.1.9). We will have to consider conditions which ensure that a spectral sequence E is biregular is a “uniform”way in p+ q = n. We will then use the following lemma:

Lemma (11.1.10). — Let (Epqr ) be a family of objects of C related by the data of (a), (b), and (c) of (11.1.1). For a

xed integer n, the following properties are equivalent:(a) There exists an integer r(n) such that for r ¾ r(n), p+ q = n or p+ q = n− 1, the morphisms d pq

r are zero.(b) There exists an integer r(n) such that for p + q = n or p + q = n + 1, we have Br(E

pq2 ) = Bs(E

pq2 ) for

s ¾ r ¾ r(n).

11. SUPPLEMENT ON HOMOLOGICAL ALGEBRA 69

(c) There exists an integer r(n) such that for p + q = n or p + q = n − 1, we have Zr(Epq2 ) = Zs(E

pq2 ) for

s ¾ r ¾ r(n).(d) There exists an integer r(n) such that for p+ q = n, we have Br(E

pq2 ) = Bs(E

pq2 ) and Zr(E

pq2 ) = Zs(E

pq2 ) for

s ¾ r ¾ r(n).

Proof. According to the conditions (a), (b), and (c) of (11.1.1), we have that saying Zr+1(Epq2 ) = Zr(E

pq2 ) is

equivalent to saying that d pqr = 0 and that saying Br(E

p+r,q−r+12 ) = Br+1(E

p+r,q−r+12 ) is equivalent to saying that

d pqr = 0; the lemma immediately follows from this remark.

11.2. The spectral sequence of a ltered complex.

(11.2.1). Given an abelian category C, we will agree to denote by notations such as K• the complexes (K i)i∈Zof objects of C whose dierential is of degree +1, and by the notations such as K• the complexes (Ki)i∈Z ofobjects of C whose dierential is of degree −1. To each complex K• = (K i) whose dierential d is of degree+1, we can associate a complex K ′• = (K

′i ) by setting K ′i = K−i , the dierential K ′i → K ′i−1 being the operator

d : K−i → K−i−1; and vice versa, which, depending on the circumstances, will allow one to consider either one ofthe types of complexes and translate any result from one type into results for the other. We similarly denote bynotations such as K•• = (K i j) (resp. K•• = (Ki j)) the bicomplexes (or double complexes) of objects of C in whichthe two dierntials are of degree +1 (resp. −1); we can still pass from one type to the other by changing thesigns of the indices, and we have similar notations and remarks for any multicomplexes. The notation K• andK• will also be used for Z-graded objects of C, which are not necessarily complexes (they can be considered assuch for the zero dierentials); for example, we write H•(K•) = (Hi(K•))i∈Z for the cohomology of a complex K•

whose dierential is of degree +1, and H•(K•) = (Hi(K•))i∈Z for the homology of a complex K• whose dierentialis of degree −1; when we pass from K• to K ′• by the method described above, we have Hi(K ′•) = H−i(K•).

Recall in this case that for a complex K• (resp. K•), we will write in general Z i(K•) = Ker(K i → K i+1)(“object of cocycles”) and Bi(K•) = Im(K i−1 → K i) (“object of coboundaries”) (resp. Zi(K•) = Ker(Ki → Ki−1)(“object of cycles”) and Bi(K•) = Im(Ki+1→ Ki) (“object of boundaries”)) so that Hi(K•) = Z i(K•)/Bi(K•) (resp.Hi(K•) = Zi(K•)/Bi(K•)).

If K• = (K i) (resp. K• = (Ki)) is a complex in C and T : C→ C′ a functor from C to an abelian category C′,then we denote by T (K•) (resp. T (K•)) the complex (T (K i)) (resp. (T (Ki))) in C′.

We will not review the definition of the ∂ -functors (T, 2.1), except to note that we also say ∂ -functor in placeof ∂ ∗-functor when the morphism ∂ decreases the degree of a unit, the context clarifying this point if there iscause for confusion.

Finally, we say that a graded object (Ai)i∈Z of C is bounded below (resp. above) if there exists an i0 such thatAi = 0 for i < i0 (resp. i > i0).

(11.2.2). Let K• be a complex in C whose dierential d is of degree +1, and suppose it is equipped with altration F(K•) = (F p(K•))p∈Z consisting of graded subobjects of K•, in other words, F p(K•) = (K i ∩ F p(K•))i∈Z; 0III | 28in addition, we assume that d(F p(K•)) ⊂ F p(K•) for every p ∈ Z. Let us quickly recall how one functoriallydefines a spectral sequence E(K•) from K• (M, XV, 4 and G, I, 4.3). For r ¾ 2, the canonical morphismF p(K•)/F p+r(K•)→ F p(K•)/F p+1(K•) defines a morphism in cohomology

Hp+q(F p(K•)/F p+r(K•)) −→ Hp+q(F p(K•)/F p+1(K•)).

We denote by Z pqr (K

•) the image of this morphism. Similarly, from the exact sequence

0 −→ F p(K•)/F p+1(K•) −→ F p−r+1(K•)/F p+1(K•) −→ F p−r+1(K•)/F p(K•) −→ 0,

we deduce from the exact sequence in cohomology a morphism

Hp+q−1(F p−r+1(K•)/F p(K•)) −→ Hp+q(F p(K•)/F p+1(K•)),

and we denote by Bpqr (K

•) the image of this morphism; we show that Bpqr (K

•) ⊂ Z pqr (K

•) and we take Epqr (K

•) =Z pq

r (K•)/Bpq

r (K•); we will not specify the definition of the d pq

r or the αpqr .

We note here that all the Z pqr (K

•) and Bpqr (K

•), for p and q fixed, are subobjects of the same objectHp+q(F p(K•)/F p+1(K•)), which we denote by Z pq

1 (K•); we set Bpq

1 (K•) = 0, so that the above definitions of

Z pqr (K

•) and Bpqr (K

•) also work for r = 1; we still set Epq1 (K

•) = Z pq1 (K

•). We define d pq1 and αpq

1 such that theconditions of (11.1.1) are satisfied for r = 1. On the other hand, we define the subobjects Z pq

∞(K•) as the imageof the morphism

Hp+q(F p(K•)) −→ Hp+q(F p(K•)/F p+1(K•)) = Epq1 (K

•),and Bpq

∞(K•) as the image of the morphism

Hp+q−1(K•/F p(K•)) −→ Hp+q(F p(K•)/F p+1(K•)) = Epq1 (K

•),

11. SUPPLEMENT ON HOMOLOGICAL ALGEBRA 70

induced as above from the exact sequence in cohomology. We set Z∞(Epq2 (K

•)) and B∞(Epq2 (K

•)) to be thecanonical images of Epq

2 (K•) in Z pq

∞(K•) and Bpq∞(K•).

Finally, we denote by F p(Hn(K•)) the image in Hn(K•) of the morphism Hn(F p(K•)) → Hn(K•) inducedfrom the canonical injection F p(K•)→ K•; by the exact sequence in cohomology, this is also the kernel of themorphism Hn(K•)→ Hn(K•/F p(K•)). This defines a filtration on En(K•) = Hn(K•); we will not give here thedefinition of the isomorphisms βpq.

(11.2.3). The functorial nature of E(K•) is understood in the following way: given two ltered complexes K• andK ′• in C and a morphism of complexes u : K•→ K ′• that is compatible with the ltrations, we induce in an evidentway the morphisms upq

r (for r ¾ 1) and un, and we show that these morphisms are compatible with the d pqr ,

αpqr , and βpq in the sense of (11.1.2), and thus given a well-defined morphism E(u) : E(K•)→ E(K ′•) of spectral

sequences. In addition, we show that if u and v are morphisms K•→ K ′• of the above type, homotopic in degree¶ k, then upq

r = vpqr for r > k and un = vn for all n (M, XV, 3.1).

0III | 29(11.2.4). Suppose that filtered inductive limits in C are exact. Then if the filtration (F p(K•)) of K• is exhaustive,then so is the filatration (F p(Hn(K•))) for all n, since by hypothesis we have K• = lim−→p→−∞

F p(K•) and since the

hypothesis on C implies that cohomology commutes with inductive limits. In addition, for the same reason, wehave B∞(E

pq2 (K

•)) = supk Bk(Epq2 (K

•)). We say that the filtration (F p(K•)) of K• is regular if for every n thereexists an integer u(n) such that Hn(F p(K•)) = 0 for p > u(n). This is particularly the case when the filtration ofK• is discrete. When the filtration of K• is regular and exhaustive, and filtered inductive limits are exact in C, wehave (M, XV, 4) that the spectral sequence E(K•) is regular.

11.3. The spectral sequences of a bicomplex.

(11.3.1). With regard the conventions for bicomplexes, we follow those of (T, 2.4) rather than those of (M), thetwo dierentials d ′ and d ′′ (of degree +1) of such a bicomplex K•• = (K i j) being thus assumed to be permutable.Suppose that one of the following two conditions is satisifed: 1st. Innite direct sums exist in C; 2nd. For all n ∈ Z,there is only a nite number of pairs (p, q) such that p+ q = n and K pq 6= 0. Then, the bicomplex K•• defines a(simple) complex (K ′n)n∈Z with K ′n =

∑

i+ j=n K i j , the dierential d (of degree +1) of this complex being given byd x = d ′x + (−1)id ′′x for x ∈ K i j . When we later speak of the spectral sequence of a (simple) complex that isdened by a bicomplex K••, it will always be understood that of the above conditions is satisfied. We adopt theanalogous conventions for multicomplexes.

We denote by K i,• (resp. K•, j) the simple complex (K i j) j∈Z (resp. (K i j)i∈Z), by Z pII(K

i,•), BpII(K

i,•), HpII(K

i,•)(resp. Z p

I (K•, j), Bp

I (K•, j), Hp

I (K•, j)) its p objects of cocycles, of coboundaries, and of cohomology, respectively;

the dierential d ′ : K i,• → K i+1,• is a morphism of complexes, which thus gives an operator on the cocycles,coboundaries, and cohomology,

d ′ : Z pII(K

i,•) −→ Z pII(K

i+1,•),

d ′ : BpII(K

i,•) −→ BpII(K

i+1,•),

d ′ : HpII(K

i,•) −→ HpII(K

i+1,•),

and it is clear that for these operators, (Z pII(K

i,•))i∈Z, (BpII(K

i,•))i∈Z, and (HpII(K

i,•))i∈Z are complexes; we denotethe complex (Hp

II(Ki,•))i∈Z by Hp

II(K••), its q objects of cocycles, coboundaries, and cohomology by Zq

I (HpII(K

••)),BqI (H

pII(K

••)), and HqI (H

pII(K

••)). We similarly define the complexes HpI (K

••) and their cohomology objectsHqII(H

pI (K

••)). Recall on the other hand that Hn(K••) denotes the n object of the cohomology of the (simple)complex defined by K••.

(11.3.2). On the complex defined by a bicomplex K••, we can consider two canonical filtrations (F pI (K

••)) and(F p

II(K••)) given by

(11.3.2.1) F pI (K

••) =

∑

i+ j=n,i¾p

K i j

!

n∈Z

and F pII(K

••) =

∑

i+ j=n, j¾p

K i j

!

n∈Z

,

which, by definition, are graded subobjects of the (simple) complex define by K••, and thus make this complex a 0III | 30filtered complex; moreover, is is clear that these filtrations are exhaustive and separated.

There corresponds to each of these filtrations a spectral sequence (11.2.2); we denote by ′E(K••) and ′′E(K••)the spectral sequences corresponding to (F p

I (K••)) and (F p

II(K••)) respectively, called the spectral sequence of the

bicomplex K••, and both having as their abutment the cohomology (Hn(K••)). We show in addition (M, XV, 6)that we have

(11.3.2.2) ′Epq2 (K

••) = HpI (H

pII(K

••)), ′′Epq2 (K

••)) = HpII(H

pI (K

••)).

14. COMBINATORIAL DIMENSION OF A TOPOLOGICAL SPACE 71

Every morphism u : K••→ K ′•• of bicomplexes is ipso facto compatible with the filtrations of the same typeof K•• and K ′••, thus define a morphism for each of the two spectral sequences; in addition, two homotopicmorphisms define a homotopy of order ¶ 1 of the corresponding filtered (simple) complexes, thus the samemorphism for each of the two spectral sequences (M, XV, 6.1).

Proposition (11.3.3). — Let K•• = (K i j) be a bicomplex in an abelian category C.(i) If there exist i0 and j0 such that K i j = 0 for i < i0 or j < j0 (resp. i > i0 or j > j0), then the two spectral

sequences ′E(K••) and ′′E(K••) are biregular.(ii) If there exist i0 and i1 such that K i j = 0 for i < i0 or i > i1 (resp. if there exist j0 and j1 such that K i j = 0 for

j < j0 or j > j1), then the two spectral sequences′E(K••) and ′′E(K••) are biregular.

(iii) If there exists i0 such that K i j = 0 for i > i0 (resp. if there exists j0 such that K i j = 0 for j < j0), then thespectral sequence ′E(K••) is regular.

(iv) If there exists i0 such that K i j = 0 for i < i0 (resp. if there exists j0 such that K i j = 0 for j > j0), then thespectral sequence ′′E(K••) is regular.

Proof. The proposition follows immediately from the definitions (11.1.3) and from (11.2.4),

§12. Supplement on sheaf cohomology

§13. Projective limits in homological algebra

§14. Combinatorial dimension of a topological space

SummaryIV-1 | 5

§14. Combinatorial dimension of a topological space.§15. M -regular sequences and F -regular sequences.§16. Dimension and depth of Noetherian local rings.§17. Regular rings.§18. Supplement on extensions of algebras.§19. Formally smooth algebras and Cohen rings.§20. Derivations and dierentials.§21. Dierentials in rings of characteristic p.§22. Dierential criteria for smoothness and regularity.§23. Japanese rings.

Almost all of the preceding sections have been focused on the exposition of ideas of commutative algebrathat will be used throughout Chapter IV. Even though a large amount of these ideas already appear in multipleworks ([CC], [Sam53a], [SZ60], [Ser55b], [Nag62]), we thought that it would be more practical for the readerto have a coherent, vaguely independent exposition. Together with §§5, 6, and 7 of Chapter IV (where we usethe language of schemes), these sections constitute, in the middle of our treatise, a miniature special treatise,somewhat independent of Chapters I to III, and one that aims to present, in a coherent manner, the propertiesof rings that “behave well” relative to operations such as completion, or integral closure, by systematicallyassociating these properties to more general ideas.11

0IV-1 | 614.1. Combinatorial dimension of a topological space.

(14.1.1). Let I be an ordered set; a chain of elements of I is, by definition, a strictly-increasing finite sequencei0 < i1 < · · ·< in of elements of I (n¾ 0); by definition, the length of this chain is n. If X is a topological space,the set of its irreducible closed subsets is ordered by inclusion, and so we have the notion of a chain of irreducibleclosed subsets of X .

Denition (14.1.2). — Let X be a topological space. We define the combinatorial dimension of X (or simplythe dimension of X , if there is no risk of confusion), denoted by dimc(X ) (or simply dim(X )), to be the upperbound of lengths of chains of irreducible closed subsets of X . For all x ∈ X , we define the combinatorialdimension of X at x (or simply the dimension of X at x), denoted by dimx(X ), to be the number infU(dim(U)),where U varies over the open neighbourhoods of x in X .

11The majority of properties which we discuss were discovered by Chevalley, Zariski, Nagata, and Serre. The method used here wasfirst developed in the autumn of 1961, in a course taught at Harvard University by A. Grothendieck.

14. COMBINATORIAL DIMENSION OF A TOPOLOGICAL SPACE 72

It follows from this definition that we have

dim(∅) = −∞

(the upper bound in R of the empty set being −∞). If (Xα) is the family of irreducible components of X , thenwe have

(14.1.2.1) dim(X ) = supα

dim(Xα),

because every chain of irreducible closed subsets of X is, by definition, contained in some irreducible componentof X , and, conversely, the irreducible components are closed in X , so every irreducible closed subset of an Xα isa irreducible closed subset of X .

Denition (14.1.3). — We say that a topological space X is equidimensional if all its irreducible componentshave the same dimension (which is thus equal to dim(X ), by (14.1.2.1)).

Proposition (14.1.4). —(i) For every closed subset Y of a topological space X , we have dim(Y )¶ dim(X ).(ii) If a topological space X is a nite union of closed subsets X i , then we have dim(X ) = supi dim(X i).

Proof. For every irreducible closed subset Z of Y , the closure Z of Z in X is irreducible (0I, 2.1.2), andZ ∩ Y = Z , whence (i). Now, if X =

⋃ni=1 X i , where the X i are closed, then every irreducible closed subset of X is

contained in one of the X i (0I, 2.1.1), and so every chain of irreducible closed subsets of X is contained in oneof the X i , whence (ii).

From (14.1.4, i), we see that, for all x ∈ X , we can also write

(14.1.4.1) dimx(X ) = limU

dim(U),

where the limit is taken over the downward-directed set of open neighbourhoods of x in X . 0IV-1 | 7

Corollary (14.1.5). — Let X be a topological space, x a point of X , U a neighbourhood of x , and Yi (1¶ i ¶ n) closedsubsets of U such that, for all i, x ∈ Yi , and such that U is the union of the Yi . Then we have

(14.1.5.1) dimx(X ) = supi(dimx(Yi)).

Proof. It follows from (14.1.4, ii) that we have dimx(X ) = infV (supi(dim(Yi ∩ V ))), where V ranges over theset of open neighbourhoods of x that are contained in U ; similarly, we have dimx(Yi) = infV (dim(Yi ∩ V )) for alli. The corollary is thus evident if

supi(dimx(Yi)) = +∞;

if this were not the case, then there would be an open neighbourhood V0 ⊂ U of x such that dim(Yi∩V ) = dimx(Yi)for 1¶ i ¶ n and for all V ⊂ V0, whence the conclusion.

Proposition (14.1.6). — For every topological space X , we have dim(X ) = supx∈X dimx(X ).

Proof. It follows from Definition (14.1.2) and Proposition (14.1.4) that dimx(X ) ¶ dim(X ) for all x ∈ X .Now, let Z0 ⊂ Z1 ⊂ . . . ⊂ Zn be a chain of irreducible closed subsets of X , and let x ∈ Z0; for every open subsetU ⊂ X that contains x , U ∩ Zi is irreducible (0I, 2.1.6) and closed in U , and since we have U ∩ Zi = Zi in X , theU ∩ Zi are pairwise distinct; thus dim(U)¾ n, which finishes the proof.

Corollary (14.1.7). — If (Xα) is an open, or closed and locally nite, cover of X , then dim(X ) = supα(dim(Xα)).

Proof. If Xα is a neighbourhood of x ∈ X , then dimx(X )¶ dim(Xα), whence the claim for open covers. Onthe other hand, if the Xα are closed, and U is a neighbourhood of x ∈ X which meets only finitely many of theXα, then

dimx(X )¶ dim(U) = supα(dim(U ∩ Xα))¶ sup

α(dim(Xα))

by (14.1.4), whence the other claim.

Corollary (14.1.8). — Let X be a Noetherian Kolmogoro space (0I, 2.1.3), and F the set of closed points of X . Thendim(X ) = supx∈F dimx(X ).

Proof. With the notation from the proof of (14.1.6), it suces to note that there exists a closed point in Z0(0I, 2.1.3).

14. COMBINATORIAL DIMENSION OF A TOPOLOGICAL SPACE 73

Proposition (14.1.9). — Let X be a nonempty Noetherian Kolmogoro space. To have dim(X ) = 0, it is necessary andsucient for X to be nite and discrete.

Proof. If a space X is separated (and a fortiori if X is a discrete space), then all the irreducible closed subsetsof X are single points, and so dim(X ) = 0. Conversely, suppose that X is a Noetherian Kolmogoro space suchthat dim(X ) = 0; since every irreducible component of X contains a closed point (0I, 2.1.3), it must be exactlythis single point. Since X has only a finite number of irreducible components, it is thus finite and discrete.

Corollary (14.1.10). — Let X be a Noetherian Kolmogoro space. For a point x ∈ X to be isolated, it is necessary andsucient to have dimx(X ) = 0.

Proof. The condition is clearly necessary (without any hypotheses on X ). It is also sucient, because it 0IV-1 | 8implies that dim(U) = 0 for any open neighbourhood U of x , and since U is a Noetherian Kolmogoro space, Uis finite and discrete.

Proposition (14.1.11). — The function x 7→ dimx(X ) is upper semi-continuous on X .

Proof. It is clear that this function is upper semi-continuous at every point where its value is +∞. Sosuppose that dimx(X ) = n < +∞; then Equation (14.1.4.1) shows that there exists an open neighbourhoodU0 of x such that dim(U) = n for every open neighbourhood U ⊂ U0 of x . So, for all y ∈ U0 and every openneighbourhood V ⊂ U0 of y, we have dim(V ) ¶ dim(U0) = n (14.1.4); we thus deduce from (14.1.4.1) thatdimy(X )¶ n.

Remark (14.1.12). — If X and Y are topological spaces, and f : X → Y a continuous map, then it can be thecase that dim( f (X )) > dim(X ); we obtain such an example by taking X to be a discrete space with 2 points,a and b, and Y to be the set a, b endowed with the topology for which12 the closed sets are ∅, a, anda, b; if f : X → Y is the identity map, then dim(Y ) = 1 and dim(X ) = 0. We note that Y is the spectrum ofa discrete valuation ring A, of which a is the unique closed point, and b the generic point; if K and k are thefield of fractions and the residue field of A (respectively), then X is the spectrum of the ring k× K , and f is thecontinuous map corresponding to the homomorphism (φ ,ψ) : A→ k×K , where φ : A→ k and ψ : A→ K are thecanonical homomorphism (cf. (IV, 5.4.3)).

14.2. Codimension of a closed subset.

Denition (14.2.1). — Given an irreducible closed subset Y of a topological space X , we define the combi-natorial codimension (or simply codimension) of Y in X , denoted by codim(Y, X ), to be the upper bound ofthe lengths of chains of irreducible closed subsets of X of which Y is the smallest element. If Y is an arbitraryclosed subset of X , then we define the codimension of Y in X , again denoted by codim(Y, X ), to be the lowerbound of the codimensions in X of the irreducible components of Y . We say that X is equicodimensional if allthe minimal irreducible closed subsets of X has the same codimension in X .

It follows from this definition that codim(∅, X ) = +∞, since the lower bound of the empty set of R is +∞.If Y is closed in X , and if (Xα) (resp. (Yα)) is the family of irreducible components of X (resp. Y ), then every Yβis contained in some Xα, and, more generally, every chain of irreducible closed subsets of X of which Yβ is thesmallest element is formed of subsets of some Xα; we thus have

(14.2.1.1) codim(Y, X ) = infβ(supα(codim(Yβ , Xα))),

where, for every β, α ranges over the set of indices such that Yβ ⊂ Xα.

Proposition (14.2.2). — Let X be a topological space.

(i) If Φ is the set of irreducible closed subsets of X , then

(14.2.2.1) dim(X ) = supY∈Φ(codim(Y, X )).

0IV-1 | 9(ii) For every nonempty closed subset Y of X , we have

(14.2.2.2) dim(Y ) + codim(Y, X )¶ dim(X ).

(iii) If Y , Z , and T are closed subsets of X such that Y ⊂ Z ⊂ T , then

(14.2.2.3) codim(Y, Z) + codim(Z , T )¶ codim(Y, T ).

12[Trans.] This is now often referred to as the Sierpiński space, or the connected two-point set.

14. COMBINATORIAL DIMENSION OF A TOPOLOGICAL SPACE 74

(iv) For a closed subset Y of X to be such that codim(Y, X ) = 0, it is necessary and sucient for Y to contain anirreducible component of X .

Proof. Claims (i) and (iv) are immediate consequences of Definition (14.2.1). To show (ii), we canrestrict to the case where Y is irreducible, and then the equation follows from Definitions (14.1.1) and (14.2.1).Finally, to show (iii), we can, by Definition (14.2.1), first restrict to the case where Y is irreducible; thencodim(Y, Z) = supα(codim(Y, Zα)) for the irreducible components Zα of Z that contain Y ; it is clear thatcodim(Y, T ) ¾ codim(Y, Z), so the inequality is true if codim(Y, Z) = +∞; but if this were not the case, thenthere would exist some α such that codim(Y, Z) = codim(Y, Zα), and by (14.2.1), we can restrict to the case whereZ itself is irreducible; but then the inequality in (14.2.2.3) is an evident consequence of Definition (14.2.1).

Proposition (14.2.3). — Let X be a topological space, and Y a closed subset of X . For every open subset U of X , we have

(14.2.3.1) codim(Y ∩ U , U)¾ codim(Y, X ).

Furthermore, for this inequality (14.2.3.1) to be an equality, it is necessary and sucient to have codim(Y, X ) =infα(codim(Yα, X )), where (Yα) is the family of irreducible components of Y that meet U .

Proof. We know (0I, 2.1.6) that Z 7→ Z is a bijection from the set of irreducible closed subsets of U tothe set of irreducible closed subsets of X that meet U , and, in particular, induces a correspondence betweenthe irreducible components of Y ∩ U and the irreducible components of Y that meet U ; if Yα is one of thelatter such components, then we have codim(Yα, X ) = codim(Yα ∩ U , U), and the proposition then follows fromDefinition (14.2.1).

Denition (14.2.4). — Let X be a topological space, Y a closed subset of X , and x a point of X . We definethe codimension of Y in X at the point x , denoted by codimx(Y, X ), to be the number supU(codim(Y ∩ U , U)),where U ranges over the set of open neighbourhoods of x in X .

By (14.2.3), we can also write

(14.2.4.1) codimx(Y, X ) = limU(codim(Y ∩ U , U)),

where the limit is taken over the downward-directed set of open neighbourhoods of x in X . We note that we have

codimx(Y, X ) = +∞ if x ∈ X − Y.0IV-1 | 10

Proposition (14.2.5). — If (Yi)1¶i¶n is a nite family of closed subsets of a topological space X , and Y is the union ofthis family, then

(14.2.5.1) codim(Y, X ) = infi(codim(Yi , X )).

Proof. Every irreducible component of one of the Yi is contained in an irreducible component of Y , and,conversely, every irreducible component of Y is also an irreducible component of ome of the Yi (0I, 2.1.1); theconclusion then follows from Definition (14.2.1) and the inequality in (14.2.2.3).

Corollary. — Let X be a topological space, and Y a locally-Noetherian closed subspace of X .

(i) For all x ∈ X , there exists only a nite number of irreducible components Yi (1 ¶ i ¶ n) of Y that contain x ,and we have codimx(Y, X ) = infi(codim(Yi , X )).

(ii) The function x 7→ codimx(Y, X ) is lower semi-continuous on X .

Proof. By hypothesis, there exists an open neighbourhood U0 of x in X such that Y ∩ U0 is Noetherian, andso U0 has only a finite number of irreducible components, which are the intersections of U0 with the irreduciblecomponents of Y ; a fortiori there are only a finite number of irreducible components Yi (1 ¶ i ¶ n) of Y thatcontain x , and we can, by replacing U0 with an open neighbourhood U ⊂ U0 of x that doesn’t meet any ofthe Yj that do not contain x , assume that the Yi ∩ U are the irreducible components of Y ∩ U ; for every openneighbourhood V ⊂ U of x in X , the Yi ∩V are thus the irreducible components of Y ∩V , and (14.2.3) then showsthat codim(Yi , X ) = codim(Yi ∩ V, V ), which proves (i). Further, for every x ′ ∈ U , the irreducible components ofY that contain x ′ are certain Yi , and so codimx ′(Y, X )¾ codimx(Y, X ), which proves (ii).

14. COMBINATORIAL DIMENSION OF A TOPOLOGICAL SPACE 75

14.3. The chain condition.

(14.3.1). In a topological space X , we say that a chain Z0 ⊂ Z1 ⊂ · · · ⊂ Zn of irreducible closed subsets if saturatedif there does not exist an irreducible closed subset Z ′, distinct from each of the Zi , such that Zk ⊂ Z ′ ⊂ Zk+1 forany k.

Proposition (14.3.2). — Let X be a topological space such that, for any two irreducible closed subsets Y and Z of Xwith Y ⊂ Z , we have codim(Y, Z)< +∞. The following two conditions are equivalent.

(a) Any two saturated chains of closed irreducible subsets of X that have the same rst and last elements as oneanother have the same length.

(b) If Y , Z , and T are irreducible closed subsets of X such that Y ⊂ Z ⊂ T , then

(14.3.2.1) codim(Y, T ) = codim(Y, Z) + codim(Z , T ).

Proof. It is immediate that (a) implies (b). Conversely, suppose that (b) is satisfied, and we will show thatif we have two saturated chains with the same first and last elements as one another, of lengths m and n¶ m(respectively), then m= n. We proceed by induction on n, with the proposition being clear for n= 1. So supposethat 1< n< m, and let Z0 ⊂ Z1 ⊂ · · · ⊂ Zn be a saturated chain such that there exists another saturated chain, 0IV-1 | 11with first element Z0 and last element Zn, of length m. Since codim(Z0, Zn)¾ m> n, and codim(Z0, Z1) = 1, itfollows from (b) that codim(Z1, Zn) = codim(Z0, Zn)− 1> n− 1, which contradicts our induction hypothesis.

When the conditions of (14.3.2) are satisfied, we say that X satisfies the chain condition, or that it is a catenaryspace. It is clear that every closed subspace of a catenary space is catenary.

Proposition (14.3.3). — Let X be a Noetherian Kolmogoro space of nite dimension. The following conditions areequivalent.

(a) Any two maximal chains of irreducible closed subsets of X have the same length.(b) X is equidimensional, equicodimensional, and catenary.(c) X is equidimensional, and, for any irreducible closed subsets Y and Z of X with Y ⊂ Z , we have

(14.3.3.1) dim(Z) = dim(Y ) + codim(Y, Z).

(d) X is equicodimensional, and, for any irreducible closed subsets Y and Z of X with Y ⊂ Z , we have

(14.3.3.2) codim(Y, Z) = codim(Y, Z) + codim(Z , X ).

Proof. The hypotheses on X imply that the first and last elements of a maximal chain of irreducible closedsubsets of X are necessarily a closed point and an irreducible component of X (respectively) (0I, 2.1.3); further,every saturated chain with first element Y and last element Z (thus Y ⊂ Z) is contained in a maximal chain whoseelements dier from those of the given chain, or are contained in Y , or contain Z . These remarks immediatelyestablish the equivalence between (a) and (b), and also show that if (a) is satisfied, then we have, for everyirreducible closed subset Y fo X ,

(14.3.3.3) dim(Y ) + codim(Y, X ) = dim(X );

from (14.3.2.1), we immediately deduce (14.3.3.1) and (14.3.3.2) from (14.3.3.3). Conversely, (14.3.3.1) implies(14.3.2.1), and so (14.3.3.1) implies the chain condition, by (14.3.2); further, by applying (14.3.3.1) to the casewhere Y is a single closed point x of X , and Z is an irreducible component of X , we get that codim(x, X ) =dim(Z); we thus conclude that (c) implies (b). Similarly, (14.3.3.2) implies (14.3.2.1), and thus the chain condition;further, with the same choice of Y and Z as above, (14.3.3.2) again implies that codim(x, X ) = dim(Z), and so(since every irreducible component of X contains a closed point, by (0I, 2.1.3)), (d) implies (b).

We say that a Noetherian Kolmogoro space is biequidimensional if it is of nite dimension and if it verifiesany of the (equivalent) conditions of (14.3.3).

Corollary (14.3.4). — Let X be a biequidimensional Noetherian Kolmogoro space; then, for every closed point x of X ,and every irreducible component Z of X , we have

(14.3.4.1) dim(X ) = dim(Z) = codim(x, X ) = dimx(X ).0IV-1 | 12

Proof. The last equality follows from the fact that, if Y0 = x ⊂ Y1 ⊂ · · · ⊂ Ym is a maximal chain ofirreducible closed subsets of X , and U an open neighbourhood of x , then the U ∩ Yi are pairwise disjointirreducible closed subsets of U (because U ∩ Yi = Yi), whence dim(U) = dim(X ), by (14.1.4).

14. COMBINATORIAL DIMENSION OF A TOPOLOGICAL SPACE 76

Corollary (14.3.5). — Let X be a Noetherian Kolmogoro space; if X is biequidimensional, then so is every union ofirreducible components of X , and every irreducible closed subset of X . In addition, for every closed subset Y of X , we have

(14.3.5.1) dim(Y ) + codim(Y, X ) = dim(X ).

Proof. Every chain of irreducible closed subsets of X is contained in an irreducible component of X , andso the first claim follows immediately from (14.3.3). Further, if X ′ is an irreducible closed subset of X , then X ′

trivially satisfies the conditions of (14.3.3, c), whence the second claim.Finally, to show (14.3.5.1), note that we have seen, in the proof of (14.3.3), that this equation is true whenever

Y is irreducible; if Yi (1 ¶ i ¶ m) are the irreducible components of Y , then the Yi for which dim(Yi) is thelargest are also those for which codim(Yi , X ) is the smallest; so (14.3.5.1) follows from the definitions of dim(Y )and codim(Y, X ).

Remark (14.3.6). — The reader will note that the proof of (14.3.2) applies to any ordered set, and the factthat we are working with the example of a set of irreducible closed subsets of a topological space is not used atall in the proof. It is the same in the proof of (14.3.3), which holds, more generally, for any ordered set E suchthat, for all x ∈ E, there exists some z ¶ x which is minimal in E, and such that the length of chains of elementsof E is bounded.

CHAPTER I

The language of schemes (EGA I)

Summary

§1. Ane schemes.§2. Preschemes and morphisms of preschemes.§3. Products of preschemes.§4. Subpreschemes and immersion morphisms.§5. Reduced preschemes; separation condition.§6. Finiteness conditions.§7. Rational maps.§8. Chevalley schemes.§9. Supplement on quasi-coherent sheaves.§10. Formal schemes.

I | 79Sections §§1–8 do little more than develop a language, which will be used in what follows. It should be

noted, however, that, in accordance with the general spirit of this treatise, §§7–8 will be used less than theothers, and in a less essential way; we have speak of Chevalley’s schemes only to make the link with the languageof Chevalley [CC] and Nagata [Nag58a]. Section §9 gives definitions and results concerning quasi-coherentsheaves, some of which are no longer limited to a translation into a “geometric” language of known notions ofcommutative algebra, but are already of a global nature; they will be indispensable, in the following chapters,for the global study of morphisms. Finally, §10 introduces a generalization of the notion of schemes, which willbe used as an intermediary in Chapter III to formulate and prove, in a convenient way, the fundamental resultsof the cohomological study of the proper morphisms; moreover, it should be noted that the notion of formalschemes seems indispensable in expressing certain facts about the “theory of modules” (classification problemsof algebraic varieties). The results of §10 will not be used before §3 of Chapter III, and it is recommended toomit their reading until then.

§1. Ane schemes

1.1. The prime spectrum of a ring.I | 80

(1.1.1). Notation. Let A be a (commutative) ring, and M an A-module. In this chapter and the following, we willconstantly use the following notation:

• Spec(A) = set of prime ideals of A, also called the prime spectrum of A; for x ∈ X = Spec(A), it will oftenbe convenient to write jx instead of x . For Spec(A) to be empty, it is necessary and sucient for thering A to be 0.

• Ax = Ajx= (local) ring of fractions S−1A, where S = A− jx .

• mx = jxAjx= maximal ideal of Ax .

• k(x) = Ax/mx = residue eld of Ax , canonically isomorphic to the field of fractions of the integral ringA/jx , with which we identify it.

• f (x) = class of f mod. jx in A/jx ⊂ k(x), for f ∈ A and x ∈ X . We also say that f (x) is the value of f ata point x ∈ Spec(A); the equations f (x) = 0 and f ∈ jx are equivalent.

• Mx = M ⊗A Ax = module of fractions with denominators in A− jx .• r(E) = radical of the ideal of A generated by a subset E of A.• V (E) = set of x ∈ X such that E ⊂ jx (or the set of x ∈ X such that f (x) = 0 for all f ∈ E), for E ⊂ A. Sowe have

(1.1.1.1) r(E) =⋂

x∈V (E)

jx .

77

1. AFFINE SCHEMES 78

• V ( f ) = V ( f ) for f ∈ A.• D( f ) = X − V ( f ) = set of x ∈ X where f (x) 6= 0.

Proposition (1.1.2). — We have the following properties:

(i) V (0) = X , V (1) =∅.(ii) The relation E ⊂ E′ implies V (E) ⊃ V (E′).(iii) For each family (Eλ) of subsets of A, V (

⋃

λ Eλ) = V (∑

λ Eλ) =⋂

λ V (Eλ).(iv) V (EE′) = V (E)∪ V (E′).(v) V (E) = V (r(E)).

Proof. The properties (i), (ii), (iii) are trivial, and (v) follows from (ii) and from equation (1.1.1.1). It isevident that V (EE′) ⊃ V (E)∩ V (E′); conversely, if x 6∈ V (E) and x 6∈ V (E′), then there exists f ∈ E and f ′ ∈ E′

such that f (x) 6= 0 and f ′(x) 6= 0 in k(x), hence f (x) f ′(x) 6= 0, i.e., x 6∈ V (EE′), which proves (iv).

Proposition (1.1.2) shows, among other things, that sets of the form V (E) (where E varies over the subsets ofA) are the closed sets of a topology on X , which we will call the spectral topology 1; unless expressly stated otherwise,we always assume that X = Spec(A) is equipped with the spectral topology.

I | 81(1.1.3). For each subset Y of X , we denote by j(Y ) the set of f ∈ A such that f (y) = 0 for all y ∈ Y ; equivalently,j(Y ) is the intersection of the prime ideals jy for y ∈ Y . It is clear that the relation Y ⊂ Y ′ implies that j(Y ) ⊃ j(Y ′)and that we have

(1.1.3.1) j

⋃

λYλ

=⋂

λj(Yλ)

for each family (Yλ) of subsets of X . Finally we have

(1.1.3.2) j(x) = jx .

Proposition (1.1.4). —

(i) For each subset E of A, we have j(V (E)) = r(E).(ii) For each subset Y of X , V (j(Y )) = Y , the closure of Y in X .

Proof. (i) is an immediate consequence of the definitions and (1.1.1.1); on the other hand, V (j(Y )) is closedand contains Y ; conversely, if Y ⊂ V (E), we have f (y) = 0 for f ∈ E and all y ∈ Y , so E ⊂ j(Y ), V (E) ⊃ V (j(Y )),which proves (ii).

Corollary (1.1.5). — The closed subsets of X = Spec(A) and the ideals of A equal to their radicals (in other words,those that are the intersection of prime ideals) correspond bijectively by the inclusion-reversing maps Y 7→ j(Y ), a 7→ V (a);the union Y1 ∪ Y2 of two closed subsets corresponds to j(Y1)∩ j(Y2), and the intersection of any family (Yλ) of closed subsetscorresponds to the radical of the sum of the j(Yλ).

Corollary (1.1.6). — If A is a Noetherian ring, X = Spec(A) is a Noetherian space.

Note that the converse of this corollary is false, as shown by any non-Noetherian integral ring with a singleprime ideal 6= 0 (for example a nondiscrete valuation ring of rank 1).

As an example of ring A whose spectrum is not a Noetherian space, one can consider the ring C (Y ) ofcontinuous real functions on an infinite compact space Y ; we know that, as a set, Y corresponds to the set ofmaximal ideals of A, and it is easy to see that the topology induced on Y by that of X = Spec(A) is the originaltopology of Y . Since Y is not a Noetherian space, the same is true for X .

Corollary (1.1.7). — For each x ∈ X , the closure of x is the set of y ∈ X such that jx ⊂ jy . For x to be closed, it isnecessary and sucient that jx is maximal.

Corollary (1.1.8). — The space X = Spec(A) is a Kolmogoro space.

Proof. If x and y are two distinct points of X , we have either jx 6⊂ jy or jy 6⊂ jx , so one of the points x , ydoes not belong to the closure of the other.

1The introduction of this topology in algebraic geometry is due to Zariski. So this topology is usually called the “Zariski topology” onX .

1. AFFINE SCHEMES 79

(1.1.9). According to Proposition (1.1.2, iv), for two elements f , g of A, we have

(1.1.9.1) D( f g) = D( f )∩ D(g).

Note also that the equality D( f ) = D(g) means, according to Proposition (1.1.4, i) and Proposition (1.1.2, v),that r( f ) = r(g), or that the minimal prime ideals containing ( f ) and (g) are the same; in particular, it is alsothe case when f = ug, where u is invertible.

Proposition (1.1.10). — I | 82

(i) When f ranges over A, the sets D( f ) forms a basis for the topology of X .(ii) For every f ∈ A, D( f ) is quasi-compact. In particular, X = D(1) is quasi-compact.

Proof.

(i) Let U be an open set in X ; by definition, we have U = X − V (E) where E is a subset of A, andV (E) =

⋂

f ∈E V ( f ), hence U =⋃

f ∈E D( f ).(ii) By (i), it suces to prove that, if ( fλ)λ∈L is a family of elements of A such that D( f ) ⊂

⋃

λ∈L D( fλ), thenthere exists a finite subset J of L such that D( f ) ⊂

⋃

λ∈J D( fλ). Let a be the ideal of A generated bythe fλ ; we have, by hypothesis, that V ( f ) ⊃ V (a), so r( f ) ⊂ r(a); since f ∈ r( f ), there exists an integern ¾ 0 such that f n ∈ a. But then f n belongs to the ideal b generated by the finite subfamily ( fλ)λ∈J ,and we have V ( f ) = V ( f n) ⊃ V (b) =

⋂

λ∈J V ( fλ), that is to say, D( f ) ⊃⋃

λ∈J D( fλ).

Proposition (1.1.11). — For each ideal a of A, Spec(A/a) is canonically identied with the closed subspace V (a) ofSpec(A).

Proof. We know there is a canonical bijective correspondence (respecting the inclusion order structure)between ideals (resp. prime ideals) of A/a and ideals (resp. prime ideals) of A containing a.

Recall that the set N of nilpotent elements of A (the nilradical of A) is an ideal equal to r(0), the intersectionof all the prime ideals of A (0, 1.1.1).

Corollary (1.1.12). — The topological spaces Spec(A) and Spec(A/N) are canonically homeomorphic.

Proposition (1.1.13). — For X = Spec(A) to be irreducible (0, 2.1.1), it is necessary and sucient that the ring A/Nis integral (or, equivalently, that the ideal N is prime).

Proof. By virtue of Corollary (1.1.12), we can restrict to the case where N= 0. If X is reducible, then thereexist two distinct closed subsets Y1 and Y2 of X such that X = Y1 ∪ Y2, so j(X ) = j(Y1)∩ j(Y2) = 0, since the idealsj(Y1) and j(Y2) are distinct from (0) (1.1.5); so A is not integral. Conversely, if there are elements f 6= 0, g 6= 0 ofA such that f g = 0, we have V ( f ) 6= X , V (g) 6= X (since the intersection of all the prime ideals of A is (0)), andX = V ( f g) = V ( f )∪ V (g).

Corollary (1.1.14). —

(i) In the bijective correspondence between closed subsets of X = Spec(A) and ideals of A equal to their radicals, theirreducible closed subsets of X correspond to the prime ideals of A. In particular, the irreducible components of Xcorrespond to the minimal prime ideals of A.

(ii) The map x 7→ x establishes a bijective correspondence between X and the set of closed irreducible subsets of X( in other words, all closed irreducible subsets of X admit exactly one generic point).

Proof. (i) follows immediately from (1.1.13) and (1.1.11); and for proving (ii), we can, by (1.1.11), restrictto the case where X is irreducible; then, according to Proposition (1.1.13), there exists a smaller prime ideal Nin A, which corresponds to the generic point of X ; in addition, X admits at most one generic point since it is a I | 83Kolmogoro space ((1.1.8) and (0, 2.1.3)).

Proposition (1.1.15). — If J is an ideal in A containing the radical N(A), the only neighborhood of V (J) inX = Spec(A) is the whole space X .

Proof. Each maximal ideal of A belongs, by definition, to V (J). As each ideal a 6= A of A is contained in amaximal ideal, we have V (a)∩ V (J) 6= 0, whence the proposition.

1. AFFINE SCHEMES 80

1.2. Functorial properties of prime spectra of rings.

(1.2.1). Let A, A′ be two rings, andφ : A′ −→ A

a homomorphism of rings. For each prime ideal x = jx ∈ Spec(A) = X , the ring A′/φ−1(jx) is canonicallyisomorphic to a subring of A/jx , and so it is integral, or, in other words, φ−1(jx) is a prime ideal of A′; we denoteit by aφ (x), and we have thus defined a map

aφ : X = Spec(A) −→ X ′ = Spec(A′)

(also denoted Spec(φ )), that we call the map associated to the homomorphism φ . We denote by φ x the injectivehomomorphism from A′/φ−1(jx) to A/jx induced by φ by passing to quotients, as well as its canonical extensionto a monomorphism of fields

φ x : k(aφ (x)) −→ k(x);for each f ′ ∈ A′, we therefore have, by definition, ErrII

(1.2.1.1) φ x( f ′(aφ (x))) = (φ ( f ′))(x) (x ∈ X ).

Proposition (1.2.2). —(i) For each subset E′ of A′, we have

(1.2.2.1) aφ−1(V (E′)) = V (φ (E′)),and in particular, for each f ′ ∈ A′,

(1.2.2.2) aφ−1(D( f ′)) = D(φ ( f ′)).(ii) For each ideal a of A, we have

(1.2.2.3) aφ (V (a)) = V (φ−1(a)).

Proof. The relation aφ (x) ∈ V (E′) is, by definition, equivalent to E′ ⊂ φ−1(jx), so φ (E′) ⊂ jx , and finallyx ∈ V (φ (E′)), hence (i). To prove (ii), we can suppose that a is equal to its radical, since V (r(a)) = V (a) (1.1.2, v)and φ−1(r(a)) = r(φ−1(a)); the relation f ′ ∈ a′ is, by definition, equivalent to f ′(x ′) = 0 for each x ∈ aφ (Y ), so,by Equation (1.2.1.1), it is also equivalent to φ ( f ′)(x) = 0 for each x ∈ Y , or to φ ( f ′) ∈ j(Y ) = a, since a is equalto its radical; hence (ii).

Corollary (1.2.3). — The map aφ is continuous.

We remark that, if A′′ is a third ring, and φ ′ a homomorphism A′′→ A′, then we have a(φ ′ φ ) = aφ aφ ′;this result, with Corollary (1.2.3), says that Spec(A) is a contravariant functor in A, from the category of rings tothat of topological spaces.

I | 84Corollary (1.2.4). — Suppose that φ is such that every f ∈ A can be written as f = hφ ( f ′), where h is invertible in A(which, in particular, is the case when φ is surjective). Then aφ is a homeomorphism from X to aφ (X ).

Proof. We show that for each subset E ⊂ A, there exists a subset E′ of A′ such that V (E) = V (φ (E′));according to the (T0) axiom (1.1.8) and the formula (1.2.2.1), this implies first of all that aφ is injective, and then,by (1.2.2.1), that aφ is a homeomorphism. But it suces, for each f ∈ E, to take f ′ ∈ A′ such that hφ ( f ′) = fwith h invertible in A; the set E′ of these elements f ′ is exactly what we are searching for.

(1.2.5). In particular, when φ is the canonical homomorphism from A to a ring quotient A/a, we again get(1.1.12), and aφ is the canonical injection of V (a), identified with Spec(A/a), into X = Spec(A).

Another particular case of (1.2.4):

Corollary (1.2.6). — If S is a multiplicative subset of A, the spectrum Spec(S−1A) is canonically identied (with itstopology) with the subspace of X = Spec(A) consisting of the x such that jx ∩ S =∅.

Proof. We know by (0, 1.2.6) that the prime ideals of S−1A are the ideals S−1jx such that jx ∩ S = ∅, andthat we have jx = (iS

A)−1(S−1jx). It then suces to apply Corollary (1.2.4) to the iS

A .

Corollary (1.2.7). — For aφ (X ) to be dense in X ′, it is necessary and sucient for each element of the kernel Kerφ tobe nilpotent.

Proof. Applying Equation (1.2.2.3) to the ideal a= (0), we have âaφ (X ) = V (Kerφ ), and for V (Kerφ ) = X ′ ErrIIto hold, it is necessary and sucient for Kerφ to be contained in all the prime ideals of A′, or, equivalently, inthe nilradical r′ of A′. ErrII

1. AFFINE SCHEMES 81

1.3. Sheaf associated to a module.

(1.3.1). Let A be a commutative ring, M an A-module, f an element of A, and S f the multiplicative set consistingof the f n, where n¾ 0. Recall that we set A f = S−1

f A, M f = S−1f M . If S′f is the saturated multiplicative subset of

A consisting of the g ∈ A which divide an element of S f , we know that A f and M f are canonically identified withS′f−1A and S′f

−1M (0, 1.4.3).

Lemma (1.3.2). — The following conditions are equivalent:

(a) g ∈ S′f ;(b) S′g ⊂ S′f ;(c) f ∈ r(g);(d) r( f ) ⊂ r(g);(e) V (g) ⊂ V ( f );(f) D( f ) ⊂ D(g).

Proof. This follows immediately from the definitions and (1.1.5).

(1.3.3). If D( f ) = D(g), then Lemma (1.3.2, b) shows that M f = Mg . More generally, if D( f ) ⊃ D(g), thenS′f ⊂ S′g , and we know (0, 1.4.1) that there exists a canonical functorial homomorphism

ρg, f : M f −→ Mg ,

and if D( f ) ⊃ D(g) ⊃ D(h), we have (0, 1.4.4)

(1.3.3.1) ρh,g ρg, f = ρh, f .I | 85

When f ranges over the elements of A− jx (for a given x in X = Spec(A)), the sets S′f constitute an increasingfiltered set of subsets of A− jx , since for elements f and g of A− jx , S′f and S′g are contained in S′f g ; since theunion of the S′f over f ∈ A− jx is A− jx , we conclude (0, 1.4.5) that the Ax -module Mx is canonically identifiedwith the inductive limit lim−→M f , relative to the family of homomorphisms (ρg, f ). We denote by

ρ fx : M f −→ Mx

the canonical homomorphism for f ∈ A− jx (or, equivalently, x ∈ D( f )).

Denition (1.3.4). — We define the structure sheaf of the prime spectrum X = Spec(A) (resp. the sheafassociated to the A-module M), denoted by eA or OX (resp. eM) as the sheaf of rings (resp. the eA-module) associatedto the presheaf D( f ) 7→ A f (resp. D( f ) 7→ M f ), defined on the basis B of X consisting of the D( f ) for f ∈ A((1.1.10), (0, 3.2.1), and (0, 3.5.6)).

We saw (0, 3.2.4) that the stalk eAx (resp. eMx) can be identied with the ring Ax (resp. the Ax -module Mx); wedenote by

θ f : A f −→ Γ(D( f ), eA)(resp. θ f : M f −→ Γ(D( f ), eM)),

the canonical map, so that, for all x ∈ D( f ) and all ξ ∈ M f , we have

(1.3.4.1) (θ f (ξ ))x = ρ fx (ξ ).

Proposition (1.3.5). — eM is an exact functor, covariant in M , from the category of A-modules to the category ofeA-modules.

Proof. Indeed, let M , N be two A-modules, and u a homomorphism M → N ; for each f ∈ A, u correspondscanonically to a homomorphism u f from the A f -module M f to the A f -module N f , and the diagram (forD(g) ⊂ D( f ))

M f

u f //

ρg, f

N f

ρg, f

Mg

ug // Ng

is commutative (1.4.1); these homomorphisms then define a homomorphism of eA-modules eu : eM → eN (0, 3.2.3).In addition, for each x ∈ X , eux is the inductive limit of the u f for x ∈ D( f ) ( f ∈ A), and as a result (0, 1.4.5), ifwe canonically identify eMx and eNx with Mx and Nx respectively, then eux is identified with the homomorphism

1. AFFINE SCHEMES 82

ux canonically induced by u. If P is a third A-module, v a homomorphism N → P, and w= v u, it is immediatethat wx = vx ux , so ew= ev eu. We have therefore clearly defined a covariant (in M ) functor eM , from the categoryof A-modules to that of eA-modules. This functor is exact, since, for each x ∈ X , Mx is an exact functor in M(0, 1.3.2); in addition, we have Supp(M) = Supp( eM), by definition ((0, 1.7.1) and (0, 3.1.6)).

I | 86

Proposition (1.3.6). — For each f ∈ A, the open subset D( f ) ⊂ X is canonically identied with the prime spectrumSpec(A f ), and the sheaf ÝM f associated to the A f -module M f is canonically identied with the restriction eM |D( f ).

Proof. The first assertion is a particular case of (1.2.6). In addition, if g ∈ A is such that D(g) ⊂ D( f ),then Mg is canonically identified with the module of fractions of M f whose denominators are the powers of thecanonical image of g in A f (0, 1.4.6). The canonical identification of ÝM f with eM |D( f ) then follows from thedefinitions.

Theorem (1.3.7). — For each A-module M and each f ∈ A, the homomorphism

θ f : M f −→ Γ(D( f ), eM)

is bijective (in other words, the presheaf D( f ) 7→ M f is a sheaf ). In particular, M can be identied with Γ(X , eM) viaθ1.

Proof. We note that, if M = A, then θ f is a homomorphism of structure rings; Theorem (1.3.7) thenimplies that, if we identify the rings A f and Γ(D( f ), eA) via θ f , the homomorphism θ f : M f → Γ(D( f ), eM) is anisomorphism of modules.

We show first that θ f is injective. Indeed, if ξ ∈ M f is such that θ f (ξ ) = 0, then, for each prime ideal p of A f ,there exists h 6∈ p such that hξ = 0; as the annihilator of ξ is not contained in any prime ideal of A f , each A fintegral, and so ξ = 0.

It remains to show that θ f is surjective; we can restrict to the case where f = 1 (the general case thenfollowing by “localizing”, using (1.3.6)). Now let s be a section of eM over X ; according to (1.3.4) and (1.1.10, ii),there exists a nite cover (D( fi))i∈I of X ( fi ∈ A) such that, for each i ∈ I , the restriction si = s|D( fi) is of theform θ fi

(ξ i), where ξ i ∈ M fi. If i, j are indices of I , and if the restrictions of si and s j to D( fi)∩ D( f j) = D( fi f j)

are equal, then it follows, by definition of M , that

(1.3.7.1) ρ fi f j , fi(ξ i) = ρ fi f j , f j

(ξ j).

By definition, we can write, for each i ∈ I , ξ i = zi/ f nii , where zi ∈ M , and. since I is finite, by multiplying each zi

by a power of fi , we can assume that all the ni are equal to one single n. Then, by definition, (1.3.7.1) impliesthat there exists an integer mi j ¾ 0 such that ( fi f j)mi j ( f n

j zi − f ni z j) = 0, and we can moreover suppose that the

mi j are equal to the one single m; then replacing the zi by f mi zi , it remains to prove the case where m= 0, in

other words, the case where we have

(1.3.7.2) f nj zi = f n

i z j

for any i, j. We have D( f ni ) = D( fi), and since the D( fi) form a cover of X , the ideal generated by the f n

i is A; inother words, there exist elements gi ∈ A such that

∑

i gi f ni = 1. Then consider the element z =

∑

i gizi of M ; in(1.3.7.2), we have f n

i z =∑

j g j f ni z j = (

∑

j g j f nj )zi = zi , where, by definition, ξ i = z/1 in M fi

. We conclude that I | 87the si are the restrictions to D( fi) of θ1(z), which proves that s = θ1(z) and finishes the proof.

Corollary (1.3.8). — Let M and N be A-modules; the canonical homomorphism u 7→ eu from HomA(M , N) toHom

eA( eM , eN) is bijective. In particular, the equations M = 0 and eM = 0 are equivalent.

Proof. Consider the canonical homomorphism v 7→ Γ(v) from HomeA( eM , eN) to HomΓ(eA)(Γ( eM),Γ(eN)); the

latter module is canonically identified with HomA(M , N), by Theorem (1.3.7). It remains to show that u 7→ eu andv 7→ Γ(v) are inverses of each other; it is evident that Γ(eu) = u by definition of eu; on the other hand, if we letu= Γ(v) for v ∈ Hom

eA( eM , eN), then the map w : Γ(D( f ), eM)→ Γ(D( f ), eN) canonically induced from v is suchthat the diagram

M u //

ρ f ,1

N

ρ f ,1

M f

w // N f

is commutative; so we necessarily have that w= u f for all f ∈ A (0, 1.2.4), which shows that ßΓ(v) = v.

1. AFFINE SCHEMES 83

Corollary (1.3.9). —(i) Let u be a homomorphism from an A-module M to an A-module N ; then the sheaves associated to Ker u, Im u,

and Coker u, are Kereu, Imeu, and Cokereu (respectively). In particular, for eu to be injective (resp. surjective,bijective), it is necessary and sucient for u to be so too.

(ii) If M is an inductive limit (resp. direct sum) of a family of A-modules (Mλ), then eM is the inductive limit (resp.direct sum) of the family (ÝMλ), via a canonical isomorphism.

Proof.

(i) It suces to apply the fact that eM is an exact functor in M (1.3.5) to the two exact sequences ofA-modules

0 −→ Ker u −→ M −→ Im u −→ 0,

0 −→ Im u −→ N −→ Coker u −→ 0.

The second claim then follows from Theorem (1.3.7).(ii) Let (Mλ , gµλ) be an inductive system of A-modules, with inductive limit M , and let gλ be the canonical

homomorphism Mλ → M . Since we have Ýgνµ Ýgµλ =Ýgνλ and fgλ =fgµ Ýgµλ for λ ¶ µ¶ ν, it follows that(ÝMλ ,Ýgµλ) is an inductive system of sheaves on X , and if we denote by hλ the canonical homomorphismÝMλ → lim−→

ÝMλ , then there is a unique homomorphism v : lim−→ÝMλ → eM such that v hλ =fgλ . To see

that v is bijective, it suces to check, for each x ∈ X , that vx is a bijection from (lim−→ÝMλ)x to eMx ; but

eMx = Mx , and(lim−→

ÝMλ)x = lim−→(ÝMλ)x = lim−→(Mλ)x = Mx (0, 1.3.3).

Conversely, it follows from the definitions that (fgλ)x and (hλ) are both equal to the canonical mapfrom (Mλ)x to Mx ; since (fgλ)x = vx (hλ)x , vx is the identity. Finally, if M is the direct sum of two I | 88A-modules N and P, it is immediate that eM = eN ⊕ eP; each direct sum being the inductive limit of finitedirect sums, the claims of (ii) are thus proved.

We note that Corollary (1.3.8) proves that the sheaves isomorphic to the associated sheaves of A-modulesform an abelian category (T, I, 1.4).

We also note that Corollary (1.3.9) implies that, if M is an A-module of nite type (that is to say, there existsa surjective homomorphism An→ M) then there exists a surjective homomorphism fAn→ eM , or, in other words,the eA-module eM is generated by a nite family of sections over X (0, 5.1.1), and vice versa.

(1.3.10). If N is a submodule of an A-module M , the canonical injection j : N → M gives, by (1.3.9), an injectivehomomorphism eN → eM , which allows us to canonically identify eN with an eA-submodule of eM ; we will alwaysassume that we have made this identification. If N and P are submodules of M , then we have

(1.3.10.1) (N + P)∼ = eN + eP,

(1.3.10.2) (N ∩ P)∼ = eN ∩ eP,

since N + P and N ∩ P are the image of the canonical homomorphism N ⊕ P → M and the kernel of the canonicalhomomorphism M → (M/N)⊕ (M/P) (respectively), and it suces to apply (1.3.9).

We conclude from (1.3.10.1) and (1.3.10.2) that, if eN = eP, then we have N = P.

Corollary (1.3.11). — On the category of sheaves isomorphic to the associated sheaves of A-modules, the functor Γ isexact.

Proof. Let eMeu−→ eN

ev−→ eP be an exact sequence corresponding to two homomorphisms u : M → N and

v : N → P of A-modules. If Q = Im u and R = Ker v, we have eQ = Imeu = Kerev = eR (Corollary (1.3.9)), henceQ = R.

Corollary (1.3.12). — Let M and N be A-modules.

(i) The sheaf associated to M ⊗A N is canonically identied with eM ⊗eAeN .

(ii) If, in addition, M admits a nite presentation, then the sheaf associated to HomA(M , N) is canonically identiedwith Hom

eA( eM , eN).

Proof.

1. AFFINE SCHEMES 84

(i) The sheaf F = eM ⊗eAeN is associated to the presheaf

U 7−→ F (U) = Γ(U , eM)⊗Γ(U ,eA) Γ(U , eN),

with U varying over the basis (1.1.10, i) of X consisting of the D( f ), where f ∈ A. We know thatF (D( f )) is canonically identified with M f ⊗A f

N f , by (1.3.7) and (1.3.6). Moreover, we know that theA f -module M f ⊗A f

N f is canonically isomorphic to (M ⊗A N) f (0, 1.3.4), which is itself canonicallyisomorphic to Γ(D( f ), (M ⊗A N)∼) (Theorem (1.3.7) and Proposition (1.3.6)). In addition, we seeimmediately that the canonical isomorphisms

F (D( f ))' Γ(D( f ), (M ⊗A N)∼)

thus obtained satisfy the compatibility conditions with respect to the restriction operations (0, 1.4.2), I | 89so they define a canonical functorial isomorphism

eM ⊗eAeN ' (M ⊗A N)∼.

(ii) The sheaf G =HomeA( eM , eN) is associated to the presheaf

U 7−→ G (U) = HomeA|U( eM |U , eN |U),

with U varying over the basis of X consisting of the D( f ). We know that G (D( f )) is canonically identifiedwith HomA f

(M f , N f ) (Proposition (1.3.6) and Corollary (1.3.8)), which, according to the hypotheseson M , is canonically identified with (HomA(M , N)) f (0, 1.3.5). Finally, (HomA(M , N)) f is canonicallyidentified with Γ(D( f ), (HomA(M , N))∼) (Proposition (1.3.6) and Theorem (1.3.7)), and the canonicalisomorphisms G (D( f ))' Γ(D( f ), (HomA(M , N))∼) thus obtained are compatible with the restrictionoperations (0, 1.4.2); they thus define a canonical isomorphism Hom

eA( eM , eN)' (HomA(M , N))∼.

(1.3.13). Now let B be a (commutative) A-algebra; this can be understood by saying that B is an A-module suchthat we have some given element e ∈ B and an A-homomorphism φ : B ⊗A B→ B, so that (a) the diagrams

B ⊗A B ⊗A Bφ⊗1 //

1⊗φ

B ⊗A B

φ

B ⊗A Bσ //

φ""

B ⊗A B

φ||

B ⊗A Bφ // B B

(σ being the canonical symmetry map) are commutative; and (b) φ (e⊗ x) =φ (x ⊗ e) = x . By Corollary (1.3.12),the homomorphism eφ : eB⊗

eAeB→ eB of eA-modules satisfies the analogous conditions, and so it defines an eA-algebra

structure on eB. In a similar way, the data of a B-module N is the same as the data of an A-module N and anA-homomorphism ψ : B ⊗A N → N such that the diagram

B ⊗A B ⊗A Nφ⊗1 //

1⊗ψ

B ⊗A N

ψ

B ⊗A Nψ // N

is commutative and ψ(e⊗ n) = n; the homomorphism eψ : eB ⊗eAeN → eN satisfies the analogous condition, and so

defines a eB-module structure on eN .In a similar way, we see that if u : B → B′ (resp. v : N → N ′) is a homomorphism of A-algebras (resp. of

B-modules), then eu (resp. ev) is a homomorphism of eA-algebras (resp. of eB-modules), and Kereu is a eB-ideal (resp.Kerev, Cokerev, and Imev are eB-modules). If N is a B-module, then eN is a eB-module of finite type if and only if Nis a B-module of finite type (0, 5.2.3). I | 90

If M , N are B-modules, then the eB-module eM ⊗eBeN is canonically identified with (M ⊗B N)∼; similarly

HomeB( eM , eN) is canonically identified with (HomB(M , N))∼ whenever M admits a finite presentation; the proofs

are similar to those for Corollary (1.3.12).If J is an ideal of B, and N is a B-module, then we have (JN)∼ = eJ · eN .Finally, if B is an A-algebra graded by the A-submodules Bn (n ∈ Z), then the eA-algebra eB, the direct sum

of the eA-modules fBn (1.3.9), is graded by these eA-submodules, the axiom of graded algebras saying that theimage of the homomorphism Bm ⊗ Bn → B is contained in Bm+n. Similarly, if M is a B-module graded by thesubmodules Mn, then eM is a eB-module graded by the ÝMn.

1. AFFINE SCHEMES 85

(1.3.14). If B is an A-algebra, and M a submodule of B, then the eA-subalgebra of eB generated by eM (0, 4.1.3) isthe eA-subalgebra eC , where we denote by C the subalgebra of B generated by M . Indeed, C is the direct sum ofthe submodules of B which are the images of the homomorphisms

⊗n M → B (n ¾ 0), so it suces to apply(1.3.9) and (1.3.12).

1.4. Quasi-coherent sheaves over a prime spectrum.

Theorem (1.4.1). — Let X be the prime spectrum of a ring A, V a quasi-compact open subset of X , and F an(OX |V )-module. The following four conditions are equivalent.

(a) There exists an A-module M such that F is isomorphic to eM |V .(b) There exists a nite open cover (Vi) of V by sets of the form D( fi) ( fi ∈ A) contained in V , such that, for each i,F|Vi is isomorphic to a sheaf of the form fMi , where Mi is an A fi

-module.(c) The sheaf F is quasi-coherent (0, 5.1.3).(d) The two following properties are satised:

(d1) For each f ∈ A such that D( f ) ⊂ V , and for each section s ∈ Γ(D( f ),F ), there exists an integer n ¾ 0such that f ns extends to a section of F over V .

(d2) For each f ∈ A such that D( f ) ⊂ V and for each section t ∈ Γ(V,F ) such that the restriction of t to D( f )is 0, there exists an integer n¾ 0 such that f n t = 0.

(In the statement of the conditions (d1) and (d2), we have tacitly identified A and Γ(eA) using Theorem (1.3.7)).

Proof. The fact that (a) implies (b) is an immediate consequence of Proposition (1.3.6) and the fact thatthe D( fi) form a basis for the topology of X (1.1.10). As each A-module is isomorphic to the cokernel of ahomomorphism of the form A(I)→ A(J), (1.3.6) implies that each sheaf associated to an A-module is quasi-coherent;so (b) implies (c). Conversely, if F is quasi-coherent, each x ∈ V has a neighborhood of the form D( f ) ⊂ V such

that F|D( f ) is isomorphic to the cokernel of a homomorphism fA f(I)→fA f

(J), so also to the sheaf eN associated

to the module N , the cokernel of the corresponding homomorphism A(I)f → A(J)f (Corollaries (1.3.8) and (1.3.9));since V is quasi-compact, it is clear that (c) implies (b). I | 91

To prove that (b) implies (d1) and (d2), we first assume that V = D(g) for some g ∈ A, and that F isisomorphic to the sheaf eN associated to an Ag -module N ; by replacing X with V and A with Ag (1.3.6), wecan reduce to the case where g = 1. Then Γ(D( f ), eN) and N f are canonically identified with one another(Proposition (1.3.6) and Theorem (1.3.7)), so a section s ∈ Γ(D( f ), eN) is identified with an element of the formz/ f n, where z ∈ N ; the section f ns is identified with the element z/1 of N f and, as a result, is the restrictionto D( f ) of the section of eN over X that is identified with the element z ∈ N ; hence (d1) in this case. Similarly,t ∈ Γ(X , eN) is identified with an element z′ ∈ N , the restriction of t to D( f ) is identified with the image z′/1of z′ in N f , and to say that this image is zero means that there exists some n ¾ 0 such that f nz′ = 0 in N , or,equivalently, f n t = 0.

To finish the proof, that (b) implies (d1) and (d2), it suces to establish the following lemma.

Lemma (1.4.1.1). — Suppose that V is the nite union of sets of the form D(gi), and that all of the sheaves F|D(gi)and F|(D(gi)∩ D(g j)) =F|D(gi g j) satisfy (d1) and (d2); then F has the following two properties:

(d’1) For each f ∈ A and for each section s ∈ Γ(D( f )∩ V,F ), there exists an integer n¾ 0 such that f ns extends toa section of F over V .

(d’2) For each f ∈ A and for each section t ∈ Γ(V,F ) such that the restriction of t to D( f )∩ V is 0, there exists aninteger n¾ 0 such that f n t = 0.

We first prove (d′2): since D( f ) ∩ D(gi) = D( f gi), there exists, for each i, an integer ni such that therestriction of ( f gi)ni t to D(gi) is zero: since the image of gi in Agi

is invertible, the restriction of f ni t to D(gi) isalso zero; taking n to be the largest of the ni , we have proved (d′2).

To show (d′1), we apply (d1) to the sheaf F|D(gi): there exists an integer ni ¾ 0 and a section s′i of Fover D(gi) extending the restriction of ( f gi)ni s to D( f gi); since the image of gi in Agi

is invertible, there isa section si of F over D(gi) such that s′i = gni

i si , and si extends the restriction of f ni s to D( f gi); in additionwe can suppose that all the ni are equal to a single integer n. By construction, the restriction of si − s j toD( f )∩ D(gi)∩ D(g j) = D( f gi g j) is zero; by (d2) applied to the sheaf F|D(gi g j), there exists an integer mi j ¾ 0such that the restriction to D(gi g j) of ( f gi g j)mi j (si − s j) is zero; since the image of gi g j in Agi g j

is invertible, therestriction of f mi j (si − s j) to D(gi g j) is zero. We can then assume that all the mi j are equal to a single integer m,and so there exists a section s′ ∈ Γ(V,F ) extending the f msi ; as a result, this section extends f n+ms, hence wehave proved (d′1).

1. AFFINE SCHEMES 86

It remains to show that (d1) and (d2) imply (a). We first show that (d1) and (d2) imply that these conditionsare satisfied for each sheaf F|D(g), where g ∈ A is such that D(g) ⊂ V . It is evident for (d1); on the other hand,if t ∈ Γ(D(g),F ) is such that its restriction to D( f ) ⊂ D(g) is zero, there exists, by (d1), an integer m¾ 0 suchthat gm t extends to a section s of F over V ; applying (d2), we see that there exists an integer n¾ 0 such that I | 92f n gm t = 0, and as the image of g in Ag is invertible, f n t = 0.

That being so, since V is quasi-compact, Lemma (1.4.1.1) proves that the conditions (d′1) and (d′2) aresatisfied. Consider then the A-module M = Γ(V,F ), and define a homomorphism of eA-modules u : eM → j∗(F ),where j is the canonical injection V → X . Since the D( f ) form a basis for the topology of X , it suces, for eachf ∈ A, to define a homomorphism u f : M f → Γ(D( f ), j∗(F )) = Γ(D( f ) ∩ V,F ), with the usual compatibilityconditions (0, 3.2.5). Since the canonical image of f in A f is invertible, the restriction homomorphism

M = Γ(V,F )→ Γ(D( f )∩ V,F ) factors as M → M f

u f−→ Γ(D( f )∩ V,F ) (0, 1.2.4), and the verification of these

compatibility conditions for D(g) ⊂ D( f ) is immediate. This being so, we show that the condition (d′1) (resp.(d′2)) implies that each of the u f are surjective (resp. injective), which proves that u is bijective, and as a resultthat F is the restriction to V of an eA-module isomorphic to eM . If s ∈ Γ(D( f )∩ V,F ), there exists, by (d′1), aninteger n¾ 0 such that f ns extends to a section z ∈ M ; we then have u f (z/ f n) = s, so u f is surjective. Similarly,if z ∈ M is such that u f (z/1) = 0, this means that the restriction to D( f )∩ V of the section z is zero; accordingto (d′2), there exists an integer n¾ 0 such that f nz = 0, hence z/1= 0 in M f , and so u f is injective.

Corollary (1.4.2). — Each quasi-coherent sheaf over a quasi-compact open subset of X is induced by a quasi-coherentsheaf on X .

Corollary (1.4.3). — Every quasi-coherent OX -algebra over X = Spec(A) is isomorphic to an OX -algebra of the formeB, where B is an algebra over A; every quasi-coherent eB-module is isomorphic to a eB-module of the form eN , where N is aB-module.

Proof. Indeed, a quasi-coherent OX -algebra is a quasi-coherent OX -module, and therefore of the form eB,where B is an A-module; the fact that B is an A-algebra follows from the characterization of the structure ofan OX -algebra using the homomorphism eB ⊗

eAeB → eB of eA-modules, as well as Corollary (1.3.12). If G is a

quasi-coherent eB-module, it suces to show, in a similar way, that it is also a quasi-coherent eA-module to concludethe proof; since the question is local, we can, by restricting to an open subset of X of the form D( f ), assume thatG is the cokernel of a homomorphism eB(I)→ eB(J) of eB-modules (and a fortiori of eA-modules); the propositionthen follows from Corollaries (1.3.8) and (1.3.9).

1.5. Coherent sheaves over a prime spectrum.

Theorem (1.5.1). — Let A be a Noetherian ring, X = Spec(A) its prime spectrum, V an open subset of X , and F an(OX |V )-module. The following conditions are equivalent.

(a) F is coherent.(b) F is of nite type and quasi-coherent.(c) There exists an A-module M of nite type such that F is isomorphic to the sheaf eM |V .

I | 93Proof. (a) trivially implies (b). To see that (b) implies (c), note that, since V is quasi-compact (0, 2.2.3), we

have previously seen that F is isomorphic to a sheaf eN |V , where N is an A-module (1.4.1). We have N = lim−→Mλ ,

where Mλ run over the set of A-submodules of N of finite type, hence (1.3.9) F = eN |V = lim−→ÝMλ |V ; but since F

is of finite type, and V is quasi-compact, there exists an index λ such that F =ÝMλ |V (0, 5.2.3).Finally, we show that (c) implies (a). It is clear that F is then of finite type ((1.3.6) and (1.3.9)); in addition,

the question being local, we can restrict to the case where V = D( f ), f ∈ A. Since A f is Noetherian, we see thatit suces to prove that the kernel of a homomorphism fAn→ eM , where M is an A-module, is of finite type. Butsuch a homomorphism is of the form eu, where u is a homomorphism An→ M (1.3.8), and if P = Ker u then wehave eP = Kereu (1.3.9). Since A is Noetherian, P is of finite type, which finishes the proof.

Corollary (1.5.2). — Under the hypotheses of (1.5.1), the sheaf OX is a quasi-coherent sheaf of rings.

Corollary (1.5.3). — Under the hypotheses of (1.5.1), every coherent sheaf over an open subset of X is induced by acoherent sheaf on X .

Corollary (1.5.4). — Under the hypotheses of (1.5.1), every quasi-coherent OX -module F is the inductive limit of thecoherent OX -submodules of F .

Proof. Indeed, F = eM , where M is an A-module, and M is the inductive limit of its submodules of finitetype; we conclude the proof by appealing to (1.3.9) and (1.5.1).

1. AFFINE SCHEMES 87

1.6. Functorial properties of quasi-coherent sheaves over a prime spectrum.

(1.6.1). Let A, A′ be rings,φ : A′ −→ A

a homomorphism, andaφ : X = Spec(A) −→ X ′ = Spec(A′)

the continuous map associated to φ (1.2.1). We will define a canonical homomorphism

eφ : OX ′ −→ aφ∗(OX )

of sheaves of rings. For each f ′ ∈ A′, we put f = φ ( f ′); we have aφ−1(D( f ′)) = D( f ) (1.2.2.2). The ringsΓ(D( f ′), eA′) and Γ(D( f ), eA) are identified with A′f ′ and A f (respectively) ((1.3.6) and (1.3.7)). The homomorphismφ canonically defines a homomorphism φ f ′ : A′f ′ → A f (0, 1.5.1), in other words, we have a homomorphism ofrings

Γ(D( f ), eA′) −→ Γ(aφ−1(D( f ′)), eA) = Γ(D( f ′), aφ∗(eA)).In addition, these homomorphisms satisfy the usual compatibility conditions: for D( f ′) ⊃ D(g ′), the diagram I | 94

Γ(D( f ′), eA′) //

Γ(D( f ′), aφ∗(eA))

Γ(D(g ′), eA′) // Γ(D(g ′), aφ∗(eA)

is commutative (0, 1.5.1); we have thus defined a homomorphism of OX ′ -algebras, as the D( f ′) form a basis forthe topology of X ′ (0, 3.2.3). The pair Φ = (aφ , eφ ) is thus a morphism of ringed spaces

Φ : (X ,OX ) −→ (X ′,OX ′),

(0, 4.1.1).We also note that, if we put x ′ = aφ (x), then the homomorphism eφ ]x (0, 3.7.1) is exactly the homomorphism

φx : A′x ′ −→ Ax

canonically induced by φ : A′→ A (0, 1.5.1). Indeed, each z′ ∈ A′x ′ can be written as g ′/ f ′, where f ′, g ′ are in A′

and f ′ 6∈ jx ′ ; D( f ′) is then a neighborhood of x ′ in X ′, and the homomorphism Γ(D( f ′), eA′)→ Γ(aφ−1(D( f ′)), eA)induced by eφ is exactly φ f ′ ; by considering the section s′ ∈ Γ(D( f ′), eA′) corresponding to g ′/ f ′ ∈ A′f ′ , we obtaineφ ]x(z

′) =φ (g ′)/φ ( f ′) in Ax .

Example (1.6.2). — Let S be a multiplicative subset of A, and φ the canonical homomorphism A→ S−1A;we have already seen (1.2.6) that aφ is a homeomorphism from Y = Spec(S−1A) to the subspace of X = Spec(A)consisting of the x such that jx ∩S =∅. In addition, for each x in this subspace, which is thus of the form aφ (y)with y ∈ Y , the homomorphism eφ ]y : Ox →Oy is bijective (0, 1.2.6); in other words, OY is identified with the sheafon Y induced by OX .

Proposition (1.6.3). — For every A-module M , there exists a canonical functorial isomorphism from the OX ′ -module(M[φ ])∼ to the direct image Φ∗( eM).

Proof. For purposes of abbreviation, we write M ′ = M[φ ], and for each f ′ ∈ A′, we put f = φ ( f ′). Themodules of sections Γ(D( f ′),ÝM ′) and Γ(D( f ), eM) are identified, respectively, with the modules M ′f ′ and M f (overA′f ′ and A f , respectively); in addition, the A′f ′ -module (M f )[φ f ′ ] is canonically isomorphic to M ′f ′ (0, 1.5.2). We

thus have a functorial isomorphism of Γ(D( f ′), eA′)-modules: Γ(D( f ′),ÝM ′) ' Γ(aφ−1(D( f ′)), eM)[φ f ′ ] and theseisomorphisms satisfy the usual compatibility conditions with the restrictions (0, 1.5.6), thus defining the desiredfunctorial isomorphism. We note that, in a precise way, if u : M1→ M2 is a homomorphism of A-modules, it canbe considered as a homomorphism (M1)[φ ]→ (M2)[φ ] of A′-modules; if we denote this homomorphism by u[φ ],then Φ∗(eu) is identified with (u[φ ])∼.

This proof also shows that, for each A-algebra B, the canonical functorial isomorphism (B[φ ])∼ ' Φ∗(eB) is an I | 95isomorphism of OX ′ -algebras; if M is a B-module, the canonical functorial isomorphism (M[φ ])∼ ' Φ∗( eM) is anisomorphism of Φ∗(eB)-modules.

Corollary (1.6.4). — The direct image functor Φ∗ is exact on the category of quasi-coherent OX -modules.

Proof. Indeed, it is clear that M[φ ] is an exact functor in M and ÝM ′ is an exact functor in M ′ (1.3.5).

1. AFFINE SCHEMES 88

Proposition (1.6.5). — Let N ′ be an A′-module, and N the A-module N ′⊗A′A[φ ]; then there exists a canonical functorialisomorphism from the OX -module Φ∗(fN ′) to eN .

Proof. We first remark that j : z′ 7→ z′⊗1 is an A′-homomorphism from N ′ to N[φ ]: indeed, by definition, forf ′ ∈ A′, we have ( f ′z′)⊗ 1 = z′ ⊗φ ( f ′) = φ ( f ′)(z′ ⊗ 1). We have (1.3.8) a homomorphism ej : fN ′ → (N[φ ])∼ ofOX ′ -modules, and, thanks to (1.6.3), we can consider ej as mapping fN ′ to Φ∗(eN). There canonically correspondsto this homomorphism ej a homomorphism h= ej] from Φ∗(fN ′) to eN (0, 4.4.3); we will see that, for each stalk,hx is bijective. Put x ′ = aφ (x) and let f ′ ∈ A′ be such that x ′ ∈ D( f ′); let f = φ ( f ′). The ring Γ(D( f ), eA) isidentified with A f , the modules Γ(D( f ), eN) and Γ(D( f ′),fN ′) with N f and N ′f ′ (respectively); let s ∈ Γ(D( f ′),fN ′),identified with n′/ f ′p (n′ ∈ N ′), and s be its image under ej in Γ(D( f ), eN); s is identified with (n′ ⊗ 1)/ f p. Onthe other hand, let t ∈ Γ(D( f ), eA), identified with g/ f q (g ∈ A); then, by definition, we have hx(s′x ⊗ t x) = t x · sx(0, 4.4.3). But we can canonically identify N f with N ′f ′ ⊗A′

f ′(A f )[φ f ′ ] (0, 1.5.4); s then corresponds to the element

(n′/ f ′p)⊗ 1, and the section y 7→ t y · sy with (n′/ f ′p)⊗ (g/ f q). The compatibility diagram of (0, 1.5.6) showthat hx is exactly the canonical isomorphism

(1.6.5.1) N ′x ′ ⊗A′x′(Ax)[φx′ ] ' Nx = (N

′ ⊗A′ A[φ ])x .

In addition, let v : N ′1→ N ′2 be a homomorphism of A′-modules; since evx ′ = vx ′ for each x ′ ∈ X ′, it followsimmediately from the above that Φ∗(ev) is canonically identified with (v ⊗ 1)∼, which finishes the proof of(1.6.5).

If B′ is an A′-algebra, the canonical isomorphism from Φ∗( eB′) to (B′ ⊗A′ A[φ ])∼ is an isomorphism of OX -algebras; if, in addition, N ′ is a B′-module, then the canonical isomorphism from Φ∗(fN ′) to (N ′ ⊗A′ A[φ ])∼ is anisomorphism of Φ∗( eB′)-modules.

Corollary (1.6.6). — The sections of Φ∗(fN ′), the canonical images of the sections s′, where s′ varies over the A′-moduleΓ(fN ′), generate the A-module Γ(Φ∗(N ′)).

Proof. Indeed, these images are identified with the elements z′ ⊗ 1 of N , when we identify N ′ and N withΓ(fN ′) and Γ(eN) (respectively) (1.3.7), and z′ varies over N ′.

(1.6.7). In the proof of (1.6.5), we had proved in passing that the canonical map (0, 4.4.3.2) ρ : fN ′→ Φ∗(Φ∗(fN ′))is exactly the homomorphism ej, where j : N ′ → N ′ ⊗A′ A[φ ] is the homomorphism z′ 7→ z′ ⊗ 1. Similarly, the I | 96canonical map (0, 4.4.3.3) σ : Φ∗(Φ∗( eM)) → eM is exactly ep, where p : M[φ ] ⊗A′ A[φ ] → M is the canonicalhomomorphism, which sends each tensor product z⊗ a (z ∈ M , a ∈ A) to a · z; this follows immediately from thedefinitions ((0, 3.7.1), (0, 4.4.3), and (1.3.7)).

We conclude ((0, 4.4.3) and (0, 3.5.4.4)) that if v : N ′→ M[φ ] is an A′-homomorphism, then ev] = (v ⊗ 1)∼.

(1.6.8). Let N ′1 and N ′2 be A′-modules, and assume N ′1 admits a nite presentation; it then follows from (1.6.7) and(1.3.12, ii) that the canonical homomorphism (0, 4.4.6)

Φ∗(HomeA′(fN

′1,fN ′2)) −→HomeA(Φ

∗(fN ′1),Φ∗(fN ′2))

is exactly eγ , where γ denotes the canonical homomorphism of A-modules HomA′(N ′1, N ′2)⊗A′ A→ HomA(N ′1 ⊗A′

A, N ′2 ⊗A′ A).

(1.6.9). Let J′ be an ideal of A′, and M an A-module; since, by definition, eJ′ eM is the image of the canonicalhomomorphism Φ∗(eJ′) ⊗

eAeM → eM , it follows from Proposition (1.6.5) and Corollary (1.3.12, i) that eJ′ eM

canonically identifies with (J′M)∼; in particular, Φ∗(eJ′)eA is identified with (J′A)∼, and, taking the right exactnessof the functor Φ∗ into account, the eA-algebra Φ∗((A′/J′)∼) is identified with (A/J′A)∼.

(1.6.10). Let A′′ be a third ring, φ ′ a homomorphism A′′ → A′, and write φ ′′ = φ φ ′. It follows immediatelyfrom the definitions that aφ ′′ = (aφ ′) (aφ ), and fφ ′′ = eφ eφ ′ (0, 1.5.7). We conclude that Φ′′ = Φ′ Φ; in otherwords, (Spec(A), eA) is a functor from the category of rings to that of ringed spaces.

1.7. Characterization of morphisms of ane schemes.

Denition (1.7.1). — We say that a ringed space (X ,OX ) is an ane scheme if it is isomorphic to a ringedspace of the form (Spec(A), eA), where A is a ring; we then say that Γ(X ,OX ), which is canonically identified withthe ring A (1.3.7), is the ring of the ane scheme (X ,OX ), and we denote it by A(X ) when there is no chance ofconfusion.

1. AFFINE SCHEMES 89

By abuse of language, when we speak of an ane scheme Spec(A); it will always be the ringed space(Spec(A), eA).

(1.7.2). Let A and B be rings, and (X ,OX ) and (Y,OY ) the ane schemes corresponding to the prime spectraX = Spec(A), Y = Spec(B). We have seen (1.6.1) that each ring homomorphism φ : B → A corresponds to amorphism Φ = (aφ , eφ ) = Spec(φ ) : (X ,OX )→ (Y,OY ). We note that φ is entirely determined by Φ, since we have,by definition, φ = Γ(eφ ) : Γ(eB)→ Γ(aφ∗(eA) = Γ(eA).

Theorem (1.7.3). — 2 Let (X ,OX ) (Y,OY ) be ane schemes. For a morphism of ringed spaces (ψ,θ) : (X ,OX )→ (Y,OY )to be of the form (aφ , eφ ), where φ is a homomorphism of rings A(Y )→ A(X ), it is necessary and sucient that, for eachx ∈ X , θ]x is a local homomorphism: Oψ(x)→Ox .

I | 97Proof. Let A= A(X ), B = A(Y ). The condition is necessary, since we saw (1.6.1) that eφ ]x is the homomorphism

from Baφ (x) to Ax canonically induced by φ , and, by definition, of aφ (x) =φ−1(jx), this homomorphism is local.We now prove that the condition is sucient. By definition, θ is a homomorphism OY → ψ∗(OX ), and we

canonically obtain a ring homomorphism

φ = Γ(θ) : B = Γ(Y,OY ) −→ Γ(Y,ψ∗(OX )) = Γ(X ,OX ) = A.

The hypotheses on θ]x mean that this homomorphism induces, by passing to quotients, a monomorphismθ x from the residue field k(ψ(x)) to the residue field k(x), such that, for each section f ∈ Γ(Y,OY ) = B, we haveθ x( f (ψ(x))) = φ ( f )(x). The relation f (ψ(x)) = 0 is therefore equivalent to φ ( f )(x) = 0, which means thatjψ(x) = jaφ (x), and we now write ψ(x) = aφ (x) for each x ∈ X , or ψ = aφ . We also know that the diagram

B = Γ(Y,OY )φ //

Γ(X ,OX ) = A

Bψ(x)

θ]x // Ax

is commutative (0, 3.7.2), which means that θ]x is equal to the homomorphism φx : Bψ(x) → Ax canonicallyinduced by φ (0, 1.5.1). As the data of the θ]x completely characterize θ], and as a result θ (0, 3.7.1), we concludethat we have θ = eφ , by the definition of eφ (1.6.1).

We say that a morphism (ψ,θ) of ringed spaces satisfying the condition of (1.7.3) is a morphism of aneschemes.

Corollary (1.7.4). — If (X ,OX ) and (Y,OY ) are ane schemes, there exists a canonical isomorphism from the set ofmorphisms of ane schemes Hom((X ,OX ), (Y,OY )) to the set of ring homomorphisms from B to A, where A= Γ(OX ) andB = Γ(OY ).

Furthermore, we can say that the functors (Spec(A), eA) in A and Γ(X ,OX ) in (X ,OX ) define an equivalencebetween the category of commutative rings and the opposite category of ane schemes (T, I, 1.2).

Corollary (1.7.5). — If φ : B→ A is surjective, then the corresponding morphism (aφ , eφ ) is a monomorphism of ringedspaces (cf. (4.1.7)).

Proof. Indeed, we know that aφ is injective (1.2.5), and, since φ is surjective, for each x ∈ X , φ ]x : Baφ (x)→ Ax ,which is induced by φ by passing to rings of fractions, is also surjective (0, 1.5.1); hence the conclusion(0, 4.1.1).

II | 2171.8. Morphisms from locally ringed spaces to ane schemes. Due to a remark by J. Tate, the statements

of Theorem (1.7.3) and Proposition (2.2.4) can be generalized as follows:3

Proposition (1.8.1). — Let (S,OS) be an ane scheme, and (X ,OX ) a locally ringed space. Then there is a canonicalbijection from the set of ring homomorphisms Γ(S,OS) → Γ(X ,OX ) to the set of morphisms of ringed spaces (ψ,θ) : II | 218(X ,OX )→ (S,OS) such that, for each x ∈ X , θ]x is a local homomorphism Oψ(x)→Ox .

Proof. We note first that if (X ,OX ) and (S,OS) are any two ringed spaces, then a morphism (ψ,θ) from(X ,OX ) to (S,OS) canonically defines a ring homomorphism Γ(θ) : Γ(S,OS)→ Γ(X ,OX ), hence a first map

(1.8.1.1) ρ : Hom((X ,OX ), (S,OS)) −→ Hom(Γ(S,OS),Γ(X ,OX )).

2[Trans.] See (1.8) and the footnote there.3[Trans.] The following section (I.1.8) was added in the errata of EGA II, hence the temporary change in page numbers, which refer to EGA II.

1. AFFINE SCHEMES 90

Conversely, under the stated hypotheses, we set A = Γ(S,OS), and consider a ring homomorphism φ : A→Γ(X ,OX ). For each x ∈ X , it is clear that the set of the f ∈ A such that φ ( f )(x) = 0 is a prime ideal of A, sinceOx/mx = k(x) is a field; it is therefore an element of S = Spec(A), which we denote by aφ (x). In addition, foreach f ∈ A, we have, by definition (0, 5.5.2), that aφ−1(D( f )) = X f , which proves that aφ is a continuous mapX → S. We then define a homomorphism

eφ : OS −→ aφ∗(OX )

of OS -modules; for each f ∈ A, we have Γ(D( f ),OS) = A f (1.3.6); for each s ∈ A, we associate to s/ f ∈ A f theelement (φ (s)|X f )(φ ( f )|X f )−1 of Γ(X f ,OX ) = Γ(D( f ), aφ∗(OX )), and we immediately see (by passing from D( f )to D( f g)) that this is a well-defined homomorphism of OS -modules, hence a morphism (aφ , eφ ) of ringed spaces.In addition, with the same notation, and setting y = aφ (x) for brevity, we immediately see (0, 3.7.1) that we haveeφ ]x(sy/ f y) = (φ (s)x)(φ ( f )x)−1; since the relation sy ∈ my is, by definition, equivalent to φ (s)x ∈ mx , we see thateφ ]x is a local homomorphism Oy →Ox , and we have thus defined a second map σ : Hom(Γ(S,OS),Γ(X ,OX ))→ L,where L is the set of the morphisms (ψ,θ) : (X ,OX )→ (S,OS) such that θ]x is local for each x ∈ X . It remains toprove that σ and ρ (restricted to L) are inverses of each other; the definition of eφ immediately shows that Γ(eφ ) =φ ,and, as a result, that ρ σ is the identity. To see that σ ρ is the identity, start with a morphism (ψ,θ) ∈ L andlet φ = Γ(θ); the hypotheses on θ]x mean that this morphism induces, by passing to quotients, a monomorphismθ x : k(ψ(x))→ k(x) such that for each section f ∈ A= Γ(S,OS), we have θ x( f (ψ(x))) =φ ( f )(x); the equationf (φ (x)) = 0 is therefore equivalent to φ ( f )(x) = 0, which proves that aφ = ψ. On the other hand, the definitionsimply that the diagram

Aφ //

Γ(X ,OX )

Aψ(x)

θ]x // Ox

is commutative, and it is the same for the analogous diagram where θ]x is replaced by eφ ]x , hence eφ]x = θ

]x (0, 1.2.4),

and, as a result, eφ = θ.

(1.8.2). When (X ,OX ) and (Y,OY ) are locally ringed spaces, we will consider the morphisms (ψ,θ) : (X ,OX )→(Y,OY ) such that, for each x ∈ X , θ]x is a local homomorphism Oψ(x) → Ox . Henceforth when we speak of a II | 219morphism of locally ringed spaces, it will always be a morphism like the above; with this definition of morphisms, itis clear that the locally ringed spaces form a category; for any two objects X and Y of this category, Hom(X , Y )thus denotes the set of morphisms of locally ringed spaces from X to Y (the set denoted L in (1.8.1)); whenwe consider the set of morphisms of ringed spaces from X to Y , we will denote it by Homrs(X , Y ) to avoid anyconfusion. The map (1.8.1.1) is then written as

(1.8.2.1) ρ : Homrs(X , Y ) −→ Hom(Γ(Y,OY ),Γ(X ,OX ))

and its restriction

(1.8.2.2) ρ′ : Hom(X , Y ) −→ Hom(Γ(Y,OY ),Γ(X ,OX ))

is a functorial map in X and Y on the category of locally ringed spaces.

Corollary (1.8.3). — Let (Y,OY ) be a locally ringed space. For Y to be an ane scheme, it is necessary and sucientthat, for each locally ringed space (X ,OX ), the map (1.8.2.2) be bijective.

Proof. Proposition (1.8.1) shows that the condition is necessary. Conversely, if we suppose that the conditionis satisfied, and if we put A= Γ(Y,OY ), then it follows from the hypotheses and from (1.8.1) that the functorsX 7→ Hom(X , Y ) and X 7→ Hom(X , Spec(A)), from the category of locally ringed spaces to that of sets, areisomorphic. We know that this implies the existence of a canonical isomorphism X → Spec(A) (cf. 0, 8).

(1.8.4). Let S = Spec(A) be an ane scheme; denote by (S′, A′) the ringed space whose underlying space is apoint and the structure sheaf A′ is the (necessarily simple) sheaf on S′ defined by the ring A. Let π : S→ S′ bethe unique map from S to S′; on the other hand, we note that, for each open subset U of S, we have a canonicalmap Γ(S′, A′) = Γ(S,OS)→ Γ(U ,OS) which thus defines a π-morphism ι : A′→ OS of sheaves of rings. We havethus canonically defined a morphism of ringed spaces i = (π, ι) : (S,OS)→ (S′, A′). For each A-module M , we denoteby M ′ the simple sheaf on S′ defined by M , which is evidently an A′-module. It is clear that i∗( eM) = M ′ (1.3.7).

1. AFFINE SCHEMES 91

Lemma (1.8.5). — With the notation of (1.8.4), for each A-module M , the canonical functorial OS -homomorphism(0, 4.3.3)

(1.8.5.1) i∗(i∗( eM)) −→ eM

is an isomorphism.

Proof. Indeed, the two parts of (1.8.5.1) are right exact (the functor M 7→ i∗( eM) evidently being exact) andcommute with direct sums; by considering M as the cokernel of a homomorphism A(I)→ A(J), we can reduce toproving the lemma for the case where M = A, and it is evident in this case.

Corollary (1.8.6). — Let (X ,OX ) be a ringed space, and u : X → S a morphism of ringed spaces. For each A-module II | 220M , we have (with the notation of (1.8.4)) a canonical functorial isomorphism of OX -modules

(1.8.6.1) u∗( eM)' u∗(i∗(M ′)).

Corollary (1.8.7). — Under the hypotheses of (1.8.6), we have, for each A-module M and each OX -module F , acanonical isomorphism, functorial in M and F ,

(1.8.7.1) HomOS( eM , u∗(F ))' HomA(M ,Γ(X ,F )).

Proof. We have, according to (0, 4.4.3) and Lemma (1.8.5), a canonical isomorphism of bifunctors

HomOS( eM , u∗(F ))' HomA′(M

′, i∗(u∗(F )))

and it is clear that the right hand side is exactly HomA(M ,Γ(X ,F )). We note that the canonical homomorphism(1.8.7.1) sends each OS -homomorphism h : eM → u∗(F ) (in other words, each u-morphism eM → F ) to theA-homomorphism Γ(h) : M → Γ(S, u∗(F )) = Γ(X ,F ).

(1.8.8). With the notation of (1.8.4), it is clear (0, 4.1.1) that each morphism of ringed spaces (ψ,θ) : X → S′

is equivalent to the data of a ring homomorphism A→ Γ(X ,OX ). We can thus interpret Proposition (1.8.1) asdefining a canonical bijection Hom(X , S) ' Hom(X , S′) (where we understand that the right-hand side is thecollection of morphisms of ringed spaces, since in general A is not a local ring). More generally, if X and Y arelocally ringed spaces, and if (Y ′, A′) is the ringed space whose underlying space is a point and whose sheaf ofrings A′ is the simple sheaf defined by the ring Γ(Y,OY ), we can interpret (1.8.2.1) as a map

(1.8.8.1) ρ : Homrs(X , Y ) −→ Hom(X , Y ′).

The result of Corollary (1.8.3) is interpreted by saying that ane schemes are characterized among locallyringed spaces as those for which the restriction of ρ to Hom(X , Y ):

(1.8.8.2) ρ′ : Hom(X , Y ) −→ Hom(X , Y ′)

is bijective for every locally ringed space X . In the following chapter, we generalize this definition, which allows usto associate to any ringed space Z (and not only to a ringed space whose underlying space is a point) a locallyringed space which we will call Spec(Z); this will be the starting point for a “relative” theory of preschemes overany ringed space, extending the results of Chapter I.

(1.8.9). We can consider the pairs (X ,F ) consisting of a locally ringed space X and an OX -module F as forminga category, a morphism in this category being a pair (u, h) consisting of a morphism of locally ringed spacesu : X → Y and a u-morphism h : G →F of modules; these morphisms (for (X ,F ) and (Y,G ) fixed) form a set II | 221which we denote by Hom((X ,F ), (Y,G )); the map (u, h) 7→ (ρ′(u),Γ(h)) is a canonical map

(1.8.9.1) Hom((X ,F ), (Y,G )) −→ Hom((Γ(Y,OY ),Γ(Y,G )), (Γ(X ,OX ),Γ(X ,F )))

functorial in (X ,F ) and (Y,G ), the right-hand side being the set of di-homomorphisms corresponding to therings and modules considered (0, 1.0.2).

Corollary (1.8.10). — Let Y be a locally ringed space, and G an OY -module. For Y to be an ane scheme and G to bea quasi-coherent OY -module, it is necessary and sucient that, for each pair (X ,F ) consisting of a locally ringed space Xand an OX -module F , the canonical map (1.8.9.1) be bijective.

We leave the proof, which is modelled on that of (1.8.3), using (1.8.1) and (1.8.7), to the reader.

Remark (1.8.11). — The statements (1.7.3), (1.7.4), and (2.2.4) are particular cases of (1.8.1), as well asthe definition in (1.6.1); similarly, (2.2.5) follows from (1.8.7). Corollary (1.8.7) also implies (1.6.3) (and,as a result, (1.6.4)) as a particular case, since if X is an ane scheme and Γ(X ,F ) = N , then the functorsM 7→ HomOS

( eM , u∗(eN)) and M 7→ HomOS( eM , (N[φ ])∼) (where φ : A→ Γ(X ,OX ) corresponds to u) are isomorphic,

by Corollaries (1.8.7) and (1.3.8). Finally, (1.6.5) (and, as a result, (1.6.6)) follow from (1.8.6), and the fact

2. PRESCHEMES AND MORPHISMS OF PRESCHEMES 92

that, for each f ∈ A, the A f -modules N ′ ⊗A′ A f and (N ′ ⊗A′ A) f (with the notation of (1.6.5)) are canonicallyisomorphic.

§2. Preschemes and morphisms of preschemes

2.1. Denition of preschemes.

(2.1.1). Given a ringed space (X ,OX ), we say that an open subset V of X is an ane open subset if the ringedspace (V,OX |V ) is an ane scheme (1.7.1).

Denition (2.1.2). — We define a prescheme to be a ringed space (X ,OX ) such that every point of X admitsan ane open neighborhood.

I | 98Proposition (2.1.3). — If (X ,OX ) is a prescheme, then its ane open subsets form a basis for the topology of X .

Proof. If V is an arbitrary open neighborhood of x ∈ X , then there exists by hypothesis an open neighborhoodW of x such that (W,OX |W ) is an ane scheme; we write A to mean its ring. In the space W , V ∩W is an openneighborhood of x ; so there exists some f ∈ A such that D( f ) is an open neighborhood of x contained insideV ∩W (1.1.10, i). The ringed space (D( f ),OX |D( f )) is thus an ane scheme, isomorphic to A f (1.3.6), whencethe proposition.

Proposition (2.1.4). — The underlying space of a prescheme is a Kolmogoro space.

Proof. If x and y are two distinct points of a prescheme X , then it is clear that there exists an openneighborhood of one of these points that does not contain the other if x and y are not in the same ane opensubset; and if they are in the same ane open subset, this is a result of (1.1.8).

Proposition (2.1.5). — If (X ,OX ) is a prescheme, then every closed irreducible subset of X admits exactly one genericpoint, and the map x 7→ x is thus a bijection of X onto its set of closed irreducible subsets.

Proof. If Y is a closed irreducible subset of X and y ∈ Y , and if U is an ane open neighborhood of y in X ,then U ∩ Y is dense in Y , and also irreducible ((0, 2.1.1) and (0, 2.1.4)); thus, by Corollary (1.1.14), U ∩ Y is theclosure in U of a point x , and so Y ⊂ U is the closure of x in X . The uniqueness of the generic point of X is aresult of Proposition (2.1.4) and of (0, 2.1.3).

(2.1.6). If Y is a closed irreducible subset of X , and y its generic point, then the local ring Oy (also writtenOX/Y ) is called the local ring of X along Y , or the local ring of Y in X .

If X itself is irreducible and x its generic point then we say that Ox is the ring of rational functions on X(cf. §7).

Proposition (2.1.7). — If (X ,OX ) is a prescheme, then the ringed space (U ,OX |U) is a prescheme for every open subsetU .

Proof. This follows directly from Definition (2.1.2) and Proposition (2.1.3).

We say that (U ,OX |U) is the prescheme induced on U by (X ,OX ), or the restriction of (X ,OX ) to U .

(2.1.8). We say that a prescheme (X ,OX ) is irreducible (resp. connected) if the underlying space X is irreducible(resp. connected). We say that a prescheme is integral if it is irreducible and reduced (cf. (5.1.4)). We say that aprescheme (X ,OX ) is locally integral if every x ∈ X admits an open neighborhood U such that the preschemeinduced on U by (X ,OX ) is integral.

2.2. Morphisms of preschemes.

Denition (2.2.1). — Given two preschemes, (X ,OX ) and (Y,OY ), we define a morphism (of preschemes) from(X ,OX ) to (Y,OY ) to be a morphism of ringed spaces (ψ,θ) such that, for all x ∈ X , θ]x is a local homomorphismOψ(x)→Ox .

I | 99By passing to quotients, the map Oψ(x)→Ox gives us a monomorphism θ x : k(ψ(x))→ k(x), which lets us

consider k(x) as an extension of the field k(ψ(x)).

(2.2.2). The composition (ψ′′,θ ′′) of two morphisms (ψ,θ), (ψ′,θ ′) of preschemes is also a morphism ofpreschemes, since it is given by the formula θ ′′] = θ] ψ∗(θ ′]) (0, 3.5.5). From this we conclude that preschemesform a category; using the usual notation, we will write Hom(X , Y ) to mean the set of morphisms from a preschemeX to a prescheme Y .

2. PRESCHEMES AND MORPHISMS OF PRESCHEMES 93

Example (2.2.3). — If U is an open subset of X , then the canonical injection (0, 4.1.2) of the inducedprescheme (U ,OX |U) into (X ,OX ) is a morphism of preschemes; it is further a monomorphism of ringed spaces(and a fortiori a monomorphism of preschemes), which follows rapidly from (0, 4.1.1).

Proposition (2.2.4). — 4 Let (X ,OX ) be a prescheme, and (S,OS) an ane scheme associated to a ring A. Then thereexists a canonical bijective correspondence between morphisms of preschemes from (X ,OX ) to (S,OS) and ring homomorphismsfrom A to Γ(X ,OX ).

Proof. First note that, if (X ,OX ) and (Y,OY ) are two arbitrary ringed spaces, a morphism (ψ,θ) from (X ,OX )to (Y,OY ) canonically defines a ring homomorphism Γ(θ) : Γ(Y,OY )→ Γ(Y,ψ∗(OX )) = Γ(X ,OX ). In the casethat we consider, everything boils down to showing that any homomorphism φ : A→ Γ(X ,OX ) is of the formΓ(θ) for exactly one θ. Now, by hypothesis, there is a covering (Vα) of X by ane open subsets; by composing φwith the restriction homomorphism Γ(X ,OX )→ Γ(Vα,OX |Vα), we obtain a homomorphism φα : A→ Γ(Vα,OX |Vα)that corresponds to a unique morphism (ψα,θα) from the prescheme (Vα,OX |Vα) to (S,OS), by Theorem (1.7.3).Furthermore, for each pair of indices (α,β), each point of Vα ∩ Vβ admits an ane open neighborhood Wcontained inside Vα ∩ Vβ (2.1.3); it is clear that, by composing φα and φβ with the restriction homomorphismsto W , we obtain the same homomorphism Γ(S,OS) → Γ(W,OX |W ), so, with the equation (θ]α)x = (φα)x forall x ∈ Vα and all α (1.6.1), the restriction to W of the morphisms (ψα,θα) and (ψβ ,θβ) coincide. From thiswe conclude that there is a morphism (ψ,θ) : (X ,OX )→ (S,OS) of ringed spaces, and only one such that itsrestriction to each Vα is (ψα,θα), and it is clear that this morphism is a morphism of preschemes and such thatΓ(θ) =φ .

Let u : A→ Γ(X ,OX ) be a ring homomorphism, and v = (ψ,θ) the corresponding morphism (X ,OX )→ (S,OS).For each f ∈ A, we have that

(2.2.4.1) ψ−1(D( f )) = Xu( f )

with the notation of (0, 5.5.2) relative to the locally free sheaf OX . In fact, it suces to verify this formula whenX itself is ane, and then this is nothing but (1.2.2.2).

Proposition (2.2.5). — Under the hypotheses of Proposition (2.2.4), let φ : A→ Γ(X ,OX ) be a ring homomorphism,f : (X ,OX ) → (S,OS) the corresponding morphism of preschemes, G (resp. F ) an OX -module (resp. OS -module), andM = Γ(S,F ). Then there exists a canonical bijective correspondence between f -morphisms F → G (0, 4.4.1) and I | 100A-homomorphisms M → (Γ(X ,G ))[φ ].

Proof. Reasoning as in Proposition (2.2.4), we reduce to the case where X is ane, and the propositionthen follows from Proposition (1.6.3) and from Corollary (1.3.8).

(2.2.6). We say that a morphism of preschemes (ψ,θ) : (X ,OX )→ (Y,OY ) is open (resp. closed) if, for all opensubsets U of X (resp. all closed subsets F of X ), ψ(U) is open (resp. ψ(F) is closed) in Y . We say that (ψ,θ) isdominant if ψ(X ) is dense in Y , and surjective if ψ is surjective. We note that these conditions rely only on thecontinuous map ψ.

Proposition (2.2.7). — Letf = (ψ,θ) : (X ,OX ) −→ (Y,OY )

andg = (ψ′,θ ′) : (Y,OY ) −→ (Z ,OZ)

be morphisms of preschemes.

(i) If f and g are both open (resp. closed, dominant, surjective), then so is g f .(ii) If f is surjective and g f closed, then g is closed.(iii) If g f is surjective, then g is surjective.

Proof. Claims (i) and (iii) are evident. Write g f = (ψ′′,θ ′′). If F is closed in Y then ψ−1(F) is closed inX , so ψ′′(ψ−1(F)) is closed in Z ; but since ψ is surjective, ψ(ψ−1(F)) = F , so ψ′′(ψ−1(F)) = ψ′(F), which proves(ii).

Proposition (2.2.8). — Let f = (ψ,θ) be a morphism (X ,OX ) → (Y,OY ), and (Uα) an open cover of Y . For fto be open (resp. closed, surjective, dominant), it is necessary and sucient for its restriction to each induced prescheme(ψ−1(Uα),OX |ψ−1(Uα)), considered as a morphism of preschemes from this induced prescheme to the induced prescheme(Uα,OY |Uα) to be open (resp. closed, surjective, dominant).

4[Trans.] See (1.8) and the footnote there.

2. PRESCHEMES AND MORPHISMS OF PRESCHEMES 94

Proof. The proposition follows immediately from the definitions, taking into account the fact that a subset Fof Y is closed (resp. open, dense) in Y if and only if each of the F ∩Uα are closed (resp. open, dense) in Uα.

(2.2.9). Let (X ,OX ) and (Y,OY ) be two preschemes; suppose that X (resp. Y ) has a finite number of irreduciblecomponents X i (resp. Yi) (1¶ i ¶ n); let ξ i (resp. ηi) be the generic point of X i (resp. Yi) (2.1.5). We say that amorphism

f = (ψ,θ) : (X ,OX ) −→ (Y,OY )

is birational if, for all i, ψ−1(ηi) = ξ i and θ]

ξ i: Oηi

→Oξ iis an isomorphism. It is clear that a birational morphism

is dominant (0, 2.1.8), and thus it is surjective if it is also closed.

Notation (2.2.10). — In all that follows, when there is no risk of confusion, we suppress the structure sheaf(resp. the morphism of structure sheaves) from the notation of a prescheme (resp. morphism of preschemes). IfU is an open subset of the underlying space X of a prescheme, then whenever we speak of U as a prescheme wealways mean the induced prescheme on U .

2.3. Gluing preschemes.I | 101

(2.3.1). It follows from Definition (2.1.2) that every ringed space obtained by gluing preschemes (0, 4.1.7) isagain a prescheme. In particular, although every prescheme admits, by definition, a cover by ane open sets,we see that every prescheme can actually be obtained by gluing ane schemes.

Example (2.3.2). — Let K be a field, B = K[s] and C = K[t] polynomial rings in one indeterminate overK , and define X1 = Spec(B) and X2 = Spec(C), which are isomorphic ane schemes. In X1 (resp. X2), let U12(resp. U21) be the ane open D(s) (resp. D(t)) where the ring Bs (resp. Ct) is formed of rational fractionsof the form f (s)/sm (resp. g(t)/tn) with f ∈ B (resp. g ∈ C). Let u12 be the isomorphism of preschemesU21 → U12 corresponding (2.2.4) to the isomorphism from Bs to Ct that, to f (s)/sm, associates the rationalfraction f (1/t)/(1/tm). We can glue X1 and X2 along U12 and U21 by using u12, because there is clearly nogluing condition. We later show that the prescheme X obtained in this manner is a particular case of a generalmethod of construction (II, 2.4.3). Here we show only that X is not an ane scheme; this will follow from the factthat the ring Γ(X ,OX ) is isomorphic to K , and so its spectrum reduces to a point. Indeed, a section of OX over Xhas a restriction over X1 (resp. X2), identified with an ane open of X , that is a polynomial f (s) (resp. g(t)),and it follows from the definitions that we should have g(t) = f (1/t), which is only possible if f = g ∈ K .

2.4. Local schemes.

(2.4.1). We say that an ane scheme is a local scheme if it is the ane scheme associated to a local ring A; therethen exists, in X = Spec(A), a single closed point a ∈ X , and for all other b ∈ X we have that a ∈ b (1.1.7).

For all preschemes Y and points y ∈ Y , the local scheme Spec(Oy) is called the local scheme of Y at thepoint y . Let V be an ane open subset of Y containing y, and B the ring of the ane scheme V ; then Oy iscanonically identified with By (1.3.4), and the canonical homomorphism B→ By thus corresponds (1.6.1) to amorphism of preschemes Spec(Oy)→ V . If we compose this morphism with the canonical injection V → Y , thenwe obtain a morphism Spec(Oy)→ Y which is independent of the ane open subset V (containing y) that wechose: indeed, if V ′ is some other ane open subset containing y , then there exists a third ane open subset Wthat contains y and is such that W ⊂ V ∩ V ′ (2.1.3); we can thus assume that V ⊂ V ′, and then if B′ is the ringof V ′, so everything relies on remarking that the diagram

B′ //

B

Oy

is commutative (0, 1.5.1). The morphismSpec(Oy) −→ Y

thus defined is said to be canonical.I | 102

Proposition (2.4.2). — Let (Y,OY ) be a prescheme; for all y ∈ Y , let (ψ,θ) be the canonical morphism (Spec(Oy), eOy)→(Y,OY ). Then ψ is a homeomorphism from Spec(Oy) to the subspace Sy of Y given by the z such that y ∈ z (or, equiva-lently, the generalizations of y (0, 2.1.2)); furthermore, if z = ψ(p), then θ]z : Oz → (Oy)p is an isomorphism; (ψ,θ) isthus a monomorphism of ringed spaces.

2. PRESCHEMES AND MORPHISMS OF PRESCHEMES 95

Proof. Since the unique closed point a of Spec(Oy) is contained in the closure of any point of this space, andsince ψ(a) = y, the image of Spec(Oy) under the continuous map ψ is contained in Sy . Since Sy is containedin every ane open containing y, one can consider just the case where Y is an ane scheme; but then thisproposition follows from (1.6.2).

We see (2.1.5) that there is a bijective correspondence between Spec(Oy) and the set of closed irreducible subsets of Ycontaining y .

Corollary (2.4.3). — For y ∈ Y to be the generic point of an irreducible component of Y , it is necessary and sucientfor the only prime ideal of the local ring Oy to be its maximal ideal ( in other words, for Oy to be of dimension zero).

Proposition (2.4.4). — Let (X ,OX ) be a local scheme of some ring A, a its unique closed point, and (Y,OY ) a prescheme.Every morphism u= (ψ,θ) : (X ,OX )→ (Y,OY ) then factors uniquely as X → Spec(Oψ(a))→ Y , where the second arrowdenotes the canonical morphism, and the rst corresponds to a local homomorphism Oψ(a)→ A. This establishes a canonicalbijective correspondence between the set of morphisms (X ,OX )→ (Y,OY ) and the set of local homomorphisms Oy → A for(y ∈ Y ).

Indeed, for all x ∈ X , we have that a ∈ x, so ψ(a) ∈ ψ(x), which shows that ψ(X ) is contained in everyane open subset that contains ψ(a). So it suces to consider the case where (Y,OY ) is an ane scheme ofring B, and then we have that u = (aφ ,φ ), where φ ∈ Hom(B, A) (1.7.3). Further, we have that φ−1(ja) = jψ(a),and hence that the image under φ of any element of B − jψ(a) is invertible in the local ring A; the factorizationin the result follows from the universal property of the ring of fractions (0, 1.2.4). Conversely, to each localhomomorphism Oy → A there is a unique corresponding morphism (ψ,θ) : X → Spec(Oy) such that ψ(a) = y(1.7.3), and, by composing with the canonical morphism Spec(Oy)→ Y , we obtain a morphism X → Y , whichproves the proposition.

(2.4.5). The ane schemes whose ring is a field K have an underlying space that is just a point. If A is a localring with maximal ideal m, then each local homomorphism A→ K has kernel equal to m, and so factors asA→ A/m→ K , where the second arrow is a monomorphism. The morphisms Spec(K)→ Spec(A) thus correspondbijectively to monomorphisms of fields A/m→ K .

Let (Y,OY ) be a prescheme; for each y ∈ Y and each ideal ay of Oy , the canonical homomorphismOy → Oy/ay defines a morphism Spec(Oy/ay)→ Spec(Oy); if we compose this with the canonical morphismSpec(Oy)→ Y , then we obtain a morphism Spec(Oy/ay)→ Y , again said to be canonical. For ay = my , this saysthat Oy/ay = k(y), and so Proposition (2.4.4) says that:

I | 103Corollary (2.4.6). — Let (X ,OX ) be a local scheme whose ring K is a eld, ξ the unique point of X , and (Y,OY ) aprescheme. Then each morphism u : (X ,OX )→ (Y,OY ) factors uniquely as X → Spec(k(ψ(ξ )))→ Y , where the secondarrow denotes the canonical morphism, and the rst corresponds to a monomorphism k(ψ(ξ ))→ K . This establishes acanonical bijective correspondence between the set of morphisms (X ,OX )→ (Y,OY ) and the set of monomorphisms k(y)→ K(for y ∈ Y ).

Corollary (2.4.7). — For all y ∈ Y , every canonical morphism Spec(Oy/ay)→ Y is a monomorphism of ringed spaces.

Proof. We have already seen this when ay = 0 (2.4.2), and it suces to apply Corollary (1.7.5).

Remark. — 2.4.8 Let X be a local scheme, and a its unique closed point. Since every ane open subsetcontaining a is necessarily equal to the whole of X , every invertible OX -module (0, 5.4.1) is necessarily isomorphicto OX (or, as we say, again, trivial). This property does not hold in general for an arbitrary ane scheme Spec(A);we will see in Chapter V that if A is a normal ring then this is true when A is a unique factorisation domain.

2.5. Preschemes over a prescheme.

Denition (2.5.1). — Given a prescheme S, we say that the data of a prescheme X and a morphism ofpreschemes φ : X → S defines a prescheme X over the prescheme S, or an S-prescheme; we say that S is the baseprescheme of the S-prescheme X . The morphism φ is called the structure morphism of the S-prescheme X . When Sis an ane scheme of ring A, we also say that X endowed with φ is a prescheme over the ring A (or an A-prescheme).

It follows from (2.2.4) that the data of a prescheme over a ring A is equivalent to the data of a prescheme(X ,OX ) whose structure sheaf OX is a sheaf of A-algebras. An arbitrary prescheme can always be considered as aZ-prescheme in a unique way.

If φ : X → S is the structure morphism of an S-prescheme X , we say that a point x ∈ X is over a point s ∈ S ifφ (x) = s. We say that X dominates S if φ is a dominant morphism (2.2.6).

3. PRODUCTS OF PRESCHEMES 96

(2.5.2). Let X and Y be S-preschemes; we say that a morphism of preschemes u : X → Y is a morphism ofpreschemes over S (or an S-morphism) if the diagram

X u //

Y

S

(where the diagonal arrows are the structure morphisms) is commutative: this ensures that, for all s ∈ S andx ∈ X over s, u(x) also lies over s.

It follows immediately from this definition that the composition of any two S-morphisms is an S-morphism;S-preschemes thus form a category.

We denote by HomS(X , Y ) the set of S-morphisms from an S-prescheme X to an S-prescheme Y ; the identitymorphism of an S-prescheme X is denoted by 1X .

When S is an ane scheme of ring A, we will also say A-morphism instead of S-morphism.I | 104

(2.5.3). If X is an S-prescheme, and v : X ′→ X a morphism of preschemes, then the composition X ′→ X → Sendows X ′ with the structure of an S-prescheme; in particular, every prescheme induced by an open set U of Xcan be considered as an S-prescheme by the canonical injection.

If u : X → Y is an S-morphism of S-preschemes, then the restriction of u to any prescheme induced by anopen subset U of X is also an S-morphism U → Y . Conversely, let (Uα) be an open cover of X , and for eachα let uα : Uα→ Y be an S-morphism; if, for all pairs of indices (α,β), the restrictions of uα and uβ to Uα ∩ Uβagree, then there exists an S-morphism X → Y , and exactly one such that the restriction to each Uα is uα.

If u : X → Y is an S-morphism such that u(X ) ⊂ V , where V is an open subset of Y , then u, considered as amorphism from X to V , is also an S-morphism.

(2.5.4). Let S′→ S be a morphism of preschemes; for all S′-preschemes, the composition X → S′→ S endows Xwith the structure of an S-prescheme. Conversely, suppose that S′ is the induced prescheme of an open subset ofS; let X be an S-prescheme and suppose that the structure morphism f : X → S is such that f (X ) ⊂ S′; then wecan consider X as an S′-prescheme. In this latter case, if Y is another S-prescheme whose structure morphismsends the underlying space to S′, then every S-morphism from X to Y is also an S′-morphism.

(2.5.5). If X is an S-prescheme, with structure morphism φ : X → S, we define an S-section of X to be anS-morphism from S to X , that is to say a morphism of preschemes ψ : S→ X such that φ ψ is the identity on S.We denote by Γ(X/S) the set of S-sections of X .

§3. Products of preschemes

3.1. Sums of preschemes. Let (Xα) be any family of preschemes; let X be a topological space which is thesum of the underlying spaces Xα; X is then the union of pairwise disjoint open subspaces X ′α, and for each αthere is a homomorphism φα from Xα to X ′α. If we equip each of the X ′α with the sheaf (φα)∗(OXα), it is clearthat X becomes a prescheme, which we call the sum of the family of preschemes (Xα) and which we denote⊔

α Xα. If Y is a prescheme, then the map f 7→ ( f φα) is a bijection from the set Hom(X , Y ) to the productset

∏

α Hom(Xα, Y ). In particular, if the Xα are S-preschemes, with structure morphisms ψα, then X is anS-prescheme by the unique morphism ψ : X → S such that ψ φα = ψα for each α. The sum of two preschemesX and Y is denoted by X t Y . It is immediate that, if X = Spec(A) and Y = Spec(B), then X t Y is canonicallyidentified with Spec(A× B).

3.2. Products of preschemes.

Denition (3.2.1). — Given S-preschemes X and Y , we say that a triple (Z , p1, p2), consisting of an S-preschemeZ , and S-morphisms p1 : Z → X and p2 : Z → Y , is a product of the S-preschemes X and Y , if, for each S- I | 105prescheme T , the map f 7→ (p1 f , p2 f ) is a bijection from the set of S-morphisms from T to Z , to the set ofpairs consisting of an S-morphism T → X and an S-morphism T → Y (in other words, a bijection

HomS(T, Z)' HomS(T, X )×HomS(T, Y )).

This is the general notion of a product of two objects in a category, applied to the category of S-preschemes(T, I, 1.1); in particular, a product of two S-preschemes is unique up to a unique S-isomorphism. Because ofthis uniqueness, most of the time we will denote a product of two S-preschemes X and Y by X ×S Y (or simplyX × Y , when there is no chance of confusion), with the morphisms p1 and p2 (the canonical projections of X ×S Yto X and to Y , respectively) being suppressed in the notation. If g : T → X and h : T → Y are S-morphisms,

3. PRODUCTS OF PRESCHEMES 97

we denote by (g, h)S , or simply (g, h), the S-morphism f : T → X ×S Y such that p1 f = g, p2 f = h. If X ′

and Y ′ ar two S-preschemes, p′1 and p′2 the canonical projections of X ′ ×S Y ′ (assumed to exist), and u : X ′→ Xand v : Y ′→ Y S-morphisms, then we write u×S v (or simply u× v) for the S-morphism (u p′1, v p′2)S fromX ′ ×S Y ′ to X ×S Y .

When S is an ane scheme given by some ring A, we often replace S by A is the above notation.

Proposition (3.2.2). — Let X , Y , and S be ane schemes, given by rings B, C , and A (respectively). Let Z =Spec(B⊗A C), and let p1 and p2 be the S-morphisms corresponding (2.2.4) to the canonical A-homomorphisms u : b 7→ b⊗1and v : c 7→ 1⊗ c (respectively) from B and C to B ⊗A C ; then (Z , p1, p2) is a product of X and Y .

Proof. According to (2.2.4), it suces to check that, if, to each A-homomorphism f : B⊗A C → L (where L isan A-algebra), we associate the pair ( f u, f v), then this defines a bijection HomA(B ⊗A C , L)' HomA(B, L)×HomA(C , L),5 which follows immediately from the definitions and the fact that b⊗ c = (b⊗ 1)(1⊗ c).

Corollary (3.2.3). — Let T be an ane scheme given by some ring D, and α = (aρ, eρ) (resp. β = (aσ, eσ)) anS-morphism T → X (resp. T → Y ), where ρ (resp. σ) is an A-homomorphism from B (resp. C ) to D; then (α,β)S = (aτ,eτ),where τ is the homomorphism B ⊗A C → D such that τ(b⊗ c) = ρ(b)σ(c).

Proposition (3.2.4). — Let f : S′ → S be a monomorphism of preschemes (T, I, 1.1), and let X and Y be S′-preschemes, also considered as S-preschemes via f . Every product of the S-preschemes X and Y is then a product of theS′-preschemes X and Y , and vice versa.

Proof. Let φ : X → S′ and ψ : Y → S′ be the structure morphisms. If T is an S-prescheme, and u : T → Xand v : T → Y are S-morphisms, then we have, by definition, that f φ u= f ψ v = θ, the structure morphismof T ; the hypotheses on f imply that φ u = ψ v = θ ′, and so we can consider T as an S′-prescheme withstructure morphism θ ′, and u and v as S′-morphisms. The conclusion of the proposition follows immediately,taking (3.2.1) into account.

Corollary (3.2.5). — Let X and Y be S-preschemes, with structure morphisms φ : X → S and ψ : Y → S, and let S′ bean open subset of S such that φ (X ) ⊂ S′ and ψ(Y ) ⊂ S′. Every product of the S-preschemes X and Y is then also a productof the S′-preschemes X and Y , and conversely.

I | 106It suces to apply (3.2.4) to the canonical injection S′→ S.

Theorem (3.2.6). — Given S-preschemes X and Y , there exists a product X ×S Y .

The proof proceeds in several steps.

Lemma (3.2.6.1). — Let (Z , p, q) be a product of X and Y , and U and V open subsets of X and Y , respectively. Ifwe let W = p−1(U)∩ q−1(V ), then the triple consisting of W and the restrictions of p and q to W (considered as themorphisms W → U and W → V , respectively) is a product of U and V .

Indeed, if T is an S-prescheme, then we can identify the S-morphisms T → W with the S-morphismsT → Z mapping T to W . Then, if g : T → U and h : T → V are any two S-morphisms, we can consider themas S-morphisms from T to X and to Y , respectively, and, by hypothesis, there is then a unique S-morphismf : T → Z such that g = p f and h= q f . Since p( f (Y )) ⊂ U , q( f (T )) ⊂ V , we have

f (T ) ⊂ p−1(U)∩ q−1(V ) =W,

hence our claim.

Lemma (3.2.6.2). — Let Z be an S-prescheme, p : Z → X and q : Z → Y both S-morphisms, (Uα) an open cover ofX , and (Vλ) an open cover of Y . Suppose that, for each pair (α,λ), the S-prescheme Wαλ = p−1(Uα)∩ q−1(Vλ) and therestrictions of p and q to Wαλ form a product of Uα and Vλ . Then (Z , p, q) is a product of X and Y .

We first show that, if f1 and f2 are S-morphisms T → Z , then the equations p f1 = p f2 and q f1 = q f2imply that f1 = f2. Indeed, Z is the union of the Wαλ , so the f −1

1 (Wαλ) form an open cover of T , and similarlyfor f −1

2 (Wαλ). In addition, we have

f −11 (Wαλ) = f −1

1 (p−1(Uα))∩ f −1

1 (q−1(Vλ)) = f −1

2 (p−1(Uα))∩ f −1

2 (q−1(Vλ)) = f −1

2 (Wαλ)

by hypothesis, and it thus reduces to noting that the the restrictions of f1 and f2 to f −11 (Wαλ) = f −1

2 (Wαλ) areidentical for each pair of indices. But since these restrictions can be considered as S-morphisms from f −1

1 (Wαλ)to Wαλ , our claim follows from the hypotheses and Definition (3.2.1).

5The notation HomA denotes here the set of homomorphisms of A-algebras.

3. PRODUCTS OF PRESCHEMES 98

Suppose now that we are given S-morphisms g : T → X and h : T → Y . Let Tαλ = g−1(Uα)∩ h−1(Vλ); thenthe Tαλ form an open cover of T . By hypothesis, there exists an S-morphism fαλ such that p fαλ and q fαλare the restrictions of g and h to Tαλ (respectively). Now, we will show that the restrictions of fαλ and fβµ toTαλ ∩ Tβµ coincide, which will finish the proof of Lemma (3.2.6.2). The images of Tαλ ∩ Tβµ under fαλ and fβµare contained in Wαλ ∩Wβµ by definition. Since

Wαλ ∩Wβµ = p−1(Uα ∩ Uβ)∩ q−1(Vλ ∩ Vµ),

it follows from Lemma (3.2.6.1) that Wαλ ∩Wβµ and the restrictions to this prescheme of p and q form a productof Uα ∩ Uβ and Vλ ∩ Vµ. Since p fαλ and p fβµ coincide on Tαλ ∩ Tβµ and similarly for q fαλ and q fβµ, wesee that fαλ and fβµ coincide on Tαλ ∩ Tβµ.

I | 107Lemma (3.2.6.3). — Let (Uα) be an open cover of X , (Vλ) an open cover of Y , and suppose that, for each pair (α,λ),there exists a product of Uα and Vλ ; then there exists a product of X and Y .

Applying Lemma (3.2.6.1) to the open sets Uα ∩ Uβ and Vλ ∩ Vµ, we see that there exists a product ofS-preschemes induced, respectively, by X and Y on these open sets; in addition, the uniqueness of the productshows that, if we set i = (α,λ) and j = (β,µ), then there is a canonical isomorphism hi j (resp. h ji) from thisproduct to an S-prescheme Wi j (resp. Wji) induced by Uα×S Vλ (resp. Uβ×S Vµ) on an open set; then fi j = hi j h−1

jiis an isomorphism from Wji to Wi j . In addition, for a third pair k = (γ ,ν), we have fik = fi j f jk on Wki ∩Wk j ,by applying Lemma (3.2.6.1) to the open sets Uα ∩Uβ ∩Uγ and Vλ ∩ Vµ ∩ Vν in Uβ and Vµ, respectively. It followsthat we have a prescheme Z , an open cover (Zi) of the underlying space of Z , and, for each i, an isomorphismgi from the induced prescheme Zi to the prescheme Uα ×S Vλ , so that, for each pair (i, j), we have fi j = gi g−1

j(2.3.1); in addition, we have gi(Zi ∩ Z j) =Wi j . If pi , qi , and θi are the projections and the structure morphismof the S-prescheme Uα ×S Vλ (respectively), we immediately see that pi gi = p j g j on Zi ∩ Z j , and similarlyfor the two other morphisms. We can thus define the morphisms of preschemes p : Z → X (resp. q : Z → Y ,θ : Z → S) by the condition that p (resp. q, θ) coincide with pi gi (resp. qi gi , θi gi) on each of the Zi ; Z ,equipped with θ, is then an S-prescheme. We now show that Z ′i = p−1(Uα)∩ q−1(Vλ) is equal to Zi . For eachindex j = (β,µ), we have Z j ∩ Z ′i = g−1

j (p−1j (Uα)∩ q−1

j (Vλ)). We have, by Lemma (3.2.6.1),

p−1j (Uα)∩ q−1

j (Vλ) = p−1j (Uα ∩ Uβ)∩ q−1

j (Vλ ∩ Vµ);

with the restrictions of p j and q j to p−1j (Uα)∩ q−1

j (Vλ) defining, on this S-prescheme, the structure of a productof Uα ∩Uβ and Vλ ∩ Vµ; but the uniqueness of the product then implies that p−1

j (Uα)∩ q−1j (Vλ) =Wji . As a result,

we have Z j ∩ Z ′i = Z j ∩ Zi for each j, hence Z ′i = Zi . We then deduce from Lemma (3.2.6.2) that (Z , p, q) is aproduct of X and Y .

Lemma (3.2.6.4). — Let φ : X → S and ψ : Y → S be the structure morphisms of X and Y , (Si) an open cover of S,and let X i =φ−1(Si), Yi = ψ−1(Si). If each of the products X i ×S Yi exists, then X ×S Y exists.

According to Lemma (3.2.6.3), everything follows from proving that the products X i×S Yi exists for any i andj. Set X i j = X i ∩ X j =φ−1(Si ∩ S j), Yi j = Yi ∩ Yj = ψ−1(Si ∩ S j); by Lemma (3.2.6.1), the product Zi j = X i j ×S Yi jexists. We now note that, if T is an S-prescheme, and if g : T → X i and h : T → Yj are S-morphisms, then wenecessarily have that φ (g(T )) = ψ(h(T )) ⊂ Si ∩ S j by the definition of an S-morphism, and thus that g(T ) ⊂ X i jand h(T ) ⊂ Yi j ; it is then immediate that Zi j is the product of X i and Yj .

(3.2.6.5). We can now complete the proof of Theorem (3.2.6). If S is an ane scheme, then there are covers (Uα)and (Vλ) of X and Y (respectively) consisting of ane open subsets; since Uα ×S Vλ exists, by (3.2.2), X ×S Yexists similarly, by Lemma (3.2.6.3). If S is any prescheme, then there is a cover (Si) of S consisting of ane opensubsets. If φ : X → S and ψ : Y → S are the structure morphisms, and if we set X i =φ−1(Si) and Yi = ψ−1(Si),then the products X i ×Si

Yi exist, by the above; but then the products X i ×S Yi also exist (3.2.5), therefore X ×S Y I | 108exists similarly, by Lemma (3.2.6.4).

Corollary (3.2.7). — Let Z = X ×S Y be the product of two S-preschemes, p and q the projections from Z to X andto Y (respectively),and φ (resp. ψ) the structure morphism of X (resp. Y ). Let S′ be an open subset of S, and U (resp. V )an open subset of X (resp. Y ) contained in φ−1(S′) (resp. ψ−1(S′)). Then the product U ×S′ V is canonically identiedwith the prescheme induced on Z by p−1(U)∩ q−1(V ) (considered as an S′-prescheme). In addition, if f : T → X andg : T → Y are S-morphisms such that f (T ) ⊂ U and g(T ) ⊂ V , then the S′-morphism ( f , g)S′ can be identied with therestriction of ( f , g)S to p−1(U)∩ q−1(V ).

Proof. This follows from Corollary (3.2.5) and Lemma (3.2.6.1).

3. PRODUCTS OF PRESCHEMES 99

(3.2.8). Let (Xα) and (Yλ) be families of S-preschemes, and X (resp. Y ) the sum of the family (Xα) (resp. (Yλ))(3.1). Then X ×S Y can be identified with the sum of the family (Xα ×S Yλ); this follows immediately fromLemma (3.2.6.3).

(3.2.9). 6 It follows from (1.8.1) that we can state (3.2.2) in the following manner: Z = Spec(B⊗A C) is not only a II | 221product of X = Spec(B) and Y = Spec(C) in the category of S-preschemes, but also in the category of locally ringedspaces over S (with a definition of S-morphisms modelled on that of (2.5.2)). The proof of (3.2.6) also proves that,for any two S-preschemes X and Y , the prescheme X ×S Y is not only the product of X and Y in the category ofS-preschemes, but also in the category of locally ringed spaces over the prescheme S.

3.3. Formal properties of the product; change of the base prescheme.

(3.3.1). The reader will notice that all the properties stated in this section, except (3.3.13) and (3.3.15), are truewithout modification in any category, whenever the products involved in the statements exist (since it is clearthat the notions of an S-object and of an S-morphism can be defined exactly as in (2.5) for any object S of thecategory).

(3.3.2). First of all, X ×S Y is a covariant bifunctor in X and Y on the category of S-preschemes: it suces in factto note that the diagram

X × Yf ×1 //

X ′ × Yf ′×1 //

X ′′ × Y

X

f // X ′f ′ // X ′′

is commutative.

Proposition (3.3.3). — For each S-prescheme X , the rst (resp. second) projection from X ×S S (resp. S ×S X ) is afunctorial isomorphism from X ×S S (resp. S ×S X ) to X , whose inverse isomorphism is (1X ,φ )S (resp. (φ , 1X )S ), where wedenote by φ the structure morphism X → S; we can therefore write, up to a canonical isomorphism,

X ×S S = S ×S X = X .

Proof. It suces to prove that the triple (X , 1X ,φ ) is a product of X and S. If T is an S-prescheme, then theonly S-morphism from T to S is necessarily the structure morphism ψ : T → S. If f is an S-morphism from T toX , we necessarily have ψ =φ f , hence our claim.

Corollary (3.3.4). — Let X and Y be S-preschemes, with structure morphisms φ : X → S and ψ : Y → S. If wecanonically identify X with X ×S S, and Y with S×S Y , then the projections X ×S Y → X and X ×S Y → Y are identiedwith 1X ×ψ and φ × 1Y (respectively).

The proof is immediate and is left to the reader.

(3.3.5). We can define, in a manner similar to (3.2), the product of a finite number n of S-preschemes, and the I | 109existence of these products follows from (3.2.6) by induction on n, and by noting that (X1×S X2×S · · ·×S Xn−1)×S Xnsatisfies the definition of a product. The uniqueness of the product implies, as in any category, its commutativityand associativity properties. If, for example, p1, p2, and p3 denote the projections from X1 ×S X2 ×S X3, and if weidentify this prescheme with (X1 ×S X2)×S X3, then the projection to X1 ×S X2 is identified with (p1, p2)S .

(3.3.6). Let S and S′ be preschemes, and φ : S′→ S a morphism, which lets us consider S′ as an S-prescheme.For each S-prescheme X , consider the product X ×S S′, and let p and π′ be the projections to X and to S′

(respectively). Equipped with π′, this product is an S′-prescheme; when we consider it as such, we denote it byX(S′) or X(φ ), and we say that this is the prescheme obtained by base change (or a change of base) from S to S′ bymeans of the morphism φ , or the inverse image of X by φ . We note that, if π is the structure morphism of X , andθ the structure morphism of X ×S S′, considered as an S-prescheme, then the diagram

X

π

X(S′)poo

θ

~~π′

S S′φoo

is commutative.

6[Trans.] (3.2.9) is from the errata of EGA II, on page 221, whence the change in page numbering.

3. PRODUCTS OF PRESCHEMES 100

(3.3.7). With the notation of (3.3.6), for each S-morphism f : X → Y , we denote by f(S′) the S′-morphismf ×S 1 : X(S′) → Y(S′), and we say that f(S′) is the base change (or inverse image) of f by φ . Therefore X(S′) is acovariant functor in X , from the category of S-preschemes to that of S′-preschemes.

(3.3.8). The prescheme X(S′) can be considered as a solution to a universal mapping problem: each S′-preschemeT is also an S-prescheme via φ ; each S-morphism g : T → X is then uniquely written as g = p f , where fis an S′-morphism T → X(S′), as follows from the definition of the product applied to the S-morphisms f andψ : T → S′ (the structure morphism of T ).

Proposition (3.3.9). — (“Transitivity of base change”). Let S′′ be a prescheme, and φ ′ : S′′→ S a morphism. Foreach S-prescheme X , there exists a canonical functorial isomorphism from the S′′-prescheme (X(φ ))(φ ′) to the S′′-preschemeX(φφ ′).

Proof. Let T be a S′′-prescheme, ψ its structure morphism, and g an S-morphism from T to X (T beingconsidered as an S-prescheme with structure morphism φ φ ′ ψ). Since T is also an S′-prescheme with structuremorphism φ ′ ψ, we can write g = p g ′, where g ′ is an S′-morphism T → X(φ ), and then g ′ = p′ g ′′, where g ′′

is an S′′-morphism T → (X(φ ))(φ ′):

X

π

X(φ )poo

π′

(X(φ ))(φ ′)p′oo

π′′

S Sφoo S′′.

φ ′oo

So the result follows by the uniqueness of the solution to a universal mapping problem. I | 110

This result can be written as the equality (up to a canonical isomorphism) (X(S′))(S′′) = X(S′′) (if there is nochance of confusion), or also as

(3.3.9.1) (X ×S S′)×S′ S′′ = X ×S S′′;

the functorial nature of the isomorphism defined in (3.3.9) can similarly be expressed by the transitivity formulafor base change morphisms

(3.3.9.2) ( f(S′))(S′′) = f(S′′)

for each S-morphism f : X → Y .

Corollary (3.3.10). — If X and Y are S-preschemes, then there exists a canonical functorial isomorphism from theS′-prescheme X(S′) ×S′ Y(S′) to the S′-prescheme (X ×S Y )(S′).

Proof. We have, up to canonical isomorphism,

(X ×S S′)×S′ (Y ×S S′) = X ×S (Y ×S S′) = (X ×S Y )×S S′

according to (3.3.9.1) and the associativity of products of S-preschemes.

The functorial nature of the isomorphism defined in Corollary (3.3.10) can be expressed by the formula

(3.3.10.1) (u(S′), v(S′))S′ = ((u, v)S)(S′)

for each pair of S-morphisms u : T → X , v : T → Y .In other words, the base change functor X(S′) commutes with products; it also commutes with sums (3.2.8).

Corollary (3.3.11). — Let Y be an S-prescheme, and f : X → Y a morphism which makes X a Y -prescheme (and,as a result, also an S-prescheme). The prescheme X(S′) is then identied with the product X ×Y Y(S′), the projectionX ×Y Y(S′)→ Y(S′) being identied with f(S′).

Proof. Let ψ : Y → S be the structure morphism of Y ; we have the commutative diagram

S′

Y(S′)oo

X(S′)f(S′)oo

S Y

ψoo X .foo

We have that Y(S′) is identified with S′(ψ), and X(S′) with S′(ψ f ); taking (3.3.9) and (3.3.4) into account, we thusdeduce the corollary.

3. PRODUCTS OF PRESCHEMES 101

(3.3.12). Let f : X → X ′ and g : Y → Y ′ be S-morphisms which are monomorphisms of preschemes (T, I, 1.1);then f ×S g is a monomorphism. Indeed, if p and q are the projections of X ×S Y , p′ and q′ the projections ofX ′ ×S Y ′, and u and v both S-morphisms T → X ×S Y , then the equation ( f ×S g) u= ( f ×S g) v implies thatp′ ( f ×S g) u= p′ ( f ×S g) v, or, in other words, that f p u= f p v, and since f is a monomorphism,p u= p v; using the fact that g is a monomorphism, we similarly obtain q u= q v, hence u= v. I | 111

It follows that, for each base change S′→ S,

f(S′) : X(S′) −→ Y(S′)

is a monomorphism.

(3.3.13). Let S and S′ be ane schemes of rings A and A′ (respectively); a morphism S′→ S then correspondsto a ring homomorphism A→ A′. If X is an S-prescheme, we denote by X(A′) or X ⊗A A′ the S′-prescheme X(S′);when X is also ane of ring B, X(A′) is ane of ring B(A′) = B ⊗A A′ obtained by extension of scalars from theA-algebra B to A′.

(3.3.14). With the notation of (3.3.6), for each S-morphism f : S′ → X , we have that f ′ = ( f , 1S′)S is an S′-morphism S′→ X ′ = X(S′) such that p f ′ = f , π′ f ′ = 1S′ , or, in other words, an S′-section of of X ′; conversely,if f ′ is such an S′-section, then f = p f ′ is an S-morphism S′ → X . We thus define a canonical bijectivecorrespondence

HomS(S′, X )' HomS′(S

′, X ′).

We say that f ′ is the graph morphism of f , and we denote it by Γf .

(3.3.15). Given a prescheme X , which we can always consider as a Z-prescheme, it follows, in particular, from(3.3.14) that the X -sections of X ⊗Z Z[T] (where T is an indeterminate) correspond bijectively with morphismsZ[T]→ X . We will show that these X -sections also correspond bijectively with sections of the structure sheaf OXover X . Indeed, let (Uα) be a cover of X by ane open subsets; let u : X → X ⊗Z Z[T] be an X -morphism, andlet uα be its restriction to Uα; if Aα is the ring of the ane scheme Uα, then Uα ⊗Z Z[T] is an ane scheme ofring Aα[T] (3.2.2), and uα canonically corresponds to an Aα-homomorphism Aα[T]→ Aα (1.7.3). Now, sincesuch a homomorphism is completely determined by the data of the image of T in Aα, let sα ∈ Aα = Γ(Uα,OX ),and if we suppose that the restrictions of uα and uβ to an ane open subset V ⊂ Uα ∩ Uβ coincide, then we seeimmediately that sα and sβ coincide on V ; thus the family (sα) consists of the restrictions to Uα of a section s ofOX over X ; conversely, it is clear that such a section defines a family (uα) of morphisms which are the restrictionsto Uα of an X -morphism X → X ⊗Z Z[T]. This result is generalized in (II, 1.7.12).

3.4. Points of a prescheme with values in a prescheme; geometric points.

(3.4.1). Let X be a prescheme; for each prescheme T , we then denote by X (T ) the set Hom(T, X ) of morphismsT → X , and the elements of this set are called the points of X with values in T . If we associate to each morphismf : T → T ′ the map u′ 7→ u′ f from X (T ′) to X (T ), we see, for fixed X , that X (T ) is a contravariant functor in T ,from the category of preschemes to that of sets. In addition, each morphism of preschemes g : X → Y defines afunctorial homomorphism X (T )→ Y (T ), which sends v ∈ X (T ) to g v.

(3.4.2). Given sets P, Q, and R, and maps φ : P → R and ψ : Q→ R, we define the bre product of P and Q over R(relative to φ and ψ) as the subset of the product set P ×Q consisting of the pairs (p, q) such that φ (p) = ψ(q); I | 112we denote it by P ×R Q. Definition (3.2.1) of the product of S-preschemes can be interpreted, with the notationof (3.4.1), via the formula

(3.4.2.1) (X ×S Y )(T ) = X (T )×S(T ) Y (T ).

the maps X (T )→ S(T ) and Y (T )→ S(T ) corresponding to the structure morphisms X → S and Y → S.

(3.4.3). If we are given a prescheme S and we consider only the S-preschemes and S-morphisms, then we willdenote by X (T )S the set HomS(T, X ) of S-morphisms T → X , and suppress the subscript S when there is nochance of confusion; we say that the elements of X (T )S are the points (or S-points, when there is a possibility ofconfusion) of the S-prescheme X with values in the S-prescheme T . In particular, an S-section of X is none other thana point of X with values in S. The formula (3.4.2.1) can then be written as

(3.4.3.1) (X ×S Y )(T )S = X (T )S × Y (T )S;

more generally, if Z is an S-prescheme, and X , Y , and T are Z -preschemes (thus ipso facto S-preschemes), thenwe have

(3.4.3.2) (X ×Z Y )(T )S = X (T )S ×Z(T )S Y (T )S .

3. PRODUCTS OF PRESCHEMES 102

We note that, to show that a triple (W, r, s) consisting of an S-prescheme W and S-morphisms r : W → Xand s : W → Y is a product of X and Y (over Z), it suces, by definition, to check that, for each S-prescheme T ,the diagram

W (T )Sr ′ //

s′

X (T )S

φ ′

Y (T )S

ψ′ // Z(T )Smakes W (T )S the fibre product of X (T )S and Y (T )S over Z(T )S , where r ′ and s′ correspond to r and s, and φ ′and ψ′ to the structure morphisms φ : X → Z and ψ : Y → Z .

(3.4.4). When T (resp. S) in the above is an ane scheme of ring B (resp. A), we replace T (resp. S) by B (resp.A) in the above notation, and we then call the elements of X (B) the points of X with values in the ring B, and theelements of X (B)A the points of the A-prescheme X with values in the A-algebra B. We note that X (B) and X (B)A arecovariant functors in B. We similarly write X (T )A for the set of points of the A-prescheme X with values in theA-prescheme T .

(3.4.5). Consider, in particular, the case where T is of the form Spec(A), where A is a local ring; the elements ofX (A) then correspond bijectively to local homomorphisms Ox → A for x ∈ X (2.2.4); we say that the point x ofthe underlying space of X is the location7 of the point of X with values in A to which it corresponds.

More specifically, we define the geometric points of a prescheme S to be the points of X with values in a eld K :the data of such a point is equivalent to the data of its location x in the underlying subspace of X , and of an I | 113extension K of k(x); K will be called the eld of values of the corresponding geometric point, and we say that thisgeometric point is located at x . We also define a map X (K)→ X , sending a geometric point with values in K toits location.

If S′ = Spec(K) is an S-prescheme (in other words, if K is considered as an extension of the residue fieldk(s), where s ∈ S), and if X is an S-prescheme, then an element of X (K)S , or, as we say, a geometric point of Xlying over s with values in K , consists of the data of a k(s)-monomorphism from the residue field k(x) to K , wherex is a point of X lying over s (therefore k(x) is an extension of k(s)).

In particular, if S = Spec(K) = ξ , then the geometric points of X with values in K can be identied with thepoints x ∈ X such that k(x) = K ; we say that these latter points are the K -rational points of the K -prescheme X ; if K ′

is an extension of K , then the geometric points of X with values in K ′ bijectively correspond to the K ′-rationalpoints of X ′ = X(K ′) (3.3.14).

Lemma (3.4.6). — Let X i (1 ¶ i ¶ n) be S-preschemes, s a point of S, and x i (1 ¶ i ¶ n) points of X i lying over s.Then there exists an extension K of k(s) and a geometric point of the product Y = X1 ×S X2 ×S · · · ×S Xn, with values inK , whose projections to the X i are localized at the x i .

Proof. There exist k(s)-monomorphisms k(x i)→ K , all in the same extension K of k(s) (Bourbaki, Alg.,chap. V, §4, prop. 2). The compositions k(s)→ k(x i)→ K are all identical, and so the morphisms Spec(K)→ X icorresponding to the k(x i)→ K are all S-morphisms, and we thus conclude that they define a unique morphismSpec(K)→ Y . If y is the corresponding point of Y , it is clear that its projection in each of the X i is x i .

Proposition (3.4.7). — Let X i (1¶ i ¶ n) be S-preschemes, and, for each index i, let x i be a point of X i . For there toexist a point y of Y = X1 ×S X2 × . . .×S Xn whose image is x i under the ith projection for each 1¶ i ¶ n, it is necessaryand sucient that the x i all lie above the same point s of S.

Proof. The condition is evidently necessary; Lemma (3.4.6) proves that it is sucient.

In other words, if we denote by (X ) the underlying set of X , we see that we have a canonical surjectivefunction (X ×S Y )→ (X )×(S) (Y ); we must point out that this function is not injective in general; in other words,there can exist multiple distinct points z in X ×S Y that have the same projections x ∈ X and y ∈ Y ; we have alreadyseen this when S, X , and Y are prime spectra of fields k, K , and K ′ (respectively), since the tensor productK ⊗k K ′ has, in general, multiple distinct prime ideals (cf. (3.4.9)).

Corollary (3.4.8). — Let f : X → Y be an S-morphism, and f(S′) : X(S′)→ Y(S′) the S′-morphism induced by f by anextension S′→ S of the base prescheme. Let p ( resp. q) be the projection X(S′)→ X ( resp. Y(S′)→ Y ); for every subset Mof X , we have

q−1( f (M)) = f(S′)(p−1(M)).

I | 1147[Trans.] We also say that the geometric point lies over this x .

3. PRODUCTS OF PRESCHEMES 103

Proof. Indeed (3.3.11), X(S′) can be identified with the product X ×Y Y(S′) thanks to the commutative diagram

X

f

X(S′)poo

f(S′)

Y Y(S′)

qoo

By (3.4.7), the equation q(y ′) = f (x) for x ∈ M and y ′ ∈ Y(S) is equivalent to the existence of some x ′ ∈ X(S′)such that p(x ′) = x and f(S′)(x ′) = y ′, whence the corollary.

Lemma (3.4.6) can be made clearer in the following manner:

Proposition (3.4.9). — Let X and Y be S-preschemes, x a point of X , and y a point of Y , with both x and y lyingabove the same point s ∈ S. The set of points of X ×S Y with projections x and y is in bijective correspondence with the setof types of extensions (?) composed of k(x) and k(y) considered as extensions of k(s) (Bourbaki, Alg., chap. VIII, §8,prop. 2).

Proof. Let p (resp. q) be the projection from X ×S Y to X (resp. Y ), and E the subspace p−1(x)∩ q−1(y)of the underlying space of X ×S Y . First, note that the morphisms Spec(k(x))→ S and Spec(k(y))→ S factoras Spec(k(x))→ Spec(k(s))→ S and Spec(k(y))→ Spec(k(s))→ S; since Spec(k(s))→ S is a monomorphism(2.4.7), it follows from (3.2.4) that we have

P = Spec(k(x))×S Spec(k(y)) = Spec(k(x))×Spec(k(s)) Spec(k(y)) = Spec(k(x)⊗k(s) k(y)).

We will define two maps, α : P0→ E and β : E→ P0, inverse to one another (where P0 denotes the underlying setof the prescheme P). If i : Spec(k(x))→ X and j : Spec(k(y))→ Y are the canonical morphisms (2.4.5), we takeα to be the map of underlying spaces corresponding to the morphism i ×S j. On the other hand, every z ∈ Edefines, by hypothesis, two k(s)-monomorphisms, k(x)→ k(z) and k(y)→ k(z), and thus a k(s)-monomorphismk(x) ⊗k(s) k(y) → k(z), and thus a morphism Spec(k(z)) → P; β(z) will be the image of z in P0 under thismorphism. The verification of the fact that α β and β α are the identity maps follows from (2.4.5) and thedefinition of the product (3.2.1). Finally, we know that P0 is in bijective correspondence with the set of types ofextensions (?) composed of k(x) and k(y) (Bourbaki, Alg., chap. VIII, §8, prop. 1).

3.5. Surjections and injections.

(3.5.1). In a general sense, consider a property P of morphisms of preschemes, and the following two propositions:(i) If f : X → X ′ and g : Y → Y ′ are S-morphisms that have property P, then f ×S g also has property P.(ii) If f : X → Y is an S-morphism that has property P, then every S′-morphism f(S′) : X(S′)→ Y(S′), induced

by f by an extension of the base prescheme, also has property P.Since f(S′) = f ×S 1S′ , we see that, if, for every prescheme X , the identity 1X has property P, then (i) implies

(ii); on the other hand, since f ×S g is the composite morphism

X ×S Yf ×1Y−−−→ X ′ ×S Y

1X ′×g−−−→ X ′ ×S Y ′,

we see that, if the composition of two morphisms has property P, then so does the product f ×S g, and so (ii)implies (i).

A first application of this remark is

Proposition (3.5.2). —(i) If f : X → X ′ and g : Y → Y ′ are surjective S-morphisms, then f ×S g is surjective.(ii) If f : X → Y is a surjective S-morphism, then f(S′) is surjective for every extension S′ of the base prescheme.

Proof. The composition of any two surjections being a surjection, it suces to prove (ii); but this propositionfollows immediately from (3.4.8) applied to M = X .

Proposition (3.5.3). — For a morphism f : X → Y to be surjective, it is necessary and sucient that, for every eldK and every morphism Spec(K)→ Y , there exist an extension K ′ of K and a morphism Spec(K ′)→ X that make thefollowing diagram commute:

X

f

Spec(K ′)oo

Y Spec(K).oo

3. PRODUCTS OF PRESCHEMES 104

Proof. The condition is sucient because, for all y ∈ Y , it suces to apply it to a morphism Spec(K)→ Ycorresponding to a monomorphism k(y)→ K , with K being an extension of k(y) (2.4.6). Conversely, supposethat f is surjective, and let y ∈ Y be the image of the unique point of Spec(K); there exists some x ∈ X suchthat f (x) = y ; we will consider the corresponding monomorphism k(y)→ k(x) (2.2.1); it then suces to takeK ′ to be the extension of k(y) such that there exist k(y)-monomorphisms from k(x) and K to K ′ (Bourbaki,Alg., chap. V, §4, prop. 2); the morphism Spec(K ′)→ X corresponding to k(x)→ K ′ is exactly that for which weare searching.

With the language introduced in (3.4.5), we can say that every geometric point of Y with values in K comes froma geometric point of X with values in an extension of K .

Denition (3.5.4). — We say that a morphism f : X → Y of preschemes is universally injective, or a radicialmorphism, if, for every field K , the corresponding map X (K)→ Y (K) is injective.

It follows also from the definitions that every monomorphism of preschemes (T, 1.1) is radicial.

(3.5.5). For a morphism f : X → Y to be radicial, it suces that the condition of Definition (3.5.4) hold forevery algebraically closed field. In fact, if K is an arbitrary field, and K ′ an algebraically-closed extension of K ,then the diagram

X (K) α //

φ

Y (K)

φ ′

X (K ′) α′ // Y (K ′)

commutes, where φ and φ ′ come from the morphism Spec(K ′)→ Spec(K), and α and α′ corresponding to f .However, φ is injective, and so too is α′, by hypothesis; hence α is necessarily injective.

Proposition (3.5.6). — Let f : X → Y and g : Y → Z be two morphisms of preschemes.(i) If f and g are radicial, then so is g f .(ii) Conversely, if g f is radicial, then so is f .

Proof. Taking into account Definition (3.5.4), the proposition reduces to the corresponding claims for themaps X (K)→ Y (K)→ Z(K), and these claims are evident.

Proposition (3.5.7). —(i) If the S-morphisms f : X → X ′ and g : X → X ′ are radicial, then so is f ×S g .(ii) If the S-morphism f : X → Y is radicial, then so is f(S′) : X(S′)→ Y(S′) for every extension S′→ S of the base

prescheme.

Proof. Given (3.5.1), it suces to prove (i). We have seen (3.4.2.1) that

(X ×S Y )(K) = X (K)×S(K) Y (K),

(X ′ ×S Y ′)(K) = X ′(K)×S(K) Y ′(K),with the map (X×S Y )(K)→ (X ′×S Y ′)(K) corresponding to f ×S g thus being identified with (u, v)→ ( f u, g v),and the proposition then follows.

Proposition (3.5.8). — For a morphism f = (ψ,θ) : X → Y to be radicial, it is necessary and sucient for ψ to beinjective and for the monomorphism θ x : k(ψ(x))→ k(x) to make k(x) a radicial extension of k(ψ(x)) for every x ∈ X .

Proof. We suppose that f is radicial and first show that the equation ψ(x1) = ψ(x2) = y necessarilyimplies that x1 = x2. Indeed, there exists a field K , and an extension of k(y), along with k(y)-monomorphismsk(x1)→ K and k(x2)→ K (Bourbaki, Alg., chap. V, §4, prop. 2); the corresponding morphisms u1 : Spec(K)→ Xand u2 : Spec(K)→ X are then such that f u1 = f u2, and so u1 = u2 by hypothesis, and this implies, inparticular, that x1 = x2. We now consider k(x) as the extension of k(ψ(x)) by means of θ x : if k(x) is nota radicial algebraically-closed extension, then there exist two distinct k(ψ(x))-monomorphisms from k(x) toan algebraically-closed extension K of k(ψ(x)), and the two corresponding morphisms Spec(K) → X wouldcontradict the hypothesis. Conversely, taking (2.4.6) into account, it is immediate that the conditions stated aresucient for f to be radicial.

Corollary (3.5.9). — If A is a ring, and S is a multiplicative set of A, then the canonical morphism Spec(S−1A)→Spec(A) is radicial.

Proof. Indeed, this morphism is a monomorphism (1.6.2).

3. PRODUCTS OF PRESCHEMES 105

Corollary (3.5.10). — Let f : X → Y be a radicial morphism, g : Y ′→ Y a morphism, and X ′ = X(Y ′) = X ×Y Y ′.Then the radicial morphism f(Y ′) (3.5.7, ii) is a bijection from the underlying space of X to g−1( f (X )); further, for everyeld K , the set X ′(K) can be identied with the subset of Y ′(K) given by the inverse image of the map Y ′(K)→ Y (K)(corresponding to g) from the subset X (K) of Y (K).

Proof. The first claim follows from (3.5.8) and (3.4.8); the second, from the commutativity of the followingdiagram: I | 117

X ′(K) //

Y ′(K)

X (K) // Y (K)

Remark (3.5.11). — We say that a morphism f = (ψ,θ) of preschemes is injective if the map ψ is injective. Fora morphism f = (ψ,θ) : X → Y to be radicial, it is necessary and sucient that, for every morphism Y ′→ Y , themorphism f(Y ′) : X(Y ′)→ Y ′ be injective (which justifies the terminology of a universally injective morphism). Infact, the condition is necessary by (3.5.7, ii) and (3.5.8). Conversely, the condition implies that ψ is injective; if,for some x ∈ X , the monomorphism θ x : k(ψ(x))→ k(x) were not radicial, then there would be an extension Kof k(ψ(x)), and two distinct morphisms Spec(K)→ X corresponding to the same morphism Spec(K)→ Y (3.5.8).But then, setting Y ′ = Spec(K), there would be two distinct Y ′-sections of X(Y ′) (3.3.14), which contradicts thehypothesis that f(Y ′) is injective.

3.6. Fibres.

Proposition (3.6.1). — Let f : X → Y be a morphism, y a point of Y , and ay an ideal of denition for Oy for themy -preadic topology. Then the projection p : X ×Y Spec(Oy/ay)→ X is a homeomorphism from the underlying space of theprescheme X ×Y Spec(Oy/ay) to the bre f −1(y) equipped with the topology induced from that of the underlying space ofX .

Proof. Since Spec(Oy/ay)→ Y is radicial ((3.5.4) and (2.4.7)), since Spec(Oy/ay) is a single point, and sincethe ideal my/ay is nilpotent by hypothesis (1.1.12), we already know ((3.5.10) and (3.3.4)) that p identifies, as sets,the underlying space of X×Y Spec(Oy/ay) with f −1(y); everything reduces to proving that p is a homeomorphism.By (3.2.7), the question is local on X and Y , and so we can suppose that X = Spec(B) and Y = Spec(A), withB being an A-algebra. The morphism p then corresponds to the homomorphism 1⊗φ : B → B ⊗A A′, whereA′ = Ay/ay and φ is the canonical map from A to A′. Then every element of B ⊗A A′ can be written as

∑

i

bi ⊗φ (ai)/φ (s) =

∑

i

(ai bi ⊗ 1)

(1⊗φ (s))−1,

where s 6∈ jy , and Proposition (1.2.4) applies.

(3.6.2). Throughout the rest of this treatise, whenever we consider a fibre f −1(y) of a morphism as having thestructure of a k(y)-prescheme, it will always be the prescheme obtained by transporting the structure of X ×Y Spec(k(y))by the projection to X . We will also write this (latter) product as X ×Y k(y), or X ⊗OY

k(y); more generally, if B isan Oy -algebra, we will denote by X ×Y B or X ⊗OY

B the product X ×Y Spec(B).

With the preceding convention, it follows from (3.5.10) that the points of X with values in an extension K ofk(y) are identified with the points of f −1(y) with values in K .

(3.6.3). Let f : X → Y and g : Y → Z be two morphisms, and h= g f their composition; for all z ∈ Z , the fibreh−1(z) is a prescheme isomorphic to

X ×Z Spec(k(z)) = (X ×Y Y )×Z Spec(k(z)) = X ×Y g−1(z).

Inparticular, if U is an open subset of X , then the prescheme induced on U ∩ f −1(y) by the prescheme f −1(y) is I | 118isomorphic to f −1

U (y) ( fU being the restriction of f to U),

Proposition (3.6.4). — (Transitivity of fibres) Let f : X → Y and g : Y ′→ Y be morphisms; let X ′ = X×Y Y ′ = X(Y ′)and f ′ = f(Y ′) : X ′ → Y ′. For every y ′ ∈ Y ′, if we let y = g(y ′), then the prescheme f ′−1(y ′) is isomorphic tof −1(y)⊗k(y) k(y ′).

Proof. Indeed, it suces to remark that the two preschemes (X ⊗Y k(y))⊗k(y) k(y ′) and (X ×Y Y ′)⊗Y ′ k(y ′)are both canonically isomorphic to X ×Y Spec(k(y ′)) by (3.3.9.1).

4. SUBPRESCHEMES AND IMMERSION MORPHISMS 106

In particular, if V is an open neighborhood of y in Y , and we denote by fV the restriction of f to the inducedprescheme on f −1(V ), then the preschemes f −1(y) and f −1

V (y) are canonically identified.

Proposition (3.6.5). — Let f : X → Y be a morphism, y a point of Y , Z the local prescheme Spec(Oy), and p = (ψ,θ)the projection X ×Y Z → X ; then p is a homeomorphism from the underlying space of X ×Y Z to the subspace f −1(Z) of X(when the underlying space of Z is identified with a subspace of Y , cf. (2.4.2)), and, for all t ∈ X ×Y Z , lettingz = ψ(t), we have that θ]t is an isomorphism from Ox to Ot .

Proof. Since Z (identified as a subspace of Y ) is contained inside every ane open containing y (2.4.2), wecan, as in (3.6.1), reduce to the case where X = Spec(A) and Y = Spec(B) are ane schemes, with A being aB-algebra. Then X ×Y Z is the prime spectrum of A⊗B By , and this ring is canonically identified with S−1A, whereS is the image of B − jy in A (0, 1.5.2); since p then corresponds to the canonical homomorphism A→ S−1A, theproposition follows from (1.6.2).

3.7. Application: reduction of a prescheme mod. J. This section, which makes use of notions and resultsfrom Chapter I and Chapter II, will not be used in what follows in this treatise, and is only intended for readers familiarwith classical algebraic geometry.

(3.7.1). Let A be a ring, X an A-prescheme, and J an ideal of A; then X0 = X ⊗A (A/J) is an (A/J)-prescheme,which we sometimes say is induced from X by reduction mod. J.

. This terminology is used foremost when A is a local ring and J its maximal ideal, in such a way that X0 is aprescheme over the residue field k = A/J of A.

When A is also integral, with field of fractions K , we can consider the K -prescheme X ′ = X ⊗A K . By anabuse of language which we will not use, it has been said, up until now, that X0 is induced by X ′ by reductionmod. J. In the case where this language was used, A was a local ring of dimension 1 (most often a discretevaluation ring) and it was implied (be it more or less explicitly) that the given K -prescheme X ′ was a closedsubprescheme of a K -prescheme P ′ (in fact, a projective space of the form (?) Pr

K , cf. (II, 4.1.1)), itself of theform P ′ = P ⊗A K , where P is a given A-prescheme (in fact, the A-scheme Pr

A, with the notation of (II, 4.1.1)). Inour language, the definition of X0 in terms of X ′ is formulated as follows:

We consider the ane scheme Y = Spec(A), formed of two points, the unique closed point y = J and the I | 119generic point (0), the singleton set U of the generic point being thus an open U = Spec(K) in Y . If X is anA-prescheme (or, in other words, a Y -prescheme), then X ⊗A K = X ′ is exactly the prescheme induced by X onψ−1(U), denoting by ψ the structure morphism X → Y . In particular, if φ is the structure morphism P → Y ,then a closed subprescheme X ′ of P ′ =φ−1(U) is a (locally closed) subprescheme of P. If P is Noetherian (forexample, if A is Noetherian and P is of finite type over A), then there exists a smaller closed subprescheme X = X ′

of G that through which X ′ factors (9.5.10), and X ′ is the prescheme induced by X on the open φ−1(U)∩ X , andso is isomorphic to X ⊗A K (9.5.10). The immersion of X ′ into P ′ = P⊗A K thus lets us canonically consider X ′ as being ofthe form X ′ = X ⊗A K , where X is an A-prescheme. We can then consider the reduced mod. J prescheme X0 = X ⊗A k,which is exactly the fibre ψ−1(y) of the closed point y . Up until now, lacking the adequate terminology, we hadavoided explicitly introducing the A-prescheme X . One ought to, however, note that all the claims normallymade about the “reduced mod. J” prescheme X0 should be seen as consequences of more complicated claimsabout X itself, and cannot be satisfactorily formulated or understood except by interpreting them as such. Itseems also that any hypotheses made on X0 always reduce to hypotheses on X itself (independent of the priordata of an immersion of X ′ in Pr

K), which lets us give more intrinsic statements.

(3.7.3). Lastly, we draw attention to a very particular fact, which has undoubtedly contributed to slowing theconceptual clarification of the situation envisaged here: if A is a discrete valuation ring, and if X is proper over A(which is indeed the case if X is a closed subprescheme of some Pr

A, cf. (II, 5.5.4)), then the points of X withvalues in A and the points of X ′ with values in k are in bijective correspondence (II, 7.3.8). This is why we oftenbelieve that facts about X ′ have been proved, when in reality we have proved facts about X , and these remainvaluable (in this form) whenever we no longer suppose that the base local ring is of dimension 1.

§4. Subpreschemes and immersion morphisms

4.1. Subpreschemes.

(4.1.1). As the notion of a quasi-coherent sheaf (0, 5.1.3) is local, a quasi-coherent OX -module F over aprescheme X can be defined by the following condition: for each ane open V of X , F|V is isomorphic to thesheaf associated to a Γ(V,OX )-module (1.4.1). It is clear that, over a prescheme X , the structure sheaf OX isquasi-coherent, and that the kernels, cokernels, and images of homomorphisms of quasi-coherent OX -modules, as

4. SUBPRESCHEMES AND IMMERSION MORPHISMS 107

well as inductive limits and direct sums of quasi-coherent OX -modules, are also quasi-coherent (Theorem (1.3.7)and Corollary (1.3.9)).

Proposition (4.1.2). — Let X be a prescheme, and I a quasi-coherent sheaf of ideals of OX . Then the support Y of thesheaf OX/I is closed, and if we denote by OY the restriction of OX/I to Y , then (Y,OY ) is a prescheme.

I | 120Proof. It evidently suces (2.1.3) to consider the case where X is an ane scheme, and to show that, in this

case, Y is closed in X , and is an ane scheme. Indeed, if X = Spec(A), then we have OX = eA and I = eI, where I isan ideal of A (1.4.1); Y is then equal to the closed subset V (I) of X and can be identified with the prime spectrumof the ring B = A/I (1.1.11); in addition, if φ is the canonical homomorphism A→ B = A/I, then the directimage aφ∗(eB) is canonically identified with the sheaf eA/eI= OX/I (Proposition (1.6.3) and Corollary (1.3.9)),which finishes the proof.

We say that (Y,OY ) is the subprescheme of (X ,OX ) dened by the sheaf of ideals I ; this is a particular case ofthe more general notion of a subprescheme:

Denition (4.1.3). — We say that a ringed space (Y,OY ) is a subprescheme of a prescheme (X ,OX ) if:1st. Y is a locally closed subspace of X ;2nd. if U denotes the largest open subset of X containing Y such that Y is closed in U (equivalently, the

complement in X of the boundary of Y with respect to Y ), then (Y,OY ) is a subprescheme of (U ,OX |U)defined by a quasi-coherent sheaf of ideals of OX |U .

We say that the subprescheme (Y,OY ) of (X ,OX ) is closed if Y is closed in X (in which case U = X ).

It follows immediately from this definition and Proposition (4.1.2) that the closed subpreschemes of X are incanonical bijective correspondence with the quasi-coherent sheaves of ideals J of OX , since if two such sheaves Jand J ′ have the same (closed) support Y , and if the restrictions of OX/J and OX/J ′ to Y are identical, thenwe have J ′ = J .

(4.1.4). Let (Y,OY ) be a subprescheme of X , U the largest open subset of X such that Y is closed (and thuscontained) in U , and V an open subset of X contained in U ; then V ∩ Y is closed in V . In addition, if Y isdefined by the quasi-coherent sheaf of ideals J of OX |U , then J |V is a quasi-coherent sheaf of ideals of OX |V ,and it is immediate that the prescheme induced by Y on Y ∩ V is the closed subprescheme of V defined by thesheaf of ideals J |V . Conversely:

Proposition (4.1.5). — Let (Y,OY ) be a ringed space such that Y is a subspace of X , and there exists a cover (Vα) of Yby open subsets of X such that, for each α, Y ∩ Vα is closed in Vα, and the ringed space (Y ∩ Vα,OY |(Y ∩ Vα)) is a closedsubprescheme of the prescheme induced on Vα by X . Then (Y,OY ) is a subprescheme of X .

Proof. The hypotheses imply that Y is locally closed in X and that the largest open U in which Y is closed(and thus contained) contains all the Vα; we can thus reduce to the case where U = X and Y is closed in X . Wethen define a quasi-coherent sheaf of ideals J of OX by taking J |Vα to be the sheaf of ideals of OX |Vα whichdefine the closed subprescheme (Y ∩ Vα,OY |(Y ∩ Vα)), and, for each open subset W of X not intersecting Y ,J |W = OX |W . We see immediately, by Definition (4.1.3) and (4.1.4), that there exists a unique sheaf of ideals Jsatisfying these conditions, and that it defines the closed subprescheme (Y,OY ).

In particular, the induced (by X ) prescheme on an open subset of X is a subprescheme of X .

Proposition (4.1.6). — A subprescheme (resp. a closed subprescheme) of a subprescheme (resp. closed subprescheme) of X I | 121is canonically identied with a subprescheme (resp. closed subprescheme) of X .

Proof. Since a locally closed subset of a locally closed subspace of X is a locally closed subspace of X , it isclear (4.1.5) that the question is local and that we can thus suppose that X is ane; the proposition then followsfrom the canonical identification of A/J′ with (A/J)/(J′/J), where A is some ring, and J and J′ are ideals of Asuch that J ⊂ J′.

In what follows, we will always make use of the above identification.

(4.1.7). Let Y be a subprescheme of a prescheme X , and denote by ψ the canonical injection Y → X of theunderlying subspaces; we know that the inverse image ψ∗(OX ) is the restriction OX |Y (0, 3.7.1). If, for each y ∈ Y ,we denote by ωy the canonical homomorphism (OX )y → (OY )y , then these homomorphisms are the restrictionsto stalks of a surjective homomorphism ω of sheaves of rings OX |Y →OY : indeed, it suces to check locally on Y ,that is to say, we can suppose that X is ane and that the subprescheme Y is closed; if in this case I is the sheafof ideals in OX which defines Y , then the ωy are none other than the restriction to stalks of the homomorphism

4. SUBPRESCHEMES AND IMMERSION MORPHISMS 108

OX |Y → (OX/I )|Y . We have thus defined a monomorphism of ringed spaces (0, 4.1.1) j = (ψ,ω[) which is evidentlya morphism Y → X of preschemes (2.2.1), and we call this the canonical injection morphism.

If f : X → Z is a morphism, we then say that the composite morphism Yj−→ X

f−→ Z is the restriction of f to

the subprescheme Y .

(4.1.8). Conforming to the general definitions (T, I, 1.1), we will say that a morphism of preschemes f : Z → X

is majoré 8 by the injection morphism j : Y → X of a subprescheme Y of X if f factors as Zg−→ Y

j−→ X , where g is

a morphism of preschemes; further, g is necessarily unique since j is a monomorphism.

Proposition (4.1.9). — For a morphism f : Z → X to be factor through an injection morphism j : Y → X , it is necessaryand sucient that f (Z) ⊂ Y and, for all z ∈ Z , letting y = f (z), that the homomorphism (OX )y →Oz corresponding to ffactor as (OZ)y → (OY )y →Oz ( or equivalently, for the kernel of (OX )y →Oz to contain the kernel of (OX )y → (OY )y ).

Proof. The conditions are evidently necessary. To see that they are sucient, we can reduce to the casewhere Y is a closed subprescheme of X , by replacing X by an open U such that Y is closed in U (4.1.3) if necessary;Y is then defined by a quasi-coherent sheaf of ideals I of OX . Let f = (ψ,θ), and let J be the sheaf of ideals ofψ∗(OX ), kernel of θ] : ψ∗(OX )→OZ ; considering the properties of the functor ψ∗ (0, 3.7.2), the hypotheses implythat, for each z ∈ Z , we have (ψ∗(I ))z ⊂ Jz , and, as a result, that ψ∗(I ) ⊂ J . Thus θ] factors as

ψ∗(OX ) −→ ψ∗(OX )/ψ∗(I ) = ψ∗(OX/I )ω−→ OZ ,

the first arrow being the canonical homomorphism. Let ψ′ be the continuous map Z → Y coinciding with ψ;it is clear that we have ψ′∗(OY ) = ψ∗(OX/J ); on the other hand, ω is evidently a local homomorphism, sog = (ψ′,ω[) is a morphism of preschemes Z → Y (2.2.1), which, according to the above, is such that f = j g, I | 122hence the proposition.

Corollary (4.1.10). — For an injection morphism Z → X to be factor through the injection morphism Y → X , it isnecessary and sucient for Z to be a subprescheme of Y .

We then write Z ¶ Y , and this condition is evidently an ordering on the set of subpreschemes of X .

4.2. Immersion morphisms.

Denition (4.2.1). — We say that a morphism f : Y → X is an immersion (resp. a closed immersion, an open

immersion) if it factors as Yg−→ Z

j−→ X , where g is an isomorphism, Z is a subprescheme of X (resp. a closed

subprescheme, a subprescheme induced by an open set), and j is the injection morphism.

The subprescheme Z and the isomorphism g are then determined in a unique way, since if Z ′ is a secondsubprescheme of X , j′ the injection Z ′→ X , and g ′ an isomorphism Y → Z ′ such that j g = j′ g ′, then wehave j′ = j g g ′−1, hence Z ′ ¶ Z (4.1.10), and we can similarly show that Z ¶ Z ′, hence Z ′ = Z , and, since jis a monomorphism of preschemes, g ′ = g.

We say that f = j g is the canonical factorization of the immersion f , and the subprescheme Z and theisomorphism g are those associated to f .

It is clear that an immersion is a monomorphism of preschemes (4.1.7) and a fortiori a radicial morphism(3.5.4).

Proposition (4.2.2). —(a) For a morphism f = (ψ,θ) : Y → X to be an open immersion, it is necessary and sucient for ψ to be a

homeomorphism from Y to some open subset of X , and, for all y ∈ Y , that the homomorphism θ]y : Oψ(y)→Oy

be bijective.(b) For a morphism f = (ψ,θ) : Y → X to be an immersion (resp. a closed immersion), it is necessary and sucient

for ψ to be a homeomorphism from Y to some locally closed (resp. closed) subset of X , and, for all y ∈ Y , that thehomomorphism θ]y : Oψ(y)→Oy be surjective.

Proof.(a) The conditions are evidently necessary. Conversely, if they are satisfied, then it is clear that θ] is an

isomorphism from OY to ψ∗(OX ), and ψ∗(OX ) is the sheaf induced by “transport of structure” via ψ−1

from OX |ψ(Y ); hence the conclusion.8[Trans.] There doesn’t seem to be an English equivalent of this, except for ‘bounded above’, which doesn’t make much sense in this context. We would

normally just say that ‘ f factors through j’, but to avoid having to entirely restructure the often-lengthy sentences in the original, we sometimes (but aslittle as we can) use ‘majoré’.

4. SUBPRESCHEMES AND IMMERSION MORPHISMS 109

(b) The conditions are evidently necessary—we prove that they are sucient. Consider first the particularcase where we suppose that X is an ane scheme, and that Z = ψ(Y ) is closed in X . We then know(0, 3.4.6) that ψ∗(OY ) has support equal to Z , and that, denoting its restriction to Z by O ′Z , the ringedspace (Z ,O ′Z) is induced from (Y,OY ) by transport of structure via the homeomorphism ψ considered asa map from Y to Z . Let us now show that f∗(OY ) = ψ∗(OY ) is a quasi-coherent OX -module. Indeed, for allx 6∈ Z , ψ∗(OY ) restricted to a suitable neighborhood of x is zero. On the contrary, if z ∈ Z , then we havex = ψ(y) for some well-defined y ∈ Y ; let V be an ane open neighborhood of y in Y ; ψ(V ) is thenopen in Z , and so equal to the intersection of Z with an open subset U of X , and the restriction of U toψ∗(OY ) is identical to the restriction of U to the direct image (ψV )∗(OY |V ), where ψV is the restriction I | 123of ψ to V . The restriction of the morphism (ψ,θ) to (V,OY |V ) is a morphism from this aforementionedprescheme to (X ,OX ), and, as a result, is of the form (aφ , eφ ), where φ is the homomorphism fromthe ring A= Γ(X ,OX ) to the ring Γ(V,OY ) (1.7.3); we conclude that (ψV )∗(OY |V ) is a quasi-coherentOX -module (1.6.3), which proves our assertion, due to the local nature of quasi-coherent sheaves. Inaddition, the hypothesis that ψ is a homeomorphism implies (0, 3.4.5) that, for all y ∈ Y , ψy is anisomorphism (ψ∗(OY ))ψ(y)→Oy ; since the diagram

Oψ(y)θψ(y) //

ψyρψ(y)

(ψ∗(OY ))ψ(y)

ψy

(ψ∗(OX ))y

θ]y // Oy

is commutative, and the vertical arrows are the isomorphisms (0, 3.7.2), the hypothesis that θ]y issurjective implies that θψ(y) is surjective as well. Since the support of ψ∗(OY ) is Z = ψ(Y ), θ is asurjective homomorphism from OX = eA to the quasi-coherent OX -module f∗(OY ). As a result, there existsa unique isomorphism ω from a sheaf quotient eA/eJ (with J being an ideal of A) to f∗(OY ) which, whencomposed with the canonical homomorphism eA→ eA/eJ, gives θ (1.3.8); if OZ denotes the restriction ofeA/eJ to Z , then (Z ,OZ) is a subprescheme of (X ,OX ), and f factors through the canonical injection ofthis subprescheme into X and the isomorphism (ψ0,ω0), where ψ0 is ψ considered as a map from Y toZ , and ω0 the restriction of ω to OZ .

We now pass to the general case. Let U be an ane open subset of X such that U ∩ψ(Y ) is closedin U and nonempty. By restricting f to the prescheme induced by Y on the open subset ψ−1(U), andby considering it as a morphism from this prescheme to the prescheme induced by X on U , we reduceto the first case; the restriction of f to ψ−1(U) is thus a closed immersion ψ−1(U)→ U , canonicallyfactoring as jU gU , where gU is an isomorphism from the prescheme ψ−1(U) to a subprescheme ZUof U , and jU is the canonical injection ZU → U . Let V be a second ane open subset of X such thatV ⊂ U ; since the restriction Z ′V of ZU to V is a subprescheme of the prescheme V , the restriction of f toψ−1(V ) factors as j′V g ′V , where j′V is the canonical injection Z ′V → V and g ′V is an isomorphism fromψ−1(V ) to Z ′V . By the uniqueness of the canonical factorization of an immersion (4.2.1), we necessarilyhave that Z ′V = ZV and g ′V = gV . We conclude (4.1.5) that there is a subprescheme Z of X whoseunderlying space is ψ(Y ), and whose restriction to each U ∩ψ(Y ) is ZU ; the gU are then the restrictionsto ψ−1(U) of an isomorphism g : Y → Z such that f = j g, where j is the canonical injection Z → X .

Corollary (4.2.3). — Let X be an ane scheme. For a morphism f = (ψ,θ) : Y → X to be a closed immersion, it isnecessary and sucient for Y to be an ane scheme, and the homomorphism Γ(ψ) : Γ(OX )→ Γ(OY ) to be surjective.

Corollary (4.2.4). —(a) Let f be a morphism Y → X , and (Vλ) a cover of f (Y ) by open subsets of X . For f to be an immersion (resp. an

open immersion), it is necessary and sucient for its restriction to each of the induced preschemes f −1(Vλ) to be I | 124an immersion (resp. an open immersion) into Vλ .

(b) Let f be a morphism Y → X , and (Vλ) an open cover of X . For f to be a closed immersion, it is necessary andsucient for its restriction to each of the induced preschemes f −1(Vλ) to be a closed immersion into Vλ .

Proof. Let f = (ψ,θ); in the case (a), θ]y is surjective (resp. bijective) for all y ∈ Y , and in the case (b), θ]yis surjective for all y ∈ Y ; it thus suces to check, in case (a), that ψ is a homeomorphism from Y to a locallyclosed (resp. open) subset of X , and, in case (b), that ψ is a homeomorphism from Y to a closed subset of X .Now ψ is evidently injective, and sends each neighborhood of y in Y to a neighborhood of ψ(y) is ψ(Y ) for all

4. SUBPRESCHEMES AND IMMERSION MORPHISMS 110

y ∈ Y , by virtue of the hypothesis; in case (a), ψ(Y )∩ Vλ is locally closed (resp. open) in Vλ , so ψ(Y ) is locallyclosed (resp. open) in the union of the Vλ , and a fortiori in X ; in case (b), ψ(Y )∩ Vλ is closed in Vλ , so ψ(Y ) isclosed in X since X =

⋃

λ Vλ .

Proposition (4.2.5). — The composition of any two immersions (resp. of two open immersions, of two closed immersions)is an immersion (resp. an open immersion, a closed immersion).

Proof. This follows easily from (4.1.6).

4.3. Products of immersions.

Proposition (4.3.1). — Let α : X ′ → X , β : Y ′ → Y be two S-morphisms; if α and β are immersions (resp. openimmersions, closed immersions), then α×S β is an immersion (resp. an open immersion, a closed immersion). In addition, ifα (resp. β) identies X ′ (resp. Y ′) with a subprescheme X ′′ (resp. Y ′′) of X (resp. Y ), then α×S β identies the underlyingspace of X ′ ×S Y ′ with the subspace p−1(X ′′) ∩ q−1(Y ′′) of the underlying space of X ×S Y , where p and q denote theprojections from X ×S Y to X and Y respectively.

Proof. According to Definition (4.2.1), we can restrict to the case where X ′ and Y ′ are subpreschemes, andα and β the injection morphisms. The proposition has already been proven for the subpreschemes induced byopen sets (3.2.7); since every subprescheme is a closed subprescheme of a prescheme induced by an open set(4.1.3), we can reduce to the case where X ′ and Y ′ are closed subpreschemes.

Let us first show that we can assume S to be ane. Let (Sλ) be a cover of S by ane open sets; if φ andψ are the structure morphisms of X and Y , then let Xλ = φ−1(Sλ) and Yλ = ψ−1(Sλ). The restriction X ′λ (resp.Y ′λ ) of X ′ (resp. Y ′) to Xλ ∩ X ′ (resp. Yλ ∩ Y ′) is a closed subprescheme of Xλ (resp. Yλ), the preschemes Xλ , Yλ ,X ′λ , and Y ′λ can be considered as Sλ -preschemes, and the products Xλ ×S Yλ and Xλ ×Sλ Yλ (resp. X ′λ ×S Y ′λ andX ′λ ×Sλ Y ′λ ) are identical (3.2.5). If the proposition is true when S is ane, then the restriction of α×S β to eachof the X ′λ ×S Y ′λ is thus an immersion (3.2.7). Since the product X ′λ ×S Y ′µ (resp. Xλ ×S Yµ) can be identified with(X ′λ ∩ X ′µ)×S (Y ′λ ∩ Y ′µ) (resp. (Xλ ∩ Xµ)×S (Yλ ∩ Yµ)) (3.2.6.4), the restriction of α×S β to each of the X ′λ ×S Y ′µ is I | 125also an immersion; the same is true for α×S β by (4.2.4).

Next, we show that we can assume X and Y to be ane. Indeed, let (Ui) (resp. (Vj)) be a cover of X (resp.Y ) by ane open sets, and let X ′i (resp. Y ′j ) be the restriction of X ′ (resp. Y ′) to X ′∩Ui (resp. Y ′∩Vj), which is aclosed subprescheme of Ui (resp. Vj); Ui ×S Vj can be identified with the restriction of X ×S Y to p−1(Ui)∩q−1(Vj)(3.2.7); similarly, if p′ and q′ are the projections from X ′×S Y ′, then X ′i ×S Y ′j can be identified with the restriction

of X ′ ×S Y ′ to p′−1(X ′i )∩ q′−1(Y ′j ). Set γ = α×S β; we have, by definition, p γ = α p′ and q γ = β q′; since

X ′i = α−1(Ui) and Y ′j = β

−1(Vj), we also have p′−1(X ′i ) = γ−1(p−1(Ui)) and q′−1(Y ′j ) = γ

−1(q−1(Vj)), hence

p′−1(X ′i )∩ q′−1(Y ′j ) = γ−1(p−1(Ui)∩ q−1(Vj)) = γ−1(Ui ×S Vj),

and we then conclude as in the previous part of the proof.So suppose X , Y , and S are ane, and let B, C , and A be their respective rings. Then B and C are A-algebras,

and X ′ and Y ′ are ane schemes whose rings are quotient algebras B′ and C ′ of B and C respectively. Inaddition, we have α = (aρ, eρ) and β = (aσ, eσ), where ρ and σ are (respectively) the canonical homomorphismsB→ B′ and C → C ′ (1.7.3). With that in mind, we know that X ×S Y (resp. X ′ ×S Y ′) is an ane scheme withring B ⊗A C (resp. B′ ⊗A C ′), and α×S β = (aτ,eτ), where τ is the homomorphism ρ⊗σ from B ⊗A C to B′ ⊗A C ′

(Proposition (3.2.2) and Corollary (3.2.3)); since this homomorphism is surjective, α ×S β is an immersion.In addition, if b (resp. c) is the kernel of ρ (resp. σ), then the kernel of τ is u(b) + v(c), where u (resp. v) isthe homomorphism b 7→ b⊗ 1 (resp. c 7→ 1⊗ c). Since p = (au,eu) and q = (a v,ev), this kernel corresponds, inthe prime spectrum of B ⊗A C , to the closed set p−1(X ′)∩ q−1(Y ′) ((1.2.2.1) and Proposition (1.1.2, iii)), whichfinishes the proof.

Corollary (4.3.2). — If f : X → Y is an immersion (resp. an open immersion, a closed immersion) and an S-morphism,then f(S′) is an immersion (resp. an open immersion, a closed immersion) for every extension S′→ S of the base prescheme.

4.4. Inverse images of a subprescheme.

Proposition (4.4.1). — Let f : X → Y be a morphism, Y ′ a subprescheme (resp. a closed subprescheme, a preschemeinduced by an open set) of Y , and j : Y ′→ Y the injection morphism. Then the projection p : X ×Y Y ′→ X is an immersion(resp. a closed immersion, an open immersion); the underlying space of the subprescheme of X associated to p is f −1(Y ′);in addition, if j′ is the injection morphism of this subprescheme into X , then for a morphism h : Z → X to be such thatf h : Z → Y factors through j, it is necessary and sucient for h to factor through j′.

4. SUBPRESCHEMES AND IMMERSION MORPHISMS 111

Proof. Since p = 1X ×Y j (3.3.4), the first claim follows from Proposition (4.3.1); the second is a particularcase of Corollary (3.5.10) (after swapping the roles of X and Y ′). Finally, if we have f h = j h′, where h′

is a morphism Z → Y ′, then it follows from the definition of the product that we have h = p u, where u is amorphism Z → X ×Y Y ′, whence the last claim.

We say that the subprescheme of X thus defined is the inverse image of the subprescheme Y ′ of Y under themorphism f , terminology which is consistent with that introduced more generally in (3.3.6). When we speak of I | 126f −1(Y ′) as a subprescheme of X , this will always be the subprescheme we mean.

When the preschemes f −1(Y ′) and X are identical, j′ is the identity and each morphism h : Z → X thus

factors through j′, so the morphism f : X → Y factors as Xg−→ Y ′

j−→ Y .

When y is a closed point of Y and Y ′ = Spec(k(y)) is the smallest closed subprescheme of Y having yas its underlying space (4.1.9), the closed subprescheme f −1(Y ′) is canonically isomorphic to the bre f −1(y)defined in (3.6.2), and we will use this identification in all that follows.

Corollary (4.4.2). — Let f : X → Y and g : Y → Z be morphisms, and h = g f their composition. For eachsubprescheme Z ′ of Z , the subpreschemes f −1(g−1(Z ′)) and h−1(Z ′) of X are identical.

Proof. This follows from the existence of the canonical isomorphism X ×Y (Y ×Z Z ′)' X ×Z Z ′ (3.3.9.1).

Corollary (4.4.3). — Let X ′ and X ′′ be subpreschemes of X , and j′ : X ′ → X , and j′′ : X ′′ → X their injectionmorphisms; then j′−1(X ′′) and j′′−1(X ′) are both equal to the greatest lower bound inf(X ′, X ′′) of X ′ and X ′′ for theordering ¶ on subpreschemes, and this is canonically isomorphic to X ′ ×X X ′′.

Proof. This follows immediately from Proposition (4.4.1) and Corollary (4.1.10).

Corollary (4.4.4). — Let f : X → Y be a morphism, and Y ′ and Y ′′ subpreschemes of Y ; then we have f −1(inf(Y ′, Y ′′)) =inf( f −1(Y ′), f −1(Y ′′)).

Proof. This follows from the existence of the canonical isomorphism between (X ×Y Y ′)×X (X ×Y Y ′′) andX ×Y (Y ′ ×Y Y ′′) (3.3.9.1).

Proposition (4.4.5). — Let f : X → Y be a morphism, and Y ′ a closed subprescheme of Y dened by a quasi-coherentsheaf of ideals K of OY (4.1.3); the closed subprescheme f −1(Y ′) of X is then dened by the quasi-coherent sheaf of idealsf ∗(K )OX of OX .

Proof. The statement is evidently local on X and Y ; it thus suces to note that if A is a B-algebra and K an ErrIIideal of B, then we have A⊗B (B/K) = A/KA, and to then apply (1.6.9).

Corollary (4.4.6). — Let X ′ be a closed subprescheme of X dened by a quasi-coherent sheaf of ideals J of OX , and ithe injection X ′→ X ; for the restriction f i of f to X ′ to factor through the injection j : Y ′→ Y (in other words, for itto factor as j g, with g a morphism X ′→ Y ′), it is necessary and sucient that f ∗(K ) ⊂ J .

Proof. It suces to apply Proposition (4.4.1) to i, taking Proposition (4.4.5) into account.

4.5. Local immersions and local isomorphisms.

Denition (4.5.1). — Let f : X → Y be a morphism of preschemes. We say that f is a local immersion at apoint x ∈ X if there exists an open neighborhood U of x in X and an open neighborhood V of f (x) in Y suchthat the restriction of f to the induced prescheme U is a closed immersion of U into the induced prescheme V .We say that f is a local immersion if f is a local immersion at each point of X .

Denition (4.5.2). — We say that a morphism f : X → Y is a local isomorphism at a point x ∈ X if there I | 127exists an open neighborhood U of x in X such that the restriction of f to the induced prescheme U is an openimmersion of U into Y . We say that f is a local isomorphism if f is a local isomorphism at each point of X .

(4.5.3). An immersion (resp. a closed immersion) f : X → Y can be characterized as a local immersion suchthat f is a homeomorphism from the underlying space of X to a subset (resp. a closed subset) of Y . An openimmersion f can be characterized as an injective local isomorphism.

Proposition (4.5.4). — Let X be an irreducible prescheme, and f : X → Y a dominant injective morphism. If f is alocal immersion, then f is an immersion, and f (X ) is open in Y .

Proof. Let x ∈ X , and let U be an open neighborhood of x , and V an open neighborhood of f (x) in Ysuch that the restriction of f to U is a closed immersion into V ; since U is dense in X , f (U) is dense in Y byhypothesis, so f (U) = V , and f is a homeomorphism from U to V ; the hypothesis that f is injective implies thatf −1(V ) = U , hence the proposition.

5. REDUCED PRESCHEMES; SEPARATION CONDITION 112

Proposition (4.5.5). —

(i) The composition of any two local immersions (resp. of two local isomorphisms) is a local immersion (resp. a localisomorphism).

(ii) Let f : X → X ′ and g : Y → Y ′ be two S-morphisms. If f and g are local immersions (resp. local isomorphisms),then so too is f ×S g .

(iii) If an S-morphism f is a local immersion (resp. a local isomorphism), then so too is f(S′) for every extensionS′→ S of the base prescheme.

Proof. According to (3.5.1), it suces to prove (i) and (ii).(i) follows immediately from the transitivity of closed (resp. open) immersions (4.2.5) and from the fact

that if f is a homeomorphism from X to a closed subset of Y , then for every open U ⊂ X , f (U) is open in f (X ),so there exists an open subset V of Y such that f (U) = V ∩ f (X ), and, as a result, f (U) is closed in V .

To prove (ii), let p and q be the projections from X ×X Y , and p′ and q′ the projections from X ′×S Y ′. Thereexists, by hypothesis, open neighborhoods U , U ′, V , and V ′ of x = p(z), x ′ = p′(z′), y = q(z), and y ′ = q′(z′)(respectively), such that the restrictions of f and g to U and V (respectively) are closed (resp. open) immersionsinto U ′ and V ′ (respectively). Since the underlying spaces of U ×S V and U ′ ×S V ′ can be identified with theopen neighborhoods p−1(U)∩ q−1(V ) and p′−1(U ′)∩ q′−1(V ′) of z and z′ (respectively) (3.2.7), the propositionfollows from Proposition (4.3.1).

§5. Reduced preschemes; separation condition

5.1. Reduced preschemes.

Proposition (5.1.1). — Let (X ,OX ) be a prescheme, and B a quasi-coherent OX -algebra. Then there exists a uniquequasi-coherent OX -module N whose stalk Nx at any x ∈ X is the nilradical of the ring Bx . When X is ane, and,consequently, B = eB, where B is an algebra over A(X ), then we have N = eN, where N is the nilradical of B.

I | 128Proof. The statement is local, so it suces to show the latter claim. We know that eN is a quasi-coherent

OX -module (1.4.1), and that its stalk at a point x ∈ X is the ideal Nx of the ring of fractions Bx ; it remains toprove that the nilradical of Bx is contained in Nx , the converse inclusion being evident. Let z/s be an elementof the nilradical of Bx , with z ∈ B, and s 6∈ jx ; by hypothesis, there exists an integer k such that (z/s)k = 0,which implies that there exists some t 6∈ jx such that tzk = 0. We conclude that (tz)k = 0, and, as a result, thatz/s = (tz)/(ts) ∈Nx .

We say that the quasi-coherent OX -module N thus defined is the nilradical of the OX -algebraB ; in particular,we denote by NX the nilradical of OX .

Corollary (5.1.2). — Let X be a prescheme; the closed subprescheme of X dened by the sheaf of ideals NX is the onlyreduced subprescheme (0, 4.1.4) of X that has X as its underlying space; it is also the smallest subprescheme of X that hasX as its underlying space.

Proof. Since the structure sheaf of the closed subprescheme of Y defined by NX is OX/NX , it is immediatethat Y is reduced and has X as its underlying space, because Nx 6= Ox for any x ∈ X . To show the other claims,note that a subprescheme Z of X that has X as its underlying space is defined by a sheaf of ideals I (4.1.3) suchthat Ix 6= Ox for any x ∈ X . We can restrict to the case where X is ane, say X = Spec(A) and I = eI, where I isan ideal of A; then, for every x ∈ X , we have Ix ⊂ jx , and so I is contained in every prime ideal of A, and soalso in their intersection N, the nilradical of A. This proves that Y is the small subprescheme of X that has X asits underlying space (4.1.9); furthermore, if Z is distinct from Y , we necessarily have Ix 6=Nx for at least onex ∈ X , and so (5.1.1) Z is not reduced.

Denition (5.1.3). — We define the reduced prescheme associated to a prescheme X , denoted by Xred, to bethe unique reduced subprescheme of X that has X as its underlying space.

Saying that a prescheme X is reduced thus implies that X = Xred.

Proposition (5.1.4). — For the prime spectrum of a ring A to be a reduced (resp. integral) prescheme (2.1.7), it isnecessary and sucient for A to be a reduced (resp. integral) ring.

Proof. Indeed, it follows immediately from (5.1.1) that the condition N = (0) is necessary and sucient forX = Spec(A) to be reduced; the claim corresponding to integral rings is then a consequence of (1.1.13).

5. REDUCED PRESCHEMES; SEPARATION CONDITION 113

Since every ring of fractions 6= 0 of an integral ring is integral, it follows from (5.1.4) that, for every locallyintegral prescheme X , Ox is an integral ring for every x ∈ X . The converse is true whenever the underlying spaceof X is locally Noetherian: indeed, X is then reduced, and if U is an ane open subset of X , which is a Noetherianspace, then U has only a finite number of irreducible components, and so its ring A has only a finite number ofminimal prime ideals (1.1.14). If two of the irreducible components of U had a common point x , then Ox wouldhave at least two distinct minimal prime ideals, and would thus not be integral; the components of U are thusopen subsets that are pairwise disjoint, and each of them is thus integral.

(5.1.5). Let f = (ψ,θ) : X → Y be a morphism of preschemes; the homomorphism θ]x : Oψ(x)→Ox sends each I | 129nilpotent element of Oψ(x) to a nilpotent element of Ox ; by passing to the quotients, θ] induces a homomorphism

ω : ψ∗(OY /NY ) −→ OX/NX .

It is clear that, for every x ∈ X , ωx : Oψ(x)/Nψ(x) → Ox/Nx is a local homomorphism, and so (ψ,ω[) is amorphism of preschemes Xred→ Yred, which we denote by fred, and call the reduced morphism associated to f .It is immediate that, for morphisms f : X → Y and g : Y → Z , we have (g f )red = gred fred, and so we havedefined Xred as a functor, covariant in X .

The preceding definition shows that the diagram

Xredfred //

Yred

X

f // Y

is commutative, where the vertical arrows are the injection morphisms; in other words, Xred→ X is a functorial

morphism. We note in particular that, if X is reduced, then every morphism f : X → Y factors as Xfred−→ Yred→ Y ;

in other words, f factors through the injection morphism Yred→ Y .

Proposition (5.1.6). — Let f : X → Y be a morphism; if f is surjective (resp. radicial, an immersion, a closedimmersion, an open immersion, a local immersion, a local isomorphism), then so too is fred. Conversely, if fred is surjective(resp. radicial), then so too is f .

Proof. The proposition is trivial if f is surjective; if f is radicial, then the proposition follows from the factthat, for every x ∈ X , the field k(x) is the same for the preschemes X and Xred (3.5.8). Finally, if f = (ψ,θ) is animmersion, a closed immersion, or a local immersion (resp. an open immersion, or a local isomorphism), thenthe proposition follows from the fact that, if θ]x is surjective (resp. bijective), then so too is the homomorphismobtained by passing to the quotients by the nilradicals Oψ(x) and Ox ((5.1.2) and (4.2.2)) (cf. (5.5.12)).

Proposition (5.1.7). — If X and Y are S-preschemes, then the preschemes Xred×SredYred and Xred×S Yred are identical,

and canonically identied with a subprescheme of X ×S Y that has the same underlying subspace as the two aforementionedproducts.

Proof. The canonical identification of Xred ×S Yred with a subprescheme of X ×S Y that has the sameunderlying space follows from (4.3.1). Furthermore, if φ and ψ are the structure morphisms Xred → S andYred→ S (respectively), then they factor through Sred (5.1.5), and since Sred→ S is a monomorphism, the firstclaim of the proposition follows from (3.2.4).

Corollary (5.1.8). — The preschemes (X ×S Y )red and (Xred ×SredYred)red are canonically identied with one another.

Proof. This follows from (5.1.2) and (5.1.7).

We note that, even if X and Y are reduced preschemes, X ×S Y might not be reduced, because the tensorproduct of two reduced algebras can have nilpotent elements.

I | 130Proposition (5.1.9). — Let X be a prescheme, and I a quasi-coherent sheaf of ideals of OX such that I n = 0 for someinteger n> 0. Let X0 be the closed subprescheme (X ,OX/I ) of X ; for X to be an ane scheme, it is necessary and sucientfor X0 to be an ane scheme.

The condition is clearly necessary, so we will show that it is sucient. If we set Xk = (X ,OX/I k+1), it isenough to prove by induction on k that Xk is ane, and so we are led to consider the base case, where I 2 = 0.We set

A= Γ(X ,OX )

A0 = Γ(X0,OX0) = Γ(X ,OX/I ).

5. REDUCED PRESCHEMES; SEPARATION CONDITION 114

The canonical homomorphism OX →OX/I induces a homomorphism of rings φ : A→ A0. We will see belowthat φ is surjective, which implies that

(5.1.9.1) 0 −→ Γ(X ,I ) −→ Γ(X ,OX ) −→ Γ(X ,OX/I ) −→ 0

is an exact sequence. We now prove, assuming that this is true, the proposition. Note that K= Γ(X ,I ) is an idealwhose square is zero in A, and thus a module over A0 = A/K. By hypothesis, we have X0 = Spec(A), and, sincethe underlying spaces of X0 and X are identical, K= Γ(X0,I ); Additionally, since I 2 = 0, I is a quasi-coherent(OX/I )-module, so we have I ∼= eK and Kx = Ix for all x ∈ X0 (1.4.1). With this in mind, let X ′ = Spec(A), andconsider the morphism f = (ψ,θ) : X → X ′ of preschemes that corresponds to the identity map A→ Γ(X ,OX )(2.2.4). For every ane open subset V of X , the diagram

A //

Γ(V,OX |V )

A0 = A/K // Γ(V,OX0

|V )

commutes, whence the diagram

X ′ Xfoo

X ′0

j′

OO

X0f0oo

j

OO

also commutes (X ′0 being the closed subprescheme of X ′ defined by the quasi-coherent sheaf of ideals eR, and j andj′ being the canonical injection morphisms). But since X0 is ane, f0 is an isomorphism, and since the underlyingcontinuous maps of j and j′ are identity maps, we see straight away that ψ : X → X ′ is a homeomorphism.Furthermore, the equation Kx = Ix shows that the restriction of θ] : ψ∗(OX ′)→OX is an isomorphism from ψ∗(eK)to I ; additionally, by passing to the quotients, θ] gives an isomorphism ψ∗(OX/eK)→ OX/I , because f0 is anisomorphism; we thus immediately conclude, by the 5 lemma (M, I, 1.1), that θ] is itself an isomorphism, andthus that f is an isomorphism, and thus that X is ane. So everything reduces to proving the exactitude of(5.1.9.1), which will follow from showing that H1(X ,I ) = 0. But H1(X ,I ) = H1(X0,I ), and we have seen that I | 131I is a quasi coherent OX0

-module. Our proof will thus follow from

Lemma (5.1.9.2). — If Y is an ane scheme, and F a quasi-coherent OY -module, then H1(Y,F ) = 0.

Proof. This lemma will be proven in Chapter III, §1, as a consequence of the more general theorem thatHi(Y,F ) = 0 for all i > 0. To give an independent proof, note that H1(Y,F ) can be identified with the moduleExt1OY(Y ;OY ,F ) of extensions classes of the OY -module OY by the OY -module F (T, 4.2.3); so everything reduces

to proving that such an extension G is trivial. But, for all y ∈ Y , there is a neighbourhood V of y in Y such thatG|V is isomorphic to F|Y ⊕OY |V (0, 5.4.9); from this we conclude that G is a quasi-coherent OY -module. If A isthe ring of Y , then we have F = eM and G = eN , where M and N are A-modules, and, by hypothesis, N is anextension of the A-module A by the A-module M (1.3.11). Since this extension is necessarily trivial, the lemma isproven, and thus so too is (5.1.9).

Corollary (5.1.10). — Let X be a prescheme such that NX is nilpotent. For X to be an ane scheme, it is necessaryand sucient for Xred to be an ane scheme.

5.2. Existence of a subprescheme with a given underlying space.

Proposition (5.2.1). — For every locally closed subspace Y of the underlying space of a prescheme X , there exists exactlyone reduced subprescheme of X that has Y as its underlying space.

Proof. The uniqueness follows from (5.1.2), so it remains only to show the existence of the prescheme inquestion.

If X is ane, given by some ring A, and Y closed in X , then the proposition is immediate: j(Y ) is the largestideal a ⊂ A such that V (a) = Y , and it is equal to its radical (1.1.4, i), so A/j(Y ) is a reduced ring.

In the general case, for every ane open U ⊂ X such that U ∩ Y is closed in U , consider the closedsubprescheme YU of U defined by the sheaf of ideals associated to the ideal j(U ∩ Y ) of A(U), which is reduced.We can show that, if V is an ane open subset of X contained in U , then YV is induced by YU on V ∩ Y ; but thisinduced prescheme is a closed subprescheme (of V ) which is reduced and has V ∩ Y as its underlying space; theuniqueness of YV thus implies our claim.

5. REDUCED PRESCHEMES; SEPARATION CONDITION 115

Proposition (5.2.2). — Let X be a reduced subprescheme of a prescheme Y ; if Z is the closed reduced subprescheme of Ythat has X as its underlying space, then X is a subprescheme induced on an open subset of Z .

I | 132Proof. There is indeed an open subset U of Y such that X = U ∩ X ; since, by (5.2.2), X is a reduced

subprescheme of Z , the subprescheme X is induced by Z on the open subspace X by uniqueness (5.2.1).

Corollary (5.2.4). — Let f : X → Y be a morphism, and X ′ (resp. Y ′) a closed subprescheme of X (resp. Y ) dened bya quasi-coherent sheaf of ideals J (resp. K ) of OX (resp. OY ). Suppose that X ′ is reduced, and that f (X ′) ⊂ Y ′. Thenf ∗(K )OX ⊂ J .

Proof. Since, by (5.2.2), the restriction of f to X ′ factors as X ′→ Y ′→ Y , it suces to apply (4.4.6).

5.3. Diagonal; graph of a morphism.

(5.3.1). Let X be an S-prescheme; we define the diagonal morphism of X in X ×S X , denoted by ∆X |S , or ∆X , oreven ∆ if no confusion may arise, to be the S-morphism (1X , 1X )S , or, in other words, the unique S-morphism ∆Xsuch that

(5.3.1.1) p1 ∆X = p2 ∆X = 1X ,

where p1 and p2 are the projections of X ×S X (Definition (3.2.1)). If f : T → X and g : T → Y are S-morphisms,we immediately have that

(5.3.1.2) ( f , g)S = ( f ×S g) ∆T |S .

The reader will note that the preceding definition and the results stated in (5.3.1) to (5.3.8) are true in anycategory, provided that the products used within exist in the category.

Proposition (5.3.2). — Let X and Y be S-preschemes; if we make the canonical identication between (X×Y )×(X×Y )and (X × X )× (Y × Y ), then the morphism ∆X×Y is identied with ∆X ×∆Y .

Proof. Indeed, if p1 : X × X → X and q1 : Y × Y → Y are the projections onto the first component, then theprojection onto the first component (X × Y )× (X × Y )→ X × Y is identified with p1 × q1, and we have

(p1 × q1) (∆X ×∆Y ) = (p1 ∆X )× (q1 ∆Y ) = 1X×Y

and we can argue similarly for the projections onto the second component.

.

Corollary (5.3.4). — For every extension S′→ S of the base prescheme, ∆XS′is canonically identied with (∆X )(S′).

Proof. It suces to remark that (X ×S X )(S′) is canonically identified with X(S′) ×S′ X(S′) (3.3.10).

Proposition (5.3.5). — Let X and Y be S-preschemes, and φ : S→ T a morphism of preschemes, which lets us considerevery S-prescheme as a T -prescheme. Let f : X → S and g : Y → S be the structure morphisms, p and q the projections ofX ×S Y , and π = f p = g q the structure morphism X ×S Y → S. Then the diagram

(5.3.5.1) X ×S Y(p,q)T //

π

X ×T Y

f ×T g

S

∆S|T // S ×T S

commutes, and identies X ×S Y with the product of the (S ×T S)-preschemes S and X ×T Y , and the projections with π I | 133and (p, q)T .

Proof. By (3.4.3), we are led to proving the corresponding proposition in the category of sets, replacing X , Y ,and S by X (Z)T , Y (Z)T , and S(Z)T (respectively), with Z being an arbitrary T -prescheme. But, in the categoryof sets, the proof is immediate and left to the reader.

Corollary (5.3.6). — The morphism (p, q)T can be identied (letting P = S ×T S) with 1X×T Y ×P ∆S .

Proof. This follows from (5.3.5) and (3.3.4).

5. REDUCED PRESCHEMES; SEPARATION CONDITION 116

Corollary (5.3.7). — If f : X → Y is an S-morphism, then the diagram

X(1X , f )S//

f

X ×S Y

f ×S1Y

Y

∆Y // Y ×S Y

commutes, and identies X with the product of the (Y ×S Y )-preschemes Y and X ×S Y .

Proof. It suces to apply (5.3.5), replacing S by Y , and T by S, and noting that X ×Y Y = X (3.3.3).

Proposition (5.3.8). — For f : X → Y to be a monomorphism of preschemes, it is necessary and sucient for ∆X |Y tobe an isomorphism from X to X ×Y X .

Proof. Indeed, to say that f is a monomorphism implies that, for every Y -prescheme Z , the correspondingmap f ′ : X (Z)Y → Y (Z)Y is an injection, and, since Y (Z)Y consists of a single element, this implies thatX (Z)Y consists of a single element as well. But this can also be expressed by saying that X (Z)Y × X (Z)Y iscanonically isomorphic to X (Z)Y ; the former is exactly the set (X ×Y X )(Z)Y (3.4.3.1), which implies that ∆X |Y isan isomorphism.

Proposition (5.3.9). — The diagonal morphism ∆X is an immersion from X to X ×S X .

Proof. Indeed, since the continuous maps p1 and ∆X from the underlying spaces are such that p1 ∆Xis the identity, ∆X is a homeomorphism from X to ∆X (X ). Similarly, the composite homomorphism Ox →O∆X (x)→Ox (composed of the homomorphisms corresponding to p1 and ∆X ) is the identity, which means thatthe homomorphism corresponding to ∆X is surjective; the proposition thus follows from (4.2.2).

We say that the subprescheme of X ×S X associated to the immersion ∆X (4.2.1) is the diagonal of X ×S X .

Corollary (5.3.10). — Under the hypotheses of (5.3.5), (p, q)T is an immersion.

Proof. This follows from (5.3.6) and (4.3.1).

We say (under the hypotheses of (5.3.5)) that (p, q)T is the canonical immersion of X ×S Y into X ×T Y .

Corollary (5.3.11). — Let X and Y be S-preschemes, and f : X → Y an S-morphism; then the graph morphismΓf = (1X , f )S of f (3.3.14) is an immersion of X into X ×S Y .

Proof. This is a particular case of Corollary (5.3.10), where we replace S by Y , and T by S (cf. (5.3.7)). I | 134

The subprescheme of X ×S Y associated to the immersion Γf (4.2.1) is called the graph of the morphism f ;the subpreschemes of X ×S Y that are graphs of morphisms X → Y are characterised by the property that therestriction to such a subprescheme G of the projection p1 : X ×S Y → X is an isomorphism g from G to X : G is thethe graph of the morphism p2 g−1, where p2 is the projection X ×S Y → Y .

When we take, in particular, X = S, then the S-morphisms S → Y (which are exactly the S-sections of Y(2.5.5)) are equal to their graph morphisms; the subpreschemes of Y that are the graphs of S-sections (in otherwords, those that are isomorphic to S by the restriction of the structure morphism Y → S) are then also calledthe images of these sections, or, by an abuse of language, the S-sections of Y .

Corollary (5.3.12). — With the hypotheses and notation of (5.3.11), for every morphism g : S′ → S, let f ′ be theinverse image of f under g (3.3.7); then Γf ′ is the inverse image of Γf under g .

Proof. This is a particular case of (3.3.10.1).

Corollary (5.3.13). — Let f : X → Y and g : Y → Z be morphisms; if g f is an immersion (resp. a local immersion),then so too is f .

Proof. Indeed, f factors as XΓf−→ X ×Z Y

p2−→ Y . Furthermore, p2 can be identified with (g f )×Z 1Y (3.3.4);if g f is an immersion (resp. a local immersion), then so too is p2 ((4.3.1) and (4.5.5)), and since Γf is animmersion (5.3.11), we are done, by (4.2.4) (resp. (4.5.5)).

Corollary (5.3.14). — Let j : X → Y and g : Z → Z be S-morphisms. If j is an immersion (resp. a local immersion),then so too is ( j, g)S .

Proof. Indeed, if p : Y ×S Z → Y is the projection onto the first component, then we have j = p ( j, g)S , andit suces to apply (5.3.13).

5. REDUCED PRESCHEMES; SEPARATION CONDITION 117

Proposition (5.3.15). — If f : X → Y is an S-morphism, then the diagram

(5.3.15.1) X∆X //

f

X ×S X

f ×S f

Y

∆Y // Y ×S Y

commutes (in other words, ∆X is a functorial morphism in the category of preschemes).

Proof. The proof is immediate and left to the reader.

Corollary (5.3.16). — If X is a subprescheme of Y , then the diagonal ∆X (X ) can be identied with a subprescheme of∆Y (Y ), and the underlying space can be identied with

∆Y (Y )∩ p−11 (X ) = ∆Y ∩ p−1

2 (X )

(p1 and p2 being the projections of Y ×S Y ).

Proof. Applying (5.3.15) to the injection morphism f : X → Y , we see that f ×S f is an immersion thatidentifies the underlying space of X ×S X with the subspace p−1

1 (X ) ∩ p−12 (X ) of Y ×S Y (4.3.1); further, if

z ∈ ∆Y (Y ) ∩ p−11 (X ), then we have z = ∆Y (y) and y = p1(z) ∈ X , so y = f (y), and z = ∆Y ( f (y)) belongs to I | 135

∆X (X ) by the commutativity of (5.3.15.1).

Corollary (5.3.17). — Let f1 : Y → X and f2 : Y → X be S-morphisms, and y a point of Y such that f1(y) = f2(y) =x , and such that the homomorphisms k(x)→ k(y) corresponding to f1 and f2 are identical. Then, if f = ( f1, f2)S , thepoint f (y) belongs to the diagonal ∆X |S(X ).

Proof. The two homomorphisms k(x) → k(y) corresponding to fi (i = 1, 2) define two S-morphismsgi : Spec(k(y))→ Spec(k(x)) such that the diagrams

Spec(k(y))gi //

Spec(k(x))

Y

fi // X

commute. The diagram

Spec(k(y))(g1,g2)S //

Spec(k(x))×S Spec(k(x))

Y

( f1, f2)S // X ×S X

thus also commutes. But it follows from the equality g1 = g2 that the image under (g1, g2)S of the unique point ofSpec(k(y)) belongs to the diagonal of Spec(k(x))×S Spec(k(x)); the conclusion then follows from (5.3.15).

5.4. Separated morphisms and separated preschemes.

Denition (5.4.1). — We say that a morphism of preschemes f : X → Y is separated if the diagonal morphismX → X ×Y X is a closed immersion; we then also say that X is a separated prescheme over Y , or a Y -scheme. We saythat a prescheme X is separated if it is separated over Spec(Z); we then also say that X is a scheme9 (cf. (5.5.7)).

By (5.3.9), for X to be separated over Y , it is necessary and sucient for ∆X (X ) to be a closed subspace of theunderlying space of X ×Y X .

Proposition (5.4.2). — Let S→ T be a separated morphism. If X and Y are S-preschemes, then the canonical immersionX ×S Y → X ×T Y (5.3.10) is closed.

Proof. Indeed, if we refer to the diagram in (5.3.5.1), we see that (p, q)T can be considered as being obtainedfrom ∆S|T by the extension f ×T g : X ×T Y → S ×T S of the base prescheme S ×T S; the proposition then followsfrom (4.3.2).

Corollary (5.4.3). — Let Y be an S-scheme, and f : X → Y an S-morphism. Then the graph morphism Γf : X → X×S Y(5.3.11) is a closed immersion.

9[Trans.] We repeat here the warning given at the very start of this translation: the early versions of the EGA use prescheme to mean is nowusually called a scheme, and scheme for what is now usually called a separated scheme.. Grothendieck himself later said that the more modernterminology was preferable, but we have decided to keep this translation ‘historically accurate’ by using the older nomenclature.

5. REDUCED PRESCHEMES; SEPARATION CONDITION 118

Proof. This is a particular case of (5.4.2), where we replace S by Y , and T by S.

Corollary (5.4.4). — Let f : X → Y and g : Y → Z be morphisms, with g separated. If g f is a closed immersion,then so too is f .

Proof. The proof using (5.4.3) is the same as that of (5.3.13) using (5.3.11). I | 136

Corollary (5.4.5). — Let Z be an S-scheme, and j : X → Y and g : X → Z S-morphisms. If j is a closed immersion,then so too is ( j, g)S : X → Y ×S Z .

Proof. The proof using (5.4.4) is the same as that of (5.3.14) using (5.3.13).

Corollary (5.4.6). — If X is an S-scheme, then every S-section of X (2.5.5) is a closed immersion.

Proof. If φ : X → S is the structure morphism, and ψ : S→ X an S-section of X , it suces to apply (5.4.5)to φ ψ = 1S .

Corollary (5.4.7). — Let X be an integral prescheme with generic point s, and X an S-scheme. If two S-sections f andg are such that f (s) = g(s), then f = g .

Proof. Indeed, if x = f (s) = g(s), then the homomorphisms k(x)→ k(s) corresponding to f and g arenecessarily identical. If h= ( f , g)S , we thus deduce (5.3.17) that h(s) belongs to the diagonal Z = ∆X (X ); butsince S = s, and since Z is closed by hypothesis, we have h(S) ⊂ Z . It then follows from (5.2.2) that h factorsas S→ Z → X ×S X , and we thus conclude that f = g, by definition of the diagonal.

Remark (5.4.8). — If we suppose, conversely, that the conclusion of (5.4.3) is true when f = 1Y , then we canconclude that Y is separated over S; similarly, if we suppose that the conclusion of (5.4.5) applies to the two

morphisms Y∆Y−→ Y ×Z Y

p1−→ Y , then we can conclude that ∆Y is a closed immersion, and thus that Y is separatedover Z ; finally, if we assume that the conclusion of (5.4.6) is true for the Y section ∆Y of the Y -preschemeY ×S Y → Y , then this implies that Y is separated over S.

5.5. Separation criteria.

Proposition (5.5.1). —

(i) Every monomorphism of preschemes (and, in particular, every immersion) is a separated morphism.(ii) The composition of any two separated morphisms is separated.(iii) If f : X → X ′ and g : Y → Y ′ are separated S-morphisms, then f ×S g is separated.(iv) If f : X → Y is a separated S-morphism, then the S′-morphism f(S′) is separated for every extension S′→ S of

the base prescheme.(v) If the composition g f is separated, then f is separated.(vi) For a morphism f to be separated, it is necessary and sucient for fred (5.1.5) to be separated.

Proof. Note that (i) is an immediate consequence of (5.3.8). If f : X → Y and g : Y → Z are morphisms,then the diagram

(5.5.1.1) X∆X |Z //

∆X |Y ""

X ×Z X

X ×Y Xj

99

commutes (where j denotes the canonical immersion (5.3.10)), as can be immediately verified. If f and g areseparated, then ∆X |Y is a closed immersion by definition, and j is a closed immersion by (5.4.2), whence ∆X |Z isa closed immersion by (4.2.4), which proves (ii). Given (i) and (ii), (iii) and (iv) are equivalent (3.5.1), so it I | 137suces to prove (iv). But X(S′) ×Y(S′) X(S′) is canonically identified with (X ×Y X )×Y Y(S′) by (3.3.11) and (3.3.9.1),and we immediately see that the diagonal morphism ∆X(S′) can then be identified with ∆X ×Y 1Y(S′) ; the propositionthen follows from (4.3.1).

To prove (v), consider, as in (5.3.13), the factorisation XΓf−→ X ×Z Y

p2−→ Y of f , noting that p2 = (g f )×Z 1Y ;the hypothesis (that g f is separated) implies that g2 is separated, by (iii) and (i), and, since Γf is an immersion,Γf is separated, by (i), whence f is separated, by (ii). Finally, to prove (vi), recall that the preschemes Xred×Yred

Xred

5. REDUCED PRESCHEMES; SEPARATION CONDITION 119

and Xred ×Y Xred are canonically identified with one another (5.1.7); if we denote by j the injection Xred → X ,then the diagram

Xred

∆Xred//

j

Xred ×Y Xred

j×Y j

X

∆X // X ×Y X

commutes (5.3.15), and the proposition follows from the fact that the vertical arrows are homeomorphisms ofthe underlying spaces (4.3.1).

Corollary (5.5.2). — If f : X → Y is separated, then the restriction of f to any subprescheme of X is separated.

Proof. This follows from (5.5.1, i and ii).

Corollary (5.5.3). — If X and Y are S-preschemes such that Y is separated over S, then X ×S Y is separated over X .

Proof. This is a particular case of (5.5.1, iv).

Proposition (5.5.4). — Let X be a prescheme, and assume that its underlying space is a finite union of closed subsetsXk (1¶ k ¶ n); for each k, consider the reduced subprescheme of X that has Xk as its underlying space (5.2.1), and denotethis again by Xk. Let f : X → Y be a morphism, and, for each k, let Yk be a closed subset of Y such that f (Xk) ⊂ Yk; weagain denote by Yk the reduced subprescheme of Y that has Yk as its underlying space, so that the restriction Xk → Y of f to

Xk factors as Xkfk−→ Yk → Y (5.2.2). For f to be separated, it is necessary and sucient for all the fk to be separated.

Proof. The necessity follows from (5.5.1, i, ii, and v). Conversely, if the condition of the statement is satisfied,then each of the restrictions Xk → Y of f is separated (5.5.1, (i) and (ii)); if p1 and p2 are the projections ofX ×Y X , then the subspace ∆Xk

(Xk) can be identified with the subspace ∆X (X )∩ p−11 (Xk) of the underlying space

of X ×Y X (5.3.16); these subspaces are closed in X ×Y X , and thus so too is their union ∆X (X ).

Suppose, in particular, that the Xk are the irreducible components of X ; then we can suppose that the Yk arethe irreducible components of Y (0, 2.1.5); Proposition (5.5.4) then, in this case, leads to the idea of separationin the case of integral preschemes (2.1.7). I | 138

Proposition (5.5.5). — Let (Yλ) be an open cover of a prescheme Y ; for a morphism f : X → Y to be separated, it isnecessary and sucient for each of its restrictions f −1(Yλ)→ Yλ to be separated.

Proof. If we set Xλ = f −1(Yλ), everything reduces, by taking (4.2.4, b) and the identification of the productsXλ×Y Xλ and Xλ×Yλ Xλ into account, to proving that the Xλ×Y Xλ form a cover of X×Y X . But if we set Yλµ = Yλ∩Yµand Xλµ = Xλ ∩ Xµ = f −1(Yλµ), then Xλ ×Y Xµ can be identified with the product Xλµ ×Yλµ Xλµ (3.2.6.4), and soalso with Xλµ ×Y Xλµ (3.2.5), and finally with an open subset of Xλ ×Y Xλ , which proves our claim (3.2.7).

Proposition (5.5.4) allows us, by taking an ane open cover of Y , to restrict our study of separated morphismsto just those that take values in ane schemes.

Proposition (5.5.6). — Let Y be an ane scheme, X a prescheme, and (Uα) a cover of X by ane open subsets. Fora morphism f : X → Y to be separated, it is necessary and sucient for Uα ∩ Uβ to be, for every pair of indices (α,β),an ane open subset, and for the ring Γ(Uα ∩ Uβ ,OX ) to be generated by the union of the canonical images of the ringsΓ(Uα,OX ) and Γ(Uβ ,OX ).

Proof. The Uα ×Y Uβ form an open cover of X ×Y X (3.2.7); denoting the projections of X ×Y X by p and q,we have

∆−1X (Uα ×Y Uβ) = ∆−1

X (p−1(Uα)∩ q−1(Uβ))

= ∆−1X (p

−1(Uα))∩∆−1X (q

−1(Uβ)) = Uα ∩ Uβ;

so everything reduces to proving that the restriction of ∆X to Uα ∩ Uβ is a closed immersion into Uα ×Y Uβ .But this restriction is exactly ( jα, jβ)Y , where jα (resp. jβ) denotes the injection morphism from Uα ∩ Uβ toUα (resp. Uβ), as follows from the definitions. Since Uα ×Y Uβ is an ane scheme whose ring is canonicallyisomorphic to Γ(Uα,OX )⊗Γ(Y,OY ) Γ(Uβ ,OX ) (3.2.2), we see that Uα ∩ Uβ must be an ane scheme, and that themap hα ⊗ hβ 7→ hαhβ from the ring A(Uα ×Y Uβ) to Γ(Uα ∩ Uβ ,OX ) must be surjective (4.2.3), which finishes theproof.

Corollary (5.5.7). — An ane scheme is separated (and is thus a scheme, which justifies the terminology of(5.4.1)).

5. REDUCED PRESCHEMES; SEPARATION CONDITION 120

Corollary (5.5.8). — Let Y be an ane scheme; for f : X → Y to be a separated morphism, it is necessary and sucientfor X to be separated (in other words, for X to be a scheme).

Proof. Indeed, we see that the criteria of (5.5.6) do not depend on f .

Corollary (5.5.9). — For a morphism f : X → Y to be separated, it is necessary and sucient for the induced preschemef −1(U) to be separated, for every open subset of U on which Y induces a separated prescheme, and it is sucient for it to bethe case for every ane open subset U ⊂ Y .

Proof. The necessity of the condition follows from (5.5.4) and (5.5.1, ii); the suciency follows from (5.5.4)and (5.5.8), taking into account the existence of ane open covers of Y .

In particular, if X and Y are ane schemes, then every morphism X → Y is separated.

Proposition (5.5.10). — Let Y be a scheme, and f : X → Y a morphism. For every ane open subset U of X , andevery ane open subset V of Y , U ∩ f −1(V ) is ane.

Proof. Let p1 and p2 be the projections of X ×Z Y ; the subspace U ∩ f −1(V ) is the image of Γf (X )∩ p−11 (U)∩

p−12 (V ) under p1. But p−1

1 (U) ∩ p−12 (V ) can be identified with the underlying space of the prescheme U ×Z V I | 139

(3.2.7), and is thus an ane scheme (3.2.2); since Γf (X ) is closed in X ×Z Y (5.4.3), Γf (X )∩ p−11 (U)∩ p−1

2 (V ) isclosed in U ×Z V , and so the prescheme induced by the subprescheme of X ×Z Y associated to Γf (4.2.1) on theopen subset Γf (X )∩ p−1

1 (U)∩ p−12 (V ) of its underlying space is a closed subprescheme of an ane scheme, and

thus an ane scheme (4.2.3). The proposition then follows from the fact that Γf is an immersion.

Example (5.5.11). — The prescheme from Example (2.3.2) (“the projective line over a field K”) is separated,because, for the cover (X1, X2) of X by ane open subsets, X1 ∩ X2 = U12 is ane, and Γ(U12,OX ), the ring ofrational fractions of the form f (s)/sm with f ∈ K[s], is generated by K[s] and 1/s, so the conditions of (5.5.6)are satisfied.

With the same choice of X1, X2, U12, and U21 as in Example (2.3.2), now take u12 to be the isomorphismwhich sends f (s) to f (t); we now obtain, by gluing, a non-separated integral prescheme X , because the firstcondition of (5.5.6) is satisfied, but not the second. It is immediate here that Γ(X ,OX )→ Γ(X1,OX ) = K[s] isan isomorphism; the inverse isomorphism defines a morphism f : X → Spec(K[s]) that is surjective, and forevery y ∈ Spec(K[s]) such that jy 6= (0), f −1(y) consists of a single point, but for jy = (0), f −1(y) consists oftwo distinct points (we say that X is the “ane line over K with the point 0 doubled”).

We can also give examples where neither of the two conditions of (5.5.6) are satisfied. First note that, in theprime spectrum Y of the ring A= K[s, t] of polynomials in two indeterminates over a field K , the open subset Ugiven by the union of D(s) and D(t) is not an ane open subset. Indeed, if z is a section of OY over U , there existtwo integers m, n¾ 0 such that smz and tnz are the restrictions of polynomials in s and t to U (1.4.1), which isclearly possible only if the section z extends to a section over the whole of Y , identified with a polynomial in sand t. If U were an ane open subset, then the injection morphism U → Y would be an isomorphism (1.7.3),which is a contradiction, since U 6= Y .

With the above in mind, take two ane schemes Y1 and Y2, prime spectra of the rings A1 = K[s1, t2] andA2 = K[s2, t2] (respectively); take U12 = D(s1)∪ D(t1) and U21 = D(s2)∪ D(t2), and take u12 to be the restrictionof an isomorphism Y2→ Y1 to U21 corresponding to the isomorphism of rings that sends f (s1, t1) to f (s2, t2);we then have an example where the conditions of (5.5.6) are not satisfied (the integral prescheme thus obtainedis called “the ane plane over K with the point 0 doubled”).

Remark (5.5.12). — Given some property P of morphisms of preschemes, consider the following propositions.(i) Every closed immersion has property P.(ii) The composition of any two morphisms that both have property P also has property P.(iii) If f : X → X ′ and g : Y → Y ′ are S-morphisms that have property P, then f ×S g has property P. I | 140(iv) If f : X → Y is an S-morphism that has property P, then every S′-morphism f(S′) obtained by an extension

S′→ S of the base prescheme also has property P.(v) If the composition g f of two morphisms f : X → Y and g : Y → Z has property P, and g is separated, then

f has property P.(vi) If a morphism f : X → Y has property P, then so too does fred (5.1.5).

If we suppose that (i) and (ii) are both true, then (iii) and (iv) are equivalent, and (v) and (vi) are consequences of(i), (ii), and (iii).

The first claim has already been shown (3.5.1). Consider the factorisation (5.3.13) XΓf−→ X ×Z Y

p2−→ Y of f ;the equation p2 = (g f )×Z 1Y shows that, if g f has property P, then so too does p2, by (iii); if g is separated,then Γf is a closed immersion (5.4.3), and so also has property P, by (i); finally, by (ii), f has property P.

6. FINITENESS CONDITIONS 121

Finally, consider the commutative diagram

Xredfred //

Yred

X

f // Y,

where the vertical arrows are the closed immersions (5.1.5), and thus have property P, by (i). The hypothesis

that f has property P implies, by (ii), that Xredfred−→ Yred→ Y has property P; finally, since a closed immersion is

separated (5.5.1, i), fred has property P, by (v).Note that, if we consider the propositions

(i’) Every immersion has property P ;(v’) If g f has property P, then so too does f ;

then the above arguments show that (v’) is a consequence of (i’), (ii), and (iii).

(5.5.13). Note that (v) and (vi) are again consequences of (i), (iii), and

(ii’) If j : X → Y is a closed immersion, and g : Y → Z is a morphism that has property P, then g j has property P.Similarly, (v’) is a consequence of (i’), (iii), and

(ii”) If j : X → Y is an immersion, and g : Y → Z is a morphism that has property P, then g j has property P.This follows immediately from the arguments of (5.5.12).

§6. Finiteness conditions

6.1. Noetherian and locally Noetherian preschemes.

Denition (6.1.1). — We say that a prescheme X is Noetherian (resp. locally Noetherian) if it is a finite union(resp. union) of ane open Vα in such a way that the ring of the of the induced scheme on each of the Vα isNoetherian.

It follows immediately from (1.5.2) that, if X is locally Noetherian, then the structure sheaf OX is a coherentsheaf of rings, the question being a local one. Every quasi-coherent OX -submodule (resp. quasi-coherent quotient I | 141OX -module) of a coherent OX -module F is coherent, as the question is once again a local one, and it suces toapply (1.5.1), (1.4.1), and (1.3.10), combined with the fact that a submodule (resp. quotient module) of a moduleof finite type over a Noetherian ring is of finite type. In particular, every quasi-coherent sheaf of ideals of OX iscoherent.

If a prescheme X is a finite union (resp. union) of open subsets Wλ in such a way that the preschemesinduced on the Wλ are Noetherian (resp. locally Noetherian), it is clear that X is Noetherian (resp. locallyNoetherian).

Proposition (6.1.2). — For a prescheme X to be Noetherian, it is necessary and sucient for it to be locally Noetherianand have a quasi-compact underlying space. The underlying space itself is then also Noetherian.

Proof. The first claim follows immediately from the definitions and (1.1.10, ii). The second follows from(1.1.6) and the fact that every space that is a finite union of Noetherian subspaces is itself Noetherian (0, 2.2.3).

Proposition (6.1.3). — Let X be an ane scheme given by a ring A. The following conditions are equivalent: (a) X isNoetherian; (b) X is locally Noetherian; (c) A is Noetherian.

Proof. The equivalence between (a) and (b) follows from (6.1.2) ant the fact that the underlying space ofevery ane scheme is quasi-compact (1.1.10); it is furthermore clear that (c) implies (a). To see that (a) implies(c), we remark that there is a finite cover (Vi) of X by ane open subsets such that the ring Ai of the preschemeinduced on Vi is Noetherian. So let (an) be an increasing sequence of ideals of A; by a canonical bijectivecorrespondence, there is a corresponding sequence (ean) of sheaves of ideals in eA= OX ; to see that the sequence(an) is stable (?), it suces to prove that the sequence (ean) is. But the restriction ean|Vi is a quasi-coherent sheafof ideals in OX |Vi , being the inverse image of ean under the canonical injection Vi → X (0, 5.1.4); ean|Vi is thusof the form eani , where ani is an ideal of Ai (1.3.7). Since Ai is Noetherian, the sequence (ani) is stable for all i,whence the proposition.

We note that the above argument proves also that if X is a Noetherian prescheme, then every increasing sequenceof coherent sheaves of ideals of OX is stable (?).

6. FINITENESS CONDITIONS 122

Proposition (6.1.4). — Every subprescheme of a Noetherian (resp. locally Noetherian) prescheme is Noetherian (resp.locally Noetherian).

Proof. If suces to give a proof for a Noetherian prescheme X ; further, by definition (6.1.1), we can alsorestrict to the case where X is an ane scheme. Since every subprescheme of X is a closed subprescheme of aprescheme induced on an open subset (4.1.3), we can restrict to the case of a subprescheme Y , either closed orinduced on an open subset of X . The proof in the case where Y is closed is immediate, since if A is the ring of X ,we know that Y is an ane scheme given by the ring A/J, where J is an ideal of A (4.2.3); since A is Noetherian(6.1.3), so too is A/J.

Now suppose that Y is open in X ; the underlying space of Y is Noetherian (6.1.2), hence quasi-compact,and thus a finite union of open subsets D( fi) ( fi ∈ A); everything reduces to showing the proposition in the casewhere Y = D( f ) with f ∈ A. But then Y is an ane scheme whose ring is isomorphic to A f (1.3.6); since A isNoetherian (6.1.3), so too is A f .

(6.1.5). We note that the product of two Noetherian S-preschemes is not necessarily Noetherian, even if thepreschemes are ane, since the tensor product of two Noetherian algebras in not necessarily a Noetherian ring(cf. (6.3.8)).

Proposition (6.1.6). — If X is a Noetherian prescheme, the nilradical NX of OX is nilpotent.

Proof. We can in fact cover X with a finite number of ane open subsets Ui , and it suces to prove thatthere exists whole numbers ni such that (NX |Ui)ni = 0; if n is the largest of the ni , then we will have N n

X = 0.We can thus restrict to the case where X = Spec(A) is ane, with A a Noetherian ring; by (5.1.1) and (1.3.13), itsuces to observe that the nilradical of A is nilpotent ([Sam53b, p. 127, cor. 4]).

Corollary (6.1.7). — Let X be a Noetherian prescheme; for X to be an ane scheme, it is necessary and sucient thatXred be ane.

Proof. This follows from (6.1.6) and (5.1.10).

Lemma (6.1.8). — Let X be a topological space, x a point of X , and U an open neighbourhood of x having only anite number of irreducible components. Then there exists a neighbourhood V of x such that every open neighbourhood of xcontained in V is connected.

Proof. Let Ui (1¶ i ¶ m) be the irreducible components of U not containing x ; the complement (in U) ofthe union of the Ui is an open neighbourhood V of X inside U , and thus so too in X ; it is also, incidentally, thecomplement (in X ) of the union of the irreducible components of X that do not contain x (0, 2.1.6). So let Wbe an open neighbourhood of X contained in V . The irreducible components of W are the intersections of Wwith the irreducible components of U (0, 2.1.6), so these components contain x ; since they are connected, sotoo is W .

Corollary (6.1.9). — A locally Noetherian topological space is locally connected (which implies, amongst other things,that its connected components are open).

Proposition (6.1.10). — Let X be a locally Noetherian topological space. The following conditions are equivalent.

(a) The irreducible components of X are open.(b) The irreducible components of X are exactly its connected components.(c) The connected components of X are irreducible.(d) Two distinct irreducible components of X have an empty intersection.

Finally, if X is a prescheme, then these conditions are also equivalent to

(e) For every x ∈ X , Spec(Ox) is irreducible (or, in other words, the nilradical of Ox is prime).

Proof. It is immediate that (a) implies (b), because an irreducible space is connected, and (a) impliesthat the irreducible components of X are the sets that are both open and closed. It is trivial that (b) implies(c); conversely, a closed set F containing a connected component C of X , with C distinct from F , cannot be I | 143irreducible, because not being connected means that F is the union of two disjoint nonempty sets that are bothopen and closed in F , and thus closed in X ; as a result, (c) implies (b). We immediately conclude from this that(c) implies (d), since two distinct connected components have no points in common.

We have not yet used the fact that X is locally Noetherian. Suppose now that this is indeed the case, and wewill show that (d) implies (a): by (0, 2.1.6), we can restrict ourselves to the case where the space X is Noetherian,and so has only a finite number of irreducible components. Since they are closed and pairwise disjoint, they areopen.

6. FINITENESS CONDITIONS 123

Finally, the equivalence between (d) and (e) holds true even without the assumption that the underlyingspace of the prescheme X is locally Noetherian. We can in fact restrict ourselves to the case where X = Spec(A)is ane, by (0, 2.1.6); to say that x is contained in only one single irreducible component of X is to say that jxcontains only one single minimal ideal of A (1.1.14), which is equivalent to saying that jxOx contains only onesingle minimal ideal of Ox , whence the conclusion.

Corollary (6.1.11). — Let X be a locally Noetherian space. For X to be irreducible, it is necessary and sucient that Xbe connected and nonempty, and that any two distinct irreducible components of X have an empty intersection. If X is aprescheme, this latter condition is equivalent to asking that Spec(Ox) be irreducible for all x ∈ X .

Proof. The second claim has already been shown in (6.1.10); the only thing thus remaining to show is that theconditions in the first claim are sucient. But by (6.1.10), these conditions imply that the irreducible componentsof X are exactly its connected components, and since X is connected and nonempty, it is irreducible.

Corollary (6.1.12). — Let X be a locally Noetherian prescheme. For X to be integral, it is necessary and sucient thatX be connected and that Ox be integral for all x ∈ X .

Proposition (6.1.13). — Let X be a locally Noetherian prescheme, and let x ∈ X be a point such that the nilradicalNx of Ox is prime (resp. such that Ox is reduced, resp. integral); then there exists an open neighbourhood U of x that isirreducible (resp. reduced, resp. integral).

Proof. It suces to consider two cases: where Nx is prime, and where Nx = 0; the third hypotheses isa combination of the first two. If Nx is prime, then x belongs to only one single irreducible component Yof X (6.1.10); the union of the irreducible components of X that do not contain x is closed (the set of thesecomponents being locally finite), and the complement U of this union is thus open and contained in Y , andthus irreducible (0, 2.1.6) If Nx = 0, we also have Ny = 0 for any y in a neighbourhood of x , because N isquasi-coherent (5.1.1), and thus coherent, since X is locally Noetherian, and the conclusion then follows from(0, 5.2.2).

6.2. Artinian preschemes.

Denition (6.2.1). — We say that a prescheme is Artinian if it is ane, and given by an Artinian ring.

Proposition (6.2.2). — Given a prescheme X , the following conditions are equivalent: I | 144(a) X is an Artinian scheme;(b) X is Noetherian and its underlying space is discrete;(c) X is Noetherian and the points of its underlying space are closed (the T1 condition).

When any of the above hold, the underlying space of X is nite, and the ring A of X is the direct sum of local (Artinian)rings of points of X .

Proof. We know that (a) implies the last claim ([SZ60, p. 205, th. 3]), so every prime ideal of A is thusmaximal and is the inverse image of a maximal ideal of one of the local components of A, and so the space X isfinite and discrete; (a) thus implies (b), and (b) clearly implies (c). To see that (c) implies (a), we first show thatX is then finite; we can indeed restrict to the case where X is ane, and we know that a Noetherian ring whoseprime ideals are all maximal is Artinian ([SZ60, p. 203]), whence our claim. The underlying space X is thendiscrete, the topological sum of a finite number of points x i , and the local rings Ox i

= Ai are Artinian; it is clearthat X is isomorphic to the prime spectrum ane scheme of the ring A (the direct sum of the Ai) (1.7.3).

6.3. Morphisms of nite type.

Denition (6.3.1). — We say that a morphism f : X → Y is of nite type if Y is the union of a family (Vα) ofane open subsets having the following property:

(P) f −1(Vα) is a finite union of ane open subsets Uαi that are such that each ring A(Uαi) is an algebra offinite type over A(Vα).

We then say that X is a prescheme of finite type over Y , or a Y -prescheme of finite type.

Proposition (6.3.2). — If f : X → Y is a morphism of nite type, then every ane open subset W of Y satises property(P) of (6.3.1).

We first show

Lemma (6.3.2.1). — If T ⊂ Y is an ane open subset, satisfying property (P), then, for every g ∈ A(T ), D(g) alsosatises property (P).

6. FINITENESS CONDITIONS 124

Proof. By hypothesis, f −1(T ) is a finite union of ane open subsets Z j , that are such that A(Z j) is analgebra of finite type over A(T ); let φ j : A(T ) → A(Z j) be the homomorphism of rings corresponding tothe restriction of f to Z j (2.2.4), and set g j = φ j(g); we then have f −1(D(g)) ∩ Z j = D(g j) (1.2.2.2). ButA(D(g j)) = A(Z j)g j

= A(Z j)[1/g j] is of finite type over A(Z j), and a fortiori over A(T ) by the hypothesis, and soalso over A(D(g)) = A(T )[1/g], which proves the lemma.

Proof. With the above lemma, since W is quasi-compact (1.1.10), there exists a finite covering of W by sets ofthe form D(gi) ⊂W , where each gi belongs to a ring A(Vα(i)). Each D(gi), being quasi-compact, is a finite union ErrIIof sets D(hik), where hik ∈ A(W ); if φi : A(W )→ A(D(gi)) is the canonical map, then we have D(hik) = D(φi(hik))by (1.2.2.2). By (6.3.2.1), each of the f −1(D(hik)) admits a finite covering by ane open subsets Ui jk, that aresuch that the A(Ui jk) are algebras of finite type over A(D(hik)) = A(W )[1/hik], whence the proposition.

We can thus say that the notion of a prescheme of finite type over Y is local on Y . I | 145

Proposition (6.3.3). — Let X and Y be ane schemes; for X to be of nite type over Y , it is necessary and sucientthat A(X ) be an algebra of nite type over A(Y ).

Proof. Since the condition clearly suces, we show that it is necessary. Set A = A(Y ) and B = A(X ); by(6.3.2), there exists a finite ane open cover (Vi) of X such that each of the rings A(Vi) is an A-algebra of finitetype. Further, since the Vi are quasi-compact, we can cover each of them with a finite number of open subsetsof the form D(gi j) ⊂ Vi , where gi j ∈ B; if φi is a homomorphism B→ A(Vi) that corresponds to the canonicalinjection Vi → X , then we have Bgi j

= (A(Vi))φi(gi j) = A(Vi)[1/φi(gi j)], so Bgi jis an A-algebra of finite type. We

can thus restrict to the case where Vi = D(gi) with gi ∈ B. By hypothesis, there exists a finite subset Fi of B andan integer ni ¾ 0 such that Bgi

is the algebra generated over A by the elements bi/gnii , where the bi run over all

of Fi . Since there are only finitely many of the gi , we can assume that all the ni are equal to the same integer n.Further, since the D(gi) form a cover of X , the ideal generated in B by the gi is equal to B, or, in other words,there exist hi ∈ B such that

∑

i hi gi = 1. So let F be the finite subset of B given by the union of the Fi , the set ofthe gi , and the set of the hi ; we will show that the subring B′ = A[F] of B is equal to B. By hypothesis, for everyb ∈ B and every i, the canonical image of b in Bgi

is of the form b′i/gmii , where b′i ∈ B′; by multiplying the b′i by

suitable powers of the gi , we can again assume that all the mi are equal to the same integer m. By the definitionof the ring of fractions, there is thus an integer N (dependant on b) such that N ¾ m and gN

i b ∈ B′ for all i; but,in the ring B′, the gN

i generate the ideal B′, because the gi do (and the hi belong to B′); there are thus ci ∈ B′

such that∑

i ci gNi = 1, whence b =

∑

i ci gNi b ∈ B′, Q.E.D.

Proposition (6.3.4). —(i) Every closed immersion is of nite type.(ii) The composition of any two morphisms of nite type is of nite type.(iii) If f : X → X ′ and g : Y → Y ′ are S-morphisms of nite type, then f ×S g is of nite type.(iv) If f : X → Y is an S-morphism of nite type, then f(S′) is of nite type for any extension g : S′→ S of the base

prescheme.(v) If the composition g f of two morphisms is of nite type, with g separated, then f is of nite type.(vi) If a morphism f is of nite type, then fred is of nite type.

Proof. By (5.5.12), it suces to prove (i), (ii), and (iv).To show (i), we can restrict to the case of a canonical injection X → Y , with X being a closed subprescheme

of Y ; further (6.3.2), we can assume that Y is ane, in which case X is also ane (4.2.3) and its ring is isomorphicto a quotient ring A/J, where A is the ring of Y and J is an ideal of A; since A/J is of finite type over A, theconclusion follows.

Now we show (ii). Let f : X → Y and g : Y → Z be two morphisms of finite type, and let U be an ane opensubset of Z ; g−1 admits a finite covering by ane open subsets Vi that are such that each A(Vi) is an algebraof finite type over A(U) (6.3.2); similarly. each of the f −1 admits a finite cover by ane open subsets Wi j that I | 146are such that each A(Wi j) is an algebra of finite type over A(Vi), and so also an algebra of finite type over A(U),whence the conclusion.

Finally, to show (iv), we can restrict to the case where S = Y ; then f(S′) is also equal to fY(S′) , where weconsider f as a Y -morphism, and the base extension is Y(S′) → Y (3.3.9). So let p and q be the projectionsX(S′)→ X and X(S′)→ S′. Let V be an ane open subset of S; f −1(V ) is a finite union of ane open subsets Wi ,each of which is such that A(Wi) is an algebra of finite type over A(V ) (6.3.2). Let V ′ be an ane open subset ofS′ contained in g−1(V ); since f p = g q, q−1(V ′) is contained in the union of the p−1(Wi); on the other hand,the intersection p−1(Wi)∩ q−1(V ′) can be identified with the product Wi × V V ′ (3.2.7), which is an ane scheme

6. FINITENESS CONDITIONS 125

whose ring is isomorphic to A(Wi)⊗A(V ) A(V ′) (3.2.2); this ring is, by hypothesis, an algebra of finite type overA(V ′), which proves the proposition.

Corollary (6.3.5). — Let f : X → Y be an immersion morphism. If the underlying space of Y (resp. X ) is locallyNoetherian (resp. Noetherian), then f is of nite type.

Proof. We can always assume that Y is ane (6.3.2); if the underlying space of Y is locally Noetherian,then we can further assume that it is Noetherian, and then the underlying space of X , which is a subspace,is also Noetherian. In other words, we can assume that Y is ane and that the underlying space of X isNoetherian; there then exists a covering of X by a finite number of ane open subsets D(gi) ⊂ Y , wheregi ∈ A(Y ), that are such that the X ∩ D(gi) are closed in D(gi) (and thus ane schemes (4.2.3)), because X islocally closed in Y (4.1.3). Then A(X ∩ D(gi)) is an algebra of finite type over A(D(gi)), by (6.3.4, i) and (6.3.3),and A(D(gi)) = A(Y )gi

= A(Y )[1/gi] is of finite type over A(Y ), which finishes the proof.

Corollary (6.3.6). — Let f : X → Y and g : Y → Z be morphisms. If g f if of nite type, with either X Noetherianor X ×Z Y locally Noetherian, then f is of nite type.

Proof. This follows immediately from the proof of (5.5.12) and from (6.3.5) applied to the immersionmorphism Γf .

Proposition (6.3.7). — Let f : X → Y be a morphism of nite type; if Y is Noetherian (resp. locally Noetherian), thenX is Noetherian (resp. locally Noetherian).

Proof. We can restrict to proving the proposition for when Y is Noetherian. Then Y is a finite union ofane open subsets Vi that are such that the A(Vi) are Noetherian rings. By (6.3.2), each of the f −1(Vi) is a finiteunion of ane open subsets Wi j that are such that the A(Wi j) are algebras of finite type over A(Vi), and thusNoetherian rings; this proves that X is Noetherian.

Corollary (6.3.8). — Let X be a prescheme of nite type over S. For every base extension S′→ S with S′ Noetherian(resp. locally Noetherian), X(S′) is Noetherian (resp. locally Noetherian).

Proof. This follows from (6.3.7), since X(S′) is of finite type over S′ by (6.3.4, iv).

We can also says that, for a product X ×S Y of S-preschemes, if one of the factors X or Y is of nite type over I | 147S and the other is Noetherian (resp. locally Noetherian), then X ×S Y is Noetherian (resp. locally Noetherian).

Corollary (6.3.9). — Let X be a prescheme of nite type over a locally Noetherian prescheme S. Then every S-morphismf : X → Y is of nite type.

Proof. In fact, we can assume that S is Noetherian; if φ : X → S and ψ : Y → S are the structure morphisms,then we have φ = ψ f , and X is Noetherian by (6.3.7); f is thus of finite type by (6.3.6).

Proposition (6.3.10). — Let f : X → Y be a morphism of nite type. For f to be surjective, it is necessary and sucientthat, for every algebraically closed eld Ω, the map X (Ω)→ Y (Ω) that corresponds to f (3.4.1) be surjective.

Proof. The condition suces, as we can see by considering, for all y ∈ Y , an algebraically closed extensionΩ of k(y), and the commutative diagram

X

f

Spec(Ω)

cc

Y

(cf. (3.5.3)). Conversely, suppose that f is surjective, and let g : ξ = Spec(Ω)→ Y be a morphism, where Ω isan algebraically closed field. If we consider the diagram

X

f

X(Ω)oo

f(Ω)

Y Spec(Ω),oo

then it suces to show that there exists a rational point over Ω in X(Ω) ((3.3.14), (3.4.3), and (3.4.4)). Since f issurjective, X(Ω) is nonempty (3.5.10), and since f is of finite type, so too is f(Ω) (6.4.3, iv); thus X(Ω) contains

6. FINITENESS CONDITIONS 126

a nonempty ane open subset Z such that A(Z) is an non-null algebra of finite type over Ω. By Hilbert’sNullstellensatz [Zar47], there exists an Ω-homomorphism A(Z)→ Ω, and thus a section of X(Ω) over Spec(Ω),which proves the proposition.

6.4. Algebraic preschemes.

Denition (6.4.1). — Given a field K , we define an algebraic K -prescheme to be a prescheme X of finite typeover K ; K is called the base field of X . If in addition X is a scheme (or if X is a K -scheme, which is equivalent(5.5.8)), we say that X is an algebraic K -scheme.

Every algebraic K -prescheme is Noetherian (6.3.7).

Proposition (6.4.2). — Let X be an algebraic K -prescheme. For a point x ∈ X to be closed, it is necessary and sucientthat k(x) be an algebraic extension of K of nite degree.

Proof. We can assume that X is ane, with the ring A of X being a K -algebra of finite type. Indeed, theane open subsets U of X such that A(U) is a K -algebra of finite type form a finite cover of X (6.3.1). The closedpoints of X are thus the points such that jx is a maximal ideal of A, or in other words, such that A/jx is a field I | 148(necessarily equal to k(x)). Since A/jx is a K -algebra of finite type, we see that if x is closed, then k(x) is a fieldthat is an algebra of finite type over K , and so necessarily a K -algebra of nite rank [Zar47]. Conversely, if k(x)is of finite rank over K , then so is A/jx ⊂ k(x), and since every integral ring that is also a K -algebra of finite rankis a field, we have that A/jx = k(x), and hence x is closed.

Corollary (6.4.3). — Let K be an algebraically-closed eld, and X an algebraic K -prescheme; the closed points of X arethen the rational points over K (3.4.4) and can be canonically identied with the points of X with values in K .

Proposition (6.4.4). — Let X be an algebraic prescheme over a eld K . The following properties are equivalent.(a) X is Artinian.(b) The underlying space of X is discrete.(c) The underlying space of X has only a nite number of closed points.(c’) The underlying space of X is nite.(d) The points of X are closed.(e) X is isomorphic to Spec(A), where A is a K -algebra of nite rank.

Proof. Since X is Noetherian, it follows from (6.2.2) that the conditions (a), (b), and (d) are equivalent,and imply (c) and (c′); it is also clear that (e) implies (a). It remains to see that (c) implies (d) and (e); we canrestrict to the case where X is ane. Then A(X ) is a K -algebra of finite type (6.3.3), and thus a Jacobson ring([CC, p. 3-11 and 3-12]), in which there are, by hypothesis, only a finite number of maximal ideals. Since a finiteintersection of prime ideals can only be a prime ideal if it is equal to one of the prime ideals being intersected,every prime ideal of A(X ) is thus maximal, whence (d). Further, we then know (6.2.2) that A(X ) is an ArtinianK -algebra of finite type, and so necessarily of nite rank [Zar47].

(6.4.5). When the conditions of (6.4.4) are satisfied, we say that X is a scheme nite over K (cf. (II, 6.1.1)), or anite K -scheme, of rank [A : K], which we also denote by rgK(X ); if X and Y are finite K -schemes, we have

(6.4.5.1) rgK(X t Y ) = rgK(X ) + rgK(Y ),

(6.4.5.2) rgK(X ×K Y ) = rgK(X ) rgK(Y ),

as a result of (3.2.2).

Corollary (6.4.6). — Let X be a nite K -scheme. For every extension K ′ of K , X ⊗K K ′ as a nite K ′-scheme, and itsrank over K ′ is equal to the rank of X over K .

Proof. If A= A(X ), then we have [A⊗K K ′ : K ′] = [A : K].

Corollary (6.4.7). — Let X be a scheme nite over a eld K ; we let n =∑

x∈X [k(X ) : K]S (we recall that if K ′ isan extension of K , then [K ′ : K]S is the separable rank of K ′ over k, the rank of the largest algebraic separableextension of K contained in K ′); then for every algebraically closed extension Ω of K , the underlying space of X ⊗K Ω has I | 149exactly n points, which can be identied with the points of X with values in Ω.

Proof. We can clearly restrict to the case where the ring A= A(X ) is local (6.2.2); let m be its maximal ideal,and L = A/m its residue field, an algebraic extension of K . The points of X with values in Ω then correspond,bijectively, to the Ω-sections of X ⊗K Ω ((3.4.1) and (3.3.14)), and also to the K -homomorphisms from L to Ω(1.7.3), whence the proposition (Bourbaki, Alg., chap. V, §7, no 5, prop. 8), taking (6.4.3) into account.

6. FINITENESS CONDITIONS 127

(6.4.8). The number n defined in (6.4.7) is called the separable rank of A (or of X ) over K , or also the geometricnumber of points of X ; it is equal to the number of elements of X (Ω)K . It follows immediately from this definitionthat, for every extension K ′ of K , X ⊗K K ′ has the same geometric number of points as X . If we denote thisnumber by n(X ), it is clear that, if X and Y are two schemes, finite over K , then

(6.4.8.1) n(X t Y ) = n(X ) + n(Y ).

Under the same hypotheses, we also have

(6.4.8.2) n(X ×K Y ) = n(X )n(Y )

because of the interpretation of n(X ) as the number of elements of X (Ω)K and Equation (3.4.3.1).

Proposition (6.4.9). — Let K be a eld, X and Y algebraic K -preschemes, f : X → Y a K -morphism, and Ω analgebraically closed extension of K of innite transcendence degree over K . For f to be surjective, it is necessary and sucientthat the map X (Ω)K → Y (Ω)K that corresponds to f (3.4.1) be surjective.

Proof. The necessity follows from (6.3.10), noting that f is necessarily of finite type (6.3.9). To see that thecondition is sucient, we argue as in (6.3.10), noting that, for every y ∈ Y , k(y) is an extension of K of finitetype, and so is K -isomorphic to a subfield of Ω.

Remark (6.4.10). — We will see in chapter IV that the conclusion of (6.4.9) still holds without the hypothesison the transcendence degree of Ω over K .

Proposition (6.4.11). — If f : X → Y is a morphism of nite type, then, for every y ∈ Y , the bre f −1(y) is analgebraic prescheme over the residue eld k(y), and for every x ∈ f −1(y), k(x) is an extension of k(x) of nite type.

Proof. Since f −1(y) = X ⊗Y k(y) (6.3.6), the proposition follows from (6.3.4, iv) and (6.3.3).

Proposition (6.4.12). — Let f : X → Y and g : Y ′→ Y be morphisms; set X ′ = X×Y Y ′, and let f ′ = f(Y ′) : X ′→ Y ′.Let y ′ ∈ Y ′ and set y = g(y ′); if the bre f −1(y) is a nite algebraic scheme over k(y), then the bre f ′−1(y) is a nitealgebraic scheme over k(y ′), and has the same rank and geometric number of points as f −1(y) does.

Proof. Taking into account the transitivity of fibres (3.6.5), this follows immediately from (6.4.6) and(6.4.8).

(6.4.13). Proposition (6.4.11) shows that the morphisms of finite type that correspond, intuitively, to the “algebraicfamilies of algebraic varieties”, with the points of Y playing the role of “parameters”, which gives these morphisms I | 150a “geometric” meaning. The morphisms which are not of finite type will show up in the following mostly inquestions of “changing the base prescheme”, by localisation or completion, for example.

6.5. Local determination of a morphism.

Proposition (6.5.1). — Let X and Y be S-preschemes, with Y of nite type over S; let x ∈ X and y ∈ Y lie over thesame point s ∈ S.

(i) If two S-morphisms f = (ψ,θ) and f ′ = (ψ′,θ ′) from X to Y are such that ψ(x) = ψ′(x) = y , and the (local)Os-homomorphisms θ]x and θ

′]x from Oy to Ox are identical, then f and f ′ agree on an open neighbourhood of x .

(ii) Suppose further that S is locally Noetherian. For every local Os-homomorphism φ : Oy →Ox , there exists an openneighbourhood U of x in X , and an S-morphism f = (ψ,θ) from U to Y such that ψ(x) = y and θ]x =φ .

Proof.

(i) Since the question is local on S, X , and Y , we can assume that S, X , and Y are ane, given by rings A,B, and C (respectively), and with f and f ′ of the form (aφ , eφ ) and (aφ ′, eφ ′) (respectively), where φand φ ′ are A-homomorphisms from C to B such that φ−1(jx) =φ ′−1(jx) = jy , and the homomorphismsφx and φ ′x from Cy to Bx , induced by φ and φ ′, are identical; we can further suppose that C is anA-algebra of nite type. Let ci (1¶ i ¶ n) be the generators of the A-algebra C , and set bi =φ (ci) andb′i = φ

′(ci); by hypothesis, we have bi/1 = b′i/1 in the ring of fractions Bx (1 ¶ i ¶ n). This impliesthat there exist elements si ∈ B − jx such that si(bi − b′i) = 0 for 1¶ i ¶ n, and we can clearly assumethat all the si are equal to a single element g ∈ B− jx . From this, we conclude that we have bi/1= b′i/1for 1¶ i ¶ n in the ring of fractions Bg ; if ig is the canonical homomorphism B→ Bg , we then haveig φ = ig φ ′; so the restrictions of f and f ′ to D(g) are identical.

6. FINITENESS CONDITIONS 128

(ii) We can restrict to the situation as in (i), and further assume that the ring A is Noetherian. Let ci(1¶ i ¶ n) be the generators of the A-algebra C , and let α : A[X1, . . . , Xn]→ C be the homomorphismof polynomial algebras that sends X i to ci for 1¶ i ¶ n. Also let iy be the canonical homomorphismC → Cy , and consider the composite homomorphism

β : A[X1, . . . , Xn]α−→ C

iy−→ Cy

φ−→ Bx .

We denote by a the kernel of β; since A is Noetherian, so too is A[X1, . . . , Xn], and so a admits a finitesystem of generators Q j(X1, . . . , Xn) (1¶ j ¶ m). Furthermore, each of the elements φ (iy(ci)) can bewritten in the form bi/si , where bi ∈ B and si 6∈ jx ; we can further assume that all of the si are equal to I | 151a single element g ∈ B − jx . With this, by hypothesis, we have Q j(b1/g, . . . , bn/g) = 0 in Bx ; set

Q j(X1/T, . . . , Xn/T ) = Pj(X1, . . . , Xn, T )/T k j

where Pj is homogeneous of degree k j . Then let d j = Pj(b1, . . . , bn, g) ∈ B. By hypothesis, we havet jd j = 0 for some t j ∈ B − jx (1 ¶ j ¶ m), and we can clearly assume that all the t j are equal to asingle element h ∈ B − jx ; from this we conclude that Pj(hb1, . . . , hbn, hg) = 0 for 1 ¶ j ¶ m. Withthis, consider the homomorphism ρ from A[X1, . . . , Xn] to the ring of fractions Bhg which sends X i tohbi/hg (1 ¶ i ¶ n); the image of a under this homomorphism is 0, and is a fortiori the same as the

image of the kernel α−1(0) under ρ. So ρ factors as A[X1, . . . , Xn]α−→ C

γ−→ Bhg , with γ(ci) = hbi/hg,

and it is clear that, if ix is the canonical homomorphism Bhg → Bx , then the diagram

(6.5.1.1) Cγ //

iy

Bhg

ix

Cy

φ // Bx

is commutative; we thus have φ = γx , and since φ is a local homomorphism, aγ(x) = y ; f = (aγ , eγ) isthus an S-morphism from the neighbourhood D(hg) of x to Y as claimed in the proposition.

Corollary (6.5.2). — Under the hypotheses of (6.5.1, ii), if, further, X is of nite type over S, then we can assume thatthe morphism f is of nite type.

Proof. This follows from Corollary (6.3.6).

Corollary (6.5.3). — Suppose that the hypotheses of Proposition (6.5.1, ii), and suppose further that Y is integral, andthat φ is an injective homomorphism. Then we can assume that f = (aγ , eγ), where γ is injective.

Proof. Indeed, we can assume C to be integral (5.1.4), hence iy injective; it then follows from the diagram(6.5.1.1) that γ is injective.

Proposition (6.5.4). — Let f = (ψ,θ) : X → Y be a morphism of nite type, x a point of X , and y = ψ(x).(i) For f to be a local immersion at the point x (4.5.1), it is necessary and sucient that θ]x : Oy →Ox be surjective.(ii) Assume further that Y is locally Noetherian. For f to be a local isomorphism at the point x (4.5.2), it is necessary

and sucient that θ]x be an isomorphism.

Proof.

(ii) By (6.5.1), there exists an open neighbourhood V of Y and a morphism g : V → X such that g f(resp. f g) is defined and agrees with the identity on a neighbourhood of x (resp. y), whence we caneasily see that f is a local isomorphism.

(i) Since the question is local on X and Y , we can assume that X and Y are ane, given by rings Aand B (respectively); we have f = (aφ , eφ ), where φ is a homomorphism of rings B → A that makesA a B-algebra of finite type; we have φ−1(jx) = jy , and the homomorphism φx : By → Ax inducedby φ is surjective. Let (t i) (1 ¶ i ¶ n) be a system of generators of the B-algebra A; the hypothesison φx implies that there exist bi ∈ B and some c ∈ B − jx such that, in the ring of fractions Ax , wehave t i/1 = φ (bi)/φ (c) for 1 ¶ i ¶ n. Then (1.3.3) there exists some a ∈ A− jx such that, if welet g = aφ (c), we also have t i/1 = aφ (bi)/g in the ring of fractions Ag . With this, there exists, byhypothesis, a polynomial Q(X1, . . . , Xn), with coecients in the ring φ (B), such that a =Q(t1, . . . , tn);let Q(X1/T, . . . , Xn/T ) = P(X1, . . . , Xn, T )/T m, where P is homogeneous of degree m. In the ring Ag , we I | 152

6. FINITENESS CONDITIONS 129

havea/1= amP(φ (b1), . . . ,φ (bn),φ (c))/gm = amφ (d)/gm

where d ∈ B. Since, in Ag , g/1= (a/1)(φ (c)/1) is invertible by definition, so too are a/1 and φ (c)/1,and we can thus write a/1= (φ (d)/1)(φ (c)/1)−m. From this we conclude that φ (d)/1 is also invertible

in Ag . So let h = cd; since φ (h)/1 is invertible in Ag , the composite homomorphism Bφ−→ A→ Ag

factors as B→ Bhγ−→ Ag (0, 1.2.4). We will show that γ is surjective; it suces to show that the image of

Bh in Ag contains the t i/1 and (g/1)−1. But we have (g/1)−1 = (φ (c)/1)m−1(φ (d)/1)−1 = γ(cm/h), anda/1= γ(dm+1/hm), so (aφ (bi))/1= γ(bid

m+1/hm), and since t i/1= (aφ (bi)/1)(g/1)−1, our claim isproved. The choice of h implies that ψ(D(g)) ⊂ D(h), and we also know that the restriction of f toD(g) is equal to (aγ , eγ); since γ is surjective, this restriction is a closed immersion of D(g) into D(h)(4.2.3).

Corollary (6.5.5). — Let f = (ψ,θ) : X → Y be a morphism of nite type. Assume that X is irreducible, and denoteby x its generic point, and let y = ψ(x).

(i) For f to be a local immersion at any point of X , it is necessary and sucient that θ]x : Oy →Ox be surjective.(ii) Assume further that Y is irreducible and locally Noetherian. For f to be a local isomorphism at any point of X ,

it is necessary and sucient that y be the generic point of Y (or, equivalently (0, 2.1.4), that f be a dominantmorphism) and that θ]x be an isomorphism (in other words, that f be birational (2.2.9)).

Proof. It is clear that (i) follows from (6.5.4, i), taking into account the fact that every nonempty opensubset of X contains x ; similarly, (ii) follows from (6.5.4, ii).

6.6. Quasi-compact morphisms and morphisms locally of nite type.

Denition (6.6.1). — We say that a morphism f : X → Y is quasi-compact if the inverse image of anyquasi-compact open subset of Y under f is quasi-compact.

Let B be a base of the topology of Y consisting of quasi-compact open subsets (for example, ane opensubsets); for f to be quasi-compact, it is necessary and sucient that the inverse image of every set of B underf be quasi-compact (or, equivalently, a nite union of ane open subsets), because every quasi-compact opensubset of Y is a finite union of sets of B. For example, if X is quasi-compact and Y ane, then every morphismf : X → Y is quasi-compact: indeed, X is a finite union of ane open subsets Ui , and for every ane open subsetV of Y , Ui ∩ f −1(V ) is ane (5.5.10), and so quasi-compact.

If f : X → Y is a quasi-compact morphism, it is clear that, for every open subset V of Y , the restriction of fto f −1(V ) is a quasi-compact morphism f −1(V )→ V . Conversely, if (Uα) is an open cover of Y , and f : X → Ya morphism such that the restrictions f −1(Uα)→ Uα are quasi-compact, then f is quasi-compact.

Denition (6.6.2). — We say that a morphism f : X → Y is locally of nite type if, for every x ∈ X , there existsan open neighbourhood U of x and an open neighbourhood V ⊃ f (U) of y such that the restriction of f to U isa morphism of finite type from U to V . We then also say that X is a prescheme locally of finite type over Y , or a I | 153Y -prescheme locally of finite type.

It follows immediately from (6.3.2) that, if f is locally of finite type, then, for every open subset W of Y , therestriction of f to f −1(W ) is a morphism f −1(W )→W that is locally of finite type.

If Y is locally Noetherian and X locally of finite type over Y , then X is locally Noetherian thanks to (6.3.7).

Proposition (6.6.3). — For a morphism f : X → Y to be of nite type, it is necessary and sucient that it bequasi-compact and locally of nite type.

Proof. The necessity of the conditions is immediate, given (6.3.1) and the remark following (6.6.1). Con-versely, suppose that the conditions are satisfied, and let U be an ane open subset of Y , given by some ring A; forall x ∈ f −1(U), there is, by hypothesis, a neighbourhood V (x) ⊂ f −1(U) of x , and a neighbourhood W (x) ⊂ Uof y = f (x) containing f (V (x)), and such that the restriction of f to V (x) is a morphism V (x)→W (x) of finitetype. Replacing W (x) with a neighbourhood W1(x) ⊂W (x) of x of the form D(g) (with g ∈ A), and V (x) withV (x)∩ f −1(W1(x)), we can assume that W (x) is of the form D(g), and thus of finite type over U (because itsring can be written as A[1/g]); so V (x) is of finite type over U . Further, f −1(U) is quasi-compact by hypothesis,and so the finite union of open subsets V (x i), which finishes the proof.

Proposition (6.6.4). —

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(i) An immersion X → Y is quasi-compact if it is closed, or if the underlying space of Y is locally Noetherian, or ifthe underlying space of X is Noetherian.

(ii) The composition of any two quasi-compact morphisms is quasi-compact.(iii) If f : X → Y is a quasi-compact S-morphism, then so too is f(S′) : X(S′)→ Y(S′) for any extension g : S→ S′ of

the base prescheme.(iv) If f : X → X ′ and g : Y → Y ′ are two quasi-compact S-morphisms, then f ×S g is quasi-compact.(v) If the composition of any two morphisms f : X → Y and g : Y → Z is quasi-compact, and if either g is separated

or the underlying space of X is locally Noetherian, then f is quasi-compact.(vi) For a morphism f to be quasi-compact, it is necessary and sucient that fred be quasi-compact.

Proof. We note that (vi) is evident because the property of being quasi-compact, for a morphism, dependsonly on the corresponding continuous map of underlying spaces. We will similarly prove the part of (v)corresponding to the case where the underlying space of X is locally Noetherian. Set h = g f , and let U bea quasi-compact open subset of Y ; g(U) is quasi-compact (but not necessarily open) in Z , and so containedin a finite union of quasi-compact open subsets Vj (2.1.3), and f −1(U) is thus contained in the union of theh−1(Vj), which are quasi-compact subspaces of X , and thus Noetherian subspaces. We thus conclude (0, 2.2.3)that f −1(U) is a Noetherian space, and a fortiori quasi-compact.

To prove the other claims, it suces to prove (i), (ii), and (iii) (5.5.12). But (ii) is evident, and (i) followsfrom (6.3.5) whenever the space Y is locally Noetherian or the space X is Noetherian, and is evident for aclosed immersion. To show (iii), we can restrict to the case where S = Y (3.3.11); let f ′ = f(S′), and let U ′ I | 154be a quasi-compact open subset of S′. For every s′ ∈ U ′, let T be an ane open neighbourhood of g(s′) in S,and let W be an ane open neighbourhood of s′ contained in U ′ ∩ g−1(T ); it will suce to show that f ′−1(W )is quasi-compact; in other words, we can restrict to showing that, when S and S′ are ane, the underlyingspace of X ×S S′ is quasi-compact. But since X is then, by hypothesis, a finite union of ane open subsets Vj ,X ×S S′ is a union of the underlying spaces of the ane schemes Vj ×S S′ ((3.2.2) and (3.2.7)), which proves theproposition.

We note also that, if X = X ′ t X ′′ is the sum of two preschemes, a morphism f : X → Y is quasi-compact ifand only if its restrictions to both X ′ and X ′′ are quasi-compact.

Proposition (6.6.5). — Let f : X → Y be a quasi-compact morphism. For f to be dominant, it is necessary andsucient that, for every generic point y of an irreducible component of Y , f −1(y) contain the generic point of an irreduciblecomponent of X .

Proof. It is immediate that the condition is sucient (even without assuming that f is quasi-compact). Tosee that it is necessary, consider an ane open neighbourhood U of y ; f −1(U) is quasi-compact, and so a niteunion of ane open subsets Vi , and the hypothesis that f be dominant implies that y belongs to the closure inU of one of the f (Vi). We can clearly assume X and Y to be reduced; since the closure in X of an irreduciblecomponent of Vi is an irreducible component on X (0, 2.1.6), we can replace X by Vi , and Y by the closedreduced subprescheme of U that has f (Vi)∩U as its underlying space (5.2.1), and we are thus led to proving theproposition when X = Spec(A) and Y = Spec(B) are ane and reduced. Since f is dominant, B is a subring of A(1.2.7), and the proposition then follows from the fact that every minimal prime ideal of B is the intersection ofB with a minimal prime ideal of A (0, 1.5.8).

Proposition (6.6.6). —(i) Every local immersion is locally of nite type.(ii) If two morphisms f : X → Y and g : Y → Z are locally of nite type, then so too is g f .(iii) If f : X → Y is an S-morphism locally of nite type, then f(S′) : X(S′) → Y(S′) is locally of nite type for any

extension S′→ S of the base prescheme.(iv) If f : X → X ′ and g : Y → Y ′ are S-morphisms locally of nite type, then f ×S g is locally of nite type.(v) If the composition g f of two morphisms is locally of nite type, then f is locally of nite type.(vi) If a morphism f is locally of nite type, then so too is fred.

Proof. By (5.5.12), it suces to prove (i), (ii), and (iii). If j : X → Y is a local immersion then, for everyx ∈ X , there is an open neighbourhood V of j(x) in Y and an open neighbourhood U of x in X such that therestriction of j to U is a closed immersion U → V (4.5.1), and so this restriction is of finite type. To prove (ii),consider a point x ∈ X ; by hypothesis, there is an open neighbourhood W of g( f (x)) and an open neighbourhoodV of f (x) such that g(V ) ⊂W and such that V is of of finite type over W ; furthermore, f −1(V ) is locally of finitetype over V (6.6.2), so there is an open neighbourhood U of x that is contained in f −1(V ) and of is finite type

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over V ; thus we have g( f (U)) ⊂W , and that U is of finite type over W (6.3.4, ii). Finally, to prove (iii), we can I | 155restrict to the case where Y = S (3.3.11); for every x ′ ∈ X ′ = X(S′), let x be the image of x ′ in X , s the image of xin S, T an open neighbourhood of s, T ′ the inverse image of T in S′, and U an open neighbourhood of x that isof finite type over T and whose image is contained in T ; then U ×S T ′ = U ×T T ′ is an open neighbourhood ofx ′ (3.2.7) that is of finite type over T ′ (6.3.4, iv).

Corollary (6.6.7). — Let X and Y be S-preschemes that are locally of nite type over S. If S is locally Noetherian, thenX ×S Y is locally Noetherian.

Proof. Indeed, X being locally of finite type over S means that it is locally Noetherian, and that X ×S Y islocally of finite type over X , and so X ×S Y is also locally Noetherian.

Remark (6.6.8). — Proposition (6.3.10) and its proof extend immediately to the case where we suppose onlythat the morphism f is locally of finite type. Similarly, propositions (6.4.2) and (6.4.9) hold true when wesuppose only that the preschemes X and Y in the claim are locally of finite type over the field K .

§7. Rational maps

7.1. Rational maps and rational functions.

(7.1.1). Let X and Y be preschemes, U and V dense open subsets of X , and f (resp. g) a morphism from U(resp. V ) to Y ; we say that f and g are equivalent if they agree on a dense open subset of U ∩ V . Since a finiteintersection of dense open subsets of X is a dense open subset of X , it is clear that this relation is an equivalencerelation.

Denition (7.1.2). — Given preschemes X and Y , we define a rational map from X to Y to be an equivalenceclass of morphisms from a dense open subset of X to a dense open subset of Y , under the equivalence relationdefined in (7.1.1). If X and Y are S-preschemes, we say that such a class is a rational S-map if there exists arepresentative of the class that is also an S-morphism. We define a rational S-section of X to be any rational S-mapfrom S to X . We define a rational function on a prescheme X to be any rational X -section on the X -preschemeX ⊗Z Z[T] (where T is an indeterminate).

By an abuse of language, whenever we are discussing only S-preschemes, we will say “rational map” insteadof “rational S-map” if no confusion may arise.

Let f be a rational map from X to Y , and U an open subset of X ; if f1 and f2 are two morphisms belongingto the class of f , defined (respectively) on dense open subsets V and W of X , then the restrictions f1|(U ∩ V )and f2|(U ∩W ) agree on U ∩ V ∩W , which is dense in U ; the class f of morphisms thus defines a rational mapfrom U to Y , called the restriction of f to U , and denoted by f |U .

If, to every S-morphism f : X → Y , we take the corresponding rational S-map to which f belongs, we obtaina canonical map from HomS(X , Y ) to the set of rational S-maps from X to Y . We denote by Γrat(X/Y ) the set ofrational Y -sections on X , and we thus have a canonical map Γ(X/Y )→ Γrat(X/Y ). It is also clear that, if X andY are S-preschemes, then the set of rational S-maps from X to Y is canonically identified with Γrat((X ×S Y )/X )(3.3.14).

(7.1.3). It also follows from (7.1.2) and (3.3.14) that the rational functions on X are canonically identified withequivalence classes of sections of the structure sheaf OX over dense open subsets of X , where two such sections areequivalent if the agree on some dense open subset of X contained inside the intersection of the subsets on whichthey are defined. In particular, it follows that the rational functions on X form a ring R(X ).

(7.1.4). When X is an irreducible prescheme, every nonempty open subset of X is dense in X ; so we can say thatthe nonempty open subsets of X are the open neighbourhoods of the generic point x of X . To say that two morphismsfrom nonempty open subsets of X to Y are equivalent thus means, in this case, that they have the same germ atthe point x . In other words, the rational maps (resp. rational S-maps) X → Y are identified with the germs ofmorphisms (resp. S-morphisms) from nonempty open subsets of X to Y at the generic point x of X . In particular:

Proposition (7.1.5). — If X is an irreducible prescheme, then the ring R(X ) of rational maps on X is canonicallyidentied with the local ring Ox of the generic point x of X . It is a local ring of dimension 0, and thus a local Artinianring when X is Noetherian; it is a eld when X is integral, and, when X is further an ane scheme, it is identied withthe eld of fractions of A(X ).

Proof. Given the above, and the identification of rational functions with sections of OX over a dense opensubset of X , the first claim is nothing but the definition of the fibre of a sheaf above a point. For the other claims,we can reduce to the case where X is ane, given by some ring A; then jx is the nilradical of A, and Ox is thus of

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dimension 0; if A is integral, then jx = (0), and Ox is thus the field of fractions of A. Finally, if A is Noetherian,we know ([Sam53b, p. 127, cor. 4]) that jx is nilpotent, and Ox = Ax Artinian.

If X is integral, then the ring Oz is integral for all z ∈ X ; every ane open subset U containing z also containsx , and R(U), being equal to the field of fractions of A(U), is identified with R(X ); we thus conclude that R(X ) canalso be identified with the eld of fractions of Oz : the canonical identification of Oz to a subring of R(X ) consistsof associating, to every germ of a section s ∈ Oz , the unique rational function on X , class of a section of OX ,(necessarily defined on a dense open subset of X ) having s as its germ at the point z.

(7.1.6). Now suppose that X has a nite number of irreducible components X i (1 ¶ i ¶ n) (which will be thecase whenever the underlying space of X is Noetherian); let X ′i be the open subset of X given by the complementof the X j ∩ X i for j 6= i inside X i ; X ′i is irreducible, its generic point x i is the generic point of X i , and the X ′i arepairwise disjoint, with their union being dense in X (0, 2.1.6). For every dense open subset U of X , Ui = U ∩ X ′iis a nonempty dense open subset of X ′i , with the Ui being pairwise disjoint, and so U ′ =

⋃

i U ′i is dense in X .Giving a morphism from U ′ to Y consists of giving (arbitrarily) a morphism from each of the Ui to Y .

Thus:

Proposition (7.1.7). — Let X and Y be two preschemes (resp. S-preschemes) such that X has a nite number ofirreducible components X i , with generic points x i (1¶ i ¶ n). If Ri is the set of germs of morphisms (resp. S-morphisms)from open subsets of X to Y at the point x i , then the set of rational maps (resp. rational S-maps) from X to Y can beidentied with the product of the Ri (1¶ i ¶ n).

Corollary (7.1.8). — Let X be a Noetherian prescheme. The ring of rational functions on X is an Artinian ring, whoselocal components are the rings Ox i

of the generic points x i of the irreducible components of X .

Corollary (7.1.9). — Let A be a Noetherian ring, and X = Spec(A). If Q is the complement of the union of the minimalprime ideals of A, then the ring of rational functions on X can be canonically identied with the ring of fractions Q−1A.

This will follow from the following lemma:

Lemma (7.1.9.1). — For an element f ∈ A to be such that D( f ) is dense in X , it is necessary and sucient that f ∈Q;every dense open subset of X contains an open subset of the form D( f ), where f ∈Q.

Proof. To show (7.1.9.1), we again denote by X i (1 ¶ i ¶ n) the irreducible components of X ; if D( f ) isdense in X then D( f )∩ X i 6=∅ for 1¶ i ¶ n, and vice-versa; but this means that f 6∈ pi for 1¶ i ¶ n, where weset pi = j(X i), and since the pi are the minimal prime ideals of A (1.1.14), the conditions f 6∈ pi (1¶ i ¶ n) areequivalent to f ∈Q, whence the first claim of the lemma. For the other claim, if U is a dense open subset of X ,the complement of U is a set of the form V (a), where a is an ideal which is not contained in any of the pi ; it isthus not contained in their union ([Nor53, p. 13]), and there thus exists some f ∈ a belonging to Q; whenceD( f ) ⊂ U , which finishes the proof.

(7.1.10). Suppose again that X is irreducible, with generic point x . Since every nonempty open subset U of Xcontains x , and thus also contains every z ∈ X such that x ∈ z, every morphism U → Y can be composed withthe canonical morphism Spec(Ox)→ X (2.4.1); and any two morphisms into Y from two nonempty open subsetsof X which agree on a nonempty open subset of X give, by composition, the same morphism Spec(Ox)→ Y . Inother words, to every rational map from X to Y there is a corresponding well-defined morphism Spec(Ox)→ Y .

Proposition (7.1.11). — Let X and Y be two S-preschemes; suppose that X is irreducible with generic point x , and thatY is of nite type over S. Any two rational S-maps from X to Y that correspond to the same S-morphism Spec(Ox)→ Yare then identical. If we further suppose S to be locally Noetherian, then every S-morphism from Spec(Ox) to Y correspondsto exactly one rational S-map from X to Y .

Proof. Taking into account that every nonempty subset of X is dense in X , this follows from (6.5.1).

Corollary (7.1.12). — Suppose that S is locally Noetherian, and that the other hypotheses of (7.1.11) are satised.The rational S-maps from X to Y can then be identied with points of the S-prescheme Y , with values in the S-preschemeSpec(Ox).

Proof. This is nothing but (7.1.11), with the terminology introduced in (3.4.1).

Corollary (7.1.13). — Suppose that the conditions of (7.1.12) are satised. Let s be the image of x in S. The dataof a rational S-map from X to Y is equivalent to the data of a point y of Y over s along with a local Os-homomorphismOy →Ox = R(X ).

7. RATIONAL MAPS 133

Proof. This follows from (7.1.11) and (2.4.4).

In particular:

Corollary (7.1.14). — Under the conditions of (7.1.12), rational S-maps from X to Y depend only (for any given Y )on the S-prescheme Spec(Ox), and, in particular, remain the same whenever X is replaced by Spec(Oz), for any z ∈ X .

Proof. Since z ∈ x, x is the generic point of Z = Spec(Oz), and OX ,x = OZ ,z .

When X is integral, R(X ) = Ox = k(x) is a field (7.1.5); the preceding corollaries then specialize to thefollowing:

Corollary (7.1.15). — Suppose that the conditions of (7.1.12) are satised, and further that X is integral. Let s be theimage of x in S. Then rational S-maps from X to Y can be identied with the geometric points of Y ⊗S k(s) with values inthe extension R(X ) of k(s), or, in other words, every such map is equivalent to the data of a point y ∈ Y above s alongwith a k(s)-monomorphism from k(y) to k(x) = R(X ).

Proof. The points of Y above s are identified with the points of Y ⊗S k(s) (3.6.3), and the local Os-homomorphisms Oy → R(X ) with the k(s)-monomorphisms k(y)→ R(X ).

More precisely:

Corollary (7.1.16). — Let k be a eld, and X and Y two algebraic preschemes over k (6.4.1); suppose further thatX is integral. Then the rational k-maps from X to Y can be identied with the geometric points of Y with values in theextension R(X ) of k (3.4.4).

7.2. Domain of denition of a rational map.

(7.2.1). Let X and Y be preschemes, and f rational map from X to Y . We say that f is dened at a point x ∈ X ifthere exists a dense open subset U of X that contains x , and a morphism U → Y belonging to the equivalenceclass of f . The set of points x ∈ X where f is defined is called the domain of denition of f ; it is clear that it isan open dense subset of X .

Proposition (7.2.2). — Let X and Y be S-preschemes such that X is reduced and Y is separated over S. Let f be a I | 159rational S-map from X to Y , with domain of denition U0. Then there exists exactly one S-morphism U0→ Y belonging tothe class of f .

Since, for every morphism U → Y belonging to the class of f , we necessarily have U ⊂ U0, it is clear thatthe proposition will be a consequence of the following:

Lemma (7.2.2.1). — Under the hypotheses of (7.2.2), let U1 and U2 be two dense open subsets of X , and fi : Ui → Y(i = 1,2) two S-morphisms such that there exists an open subset V ⊂ U1 ∩ U2, dense in X , and on which f1 and f2 agree.Then f1 and f2 agree on U1 ∩ U2.

Proof. We can clearly restrict to the case where X = U1 = U2. Since X (and thus V ) is reduced, X is thesmallest closed subprescheme of X containing V (5.2.2). Let g = ( f1, f2)S : X → Y ×S Y ; since, by hypothesis,the diagonal T = ∆Y (Y ) is a closed subprescheme of Y ×S Y , Z = g−1(T ) is a closed subprescheme of X (4.4.1).If h : V → Y is the common restriction of f1 and f2 to V , then the restriction of g to V is g ′ = (h, h)S , whichfactors as g ′ = ∆Y h; since ∆−1

Y (T ) = Y , we have that g ′−1(T ) = V , and so Z is a closed subprescheme of Xinducing V , thus containing V , which implies that Z = X . From the equation g−1(T ) = X , we deduce (4.4.1) thatg factors as ∆Y f , where f is a morphism X → Y , which implies, by the definition of the diagonal morphism,that f1 = f2 = f .

It is clear that the morphism U0→ Y defined in (7.2.2) is the unique morphism of the class f that cannot beextended to a morphism from an open subset of X that strictly contains U0. Under the hypotheses of (7.2.2), we canthus identify the rational maps from X to Y with the non-extendible (to strictly larger open subsets) morphismsfrom dense open subsets of X to Y . With this identification, Proposition (7.2.2) implies:

Corollary (7.2.3). — With the hypotheses from (7.2.2) on X and Y , let U be a dense open subset of X . Then there existsa canonical bijective correspondence between S-morphisms from U to Y and rational S-maps from X to Y that are denedat all points of U .

Proof. By (7.2.2), for every S-morphism f from U to Y , there exists exactly one rational S-map f from X toY which extends f .

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Corollary (7.2.4). — Let S be a scheme, X a reduced S-prescheme, Y an S-scheme, and f : U → Y an S-morphism froma dense open subset U of X to Y . If f is the rational Z-map from X to Y that extends f , then f is an S-morphism (andthus the rational S-map from X to Y that extends f ).

Proof. Indeed, if φ : X → S and ψ : Y → S are the structure morphisms, U0 the domain of definition of f ,and j the injection U0→ X , then it suces to show that ψ f =φ j, but this follows from (7.2.2.1), since f isan S-morphism.

Corollary (7.2.5). — Let X and Y be two S-preschemes; suppose that X is reduced, and that X and Y are separatedover S. Let p : Y → X be an S-morphism (making Y an X -prescheme), U a dense open subset of X , and f a U -section of Y ;then the rational map f from X to Y extending f is a rational X -section of Y .

Proof. We have to show that p f is the identity on the domain of definition of f ; since X is separated over I | 160S, this again follows from (7.2.2.1).

Corollary (7.2.6). — Let X be a reduced prescheme, and U a dense open subset of X . Then there is a canonical bijectivecorrespondence between sections of OX over U and rational functions on X dened at every point of U .

Proof. Taking (7.2.3), (7.1.2), and (7.1.3) into account, it suces to note that the X -prescheme X ⊗Z Z[T] isseparated over X (5.5.1, iv).

Corollary (7.2.7). — Let Y be a reduced prescheme, f : X → Y a separated morphism, U a dense open subset of Y ,g : U → f −1(U) a U -section of f −1(U), and Z the reduced subprescheme of X that has g(U) as its underlying space(5.2.1). For g to be the restriction of a Y -section of X (in other words (7.2.5), for the rational map from Y to Xextending g to be defined everywhere), it is necessary and sucient for the restriction of f to Z to be an isomorphismfrom Z to Y .

Proof. The restriction of f to f −1(U) is a separated morphism (5.5.1, i), so g is a closed immersion (5.4.6),and so g(U) = Z ∩ f −1(U), and the subprescheme induced by Z on the open subset g(U) of Z is identical to theclosed subprescheme of f −1(U) associated to g (5.2.1). It is then clear that the stated condition is sucient,because, if satisfied, and if fZ : Z → Y is the restriction of f to Z , and g : Y → Z is the inverse isomorphism,then g extends g. Conversely, if g is the restriction to U of a Y -section h of X , then h is a closed immersion(5.4.6), and so h(Y ) is closed, and, since it is contained in Z , is equal to Z , and it follows from (5.2.1) that h isnecessarily an isomorphism from Y to the closed subprescheme Z of X .

(7.2.8). Let X and Y be two S-preschemes, with X reduced, and Y separated over S. Let f be a rational S-mapfrom X to Y , and let x be a point of X ; we can compose f with the canonical S-morphism Spec(Ox)→ X (2.4.1)provided that the intersection of Spec(Ox) with the domain of definition of f is dense in Spec(Ox) (identifiedwith the set of z ∈ X such that x ∈ z (2.4.2)). This will happen in the follow cases:

1st. X is irreducible (and thus integral), because then the generic point ξ of X is the generic point ofSpec(Ox); since the domain of definition U of f contains ξ , U ∩ Spec(Ox) contains ξ , and so is densein Spec(Ox).

2nd. X is locally Noetherian; our claim then follows from:

Lemma (7.2.8.1). — Let X be a prescheme whose underlying space is locally Noetherian, and x a point of X . Theirreducible components of Spec(Ox) are the intersections of Spec(Ox) with the irreducible components of X containing x .For an open subset U ⊂ X to be such that U ∩ Spec(Ox) is dense Spec(Ox), it is necessary and sucient for it to have anonempty intersection with the irreducible components of X that contain x (which will be the case whenever U is densein X ).

Proof. It suces to show just the first claim, since the second then follows. Since Spec(Ox) is containedin every ane open subset U that contains x , and since the irreducible components of U that contain x arethe intersections of U with the irreducible components of X containing x (0, 2.1.6), we can suppose that X isane, given by some ring A. Since the prime ideals of Ax correspond bijectively to the prime ideals of A that arecontained in jx (2.1.6), the minimal prime ideals of Ax correspond to the minimal prime ideals of A that are I | 161contained in jx , hence the lemma.

With this in mind, suppose that we are in one of the two cases mentioned in (7.2.8). If U is the domainof definition of the rational S-map f , then we denote by f ′ the rational map from Spec(Ox) to Y which agrees(taking (2.4.2) into account) with f on U ∩ Spec(Ox); we say that this rational map is induced by f .

7. RATIONAL MAPS 135

Proposition (7.2.9). — Let S be a locally Noetherian prescheme, X a reduced S-prescheme, and Y an S-scheme of nitetype. Suppose further that X is either irreducible or locally Noetherian. Then let f be a rational S-map from X to Y , and xa point of X . For f to be dened at a point x , it is necessary and sucient for the rational map f ′ from Spec(Ox) to Y ,induced by f (7.2.8), to be a morphism.

Proof. The condition clearly being necessary (since Spec(Ox) is contained in every open subset containingx), we show that it is sucient. By (6.5.1), there exists an open neighbourhood U of x in X , and an S-morphismg from U to Y that induces f ′ on Spec(Ox). If X is irreducible, then U is dense in X , and, by (7.2.3), we cansuppose that g is a rational S-map. Further, the generic point of X belongs to both Spec(Ox) and the domain ofdefinition of f , and so s and g agree at this point, and thus on a nonempty open subset of X (6.5.1). But since fand g are rational S-maps, they are identical (7.2.3), and so f is defined at x .

If we now suppose that X is locally Noetherian, then we can suppose that U is Noetherian; then there areonly a finite number of irreducible components X i of X that contain x (7.2.8.1), and we can suppose that theyare the only ones that have a nonempty intersection with U , by replacing, if needed, U with a smaller opensubset (since there are only a finite number of irreducible components of X that have a nonempty intersectionwith U , because U is Noetherian). We then have, as above, that f and g agree on a nonempty open subset ofeach of the X i . Taking into account the fact that each of the X i is contained in U , we consider the morphism f1,defined on a dense open subset of U ∪ (X − U), equal to g on U , and to f on the intersection of X − U with thedomain of definition of f . Since U ∪ (X − U) is dense in X , f1 and f agree on a dense open subset of X , andsince f is a rational map, f is an extension of f1 (7.2.3), and is thus defined at the point x .

7.3. Sheaf of rational functions.

(7.3.1). Let X be a prescheme. For every open subset U ⊂ X , we denote by R(U) the ring of rational functionson U (7.1.3); this is a Γ(U ,OX )-algebra. Further, if V ⊂ U is a second open subset of X , then every section of OXover a dense (in X ) open subset of V gives, by restriction to V , a section over a dense (in X ) open subset of V ,and if two sections agree on a dense (in X ) open subset of U , then their restrictions to V agree on a dense (inX ) open subset of V . We can thus define a di-homomorphism of algebras ρV,U : R(U)→ R(V ), and it is clearthat, if U ⊃ V ⊃W are open subsets of X , then we have ρW,U = ρW,V ρV,U ; the R(U) thus define a presheaf ofalgebras on X .

Denition (7.3.2). — We define the sheaf of rational functions on a prescheme X , denoted by R(X ), to be I | 162the OX -algebra associated to the presheaf defined by the R(U).

For every prescheme X and open subset U ⊂ X , it is clear that the induced sheaf R(X )|U is exactly R(U).

Proposition (7.3.3). — Let X be a prescheme such that the family (Xλ) of its irreducible components is locally nite(which is the case whenever the underlying space of X is locally Noetherian). Then the OX -module R(X ) is quasi-coherent,and for every open subset U of X that has a nonempty intersection with only nitely many of the components Xλ , R(U) isequal to Γ(U ,R(X )), and can be canonically identied with the direct sum of the local rings of the generic points of theXλ such that U ∩ Xλ 6=∅.

Proof. We can evidently restrict to the case where X has only a finite number of irreducible componentsX i , with generic points x i (1 ¶ i ¶ n). The fact that R(U) can be canonically identified with the direct sumof the Ox i

= R(X i) such that U ∩ X i 6= ∅ then follows from (7.1.7). We will show that the presheaf U → R(U)satisfies the sheaf axioms, which will prove that R(U) = Γ(U ,R(X )). Indeed, it satisfies (F1) by what has alreadybeen discussed. To see that it satisfies (F2), consider a cover of an open subset U of X by open subsets Vα ⊂ U ;if the sα ∈ R(Vα) are such that the restrictions of sα and sβ to Vα ∩ Vβ agree for every pair of indices, then wecan conclude that, for every index i such that U ∩ X i 6= ∅, the components in R(X i) of all the sα such thatVα ∩ X i 6=∅ are all the same; denoting this component by t i , it is clear that the element of R(U) that has the t ias its components has sα as its restriction to each Vα. Finally, to see that R(X ) is quasi-coherent, we can restrictto the case where X = Spec(A) is ane; by taking U to be an ane open subset of the form D( f ), where f ∈ A,it follows from the above and from Definition (1.3.4) that we have R(X ) = eM , where M is the direct sum of theA-modules Ax i

.

Corollary (7.3.4). — Let X be a reduced prescheme that has only a nite number of irreducible components, and letX i (1¶ i ¶ n) be the closed reduced preschemes of X that have the irreducible components of X as their underlying spaces(5.2.1). If hi is the canonical injection X i → X , then R(X ) is the direct sum of the OX -algebras (hi)∗(R(X i)).

Corollary (7.3.5). — If X is irreducible, then every quasi-coherent R(X )-module F is a simple sheaf.

7. RATIONAL MAPS 136

Proof. It suces to show that every x ∈ X admits a neighbourhood U such that F|U is a simple sheaf(0, 3.6.2); in other words, we are led to considering the case where X is ane; we can further suppose that Fis the cokernel of a homomorphism (R(X ))I → (R(X ))J (0, 5.1.3), and everything then follows from showingthat R(X ) is a simple sheaf; but this is evident, because Γ(U ,R(X )) = R(X ) for every nonempty open subset U ,where U contains the generic point of X .

Corollary (7.3.6). — If X is irreducible, then, for every quasi-coherent OX -module F , F ⊗OXR(X ) is a simple sheaf;

if, further, X is reduced (and thus integral), then F ⊗OXR(X ) is isomorphic to a sheaf of the form (R(X ))(I).

Proof. The second claim follows from the fact that R(X ) is a field.

Proposition (7.3.7). — Suppose that the prescheme X is locally integral or locally Noetherian. Then R(X ) is a I | 163quasi-coherent OX -algebra; if, further, X is reduced (which will be the case whenever X is locally integral), then the canonicalhomomorphism OX →R(X ) is injective.

Proof. The question being local, the first claim follows from (7.3.3); the second follows from (7.2.3).

(7.3.8). 10 Let X and Y be two integral preschemes, which implies that R(X ) (resp. R(Y )) is a quasi-coherentOX -module (resp. OY -module) (7.3.3). Let f : X → Y be a dominant morphism; then there exists a canonicalhomomorphism of OX -modules

(7.3.8.1) τ : f ∗(R(Y )) −→R(X ).

Proof. Suppose first that X = Spec(A) and Y = Spec(B) are ane, given by integral rings A and B, with fthus corresponding to an injective homomorphism B→ A which extends to a monomorphism L→ K from thefield of fractions L of B to the field of fractions K of A. The homomorphism (7.3.8.1) then corresponds to thecanonical homomorphism L ⊗B A→ K (1.6.5). In the general case, for each pair of nonempty ane open setsU ⊂ X and V ⊂ Y such that f (U) ⊂ V , we define, as above, a homomorphism τU ,V and we immediately havethat, if U ′ ⊂ U , V ′ ⊂ V , f (U ′) ⊂ V ′, then τU ,V extends τU ′,V ′ , and hence our assertion. If x and y are the genericpoints of X and Y respectively, then we have f (x) = y ,

( f ∗(R(Y )))x = Oy ⊗OyOx = Ox

(0, 4.3.1) and τx is thus an isomorphism.

7.4. Torsion sheaves and torsion-free sheaves.

(7.4.1). Let X be an integral scheme. For every OX -module F , the canonical homomorphism OX →R(X ) defines,by tensoring, a homomorphism (again said to be canonical) F →F ⊗OX

R(X ), which, on each fibre, is exactlythe homomorphism z→ z ⊗ 1 from Fx to Fx ⊗OX

R(X ). The kernel T of this homomorphism is a OX -submoduleof F , called the torsion sheaf of F ; it is quasi-coherent if F is quasi-coherent ((4.1.1) and (7.3.6)). We say thatF is torsion free if T = 0, and that F is a torsion sheaf if T =F . For every OX -module F , F/T is torsion free.We deduce from (7.3.5) that:

Proposition (7.4.2). — If X is an integral prescheme, then every torsion-free quasi-coherent OX -module F is isomorphicto a subsheaf G of a simple sheaf of the form (R(X ))(I), generated (as a R(X )-module) by G .

The cardinality of I is called the rank of F ; for every nonempty ane open subset U of X , the rank of F isequal to the rank of Γ(U ,F ) as a Γ(U ,OX )-module, as we see by considering the generic point of X , containedin U . In particular:

Corollary (7.4.3). — On an integral prescheme X , every torsion-free quasi-coherent OX -module of rank 1 (in particular,every invertible OX -module) is isomorphic to a OX -submodule of R(X ), and vice versa.

Corollary (7.4.4). — Let X be an integral prescheme, L and L ′ torsion-free OX -modules, and f (resp. f ′) a section ofL (resp. L ′) over X . In order to have f ⊗ f ′ = 0, it is necessary and sucient for one of the sections f and f ′ to be zero.

Proof. Let x be the generic point of X ; we have, by hypothesis, that ( f ⊗ f ′)x = fx ⊗ f ′x = 0. Since Lx andL ′x can be identified with OX -submodules of the field Ox , the above equation leads to fx = 0 or f ′x = 0, and thusf = 0 or f ′ = 0, since L and L ′ are torsion free (7.3.5).

Proposition (7.4.5). — Let X and Y be integral preschemes, and f : X → Y a dominant morphism. For everytorsion-free quasi-coherent OX -module F , f∗(F ) is a torsion-free OY -module.

8. CHEVALLEY SCHEMES 137

Proof. Since f∗ is left exact (0, 4.2.1), it suces, by (7.4.2), to prove the proposition in the case where I | 164F = (R(X ))(I). But every nonempty open subset U of Y contains the generic point of Y , so f −1(U) contains thegeneric point of X (0, 2.1.5), so we have that Γ(U , f∗(F )) = Γ( f −1(U),F ) = (R(X ))(I); in other words, f∗(F ) isthe simple sheaf with fibre (R(X ))(I), considered as a R(Y )-module, and it is clearly torsion free.

Proposition (7.4.6). — Let X be an integral prescheme, and x its generic point. For every quasi-coherent OX -module Fof nite type, the following conditions are equivalent: (a) F is a torsion sheaf; (b) Fx = 0; (c) Supp(F ) 6= X .

Proof. By (7.3.5) and (7.4.1), the equations Fx = 0 and F ⊗OXR(X ) = 0 are equivalent, so (a) and (b) are

equivalent; then Supp(F ) is closed in X (0, 5.2.2), and since every nonempty open subset of X contains x , (b)and (c) are equivalent.

(7.4.7). We generalise (by an abuse of language) the definitions of (7.4.1) to the case where X is a reducedprescheme having only a nite number of irreducible components; it then follows from (7.3.4) that the equivalencebetween a) and c) in (7.4.6) still holds true for such a prescheme.

§8. Chevalley schemes

8.1. Allied local rings. For each local ring A, we denote by m(A) the maximal ideal of A.

Lemma (8.1.1). — Let A and B be two local rings such that A⊂ B; Then the following conditions are equivalent.(i) m(B)∩ A= m(A).(ii) m(A) ⊂ m(B).(iii) 1 is not an element of the ideal of B generated by m(A).

Proof. It is evident that (i) implies (ii), and (ii) implies (iii); lastly, if (iii) is true, then m(B)∩ A containsm(A), and does not contain 1, and is thus equal to m(A).

When the equivalent conditions of (8.1.1) are satisfied, we say that B dominates A; this is equivalent to sayingthat the injection A→ B is a local homomorphism. It is clear that, in the set of local subrings of a ring R, therelation given by domination is an order.

(8.1.2). Now consider a eld R. For all subrings A of R, we denote by L(A) the set of local rings Ap, where pranges over the prime spectrum of A; such local rings are identified with the subrings of R containing A. Sincep= (pAp)∩ A, the map p 7→ Ap from Spec(A) to L(A) is bijective.

Lemma (8.1.3). — Let R be a eld, and A a subring of R. For a local subring M of R to dominate a ring Ap ∈ L(A), itis necessary and sucient that A⊂ M ; the local ring Ap dominated by M is then unique, and corresponds to p= m(M)∩A.

Proof. If M dominates Ap, then m(M)∩Ap = pAp, by (8.1.1), whence the uniqueness of p; on the other hand,if A ⊂ M , then mM ∩ A= p is prime in A, and since A− p ⊂ M , we have that Ap ⊂ M and pAp ⊂ m(M), so Mdominates Ap.

I | 165Lemma (8.1.4). — Let R be a eld, M and N local subrings of R, and P the subring of R generated by M ∪ N . Thenthe following conditions are equivalent.

(i) There exists a prime ideal p of P such that m(M) = p∩M and m(N) = p∩ N .(ii) The ideal a generated in P by m(M)∪m(N) is distinct from P .(iii) There exists a local subring Q of R simultaneously dominating both M and N .

Proof. It is clear that (i) implies (ii); conversely, if a 6= P, then a is contained in a maximal ideal n of P,and since 1 6∈ n, n∩M contains m(M) and is distinct from M , so n∩M = m(M), and similarly n∩ N = m(N).It is clear that, if Q dominates both M and N , then P ⊂ Q and m(M) = m(Q) ∩ M = (m(Q) ∩ P) ∩ M , andm(N) = (m(Q)∩ P)∩ N , so (iii) implies (i); the converse is evident when we take Q = Pp.

When the conditions of (8.1.4) are satisfied, we say, with C. Chevalley, that the local rings M and N areallied.

Proposition (8.1.5). — Let A and B be subrings of a eld R, and C the subring of R generated by A∪ B. Then thefollowing conditions are equivalent.

(i) For every local ring Q containing A and B, we have that Ap = Bq, where p= m(Q)∩ A and q= m(Q)∩ B.(ii) For all prime ideals r of C , we have that Ap = Bq, where p= r∩ A and q= r∩ B.(iii) If M ∈ L(A) and N ∈ L(B) are allied, then they are identical.(iv) L(A)∩ L(B) = L(C).10[Trans.] This paragraph was changed entirely in the Errata of EGA II.

8. CHEVALLEY SCHEMES 138

Proof. Lemmas (8.1.3) and (8.1.4) prove that (i) and (iii) are equivalent; it is clear that (i) implies (ii) bytaking Q = Cr; conversely, (ii) implies (i), because if Q contains A∪ B then it contains C , and if r = m(Q)∩ C ,then p = r∩ A and q = r∩ B, by (8.1.3). It is immediate that (iv) implies (i), because if Q contains A∪ B thenit dominates a local ring Cr ∈ L(C) by (8.1.3); by hypothesis we have that Cr ∈ L(A) ∩ L(B), and (8.1.1) and(8.1.3) prove that Cr = Ap = Bq. We prove finally that (iii) implies (iv). Let Q ∈ L(C); Q dominates someM ∈ L(A) and some N ∈ L(B) (8.1.3), so M and N , being allied, are identical by hypothesis. As we then havethat C ⊂ M , we know that M dominates some Q′ ∈ L(C) (8.1.3), so Q dominates Q′, whence necessarily (8.1.3)Q = Q′ = M , so Q ∈ L(A) ∩ L(B). Conversely, if Q ∈ L(A) ∩ L(B), then C ⊂ Q, so (8.1.3) Q dominates someQ′′ ∈ L(C) ⊂ L(A)∩ L(B); Q and Q′′, being allied, are identical, so Q′′ =Q ∈ L(C), which completes the proof.

8.2. Local rings of an integral scheme.

(8.2.1). Let X be an integral prescheme, and R its field of rational functions, identical to the local ring of thegeneric point a of X ; for all x ∈ X , we know that Ox can be canonically identified with a subring of R (7.1.5), andfor every rational function f ∈ R, the domain of definition δ( f ) of f is the open set of x ∈ X such that f ∈ Ox .It thus follows, from (7.2.6), that, for every open U ⊂ X , we have

(8.2.1.1) Γ(U ,OX ) =⋂

x∈U

Ox .

I | 166Proposition (8.2.2). — Let X be an integral prescheme, and R its eld of rational fractions. For X to be a scheme, it isnecessary and sucient for the relation “Ox and Oy are allied” (8.1.4), for points x and y of X , to imply that x = y .

Proof. We suppose that this condition is satisfied, and aim to show that X is separated. Let U and V betwo distinct ane open subsets of X , given by rings A and B (respectively), identified with subrings of R; U(resp. V ) is thus identified (8.1.2) with L(A) (resp. L(B)), and the hypotheses tell us (8.1.5) that C is the subringof R generated by A∪ B, and W = U ∩ V is identified with L(A)∩ L(B) = L(C). Furthermore, we know ([CC],p. 5-03, 4 bis) that every subring E of R is equal to the intersection of the local rings belonging to L(E); C is thusidentified with the intersection of the rings Oz for z ∈W , or, equivalently (8.2.1.1), with Γ(W,OX ). So considerthe subprescheme induced by X on W ; to the identity morphism φ : C → Γ(W,OX ) there corresponds (2.2.4)a morphism Φ = (ψ,θ) : W → Spec(C); we will see that Φ is an isomorphism of preschemes, whence W is anane open subset. The identification of W with L(C) = Spec(C) shows that ψ is bijective. On the other hand, forall x ∈W , θ]x is the injection Cr → Ox , where r = mx ∩ C , and, by definition, Cr is identified with Ox , so θ]x isbijective. It thus remains to show that ψ is a homeomorphism, or, in other words, that for every closed subsetF ⊂W , ψ(F) is closed in Spec(C). But F is the intersection of W with a closed subspace of U of the form V (a),where a is an ideal of A; we will show that ψ(F) = V (aC), which proves our claim. In fact, the prime idealsof C containing aC are the prime ideals of C containing a, and so are the ideals of the form ψ(x) = mx ∩ C ,where a ⊂ mx and x ∈W ; since a ⊂ mx is equivalent to x ∈ V (a) =W ∩ F for x ∈ U , we do indeed have thatψ(F) = V (aC).

It follows that X is separated, because U ∩ V is ane and its ring C is generated by the union A∪ B of therings of U and V (5.5.6).

Conversely, suppose that X is separated, and let x and y be points of X such that Ox and Oy are allied. LetU (resp. V ) be an ane open subset containing x (resp. y), of ring A (resp. B); we then know that U ∩ V isane and that its ring C is generated by A∪ B (5.5.6). If p= mx ∩ A and q= my ∩ B, then Ap = Ox and Bq = Oy ,and since Ap and Bq are allied, there exists a prime ideal r of C such that p = r∩ A and q = r∩ B (8.1.4). Butthen there exists a point z ∈ U ∩ V such that r= mz ∩ C , since U ∩ V is ane, and so evidently x = z and y = z,whence x = y .

Corollary (8.2.3). — Let X be an integral scheme, and x and y points of X . In order for x ∈ y, it is necessary andsucient for Ox ⊂ Oy , or, equivalently, for every rational function dened at x to also be dened at y .

Proof. The condition is evidently necessary because the domain of definition δ( f ) of a rational functionf ∈ R is open; we now show that it is sucient. If Ox ⊂ Oy , then there exists a prime ideal p of Ox such that Oy

dominates (Ox)p (8.1.3); but (2.4.2) there exists some z ∈ X such that x ∈ z and Oz = (Ox)p; since Oz and Oyare allied, we have that z = y by (8.2.2), whence the corollary.

Corollary (8.2.4). — If X is an integral scheme then the map x →Ox is injective; equivalently, if x and y are twodistinct points of X , then there exists a rational function dened at one of these points but not the other.

I | 167Proof. This follows from (8.2.3) and the axiom (T0) (2.1.4).

Corollary (8.2.5). — Let X be an integral scheme whose underlying space is Noetherian; letting f range over the eld Rof rational functions on X , the sets δ( f ) generate the topology of X .

8. CHEVALLEY SCHEMES 139

In fact, every closed subset of X is thus a finite union of irreducible closed subsets, or, in other words, of theform y (2.1.5). But, if x 6∈ y, then there exists a rational function f defined at x but not at y (8.2.3), or,equivalently, we have that x ∈ δ( f ) and that δ( f ) is not contained in y. The complement of y is thus aunion of sets of the form δ( f ), and, by virtue of the first remark, every open subset of X is the union of finiteintersections of open sets of the form δ( f ).

(8.2.6). Corollary (8.2.5) shows that the topology of X is entirely characterised by the data of the local rings(Ox)x∈X that have R as their field of fractions. It is equivalent to say that the closed subsets of X are definedin the following manner: given a finite subset x1, . . . , xn of X , consider the set of y ∈ X such that Oy ⊂ Ox i

for at least one index i, and these sets (over all choices of x1, . . . , xn) are the closed subsets of X . Further,once the topology on X is known, the structure sheaf OX is also determined by the family of the Ox , sinceΓ(U ,OX ) =

⋂

x∈U Ox , by (8.2.1.1). The family (OX )x∈X thus completely determines the prescheme X when X isan integral scheme whose underlying space is Noetherian.

Proposition (8.2.7). — Let X and Y be integral schemes, f : X → Y a dominant morphism (2.2.6), and K (resp.L)the eld of rational functions on X (resp.Y ). Then L can be identied with a subeld of K , and, for all x ∈ X , O f (x) is theunique local ring of Y dominated by Ox .

Proof. If f = (ψ,θ) and a is the generic point of X , then ψ(a) is the generic point of Y (0, 2.1.5); θ]a isthen a monomorphism of fields, from L = Oψ(a) to K = Oa. Since every nonempty ane open subset U of Ycontains ψ(a), it follows from (2.2.4) that the homomorphism Γ(U ,OY ) → Γ(ψ−1(U),OX ) corresponding tof is the restriction of θ]a to Γ(U ,OY ). So, for every x ∈ X , θ]x is the restriction to Oψ(a) of θ]a, and is thus amonomorphism. We also know that θ]x is a local homomorphism, so, if we identify L with a subfield of K by θ]a,Oψ(x) is dominated by Ox (8.1.1); it is also the only local ring of Y dominated by Ox , since two local rings of Ythat are allied are identical (8.2.2).

Proposition (8.2.8). — Let X be an irreducible prescheme, f : X → Y a local immersion (resp. local isomorphism),and suppose further that f is separated. Then f is an immersion (resp. an open immersion).

Proof. Let f = (ψ,θ); it suces, in both cases, to prove that ψ is a homeomorphism from X to ψ(X ) (4.5.3).Replacing f by fred ((5.1.6) and (5.5.1, vi)), we can assume that X and Y are reduced. If Y ′ is the closed reduced

subprescheme of Y that has ψ(X ) as its underlying space, then f factors as Xf ′−→ Y ′

j−→ Y , where j is the canonical

injection (5.2.2). It follows from (5.5.1, v) that f ′ is again a separated morphism; further, f ′ is again a local I | 168immersion (resp. a local isomorphism), because, since the condition is local on X and Y , we can restrict tothe case where f is a closed immersion (resp. open immersion), and our claim then follows immediately from(4.2.2).

We can thus suppose that f is a dominant morphism, which leads to the fact that Y is, itself, irreducible(0, 2.1.5), and so X and Y are both integral. Further, the condition being local on Y , we can suppose thatY is an ane scheme; since f is separated, X is a scheme (5.5.1, ii), and we are finally at the hypotheses ofProposition (8.2.7). Then, for all x ∈ X , θ]x is injective; but the hypothesis that f is a local immersion impliesthat θ]x is surjective (4.2.2), so θ]x is bijective, or, equivalently (with the identification of Proposition (8.2.7)) wehave that Oψ(x) = Ox . This implies, by Corollary (8.2.4), that ψ is an injective map, which already proves theproposition when f is a local isomorphism (4.5.3). When we suppose that f is only a local immersion, for allx ∈ X there exists an open neighborhood U of x in X and an open neighborhood V of ψ(x) in Y such that therestriction of ψ to U is a homeomorphism from U to a closed subset of V . But U is dense in X , so ψ(U) is densein Y and a fortiori in V , which proves that ψ(U) = V ; since ψ is injective, ψ−1(V ) = U and this proves that ψ is ahomeomorphism from X to ψ(X ).

8.3. Chevalley schemes.

(8.3.1). Let X be a Noetherian integral scheme, and R its field of rational functions; we denote by X ′ the set oflocal subrings Ox ⊂ R, where x ranges over all points of X . The set X ′ satisfies the following three conditions.(Sch. 1) For all M ∈ X ′, R is the field of fractions of M .(Sch. 2) There exists a finite set of Noetherian subrings Ai of R such that X ′ =

⋃

i L(Ai), and, for all pairs ofindices i, j, the subring Ai j of R generated by Ai ∪ A j is an algebra of finite type over Ai .

(Sch. 3) Any two elements M and N of X ′ that are allied are identical.

We have seen in (8.2.1) that (Sch. 1) is satisfied, and (Sch. 3) follows from (8.2.2). To show (Sch. 2), it sucesto cover X by a finite number of ane open subsets Ui whose rings are Noetherian, and to take Ai = Γ(Ui ,OX );the hypothesis that X is a scheme implies that Ui ∩U j is ane, and also that Γ(Ui ∩U j ,OX ) = Ai j (5.5.6); further,

9. SUPPLEMENT ON QUASI-COHERENT SHEAVES 140

since the space Ui is Noetherian, the immersion Ui ∩ U j → Ui is of finite type (6.3.5), so Ai j is an Ai -algebra offinite type (6.3.3).

(8.3.2). The structures whose axioms are (Sch. 1), (Sch. 2), and (Sch. 3) generalise “schemes”, in the senseof C. Chevalley, who additionally supposes that R is an extension of finite type of a field K , and that the Aiare K -algebras of finite type (which renders a part of (Sch. 2) useless) [CC]. Conversely, if we have such astructure on a set X ′, then we can associate to it an integral scheme X by using the remarks from (8.2.6): theunderlying space of X is equal to X ′ endowed with the topology defined in (8.2.6), and with the sheaf OX suchthat Γ(U ,OX ) =

⋂

x∈U Ox for all open U ⊂ X , with the evident definition of restriction homomorphisms. We leaveto the reader the task of verifying that we thus obtain an integral scheme, whose local rings are the elements ofX ′; we will not use this result in what follows.

§9. Supplement on quasi-coherent sheaves

9.1. Tensor product of quasi-coherent sheaves.I | 169

Proposition (9.1.1). — Let X be a prescheme (resp. a locally Noetherian prescheme). Let F and G be quasi-coherent(resp. coherent) OX -modules; then F ⊗OX

G is quasi-coherent (resp. coherent); it is further of nite type if both F andG are of nite type. If F admits a nite presentation and if G is quasi-coherent (resp. coherent), then Hom(F ,G ) isquasi-coherent (resp. coherent).

Proof. Being a local proposition, we can suppose that X is ane (resp. Noetherian ane); further, if Fis coherent, then we can assume that it is the cokernel of a homomorphism O m

X →OnX . The claims pertaining

to quasi-coherent sheaves then follow from Corollaries (1.3.12) and (1.3.9); the claims pertaining to coherentsheaves follow from Theorem (1.5.1) and from the fact that if M and N are modules of finite type over aNoetherian ring A then M ⊗A N and HomA(M , N) are both A-modules of finite type.

Denition (9.1.2). — Let X and Y be S-preschemes, p and q the projections of X ×S Y , and F (resp.G ) aquasi-coherent OX -module (resp. quasi-coherent OY -module). We define the tensor product of F and G over OS(or over S), denoted by F ⊗OS

G (or F ⊗S G ) to be the tensor product p∗(F )⊗OX×S Yq∗(G ) over the prescheme

X ×S Y .

If X i (1 ¶ i ¶ n) are S-preschemes, and Fi (1 ¶ i ¶ n) are quasi-coherent OX i-modules, then we similarly

define the tensor product F1⊗SF2⊗S · · ·⊗SFn over the prescheme Z = X1×S X2×S · · ·×S Xn; it is a quasi-coherentOZ -module by virtue of (9.1.1) and (0, 5.1.4); it is further coherent if all the Fi are coherent and Z is locallyNoetherian, by virtue of (9.1.1), (0, 5.3.11), and (6.1.1).

Note that, if we take X = Y = S, then Definition (9.1.2) gives us back the tensor product of OS -modules.Furthermore, since q∗(OY ) = OX×S Y (0, 4.3.4), the product F ⊗S OY is canonically identified with p∗(F ), and, inthe same way, OX ⊗S G is canonically identified with q∗(G ). In particular, if we take Y = S and denote by f thestructure morphism X → Y , then we have that OX ⊗Y G = f ∗(G ): the ordinary tensor product and the inverseimage thus appear as particular cases of the general tensor product.

Definition (9.1.2) leads immediately to the fact that, for fixed X and Y , F ⊗S G is a right-exact additivecovariant bifunctor in F and G .

Proposition (9.1.3). — Let S, X , and Y be ane schemes of rings A, B, and C (respectively), with B and C beingA-algebras. Let M (resp. N ) be a B-module (resp. C -module), and F = eM (resp. G = eN ) the associated quasi-coherent sheaf;then F ⊗S G is canonically isomorphic to the sheaf associated to the (B ⊗A C)-module M ⊗A N .

I | 170Proof. According to Proposition (1.6.5), F ⊗S G is canonically isomorphic to the sheaf associated to the

(B ⊗A C)-module

M ⊗B (B ⊗A C)

⊗B⊗AC

(B ⊗A C)⊗C N

and, by the canonical isomorphisms between tensor products, this latter module is isomorphic to

M ⊗B (B ⊗A C)⊗C N = (M ⊗B B)⊗A (C ⊗C N) = M ⊗A N .

Proposition (9.1.4). — Let f : T → X and g : T → Y be S-morphisms, and F (resp. G ) a quasi-coherent OX -module(resp. quasi-coherent OY -module). Then

( f , g)∗S(F ⊗S G ) = f ∗(F )⊗OTg∗(G ).

9. SUPPLEMENT ON QUASI-COHERENT SHEAVES 141

Proof. If p, q are the projections of X ×S Y , then the formula follows from the equalities ( f , g)∗S p∗ = f ∗

and ( f , g)∗S q∗ = g∗ (0, 3.5.5), and the fact that the inverse image of a tensor product of algebraic sheaves is thetensor product of their inverse images (0, 4.3.3).

Corollary (9.1.5). — Let f : X → X ′ and g : Y → Y ′ be S-morphisms, and F ′ (resp. G ′) a quasi-coherent OX ′ -module(resp. quasi-coherent OY ′ -module). Then

( f , g)∗S(F′ ⊗S G ′) = f ∗(F ′)⊗S g∗(G ′)

Proof. This follows from (9.1.4) and the fact that f ×S g = ( f p, g q)S , where p and q are the projectionsof X ×S Y .

Corollary (9.1.6). — Let X , Y , and Z be S-preschemes, and F (resp. G , H ) a quasi-coherent OX -module (resp.quasi-coherent OY -module, quasi-coherent OZ -module); then the sheaf F ⊗S G ⊗SH is the inverse image of (F ⊗S G )⊗SHby the canonical isomorphism from X ×S Y ×S Z to (X ×S Y )×S Z .

Proof. This isomorphism is given by (p1, p2)S ×S p3, where p1, p2, and p3 are the projections of X ×S Y ×S Z .Similarly, the inverse image of G⊗SF under the canonical isomorphism from X×S Y to Y ×S X isF⊗SG .

Corollary (9.1.7). — If X is an S-prescheme, then every quasi-coherent OX -module F is the inverse image of F ⊗S OSby the canonical isomorphism from X to X ×S S (3.3.3).

Proof. This isomorphism is (1X ,φ )S , where φ is the structure morphism X → S, and the corollary followsfrom (9.1.4) and the fact that φ ∗(OS) = OX .

(9.1.8). Let X be an S-prescheme, F a quasi-coherent OX -module, and φ : S′→ S a morphism; we denote byF(φ ) or F(S′) the quasi-coherent sheaf F ⊗S OS′ over X ×S S′ = X(φ ) = X(S′); so F(S′) = p∗(F ), where p is theprojection X(S′)→ X .

Proposition (9.1.9). — Let φ ′′ : S′′→ S′ be a morphism. For every quasi-coherent OX -module F on the S-prescheme X ,(F(φ ))(φ ′) is the inverse image of F(φφ ′) by the canonical isomorphism (X(φ ))(φ ′) ' X(φφ ′) (3.3.9).

Proof. This follows immediately from the definitions and from (3.3.9), and is written

(9.1.9.1) (F ⊗S OS′)⊗S′ OS′′ =F ⊗S OS′′ .

Proposition (9.1.10). — Let Y be an S-prescheme, and f : X → Y an S-morphism. For every quasi-coherent OY -moduleG and every morphism S′→ S, we have that ( f(S′))∗(G(S′)) = ( f ∗(G ))(S′).

Proof. This follows immediately from the commutativity of the diagram I | 171

X(S′)f(S′) //

Y(S′)

X

f // Y.

Corollary (9.1.11). — Let X and Y be S-preschemes, and F (resp. G ) a quasi-coherent OX -module (resp. quasi-coherent OY -module). Then the inverse image of the sheaf (F(S′))⊗(S′) (G(S′)) by the canonical isomorphism (X ×S Y )(S′) '(X(S′))×S′ (Y(S′)) (3.3.10) is equal to (F ⊗S G )(S′).

Proof. If p and q are the projections of X ×S Y , then the isomorphism in question is nothing but (p(S′), q(S′))S′ ;the corollary then follows from Propositions (9.1.4) and (9.1.10).

Proposition (9.1.12). — With the notation from Denition (9.1.2), let z be a point of X ×S Y , and let x = p(z), andy = q(z); the stalk (F ⊗S G )z is isomorphic to (Fx ⊗Ox

Oz)⊗Oz(Gy ⊗Oy

Oz) =Fx ⊗OxOz ⊗Oy

⊗Gy .

Proof. Since we can reduce to the ane case, the proposition follows from Equation (1.6.5.1).

Corollary (9.1.13). — If F and G are of nite type, then

Supp(F ⊗S G ) = p−1(Supp(F ))∩ q−1(Supp(G )).

9. SUPPLEMENT ON QUASI-COHERENT SHEAVES 142

Proof. Since p∗(F ) and q∗(G ) are both of finite type over OX×S Y , we reduce, by Proposition (9.1.12) and by(0, 1.7.5), to the case where G = OY , that is, it remains to prove the following equation:

(9.1.13.1) Supp(p∗(F )) = p−1(Supp(F )).

The same reasoning as in (0, 1.7.5) leads us to prove that, for all z ∈ X ×S Y , we have Oz/mxOz 6= 0 (withx = p(z)), which follows from the fact that the homomorphism Ox →Oz is local, by hypothesis.

We leave it to the reader to extend the results in this section to the more general case of an arbitrary (butfinite) number of factors, instead of just two.

9.2. Direct image of a quasi-coherent sheaf.

Proposition (9.2.1). — Let f : X → Y be a morphism of preschemes. We suppose that there exists a cover (Yα) of Y byane opens having the following property: every f −1(Yα) admits a finite cover (Xαi) by ane opens that are contained inf −1(Yα) and that are such that every intersection Xαi ∩ Xα j is itself a finite union of ane opens. With these hypotheses,for every quasi-coherent OX -module F , f∗(F ) is a quasi-coherent OY -module.

Proof. Since this is a local condition on Y , we can assume that Y is equal to one of the Yα, and thus omitthe indices α.

(a) First, suppose that the X i ∩ X j are themselves ane opens. We set Fi =F|X i and Fi j =F|(X i ∩ X j),and let F ′i and F ′i j be the images of Fi and Fi j (respectively) by the restriction of f to X i and toX i ∩ X j (respectively); we know that the F ′i and F

′i j are quasi-coherent (1.6.3). Set G =

⊕

iF ′i andH =

⊕

i, jF ′i j ; G and H are quasi-coherent OY -modules; we will define a homomorphism u : G →Hsuch that f∗(F ) is the kernel of u; it will follow from this that f∗(F ) is quasi-coherent (1.3.9). It sucesto define u as a homomorphism of presheaves; taking into account the definitions of G and H , it thus I | 172suces, for every open subset W ⊂ Y , to define a homomorphism

uW :⊕

i

Γ( f −1(W )∩ X i ,F ) −→⊕

i, j

Γ( f −1(W )∩ X i ∩ X j ,F )

that satisfies the usual compatibility conditions when we let W vary. If, for every section si ∈Γ( f −1(W )∩ X i ,F ), we denote by si| j its restriction to f −1(W )∩ X i ∩ X j , then we set

uW

(si)

= (si| j − s j|i)

and the compatibility conditions are clearly satisfied. To prove that the kernel R of u is f∗(F ), wedefine a homomorphism from f∗(F ) to R by sending each section s ∈ Γ( f −1(W ),F ) to the family (si),where si is the restriction of s to f −1(W )∩ X i ; axioms (F1) and (F2) of sheaves (G, II, 1.1) tell us thatthis homomorphism is bijective, which finishes the proof in this case.

(b) In the general case, the same reasoning applies once we have established that theFi j are quasi-coherent.But, by hypothesis, X i ∩ X j is a finite union of ane opens X i jk; and since the X i jk are ane opens in ascheme, the intersection of any two of them is again an ane open (5.5.6). We are thus led to the firstcase, and so we have proved Proposition (9.2.1).

Corollary (9.2.2). — The conclusion of Proposition (9.2.1) holds true in each of the following cases:(a) f is separated and quasi-compact;(b) f is separated and of nite type;(c) f is quasi-compact, and the underlying space of X is locally Noetherian.

Proof. In case (a), the Xαi ∩ Xα j are ane (5.5.6). Case (b) is a particular example of case (a) (6.6.3).Finally, in case (c), we can reduce to the case where Y is ane and the underlying space of X is Noetherian;then X admits a finite cover of ane opens (X i), and the X i ∩ X j , being quasi-compact, are finite unions of aneopens (2.1.3).

9.3. Extension of sections of quasi-coherent sheaves.

Theorem (9.3.1). — Let X be a prescheme whose underlying space is Noetherian, or a scheme whose underlying space isquasi-compact. Let L be an invertible OX -module (0, 5.4.1), f a section of L over X , X f the open set of x ∈ X such thatf (x) 6= 0 (0, 5.5.1), and F a quasi-coherent OX -module.

(i) If s ∈ Γ(X ,F ) is such that s|X f = 0, then there exists an integer n> 0 such that s⊗ f ⊗n = 0.(ii) For every section s ∈ Γ(X f ,F ), there exists an integer n> 0 such that s⊗ f ⊗n extends to a section of F ⊗L ⊗n

over X .

9. SUPPLEMENT ON QUASI-COHERENT SHEAVES 143

Proof.(i) Since the underlying space of X is quasi-compact, and thus the union of finitely-many ane opens Ui

with L|Ui isomorphic to OX |Ui , we can reduce to the case where X is ane and L = OX . In this case,f can be identified with an element of A(X ), and we have that X f = D( f ); s can be identified with anelement of an A(X )-module M , and s|X f to the corresponding element of M f , and the result is thentrivial, recalling the definition of a module of fractions. I | 173

(ii) Again, X is a finite union of ane opens Ui (1 ¶ i ¶ r) such that L|Ui∼= OX |Ui , and, for every i,

(s⊗ f ⊗n)|(Ui∩X f ) can be identified (by the aforementioned isomorphism) with ( f |(Ui∩X f ))n(s|(Ui∩X f )).We then know (1.4.1) that there exists an integer n> 0 such that, for all i, (s⊗ f ⊗n)|(Ui ∩ X f ) extendsto a section si of F ⊗L ⊗n over Ui . Let si| j be the restriction of si to Ui ∩U j ; by definition we have thatsi| j − s j|i = 0 on X f ∩ Ui ∩ U j . But, if X is a Noetherian space, then Ui ∩ U j is quasi-compact; if X is ascheme, then Ui ∩ U j is an ane open (5.5.6), and so again quasi-compact. By virtue of (i), there thusexists an integer m (independent of i and j) such that (si| j − s j|i)⊗ f ⊗m = 0. It immediately followsthat there exists a section s′ of F ⊗L ⊗(n+m) over X that restricts to si ⊗ f ⊗m over each Ui , and restrictsto s⊗ f ⊗(n+m) over X f .

The following corollaries give an interpretation of Theorem (9.3.1) in a more algebraic language:

Corollary (9.3.2). — With the hypotheses of (9.3.1), consider the graded ring A∗ = Γ∗(L ) and the graded A∗-moduleM∗ = Γ∗(L ,F ) (0, 5.4.6). If f ∈ An, where n ∈ Z, then there is a canonical isomorphism Γ(X f ,F ) ' ((M∗) f )0 ( thesubgroup of the module of fractions (M∗) f consisting of elements of degree 0).

Corollary (9.3.3). — Suppose that the hypotheses of (9.3.1) are satised, and suppose further that L = OX . Then,setting A= Γ(X ,OX ) and M = Γ(X ,F ), the A f -module Γ(X f ,F ) is canonically isomorphic to M f .

Proposition (9.3.4). — Let X be a Noetherian prescheme, F a coherent OX -module, and J a coherent sheaf of idealsin OX , such that the support of F is contained in that of OX |J . Then there exists an integer n> 0 such that J nF = 0.

Proof. Since X is a union of finitely-many ane opens whose rings are Noetherian, we can suppose thatX is ane, given by some Noetherian ring A; then F = eM , where M = Γ(X ,F ) is an A-module of finite type,and J = eJ, where J = Γ(X ,J ) is an ideal of A ((1.4.1) and (1.5.1)). Since A is Noetherian, J admits a finitesystem of generators fi (1 ¶ i ¶ m). By hypothesis, every section of F over X is zero on each of the D( fi); ifs j (1 ¶ j ¶ q) are sections of F that generate M , then there exists an integer h, independent of i and j, suchthat f h

i s j = 0 (1.4.1), whence f hi s = 0 for all s ∈ M . We thus conclude that, if n= mh, then JnM = 0, and so the

corresponding OX -module J nF =ßJnM (1.3.13) is zero.

Corollary (9.3.5). — With the hypotheses of (9.3.4), there exists a closed subprescheme Y of X , whose underlying spaceis the support of OX/J , such that, if j : Y → X is the canonical injection, then F = j∗( j∗(F )).

Proof. First, note that the supports of OX/J and OX/J n are the same, since, if Jx = Ox , then J nx = Ox ,

and we also have that J nx ⊂ Jx for all x ∈ X . We can, thanks to (9.3.4), thus suppose that JF = 0; we can

then take Y to be the closed subprescheme of X defined by J , and since F is then an (OX/J )-module, theconclusion follows immediately.

9.4. Extension of quasi-coherent sheaves.

(9.4.1). Let X be a topological space, F a sheaf of sets (resp. of groups, of rings) on X , U an open subset of X , I | 174ψ : U → X the canonical injection, and G a subsheaf of F|U = ψ∗(F ). Since ψ∗ is left exact, ψ∗(G ) is a subsheafof ψ∗(ψ∗(F )); we denote by ρ the canonical homomorphism F → ψ∗(ψ∗(F )) (0, 3.5.3), and we denote by Gthe subsheaf ρ−1(ψ∗(G )) of F . It follows immediately from the definitions that, for every open subset V of X ,Γ(V,G ) consists of sections s ∈ Γ(V,F ) whose restriction to V ∩ U is a section of G over V ∩ U . We thus havethat G|U = ψ∗(G ) = G , and that G is the largest subsheaf of F that restricts to G over U ; we say that G is thecanonical extension of the subsheaf G of F|U to a subsheaf of F .

Proposition (9.4.2). — Let X be a prescheme, and U an open subset of X such that the canonical injection j : U → Xis a quasi-compact morphism (which will be the case for all U if the underlying space of X is locally Noetherian(6.6.4, i)). Then:

(i) for every quasi-coherent (OX |U)-module G , j∗(G ) is a quasi-coherent OX -module, and j∗(G )|U = j∗( j∗(G )) = G ;(ii) for every quasi-coherent OX -module F and every quasi-coherent (OX |U)-submodule G , the canonical extension G

of G (9.4.1) is a quasi-coherent OX -submodule of F .

9. SUPPLEMENT ON QUASI-COHERENT SHEAVES 144

Proof. If j = (ψ,θ) (ψ being the injection U → X of underlying spaces), then, by definition, we have thatj∗(G ) = ψ∗(G ) for every (OX |U)-module G , and, further, that j∗(H ) = ψ∗(H ) =H |U for every OX -module H ,by definition of the prescheme induced over an open subset. So (i) is thus a particular case of ((9.2.2, a)); forthe same reason, j∗( j∗(F )) is quasi-coherent, and since G is the inverse image of j∗(G ) by the homomorphismρ :F → j∗( j∗(F )), (ii) follows from (4.1.1).

Note that the hypothesis that the morphism j : U → X is quasi-compact holds whenever the open subset Uis quasi-compact and X is a scheme: indeed, U is then a union of finitely-many ane opens Ui , and, for every aneopen V of X , V ∩ Ui is an ane open (5.5.6), and thus quasi-compact.

Corollary (9.4.3). — Let X be a prescheme, and U a quasi-compact open subset of X such that the injection morphismj : U → X is quasi-compact. Suppose as well that every quasi-coherent OX -module is the inductive limit of its quasi-coherentOX -submodules of nite type (which will be the case if X is an ane scheme). Then let F be a quasi-coherent OX -module,and G a quasi-coherent (OX |U)-submodule of F|U of finite type. Then there exists a quasi-coherent OX -submodule G ′ ofF of finite type such that G ′|U = G .

Proof. We have G = G|U , and G is quasi-coherent, from (9.4.2), so the inductive limit of its quasi-coherentOX -submodules Hλ of finite type. It follows that G is the inductive limit of the Hλ |U , and thus equal to one ofthe Hλ |U , since it is of finite type (0, 5.2.3).

Remark (9.4.4). — Suppose that for every ane open U ⊂ X , the injection morphism U → X is quasi-compact.Then, if the conclusion of (9.4.3) holds for every ane open U and for every quasi-coherent (OX |U)-submoduleG of F|U of finite type, it follows that F is the inductive limit of its quasi-coherent OX -submodules of finite I | 175type. Indeed, for every ane open U ⊂ X , we have that F|U = eM , where M is an A(U)-module, and sincethe latter is the inductive limit of its quasi-coherent submodules of finite type, F|U is the inductive limit ofits (OX |U)-submodules of finite type (1.3.9). But, by hypothesis, each of these submodules is induced on U bya quasi-coherent OX -submodule Gλ,U of F of finite type. The finite sums of the Gλ,U are again quasi-coherentOX -modules of finite type, because this is a local property, and the case where X is ane was covered in (1.3.10);it is clear then that F is the inductive limit of these finite sums, whence our claim.

Corollary (9.4.5). — Under the hypotheses of Corollary (9.4.3), for every quasi-coherent (OX |U)-module G of nitetype, there exists a quasi-coherent OX -module G ′ of nite type such that G ′|U = G .

Proof. Since F = j∗(G ) is quasi-coherent (9.4.2) and F|U = G , it suces to apply Corollary (9.4.3) toF .

Lemma (9.4.6). — Let X be a prescheme, L a well-ordered set, (Vλ)λ∈L a cover of X by ane opens, and U an opensubset of X ; for all λ ∈ L, we set Wλ =

⋃

µ<λ Vµ. Suppose that: (1) for every λ ∈ L, Vλ ∩Wλ is quasi-compact; and (2)the immersion morphism U → X is quasi-compact. Then, for every quasi-coherent OX -module F and every quasi-coherent(OX |U)-submodule G of F|U of finite type, there exists a quasi-coherent OX -submodule G ′ of F of finite type such thatG ′|U = G .

Proof. Let Uλ = U ∪Wλ ; we will define a family (G ′λ) by induction, where G ′λ is a quasi-coherent (OX |Uλ)-submodule of F|Uλ of finite type, such that G ′λ |Uµ = G

′µ for µ< λ and G ′λ |U = G . The unique OX -submodule G ′

of F such that G ′|Uλ = G ′ for all λ ∈ L (0, 3.3.1) will then give us what we want. So suppose that the G ′µ aredefined and have the preceding properties for µ< λ; if λ does not have a predecessor then we take G ′λ to be theunique (OX |Uλ)-submodule of F|Uλ such that G ′λ |Uµ = G

′µ for all µ< λ, which is allowed since the Uµ with µ< λ

then form a cover of Uλ . If, conversely, λ = µ+ 1, then Uλ = Uµ ∪ Vµ, and it suces to define a quasi-coherent(OX |Vµ)-submodule G ′′µ of F|Vµ of finite type such that

G ′′µ |(Uµ ∩ Vµ) = G ′µ|(Uµ ∩ Vµ);

and then to take G ′λ to be the (OX |Uλ)-submodule of F|Uλ such that G ′λ |Uµ = G′µ and G ′λ |Vµ = G

′′µ (0, 3.3.1). But,

since Vµ is ane, the existence of G ′′µ is guaranteed by (9.4.3) as soon as we show that Uµ ∩ Vµ is quasi-compact;but Uµ ∩ Vµ is the union of U ∩ Vµ and Wµ ∩ Vµ, which are both quasi-compact by virtue of the hypotheses.

Theorem (9.4.7). — Let X be a prescheme, and U an open subset of X . Suppose that one of the following conditions isveried:

(a) the underlying space of X is locally Noetherian;(b) X is a quasi-compact scheme and U is a quasi-compact open.

9. SUPPLEMENT ON QUASI-COHERENT SHEAVES 145

Then, for every quasi-coherent OX -module F and every quasi-coherent (OX |U)-submodule G of F|U of finite type, thereexists a quasi-coherent OX -submodule G ′ of F of finite type such that G ′|U = G .

Proof. Let (Vλ)λ∈L be a cover of X by ane opens, with L assumed to be finite in case (b); since L is I | 176equipped with the structure of a well-ordered set, it suces to check that the conditions of (9.4.6) are satisfied.It is clear in the case of (a), since the spaces Vλ are Noetherian. For case (b), the Vλ ∩λµ are ane (5.5.6), andthus quasi-compact, and, since L is finite, Vλ ∩Wλ is quasi-compact. Whence the theorem.

Corollary (9.4.8). — Under the hypotheses of (9.4.7), for every quasi-coherent (OX |U)-module G of nite type, thereexists a quasi-coherent OX -module G ′ of nite type such that G ′|U = G .

Proof. It suces to apply (9.4.7) to F = j∗(G ), which is quasi-coherent (9.4.2) and such that F|U = G .

Corollary (9.4.9). — Let X be a prescheme whose underlying space is locally Noetherian, or a quasi-compact scheme.Then every quasi-coherent OX -module is the inductive limit of its quasi-coherent OX -submodules of nite type.

Proof. This follows from Theorem (9.4.7) and Remark (9.4.4).

Corollary (9.4.10). — Under the hypotheses of (9.4.9), if a quasi-coherent OX -moduleF is such that every quasi-coherentOX -submodule of nite type of F is generated by its sections over X , then F is generated by its sections over X .

Proof. Let U be an ane open neighborhood of a point x ∈ X , and let s be a section of F over U ; theOX -submodule G of F|U generated by s is quasi-coherent and of finite type, so there exists a quasi-coherentOX -submodule G ′ of F of finite type such that G ′|U = G (9.4.7). By hypothesis, there thus exists a finite numberof sections t i of G ′ over X and of sections ai of OX over a neighborhood V ⊂ U of x such that s|V =

∑

i ai(t i |V ),which proves the corollary.

9.5. Closed image of a prescheme; closure of a subprescheme.

Proposition (9.5.1). — Let f : X → Y be a morphism of preschemes such that f∗(OX ) is a quasi-coherent OY -module(which will be the case if f is quasi-compact and, in addition, either f is separated or X is locally Noetherian (9.2.2)).Then there exists a smaller subprescheme Y ′ of Y such that f factors through the canonical injection j : Y ′ → Y ( or,equivalently (4.4.1), such that the subprescheme f −1(Y ′) of X is identical to X ).

More precisely:

Corollary (9.5.2). — Under the conditions of (9.5.1), let f = (ψ,θ), and let J be the (quasi-coherent) kernel of thehomomorphism θ : OY → f∗(OX ). Then the closed subprescheme Y ′ of Y dened by J satises the conditions of (9.5.1).

Proof. Since the functor ψ∗ is exact, the canonical factorization θ : OY →OY /Jθ ′−→ ψ∗(OX ) gives (0, 3.5.4.3)

a factorization θ] : ψ∗(OY )→ ψ∗(OY )/ψ∗(J )θ ′]−→ OX ; since θ]x is a local homomorphism for every x ∈ X , the

same is true of θ ′x]; if we denote by ψ0 the continuous map ψ considered as a map from X to Y ′, and by θ0 ErrII

the restriction θ ′|X ′ : (OY /J )|Y ′ → ψ∗(OX )|Y ′ = (ψ0)∗(OX ), then we see that f0 = (ψ0,θ0) is a morphism ofpreschemes X → Y ′ (2.2.1) such that f = j f0. Now, if Y ′′ is a second closed subprescheme of Y , defined by a I | 177quasi-coherent sheaf of ideals J ′ of OY , such that f factors through the injection j′ : Y ′′→ Y , then we shouldimmediately have that ψ(X ) ⊂ Y ′′, and thus that Y ′ ⊂ Y ′′, since Y ′′ is closed. Furthermore, for all y ∈ Y ′′, θshould factor as Oy → Oy/J ′y → (ψ∗(OX ))y , which, by definition, leads to J ′y ⊂ Jy , and thus X ′ is a closedsubprescheme of Y ′′ (4.1.10).

Denition (9.5.3). — Whenever there exists a smaller subprescheme Y ′ of Y such that f factors through thecanonical injection j : Y ′→ Y , we say that Y ′ is the closed image prescheme of X under the morphism f .

Proposition (9.5.4). — If f∗(OX ) is a quasi-coherent OY -module, then the underlying space of the closed image of Xunder f is the closure f (X ) in Y .

Proof. As the support of f∗(OX ) is contained in f (X ), we have (with the notation of (9.5.2)) Jy = Oy for

y 6∈ f (X ), thus the support of OY /J is contained in f (X ). In addition, this support is closed and contains f (X ):indeed, if y ∈ f (X ), the unit element of the ring (ψ∗(OX ))y is not zero, being the germ at y of the section

1 ∈ Γ(X ,OX ) = Γ(Y,ψ∗(OX ));

since it is the image under θ of the unit element of Oy , the latter does not belong to Jy , hence Oy/Jy 6= 0; thisfinishes the proof.

9. SUPPLEMENT ON QUASI-COHERENT SHEAVES 146

Proposition (9.5.5). — (Transitivity of closed images). Let f : X → Y and g : Y → Z be two morphisms ofpreschemes; we suppose that the closed image Y ′ of X under f exists, and that, if g ′ is the restriction of g to Y ′, then theclosed image Z ′ of Y ′ under g ′ exists. Then the closed image of X under g f exists and is equal to Z ′.

Proof. It suces (9.5.1) to show that Z ′ is the smallest closed subprescheme Z1 of Z such that the closedsubprescheme (g f )−1(Z1) of X (equal to f −1(g−1(Z1)) by Corollary (4.4.2)) is equal to X ; it is equivalent to saythat Z ′ is the smallest closed subprescheme of Z such that f factors through the injection g−1(Z1)→ Y (4.4.1).By virtue of the existence of the closed image Y ′, every Z1 with this property is such that g−1(Z1) factors throughY ′, which is equivalent to saying that j−1(g−1(Z1)) = g ′−1(Z1) = Y ′, denoting by j the injection Y ′→ Y . By thedefinition of Z ′, we indeed conclude that Z ′ is the smallest closed subprescheme of Z satisfying the precedingcondition.

Corollary (9.5.6). — Let f : X → Y be an S-morphism such that Y is the closed image of X under f . Let Z be anS-scheme; if two S-morphisms g1, g2 from Y to Z are such that g1 f = g2 f , then g1 = g2.

Proof. Let h = (g1, g2)S : Y → Z ×S Z ; since the diagonal T = ∆Z(Z) is a closed subprescheme of Z ×S Z ,Y ′ = h−1(T ) is a closed subprescheme of Y (4.4.1). Let u = g1 f = g2 f ; we then have, by definition of theproduct, h′ = h f = (u, u)S , so h f = ∆Z u; since ∆−1

Z (T ) = Z , we have h′−1(T ) = u−1(Z) = X , so f −1(Y ′) = X .From this, we conclude (4.4.1) that the f factors through the canonical injection Y ′→ Y , so Y ′ = Y by hypothesis;it then follows (4.4.1) that h factors as ∆Z v, where v is a morphism Y → Z , which implies that g1 = g2 = v.

Remark (9.5.7). — If X and Y are S-schemes, Proposition (9.5.6) implies that, when Y is the closed image of I | 178X under f , f is an epimorphism in the category of S-schemes (T, 1.1). We will show in Chapter V that, conversely,if the closed image Y ′ of X under f exists and if f is an epimorphism of S-schemes, then we necessarily havethat Y ′ = Y .

Proposition (9.5.8). — Suppose that the hypotheses of (9.5.1) are satised, and let Y ′ be the closed image of X under f .For every open V of Y , let fV : f −1(V )→ V be the restriction of f ; then the closed image of f −1(V ) under fV in V existsand is equal to the prescheme induced by Y ′ on the open V ∩ Y ′ of Y ′ (in other words, to the subprescheme inf(V, Y ) ofY (4.4.3)).

Proof. Let X ′ = f −1(V ); since the direct image of OX ′ by fV is exactly the restriction of f∗(OX ) to V , it isclear that the kernel J ′ of the homomorphism OV → ( fV )∗(OX ′) is the restriction of J to V , from which theproposition quickly follows.

We will see that this result can be understood as saying that taking the closed image commutes with anextension Y1→ Y of the base prescheme, which is an open immersion. We will see in Chapter IV that it is thesame for an extension Y1→ Y which is a at morphism, provided that f is separated and quasi-compact.

Proposition (9.5.9). — Let f : X → Y be a morphism such that the closed image Y ′ of X under f exists.(i) If X is reduced, then so too is Y ′.(ii) If the hypotheses of Proposition (9.5.1) are satised and X is irreducible (resp. integral), then so too is Y ′.

Proof. By hypothesis, the morphism f factors as Xg−→ Y ′

j−→ Y , where j is the canonical injection. Since X is

reduced, g factors as Xh−→ Y ′red

j′−→ Y ′, where j′ is the canonical injection (5.2.2), and it then follows from the

definition of Y ′ that Y ′red = Y ′. If, moreover, the conditions of Proposition (9.5.1) are satisfied, then it followsfrom (9.5.4) that f (X ) is dense in Y ′; if X is irreducible, then so is Y ′ (0, 2.1.5). The claim about integralpreschemes follows from the conjunction of the two others.

Proposition (9.5.10). — Let Y be a subprescheme of a prescheme X , such that the canonical injection i : Y → X is aquasi-compact morphism. Then there exists a smaller closed subprescheme Y of X containing Y ; its underlying space is theclosure of that of Y ; the latter is open in its closure, and the prescheme Y is induced on this open by Y .

Proof. It suces to apply Proposition (9.5.1) to the injection j, which is separated (5.5.1) and quasi-compactby hypothesis; (9.5.1) thus proves the existence of Y , and (9.5.4) shows that its underlying space is the closure ofY in X ; since Y is locally closed in X , it is open in Y , and the last claim comes from (9.5.8) applied to an opensubset V of X such that Y is closed in V .

With the above notation, if the injection V → X is quasi-compact, and if J is the quasi-coherent sheaf ofideals of OX |V defining the closed subprescheme Y of V , it follows from Proposition (9.5.1) that the quasi-coherentsheaf of ideals of OX defining Y is the canonical extension (9.4.1) J of J , because it is clearly the largestquasi-coherent subsheaf of ideals of OX inducing J on V .

9. SUPPLEMENT ON QUASI-COHERENT SHEAVES 147

Corollary (9.5.11). — Under the hypotheses of Proposition (9.5.10), every section of OY over an open V of Y that is I | 179zero on V ∩ Y is zero.

Proof. By Proposition (9.5.8), we can reduce to the case where V = Y . If we take into account that thesections of OY over Y canonically correspond to the Y -sections of Y ⊗Z Z[T] (3.3.15) and that the latter isseparated over Y , then the corollary appears as a specific case of (9.5.6).

When there exists a smaller closed subprescheme Y ′ of X containing a subprescheme Y of X , we say that Y ′

is the closure of Y in X , when there is little cause for confusion.

9.6. Quasi-coherent sheaves of algebras; change of structure sheaf.

Proposition (9.6.1). — Let X be a prescheme, and B a quasi-coherent OX -algebra (0, 5.1.3). For a B -module F to bequasi-coherent (on the ringed space (X ,B)) it is necessary and sucient that F be a quasi-coherent OX -module.

Proof. Since the question is local, we can assume X to be ane, given by some ring A, and thus B = eB,where B is an A-algebra (1.4.3). If F is quasi-coherent on the ringed space (X ,B) then we can also assume that Fis the cokernel of a B -homomorphism B (I)→B (J); since this homomorphism is also an OX -homomorphism ofOX -modules, and B (I) and B (J) are quasi-coherent OX -modules (1.3.9, ii), F is also a quasi-coherent OX -module(1.3.9, i).

Conversely, if F is a quasi-coherent OX -module, then F = eM , where M is a B-module (1.4.3); M isisomorphic to the cokernel of a B-homomorphism B(I)→ B(J), so F is a B -module isomorphic to the cokernelof the corresponding homomorphism B (I)→B (J) (1.3.13), which finishes the proof.

In particular, if F and G are two quasi-coherent B -modules, then F ⊗B G is a quasi-coherent B -module;similarly for Hom(F ,G ) whenever we further suppose that F admits a finite presentation (1.3.13).

(9.6.2). Given a prescheme X , we say that a quasi-coherent OX -algebra B is of nite type if, for all x ∈ X , thereexists an open ane neighborhood U of x such that Γ(U ,B) = B is an algebra of finite type over Γ(U ,OX ) = A.We then have that B|U = eB and, for all f ∈ A, the induced (OX |D( f ))-algebra B|D( f ) is of finite type, becauseit is isomorphic to (B f )∼, and B f = B ⊗A A f is clearly an algebra of finite type over A f . Since the D( f ) form abasis for the topology of U , we thus conclude that if B is a quasi-coherent OX -algebra of finite type then, forevery open V of X , B|V is a quasi-coherent (OX |V )-algebra of finite type.

Proposition (9.6.3). — Let X be a locally Noetherian prescheme. Then every quasi-coherent OX -algebraB of nite typeis a coherent sheaf of rings (0, 5.3.7).

Proof. We can once again restrict to the case where X is an ane scheme given by a Noetherian ring A,and where B = eB, with B being an A-algebra of finite type; B is then a Noetherian ring. With this, it remains toprove that the kernel N of a B -homomorphism Bm→B is a B -module of finite type; but it is isomorphic (as I | 180a B -module) to eN , where N is the kernel of the corresponding homomorphism of B-modules Bm→ B (1.3.13).Since B is Noetherian, the submodule N of Bm is a B-module of finite type, so there exists a homomorphismBp → Bm with image N ; since the sequence Bp → Bm → B is exact, so too is the corresponding sequenceB p →Bm→B (1.3.5), and since N is the image of B p →Bm (1.3.9, i), this proves the proposition.

Corollary (9.6.4). — Under the hypotheses of (9.6.3), for a B -module F to be coherent, it is necessary and sucientthat it be a quasi-coherent OX -module and a B -module of nite type. If this is the case, and if G is a B -submodule or aquotient module ofF , then in order for G to be a coherentB -module, it is necessary and sucient that it be a quasi-coherentOX -module.

Proof. Taking (9.6.1) into account, the conditions on F are clearly necessary; we will show that they aresucient. We can restrict to the case where X is ane, given by some Noetherian ring A, whereB = eB, with B anA-algebra of finite type, where F = eM , with M a B-module, and where there exists a surjectiveB -homomorphismBm→F → 0. We then have the corresponding exact sequence Bm→ M → 0, so M is a B-module of finite type;further, the kernel P of the homomorphism Bm→ M is then a B-module of finite type, since B is Noetherian. Wethus conclude (1.3.13) that F is the cokernel of a B -homomorphism Bm→Bn, and is thus coherent, since Bis a coherent sheaf of rings (0, 5.3.4). The same reasoning shows that a quasi-coherent B -submodule (resp. aquotient B -module) of F is of finite type, from whence the second part of the corollary.

Proposition (9.6.5). — Let X be a quasi-compact scheme, or a prescheme whose underlying space is Noetherian. For allquasi-compact OX -algebras B of nite type, there exists a quasi-coherent OX -submodule E of B of nite type such that Egenerates (0, 4.1.3) the OX -algebra B .

10. FORMAL SCHEMES 148

Proof. In fact, by hypothesis, there exists a finite cover (Ui) of X consisting of ane opens such thatΓ(Ui ,B) = Bi is an algebra of finite type over Γ(Ui ,OX ) = Ai ; let Ei be a Ai -submodule of Bi of finite type thatgenerates the Ai -algebra Bi ; thanks to (9.4.7), there exists a OX -submodule Ei of B , quasi-coherent and of finitetype, such that Ei |Ui = eEi . It is clear that the sum E of the Ei is the desired object.

Proposition (9.6.6). — Let X be a prescheme whose underlying space is locally Noetherian, or a quasi-compact scheme.Then every quasi-coherent OX -algebra B is the inductive limit of its quasi-coherent OX -subalgebras of nite type.

Proof. In fact, it follows from (9.4.9) that B is the inductive limit (as an OX -module) of its quasi-coherentOX -submodules of finite type; the latter generating quasi-coherent OX -subalgebras of B of finite type (1.3.14), andso B is a fortiori their inductive limit.

§10. Formal schemes

10.1. Formal ane schemes.

(10.1.1). Let A be an admissible topological ring (0, 7.1.2); for each ideal of definition J of A, Spec(A/J) can beidentified with the closed subspace V (J) of Spec(A) (1.1.11), the set of open prime ideals of A; this topologicalspace does not depend on the ideal of definition J considered; we denote this topological space by X. Let (Jλ) I | 181be a fundamental system of neighborhoods of 0 in A, consisting of ideals of definition, and for each λ, let Oλ bethe structure sheaf of Spec(A/Jλ); this sheaf is induced on X by eA/ eJλ (which is zero outside of X). For Jµ ⊂ Jλ ,the canonical homomorphism A/Jµ → A/Jλ thus defines a homomorphism uλµ : Oµ → Oλ of sheaves of rings(1.6.1), and (Oλ) is a projective system of sheaves of rings for these homomorphisms. Since the topology of X admitsa basis consisting of quasi-compact open subsets, we can associate to each Oλ a pseudo-discrete sheaf of topologicalrings (0, 3.8.1) which have Oλ as the underlying sheaf of rings (without topologies), and that we denote also byOλ ; and the Oλ give again a projective system of sheaves of topological rings (0, 3.8.2). We denote by OX the sheaf oftopological rings on X, the projective limit of the system (Oλ); for each quasi-compact open subset U of X, Γ(U ,OX)is a topological ring, the projective limit of the system of discrete rings Γ(U ,Oλ) (0, 3.2.6).

Denition (10.1.2). — Given an admissible topological ring A, we define the formal spectrum of A, denotedby Spf(A), to be the closed subspace X of Spec(A) consisting of the open prime ideals of A. We say that atopologically ringed space is a formal ane scheme if it is isomorphic to a formal spectrum Spf(A) = X equippedwith a sheaf of topological rings OX which is the projective limit of sheaves of pseudo-discrete topological rings(eA/ eJλ)|X, where Jλ varies over the filtered set of ideals of definition of A.

When we speak of a formal spectrum X= Spf(A) as a formal ane scheme, it will always be as the topologicallyringed space (X,OX) where OX is defined as above.

We note that every ane scheme X = Spec(A) can be considered as a formal ane scheme in only one way,by considering A as a discrete topological ring: the topological rings Γ(U ,OX ) are then discrete whenever U isquasi-compact (but not, in general, when U is an arbitrary open subset of X ).

Proposition (10.1.3). — If X= Spf(A), where A is an admissible ring, then Γ(X,OX ) is topologically isomorphic to A.

Proof. Indeed, since X is closed in Spec(A), it is quasi-compact, and so Γ(X,OX) is topologically isomorphicto the projective limit of the discrete rings Γ(X,Oλ); but Γ(X,Oλ) is isomorphic to A/Jλ (1.3.7); since A isseparated and complete, it is topologically isomorphic to lim←−A/Jλ (0, 7.2.1), whence the proposition.

Proposition (10.1.4). — Let A be an admissible ring, X= Spf(A), and, for every f ∈ A, let D( f ) = D( f )∩X; thenthe topologically ringed space (D( f ),OX|D( f )) is isomorphic to the formal ane spectrum Spf(A f ) (0, 7.6.15).

Proof. For every ideal of definition J of A, the discrete ring S−1f A/S−1

f J is canonically identified with A f /J f (0, 7.6.9), so, by (1.2.5) and (1.2.6), the topological space Spf(A f ) is canonically identified with D( f ). Further,for every quasi-compact open subset U of X contained in D( f ), Γ(U ,Oλ) can be identified with the module ofsections of the structure sheaf of Spec(S−1

f A/S−1f Jλ) over U (1.3.6), so, setting Y= Spf(A f ), Γ(U ,OX) can be

identified with the module of sections Γ(U ,OY), which proves the proposition.

(10.1.5). As a sheaf of rings without topology, the structure sheaf OX of Spf(A) admits, for every x ∈ X, a fibrewhich, by (10.1.4), can be identified with the inductive limit lim−→A f for the f 6∈ jx . Then, by (0, 7.6.17) and(0, 7.6.18):

Proposition (10.1.6). — For every x ∈ X = Spf(A), the bre Ox is a local ring whose residue eld is isomorphic tok(x) = Ax/jxAx . If, further, A is adic and Noetherian, then Ox is a Noetherian ring.

Since k(x) is not reduced at 0, we conclude from this that the support of the ring of sheaves OX is equal to X.

10. FORMAL SCHEMES 149

10.2. Morphisms of formal ane schemes.

(10.2.1). Let A, B be two admissible rings, and let φ : B→ A be a continuous morphism. The continuous mapaφ : Spec(A)→ Spec(B) (1.2.1) then maps X= Spf(A) to Y= Spf(B), since the inverse image under φ of an openprime ideal of A is an open prime ideal of B. Conversely, for all g ∈ B, φ defines a continuous homomorphismΓ(D(g),OY) → Γ(D(φ (g)),OX) according to (10.1.4), (10.1.3), and (0, 7.6.7); since these homomorphismssatisfy the compatibility conditions for the restrictions corresponding to the change from g to a multiple ofg, and since D(φ (g)) = aφ−1(D(g)), they define a continuous homomorphism of sheaves of topological ringsOY → aφ∗(OX) (0, 3.2.5) that we denote by eφ ; we have thus defined a morphism Φ = (aφ , eφ ) of topologicallyringed spaces X→Y. We note that, as a homomorphism of sheaves without topology, eφ defines a homomorphismeφ ]x : Oaφ (x)→Ox on the stalks, for all x ∈ X.

Proposition (10.2.2). — Let A and B be admissible topological rings, and let X = Spf(A) and Y = Spf(B). Fora morphism u = (ψ,θ) : X→ Y of topologically ringed spaces to be of the form (aφ , eφ ), where φ is a continuous ringhomomorphism B→ A, it is necessary and sucient that θ]x be a local homomorphism Oφ (x)→Ox for all x ∈ X.

Proof. The condition is necessary: let p= jx ∈ Spf(A), and let q=φ−1(jx); if g 6∈ q, then we have φ (g) 6∈ p,and it is immediate that the homomorphism Bg → Aφ (g) induced by φ (0, 7.6.7) sends qg to a subset ofpφ (g); by passing to the inductive limit, we see (taking (10.1.5) and (0, 7.6.17) into account) that eφ ]x is a localhomomorphism.

Conversely, let (ψ,θ) be a morphism satisfying the condition of the proposition; by (10.1.3), θ defines acontinuous ring homomorphism

φ = Γ(θ) : B = Γ(Y,OY) −→ Γ(X,OX) = A.

By virtue of the hypothesis on θ, for the section φ (g) of OX over X to be an invertible germ at the point x , it isnecessary and sucient that g be an invertible germ at the point ψ(x). But, by (0, 7.6.17), the sections of OX(resp. OY) over X (resp. Y) that have a non-invertible germ at the point x (resp. ψ(x)) are exactly the elementsof jx (resp. jψ(x)); the above remark thus shows that aφ = ψ. Finally, for all g ∈ B the diagram I | 183

B = Γ(Y,OY)φ //

Γ(X,OX) = A

Bg = Γ(D(g),OY)

Γ(θD(g))// Γ(D(φ (g)),OX) = Aφ (g)

is commutative; by the universal property of completed rings of fractions (0, 7.6.6), θD(g) is equal to eφD(g) forall g ∈ B, and so (0, 3.2.5) we have θ = eφ .

We say that a morphism (ψ,θ) of topologically ringed spaces satisfying the condition of Proposition (10.2.2)is a morphism of formal ane schemes. We can say that the functors Spf(A) in A and Γ(X,OX) in X define anequivalence between the category of admissible rings and the opposite category of formal ane schemes (T, I,1.2).

(10.2.3). As a particular case of (10.2.2), note that, for f ∈ A, the canonical injection of the formal ane schemeinduced by X on D( f ) corresponds to the continuous canonical homomorphism A→ A f . Under the hypothesesof Proposition (10.2.2), let h be an element of B, and g an element of A that is a multiple of φ (h); we thenhave ψ(D(g)) ⊂D(h); the restriction of u to D(g), considered as a morphism from D(g) to D(h), is the uniquemorphism v making the diagram

D(g) v //

D(h)

X

u // Y

commutate.This morphism corresponds to the unique continuous homomorphism φ ′ : Bh→ Ag (0, 7.6.7) making the

diagram

A

Bφoo

Ag Bh

φ ′oo

10. FORMAL SCHEMES 150

commutate.

10.3. Ideals of denition for a formal ane scheme.

(10.3.1). Let A be an admissible ring, J an open ideal of A, and X the formal ane scheme Spf(A). Let (Jλ) bethe set of those ideals of definition for A that are contained in J; then eJ/eJλ is a sheaf of ideals of eA/eJλ . Denoteby J∆ the projective limit of the sheaves on X induced by eJ/eJλ , which is identified with a sheaf of ideals of OX(0, 3.2.6). For every f ∈ A, Γ(D( f ),J∆) is the projective limit of the S−1

f J/S−1f Jλ , or, in other words, can be

identified with the open ideal J f of the ring A f (0, 7.6.9), and, in particular, Γ(X,J∆) = J; we conclude (theD( f ) forming a basis for the topology of X) that

(10.3.1.1) J∆|D( f ) = (J f )∆.I | 184

(10.3.2). With the notation of (10.3.1), for all f ∈ A, the canonical map from A f = Γ(D( f ),OX) to Γ(D( f ), (eA/eJ)|X) =S−1

f A/S−1f J is surjective and has Γ(D( f ),J∆) = J f as its kernel (0, 7.6.9); these maps thus define a surjective

continuous homomorphism, said to be canonical, from the sheaf of topological rings OX to the sheaf of discreterings (eA/eJ)|X, whose kernel is J∆; this homomorphism is none other than eφ (10.2.1), where φ is the continuoushomomorphism A→ A/J; the morphism (aφ , eφ ) : Spec(A/J) → X of formal ane schemes (where aφ is theidentity homeomorphism from X to itself) is also said to be canonical. We thus have, according to the above, acanonical isomorphism

(10.3.2.1) OX/J∆ ' (eA/eJ)|X.

It is clear (since Γ(X,J∆) = J) that the map J→ J∆ is strictly increasing ; according to the above, for J ⊂ J′,the sheaf J′∆/J∆ is canonically isomorphic to eJ′/eJ= (J′/J)∼.

(10.3.3). The hypotheses and notation being the same as in (10.3.1), we say that a sheaf of ideals J of OXis a sheaf of ideals of denition for X (or an ideal sheaf of denition for X) if, for all x ∈ X, there exists an openneighborhood of x of the form D( f ), where f ∈ A, such that J |D( f ) is of the form H∆, where H is an ideal ofdefinition for A f .

Proposition (10.3.4). — For all f ∈ A, each sheaf of ideals of denition for X induces a sheaf of ideals of denitionfor D( f ).

Proof. This follows from (10.3.1.1).

Proposition (10.3.5). — If A is an admissible ring, then every sheaf of ideals of denition for X = Spf(A) is of theform J∆, where J is a uniquely determined ideal of denition for A.

Proof. Let J be a sheaf of ideals of definition of X; by hypothesis, and since X is quasi-compact, there arefinitely-many elements fi ∈ A such that the D( fi) cover X and such that J |D( fi) = H∆

i , where Hi is an ideal ofdefinition for A fi. For each i, there exists an open ideal Ki of A such that (Ki) fi = Hi (0, 7.6.9); let K be anideal of definition for A containing all the Ki . The canonical image of J /K∆ in the structure sheaf (A/K)∼ ofSpec(A/K) (10.3.2) is thus such that its restriction to D( fi) is equal to its restriction to (Ki/K)∼; we conclude thatthis canonical image is a quasi-coherent sheaf on Spec(A/K), and so is of the form (J/K)∼, where J is an ideal ofdefinition for A containing K (1.4.1), whence J = J∆ (10.3.2); in addition, since for each i there exists an integerni such that Hni

i ⊂ K fi, we will have, by setting n to be the largest of the ni , that (J /K∆)n = 0, and, as a result(10.3.2), that ((J/K)∼)n = 0, whence finally that (J/K)n = 0 (1.3.13), which proves that J is an ideal of definitionfor A (0, 7.1.4).

Proposition (10.3.6). — Let A be an adic ring, and J an ideal of denition for A such that J/J2 is an (A/J)-moduleof nite type. For any integer n> 0, we then have (J∆)n = (Jn)∆.

Proof. For all f ∈ A, we have (since Jn is an open ideal)

(Γ(D( f ),J∆))n = (J f )n = (Jn) f = Γ(D( f n), (Jn)∆)

by (10.3.1.1) and (0, 7.6.12). The result then follows from the fact that (J∆)n is associated to the presheaf I | 185U 7→ (Γ(U ,J∆))n (0, 4.1.6), since the D( f ) form a basis for the topology of X.

(10.3.7). We say that a family (Jλ) of sheaves of ideals of definition for X is a fundamental system of sheaves ofideals of denition if each sheaf of ideals of definition for X contains one of the Jλ ; since Jλ = J∆

λ , it is equivalentto say that the Jλ form a fundamental system of neighborhoods of 0 in A. Let ( fα) be a family of elements of A suchthat the D( fα) cover X. If (Jλ) is a filtered decreasing family of sheaves of ideals of OX such that, for each

10. FORMAL SCHEMES 151

α, the family (Jλ |D( fα)) is a fundamental system of sheaves of ideals of definition for D( fα), then (Jλ) is afundamental system of sheaves of ideals of definition for X. Indeed, for each sheaf of ideals of definition J forX, there is a finite cover of X by D( fi) such that, for each i, Jλ i

|D( fi) is a sheaf of ideals of definition for D( fi)that is contained in J |D( fi). If µ is an index such that Jµ ⊂ Jλ i

for all i, then it follows from (10.3.3) that Jµ isa sheaf of ideals of definition for X, evidently contained in J , whence our claim.

10.4. Formal preschemes and morphisms of formal preschemes.

(10.4.1). Given a topologically ringed space X, we say that an open U ⊂ X is a formal ane open (resp. a formaladic ane open, resp. a formal Noetherian ane open) if the topologically ringed space induced on U by X is aformal ane scheme (resp. a scheme whose ring is adic, resp. adic and Noetherian).

Denition (10.4.2). — A formal prescheme is a topologically ringed space X which admits a formal ane openneighborhood for each point. We say that the formal prescheme X is adic (resp. locally Noetherian) if eachpoint of X admits a formal adic (resp. Noetherian) open neighborhood. We say that X is Noetherian if it islocally Noetherian and its underlying space is quasi-compact (and hence Noetherian).

Proposition (10.4.3). — If X is a formal prescheme (resp. a locally Noetherian formal prescheme), then the formalane (resp. Noetherian ane) open sets form a basis for the topology of X.

Proof. This follows from Definition (10.4.2) and Proposition (10.1.4) by taking into account that, if A is anadic Noetherian ring, then so too is A f for all f ∈ A (0, 7.6.11).

Corollary (10.4.4). — If X is a formal prescheme (resp. a locally Noetherian formal prescheme, resp. a Noetherianformal prescheme), then the topologically ringed space induced on each open set of X is also a formal prescheme (resp. alocally Noetherian formal prescheme, resp. a Noetherian formal prescheme).

Denition (10.4.5). — Given two formal preschemes X and Y, a morphism (of formal preschemes) from X

to Y is a morphism (ψ,θ) of topologically ringed spaces such that, for all x ∈ X, θ]x is a local homomorphismOψ(x)→Ox .

It is immediate that the composition of any two morphisms of formal preschemes is again a morphismof formal preschemes; the formal preschemes thus form a category, and we denote by Hom(X,Y) the set ofmorphisms from a formal prescheme X to a formal prescheme Y. I | 186

If U is an open subset of X, then the canonical injection into X of the formal prescheme induced on U by Xis a morphism of formal preschemes (and in fact a monomorphism of topologically ringed spaces (0, 4.1.1)).

Proposition (10.4.6). — Let X be a formal prescheme, andS= Spf(A) a formal ane scheme. There exists a canonicalbijective equivalence between the morphisms from the formal prescheme X to the formal prescheme S and the continuoushomomorphisms from the ring A to the topological ring Γ(X,OX).

Proof. The proof is similar to that of (2.2.4), by replacing “homomorphism” with “continuous homomor-phism”, “ane open” with “formal ane open”, and by using Proposition (10.2.2) instead of Theorem (1.7.3);we leave the details to the reader.

(10.4.7). Given a formal prescheme S, we say that the data of a formal prescheme X and a morphism φ : X→Sdefines a formal prescheme X over S or an formal S-prescheme, φ being called the structure morphism of theS-prescheme X. If S= Spf(A), where A is an admissible ring, then we also say that the formal S-prescheme X isa formal A-prescheme or a formal prescheme over A. An arbitrary formal prescheme can be considered as a formalprescheme over Z (equipped with the discrete topology).

If X and Y are formal S-preschemes, then we say that a morphism u : X→Y is a S-morphism if the diagram

Xu //

Y

S

(where the downwards arrows are the structure morphisms) is commutative. With this definition, the formalS-preschemes (for some fixed S) form a category. We denote by HomS(X,Y) the set of S-morphisms from aformal S-prescheme X to a formal S-prescheme Y. When S= Spf(A), we sometimes say A-morphism instead ofS-morphism.

(10.4.8). Since every ane scheme can be considered as a formal ane scheme (10.1.2), every (usual) preschemecan be considered as a formal prescheme. In addition, it follows from Definition (10.4.5) that, for the usualpreschemes, the morphisms (resp. S-morphisms) of formal preschemes coincide with the morphisms (resp.S-morphisms) defined in §2.

10. FORMAL SCHEMES 152

10.5. Sheaves of ideals of denition for formal preschemes.

(10.5.1). Let X be a formal prescheme; we say that an OX-ideal J is a sheaf of ideals of denition (or an ideal sheafof denition) for X if every x ∈ X has a formal ane open neighborhood U such that J |U is a sheaf of ideals ofdefinition for the formal ane scheme induced on U by X (10.3.3); by (10.3.1.1) and Proposition (10.4.3), foreach open V ⊂ X, J |V is then a sheaf of ideals of definition for the formal prescheme induced on V by X.

We say that a family (Jλ) of sheaves of ideals of definition for X is a fundamental system of sheaves of ideals of I | 187denition if there exists a cover (Uα) of X by formal ane open sets such that, for each α, the family of the Jλ |Uαis a fundamental system of sheaves of ideals of definition (10.3.6) for the formal ane scheme induced on Uα byX. It follows from the last remark of (10.3.7) that, when X is a formal ane scheme, this definition coincideswith the definition given in (10.3.7). For an open subset V of X, the restrictions Jλ |V then form a fundamentalsystem of sheaves of ideals of definition for the formal prescheme induced on V , according to (10.3.1.1). IfX is a locally Noetherian formal prescheme, and J is a sheaf of ideals of definition for X, then it follows fromProposition (10.3.6) that the powers J n form a fundamental system of sheaves of ideals of definition for X.

(10.5.2). Let X be a formal prescheme, and J a sheaf of ideals of definition for X. Then the ringed space(X,OX/J ) is a (usual) prescheme, which is ane (resp. locally Noetherian, resp. Noetherian) when X is a formalane scheme (resp. a locally Noetherian formal scheme, resp. a Noetherian formal scheme); we can reduce tothe ane case, and then the proposition has already been proved in (10.3.2). In addition, if θ : OX→ OX/Jis the canonical homomorphism, then u = (1X,θ) is a morphism (said to be canonical) of formal preschemes(X,OX/J )→ (X,OX), because, again, this was proved in the ane case (10.3.2), to which it is immediatelyreduced.

Proposition (10.5.3). — Let X be a formal prescheme, and (Jλ) a fundamental system of sheaves of ideals of denitionfor X. Then the sheaf of topological rings OX is the projective limit of the pseudo-discrete sheaves of rings (0, 3.8.1) OX/Jλ .

Proof. Since the topology of X admits a basis of formal quasi-compact ane open sets (10.4.3), we reduceto the ane case, where the proposition is a consequence of Proposition (10.3.5), (10.3.2), and Definition(10.1.1).

It is not true that any formal prescheme admits a sheaf of ideals of definition. However:

Proposition (10.5.4). — Let X be a locally Noetherian formal prescheme. There exists a largest sheaf of ideals ofdenition T for X; this is the unique sheaf of ideals of denition J such that the prescheme (X,OX/J ) is reduced. If Jis a sheaf of ideals of denition for X, then T is the inverse image under OX→OX/J of the nilradical of OX/J .

Proof. Suppose first that X= Spf(A), where A is an adic Noetherian ring. The existence and the propertiesof T follow immediately from Propositions (10.3.5) and (5.1.1), taking into account the existence and the ErrIIproperties of the largest ideal of definition for A ((0, 7.1.6) and (0, 7.1.7)).

To prove the existence and the properties of T in the general case, it suces to show that, if U ⊃ V aretwo Noetherian formal ane open subsets of X , then the largest sheaf of ideals of definition TU for U inducesthe largest sheaf of ideals of definition TV for V ; but as (V, (OX|V )/(TU |V )) is reduced, this follows from theabove.

We denote by Xred the (usual) reduced prescheme (X,OX/T ).

Corollary (10.5.5). — Let X be a locally Noetherian formal prescheme, and T the largest sheaf of ideals of denitionfor X; for each open subset V of X, T |V is the largest sheaf of ideals of denition for the formal prescheme induced on V byX.

Proposition (10.5.6). — Let X and Y be formal preschemes, J (resp. K ) be a sheaf of ideals of denition for X (resp.Y), and f : X→Y a morphism of formal preschemes.

(i) If f ∗(K )OX ⊂ J , then there exists a unique morphism f ′ : (X,OX/J )→ (Y,OY/K ) of usual preschemesmaking the diagram

(10.5.6.1) (X,OX)f // (Y,OY)

(X,OX/J )f ′ //

OO

(Y,OY/K )

OO

commutate, where the vertical arrows are the canonical morphisms.

10. FORMAL SCHEMES 153

(ii) Suppose that X= Spf(A) and Y= Spf(B) are formal ane schemes, J = J∆ and K = K∆, where J (resp. K)is an ideal of denition for A (resp. B), and f = (aφ , eφ ), where φ : B→ A is a continuous homomorphism; forf ∗(K )OX ⊂ J to hold, it is necessary and sucient that φ (K) ⊂ J, and, in this case, f ′ is then the morphism(aφ ′, eφ ′), where φ ′ : B/K→ A/J is the homomorphism induced from φ by passing to quotients.

Proof.(i) If f = (ψ,θ), then the hypotheses imply that the image under θ] : ψ∗(OY)→OX of the sheaf of ideals

ψ∗(K ) of ψ∗(OY) is contained in J (0, 4.3.5). By passing to quotients, we thus obtain from θ] ahomomorphism of sheaves of rings

ω : ψ∗(OY/K ) = ψ∗(OY)/ψ∗(K ) −→ OX/J ;

furthermore, since, for all x ∈ X, θ]x is a local homomorphism, so too is ωx . The morphism of ringedspaces (ψ,ω[) is thus (2.2.1) the unique morphism f ′ of ringed spaces whose existence was claimed.

(ii) The canonical functorial correspondence between morphisms of formal ane schemes and continuoushomomorphisms of rings (10.2.2) shows that, in the case considered, the relation f ∗(K )OX ⊂ J impliesthat we have f ′ = (aφ ′, eφ ′), where φ ′ : B/K→ A/J is the unique homomorphism making the diagram

(10.5.6.2) Bφ //

A

B/K

φ ′ // A/J

commutate. The existence of φ ′ thus implies that φ (K) ⊂ J. Conversely, if this condition is satisfied,then, denoting by φ ′ the unique homomorphism making the diagram (10.5.6.2) commutate and settingf ′ = (aφ ′, eφ ′), it is clear that the diagram (10.5.6.1) is commutative; considering the homomorphismsaφ ∗(OY)→ OX and aφ ′∗(OY/K )→ OX/J corresponding to f and f ′ respectively then leads to thefact that this implies the relation f ∗(K )OX ⊂ J .

It is clear that the correspondence f 7→ f ′ defined above is functorial.

10.6. Formal preschemes as inductive limits of preschemes.

(10.6.1). Let X be a formal prescheme, and (Jλ) a fundamental system of sheaves of ideals of definition for X;for each λ, let fλ be the canonical morphism (X,OX/Jλ)→ X (10.5.2); for Jµ ⊂ Jλ , the canonical morphismOX/Jµ→OX/Jλ defines a canonical morphism fµλ : (X,OX/Jλ)→ (X,OX/Jµ) of (usual) preschemes such that I | 189fλ = fµ fµλ . The preschemes Xλ = (X,OX/Jλ) and the morphisms fµλ thus form (by (10.4.8)) an inductive systemin the category of formal preschemes.

Proposition (10.6.2). — With the notation of (10.6.1), the formal prescheme X and the morphisms fλ form an inductivelimit (T, I, 1.8) of the system (Xλ , fµλ) in the category of formal preschemes.

Proof. Let Y be a formal prescheme, and, for each index λ, letgλ = (ψλ ,θλ) : Xλ −→Y

be a morphism such that gλ = gµ fµλ for Jµ ⊂ Jλ . This latter condition and the definition of the Xλ imply firstof all that the ψλ are identical to a single continuous map ψ : X→Y of the underlying spaces; in addition, thehomomorphisms θ]λ : ψ∗(OY)→OX i

= OX/Jλ form a projective system of homomorphisms of sheaves of rings. Bypassing to the projective limit, there is an induced homomorphism ω : ψ∗(OY)→ lim←−OX/Jλ = OX, and it is clear

that the morphism g = (ψ,ω[) of ringed spaces is the unique morphism making the diagrams

(10.6.2.1) Xλgλ //

fλ

Y

X

g

??

commutative. It remains to prove that g is a morphism of formal preschemes; the question is local on X and Y, sowe can assume that X= Spf(A) and Y= Spf(B), with A and B admissible rings, and with Jλ = J∆

λ , where (Jλ) isa fundamental system of ideals of definition for A (10.3.5); since A= lim←−A/Jλ , the existence of a morphism g offormal ane schemes making the diagrams (10.6.2.1) commutate then follows from the bijective correspondence

10. FORMAL SCHEMES 154

(10.2.2) between morphisms of formal ane schemes and continuous ring homomorphisms, and from thedefinition of the projective limit. But the uniqueness of g as a morphism of ringed spaces shows that it coincideswith the morphism in the beginning of the proof.

The following proposition establishes, under certain additional conditions, the existence of the inductivelimit of a given inductive system of (usual) preschemes in the category of formal preschemes:

Proposition (10.6.3). — Let X be a topological space, and (Oi , u ji) a projective system of sheaves of rings on X, with Nfor its set of indices. Let Ji be the kernel of u0i : Oi →O0. Suppose that:

(a) the ringed space (X,Oi) is a prescheme X i ;(b) for all x ∈ X and all i, there exists an open neighborhood Ui of x in X such that the restriction Ji |Ui is nilpotent; ErrII

and(c) the homomorphisms u ji are surjective. I | 190Let OX be the sheaf of topological rings given by the projective limit of the pseudo-discrete sheaves of rings Oi , and let

ui : OX → Oi be the canonical homomorphism. Then the topologically ringed space (X,OX) is a formal prescheme; thehomomorphisms ui are surjective; their kernels J (i) form a fundamental system of sheaves of ideals of denition for X, andJ (0) is the projective limit of the sheaves of ideals Ji .

Proof. We first note that, on each stalk, u ji is a surjective homomorphism and a fortiori a local homomor-phism; thus vi j = (1X, u ji) is a morphism of preschemes X j → X i (i ¾ j) (2.2.1). Suppose first that each X i isan ane scheme of ring Ai . There exists a ring homomorphism φ ji : Ai → A j such that u ji = fφ ji (1.7.3); as aresult (1.6.3), the sheaf O j is a quasi-coherent Oi -module over X i (for the external law defined by u ji), associatedto A j considered as an Ai -module by means of φ ji . For all f ∈ Ai , let f ′ =φ ji( f ); by hypothesis, the open setsD( f ) and D( f ′) are identical in X, and the homomorphism from Γ(D( f ),Oi) = (Ai) f to Γ(D( f ),O j) = (A j) f ′corresponding to u ji is exactly (φ ji) f (1.6.1). But when we consider A j as an Ai -module, (A j) f ′ is the (Ai) f -module(A j) f , so we also have u ji = fφ ji , where φ ji is now considered as a homomorphism of Ai -modules. Then, since u jiis surjective, we conclude that φ ji is also surjective (1.3.9), and if J ji is the kernel of φ ji , then the kernel of u ji isa quasi-coherent Oi -module equal to fJ ji . In particular, we have Ji = eJi , where Ji is the kernel of φ0i : Ai → A0.Hypothesis (b) implies that Ji is nilpotent: indeed, since X is quasi-compact, we can cover X by a finite numberof open sets Uk such that (Ji |Uk)nk = 0, and, by setting n to be the largest of the nk, we have J n

i = 0. Wethus conclude that Ji is nilpotent (1.3.13). Then the ring A = lim←−Ai is admissible (0, 7.2.2), the canonical

homomorphism φi : A→ Ai is surjective, and its kernel J(i) is equal to the projective limit of the Jik for k ¾ i; theJ(i) form a fundamental system of neighborhoods of 0 in A. The claims of Proposition (10.6.3) follow in thiscase from (10.1.1) and (10.3.2), with (X,OX) being Spf(A).

Again, in this particular case, we note that, if f = ( fi) is an element of the projective limit A= lim←−Ai , thenall the open sets D( fi) (which are ane open sets in X i) can be identified with the open subset D( f ) of X, andthe prescheme induced on D( f ) by X i thus being identified with the ane scheme Spec((Ai) fi

).In the general case, we remark first that, for every quasi-compact open subset U of X, each of the Ji |U

is nilpotent, as shown by the above reasoning. We will show that, for every x ∈ X, there exists an openneighborhood U of x in X which is an ane open set for all the X i . Indeed, we take U to be an ane open set forX0, and observe that OX0

= OX i/Ji . Since, by the above, Ji |U is nilpotent, U is also an ane open set for each

X i , by Proposition (5.1.9). This being so, for each U satisfying the preceding conditions, the study of the anecase as above shows that (U ,OX |U) is a formal prescheme whose J (i)|U form a fundamental system of sheavesof ideals of definition, and J (0)|U is the projective limit of the Ji |U ; whence the conclusion.

Corollary (10.6.4). — Suppose that, for i ¾ j, the kernel of u ji is Jj+1

i and that J1/J 21 is of nite type over I | 191

O0 = O1/J1. Then X is an adic formal prescheme, and if J (n) is the kernel of OX → On, then J (n) = J n+1 andJ /J 2 is isomorphic to J1. If, in addition, X0 is locally Noetherian (resp. Noetherian), then X is locally Noetherian (resp.Noetherian).

Proof. Since the underlying spaces of X and X0 are the same, the question is local, and we can supposethat all the X i are ane; taking into account the fact that Ji j = fJ ji (with the notation of Proposition (10.3.6)),we can immediately reduce the problem to the corresponding claims of Proposition (0, 7.2.7) and Corollary(0, 7.2.8), by noting that J1/J

21 is then an A0-module of finite type (1.3.9).

In particular, every locally Noetherian formal prescheme X is the inductive limit of a sequence (Xn) of locallyNoetherian (usual) preschemes satisfying the conditions of Proposition (10.3.6) and Corollary (10.6.4): itsuces to consider a sheaf of ideals of definition J for X (10.5.4) and to take Xn = (X,OX/J n+1) ((10.5.1) andProposition (10.6.2)).

10. FORMAL SCHEMES 155

Corollary (10.6.5). — Let A be an admissible ring. For the formal ane scheme X = Spf(A) to be Noetherian, it isnecessary and sucient for A to be adic and Noetherian.

Proof. The condition is evidently sucient. Conversely, suppose that X is Noetherian, and let J be an idealof definition for A, and J = J∆ the corresponding sheaf of ideals of definition for X. The (usual) preschemesXn = (X,OX/J n+1) are then ane and Noetherian, so the rings An = A/Jn+1 are Noetherian (6.1.3), whence weconclude that J/J2 is an A/J-module of finite type. Since the J n form a fundamental system of sheaves of idealsof definition for X (10.5.1), we have OX = lim←−OX/J

n (10.5.3); we thus conclude (10.1.3) that A is topologicallyisomorphic to lim←−A/Jn, which is adic and Noetherian (0, 7.2.8).

Remark (10.6.6). — With the notation of Proposition (10.6.3), let Fi be an Oi -module, and suppose we aregiven, for i ¾ i, a vi j -morphism θ ji : Fi → F j such that θk j θ ji = θki for k ¶ j ¶ i. Since the underlyingcontinuous map of vi j is the identity, θ ji is a homomorphism of sheaves of abelian groups to the space X; inaddition, if F is the projective limit of the projective system (Fi) of sheaves of abelian groups, then the factthat the θ ji are vi j -morphisms lets us define an OX-module structure on F by passing to the projective limit;when equipped with this structure, we say that F is the projective limit (with respect to the θ ji) of the system ofOi -modules (Fi). In the particular case where v∗i j(Fi) =F j and θ ji is the identity, we say that F is the projectivelimit of a system (Fi) such that v∗i j(Fi) =F j for j ¶ i (without mentioning the θ ji).

(10.6.7). Let X and Y be formal preschemes, J (resp. K ) a sheaf of ideals of definition for X (resp. Y), andf : X → Y a morphism such that f ∗(K )OX ⊂ J . We then have, for every integer n > 0, that f ∗(K n)OX =( f ∗(K )OX)n ⊂ J n; f thus induces (10.5.6) a morphism of (usual) preschemes fn : Xn → Yn by setting Xn =(X,OX/J n+1) and Yn = (Y ,OY/K n+1), and it immediately follows from the definitions that the diagrams

(10.6.7.1) Xmfm //

Ym

Xn

fn // Yn

commute for m¶ n; in other words, ( fn) is an inductive system of morphisms. I | 192

(10.6.8). Conversely, let (Xn) (resp. (Yn)) be an inductive system of (usual) preschemes satisfying conditions (b)and (c) of Proposition (10.6.3), and let X (resp. Y) be its inductive limit. By definition of the inductive limit,each sequence ( fn) of morphisms Xn→ Yn forms an inductive system that admits an inductive limit f : X→Y,which is the unique morphism of formal preschemes that makes the diagrams

Xnfn //

Yn

X

f // Y

commutate.

Proposition (10.6.9). — Let X and Y be locally Noetherian formal preschemes, and J (resp. K ) be a sheaf of idealsof denition for X (resp. Y); the map f 7→ ( fn) dened in (10.6.7) is a bijection from the set of morphisms f : X→Ysuch that f ∗(K )OX ⊂ J to the set of sequences ( fn) of morphisms that make the diagrams (10.6.7.1) commutate.

Proof. If f is the inductive limit of this sequence, then it remains to show that f ∗(K )OX ⊂ J . The statement,being local on X and Y, can be reduced to the case where X= Spf(A) and Y= Spf(B) are ane, with A and Badic Noetherian rings, and with J = J∆ and K = K∆, where J (resp. K) is an ideal of definition for A (resp. B).We then have that Xn = Spec(An) and Yn = Spec(Bn), with An = A/Jn+1 and Bn = B/Kn+1, by Proposition (10.3.6)and (10.3.2); fn = (aφn,fφn), where the homomorphisms φn : Bn→ An forms a projective system, thus f = (aφ , eφ ),and so f = (aφ , eφ ), where φ = lim←−φn. The commutativity of the diagram (10.6.7.1) for m = 0 then gives the

condition φn(K/Kn+1) ⊂ J/Jn+1 for all n, so, by passing to the projective limit, we have φ (K) ⊂ J, which impliesthat f ∗(K )OX ⊂ J (10.5.6, ii).

Corollary (10.6.10). — Let X and Y be locally Noetherian formal preschemes, and T the largest sheaf of ideals ofdenition for X (10.5.4).

(i) For every sheaf of ideals of denition K for Y and every morphism f : X→Y, we have f ∗(K )OX ⊂ T .(ii) There is a canonical bijective correspondence between Hom(X,Y) and the set of sequences ( fn) of morphisms

making the diagrams (10.6.7.1) commute, where Xn = (X,OX/T n+1) and Yn = (Y,OY/K n+1).

10. FORMAL SCHEMES 156

Proof. (ii) follows immediately from (i) and Proposition (10.6.9). To prove (i), we can reduce to the casewhere X = Spf(A) and Y = Spf(B), with A and B Noetherian, and with T = T∆ and K = K∆, where T is thelargest ideal of definition for A and K is an ideal of definition for B. Let f = (aφ , eφ ), where φ : B → A is acontinuous homomorphism; since the elements of K are topologically nilpotent (0, 7.1.4, ii), so too are those ofφ (K), and so φ (K) ⊂ T, since T is the set of topologically nilpotent elements of A (0, 7.1.6); hence, by Proposition(10.5.6, ii), we are done.

Corollary (10.6.11). — Let S, X, Y be locally Noetherian formal preschemes, and f : X → S and g : Y → Sthe morphisms that make X and Y formal S-preschemes. Let J (resp. K , L ) be a sheaf of ideals of denition for S(resp. X, Y), and suppose that f ∗(J )OX ⊂K and g∗(J )OY =L ; set Sn = (S,OS/J n+1), Xn = (X,OX/K n+1), andYn = (Y,OY/L n+1). Then there exists a canonical bijective correspondence between HomS(X,Y) and the set of sequences I | 193(un) of Sn-morphisms un : Xn→ Yn making the diagrams (10.6.7.1) commute.

Proof. For each S-morphism u : X→Y, we have by definition that f = g u, whence

u∗(L )OX = u∗(g∗(J OY)OX = f ∗(J )OX ⊂K ,

and the corollary then follows from Proposition (10.6.9).

We note that, for m¶ n, the data of a morphism fn : Xn→ Yn determines a unique morphism fm : Xm→ Ymmaking the diagram (10.6.7.1) commutate, since we immediately see that we can reduce to the ane case; wehave thus defined a map φmn : HomSn

(Xn, Yn)→ HomSm(Xm, Ym), and the HomSn

(Xn, Yn) form, with the φmn, aprojective system of sets; Corollary (10.6.11) then says that there is a canonical bijection

HomS(X,Y)' lim←−n

HomSn(Xn, Yn).

10.7. Products of formal preschemes.

(10.7.1). Let S be a formal prescheme; formal S-preschemes form a category, and we can define a notion of aproduct of formal S-preschemes.

Proposition (10.7.2). — Let X = Spf(B) and Y = Spf(C) be formal ane schemes over a formal ane schemeS = Spf(A). Let Z = Spf(Bb⊗AC), and let p1 and p2 be the S-morphisms corresponding (10.2.2) to the canonical(continuous) A-homomorphisms ρ : B → Bb⊗AC and σ : C → Bb⊗AC ; then (Z, p1, p2) is a product of the formal aneS-schemes X and Y.

Proof. By Proposition (10.4.6), it suces to check that, if we associate, to each continuous A-homomorphismφ : Bb⊗AC → D (where D is an admissible ring which is a topological A-algebra), the pair (φ ρ,φ σ), then thisdefines a bijection

HomA(Bb⊗AC , D)' HomA(B, D)×HomA(C , D),which is exactly the universal property of the completed tensor product (0, 7.7.6).

Proposition (10.7.3). — Given formal S-preschemes X and Y, their product X×S Y exists.

Proof. The proof is similar to that of Theorem (3.2.6), replacing ane schemes (resp. ane open sets)by formal ane schemes (resp. formal ane open sets), and replacing Proposition (3.2.2) by Proposition(10.7.2).

All the formal properties of the product of preschemes ((3.2.7) and (3.2.8), (3.3.1) and (3.3.12)) hold truewithout modification for the product of formal preschemes.

(10.7.4). Let S, X, and Y be formal preschemes, and let f : X→S and g : Y→S be morphisms. Suppose thatthere exist, in S, X, and Y respectively, three fundamental systems of sheaves of ideals of definitions (Jλ), (Kλ),and (Lλ), all having the same set I of indices, and such that f ∗(Jλ)OX ⊂Kλ and g∗(Jλ)OY ⊂Lλ for all λ. SetSλ = (S,OS/Jλ), Xλ = (X,OX/Kλ), and Yλ = (Y,OY/Lλ); for Jµ ⊂ Jλ , Kµ ⊂ Kλ , and Lµ ⊂ Lλ , note that Sλ(resp. Xλ , Yλ) is a closed subprescheme of Sµ (resp. Xµ, Yµ) that has the same underlying space (10.6.1). Since I | 194Sλ → Sµ is a monomorphism of preschemes, we see that the products Xλ×Sλ Yλ and Xλ×Sµ Yλ are identical (3.2.4),since Xλ ×Sµ Yλ can be identified with a closed subprescheme of Xµ ×Sµ Yµ that has the same underlying space(4.3.1). With this in mind, the product X×SY is the inductive limit of the usual preschemes Xλ ×Sλ Yλ : indeed, aswe see in Proposition (10.6.2), we can reduce to the case where S, X, and Y are formal ane schemes. Takinginto account both Proposition (10.5.6, ii) and the hypotheses on the fundamental systems of sheaves of idealsof definition for S, X, and Y, we immediately see that our claim follows from the definition of the completedtensor product of two algebras (0, 7.7.1).

10. FORMAL SCHEMES 157

Furthermore, let Z be a formal S-prescheme, (Mλ) a fundamental system of ideals of definition for Zhaving I for its set of indices, and let u : Z → X and v : Z → Y be S-morphisms such that u∗(Kλ)OZ ⊂ Mλand v∗(Lλ)OZ ⊂ Mλ for all λ. If we set Zλ = (Z,OZ/Mλ), and if uλ : Zλ → Xλ and vλ : Zλ → Yλ are theSλ -morphisms corresponding to u and v (10.5.6), then we immediately have that (u, v)S is the inductive limit ofthe Sλ -morphisms (uλ , vλ)Sλ .

The ideas of this section apply, in particular, to the case where S, X, and Y are locally Noetherian, takingthe systems consisting of the powers of a sheaf of ideals of definition (10.5.1) as the fundamental systems ofsheaves of ideals of definition . However, we note that X×SY is not necessarily locally Noetherian (see however(10.13.5)).

10.8. Formal completion of a prescheme along a closed subset.

(10.8.1). Let X be a locally Noetherian (usual) prescheme, and X ′ a closed subset of the underlying space of X ;we denote by Φ the set of coherent sheaves of ideals J of OX such that the support of OX/J is X ′. The set Φ isnonempty ((5.2.1), (4.1.4), (6.1.1)); we order it by the relation ⊃.

Lemma (10.8.2). — The ordered set Φ is ltered; if X is Noetherian, then, for all J0 ∈ Φ, the set of powers J n0 (n> 0)

is conal in Φ.

Proof. If J1 and J2 are in Φ, and if we set J = J1 ∩J2, then J is coherent since OX is coherent ((6.1.1)and (0, 5.3.4)), and we have Jx = (J1)x ∩ (J2)x for all x ∈ X , whence Jx = Ox for x 6∈ X ′, and Jx 6= Ox forx ∈ X ′, which proves that J ∈ Φ. On the other hand, if X is Noetherian and if J0 and J are in Φ, then thereexists an integer n> 0 such that J n

0 (OX/J ) = 0 (9.3.4), which implies that J n0 ⊂ J .

(10.8.3). Now let F be a coherent OX -module; for all J ∈ Φ, we have that F ⊗OX(OX/J ) is a coherent OX -module

(9.1.1) with support contained in X ′, and we will usually identify it with its restriction to X ′. When J variesover Φ, these sheaves form a projective system of sheaves of abelian groups.

Denition (10.8.4). — Given a closed subset X ′ of a locally Noetherian prescheme X and a coherent OX -moduleF , we define the completion of F along X ′, denoted by F/X ′ (or ÒF when there is little chance of confusion), to bethe restriction to X ′ of the sheaf lim←−Φ

(F ⊗OX(OX/J )); we say that its sections over X ′ are the formal sections of I | 195

F along X ′.

It is immediate that, for every open U ⊂ X , we have (F|U)/(U∩X ′) = (F/X ′)|(U ∩ X ′).By passing to the projective limit, it is clear that (OX )/X ′ is a sheaf of rings, and that F/X ′ can be considered

as an (OX )/X ′ -module. In addition, since there exists a basis for the topology of X ′ consisting of quasi-compactopen sets, we can consider (OX )/X ′ (resp. F/X ′) as a sheaf of topological rings (resp. of topological groups), theprojective limit of the pseudo-discrete sheaves of rings (resp. groups) OX/F (resp. F ⊗OX

(OX/F ) =F/JF ), and,by passing to the projective limit, F/X ′ then becomes a topological (OX )/X ′ -module ((0, 3.8.1) and (0, 8.2)); recallthat, for every quasi-compact open U ⊂ X , Γ(U ∩ X ′, (OX )/X ′) (resp. Γ(U ∩ X ′,F/X ′)) is then the projective limit ofthe discrete rings (resp. groups) Γ(U ,OX/J ) (resp. Γ(U ,F/JF )).

Now, if u :F →G is a homomorphism of OX -modules, then there are canonically induced homomorphismsuJ : F ⊗OX

(OX/J )→ G ⊗OX(OX/J ) for all J ∈ Φ, and these homomorphisms form a projective system. By

passing to the projective limit and restricting to X ′, these give a continuous (OX )/X ′ -homomorphism F/X ′ →G/X ′ ,denoted u/X ′ or bu, and called the completion of the homomorphism u along X ′. It is clear that, if v : G →H is asecond homomorphism of OX -modules, then we have (v u)/X ′ = (v/X ′) (u/X ′), hence F/X ′ is a covariant additivefunctor in F from the category of coherent OX -modules to the category of topological (OX )/X ′ -modules.

Proposition (10.8.5). — The support of (OX )/X ′ is X ′; the topologically ringed space (X ′, (OX )/X ′) is a locally Noetherianformal prescheme, and, if J ∈ Φ, then J/X ′ is a sheaf of ideals of denition for this formal prescheme. If X = Spec(A) isan ane scheme with Noetherian ring A, J = eJ for some ideal J of A, and X ′ = V (J), then (X ′, (OX )/X ′) is canonicallyidentied with Spf(bA), where bA is the separated completion of A with respect to the J-preadic topology.

Proof. We can evidently reduce to proving the latter claim. We know (0, 7.3.3) that the separated completionbJ of J with respect to the J-preadic topology can be identified with the ideal JbA of bA, where bA is the NoetherianbJ-adic ring such that bA/bJn = A/Jn (0, 7.2.6). This latter equation shows that the open prime ideals of bA are theideals bp = pbJ, where p is a prime ideal of A containing J, and that we have bp∩ A= p, and hence Spf(bA) = X ′.Since OX/J n = (A/Jn)∼, the proposition follows immediately from the definitions.

We say that the formal prescheme defined above is the completion of X along X ′, and we denote it by X/X ′ orbX when there is little chance of confusion. When we take X ′ = X , we can set J = 0, and we thus have X/X = X .

10. FORMAL SCHEMES 158

It is clear that, if U is a subprescheme induced on an open subset of X , then U/(U∩X ′) is canonically identifiedwith the formal subprescheme induced on X/X ′ by the open subset U ∩ X ′ of X ′.

Corollary (10.8.6). — The (usual) prescheme bXred is the unique reduced subprescheme of X having X ′ as its underlyingspace (5.2.1). For bX to be Noetherian, it is necessary and sucient for bXred tob e Noetherian, and it suces that X beNoetherian.

Proof. Since bXred is determined locally (10.5.4), we can assume that X is an ane scheme of some Noetherianring A; with the notation of Proposition (10.8.5), the ideal T of topologically nilpotent elements of bA is the inverseimage under the canonical map bA→ bA/bJ= A/J of the nilradical of A/J (0, 7.1.3), so bA/T is isomorphic to thequotient of A/J by its nilradical. The first claim then follows from Propositions (10.5.4) and (5.1.1). If bXred isNoetherian, then so too is its underlying space X ′, and so the X ′n = Spec(OX/J n) are Noetherian (6.1.2), andthus so too is bX (10.6.4); the converse is immediate, by Proposition (6.1.2).

(10.8.7). The canonical homomorphisms OX →OX/J (for J ∈ Φ) form a projective system, and give, by passingto the projective limit, a homomorphism of sheaves of rings θ : OX → ψ∗((OX )/X ′) = lim←−Φ

(OX/J ), where ψdenotes the canonical injection X ′→ X of the underlying spaces. We denote by i (or iX ) the morphism (said tobe canonical)

(ψ,θ) : X/X ′ −→ Xof ringed spaces.

By taking tensor products, for every coherent OX -module F , the canonical homomorphisms OX →OX/Jgive homomorphisms F → F ⊗OX

(OX/J ) of OX -modules which form a projective system, and thus give, bypassing to the projective limit, a canonical functorial homomorphism γ :F → ψ∗(F/X ′) of OX -modules.

Proposition (10.8.8). —(i) The functor F/X ′ (in F ) is exact.(ii) The functorial homomorphism γ ] : i∗(F )→F/X ′ of (OX )/X ′ -modules is an isomorphism.

Proof.(i) It suces to prove that, if 0→F ′→F →F ′′→ 0 is an exact sequence of coherent OX -modules, and

if U is an ane open subset of X with Noetherian ring A, then the sequence

0 −→ Γ(U ∩ X ′,F ′/X ′) −→ Γ(U ∩ X ′,F/X ′) −→ Γ(U ∩ X ′,F ′′/X ′) −→ 0

is exact. We have that F|U = eM , F ′|U =ÝM ′, and F ′′|U =gM ′′, where M , M ′, and M ′′ are threeA-modules of finite type such that the sequence 0→ M ′→ M → M ′′→ 0 is exact ((1.5.1) and (1.3.11));let J ∈ Φ, and let J be an ideal of A such that J |U = eJ. We then have

Γ(U ∩ X ′,F ⊗OXOX/J n) = M ⊗A (A/J

n)

(3.12); so, by definition of the projective limit, we have

Γ(U ∩ X ′,F/X ′) = lim←−n

(M ⊗A (A/Jn)) = ÒM ,

the separated completion of M with respect to the J-preadic topology, and similarly

Γ(U ∩ X ′,F ′/X ′) =ÓM ′, Γ(U ∩ X ′,F ′′/X ′) =dM ′′;

our claim then follows, since A is Noetherian, and since the functor ÒM in M is exact on the category ofA-modules of finite type (0, 7.3.3). I | 197

(ii) The question is local, so we can assume that we have an exact sequence O mX →O

nX →F → 0 (0, 5.3.2);

since γ ] is functorial, and the functors i∗(F ) and F/X ′ are right exact (by (i) and (0, 4.3.1)), we havethe commutative diagram

(10.8.8.1) i∗(O mX )

//

γ ]

i∗(O nX )

//

γ ]

i∗(F ) //

γ ]

0

(O mX )/X ′

// (O nX )/X ′

// F/X ′ // 0

whose rows are exact. Furthermore, the functors i∗(F ) and F/X ′ commute with finite direct sums((0, 3.2.6) and (0, 4.3.2)), and we thus reduce to proving our claim for F = OX . We have i∗(OX ) =(OX )/X ′ = ObX (0, 4.3.4), and that γ ] is a homomorphism of O

bX -modules; so it suces to check that γ ]

10. FORMAL SCHEMES 159

sends the unit section of ObX over an open subset of X ′ to itself, which is immediate, and shows, in this

case, that γ ] is the identity.

Corollary (10.8.9). — The morphism i : X/X ′ → X of ringed spaces is at.

Proof. This follows from (0, 6.7.3) and Proposition (10.8.8, i).

Corollary (10.8.10). — If F and G are coherent OX -modules, then there exist canonical functorial (in F and G )isomorphisms

(10.8.10.1) (F/X ′)⊗(OX )/X ′ (G/X ′)' (F ⊗OXG )/X ′ ,

(10.8.10.2) (HomOX(F ,G ))/X ′ 'Hom(OX )/X ′ (F/X ′ ,G/X ′).

Proof. This follows from the canonical identification of i∗(F ) withF/X ′ ; the existence of the first isomorphismis then a result which holds for all morphisms of ringed spaces (0, 4.3.3.1), and the second is a result whichholds for all flat morphisms (0, 6.7.6), by Corollary (10.8.9).

Proposition (10.8.11). — For every coherent OX -module F , the canonical homomorphism Γ(X ,F )→ Γ(X ′,F/X ′)induced by F →F/X ′ has kernel consisting of the zero sections in some neighborhood of X ′.

Proof. It follows from the definition of F/X ′ that the canonical image of such a section is zero. Conversely,if s ∈ Γ(X ,F ) has a zero image in Γ(X ′,F/X ′), then it suces to see that every x ∈ X ′ admits a neighborhood inX in which s is zero, and we can thus reduce to the case where X = Spec(A) is ane, A Noetherian, X ′ = V (J)for some ideal J of A, and F = eM for some A-module M of finite type. Then Γ(X ′,F/X ′) is the separatedcompletion ÒM of M for the J-preadic topology, and the homomorphism Γ(X ,F )→ Γ(X ′,F/X ′) is the canonicalhomomorphism M → ÒM . We know (0, 7.3.7) that the kernel of this homomorphism is the set of the z ∈ M killedby an element of 1+ J. So we have (1+ f )s = 0 for some f ∈ J; for every x ∈ X ′ we have (1x + fx)sx = 0, and,since 1x + fx is invertible in Ox (JxOx being contained in the maximal ideal of Ox), we have sx = 0, which provesthe proposition.

Corollary (10.8.12). — The support of F/X ′ is equal to Supp(F )∩ X ′.

Proof. It is clear that F/X ′ is an (OX )/X ′ -module of finite type ((10.8.8, ii) and (0, 5.2.4)), so its support I | 198is closed (0, 5.2.2) and evidently contained in Supp(F ) ∩ X ′. To show that it is equal to the latter set, weimmediately reduce to proving that the equation Γ(X ′,F/X ′) = 0 implies that Supp(F )∩ X ′ = ∅; this followsfrom Proposition (10.8.11) and Theorem (1.4.1).

Corollary (10.8.13). — Let u : F → G be a homomorphism of coherent OX -modules. For u/X ′ : F/X ′ → G/X ′ to bezero, it is necessary and sucient for u to be zero on a neighborhood of X ′.

Proof. By Proposition (10.8.8, ii), u/X ′ can be identified with i∗(u), so, if we consider u as a section overX of the sheaf H = HomOX

(F ,G ), then u/X ′ is the section over X ′ of i∗(H ) = H/X ′ to which it canonicallycorresponds ((10.8.10.2) and (0, 4.4.6)). It thus suces to apply Proposition (10.8.11) to the coherent OX -moduleH .

Corollary (10.8.14). — Let u :F →G be a homomorphism of coherent OX -modules. For u/X ′ to be a monomorphism(resp. an epimorphism), it is necessary and sucient for u to be a monomorphism (resp. an epimorphism) on a neighborhoodof X ′.

Proof. Let P and N be the cokernel and kernel (respectively) of u, so that we have the exact sequence0→N

v−→F

u−→G

w−→P → 0, hence (10.8.8, i) the exact sequence

0 −→N/X ′v/X ′−−→F/X ′

u/X ′−−→G/X ′

w/X ′−−→P/X ′ −→ 0.

If u/X ′ is a monomorphism (resp. an epimorphism), then we have v/X ′ = 0 (resp. w/X ′ = 0), so there exists aneighborhood of X ′ on which v = 0 (resp. w= 0) by Corollary (10.8.13).

10. FORMAL SCHEMES 160

10.9. Extension of morphisms to completions.

(10.9.1). Let X and Y be locally Noetherian (usual) preschemes, f : X → Y a morphism, and X ′ (resp. Y ′) aclosed subset of the underlying space X (resp. Y ) such that f (X ′) ⊂ Y ′. Let J (resp. K ) be a sheaf of ideals ofOX (resp. OY ) such that the support of OX/J (resp. OY /K ) is X ′ (resp. Y ′) and f ∗(K )OX ⊂ J ; we note thatthere always exist such sheaves of ideals, since, for example, we can take J to be the largest sheaf of idealsof OX defining a subprescheme of X with underlying space X ′ (5.2.1), and the hypothesis f (X ′) ⊂ Y ′ impliesthat f ∗(K )OX ⊂ J (5.2.4). For every integer n> 0 we have f ∗(K n)OX ⊂ J n (0, 4.3.5); as a result (4.4.6), if weset X ′n = (X

′,OX/J n+1) and Y ′n = (Y′,OY /K n+1), then f induces a morphism fn : X ′n→ Y ′n, and it is immediate

that the fn form an inductive system. We denote its inductive limit (10.6.8) by bf : X/X ′ → Y/Y ′ , and we say (byabuse of language) that bf is the extension of f to the completions of X and Y along X ′ and Y ′. It can be checkedimmediately that this morphism does not depend on the choice of sheaves of ideals J and K satisfying theabove conditions. It suces to consider the case where X and Y are Noetherian ane schemes with rings A andB (respectively); then J = eJ and K = eK, where J (resp. K) is an ideal of A (resp. B), f corresponds to a ringhomomorphism φ : B→ A such that φ (K) ⊂ J ((4.4.6) and (1.7.4)); bf is then the morphism corresponding (10.2.2) tothe continuous homomorphism bφ : bB→ bA, where bA (resp. bB) is the separated completion of A (resp. B) with respectto the J-preadic (resp. K-preadic) topology (10.6.8); we know that, if we replace J by another sheaf of ideals I | 199J ′ = eJ′ such that the support of OX/J ′ is X ′, then the J-preadic and J′-preadic topologies on A are the same(10.82).

We note that, by definition, the continuous map X ′ → Y ′ of the underlying spaces of X/X ′ and Y/Y ′corresponding to bf is exactly the restriction to X ′ of f .

(10.9.2). It follows immediately from the above definition that the diagram of morphisms of ringed spaces

bXbf //

iX

bY

iY

Xf // Y

commutes, with the vertical arrows being the canonical morphisms (10.8.7).

(10.9.3). Let Z be a third prescheme, g : Y → Z a morphism, and Z ′ a closed subset of Z such that g(Y ′) ⊂ Z ′.If bg denotes the completion of the morphism g along Y ′ and Z ′, then it immediately follows from (10.9.1) thatwe have (g f )∧ = bg bf .

Proposition (10.9.4). — Let X and Y be locally Noetherian S-preschemes, with Y of nite type over S. Let f and g beS-morphisms from X to Y such that f (X ′) ⊂ Y ′ and g(X ′) ⊂ Y ′. For bf = bg to hold, it is necessary and sucient for fand g to coincide on a neighborhood of X ′.

Proof. The condition is evidently sucient (even without the finiteness hypothesis on Y ). To see that it isnecessary, we remark first that the hypothesis bf = bg implies that f (x) = g(x) for all x ∈ X ′. Also, the questionbeing local, we can assume that X and Y are ane open neighborhoods of x and y = f (x) = g(x) respectively(with Noetherian rings), that S is ane, and that Γ(Y,OY ) is a Γ(S,OS)-algebra of finite type (6.3.3). Then fand g correspond to Γ(S,OS)-homomorphisms ρ and σ (respectively) from Γ(Y,OY ) to Γ(X ,OX ) (1.7.3), and,by hypothesis, the extensions by continuity of these homomorphisms to the separated completion of Γ(Y,OY )are the same. We conclude from Proposition (10.8.11) that, for every section s ∈ Γ(Y,OY ), the sections ρ(s)and σ(s) coincide on a neighborhood (depending on s) of X ′; since Γ(Y,OY ) is an algebra of finite type overΓ(S,OS), we have that there exists a neighborhood V of X ′ such that ρ(s) and σ(s) coincide on V for every sections ∈ Γ(Y,OY ). If h ∈ Γ(X ,OX ) is such that D(h) is a neighborhood of x contained in V , then we conclude fromthe above and from Theorem (1.4.1, d) that f and g coincide on D(h).

Proposition (10.9.5). — Under the hypotheses of (10.9.1), for every coherent OY -module G , there exists a canonicalfunctorial isomorphism of (OX )/X ′ -modules

( f ∗(G ))/X ′ ' bf ∗(G/Y ′).

Proof. If we canonically identify ( f ∗(G ))/X ′ with i∗X ( f∗(G )), and bf ∗(G/Y ′) with bf ∗(i∗Y (G )) (10.8.8), then the

proposition follows immediately from the commutativity of the diagram in (10.9.2).

10. FORMAL SCHEMES 161

(10.9.6). Now letF be a coherent OX -module, and let G be a coherent OY -module. If u : G →F is an f -morphism,then it corresponds to an OX -homomorphism u] : f ∗(G )→ F , thus, by completion, to a continuous (OX )/X ′ -homomorphism (u])/X ′ : ( f ∗(G ))/X ′ → F/X ′ , and, by Proposition (10.9.5), there exists a unique bf -morphismv : G/Y ′ →F/X ′ such that v] = (u])/X ′ . If we consider the triples (F , X , X ′) (F being a coherent OX -module, and I | 200X ′ a closed subset of X ) as a category, with the morphisms (F , X , X ′)→ (G , Y, Y ′) consisting of a morphism ofpreschemes f : X → Y such that f (X ′) ⊂ Y ′ and an f -morphism u : G →F , then we can say that (X/X ′ ,F/X ′)is a functor in (F , X , X ′) with values in the category of pairs (Z,H ) consisting of a locally Noetherian formalprescheme Z and an OZ-module H , with the morphisms of the latter category being the pairs consisting of amorphism g of formal preschemes and a g -morphism.

Proposition (10.9.7). — Let S, X , and Y be locally Noetherian preschemes, g : X → S and h : Y → S morphisms, S′ aclosed subset of S, and X ′ (resp. Y ′) a closed subset of X (resp. Y ) such that g(X ′) ⊂ S′ (resp. h(Y ′) ⊂ S′); let Z = X ×S Y ;suppose that Z is locally Noetherian, and let Z ′ = p−1(X ′)∩ q−1(Y ′), where p and q are the projections of X ×S Y . Withthese conditions, the completion Z/Z ′ can be identied with the product (X/X ′)×S/S′ (Y/Y ′) of formal S/S′ -preschemes, where

the structure morphisms are identied with bg and bh, and the projections with bp and bq.

Proof. It is immediate that the question is local for S, X , and Y , and we thus reduce to the case whereS = Spec(A), X = Spec(B), Y = Spec(C), S′ = V (J), X ′ = V (K), and Y ′ = V (L), with J, K, and L ideals suchthat φ (J) ⊂ K and ψ(J) ⊂ L, where we denote by φ and ψ the homomorphisms A → B and A → C whichcorrespond to g and h (respectively). We know that Z = Spec(B ⊗A C) and that Z ′ = V (M), where M is theideal Im(K⊗A C) + Im(B ⊗A L). The conclusion follows (10.7.2) from the fact that the completed tensor product(bB ⊗

bAbC)∧ (where bA, bB, and bC are, respectively, the separated completions of A, B, and C with respect to the

J-, K-, and L-preadic topologies) is the separated completion of the tensor product B ⊗A C with respect to theM-preadic topology (0, 7.7.2).

In addition, we note that, if T is a locally Noetherian S-prescheme, u : T → X and v : T → Y bothS-morphisms, and T ′ a closed subset of T such that u(T ′) ⊂ X ′ and v(T ′) ⊂ Y ′, then the extension to thecompletion ((u, v)S)∧ can be identified with (bu,bv)S/S′ .

Corollary (10.9.8). — Let X and Y be locally Noetherian S-preschemes such that X ×S Y is locally Noetherian; letS′ be a closed subset of S, and X ′ (resp. Y ′) a closed subset of X (resp. Y ) whose image in S is contained in S′. For everyS-morphism f : X → Y such that f (X ′) ⊂ Y ′, the graph morphism Γ

bf can be identied with the extension (Γf )∧ of thegraph morphism of f .

Corollary (10.9.9). — Let X and Y be locally Noetherian preschemes, f : X → Y a morphism, Y ′ a closed subset of Y ,and X ′ = f −1(Y ′). Then the prescheme X/X ′ can be identied, by the commutative diagram

X

f

X/X ′oo

bf

Y Y/Y ′oo

with the product X ×Y (Y/Y ′) of formal preschemes.

Proof. It suces to apply Proposition (10.9.7), replacing S and S′ by Y , and X and X ′ by X . I | 201

Remark (10.9.10). — If S is the sum X1 t X2 (3.1), X ′ the union X ′1 ∪ X ′2, where X ′i is a closed subset of X i(i = 1,2), then we have X/X ′ = X1/X ′1

t X2/X ′2.

10.10. Application to coherent sheaves on formal ane schemes.

(10.10.1). In this paragraph, A denotes an adic Noetherian ring, and J an ideal of definition for A. Let X = Spec(A),and X = Spf(A), which can be identified with the closed subset V (J) of X (10.1.2). In addition, Definitions(10.1.2) and (10.8.4) show that the formal ane scheme X is identical the completion X/X of the ane scheme Xalong the closed subset X of its underling space. To every coherent OX -module F , corresponds an OX-module offinite type F/X, which is a sheaf of topological modules over the sheaf of topological rings OX. Every coherentOX -module F is of the form eM , where M is an A-module of finite type (1.5.1); we set ( eM)/X = M∆. In addition,if u : M → N is an A-homomorphism of A-modules of finite type, then it corresponds to a homomorphismeu : eM → eN , and, as a result, to a continuous homomorphism eu/X ′ : ( eM)/X ′ → (eN)/X ′ , which we denote by u∆. Itis immediate that (v u)∆ = v∆ u∆; we have thus defined a covariant additive functor M∆ from the category ofA-modules of finite type to the category of OX-modules of finite type. When A is a discrete ring, we have M∆ = eM .

10. FORMAL SCHEMES 162

Proposition (10.10.2). —

(i) M∆ is an exact functor in M , and there exists a canonical functorial isomorphism of A-modules Γ(X, M∆)' M .(ii) If M and N are A-modules of nite type, then there exist canonical functorial isomorphisms

(10.10.2.1) (M ⊗A N)∆ ' M∆ ⊗OX N∆,

(10.10.2.2) (HomA(M , N))∆ 'HomOX(M∆, N∆).

(iii) The map u 7→ u∆ is a functorial isomorphism

(10.10.2.3) HomA(M , N)' HomOX(M∆, N∆).

Proof. The exactness of M∆ follows from the exactness of the functors eM (1.3.5) and F/X ′ (10.8.8). Bydefinition, Γ(X , M∆) is the separated completion of the A-module Γ(X , eM) = M with respect to the J-preadictopology; but, since A is complete and M is of finite type, we know (0, 7.3.6) that M is separated and com-plete, which proves (i). The isomorphism (10.10.2.1) (resp. (10.10.2.2)) comes from the composition of theisomorphisms (1.3.12, i) and (10.8.10.1) (resp. (1.3.12, ii) and (10.8.10.2)). Finally, since HomA(M , N) is anA-module of finite type, we can apply (i), which identifies Γ(X, (HomA(M , N))∆) with HomA(M , N), and we canuse (10.10.2.2), which proves that the homomorphism (10.10.2.3) is an isomorphism.

We deduce from Proposition (10.10.2) a series of results analogous to those of Theorem (1.3.7) andCorollary (1.3.12), whose formulation we leave to the reader. I | 202

We note that the exactness property of M∆, applied to the exact sequence 0→ J→ A→ A/J→ 0, shows thatthe sheaf of ideals of OX denoted here by J∆ coincides with the one denoted also by J∆ in (10.3.1), by (10.3.2).

Proposition (10.10.3). — Under the hypotheses of (10.10.1), OX is a coherent sheaf of rings.

Proof. If f ∈ A, then we know that A f is an adic Noetherian ring (0, 7.6.11), and, since the question islocal, we reduce (10.1.4) to proving that the kernel of the homomorphism v : O n

X→OX is an OX-module of finitetype. We then have v = u∆, where u is an A-homomorphism An→ A (10.10.2); since A is Noetherian, the kernelof u is of finite type, or, equivalently, we have a homomorphism Am w

−→ An such that the sequence Am w−→ An u

−→ A is

exact. We conclude (10.10.2) that the sequence O mX

w∆

−→ O nX

v−→ OX is exact, which proves that the kernel of v is of

finite type.

(10.10.4). With the above notation, set An = A/Jn+1, and let Sn be the ane scheme Spec(An) = (X,OX/J n+1),with J = J∆ the sheaf of ideals of definition for OX corresponding to the ideal J. Let umn be the morphism ofpreschemes Xm→ Xn corresponding to the canonical homomorphism An→ Am for m¶ n; the formal scheme Xis the inductive limit of the Xn with respect to the umn (10.6.3).

Proposition (10.10.5). — Under the hypothesis of (10.10.1), let F be an OX-module. The following conditions areequivalent:

(a) F is a coherent OX-module;(b) F is isomorphic to the projective limit (10.6.6) of a sequence (Fn) of coherent OXn

-modules such that u∗mn(Fn) =Fm; and

(c) there exists an A-module M of nite type (determined up to canonical isomorphism by Proposition (10.10.2, i))such that F is isomorphic to M∆.

Proof. We first show that (b) implies (c). We have Fn =ÝMn, where Mn is an An-module of finite type, andthe hypotheses imply that Mm = Mn ⊗An

Am for m ¶ n (1.6.5); the Mn thus form a projective system for thecanonical di-homomorphisms Mn→ Mm (m¶ n), and it follows immediately from the definition of the An thatthis projective system satisfies the conditions of (0, 7.2.9); as a result, its projective limit M is an A-module offinite type such that Mn = M ⊗A An for all n. We deduce that Fn is induced over Xn by eM ⊗OX

(OX/eJn+1), and so

F = M∆ by Definition (10.8.4).Conversely, (c) implies (b); indeed, if un is the immersion morphism Xn→ X , then u∗n( eM) = (M ⊗A An)∼ is

induced over Xn by eM ⊗OX(OX/eJ

n+1), and M∆ = lim←−u∗n( eM) by Definition (10.8.4); since um = un umn for m¶ n,

the Fn = u∗n( eM) satisfy the conditions of (b), whence our claim.We now show that (c) implies (a): indeed, we have, by definition, that OX = A∆; since M is the cokernel of

a homomorphism Am→ An, it follows from Proposition (10.10.2) that M∆ is the cokernel of a homomorphismO mX →O

nX, and, since the sheaf of rings OX is coherent (10.10.3), so too is M∆ (0, 5.3.4). I | 203

10. FORMAL SCHEMES 163

Finally, (a) implies (b). Considered as an OX-module, we have that OXn= OX/J n+1 = A∆

n; butFn =F⊗OXOXn

is a coherent OX-module (0, 5.3.5), and, since it is also an OXn-module, and J n+1 is coherent, we conclude

that Fn is a coherent OXn-module (0, 5.3.10), and it is immediate that u∗mn(Fn) = Fm for m ¶ n (recalling

that the continuous map Xm → Xn of the underlying spaces is the identity on X). The sheaf G = lim←−Fn isthus a coherent OX-module, since we have seen that (b) implies (a). The canonical homomorphisms F →Fnform a projective system, which, by passing to the limit, gives a canonical homomorphism w : F → G , andit remains only to prove that w is bijective. The question is now local, so we can reduce to the case whereF is the cokernel of a homomorphism O p

X → OqX; since this homomorphism is of the form v∆, where v is a

homomorphism Am → An (10.10.2), F is isomorphic to M∆, where M = Coker v (10.10.2). We then have, byProposition (10.10.2), that Fn = M∆ ⊗OX A∆

n = (M ⊗A An)∆, and, since the J-adic topology on M ⊗A An is discrete,we have (M ⊗A An)∆ = (M ⊗A An)∼ (as an OXn

-module); we have seen above that M∆ = lim←−Fn, and w is thus theidentity in this case.

Corollary (10.10.6). — If F satises condition (b) of Proposition (10.10.5), then the projective system (Fn) isisomorphic to the system of the F ⊗OX OXn

.

(10.10.7). Now let A and B be adic Noetherian rings, and φ : B→ A a continuous homomorphism; we denoteby J (resp. K) an ideal of definition for A (resp. B) such that φ (K) ⊂ J, and we set X = Spec(A), Y = Spec(B),X = Spf(A), and Y = Spf(B). Let f : X → Y be the morphism of preschemes corresponding to φ (1.6.1), andbf : X → Y its extension to the completions (10.9.1), which is also a morphism of formal preschemes thatcorresponds to φ (10.2.2).

Proposition (10.10.8). — For every B-module N of nite type, there exists a canonical functorial isomorphism ofOX-modules

bf ∗(N∆)' (N ⊗B A)∆.

Proof. Denoting by iX : X→ X and iY : Y→ Y the canonical morphisms, we have (10.8.8), up to canonicalfunctorial isomorphisms, N∆ = i∗Y (eN) and

(N ⊗B A)∆ = i∗X ((N ⊗B A)∼) = i∗X ( f∗(eN))

(1.6.5); the proposition then follows from the commutativity of the diagram in (10.9.2).

Corollary. — For every ideal b of B, we have bf ∗(b∆)OX = (bA)∆.

Proof. Let j be the canonical injection b→ B, to which corresponds the canonical injection j∆ : b∆→OYof sheaves of OY-modules; by definition, bf ∗(b∆)OX is the image of the homomorphism bf ∗( j∆) : bf ∗(b∆)→OX =bf ∗(OY); but this homomorphism can be identified with ( j ⊗ 1)∆ : (b⊗B A)∆ → OX = (B ⊗B A)∆ by Proposition(10.10.8). Since the image of j⊗1 is the ideal bA of A, the image of ( j⊗1)∆ is thus (bA)∆, by Proposition (10.10.2),whence the conclusion.

10.11. Coherent sheaves on formal preschemes.I | 204

Proposition (10.11.1). — If X is a locally Noetherian formal prescheme, then the sheaf of rings OX is coherent, andevery sheaf of ideals of denition for X is coherent.

Proof. The question is local, so we can reduce to the case of a Noetherian ane formal scheme, and theproposition then follows from Propositions (10.10.3) and (10.10.5).

(10.11.2). Let X be a locally Noetherian formal prescheme, J a sheaf of ideals of definition for X, and Xn thelocally Noetherian (usual) prescheme (X,OX/J n+1), so that X is the inductive limit of the sequence (Xn) withrespect to the canonical morphisms umn : Xm→ Xn (10.6.3). With this notation:

Theorem (10.11.3). — For an OX-module F to be coherent, it is necessary and sucient for it to be isomorphic to aprojective limit of a sequence (Fn), where the Fn are coherent OXn

-modules such that u∗mn(Fn) =Fm for m¶ n (10.6.6).The projective system (Fn) is then isomorphic to the system of the u∗n(F ) =F ⊗OX OXn

, where un is the canonical morphismXn→ X.

Proof. The question is local, so we can reduce to the case where X is a Noetherian ane formal scheme,and the theorem then is a consequence of Proposition (10.10.5) and Corollary (10.10.6).

We can thus say that the data of a coherent OX-module is equivalent to the data of a projective system (Fn) ofcoherent OXn

-modules such that umn(Fn) =Fm for m¶ n.

10. FORMAL SCHEMES 164

Corollary (10.11.4). — If F and G are coherent OX-modules, then we can (with the notation of Theorem (10.11.3))dene a canonical functorial isomorphism

(10.11.4.1) HomOX(F ,G )' lim←−n

HomOXn(Fn,Gn).

Proof. The projective limit on the right-hand side is understood to be taken with respect to the mapsθn 7→ u∗mn(θn) (m ¶ n) from HomOXn

(Fn,Gn) to HomOXm(Fm,Fm). The homomorphism (10.11.4.1) sends an

element θ ∈ HomOX(F ,G ) to the sequence (u∗n(θ)); we see that we can define an inverse homomorphism of theabove by sending a projective system (θn) ∈ lim←−n

HomOXn(Fn,Gn) to its projective limit in HomOX(F ,G ), taking

into account Theorem (10.11.3).

Corollary (10.11.5). — For a homomorphism θ : F → G to be surjective, it is necessary and sucient for thecorresponding homomorphism θ0 = u∗0(θ) :F0→G0 to be surjective.

Proof. The question is local, so we reduce to the case where X = Spf(A) with A an adic Noetherian ring,F = M∆, G = N∆, and θ = u∆, where M and N are A-modules of finite type, and u is a homomorphism M → N ;we then have that θ0 = eu0, where u0 is the homomorphism u⊗ 1 : M ⊗A A/J→ N ⊗A A/J; the conclusion followsfrom the fact θ and u (resp. θ0 and u0) are simultaneously surjective ((1.3.9) and (10.10.2)) and that u and u0are simultaneously surjective (0, 7.1.14).

(10.11.6). Theorem (10.11.3) shows that we can consider every coherent OX-module F as a topological OX-module,considering it as a projective limit of pseudo-discrete sheaves of groups Fn (0, 3.8.1). It then follows from Corollary(10.11.4) that every homomorphism u :F →G of coherent OX-modules is automatically continuous (0, 3.8.2). I | 205Furthermore, ifH is a coherent OX-submodule of a coherent OX-module F , then, for every open U ⊂ X, Γ(U ,H )is a closed subgroup of the topological group Γ(U ,F ), since the functor Γ is left exact, and Γ(U ,H ) is thekernel of the homomorphism Γ(U ,F )→ Γ(U ,F/H ), which is continuous by the above, since F/G is coherent(0, 5.3.4); our claim follows from the fact that Γ(U ,F/H ) is a separated topological group.

Proposition (10.11.7). — Let F and G be coherent OX-modules. We can dene (with the notation of Theorem(10.11.3)) canonical functorial isomorphisms of topological OX-modules (10.11.6)

(10.11.7.1) F ⊗OX G ' lim←−n

(Fn ⊗OXnGn),

(10.11.7.2) HomOX(F ,G )' lim←−n

HomOXn(Fn,Gn).

Proof. The existence of the isomorphism (10.11.7.1) follows from the formula

Fn ⊗OXnGn = (F ⊗OX OXn

)⊗OXn(G ⊗OX OXn

) = (F ⊗OX G )⊗OX OXn

and from Theorem (10.11.3). The isomorphism (10.11.7.2), where both sides are considered as sheaves ofmodules without topology, follows from the definition of the sections of HomOX(F ,G ) and HomOXn

(Fn,Gn), andfrom the existence of the isomorphism (10.11.4.1), mapping a prescheme induced on an arbitrary Noetherianformal ane open set to X. It remains to prove that the isomorphism (10.11.7.2) is bicontinuous over a quasi-compact set, and we can thus reduce to the case where X= Spf(A) with A an adic Noetherian ring, and hence(10.10.5) to the case where F = M∆ and G = N∆, with M and N both A-modules of finite type; taking (10.10.2.1),(10.10.2.3), and Corollary (1.3.12, ii) into account, we reduce to showing that the canonical isomorphismHomA(M , N) ' lim←−n

HomAn(Mn, Nn) (with Mn = M ⊗A An and Nn = N ⊗A An) is continuous, which has already

been proved in (0, 7.8.2).

(10.11.8). Since HomOX(F ,G ) is the group of sections of the sheaf of topological groups HomOX(F ,G ), itis equipped with a group topology. If X is Noetherian, then it follows from (10.11.7.2) that the subgroupsHomOX(F ,J nG ) (for arbitrary n) form a fundamental system of neighborhoods of 0 in this group.

Proposition (10.11.9). — Let X be a Noetherian formal prescheme, and F and G coherent OX-modules. In thetopological group HomOX(F ,G ), the surjective (resp. injective, bijective) homomorphisms form an open set.

Proof. By Corollary (10.11.5), the set of surjective homomorphisms in HomOX(F ,G ) is the inverse imageunder the continuous map HomOX(F ,G )→ HomOX0

(F0,G0) of a subset of the discrete group HomOX0(F0,G0),

whence the first claim. To show the second, we cover X by a finite number of Noetherian formal ane subsets Ui .For θ ∈ HomOX(F ,G ) to be injective, it is necessary and sucient for all of the images under the (continuous)restriction maps HomOX(F ,G )→ HomOX|Ui)(F|Ui ,G|Ui) to be injective; we can thus reduce to the ane case,and then this has already been proved in (0, 7.8.3).

10. FORMAL SCHEMES 165

10.12. Adic morphisms of formal preschemes.I | 206

(10.12.1). Let X and S be locally Noetherian formal preschemes; we say that a morphism f : X → S is adicif there exists an ideal of definition J of S such that K = f ∗(J )OX is an ideal of definition of X; we thenalso say that X is an adic S-prescheme (for f ). Whenever this is the case, for every ideal of definition J1 of S,K1 = f ∗(J1)OX is an ideal of definition of X. Indeed, the question being local, we can assume that X and Sare Noetherian and ane; there then exists a whole number n such that J n ⊂ J1 and J n

1 ⊂ J ((10.3.6) and(0, 7.1.4)), whence K n ⊂K1 and K n

1 ⊂K . The first of these relations shows that K1 = K∆1 , where K1 is an open

ideal of A= Γ(X,OX), and the second shows that K1 is an ideal of definition of A (0, 7.1.4), whence our claim.

It follows immediately from the above that, if X and Y are adic S-preschemes, then every S-morphismu : X → Y is adic: indeed, if f : X → S and g : Y → S are the structure morphisms, and J is an ideal ofdefinition of S, then we have f = g u, and so u∗(g∗(J )OY)OX = f ∗(J )OX is an ideal of definition of X, and,by hypothesis, g∗(J )OY is an ideal of definition of Y.

(10.12.2). In what follows, we suppose that we have some fixed locally Noetherian formal prescheme S, andsome ideal of definition J of S; we set Sn = (S,OS/J n+1). The (locally Noetherian) adic S-preschemes clearlyform a category. We say that an inductive system (Xn) of locally Noetherian (usual) Sn-preschemes is an adicinductive (Sn)-system if the structure morphisms fn : Xn→ Sn are such that, for m¶ n, the diagrams

Xn

fn

Xmoo

fm

Sn Smoo

commute and identify Xm with the product Xn ×SnSm = (Xn)(Sm). The adic inductive systems form a category:

it suces in fact to define a morphism (Xn)→ (Yn) of such systems to be an inductive system of Sn-morphismsun : Xn→ Yn such that um is identified with (un)(Sm) for m¶ n. With this in mind:

Theorem (10.12.3). — There is a canonical equivalence between the category of adic S-preschemes and the category ofadic inductive (Sn)-systems.

The equivalence in question is obtained in the following way: if X is an adic S-prescheme, and f : X→S isthe structure morphism, then K = f ∗(J )OX is an ideal of definition of X, and we associate to X the inductivesystem of the Xn = (X,OX/K n+1), with the structure morphism fn : Xn→ Sn corresponding to f (10.5.6). Wefirst show that (Xn) is an adic inductive system: if f = (ψ,θ), then ψ∗(J )OX = K , so ψ∗(J n)OX = K n for alln, and (by exactness of the functor ψ∗) K m+1/K n+1 = ψ∗(J m+1/J n+1)(OX/K n+1) for m¶ n; our conclusionthus follows from (4.4.5). Furthermore, it can be immediately verified that a S-morphism u : X→ Y of adicS-preschemes corresponds (with the obvious notation) to an inductive system of Sn-morphisms un : Xn → Yn I | 207such that um is identified with (un)(Sm) for m¶ n.

The fact that this equivalence is well defined will follow from the more-precise following proposition.

Proposition (10.12.3.1). — Let (Xn) be an inductive system of Sn-preschemes; suppose that the structure morphismsfn : Xn → Sn are such that the diagrams in (10.12.2.1) commute and identify Xm with Xn ×Sn

Sm for m ¶ n. Thenthe inductive system (Xn) satises conditions (b) and (c) of (10.6.3); let X be the inductive limit, and f : X→ S themorphism given by the inductive limit of the inductive system ( fn). Then, if X0 is locally Noetherian, X is locally Noetherian,and f is an adic morphism.

Proof. Since the sheaf of ideals of OSnthat defines the subprescheme Sm of Sn is nilpotent, by (4.4.5), so

too is the sheaf of ideals of OXnthat defines the subprescheme Xm of Xn, and so the conditions of (10.6.3) are

satisfied. The question being local on X and S, we can assume that S= Spf(A), J = J∆ (with A a NoetherianJ-adic ring), and Xn = Spec(Bn); if An = A/Jn+1, then the hypothesis implies that B0 is Noetherian, and if we setJn = J/Jn+1, then Bm = Bn/J

m+1n Bn. The kernel of Bn→ B0 is thus Kn = JnBn, and the kernel of Bn→ Bm is Km+1

nfor m¶ n; further, since A1 is Noetherian, J1 is of finite type over A1, and so K1 = K1/K

21 is of finite type over B1,

and a fortiori of finite type over B0 = B1/K1; the fact that X is Noetherian then follows from (10.6.4); if B = lim←−Bn,

then we have X = Spf(B), and, if K is the kernel of B→ B0, then Bn = B/Kn+1. If ρn : A/Jn+1 → B/Kn+1 is thehomomorphism corresponding to fn, then we have that

K/Kn+1 = (B/Kn+1)ρn(J/Jn+1)

since the homomorphism ρ : A→ B corresponding to f is equal to lim←−ρn, and that the ideal JB of B is dense in K,

and, since every ideal of B is closed (0, 7.3.5), we also have that K= JB. If K = K∆, the equality f ∗(J )OX =Kthen follows from (10.10.9), and finishes the proof.

10. FORMAL SCHEMES 166

(10.12.3.2). The above equivalence gives, for adic S-preschemes X and Y, a canonical bijection

HomS(X,Y)' lim←−n

HomSn(Xn, Yn)

where the projective limit is relative to the maps un→ (un)(Sm) for m¶ n.

10.13. Morphisms of nite type.

Proposition (10.13.1). — Let Y be a locally Noetherian formal prescheme, K an ideal of denition of Y, andf : X→Y a morphism of formal preschemes. Then the following conditions are equivalent.

(a) X is locally Noetherian, f is an adic morphism (10.12.1), and, if we set J = f ∗(K )OX, then the morphismf0 : (X,OX/J )→ (Y,OY/K ) induced by f is of nite type.

(b) X is locally Noetherian, and is the inductive limit of an adic inductive (Yn)-system (Xn) such that the morphismX0→ Y0 is of nite type. I | 208

(1) Every point of Y has a Noetherian formal ane open neighbourhood V which satises the following property:(Q) f −1(V ) is a nite union of Noetherian formal ane open subsets Ui such that the Noetherian adic ring

Γ(Ui ,OX) is topologically isomorphic to the quotient of a formal series algebra, restricted (0, 7.5.1) toΓ(V,OY), by an ideal (which is necessarily closed).

Proof. It is immediate that (a) implies (b), by (10.12.3). To show that (b) implies (c), we can, since thequestion is local on Y, assume that Y= Spf(B), where B is Noetherian and adic; let K = K∆, with K an idealof definition of B. Since, by hypothesis, X0 is of finite type over Y0, X0 is a finite union of ane open subsetsUi such that the ring Ai0 of the ane scheme induced by X0 on Ui is an algebra of finite type over the ringB/K of Y0 (6.3.2). By (5.1.9), Ui is also an ane open subset in each of the Noetherian preschemes Xn, and,if Ain is the ring of the ane scheme induced by Xn on Ui , then hypothesis (b) implies, for m¶ n, that Aim isisomorphic to Ain/K

m+1Ain. Consequently, the formal prescheme induced by X on Ui is isomorphic to Spf(Ai),where Ai = lim←−n

Ain (10.6.4); Ai is a KAi -adic ring, and Ai/KAi , being isomorphic to Ai0, is an algebra of finitetype over B/K. We thus conclude (0, 7.5.5) that Ai is topologically isomorphic to a quotient of a formal seriesalgebra restricted to B (by a necessarily closed ideal, because such an algebra is Noetherian (0, 7.5.4)).

To show that (c) implies (a), we can restrict to the case where X= Spf(A) is also ane, with A a Noetherianadic ring isomorphic to a quotient of a formal series algebra, restricted to B, by a closed ideal. Then (0, 7.5.5)A/KA is an algebra of finite type over B/K, and KA= J is an ideal of definition of A, and so, by (10.10.9), theconditions of (a) are satisfied.

We note that, if the conditions of Proposition (10.13.1) are satisfied, then property (a) holds true for anyideal of definition K of Y (by (c)), and so, in property (b), all the fn are morphisms of finite type.

Corollary (10.13.2). — If the conditions of (10.13.1) are satised, then every Noetherian formal ane open subset Vof Y has property (Q), and, if Y is Noetherian, then so too is X.

Proof. This follows immediately from (10.13.1) and (6.3.2).

Denition (10.13.3). — When the equivalent properties (a), (b), and (c) of (10.13.1) are satisfied, we say thatthe morphism f is of finite type, or that X is a formal Y-prescheme of finite type, or a formal prescheme of finitetype over Y.

Corollary (10.13.4). — Let X = Spf(A) and Y = Spf(B) be Noetherian formal ane schemes; for X to be of nitetype over Y, it is necessary and sucient for the Noetherian adic ring A to be isomorphic to the quotient of a formal seriesalgebra, restricted to B, by some closed ideal.

Proof. With the notation of (10.13.1), if X is of finite type over Y, then A/KA is a (B/K)-algebra of finitetype by (6.3.3), and KA is an ideal of definition of A (10.10.9). We are then done, by (0, 7.5.5).

Proposition (10.13.5). —

(i) The composition of any two morphisms (of formal preschemes) of nite type is again of nite type. I | 209(ii) Let X, S, and S′ be locally Noetherian (resp. Noetherian) formal preschemes, and f : X→ S and X→ S′

morphisms. If f is of nite type, then X×SS′ is locally Noetherian (resp. Noetherian) and of nite type over S′.(iii) Let S be a locally Noetherian formal prescheme, and X′ and Y′ formal S-preschemes such that X′ ×S Y′ is

locally Noetherian. If X and Y are locally Noetherian formal S-preschemes, and f : X→ X′ and g : Y→Y′

are S-morphisms of nite type, then X×S Y is locally Noetherian, and f ×S g is a S-morphism of nite type.

10. FORMAL SCHEMES 167

Proof. By the formal argument of (3.5.1), (iii) follows from (i) and (ii), so it suces to prove (i) and (ii).Let X, Y, and Z be locally Noetherian formal preschemes, and f : X→Y and g : Y→ Z morphisms of finite

type. If L is an ideal of definition of Z, thenK = g∗(L )OY is an ideal of definition of Y, and J = f ∗(g∗(L ))OXis an ideal of definition for X. Let X0 = (X,OX/J ), Y0 = (Y,OY/K ), and Z0 = (Z,OZ/L ), and let f0 : X0→ Y0and g0 : Y0→ Z0 be the morphisms corresponding to f and g (respectively). Since, by hypothesis, f0 and g0 areof finite type, so too is g0 f0 (6.3.4), which corresponds to g f ; thus g f is of finite type, by (10.13.1).

Under the conditions of (ii), S (resp. X, S′) is the inductive limit of a sequence (Sn) (resp. (Xn), (S′n))of locally Noetherian preschemes, and we can assume (10.13.1) that Xm = Xn ×Sn

Sm for m ¶ n. The formalprescheme X×S S′ is then the inductive limit of the preschemes Xn ×Sn

S′n (10.7.4), and we have that

Xm ×SmS′m = (Xn ×Sn

Sm)×SmS′m = (Xn ×Sn

S′n)×S′nS′m.

Furthermore, X0×S0S′0 is locally Noetherian, since X0 is of finite type over S0 (6.3.8). We thus conclude (10.12.3.1),

first of all, that X×SS′ is locally Noetherian; then, since X0×S0S′0 is of finite type over S′0 (6.3.8), it follows from

(10.12.3.1) and (10.13.1) that X×S S′ is of finite type over S′, which proves (ii) (the claim about Noetherianpreschemes being an immediate consequence of (6.3.8)).

Corollary (10.13.6). — Under the hypotheses of (10.9.9), if f is a morphism of nite type, then so too is its extensionbf to the completions.

10.14. Closed subpreschemes of formal preschemes.

Proposition (10.14.1). — Let X be a locally Noetherian formal prescheme, and A a coherent sheaf of ideals of OX. IfY if the (closed) support of OX/A , then the topologically ringed space (Y, (OX/A )|Y) is a locally Noetherian formalprescheme that is Noetherian if X is.

Proof. Note that OX/A is coherent by (10.10.3) and (0, 5.3.4), so its support Y is closed (0, 5.2.2). LetJ be an ideal of definition of X, and let Xn = (X/OX/J n+1); the sheaf of rings OX/A is the projective limitof the sheaves OX/(A +J n+1) = (OX/A )⊗OX (OX/J

n+1) (10.11.3), all of which have support Y. The sheaf(A +J n+1)/J n+1 is a coherent OX-module, since J n+1 is coherent, and so (A +J n+1)/J n+1 is also a coherent(OX/J n+1)-module (0, 5.3.10); if Yn is the closed subprescheme of Xn defined by this sheaf of ideals, it isimmediate that (Y, (OX/A )|Y) is the formal prescheme given by the inductive limit of the Yn, and, since the I | 210conditions of (10.6.4) are satisfied, this proves that this formal prescheme is locally Noetherian, and furtherNoetherian if X is (since then Y0 is, by (6.1.4)).

Denition (10.14.2). — We define a closed subprescheme of a formal prescheme X to be any formal preschemeof the form (Y, (OX/A )|Y) with A a coherent ideal of OX; we say that this prescheme is the subprescheme ErrIIdefined by A .

It is clear that the correspondence thus defined between coherent ideals of OX and closed subpreschemes of ErrIIX is bijective.

The morphism of topologically ringed spaces j = (ψ,θ) : Y→ X, where ψ is the injection Y→ X and θ]the canonical homomorphism OX → OX/A , is evidently (10.4.5) a morphism of formal preschemes, and wecall it the canonical injection from Y to X. Note that, if X= Spf(A), or if A is Noetherian and adic, then A = a∆,where a is an ideal of A (10.10.5), and it then follows immediately from the above that Y = Spf(A/a), up toisomorphism, and that j corresponds (10.2.2) to the canonical homomorphism A→ A/a.

We say that a morphism f : Z→ X of locally Noetherian formal preschemes is a closed immersion if it factors

as Zg−→ Y

j−→ X, where g is an isomorphism from Z to a closed subprescheme Y of X, and j is the canonical

injection. Since j is a monomorphism of ringed spaces, g and Y are necessarily unique.

Proposition (10.14.3). — A closed immersion is a morphism of nite type.

Proof. We can immediately restrict to the case where X is a formal ane scheme Spf(A), and Y= Spf(A/a);the proposition then follows from Proposition (10.13.1, c).

Lemma (10.14.4). — Let f : Y→ X be a morphism of locally Noetherian formal preschemes, and let (Uα) be a coverof f (Y) by Noetherian formal ane open subsets of X such that the f −1(Uα) are Noetherian formal ane open subsets ofY. For f to be a closed immersion, it is necessary and sucient for f (Y) to be a closed subset of X and, for all α, for therestriction of f to f −1(Uα) to correspond (10.4.6) to a surjective homomorphism Γ(Uα,OX)→ Γ( f −1(Uα),OY).

Proof. The conditions are clearly necessary. Conversely, if the conditions are satisfied, and if we denoteby aα the kernel of Γ(Uα,OX)→ Γ( f −1(Uα),OY), then we can define a coherent sheaf of ideals A of OX by

10. FORMAL SCHEMES 168

setting A|Uα = a∆α and taking A to be zero on the complement of the union of the Uα. Since f (Y) is closed,

and since the support of a∆α is Uα ∩ f (Y), everything reduces to proving that a∆

α and a∆β induce the same sheaf

on any Noetherian formal ane open subset V ⊂ Uα ∩ Uβ . But the restriction to f −1(Uα) of f is a closedimmersion of this formal prescheme into Uα, f −1(V ) is a Noetherian formal ane open subsets of f −1(Uα),and the restriction of f to f −1(V ) is a closed immersion; if b is the kernel of the surjective homomorphismΓ(V,OX)→ Γ( f −1(V ),OY) corresponding to this restriction, then it is immediate (10.10.2) that a∆

α induces b∆ onV . The sheaf of ideals A being thus defined, it is then clear that f = g j, where j : Z→ X is the canonicalinjection of the closed subprescheme Z of X defined by A , and that g is an isomorphism from Y to Z.

Proposition (10.14.5). —

(i) If f : Z→ Y and g : Y→ X are closed immersions of locally Noetherian formal preschemes, then g f is aclosed immersion. I | 211

(ii) Let X, Y, and S be locally Noetherian formal preschemes, f : X→S a closed immersion, and g : Y→S amorphism. Then the morphism X×S Y→Y is a closed immersion.

(iii) Let S be a locally Noetherian formal prescheme, and X′ and Y′ locally Noetherian formal S-preschemes suchthat X′ ×S Y′ is locally Noetherian. If X and Y are locally Noetherian S-preschemes, and f : X→ X′ andg : Y→Y′ are S-morphisms that are closed immersions, then f ×S g is a closed immersion.

Proof. By (3.5.1), it again suces to prove (i) and (ii).To prove (i), we can assume that Y (resp. Z) is a closed subprescheme of X (resp. Y) defined by a coherent

sheaf J (resp. K ) of ideals of OX (resp. OY); if ψ is the injection Y→ X of underlying spaces, then ψ∗(K ) isa coherent sheaf of ideals of ψ∗(OY) = OX/J (0, 5.3.12), and thus also a coherent OX-module (0, 5.3.10); thekernel K1 of OX→ (OX/J )/ψ∗(K ) is thus a coherent sheaf of ideals of OX (0, 5.3.4), and OX/K1 is isomorphicto ψ∗(OY/K ), which proves that Z is an isomorphism to a closed subprescheme of X.

To prove (ii), we can immediately restrict to the case where S= Spf(A), X= Spf(B), and Y= Spf(C), withA a Noetherian J-adic ring, B = A/a (where a is an ideal of A), and C a Noetherian topological adic A-algebra.Everything then reduces to proving that the homomorphism C → C b⊗A(A/a) is surjective: but A/a is an A-moduleof finite type, and its topology is the J-adic topology; it then follows from (0, 7.7.8) that C b⊗A(A/a) can beidentified with C ⊗A (A/a) = C/aC , whence our claim.

Corollary (10.14.6). — Under the hypotheses of (10.14.5, ii), let p : X ×S Y → X and q : X ×S Y → Y be theprojections, so that the diagram

X

f

X×S Ypoo

q

S Y

goo

commutes. For every coherent OX module F , we then have a canonical isomorphism of OY-modules

(10.14.6.1) u : g∗ f∗(F )' q∗p∗(F ).

Proof. We know that defining a homomorphism g∗ f∗(F )→ q∗p∗(F ) is equivalent to defining a homomor-

phism f∗(F )→ g∗q∗p∗(F ) = f∗p∗p

∗(F ) (0, 4.4.3): we take u= f∗(ρ), where ρ is the canonical homomorphismF → p∗p

∗(F ) (0, 4.4.3). To see that u is an isomorphism, we can immediately restrict to the case where S,X, and Y are formal spectra of Noetherian adic rings A, B, and C (respectively), satisfying the conditions in(10.14.5, ii) above; we then have F = M∆, where M is an (A/a)-module of finite type (10.10.5), and the two sidesof (10.14.6.1) can then be identified, respectively, by (10.10.8), with (C ⊗A M)∆ and ((C/aC)⊗A/a M)∆, whencethe corollary, since (C/aC)⊗A/a M = (C ⊗A (A/a))⊗A/a M is canonically identified with C ⊗A M .

I | 212

Corollary (10.14.7). — Let X be a locally Noetherian usual prescheme, Y a closed subprescheme of X , j the canonicalinjection Y → X , X ′ a closed subset of X , and Y ′ = Y ∩ X ′; then bj : Y/Y ′ → X/X ′ is a closed immersion, and, for everycoherent OY -module F , we have

bj∗(F/Y ′) = ( j∗(F ))/X ′ .

Proof. Since Y ′ = j−1(X ′), it suces to use (10.9.9) and to apply (10.14.5) and (10.14.6).

10. FORMAL SCHEMES 169

10.15. Separated formal preschemes.

Denition (10.15.1). — Let S be a formal prescheme, X a formal S-prescheme, and f : X→S the structuremorphism. We define the diagonal morphism ∆X|S : X→ X×S X (also denoted by ∆X) to be the morphism(1X, 1X)S. We say that X is separated over S, or is a formal S-scheme, or that f is a separated morphism, if theimage of the underlying space of X under ∆X is a closed subset of the underlying space of X×S X. We say thata formal prescheme X is separated, or is a formal scheme, if it is separated over Z.

Proposition (10.15.2). — Suppose that the formal preschemes S and X are inductive limits of sequences (Sn) and(Xn) (respectively) of usual preschemes, and that the morphism f : X→S is the inductive limit of a sequence of morphismsfn : Xn→ Sn. For f to be separated, it is necessary and sucient for the morphism f0 : X0→ S0 to be separated.

Proof. Indeed, ∆X|S is then the inductive limit of the sequence of morphisms ∆Xn|Sn(10.7.4), and the image

of the underlying space of X (resp. of X×S X under ∆X|S) is identical to the image of the underlying space ofX0 (resp. of X0 ×S0

X0) under ∆X0|S0; whence the conclusion.

Proposition (10.15.3). — Suppose that all the formal preschemes (resp. morphisms of formal preschemes) in whatfollows are inductive limits of sequences of usual preschemes (resp. of morphisms of usual preschemes).

(i) The composition of any two separated morphisms is separated.(ii) If f : X→ X′ and g : Y→Y′ are separated S-morphisms, then f ×S g is separated.(iii) If f : X→Y is a separated S-morphism, then the S′-morphism f(S′) is separated for every extension S′→S

of the base formal prescheme.(iv) If the composition g f of two morphisms is separated, then f is separated.

(In the above, it is implicit that if the same formal prescheme Z is mentioned more than once in the sameproposition, we consider it as the inductive limit of the same sequence (Zn) of usual preschemes wherever it ismentioned, and the morphisms from Z to another formal prescheme (resp. from a formal prescheme to Z) asinductive limits of morphisms from Zn to some usual preschemes (resp. from some usual preschemes to Zn)).

Proof. With the notation of (10.15.2), we have, in fact, that (g f )0 = g0 f0 and ( f ×S g)0 = f0 ×S0g0; the

claims of (10.15.3) are then immediate consequences of (10.15.2) and the corresponding claims in (5.5.1) forusual preschemes.

We leave it to the reader to state, for the same type of formal preschemes and morphisms as in (10.15.3),the propositions corresponding to (5.5.5), (5.5.9), and (5.5.10) (by replacing “ane open subset” by “formal I | 213ane open subset satisfying condition (b) of (10.6.3)”).

A similar argument also shows that every Noetherian formal ane scheme is separated, which justifies theterminology.

Proposition (10.15.4). — Let S be a locally Noetherian formal prescheme, and X and Y locally Noetherian formalS-preschemes such that X or Y is of nite type over S (so that X×S Y is locally Noetherian) and such that Y is separatedover S. Let f : X→Y be an S-morphism; then the graph morphism Γf (1X, f )S : X→ X×S Y is a closed immersion.

Proof. We can assume that S is the inductive limit of a sequence (Sn) of locally Noetherian preschemes,X (resp. Y) the inductive limit of a sequence (Xn) (resp. (Yn)) of Sn-preschemes, and f the inductive limitof a sequence ( fn : Xn→ Yn) of Sn-morphisms; then X×S Y is the inductive limit of the sequence (Xn ×Sn

Yn),and Γf the inductive limit of the sequence (Γfn

) (10.7.4); by hypothesis, Y0 is separated over S0 (10.15.2), sothe space Γf0(X0) is a closed subspace of X0 ×S0

Y0; since the underlying spaces of X×S Y (resp. Γf (X)) andX0 ×S0

Y0 (resp. Γf0(X0)) are identical, we already see that Γf (X) is a closed subspace of X×S Y. Now note that,when (U , V ) runs over the set of pairs consisting of a Noetherian formal ane open subset U (resp. V ) of X(resp Y) such that f (U) ⊂ V , the open subsets U ×S V form a cover of Γf (X) in X×S Y, and, if fU : U → V isthe restriction of f to U , then ΓfU

: U → U ×S V is the restriction of Γf to U . If we show that ΓfUis a closed

immersion, then Γf will be a closed immersion (10.14.4), or, in other words, we are led to consider the case whereS= Spf(A), X= Spf(B), and Y= Spf(C) are ane (with A, B, and C Noetherian adics), with f correspondingto a continuous A-homomorphism φ : C → B; then Γf corresponds to the unique continuous homomorphismω : Bb⊗AC → B which, when composed with the canonical homomorphisms B→ Bb⊗AC and C → Bb⊗AC , gives theidentity and φ (respectively). But it is clear that ω is surjective, whence our claim.

Corollary (10.15.5). — Let S be a locally Noetherian formal prescheme, and X an S-prescheme of nite type; for X tobe separated over S, it is necessary and sucient for the diagonal morphism X→ X×S X to be a closed immersion.

Proposition (10.15.6). — A closed immersion j : Y → X of locally Noetherian formal preschemes is a separatedmorphism.

10. FORMAL SCHEMES 170

Proof. With the notation of (10.14.2), j0 : Y0→ X0 is a closed immersion, thus a separated morphism, andso it suces to apply (10.15.2).

Proposition (10.15.7). — Let X be a locally Noetherian (usual) prescheme, X ′ a closed subset of X , and bX = X/X ′ . ForbX to be separated, it is necessary and sucient that bXred be separated, and it is sucient that X be separated.

Proof. With the notation of (10.8.5), for bX to be separated, it is necessary and sucient for X ′0 to beseparated (10.15.2), and since bXred = (X ′0)red, it is equivalent to ask for bXred to be separated (5.5.1, vi).

CHAPTER II

Elementary global study of some classes of morphisms (EGA II)

Summary

§1. Ane morphisms.§2. Homogeneous prime spectra.§3. Homogeneous prime spectrum of a sheaf of graded algebras.§4. Projective bundles; ample sheaves.§5. Quasi-ane morphisms; quasi-projective morphisms; proper morphisms; projective morphisms.§6. Integral morphisms and finite morphisms.§7. Valuative criteria.§8. Blowup schemes; projective cones; projective closure.

II | 5The various classes of morphisms studied in this chapter are used extensively in cohomological methods;

further study, using these methods, will be done in Chapter III, where we use especially §§2, 4, and 5 of Chapter II.Section §8 can be omitted on a first reading: it gives some supplements to the formalism developed in §§1 and 3,reducing to easy applications of this formalism, and we will use it less consistently than the other results of thischapter.

§1. Ane morphisms

1.1. S-preschemes and OS -algebras.

(1.1.1). Let S be a prescheme, X an S-prescheme, and f : X → S its structure morphism. We know (0, 4.2.4)that the direct image f∗(OX ) is an OS -algebra, which we denote A (X ) when there is little chance of confusion; if II | 6U is an open subset of S, then we have

A ( f −1(U)) =A (X )|U .

Similarly, for every OX -module F (resp. every OX -algebra B), we write A (F ) (resp. A (B)) for the directimage f∗(F ) (resp. f∗(B)) which is an A (X )-module (resp. an A (X )-algebra) and not only an OS -module (resp.an OS -algebra).

(1.1.2). Let Y be a second S-prescheme, g : Y → S its structure morphism, and h : X → Y an S-morphism; wethen have the commutative diagram

(1.1.2.1) X h //

f

Y

gS.

We have by definition h= (ψ,θ), where θ : OY → h∗(OX ) = ψ∗(OX ) is a homomorphism of sheaves of rings;we induce (0, 4.2.2) a homomorphism of OS -algebras g∗(θ) : g∗(OY )→ g∗(h∗(OX )) = f∗(OX ), in other words, ahomomorphism of OS -algebrasA (Y )→A (X ), which we denote byA (h). If h′ : Y → Z is a second S-morphism,then it is immediate that A (h′ h) =A (h) A (h′). We havve thus define a contravariant functor A (X ) from thecategory of S-preschemes to the category of OS -algebras.

Now let F be an OX -module, G an OY -module, and u : G → F an h-morphism, that is (0, 4.4.1) ahomomorphism of OY -modules G → h∗(F ). Then g∗(u) : g∗(G ) → g∗(h∗(F )) = f∗(F ) is a homomorphismA (G ) → A (F ) of OS -modules, which we denote by A (u); in addition, the pair (A (h),A (u)) form a di-homomorphism from the A (Y )-module A (G ) to the A (X )-module A (F ).

171

1. AFFINE MORPHISMS 172

(1.1.3). If we fix the prescheme S, then we can consider the pairs (X ,F ), where X is an S-prescheme and F isan OX -module, as forming a category, by defining a morphism (X ,F )→ (Y,G ) as a pair (h, u), where h : X → Y isan S-morphism and u : G →F is an h-morphism. We can then say that (A (X ),A (F )) is a contravariant functorwith values in the category whose objects are pairs consisting of an OS -algebra and a module over that algebra,and the morphisms are the di-homomorphisms.

1.2. Ane preschemes over a prescheme.

Denition (1.2.1). — Let X be an S-prescheme, f : X → S its structure morphism. We say that X is ane overS if there exists a cover (Sα) of S by ane open sets such that for all α, the induced prescheme on X by theopen set f −1(Sα) is ane.

Example (1.2.2). — Every closed subprescheme of S is an ane S-prescheme over S ((I, 4.2.3) and (I, 4.2.4)).

Remark (1.2.3). — An ane prescheme X over S is not necessarily an ane scheme, as the example X = Sshows (1.2.2). On the other hand, if an ane scheme X is an S-prescheme, then X is not necessarily ane overS (see Example (1.3.3)). However, remember that if S is a scheme, then every S-prescheme which is an ane II | 7scheme is ane over S (I, 5.5.10).

Proposition (1.2.4). — Every S-prescheme which is ane over S is separated over S (in other words, it is an S-scheme).

Proof. This follows immediately from (I, 5.5.5) and (I, 5.5.8).

Proposition (1.2.5). — Let X be an S-scheme ane over S, f : X → S its structure morphism. For every open U ⊂ S,f −1(U) is ane over U .

Proof. By Definition (1.2.1), we can reduce to the case where S = Spec(A) and X = Spec(B) are ane; thenf = (aφ , eφ ), where φ : A→ B is a homomorphism. As the D(g) for g ∈ A form a basis for S, we reduce to thecase where U = D(g); but we then know that f −1(U) = D(φ (g)) (I, 1.2.2.2), hence the proposition.

Proposition (1.2.6). — Let X be an S-scheme ane over S, f : X → S its structure morphism. For every quasi-coherentOX -module F , f∗(F ) is a quasi-coherent OS -module.

Proof. Taking into account Proposition (1.2.4), this follows from (I, 9.2.2, a).

In particular, the OS -algebra A (X ) = f∗(OX ) is quasi-coherent.

Proposition (1.2.7). — Let X be an S-scheme ane over S. For every S-prescheme Y , the map h 7→ A (h) from the setHomS(Y, X ) to the set Hom(A (X ),A (Y )) (1.1.2) is bijective.

Proof. Let f : X → S and g : Y → S be the structure morphisms. First, suppose that S = Spec(A) andX = Spec(B) are ane; we must prove that for every homomorphism ω : f∗(OX )→ g∗(OY ) of OS -algebras, thereexists a unique S-morphism h : Y → X such that A (h) = ω. By definition, for every open U ⊂ S, ω defines ahomomorphism ωU = Γ(U ,ω) : Γ( f −1(U),OX )→ Γ(g−1(U),OY ) of Γ(U ,OS)-algebras. In particular, for U = S,this gives a homomorphism φ : Γ(X ,OX )→ Γ(Y,OY ) of Γ(S,OS)-algebras, to which corresponds a well-definedS-morphism h : Y → X , since X is ane (I, 2.2.4). It remains to prove that A (h) = ω, in other words, forevery open set U of a basis for S, ωU coincides with the homomorphism of algebras φU corresponding tothe S-morphism g−1(U)→ f −1(U), a restiction of h. We can reduce to the case where U = D(λ), with λ ∈ S;then, if f = (aρ, eρ), where ρ : A→ B is a ring homomorphism, we have f −1(U) = D(µ), where µ = ρ(λ), andΓ( f −1(U),OX ) is the ring of fractions Bµ; the diagram

Bφ //

Γ(Y,OY )

Bµ

φU// Γ(g−1(U),OY )

is commutative, and so is the analogous diagram where φU is replaced by ωU ; the equality φU = ωU then followsfrom the universal property of rings of fractions (0, 1.2.4).

We now pass to the general case; let (Sα) be a cover of S by ane open sets such that the f −1(Sα) are ane. II | 8Then every homomorphism ω :A (X )→A (Y ) of OS -algebras gives by restriction a family of homomorphisms

ωα :A ( f −1(Sα)) −→A (g−1(Sα))

of OSα -algebras, hence a family of Sα-morphisms hα : g−1(Sα)→ f −1(Sα) by the above. It remains to see that forevery ane open set of a basis for Sα ∩ Sβ , the restriction of hα and hβ to g−1(U) coincide, which is evident

1. AFFINE MORPHISMS 173

since by the above, these restrictions both correspond to the homomorphism A (X )|U →A (Y )|U , a restrictionof ω.

Corollary (1.2.8). — Let X and Y be two S-schemes which are ane over S. For an S-morphism h : Y → X to be anisomorphism, it is necessary and sucient for A (h) :A (X )→A (Y ) to be an isomorphism.

Proof. This follows immediately from Proposition (1.2.7) and from the functorial nature of A (X ).

1.3. Ane preschemes over S associated to an OS -algebra.

Proposition (1.3.1). — Let S be a prescheme. For every quasi-coherent OS -algebra B , there exists a prescheme X aneover S, dened up to unique S-isomorphism, such that A (X ) =B .

Proof. The uniqueness follows from Corollary (1.2.8); we prove the existence of X . For every ane openU ⊂ S, let XU be the prescheme Spec(Γ(U ,B)); as Γ(U ,B) is a Γ(U ,OS)-algebras, XU is an S-prescheme(I, 1.6.1). In addition, as B is quasi-coherent, the OS -algebra A (XU) canonically identifies with B|U ((I, 1.3.7),(I, 1.3.13), (I, 1.6.3)). Let V be a second ane open subset of S, and let XU ,V be the prescheme induced by XU

on f −1U (U ∩ V ), where fU denotes the structure morphism XU → S; XU ,V and XV,U are ane over U ∩ V (1.2.5),

and by definition A (XU ,V ) and A (XV,U) canonically identify with B|(U ∩ V ). Hence there is (1.2.8) a canonicalS-isomorphism θU ,V : XU ,V → XU ,V ; in addition, if W is a third ane open subset of S, and if θ ′U ,V , θ

′V,W , and θ ′U ,W

are the restrictions of θU ,V , θV,W , and θU ,W to the inverse images of U ∩ V ∩W in XV , XW , and XW respectivelyunder the structure morphisms, then we have θ ′U ,V θ

′V,W = θ

′U ,W . As a result, there exists a prescheme X , a cover

(TU) of X by ane open sets, and for every U an isomorphism φU : XU → TU , such that φU maps f −1U (U ∩ V ) to

TU ∩ TV , and we have θU ,V =φ−1U φV (I, 2.3.1). The morphism gU = fU φ−1

U makes TU an S-prescheme, andthe morphisms gU and gV coincide on TU ∩ TV , hence X is an S-prescheme. In addition, it is clear by definitionthat X is ane over S and that A (TU) =B|U , hence A (X ) =B .

We say that the S-scheme X defined in this way is associated to the OS -algebra B , or is the spectrum of B , andwe denote it by Spec(B).

Corollary (1.3.2). — Let X be a prescheme ane over S, f : X → S the structure morphism. For every ane openU ⊂ S, the induced prescheme on f −1(U) is the ane scheme with ring Γ(U ,A (X )).

II | 9Proof. As we can suppose that X is associated to an OS -algebra by Propositions (1.2.6) and (1.3.1), the

corollary follows from the construction of X described in Proposition (1.3.1).

Example (1.3.3). — Let S be the ane plane over a field K , where the point 0 has been doubled (I, 5.5.11);with the notation of (I, 5.5.11), S is the union of two ane open sets Y1 and Y2; if f is the open immersionY1→ S, then f −1(Y2) = Y1 ∩ Y2 is not an ane open set in Y1 (loc. cit.), hence we have an example of an anescheme which is not ane over S.

Corollary (1.3.4). — Let S be an ane scheme; for an S-prescheme X to be ane over S, it is necessary and sucientfor X to be an ane scheme.

Corollary (1.3.5). — Let X be a prescheme ane over a prescheme S, and let Y be an X -prescheme. For Y to be aneover S, it is necessary and sucient for Y to be ane over X .

Proof. We immediately reduce to the case where S is an ane scheme, and then we can reduce to the casewhere X is an ane scheme (1.3.4); the two conditions of the statement then give that Y is an ane scheme(1.3.4).

(1.3.6). Let X be a prescheme ane over S. To define a prescheme Y ane over X , it is equivalent, by Corol-lary (1.3.5), to give a prescheme Y ane over S, and an S-morphism g : Y → X ; in other words (Proposition (1.3.1)and (1.2.7)), it is equivalent to give a quasi-coherent OS -algebra B and a homomorphism A (X )→B of OS -algebras (which can be considered as defining on B an A (X )-algebra structure). If f : X → S is the structuremorphism, then we have B = f∗(g∗(OY )).

Corollary (1.3.7). — Let X be a prescheme ane over S; for X to be of nite type over S, it is necessary and sucientfor the quasi-coherent OS -algebra A (X ) to be of nite type (I, 9.6.2).

Proof. By definition (I, 9.6.2), we can reduce to the case where S is ane; then X is an ane scheme (1.3.4),and if S = Spec(A), X = Spec(B), then A (X ) is the OS -algebra eB; as Γ(U , eB) = B, the corollary follows from(I, 9.6.2) and (I, 6.3.3).

1. AFFINE MORPHISMS 174

Corollary (1.3.8). — Let X be a prescheme ane over S; for X to be reduced, it is necessary and sucient for thequasi-coherent OX -algebra A (X ) to be reduced (0, 4.1.4).

Proof. The question is local on S; by Corollary (1.3.2), the corollary follows from (I, 5.1.1) and (I, 5.1.4).

1.4. Quasi-coherent sheaves over a prescheme ane over S.

Proposition (1.4.1). — Let X be a prescheme ane over S, Y an S-prescheme, and F (resp. G ) a quasi-coherentOX -module (resp. an OY -module). Then the map (h, u) 7→ (A (h),A (u)) from the set of morphism (Y,G )→ (X ,F ) to theset of di-homomorphisms (A (X ),A (F ))→ (A (Y ),A (G )) ((1.1.2) and (1.1.3)) is bijective.

Proof. The proof follows exactly as that of Proposition (1.2.7) by using (I, 2.2.5) and (I, 2.2.4), and thedetails are left to the reader.

Corollary (1.4.2). — If, in addition to the hypotheses of Proposition (1.4.1), we suppose that Y is ane over S, thenfor (h, u) to be an isomorphism, it is necessary and sucient for (A (h),A (u)) to be a di-isomorphism.

II | 10Proposition (1.4.3). — For every pair (B ,M ) consisting of a quasi-coherent OS -algebra B and a quasi-coherentB -module M (considered as an OS -module or as a B -module, which are equivalent (I, 9.6.1)), there exists apair (X ,F ) consisting of a prescheme X ane over S and of a quasi-coherent OX -module F , such that A (X ) =B andA (F ) =M ; in addition, this couple is determined up to unique isomorphism.

Proof. The uniqueness follows from Proposition (1.4.1) and Corollary (1.4.2); the existence is proved as inProposition (1.3.1), and we leave the details to the reader.

We denote by ÝM the OX -module F , and we say that it is associated to the quasi-coherent B -moduleM ; forevery ane open U ⊂ S,M|p−1(U) (where p is the structure morphism X → S) is canonically isomorphic to(Γ(U ,M ))∼.

Corollary (1.4.4). — On the category of quasi-coherent B -modules, ÝM is an additive covariant exact functor inM ,which commutes with inductive limitd and direct sums.

Proof. We immediately reduce to the case where S is ane, and the corollary then follows from (I, 1.3.5),(I, 1.3.9), and (I, 1.3.11).

Corollary (1.4.5). — Under the hypotheses of Proposition (1.4.3), for ÝM to be an OX -module of nite type, it is necessaryand sucient forM to be a B -module of nite type.

Proof. We immediately reduce to the case where S = Spec(A) is an ane scheme. Then B = eB, where B isan A-algebra of finite type (I, 9.6.2), andM = eM , where M is a B-module (I, 1.3.13); over the prescheme X , OX isassociated to the ring B and ÝM to the B-module M ; for ÝM to be of finite type, it is therefore necessary andsucient for M to be of finite type (I, 1.3.13), hence our assertion.

Proposition (1.4.6). — Let Y be a prescheme ane over S, X and X ′ two preschemes ane over Y (hence also overS (1.3.5)). Let B =A (Y ), A =A (X ), and A ′ =A (X ′). Then X ×Y X ′ is ane over Y (thus also over S), andA (X ×Y X ′) canonically identies with A ⊗B A ′.

Proof. By (I, 9.6.1),A ⊗BA ′ is a quasi-coherentB -algebra, thus also a quasi-coherent OS -algebra (I, 9.6.1);let Z be the spectrum ofA⊗BA ′ (1.3.1). The canonicalB -homomorphismsA →A⊗BA ′ andA ′→A⊗BA ′correspond (1.2.7) to Y -morphisms Z → X and p′ : Z → X ′. To see that the triple (Z , p, p′) is a product X ×Y X ′,we can reduce to the case where S is an ane scheme with ring C (I, 3.2.6.4). But then Y , X , and X ′ are aneschemes (1.3.4) whose rings B, A, and A′ are C -algebras such that B = eB, A = eA, and A ′ = eA′. We then know(I, 1.3.13) thatA ⊗BA ′ canonically identifies with the OS -algebra (A⊗B A′)∼, hence the ring A(Z) identifies withA⊗B A′ and the morphisms p and p′ correspond to the canonical homomorphisms A→ A⊗B A′ and A′→ A⊗B A′.The proposition then follows from (I, 3.2.2).

Corollary (1.4.7). — LetF (resp.F ′) be a quasi-coherent OX -module (resp. OX ′ -module); thenA (F⊗YF ′) canonicallyidenties with A (F )⊗A (Y )A (F ′).

Proof. We know that F ⊗Y F ′ is quasi-coherent over X ×Y X ′ (I, 9.1.2). Let g : Y → S, f : X → Y , andf ′ : X ′→ Y be the structure morphisms, such that the structure morphism h : Z → S is equal to g f p and to II | 11g f ′ p′. We define a canonical homomorphism

A (F )⊗A (Y )A (F ′) −→A (F ⊗Y F ′)

1. AFFINE MORPHISMS 175

in the following way: for every open U ⊂ S, we have canonical homomorphisms Γ( f −1(g−1(U)),F ) →Γ(h−1(U), p∗(F )) and Γ( f ′−1(g−1(U)),F ′) → Γ(h−1(U), p′∗(F ′)) (0, 4.4.3), thus we obtain a canonical ho-momorphism

Γ( f −1(g−1(U)),F )⊗Γ(g−1(U),OY ) Γ( f′−1(g−1(U)),F ′) −→ Γ(h−1(U), p∗(F ))⊗Γ(h−1(U),OZ ) Γ(h

−1(U), p′∗(F ′)).

To see that we have defined an isomorphism of A (Z)-modules, we can reduce to the case where S (and as aresult X , X ′, Y , and X ×Y X ′) are ane scheme, and (with the notation of Proposition (1.4.6)), F = eM , F ′ =ÝM ′,where M (resp. M ′) is an A-module (resp. an A′-module). Then F ⊗Y F ′ identifies with the sheaf on X ×Y X ′

associated to the (A⊗B A′)-module M ⊗B M ′ (I, 9.1.3), and the corollary follows from the canonical identificationof the OS -modules (M ⊗B M ′)∼ and eM ⊗

eBÝM ′ (where M , M ′, and B are considered as C -modules) ((I, 1.3.12)

and (I, 1.6.3)).

If we apply Corollary (1.4.7) in particular to the case where X = Y and F ′ = OX ′ , then we see that theA ′-module A ( f ′∗(F )) identifies with A (F )⊗B A ′.

(1.4.8). In particular, when X = X ′ = Y (X being ane over S), we see that if F and G are two quasi-coherentOX -modules, then we have

(1.4.8.1) A (F ⊗OXG ) =A (F )⊗A (X )A (G )

up to canonical functorial isomorphism. If in addition F admits a finite presentation, then it follows from(I, 1.6.3) and (I, 1.3.12) that

(1.4.8.2) A (HomX (F ,G )) =HomA (X )(A (F ),A (G ))up to canonical isomorphism.

Remark (1.4.9). — If X and X ′ are two preschemes ane over S, then the sum X t X ′ is also ane over S, asthe sum of two ane schemes is an ane scheme.

Proposition (1.4.10). — Let S be a prescheme,B a quasi-coherent OS -algebra, and X = Spec(B). For a quasi-coherentsheaf of ideals J of B , eJ is quasi-coherent sheaf of ideals of OX , and the closed subprescheme Y of X dened by eJ iscanonically isomorphic to Spec(B/J ).

Proof. It follows immediately from (I, 4.2.3) that Y is ane over S; by Proposition (1.3.1), we reduce to thecase where S is ane, and the proposition then follows immediately from (I, 4.1.2).

We can also express the result of Proposition (1.4.10) by saying that if h :B →B ′ is a surjective homomor-phism of quasi-coherent OS -algebras, A (h) : Spec(B ′)→ Spec(B) is a closed immersion.

II | 12Proposition (1.4.11). — Let S be a prescheme, B a quasi-coherent OS -algebra, and X = Spec(B). For every quasi-coherent sheaf of ideals K of OS , we have (denoting by f the structure morphism X → S) f ∗(K )OX = (KB)∼ up tocanonical isomorphism.

Proof. The question being local on S, we can reduce to the case where S = Spec(A) is ane, and in thiscase the proposition is none other than (I, 1.6.9).

1.5. Change of base prescheme.

Proposition (1.5.1). — Let X be a prescheme ane over S. For every extension g : S′ → S of the base prescheme,X ′ = X(S′) = X ×S S′ is ane over S′.

Proof. If f ′ is the projection X ′→ S′, then it suces to prove that f ′−1(U ′) is an ane open set for everyane open subset U ′ of S′ such that g(U ′) is contained in an ane open subset U of S (1.2.1); we can thusreduce to the case where S and S′ are ane, and it suces to prove that X ′ is then an ane scheme (1.3.4). Butthen (1.3.4) X is an ane scheme, and if A, A′, and B are the rings of S, S′, and X respectively, then we knowthat X ′ is the ane scheme with ring A′ ⊗A B (I, 3.2.2).

Corollary (1.5.2). — Under the hypotheses of Proposition (1.5.1), let f : X → S be the structure morphism, f ′ : X ′→ S′

and g ′ : X ′→ X the projections, such that the diagram

X

f

X ′g ′oo

f ′

S S′

goo

1. AFFINE MORPHISMS 176

is commutative. For every quasi-coherent OX -module F , there exists a canonical isomorphism of OS′ -modules

(1.5.2.1) u : g∗( f∗(F ))' f ′∗ (g′∗(F )).

In particular, there exists a canonical isomorphism from A (X ′) to g∗(A (X )).

Proof. To define u, it suces to define a homomorphism

v : f∗(F ) −→ g∗( f′∗ (g

′∗(F ))) = f∗(g′∗(g

′∗(F )))

and to set u = v] (0, 4.4.3). We take v = f∗(ρ), where ρ is the canonical homomorphism F → g ′∗(g′∗(F ))

(0, 4.4.3). To prove that u is an isomorphism, we can reduce to the case where S and S′, hence X and X ′, areane; with the notation of Proposition (1.5.1), we then have F = eM , where M is a B-module. We then noteimmediately that g∗( f∗(F )) and f ′∗ (g

′∗(F )) are both equal to the OS′ -module associated to the A′-module A′⊗A M(where M is considered as an A-module), and that u is the homomorphism associated to the identity ((I, 1.6.3),(I, 1.6.5), (I, 1.6.7)).

Remark (1.5.3). — We do not have that Corollary (1.5.2) remains true when X is not assumed ane over S,even when S′ = Spec(k(s)) (s ∈ S) and S′→ S is the canonical morphism (I, 2.4.5)—in which case X ′ is noneother than the bre f −1(s) (I, 3.6.2). In other words, when X is not ane over S, the operation “direct image of II | 13quasi-coherent sheaves” does not commute with the operation of “passing to fibres”. However, we will see inChapter III (III, 4.2.4) a result in this sense, of an “asymptotic” nature, valid for coherent sheaves on X when fis proper (5.4) and S is Noetherian.

Corollary (1.5.4). — For every prescheme X ane over S and every s ∈ S, the bre f −1(s) (where f denoted thestructure morphism X → S) is an ane scheme.

Proof. It suces to apply Proposition (1.5.1) with S′ = Spec(k(s)) and to use Corollary (1.3.4).

Corollary (1.5.5). — Let X be an S-prescheme, S′ a prescheme ane over S; then X ′ = X(S′) is a prescheme aneover X . In addition, if f : X → S is the structure morphism, then there is a canonical isomorphism of OX -algebrasA (X ′)' f ∗(A (S′)), and for every quasi-coherentA (S′)-moduleM , a canonical di-isomorphism f ∗(M )'A ( f ′∗( ÝM )),denoting by f ′ = f(S′) the structure morphism X ′→ S′.

Proof. It suces to swap the roles of X and S′ in (1.5.1) and (1.5.2).

(1.5.6). Now let S, S′ be two preschemes, q : S′→ S a morphism,B (resp. B ′) a quasi-coherent OS -algebra (resp.OS′ -algebra), u :B →B ′ a q-morphism (that is, a homomorphismB → q∗(B ′) of OS -algebras). If X = Spec(B),X ′ = Spec(B ′), then we canonically obtain a morphism

v = Spec(u) : X ′ −→ X

such that the diagram

(1.5.6.1) X ′ v′ //

X

S′

q // S

is commutative (the vertical arrows being the structure morphisms). Indeed, the data of u is equivalent to thatof a homomorphism of quasi-coheren OS′ -algebras u] : q∗(B)→B ′ (0, 4.4.3); this thus canonically defines anS′-morphism

w : Spec(B ′) −→ Spec(q∗(B))such that A (w) = u] (1.2.7). On the other hand, it follows from (1.5.2) that Spec(q∗(B)) canonically identifies

with X ×S S′; the morphism v is the composition X ′w−→ X ×S S′

p1−→ X of w with the first projection, and thecommutativity of (1.5.6.1) follows from the definitions. Let U (resp. U ′) be an ane open of S (resp. S′) suchthat q(U ′) ⊂ U , A= Γ(U ,OS), A′ = Γ(U ′,OS′) their rings, B = Γ(U ,B), B′ = Γ(U ′,B ′); the restriction of u to a(q|U ′)-morphism: B|U →B ′|U ′ corresponds to a di-homomorphism of algebras B→ B′; if V , V ′ are the inverseimages of U , U ′ in X , X ′ respectively, under the structure morphisms, then the morphism V ′→ V , the restrictionof v, corresponds (I, 1.7.3) to the above di-homomorphism.

(1.5.7). Under the same hypotheses as in (1.5.6), letM be a quasi-coherent B -module; there is then a canonicalisomorphism of OX ′ -modules

(1.5.7.1) v∗( ÝM )' (q∗(M )⊗q∗(B)B ′)∼.II | 14

1. AFFINE MORPHISMS 177

Indeed, the canonical isomorphism (1.5.2.1) gives a canonical isomorphism from p∗1( ÝM ) to the sheaf onSpec(q∗(B)) associated to the q∗(B)-module q∗(M ), and it then suces to apply (1.4.7).

1.6. Ane morphisms.

(1.6.1). We say that a morphism f : X → Y of preschemes is ane if it defines X as a prescheme ane over Y .The properties of preschemes ane over another translates as follows in this language:

Proposition (1.6.2). —

(i) A closed immersion is ane.(ii) The composition of two ane morphisms is ane.(iii) If f : X → Y is an ane S-morphism, then f(S′) : X(S′)→ Y(Y ′) is ane for every base change S′→ S.(iv) If f : X → Y and f ′ : X ′→ Y ′ are two ane S-morphisms, then

f ×S f ′ : X ×S X ′ −→ Y ×S Y ′

is ane.(v) If f : X → Y and g : Y → Z are two morphisms such that g f is ane and g is separated, then f is ane.(vi) If f is ane, then so if fred.

Proof. By (I, 5.5.12), it suces to prove (i), (ii), and (iii). But (i) is none other than Example (1.2.2), and(ii) is none other than Corollary (1.3.5); finally, (iii) follows from Proposition (1.5.1), since X(S′) identifies withthe product X ×Y Y(S′) (I, 3.3.11).

Corollary (1.6.3). — If X is an ane scheme and Y is a scheme, then every morphism f : X → Y is ane.

Proposition (1.6.4). — Let Y be a locally Noetherian prescheme, f : X → Y a morphism of nite type. For f to beane, it is necessary and sucient for fred to be.

Proof. By (1.6.2, vi), we see only need to prove the suciency of the condition. It suces to prove that if Yis ane and Noetherian, then X is ane; but Yred is then ane, so the same is true for Xred by hypothesis. NowX is Noetherian, so the conclusion follows from (I, 6.1.7).

1.7. Vector bundle associated to a sheaf of modules.

(1.7.1). Let A be a ring, E an A-module. Recall that we call the symmetric algebra on E and denote by S(E)(or SA(E)) the quotient algebra of the tensor algebra T(E) by the two-sided ideal generated by the elementsx ⊗ y − y ⊗ x , where x and y vary over E. The algebra S(E) is characterized by the following universal property:if σ is the canonical map E→ S(E) (obtained by composing E→ T(E) with the canonical map T(E)→ S(E)),then every A-linear map E→ B, where B is a commutative A-algebra, factors uniquely as E

σ−→ S(E)

g−→ B, where g

is an A-homomorphism of algebras. We immediately deduce from this characterization that for two A-modules Eand F , we have

S(E ⊕ F) = S(E)⊕ S(F)

up to canonical isomorphism; in addition, S(E) is a covariant functor in E, from the category of A-modules to II | 15that of commutative A-algebras; finally, the above characterization also shows that if E = lim−→ Eλ , then we haveS(E) = lim−→S(Eλ) up to canonical isomorphism. By abuse of language, a product σ(x1)σ(x2) · · ·σ(xn), wherex i ∈ E, is often denoted by x1 x2 · · · xn if no confusion follows. The algebra S(E) is graded, Sn(E) being the set oflinear combinations of n elements of E (n¾ 0); the algebra S(A) is canonically isomorphic to the polynomialalgebra A[T] is an indeterminate, and the algebra S(An) with the polynomial algebra in n indeterminatesA[T1, . . . , Tn].

(1.7.2). Let φ be a ring homomorphism A→ B. If F is a B-module, then the canonical map F → S(F) gives acanonical map F[φ ] → S(F)[φ ], which thus factors as F[φ ] → S(F[φ ])→ S(F)[φ ]; the canonical homomorphismS(F[φ ])→ S(F)[φ ] is surjective, but not necessarily bijective. If E is an A-module, then every di-homomorphismE→ F (that is to say, every A-homomorphism E→ F[φ ]) thus canonically gives an A-homomorphism of algebrasS(E)→ S(F[φ ])→ S(F)[φ ], that is to say a di-homomorphism of algebras S(E)→ S(F).

With the same notations, for every A-module E, S(E ⊗A B) canonically identifies with the algebra S(E)⊗A B;this follows immediately from the universal property of S(E) (1.7.1).

1. AFFINE MORPHISMS 178

(1.7.3). Let R be a multiplicative subset of the ring A; apply (1.7.2) to the ring B = R−1A, and remembering thatR−1E = E ⊗A R−1A, we see that we have S(R−1E) = R−1S(E) up to canonical isomorphism. In addition, if R′ ⊃ Ris a second multiplicative subset of A, then the diagram

R−1E //

R′−1E

S(R−1E) // S(R′−1E)

is commutative.

(1.7.4). Now let (S,A ) be a ringed space, and let E be a A -module over S. If to any open U ⊂ S we associatethe Γ(U ,A )-module S(Γ(U ,E )), then we define (see the functorial nature of S(E) (1.7.2)) a presheaf of algebras;we say that the associated sheaf, which we denote by S(E ) or SA (E ) is the symmetricA -algebra on the A -moduleE . It follows immediately from (1.7.1) that S(E ) is a solution to a universal problem: every homomorphism ofA -modules E →B , where B is an A -algebra, factors uniquely as E → S(E )→B , the second arrow being ahomomorphism of A -algebras. There is thus a bijective correspondence between homomorphisms E →B ofA -modules and homomorphisms S(E )→B of A -algebras. In particular, every homomorphism u : E →F ofA -modules defines a homomorphism S(u) : S(E )→ S(F ) of A -algebras, and S(E ) is thus a covariant functorin E . II | 16

By (1.7.2) and the commutativity of S with inductive limts, we have (S(E ))x = S(Ex) for every point x ∈ S. IfE , F are twoA -modules, then S(E⊕F ) canonically identifies with S(E )⊗AS(F ), as we see for the correspondingpresheaves.

We also note that S(E ) is a graded A -algebra, the infinite direct sum of the Sn(E ), where the A -moduleSn(E ) is the sheaf associated to the presheaf U 7→ Sn(Γ(U ,E )). If we take in particular E =A , then we see thatSA (A ) identifies with A [T] =A ⊗Z Z[T] (T indeterminate, Z being considered as a simple sheaf).

(1.7.5). Let (T,B) be a second ringed space, f a morphism (S,A )→ (T,B). IfF is aB -module, then S( f ∗(F ))canonically identifies with f ∗(S(F )); indeed, if f = (ψ,θ), then by definition (0, 4.3.1),

S( f ∗(F )) = S(ψ∗(F )⊗ψ∗(B)A ) = S(ψ∗(F ))⊗ψ∗(B)A

(1.7.2); for every open U of S and every section h of S(ψ∗(F )) over U , h coincides, in a neighborhood V ofevery point s ∈ U , with an element of S(Γ(V,ψ∗(F ))); if we refer to the definition of ψ∗(F ) (0, 3.7.1) and takeinto account that every element of S(E) for a module E is a linear combination of a finite number of products ofelements of E, then we see that there is a neighborhood W of ψ(s) in T , a section h′ of S(F ) over W , and aneighborhood V ′ ⊂ V ∩ψ−1(W ) of s such that h coincises with t 7→ h′(ψ(t)) over V ′; hence out assertion.

Proposition (1.7.6). — Let A be a ring, S = Spec(A) its prime spectrum, E = eM the OS -module associated to anA-module M ; then the OS -algebra S(E ) is associated to the A-algebra S(M).

Proof. For every f ∈ A, S(M f ) = (S(M)) f (1.7.3), and the proposition thus folloes from the definition(I, 1.3.4).

Corollary (1.7.7). — If S is a prescheme, E a quasi-coherent OS -module, then the OS -algebra S(E ) is quasi-coherent. Ifin addition E is of nite type, then each of the OS -modules Sn(E ) is of nite type.

Proof. The first assertion is an immediate consequence of (1.7.6) and of (I, 1.4.1); the second followsfrom the fact that if E is an A-module of finite type, then Sn(E) is an A-module of finite type; we then apply(I, 1.3.13).

Denition (1.7.8). — Let E be a quasi-coherent OS -module. We call the vector bundle over S dened by E anddenote by V(E ) the spectrum (1.3.1) of the quasi-coherent OS -algebra S(E ).

By (1.2.7), for every S-prescheme X , there is a canonical bijective correspondence between the S-morphismsX → V(E ) and the homomorphisms of OS -algebras S(E )→A (X ), thus also between these S-morphisms and thehomomorphisms of OS -modules E →A (X ) = f∗(OX ) (where f is the structure morphism X → S). In particular:

(1.7.9). Take for X a subprescheme induced by S on an open U ⊂ S. Then the S-morphisms U → V(E ) arenone other that the U -sections (I, 2.5.5) of the U -prescheme induces by V(E ) on the open p−1(U) (where pis the structure morphism V(E )→ S). From what we have just seen, these U -sections bijectively correspondto homomorphisms of OS -modules E → j∗(OS |U) (where j is the canonical injection U → S), or equivalently II | 17(0, 4.4.3) with the (OS |U)-homomorphisms j∗(E ) = E|U →OS |U . In addition, it is immediate that the restriction

1. AFFINE MORPHISMS 179

to an open U ′ ⊂ U of an S-morphism U → V(E ) corresponds to the restriction to U ′ of the correspondinghomomorphism E|U →OS |U . We conclude that the sheaf of germs of S-sections of V(E ) canonically identifies withthe dual E∨ of E .

In particular, if we set X = U = S, then the zero homomorphism E → OS corrresponds to a canonicalS-section of V(E ), called the zero S-section (cf. (8.3.3)).

(1.7.10). Now take X to be the spectrum ξ of a field K ; the structure morphism f : X → S then corresponfsto a monomorphism k(s)→ K , where s = f (ξ ) (I, 2.4.6); the S-morphisms ξ → V(E ) are none other thanthe geometric points of V(E ) with values in the extension K of k(s) (I, 3.4.5), points which are localized at thepoints of p−1(s). The set of these points, which we can call the rational geometric bre over K of V(E ) over thepoint s, is identified by (1.7.8) with the set of homomorphisms of OS -modules E → f∗(OX ), or, equivalently(0, 4.4.3) with the set of homomorphisms of OX -modules f ∗(E )→OX = K . But we have by definition (0, 4.3.1)f ∗(E ) = Es ⊗Os

K = E s ⊗k(s) K , setting E s = Es/msEs; the geometric fibre of V(E ) rational over K over s thusidentifies with the dual of the K -vector space E s ⊗k(s) K ; if E s or K is of finite dimension over k(s), then this dualalso identifies with (E s)∨ ⊗k(s) K , denoting by (E s)∨ the dual of the k(s)-vector space E s.

Proposition (1.7.11). —

(i) V(E ) is a contravariant functor in E from the category of quasi-coherent OS -modules to the category of aneS-schemes.

(ii) If E is an OS -module of nite type, then V(E ) is of nite type over S.(iii) If E and F are two quasi-coherent OS -modules, then V(E ⊕F ) canonically identies with V(E )×S V(F ).(iv) Let g : S′ → S be a morphism; for every quasi-coherent OS -module E , V(g∗(E )) canonically identies with

V(E )(S′) = V(E )×S S′.(v) A surjective homomorphism E →F of quasi-coherent OS -modules corresponds to a closed immersion V(F )→

V(E ).

Proof. (i) is an immediate consequence of (1.2.7), taking into account that every homomorphism of OS -modules E → F canonically defines a homomorphism of OS -algebras S(E )→ S(F ). (ii) follows immediatelyfrom the definition (I, 6.3.1) and the fact that if E is an A-module of finite type, then S(E) is an A-algebra of finitetype. To prove (iii), it suces to start with the canonical isomorphism S(E ⊕F )' S(E )⊗OS

S(F ) (1.7.4) and toapply (1.4.6). Similarly, to prove (iv), it suces to start with the canonical isomorphism S(g∗(E ))' g∗(S(E ))(1.7.5) and to apply (1.5.2). Finally, to establish (v), it suces to remark that if the homomorphism E →F issurjective, then so is the corresponding homomorphism S(E )→ S(F ) of OS -algebras, and the conclusion followsfrom (1.4.10).

(1.7.12). Take in particular E = OS ; the prescheme V(OS) is the ane S-scheme, spectrum of the OS -algebra S(OS)which identifies with the OS -algebra OS[T] = OS ⊗Z Z[T] (T indeterminate); this is evident when S = Spec(Z), by II | 18virtue of (1.7.6), and we pass from there to the general case by considering the structure morphism S→ Spec(Z)and using (1.7.11, iv). Because of this result, we set V(OS) = S[T], and we thus have the formula

(1.7.12.1) S[T] = S ⊗Z Z[T].

The identification of the sheaf of germs of S-sections of S[T] with OS , already seen in (I, 3.3.15), here in amore general context, as a special case of (1.7.9).

(1.7.13). For every S-prescheme X , we have seen (1.7.8) that HomS(X , S[T]) canonically identifies with HomOS(OS ,A (X )),

which is canonically isomorphic to Γ(S,A (X )), and as a result is equipped with the structure of a ring; inaddition, to every S-morphism h : X → Y there corresponds a morphism Γ(A (h)) : Γ(S,A (Y ))→ Γ(S,A (X ))for the ring structures (1.1.2). When we equip HomS(X , S[T]) with a ring structure as defined, then we cansee that Hom(X , S[T]) can be considered as a contravariant functor in X , from the category of S-preschemes tothat of rings. On the other hand, HomS(X ,V(E )) is likewise identified with HomOS

(E ,A (X )) (where A (X ) isconsidered as an OS -module); as a result, we can canonically give a module structure on the ring HomS(X , S[T]),and we see as above that the pair

(HomS(X , S[T]), Hom(X ,V(E )))

is a contravariant functor in X , with values in the category whose elements are the pairs (A, M) consisting of aring A and an A-module M , the morphisms being di-homomorphisms.

We will interpret these facts by saying that S[T] is an S-scheme of rings and that V(E ) is an S-scheme of moduleson the S-scheme of rings S[T] (cf. Chapter 0, §8).

2. HOMOGENEOUS PRIME SPECTRA 180

(1.7.14). We will see that the structure of an S-scheme of modules defined on the S-scheme V(E ) allows us toreconstruct the OS -module E up to unique isomorphism: for this, we will show that E is canonically isomorphic to aOS -submodule of S(E ) =A (V(E )), defined by means of this structure. Indeed (1.7.4) the set HomOS

(S(E ),A (X ))of homomorphisms of OS -algebras canonical identifies with HomOS

(E ,A (X )), the set of homomorphisms ofOS -modules: if h and h′ are two elements of this latter set, si (1 ¶ i ¶ n) sections of E over an open U ⊂ S, t asection of A (X ) over U , then we have by definition

(h+ h′)(s1s2 · · · sn) =n∏

i=1

(h(si) + h′(si))

and

(t · h)(s1s2 · · · sn) = tnn∏

i=1

h(si).

This being so, if z is a section of S(E ) over U , then h 7→ h(z) is a map from HomS(X ,V(E )) = HomOS(S(E ),A (X ))

to Γ(U ,A (X )). We will show that E identifies with a submodule of S(E ) such that, for every open U ⊂ S, every II | 19section z of this OS -submodule of U , and every S-prescheme X , the map h 7→ h(z) from HomOS

(S(E )|U ,A (X )|U) toΓ(U ,A (X )) is a homomorphism of Γ(U ,A (X ))-modules.

It is immediate that E has this property; to show the converse, we can reduce to proving that whenS = Spec(A), E = eM , a section z of S(E ) over S that (for U = S) has the property stated above is necessarily asection of E ; we then have z =

∑∞n=0 zn, where zn ∈ Sn(M), and it is a question of proving that zn = 0 for n 6= 1. Set

B = S(M) and take for X the prescheme Spec(B[T]), where T is an indeterminate. The set HomOS(S(E ),A (X ))

identifies with the set of ring homomorphisms h : B→ B[T] (I, 1.3.13), and from what we saw above, we have(T · h)(z) =

∑∞n=0 T nh(zn): the hypothesis on z implies that we have

∑∞n=0 T nh(zn) = T ·

∑∞n=0 h(zn) for every

homomorphism h. In in particular we take for h the canonical injection, then∑∞

n=0 T nzn = T ·∑∞

n=0 zn, whichimplies the conclusion zn = 0 for n 6= 1.

Proposition (1.7.15). — Let Y be a prescheme whose underlying space is Noetherian, or a quasi-compact scheme. Everyane Y -scheme X of nite type over Y is Y -isomorphic to a closed Y -subscheme of the form V(E ), where E is a quasi-coherentOY -module of nite type.

Proof. The quasi-coherent OY -algebra A (X ) is of finite type (1.3.7). The hypotheses imply that A (X ) isgenerated by a quasi-coherent OY -submodule of finite type E (I, 9.6.5); by definition, this implies that the canonicalhomomorphism S(E )→A (X ) canonically extending the injection E →A (X ) is surjective; the conclusion thenfollows from (1.4.10).

§2. Homogeneous prime spectra

2.1. Generalities on graded rings and modules.

Notation (2.1.1). — Given a ring S graded in positive degrees, we denote by Sn the subset of S consisting ofhomogeneous elements of degree n (n¾ 0), by S+ the (direct) sum of the Sn for n> 0; we have 1 ∈ S0, S0 is asubring of S, S+ is a graded ideal of S, and S is the direct sum of S0 and S+. If M is a graded module over S (withpositive or negative degrees), we similarly denote by Mn the S0-module consisting of homogeneous elements ofM of degree n (with n ∈ Z).

For every integer d > 0, we denote by S(d) the direct sum of the Snd ; by considering the elements of Snd ashomogeneous of degree n, the Snd define on S(d) a graded ring structure.

For every integer k such that 0¶ k ¶ d − 1, we denote by M (d,k) the direct sum of the Mnd+k (n ∈ Z); this is II | 20a graded S(d)-module when we consider the elements of Mnd+k as homogeneous of degree n. We write M (d) inplace of M (d,0).

With the above notation, for every integer n (positive or negative), we denote by M(n) the graded S-moduledefined by (M(n))k = Mn+k for every k ∈ Z. In particular, S(n) will be a graded S-module such that (S(n))k = Sn+k,by agreeing to set Sn = 0 for n< 0. We say that a graded S-module M is free if it is isomorphic, considered as agraded module, to a direct sum of modules of the form S(n); as S(n) is a monogeneous S-module, generated bythe element 1 of S considered as an element of degree −n, it is equivalent to say that M admits a basis over Sconsisting of homogeneous elements.

We say that a graded S-module M admits a nite presentation if there exists an exact sequence P →Q→ M → 0,where P and Q are finite direct sums of modules of the form S(n) and the homomorphisms are of degree 0(cf. (2.1.2)).

2. HOMOGENEOUS PRIME SPECTRA 181

(2.1.2). Let M and N be two graded S-modules; we define on M⊗S N a graded S-module structure in the followingway. On the tensor product M ⊗Z N , we can define a graded Z-module structure (where Z is graded by Z0 = Z,Zn = 0 for n 6= 0) by setting (M ⊗Z N)q =

⊕

m+n=q Mm⊗Z Nn (as M and N are respectively direct sums of the Mmand the Nn, we know that we can canonically identify M ⊗Z N with the direct sum of all the Mm ⊗Z Nn). Thisbeing so, we have M ⊗S N = (M ⊗Z N)/P, where P is the Z-submodule of M ⊗Z N generated by the elements(xs)⊗ y − x ⊗ (s y) for x ∈ M , y ∈ N , s ∈ S; it is clear that P is a graded Z-submodule of M ⊗Z N , and we seeimmediately that we obtain a graded S-module structure on M ⊗S N by passing to the quotient.

For two graded S-modules M and N , recall that a homomorphism u : M → N of S-modules is said to beof degree k if u(M j) ⊂ N j+k for all j ∈ Z. If Hn denotes the set of all the homomorphisms of degree n fromM to N , then we denote by HomS(M , N) the (direct) sum of the Hn (n ∈ Z) in the S-module H of all thehomomorphisms (of S-modules) from M to N ; in general, HomS(M , N) is not equal to the later. However, wehave H = HomS(M , N) when M is of nite type; indeed, we can then suppose that M is generated by a finitenumber of homogeneous elements x i (1¶ i ¶ n), and every homomorphism u ∈ H can be written in a uniqueway as

∑

k∈Z uk, where for each k, uk(x i) is equal to the homogeneous component of degree k+ deg(x i) of u(x i)(1¶ i ¶ n), which implies that uk = 0 except for a finite number of indices; we have by definition that uk ∈ Hk,hence the conclusion.

We say that the elements of degree 0 of HomS(M , N) are the homomorphisms of graded S-modules. It is clearthat SmHn ⊂ Hm+n, so the Hn define on HomS(M , N) a graded S-module structure.

It follows immediately from these definitions that we have

(2.1.2.1) M(m)⊗S N(n) = (M ⊗S N)(m+ n),

(2.1.2.2) HomS(M(m), N(n)) = (HomS(M , N))(n−m),

for two graded S-modules M and N . II | 21Let S and S′ be two graded rings; a homomorphism of graded rings φ : S→ S′ is a homomorphism of rings

such that φ (Sn) ⊂ S′n for all n ∈ Z (in other words, φ must be a homomorphism of degree 0 of graded Z-modules).The data of such a homomorphism defines on S′ a graded S′-module structure; equipped with this structure andits graded ring structure, we say that S′ is a graded S′-algebra.

If M is also a graded S-module, then the tensor product M ⊗S S′ of graded S-modules is equipped in a naturalway with a graded S′-module structure, the grading being defined as above.

Lemma (2.1.3). — Let S be a ring graded in positive degrees. For a subset E of S+ consisting of homogeneous elementsto generate S+ as an S-module, it is necessary and sucient for E to generate S an an S0-algebra.

Proof. The condition is evidently sucient; we show that it is necessary. Let En (resp. En) be the set ofelements of E equal to n (resp. ¶ n); it suces to show, by induction on n> 0, that Sn is the S0-module generatedby the elements of degree n which are products of elements of En. This is evident for n = 1 by virtue of thehypothesis; the latter also shows that Sn =

∑n−1p=0 Sp En−p, and the induction argument is then immediate.

Corollary (2.1.4). — For S+ to be an ideal of nite type, it is necessary and sucient for S to be an S0-algebra of nitetype.

Proof. We can always assume that a finite system of generators of the S0-algebra S (resp. of the S-idealS+) consists of homogeneous elements, by replacing each of the generators considered by its homogeneouscomponents.

Corollary (2.1.5). — For S to be Noetherian, it is necessary and sucient for S0 to be Noetherian and for S to be anS0-algebra of nite type.

Proof. The condition is evidently sucient; it is necessary, since S0 is isomorphic to S/S+ and S+ must bean ideal of finite type (2.1.4).

Lemma (2.1.6). — Let S be a ring graded in positive degrees, which is an S0-algebra of nite type. Let M be a gradedS-module of nite type. Then:

(i) The Mn are S0-modules of nite type, and there exists an integer n0 such that Mn = 0 for n¶ n0.(ii) There exists an integer n1 and an integer h> 0 such that, for every integer n¾ n1, we have Mn+h = ShMn.(iii) For every pair of integers (d, k) such that d > 0, 0¶ k ¶ d − 1, M (d,k) is an S(d)-module of nite type.(iv) For every integer d > 0, S(d) is an S0-algebra of nite type.(v) There exists an integer h> 0 such that Smh = (Sh)m for all m> 0.(vi) For every integer n> 0, there exists an integer m0 such that Sm ⊂ Sn

+ for all m¾ m0.

2. HOMOGENEOUS PRIME SPECTRA 182

Proof. We can assume that S is generated (as an S0-algebra) by homogeneous elements fi , of degrees hi(1¶ i ¶ r), and M is generated (as an S-module) by homogeneous elements x j of degrees k j (1¶ j ¶ s). It isclear that Mn is formed by linear combinations, with coecients in S0, of elements f α1

1 · · · fαrr x j such that the αi II | 22

are integers ¾ 0 satisfying k j +∑

i αihi = n; for each j, there are only finitly many systems (αi) satisfying thisequation, since the hi are > 0, hence the first assertion of (i); the second is evident. On the other hand, let h bethe l.c.m. of the hi and set gi = f h/hi

i (1¶ i ¶ r) such that all the gi are of degree h; let zµ be the elements ofM of the form f α1

1 · · · fαrr x j with 0 ¶ αi < h/hi for 1 ¶ i ¶ r; there are finitely many of these elements, so let

n1 be the largest of their degrees. It is clear that for n¾ n1, every element of Mn+h is a linear combination ofthe zµ whose cocients are monomials of degree > 0 with respect to the gi , so we have Mn+h = ShMn, whichestablishes (ii). In a similar way, we see (for all d > 0) that an element of M (d,k) is a linear combinations, withcoeents in S0, of elements of the form gd f α1

1 · · · fαrr x j with 0¶ αi < d, g being a homogeneous element of S;

hence (iii); (iv) then follows from (iii) and from Lemma (2.1.3), by taking M = S+, since (S+)(d) = (S(d))+. Theassertion of (v) is deduced from (ii) by taking M = S. Finally, for a given n, there are finitely many systems (αi)such that αi ¾ 0 and

∑

i αi < n, so if m0 is the largest value of the sum∑

i αihi of these systems, then we haveSm ⊂ Sn

+ for m> m0, which proves (vi).

Corollary (2.1.7). — If S is Noetherian, then so is S(d) for every integer d > 0.

Proof. This follows from (2.1.5) and (2.1.6, iv).

(2.1.8). Let p be a graded prime ideal of the graded ring S; p is thus a direct sum of the subgroups pn = p∩ Sn.Suppose that p does not contain S+. Then if f ∈ S+ is not in p, the relation f n x ∈ p is equivalent to x ∈ p; inparticular, if f ∈ Sd (d > 0), for all x ∈ Sm−nd , then the relation f n x ∈ pm is equivalent to x ∈ pm−nd .

Proposition (2.1.9). — Let n0 be an integer > 0; for all n¾ n0, let pn be a subgroup of Sn. For there to exist a gradedprime ideal p of S not containing S+ and such that p ∩ Sn = pn for all n ¾ n0, it is necessary and sucient for thefollowing coniditions to be satised:

(1st) Smpn ⊂ pm+n for all m¾ 0 and all n¾ n0.(2nd) For m¾ n0, n¾ n0, f ∈ Sm, g ∈ Sn, the relation f g ∈ pm+n implies f ∈ pm or g ∈ pn.(3rd) pn 6= Sn for at least one n¾ n0.

In addition, the graded prime ideal p is then unique.

Proof. It is evident that the conditions (1st) and (2nd) are necessary. In addition, if p 6⊃ S+, then there existsat least one k > 0 such that p∩ Sk 6= Sk; if f ∈ Sk is not in p, the relation p∩ Sn = Sn implies p∩ Sn−mk = Sn−mkaccording to (2.1.8); therefore, if p ∩ Sn = Sn for a certain value of n, we would have p ⊃ S+ contrary to thehypothesis, which proves that (3rd) is necessary. Conversely, suppose that the conditions (1st), (2nd), and (3rd)are satisfied. Note that if for an integer d ¾ n0, f ∈ Sd is not in pd , then, if p exists, pm, for m< n0, is necessarilyequal to the set of the x ∈ Sm such that f r x ∈ pm+rd , except for a finite number of values of r. This alreadyproves that if p exists, then it is unique. It remains to show that if we define the pm for m< n0 by the previouscondition, then p=

∑∞n=0 pn is a prime ideal. First, note that by virtue of (2nd), for m¾ n0, pm is also defined as

the set of the x ∈ Sm such that f r x ∈ pm+rd except for a finite number of values of r. This being so, if g ∈ Sm, II | 23x ∈ pn, then we have f r g x ∈ pm+n+rd except for a finite number of values of r, so g x ∈ pm+n, which proves that pis an ideal of S. To establish that this ideal is prime, in other words that the ring S/p, graded by the subgroupsSn/pn, is an integral domain, it suces (by considering the components of higher degree of two elements ofS/p) to prove that if x ∈ Sm and y ∈ Sn are such that x 6∈ pm and y 6∈ pn, then x y 6∈ pm+n. If not, for r largeenough, we would have f 2r x y ∈ pm+n+2rd ; but we have f r y 6∈ pn+rd for all r > 0; it then follows from (2nd) that,except for a finite number of values of r, we have f r x ∈ pm+rd , and we conclude that x ∈ pm contrary to thehypothesis.

(2.1.10). We say that a subset J of S+ is an ideal of S+ if it is an ideal of S, and J is a graded prime ideal of S+if it is the intersection of S+ and a graded prime ideal of S not containing S+ (this prime ideal is also uniqueaccording to Proposition (2.1.9)). If J is an ideal of S+, the radical of J in S+ is the set of elements of S+ whichhave a power in J, in other words the set r+(J) = r(J)∩ S+; in particular, the radical of 0 in S+ is then calledthe nilradical of S+ and denoted by N+: this is the set of nilpotent elements of S+. If J is an graded ideal of S+,then its radical r+(J) is a graded ideal: by passing to the quotient ring S/J, we can reduce to the case J = 0,and it remains to see that if x = xh + xh+1 + · · ·+ xk is nilpotent, then so are the x i ∈ Si (1¶ h¶ i ¶ k); we canassume xk 6= 0 and the component of highest degree of xn is then xn

k , hence xk is nilpotent, and we then argueby induction on k. We say that the graded ring S is essentially reduced if N+ = 0, in other words, if S+ does notcontain nilpotent elements 6= 0.

2. HOMOGENEOUS PRIME SPECTRA 183

(2.1.11). We note that if, in the graded ring S, an element x is a zero-divisor, then so is its component of highestdegree. We say that a ring S is essentially integral if the ring S+ (without the unit element) does not contain azero-divisor and is 6= 0; it suces that a homogeneous element 6= 0 in S+ is not a zero-divisor in this ring. It isclear that if p is a graded prime ideal of S+, then S/p is essentially integral.

Let S be an essentially integral graded ring, and let x0 ∈ S0: if there then exists a homogeneous elementf 6= 0 of S+ such that x0 f = 0, then we have x0S+ = 0, since we have (x0 g) f = (x0 f )g = 0 for all g ∈ S+, andthe hypothesis thus implies x0 g = 0. For S to be integral, it is necessary and sucient for S0 to be integral andthe annihilator of S+ in S0 to be 0.

2.2. Rings of fractions of a graded ring.

(2.2.1). Let S be a graded ring, in positive degrees, f a homogeneous element of S, of degree d > 0; then the ringof fractions S′ = S f is graded, taking for S′n the set of the x/ f k, where x ∈ Sn+kd with k ¾ 0 (we observe herethat n can take arbitrary negative values); we denote the subring S′0 = (S f )0 of S′ consisting of elements of degree0 by the notation S( f ).

If f ∈ Sd , then the monomials ( f /1)h in S f (h a positive or negative integer) form a free system over thering S( f ), and the set of their linear combinations is none other than the ring (S(d)) f , which is thus isomorphic to II | 24

S( f )[T, T−1] = S( f ) ⊗Z Z[T, T−1] (where T is an indeterminate). Indeed, if we have a relation∑b

h=−a zh( f /1)h = 0with zh = xh/ f m, where the xh are in Smd , then this relation is equivalent by definition to the existence of ak > −a such that

∑bh=−a f h+k xh = 0, and as the degrees of the terms of this sum are distinct, we have f h+k xh = 0

for all h, hence zh = 0 for all h.If M is a graded S-module, then M ′ = M f is a graded S f -module, M ′n being the set of the z/ f k with z ∈ Mn+kd

(k ¾ 0); we denote by M( f ) the set of the homomogenous elements of degree 0 of M ′; it is immediate that M( f )is an S( f )-module and that we have (M (d)) f = M( f ) ⊗S( f ) (S

(d)) f .

Lemma (2.2.2). — Let d and e be integers > 0, f ∈ Sd , g ∈ Se. There exists a canonical ring isomorphism

S( f g) ' (S( f ))gd/ f e ;

if we canonically identify these two rings, then there exists a canonical module isomorphism

M( f g) ' (M( f ))gd/ f e .

Proof. Indeed, f g divides f e gd , and this latter element divides ( f g)de, so the graded rings S f g and S f e gd arecanonically identified; on the other hand, S f e gd also identifies with (S f e)gd/1 (0, 1.4.6), and as f e/1 is invertible inS f e , S f e gd also identifies with (S f e)gd/ f e . The element gd/ f e is of degree 0 in S f e ; we immediately conclude thatthe subring of (S f e)gd/ f e consisting of elements of degree 0 is (S( f e))gd/ f e , and as we evidently have S( f e) = S( f ),this proves the first part of the proposition; the second is established in a similar way.

(2.2.3). Under the hypotheses of (2.2.2), it is clear that the canonical homomorphism S f → S f g (0, 1.4.1), whichsends x/ f k to gk x/( f g)k, is of degree 0, thus gives by restriction a canonical homomorphism S( f )→ S( f g), suchthat the diagram

S( f )

$$S( f g)

∼ // (S( f ))gd/ f e

is commutative. We similarly define a canonical homomorphism M( f )→ M( f g).

Lemma (2.2.4). — If f and g are two homogeneous elements of S+, then the ring S( f g) is generated by the union of thecanonical images of S( f ) and S(g).

Proof. By virtue of Lemma (2.2.2), it suces to see that 1/(gd/ f e) = f d+e/( f g)d belongs to the canonicalimage of S(g) in S( f g), which is evident by definition.

Proposition (2.2.5). — Let d be an integer > 0 and let f ∈ Sd . Then there exists a canonical ring isomorphismsS( f ) ' S(d)/( f −1)S(d); if we identify these two rings by this isomorphism, then there exists a canonical module isomorphismM( f ) ' M (d)/( f − 1)M (d).

Proof. The first of these isomorphisms is defined by sending x/ f n, where x ∈ Snd , to the element x , the classof x mod. ( f −1)S(d); this map is well-defined, because we have the congruence f h x ≡ x (mod. ( f −1)S(d)) for allx ∈ S(d), so if f h x = 0 for an h> 0, then we have x = 0. On the other hand, if x ∈ Snd is such that x = ( f − 1)y II | 25

4. PROJECTIVE BUNDLES; AMPLE SHEAVES 184

with y = yhd + y(h+1)d + · · ·+ ykd with y jd ∈ S jd and yhd 6= 0, then we necessarily have h= n and x = −yhd , aswell as the relations y( j+1)d = f y jd for h¶ j ¶ k− 1, f ykd = 0, which ultimately gives f k−n x = 0; we send everyclass x mod. ( f − 1)S(d) of an element x ∈ Snd to the element x/ f n of S( f ), since the preceding remark showsthat this map is well-defined. It is immediate that these two maps thus defined are ring homomorphisms, eachthe reciprocal of the other. We proceed exactly the same way for M .

Corollary (2.2.6). — If S is Noetherian, then so is S( f ) for f homogeneous of degree > 0.

Proof. This follows immediately from Corollary (2.1.7) and Proposition (2.2.5).

(2.2.7). Let T be a multiplicative subset of S+ consisting of homogeneous elements; T0 = T ∪ 1 is then amultiplicative subset of S; as the elements of T0 are homogeneous, the ring T−1

0 S is still graded in the evidentway; we denote by S(T ) the subring of T−1

0 S consisting of elements of order 0, that is to say, the elements of theform x/h, where h ∈ T and x is homogeneous of degree equal to that of h. We know (0, 1.4.5) that T−1

0 S iscanonically identified with the inductive limit of the rings S f , where f varies over T (with respect to the canonicalhomomorphisms S f → S f g); as this identification respects the degrees, it identifies S(T ) with the inductive limitof the S( f ) for f ∈ T . For every graded S-module M , we similarly define the module M(T ) (over the ring S(T ))consisting of elements of degree 0 of T−1

0 M , and we see that this module is the inductive limit of the M( f ) forf ∈ T .

If p is a graded prime ideal of S+, then we denote by S(p) and M(p) the ring S(T ) and the module M(T )respectively, where T is the set of homogeneous elements of S+ which do not belong to p.

2.3. Homogeneous prime spectrum of a graded ring.

(2.3.1). Given a graded ring S, in positive degrees, we call the homogeneous prime spectrum of S and denote itby Proj(S) the set of graded prime ideals of S+ (2.1.10), or equivalently the set of graded prime ideals of S notcontaining S+; we will define a scheme structure having Proj(S) as the underlying set.

(2.3.2). For every subset E of S, let V+(E) be the set of graded prime ideals of S containing S and not containingS+; this is thus the subset V (E)∩ Proj(S) of Spec(S). From (I, 1.1.2) we deduce:

(2.3.2.1) V+(0) = Proj(S), V+(S) = V+(S+) =∅,

(2.3.2.2) V+⋃

λ Eλ

=⋂

λ V+(Eλ),

(2.3.2.3) V+(EE′) = V+(E)∪ V+(E′).

We do not change V+(E) by replacing E with the graded ideal generated by E; in addition, if J is a gradedideal of S, then we have

(2.3.2.4) V+(J) = V+⋃

q¾n(J∩ Sq)

for all n> 0: indeed, if p ∈ Proj(S) contains the homogeneous elements of J of degree ¾ n, then as by hypothesis II | 26there exists a homogeneous element f ∈ Sd not contained in p, for every m¾ 0 and every x ∈ Sm ∩ J, we havef r x ∈ J∩ Sm+rd for all but finitely many values of r, so f r x ∈ p∩ Sm+rd , which implies that x ∈ p∩ Sm (2.1.9).

Finally, we have, for every graded ideal J of S,

(2.3.2.5) V+(J) = V+(r+(J)).

§3. Homogeneous spectrum of a sheaf of graded algebras

3.1. Homogeneous spectrum of a quasi-coherent graded OY -algebra.

§4. Projective bundles; ample sheaves

4.1. Denition of projective bundles.

5. QUASI-AFFINE MORPHISMS; QUASI-PROJECTIVE MORPHISMS; PROPER MORPHISMS; PROJECTIVE MORPHISMS 185

§5. Quasi-ane morphisms; quasi-projective morphisms; proper morphisms;projective morphisms

5.1. Quasi-ane morphisms.

Denition (5.1.1). — We define a quasi-ane scheme to be a scheme isomorphic to some subscheme inducedon some quasi-compact open subset of an ane scheme. We say that a morphism f : X → Y is quasi-ane, orthat X (considered as a Y -prescheme via f ) is a quasi-ane Y -scheme, if there exists a cover (Uα) of Y by aneopen subsets such that the f −1(Uα) are quasi-ane schemes.

It is clear that a quasi-ane morphism is separated ((I, 5.5.5) and (I, 5.5.8)) and quasi-compact (I, 6.6.1);every ane morphisms is evidently quasi-ane.

Recall that, for any prescheme X , setting A = Γ(X ,OX ), the identity homomorphism A→ A = Γ(X ,OX )defines a morphism X → Spec(A), said to be canonical (I, 2.2.4); this is nothing but the canonical morphismdefined in (4.5.1) for the specific case where L = OX , if we remember that Proj(A[T]) is canonically identifiedwith Spec(A) (3.1.7).

Proposition (5.1.2). — Let X be a quasi-compact scheme or a prescheme whose underlying space is Noetherian, and Athe ring Γ(X ,OX ). The following conditions are equivalent.

(a) X is a quasi-ane scheme.(b) The canonical morphism u : X → Spec(A) is an open immersion.(b’) The canonical morphism u : X → Spec(A) is a homeomorphism from X to some subspace of the underlying space

of Spec(A).(c) The OX -module OX is very ample relative to u (4.4.2).(c’) The OX -module OX is ample (4.5.1).(d) When f ranges over A, the X f form a basis for the topology of X .(d’) When f varies over A, the X f that are ane form a cover of X . II | 95(e) Every quasi-coherent OX -module is generated by its sections over X .(e’) Every quasi-coherent sheaf of ideals of OX of nite type is generated by its sections over X .

Proof. It is clear that (b) implies (a), and (a) implies (c) by (4.4.4, b) applied to the identity morphism(taking into account the remark preceding this proposition); Furthermore, (c) implies (c′) (4.5.10, i), and (c′),(b), and (b′) are all equivalent by (4.5.2, b) and (4.5.2, b′). Finally, (c′) is the same as each of (d), (d′), (e), and(e′) by (4.5.2, a), (4.5.2, a′), (4.5.2, c), and (4.5.5, d′′).

We further observe that, with the previous notation, the X f that are ane form a basis for the topology of X ,and that the canonical morphism u is dominant (4.5.2).

Corollary (5.1.3). — Let X be a quasi-compact prescheme. If there exists a morphism v : X → Y from X to some anescheme Y (which would be a homeomorphism from X to some open subspace of Y ), then X is quasi-ane.

Proof. There exists a family (gα) of sections of OY over Y such that the D(gα) form a basis for the topologyof v(X ); if v = (ψ,θ) and we set fα = θ(gα), then we have X fα = ψ

−1(D(gα)) (I, 2.2.4.1), so the X fα form a basisfor the topology of X , and the criterion (5.1.2, d) is satisfied.

Corollary (5.1.4). — If X is a quasi-ane scheme, then every invertible OX -module is very ample (relative to thecanonical morphism), and a fortiori ample.

Proof. Such a module L is generated by its sections over X (5.1.2, e), so L ⊗ OX = L is very ample(4.4.8).

Corollary (5.1.5). — Let X be a quasi-compact prescheme. If there exists an invertible OX -module L such that L andL −1 are ample, then X is a quasi-ane scheme.

Proof. Indeed, OX =L ⊗L −1 is then ample (4.5.7).

Proposition (5.1.6). — Let f : X → Y be a quasi-compact morphism. Then the following conditions are equivalent.

(a) The morphism f is quasi-ane.(b) The OY -algebra f∗(OX ) =A (X ) is quasi-coherent, and the canonical morphism X → Spec(A (X )) corresponding

to the identity morphism A (X )→A (X ) (1.2.7) is an open immersion.(b’) The OY -algebra A (X ) is quasi-coherent, and the canonical morphism X → Spec(A (X )) is a homeomorphism

from X to some subspace of Spec(A (X )).(c) The OX -module OX is very ample for f .

5. QUASI-AFFINE MORPHISMS; QUASI-PROJECTIVE MORPHISMS; PROPER MORPHISMS; PROJECTIVE MORPHISMS 186

(c’) The OX -module OX is ample for f .(d) The morphism f is separated, and, for every quasi-coherent OX -module F , the canonical homomorphism

σ : f ∗( f∗(F ))→F (0, 4.4.3) is surjective.

Furthermore, whenever f is quasi-ane, every invertible OX -module L is very ample relative to f .

Proof. The equivalence between (a) and (c′) follows from the local (on Y ) character of the f -ampleness(4.6.4), Definition (5.1.1), and the criterion (5.1.2, c′). The other properties are local on Y and thus follow II | 96immediately from (5.1.2) and (5.1.4), taking into account the fact that f∗(F ) is quasi-coherent whenever f isseparated (I, 9.2.2, a).

Corollary (5.1.7). — Let f : X → Y be a quasi-ane morphism. For every open subset U of Y , the restrictionf −1(U)→ U of f is quasi-ane.

Corollary (5.1.8). — Let Y be an ane scheme, and f : X → Y a quasi-compact morphism. For f to be quasi-ane, itis necessary and sucient for X to be a quasi-ane scheme.

Proof. This is an immediate consequence of (5.1.6) and (4.6.6).

Corollary (5.1.9). — Let Y be a quasi-compact scheme or a prescheme whose underlying space is Noetherian, andf : X → Y a morphism of finite type. If f is quasi-ane, then there exists a quasi-coherent OY -subalgebra B ofA (X ) = f∗(OX ) of finite type (I, 9.6.2) such that the morphism X → Spec(B) corresponding to the canonical injectionB →A (X ) is an immersion. Further, every quasi-coherent OY -subalgebra B ′ of nite type over A (X ) containing B hasthe same property.

Proof. Indeed, A (X ) is the inductive limit of its quasi-coherent OY -subalgebras of finite type (I, 9.6.5); theresult is then a particular case of (3.8.4), taking into account the identification of Spec(A (X )) with Proj(A (X )[T])(3.1.7).

Proposition (5.1.10). —

(i) A quasi-compact morphism X → Y that is a homeomorphism from the underlying space of X to some subspace ofthe underlying space of Y (so, in particular, any closed immersion) is quasi-ane.

(ii) The composition of any two quasi-ane morphisms is quasi-ane.(iii) If f : X → Y is a quasi-ane S-morphism, then f(S′) : X(S′)→ Y(S′) is a quasi-ane morphism for any extension

S′→ S of the base prescheme.(iv) If f : X → Y and g : X ′→ Y ′ are quasi-ane S-morphisms, then f ×S g is quasi-ane.(v) If f : X → Y and g : Y → Z are morphisms such that g f is quasi-ane, and if g is separated or the

underlying space of X is locally Noetherian, then f is quasi-ane.(vi) If f is a quasi-ane morphism, then so is fred.

Proof. Taking into account the criterion (5.1.6, c′), all of (i), (iii), (iv), (v), and (vi) follow immediatelyfrom (4.6.13, i bis), (4.6.13, iii), (4.6.13, iv), (4.6.13, v), and (4.6.13, vi) (respectively). To prove (ii), we canrestrict to the case where Z is ane, and then the claim follows directly from applying (4.6.13, ii) to L = OXand K = OY .

Remark (5.1.11). — Let f : X → Y and g : Y → Z be morphisms such that X ×Z Y is locally Noetherian. Thenthe graph immersion Γf : X → X ×Z Y is quasi-ane, since it is quasi-compact (I, 6.3.5), and since (I, 5.5.12)shows that, in (v), the conclusion still holds true if we remove the hypothesis that g is separated.

Proposition (5.1.12). — Let f : X → Y be a quasi-compact morphism, and g : X ′→ X a quasi-ane morphism. If Lis an ample (for f ) OX -module, then g∗(L ) is an ample (for f g) OX ′ -module.

Proof. Since OX ′ is very ample for g, and the question is local on Y (4.6.4), it follows from (4.6.13, ii) thatthere exists (for Y ane) an integer n such that

g∗(L ⊗n) = (g∗(L ))⊗n

is ample for f g, and so g∗(L ) is ample for f g (4.6.9)

5. QUASI-AFFINE MORPHISMS; QUASI-PROJECTIVE MORPHISMS; PROPER MORPHISMS; PROJECTIVE MORPHISMS 187

5.2. Serre’s criterion.

Theorem (5.2.1). — (Serre’s criterion). Let X be a quasi-compact scheme or a prescheme whose underlying space isNoetherian. The following conditions are equivalent.

(a) X is an ane scheme.(b) There exists a family of elements fα ∈ A = Γ(X ,OX ) such that the X fα are ane, and such that the ideal

generated by the fα in A is equal to A itself.(c) The functor Γ(X ,F ) is exact in F on the category of quasi-coherent OX -modules, or, in other words, if

(*) 0 −→F ′ −→F −→F ′′ −→ 0

is an exact sequence of quasi-coherent OX -modules, then the sequence

0 −→ Γ(X ,F ′) −→ Γ(X ,F ) −→ Γ(X ,F ′′) −→ 0

is also exact.(c’) Condition (c) holds for every exact sequence (∗) of quasi-coherent OX -modules such that F is isomorphic to a

OX -submodule of O nX for some nite n.

(d) H1(X ,F ) = 0 for every quasi-coherent OX -module F .(d′) H1(X ,J ) = 0 for every quasi-coherent sheaf of ideals J of OX .

Proof. It is evident that (a) implies (b); furthermore, (b) implies that the X fα cover X , because, by hypothesis,the section 1 is a linear combination of the fα, and the D( fα) thus cover Spec(A). The final claim of (4.5.2) thusimplies that X → Spec(A) is an isomorphism.

We know tha (a) implies (c) (I, 1.3.11), and (c) trivially implies (c′). We now prove that (c′) implies (b).First of all, (c′) implies that, for every closed point x ∈ X and every open neighbourhood U of x , there existssome f ∈ A such that x ∈ X f ⊂ X − U . Let J (resp. J ′) be the quasi-coherent sheaf of ideals of OX defining thereduced closed subprescheme of X that has X − U (resp. (X − U)∪ x) as its underlying space (I, 5.2.1); it isclear that we have J ′ ⊂ J , and that J ′′ = J /J ′ is a quasi-coherent OX module that has support equal to x,and such that J ′′x = k(x). Hypothesis (c′) applied to the exact sequence 0→J ′→J →J ′′→ 0 shows thatΓ(X ,J )→ Γ(X ,J ′′) is surjective. The section of J ′′ whose germ at x is 1x is thus the image of some sectionf ∈ Γ(X ,J ) ⊂ Γ(X ,OX ), and we have, by definition, that f (x) = 1x and f (y) = 0 in X − U , which establishesour claim. Now, if U is ane, then so is X f (I, 1.3.6), so the union of the X f that are ane ( f ∈ A) is an openset Z that contains all the closed points of X ; since X is a quasi-compact Kolmogoro space, we necessarily haveZ = X (0, 2.1.3). Because X is quasi-compact, there are a nite number of elements fi ∈ A (1¶ i ¶ n) such thatthe X fi

are ane and cover X . So consider the homomorphism O nX →OX defined by the sections fi (0, 5.1.1);

since, for all x ∈ X , at least one of the ( fi)x is invertible, this homomorphism is surjective, and we thus have anexact sequence 0→R →O n

X →OX → 0, where R is a quasi-coherent OX -submodule of OX . It then follows from II | 98(c′) that the corresponding homomorphism Γ(X ,O n

X )→ Γ(X ,OX ) is surjective, which proves (b).Finally, (a) implies (d) (I, 5.1.9.2), and (d) trivially implies (d′). It remains to show that (d′) implies

(c′). But if F ′ is a quasi-coherent OX -submodule of O nX , then the filtration 0 ⊂ OX ⊂ O 2

X ⊂ . . . ⊂ O nX defines a

filtration of F ′ given by the F ′k = F ∩ OkX (0 ¶ k ¶ n), which are quasi-coherent OX -modules (I, 4.1.1), and

F ′k+1/F′k is isomorphic to a quasi-coherent OX -submodule of O k+1

X /O kX = OX , which is to say, a quasi-coherent

sheaf of ideals of OX . Hypothesis (d′) thus implies that H1(X ,F ′k+1/F′k) = 0; the exact cohomology sequence

H1(X ,F ′k)→ H1(X ,F ′k+1)→ H1(X ,F ′k+1/F′k) = 0 then lets us prove by induction on k that H1(X ,F ′k) = 0 for

all k.

Remark (5.2.1.1). — When X is a Noetherian prescheme, we can replace “quasi-coherent” by “coherent” in thestatements of (c′) and (d′). Indeed, in the proof of the fact that (c′) implies (b), J and J ′ are then coherentsheaves of ideals, and, furthermore, every quasi-coherent submodule of a coherent module is coherent (I, 6.1.1);whence the conclusion.

Corollary (5.2.2). — Let f : X → Y be a separated quasi-compact morphism. The following conditions are equivalent.

(a) The morphism f is an ane morphism.(b) The functor f∗ is exact on the category of quasi-coherent OX -modules.(c) For every quasi-coherent OX -module F , we have R1 f∗(F ) = 0.(c’) for every quasi-coherent sheaf of ideals J of OX , we have R1 f∗(J ) = 0.

Proof. All these conditions are local on Y , by definition of the functor R1 f∗ (T, 3.7.3), and so we can assumethat Y is ane. If f is ane, then X is ane, and property (b) is nothing more than (I, 1.6.4). Conversely, wenow show that (b) implies (a): for every quasi-coherent OX -module F , we have that f∗(F ) is a quasi-coherent

5. QUASI-AFFINE MORPHISMS; QUASI-PROJECTIVE MORPHISMS; PROPER MORPHISMS; PROJECTIVE MORPHISMS 188

OY -module (I, 9.2.2, a). By hypothesis, the functor f∗(F ) is exact in F , and the functor Γ(Y,G ) is exact in G(in the category of quasi-coherent OY -modules) because Y is ane (I, 1.3.11); so Γ(Y, f∗(F )) = Γ(X ,F ) is exactin F , which proves our claim, by (5.2.1, c).

If f is ane, then f −1(U) is ane for every ane open subset U of Y (1.3.2), and so H1( f −1(U),F ) = 0(5.2.1, d), which, by definition, implies that R1 f∗(F ) = 0. Finally, suppose that condition (c′) is satisfied; theexact sequence of terms of low degree in the Leray spectral sequence (G, II, 4.17.1 and I, 4.5.1) give, in particular,the exact sequence

0 −→ H1(Y, f∗(J )) −→ H1(X ,J ) −→ H0(Y, R1 f∗(J )).Since Y is ane, and f∗(J ) quasi-coherent (I, 9.2.2, a), we have that H1(Y, f∗(J )) = 0 (5.2.1); hypothesis (c′)thus implies that H1(X ,J ) = 0, and we conclude, by (5.2.1), that X is an ane scheme.

Corollary (5.2.3). — If f : X → Y is an ane morphism, then, for every quasi-coherent OX -module F , the canonicalhomomorphism H1(Y, f∗(F ))→ H1(X ,F ) is bijective.

II | 99

Proof. We have the exact sequence

0 −→ H1(Y, f∗(F )) −→ H1(X ,F ) −→ H0(Y, R1 f∗(F ))

of terms of low degree in the Leray spectral sequence, and the conclusion follows from (5.2.2).

Remark (5.2.4). — In Chapter III, §1, we prove that, if X is ane, then we have Hi(X ,F ) = 0 for all i > 0and all quasi-coherent OX -modules F .

5.3. Quasi-projective morphisms.

Denition (5.3.1). — We say that a morphism f : X → Y is quasi-projective, or that X (considered as aY -prescheme via f ) is quasi-projective over Y , or that X is a quasi-projective Y -scheme, if f is of finite type and thereexists an invertible f -ample OX -module.

We note that this notion is not local on Y : the counterexamples of Nagata [Nag58b] and Hironaka show that,even if X and Y are non-singular algebraic schemes over an algebraically closed field, every point of Y can havean ane neighbourhood U such that f −1(U) is quasi-projective over U , without f being quasi-projective.

We note that a quasi-projective morphism is necessarily separated (4.6.1). When Y is quasi-compact, it isequivalent to say either that f is quasi-projective, or that f is of finite type and there exists a very ample (relativeto f ) OX -module ((4.6.2) and (4.6.11)). Further:

Proposition (5.3.2). — Let Y be a quasi-compact scheme or a prescheme whose underlying space is Noetherian, and letX be a Y -prescheme. The following conditions are equivalent.

(a) X is a quasi-projective Y -scheme.(b) X is of nite type over Y , and there exists some quasi-coherent OY -module E of nite type such that X is

Y -isomorphic to a subprescheme of P(E ).(c) X is of nite type over Y , and there exists some quasi-coherent graded OY -algebra S such that S1 is of nite type

and generates S , and such that X is Y -isomorphic to a induced subprescheme on some everywhere-dense opensubset of Proj(S ).

Proof. This follows immediately from the previous remark and from (4.4.3), (4.4.6), and (4.4.7).

We note that, whenever Y is a Noetherian prescheme, we can, in conditions (b) and (c) of (5.3.2), removethe hypothesis that X is of finite type over Y , since this automatically satisfied (I, 6.3.5).

Corollary (5.3.3). — Let Y be a quasi-compact scheme such that there exists an ample OY -module L (4.5.3). For aY -scheme X to be quasi-projective, it is necessary and sucient for it to be of nite type over Y and also isomorphic to aY -subscheme of a projective bundle of the form Pr

Y .

Proof. If E is a quasi-coherent OY -module of finite type, then E is isomorphic to a quotient of an OY -moduleL ⊗(−n) ⊗OY

O kY (4.5.5), and so P(E ) is isomorphic to a closed subscheme of Pk−1

Y ((4.1.2) and (4.1.4)).

Proposition (5.3.4). —(i) A quasi-ane morphism of nite type (and, in particular, a quasi-compact immersion, or an ane morphism of

nite type) is quasi-projective.(ii) If f : X → Y and g : Y → Z are quasi-projective, and if Z is quasi-compact, then g f is quasi-projective. II | 100(iii) If f : X → Y is a quasi-projective S-morphism, then f(S′) : X(S′)→ Y(S′) is quasi-projective for every extension

S′→ S of the base prescheme.

5. QUASI-AFFINE MORPHISMS; QUASI-PROJECTIVE MORPHISMS; PROPER MORPHISMS; PROJECTIVE MORPHISMS 189

(iv) If f : X → Y and g : X ′→ Y ′ are quasi-projective S-morphisms, then f ×S g is quasi-projective.(v) If f : X → Y and g : Y → Z are morphisms such that g f is quasi-projective, and if g is separated or X locally

Noetherian, then f is quasi-projective.(vi) If f is a quasi-projective morphism, then so is fred.

Proof. (i) follows from (5.1.6) and (5.1.10, i). The other claims are immediate consequences of Defini-tion (5.3.1), of the properties of morphisms of finite type (I, 6.3.4), and of (4.6.13).

Remark (5.3.5). — We note that we can have fred being quasi-projective without f being quasi-projective, evenif we assume that Y is the spectrum of an algebra of finite rank over C and that f is proper.

Corollary (5.3.6). — If X and X ′ are quasi-projective Y -schemes, then X t X ′ is a quasi-projective Y -scheme.

Proof. This follows from (4.6.18).

5.4. Proper morphisms and universally closed morphisms.

Denition (5.4.1). — We say that a morphism of preschemes f : X → Y is proper if it satisfies the followingtwo conditions:

(a) f is separated and of finite type; and(b) for every prescheme Y ′ and every morphism Y ′ → Y , the projection f(Y ′) : X ×Y Y ′ → Y ′ is a closed

morphism (I, 2.2.6).When this is the case, we also say that X (considered as a Y -prescheme with structure morphism f ) is

proper over Y .

It is immediate that conditions (a) and (b) are local on Y . To show that the image of a closed subset Zof X ×Y Y ′ under the projection q : X ×Y Y ′→ Y ′ is closed in Y , it suces to see that q(Z)∩ U ′ is closed in U ′

for every ane open subset U ′ of Y ′; since q(Z)∩ U ′ = q(Z ∩ q−1(U ′)), and since q−1(U ′) can be identified withX ×Y U ′ (I, 4.4.1), we see that to satisfy condition (b) of Definition (5.4.1), we can restrict to the case where Y isan ane scheme. We further see (5.3.6) that, if Y is locally Noetherian, then we can even restrict to proving (b)in the case where Y ′ is of finite type over Y .

It is clear that every proper morphism is closed.

Proposition (5.4.2). —(i) A closed immersion is a proper morphism.(ii) The composition of two proper morphisms is proper.(iii) If X and Y are S-preschemes, and f : X → Y a proper S-morphism, then f(S′) : X(S′)→ Y(S′) is proper for every

extension S′→ S of the base prescheme.(iv) If f : X → Y and g : X ′→ Y ′ are proper S-morphisms, then f ×S g : X ×S Y → X ′×S Y ′ is a proper S-morphism.

II | 101

Proof. It suces to prove (i), (ii), and (iii) (I, 3.5.1). In each of the three cases, verifying condition (a) of(5.4.1) follows from previous results ((I, 5.5.1) and (6.4.3)); it remains to verify condition (b). It is immediatein case (i), because if X → Y is a closed immersion, then so is X ×Y Y ′→ Y ×Y Y ′ = Y ′ ((I, 4.3.2) and (3.3.3)).To prove (ii), consider two proper morphisms X → Y and Y → Z , and a morphism Z ′ → Z . We can writeX ×Z Z ′ = X ×Y (Y ×Z Z ′) (I, 3.3.9.1), and so the projection X ×Z Z ′→ Z ′ factors as X ×Y (Y ×Z Z ′)→ Y ×Z Z ′→ Z ′.Taking the initial remark into account, (ii) follows from the fact that the composition of two closed morphisms isclosed. Finally, for every morphism S′→ S, we can identify X(S′) with X ×Y Y(S′) (I, 3.3.11); for every morphismZ → Y(S′), we can write

X(S′) ×Y(S′) Z = (X ×Y Y(S′))×Y(S′) Z = X ×Y Z;

since by hypothesis X ×Y Z → Z is closed, this proves (iii).

Corollary (5.4.3). — Let f : X → Y and g : Y → Z be morphisms such that g f is proper.(i) If g is separated, then f is proper.(ii) If g is separated and of nite type, and if f is surjective, then g is proper.

Proof. (i) follows from (5.4.2) by the general procedure (I, 5.5.12). To prove (ii), we need only verify thatcondition (b) of Definition (5.4.1) is satisfied. For every morphism Z ′→ Z , the diagram

X ×Z Z ′f ×1Z′ //

p%%

Y ×Z Z ′

p′

Z ′

5. QUASI-AFFINE MORPHISMS; QUASI-PROJECTIVE MORPHISMS; PROPER MORPHISMS; PROJECTIVE MORPHISMS 190

(where p and p′ are the projections) commutes (I, 3.2.1); furthermore, f ×1Z ′ is surjective because f is surjective(I, 3.5.2), and p is a closed morphism by hypothesis. Every closed subset F of Y ×Z Z ′ is thus the imageunder f × 1Z ′ of some closed subset E of X ×Z Z ′, so p′(F) = p(E) is closed in Z ′ by hypothesis, whence thecorollary.

Corollary (5.4.4). — If X is a proper prescheme over Y , and S a quasi-coherent OY -algebra, then every Y -morphismf : X → Proj(S ) is proper (and a fortiori closed).

Proof. The structure morphism p : Proj(S )→ Y is separated, and p f is proper by hypothesis.

Corollary (5.4.5). — Let f : X → Y be a separated morphism of nite type. Let (X i)1¶i¶n (resp. (Yi)1¶i¶n) be a nitefamily of closed subpreschemes of X (resp. Y ), and ji (resp. hi ) the canonical injection X i → X (resp. Yi → Y ). Suppose thatthe underlying space of X is the union of the X i , and that, for all i, there is a morphism fi : X i → Yi , such that the diagram

X ifi //

ji

Yi

hi

X

f // Y

commutes. Then, for f to be proper, it is necessary and sucient for all of the fi to be proper.II | 102

Proof. If f is proper, then so is f ji , because ji is a closed immersion (5.4.2); since hi is a closed immersion,and thus a separated morphism, fi is proper, by (5.4.3). Conversely, suppose that all of the fi are proper, andconsider the prescheme Z given by the sum of the X i ; let u be the morphism Z → X which reduces to ji on eachX i . The restriction of f u to each X i is equal to f ji = hi fi , and is thus proper, because both the hi and thefi are (5.4.2); it then follows immediately from Definition (5.4.1) that u is proper. But since by hypothesis u issurjective, we conclude that f is proper by (5.4.3).

Corollary (5.4.6). — Let f : X → Y be a separated morphism of nite type; for f to be proper, it is necessary andsucient for fred : Xred→ Yred to be proper.

Proof. This is a particular case of (5.4.5), with n= 1, X1 = Xred, and Y1 = Yred (I, 5.1.5).

(5.4.7). If X and Y are Noetherian preschemes, and f : X → Y a separated morphism of finite type, then wecan, to show that f is proper, restrict to the the case of dominant morphisms and integral preschemes. Indeed,let X i (1 ¶ i ¶ n) be the (finitely many) irreducible components of X , and consider, for each i, the uniquereduced closed subprescheme of X that has X i as its underlying space, which we again denote by X i (I, 5.2.1).Let Yi be the unique reduced closed subprescheme of Y that has f (X i) as its underlying space. If gi (resp. hi) isthe injection morphism X i → X (resp. Yi → Y ), then we conclude that f gi = hi fi , where fi is a dominantmorphism X i → Yi (I, 5.2.2); we are then under the right conditions to apply (5.4.5), and for f to be proper, it isnecessary and sucient for all the fi to be proper.

Corollary (5.4.8). — Let X and Y be separated S-preschemes of nite type over S, and f : X → Y an S-morphism. Forf to be proper, it is necessary and sucient that, for every S-prescheme S′, the morphism f ×S 1S′ : X ×S S′→ Y ×S S′ beclosed.

Proof. First note that, if g : X → S and h : Y → S are the structure morphisms, then we have, by definition,g = h f , and so f is separated and of finite type ((I, 5.5.1) and (6.3.4)). If f is proper, then so is f ×S 1S′

(5.4.2); a fortiori, f ×S 1S′ is closed. Conversely, suppose that the conditions of the statement are satisfied, andlet Y ′ be a Y -prescheme; Y ′ can also be considered as an S-prescheme, and since Y → S is separated, X ×Y Y ′

can be identified with a closed subprescheme of X ×S Y ′ (I, 5.4.2). In the commutative diagram

X ×Y Y ′f ×1Y ′//

Y ×Y Y ′ = Y ′

X ×S Y ′

f ×1S′ // Y ×S Y ′,

the vertical arrows are closed immersions; it thus immediately follows that if f × 1S′ is a closed morphism, thenso is f × 1Y ′

5. QUASI-AFFINE MORPHISMS; QUASI-PROJECTIVE MORPHISMS; PROPER MORPHISMS; PROJECTIVE MORPHISMS 191

Remark (5.4.9). — We say that a morphism f : X → Y is universally closed if it satisfies condition (b) ofDefinition (5.4.1). The reader will observe that, in (5.4.2) to (5.4.8), we can replace every occurrence of “proper” II | 103with “universally closed” without changing the validity of the results (and in the hypotheses of (5.4.3), (5.4.5),(5.4.6), and (5.4.8), we can omit the finiteness conditions).

(5.4.10). Let f : X → Y be a morphism of finite type. We say that a closed subset Z of X is proper on Y (orY -proper, or proper for f ) if the restriction of f to a closed subprescheme of X , with underlying space Z (1, 5.2.1),is proper. Since this restriction is then separated, it follows from (5.4.6) and (I, 5.5.1, vi) that the precedingproperty does not depend on the closed subprescheme of X that has Z as its underlying space. If g : X ′→ X is aproper morphism, then g−1(Z) is a proper subset of X ′: if T is a subprescheme of X that has Z as its underlyingspace, it suces to note that the restriction of g to the closed subprescheme g−1(T ) of X ′ is a proper morphismg−1(T )→ T , by (5.4.2, iii), and to then apply (5.4.2, ii). Further, if X ′′ is a Y -scheme of finite type, and u : X → X ′′

a Y -morphism, then u(Z) is a proper subset of X ′′; indeed, let us take T to be the reduced closed subpreschemeof X having Z as its underlying space; then the restriction of f to T is proper, and thus so is the restriction of uto T (5.4.3, i), thus u(Z) is closed in X ′′; let T ′′ be a closed subprescheme of X ′′ having u(Z) as its underlying

space (I, 5.2.1), such that u|T factors as Tv−→ T ′′

j−→ X ′′, where j is the canonical injection (I, 5.2.2), and v is

thus proper and surjective (5.4.5); if g is the restriction to T ′′ of the structure morphism X ′′ → Y , then g isseparated and of finite type, and we have that f |T = g v; it thus follows from (5.4.3, ii) that g is proper, whenceour assertion.

It follows, in particular, from these remarks that, if Z is a Y -proper subset of X , then(1) for every closed subprescheme X ′ of X , Z ∩ X ′ is a Y -proper subset of X ′; and(2) if X is a subprescheme of a Y -scheme of finite type X ′′, then Z is also a Y -proper subset of X ′′ (and so,

in particular, is closed in X ′′).

5.5. Projective morphisms.

Proposition (5.5.1). — Let X be a Y -prescheme. The following conditions are equivalent.(a) X is Y -isomorphic to a closed subprescheme of a projective bundle P(E ), where E is a quasi-coherent OY -module

of nite type.(b) There exists a quasi-coherent graded OY -algebra S such that S1 is of nite type and generates S , and such that

X is Y -isomorphic to Proj(S ).

Proof. Condition (a) implies (b), by (3.6.2, ii): if J is a quasi-coherent graded sheaf of ideals of S(E ), thenthe quasi-coherent graded OY -algebra S = S(E )/J is generated by S1, and S1, the canonical image of E , is anOY -module of finite type. Condition (b) implies (a) by (3.6.2) applied to the case whereM →S1 is the identitymap.

II | 104

Denition (5.5.2). — We say that a Y -prescheme X is projective on Y , or is a projective Y -scheme, if it satisfieseither of the (equivalent) conditions (a) and (b) of (5.5.1). We say that a morphism f : X → Y is projective if itmakes X a projective Y -scheme.

It is clear that if f : X → Y is projective, then there exists a very ample (relative to f ) OX -module (4.4.2).

Theorem (5.5.3). —(i) Every projective morphism is quasi-projective and proper.(ii) Conversely, let Y be a quasi-compact scheme or a prescheme whose underlying space is Noetherian; then every

morphism f : X → Y that is quasi-projective and proper is projective.

Proof.(i) It is clear that if f : X → Y is projective, then it is of finite type and quasi-projective (thus, in particular,

separated); furthermore, it follows immediately from (5.5.1, b) and (3.5.3) that if f is projective, thenso is f ×Y 1Y ′ : X ×Y Y ′→ Y ′ for every morphism Y ′→ Y . To show that f is universally closed, it isthus enough to show that a projective morphism f is closed. Since the question is local on Y , we cansuppose that Y = Spec(A), thus (5.5.1) X = Proj(S), where S is a graded A-algebra generated by a finitenumber of elements of S1. For all y ∈ Y , the fibre f −1(y) can be identified with Proj(S)×Y Spec(k(y))(I, 3.6.1), and so also with Proj(S⊗A k(y)) (2.8.10); so f −1(y) is empty if and only if S⊗A k(y) satisfiescondition (TN) (2.7.4), or, in other words, if Sn ⊗A k(y) = 0 for suciently large n. But since (Sn)y isan Oy -module of finite type, the preceding condition implies that (Sn)y = 0 for suciently large N , by

5. QUASI-AFFINE MORPHISMS; QUASI-PROJECTIVE MORPHISMS; PROPER MORPHISMS; PROJECTIVE MORPHISMS 192

Nakayama’s lemma. If an is the annihilator in A of the A-module Sn, then the preceding condition alsoimplies that an ⊂ jn for suciently large n (0, 1.7.4). But since SnS1 = Sn+1, by hypothesis, we havethat an ⊂ an+1, and if a is the union of the an, then we see that f (X ) = V (a), which proves that f (X ) isclosed in Y . If now X ′ is an arbitrary closed subset of X , then there exists a closed subprescheme of X

that has X ′ as its underlying space (I, 5.2.1), and it is clear (5.5.1, a) that the morphism X ′→ Xf−→ Y

is projective, and so f (X ′) is closed in Y .(ii) The hypothesis on Y and the fact that f is quasi-projective implies the existence of a quasi-coherentOY -module E of finite type, as well as a Y -immersion j : X → P(E ) (5.3.2). But since f is proper, j isclosed, by (5.4.4), and so f is projective.

Remark (5.5.4). —(i) Let f : X → Y be a morphism such that f is proper, such that there exists a very ample (relative to

f ) OX -module L , and such that the quasi-coherent OY -module E = f∗(L ) is of nite type. Then f is aprojective morphism: indeed (4.4.4), there is then a Y -immersion r : X → P(E ), and, since f is proper, ris a closed immersion (5.4.4). We will see in Chapter III, §3, that when Y is locally Noetherian, the thirdcondition above (E being of finite type) is a consequence of the first two, and so the first two conditionscharacterise, in this case, the projective morphisms, and if Y is quasi-compact, then we can replace thesecond condition (the existence of a very ample (relative to f ) OX -module L ) by the hypothesis thatthere exists an ample (relative to f ) OX -module (4.6.11).

(ii) Let Y be a quasi-compact scheme such that there exists an ample OY -module. For a Y -scheme X to beprojective, it is necessary and sucient for it to be Y -isomorphic to a closed Y -subscheme of a projectivebundle of the form Pr

Y . The condition is clearly sucient. Conversely, if X is projective over Y , then II | 105it is quasi-projective, and so there exists a Y -immersion j of X into some Pr

Y (5.3.3) that is closed, by(5.4.4) and (5.5.3).

(iii) The argument of (5.5.3) shows that, for every prescheme Y and every integer r ¾ 0, the structuremorphism Pr

Y → Y is surjective, because if we set S = SOY(O r+1

Y ), then we evidently have Sy =Sk(y)(k(y)r+1) (1.7.3), and so (Sn)y 6= 0 for any y ∈ Y or any n¾ 0.

(iv) It follows from the examples of Nagata [Nag58b] that there exist proper morphisms that are notquasi-projective.

Proposition (5.5.5). —(i) A closed immersion is a projective morphism.(ii) If f : X → Y and g : Y → Z are projective morphisms, and if Z is a quasi-compact scheme or a prescheme whose

underlying space is Noetherian, then g f is projective.(iii) If f : X → Y is a projective S-morphism, then f(S′) : X(S′)→ Y(S′) is projective for every extension S′→ S of the

base prescheme.(iv) If f : X → Y and g : X ′→ Y ′ are projective S-morphisms, then so is f ×S g .(v) If g f is a projective morphism, and if g is separated, then f is projective.(vi) If f is projective, then so is fred.

Proof. (i) follows immediately from (3.1.7). We have to show (iii) and (iv) separately, because of therestriction introduced on Z in (ii) (cf. (I, 3.5.1)). To show (iii), we restrict to the case where S = Y (I, 3.3.11),and the claim then immediately follows from (5.5.1, b) and (3.5.3). To show (iv), we are immediately led to thecase where X = P(E ) and X = P(E ′), where E (resp. E ′) is a quasi-coherent OY -module (resp. quasi-coherentOY ′ -module) of finite type. Let p and p′ be the canonical projections of T = Y ×S Y ′ to Y and Y ′ (respectively);by (4.1.3.1), we have P(p∗(E )) = P(E )×Y T and P(p′∗(E ′)) = P(E ′)×Y ′ T ; whence

P(p∗(E ))×T P(p′∗(E ′)) = (P(E )×Y T )×T (T ×Y ′ P(E ′))

= P(E )×Y (T ×Y ′ P(E ′)) = P(E )×S P(E ′)

by replacing T with Y ×S Y ′, and using (I, 3.3.9.1). But p∗(E ) and p′∗(E ′) are of finite type over T (0, 5.2.4),and thus so is p∗(E )⊗OT

p′∗(E ′); since P(p∗(E ))×T P(p′∗(E ′)) can be identified with a closed subprescheme ofp∗(E )⊗OT

p′∗(E ′) (4.3.3), this proves (iv). To show (v) and (vi), we can apply (I, 5.5.13), because every closedsubprescheme of a projective Y -scheme is a projective Y -scheme, by (5.5.1, a).

It remains to prove (ii); by the hypothesis on Z , this follows from (5.5.3), (5.3.4, ii), and (5.4.2, ii).

Proposition (5.5.6). — If X and X ′ are projective Y -schemes, then X t X ′ is a projective Y -scheme.

5. QUASI-AFFINE MORPHISMS; QUASI-PROJECTIVE MORPHISMS; PROPER MORPHISMS; PROJECTIVE MORPHISMS 193

Proof. This is an evident consequence of (5.5.2) and (4.3.6).

Proposition (5.5.7). — Let X be a projective Y -scheme, and L a Y -ample OX -module; then, for every section f of Lover X , X f is ane over Y .

II | 106

Proof. Since the question is local on Y , we can assume that Y = Spec(A); furthermore, X f ⊗n = X f , so byreplacing L with some suitable L ⊗n, we can assume that L is very ample relative to the structure morphismq : X → Y (4.6.11). The canonical homomorphism σ : q∗(q∗(L ))→L is thus surjective, and the correspondingmorphism

r = rL ,σ : X −→ P = P(q∗(L ))

is an immersion such thatL = r∗(OP(1)) (4.4.4); furthermore, since X is proper over Y , the immersion r is closed(5.4.4). But by definition, f ∈ Γ(Y, q∗(L )), and σ[ is the identity of q∗(L ); it then follows from Equation (3.7.3.1)that we have X f = r−1(D+( f )); so X f is a closed subprescheme of the ane scheme D+( f ), and is thus also anane scheme.

In the particular case where Y = X , we obtain (taking (4.6.13, i) into account) the following corollary, whosedirect proof is immediate anyway:

Corollary (5.5.8). — Let X be a prescheme, and L an invertible OX -module. For every section f of L over X , X f isane over X (and thus also an ane scheme whenever X is an ane scheme).

5.6. Chow’s lemma.

Theorem (5.6.1). — (Chow’s lemma). Let S be a prescheme, and X an S-scheme of nite type. Suppose that thefollowing conditions are satised:

(a) S is Noetherian;(b) S is a quasi-compact scheme, and X has a nite number of irreducible components.

Under these hypotheses,

(i) there exists a quasi-projective S-scheme X ′, and an S-morphism f : X ′ → X that is bothprojective andsurjective;

(ii) we can take X ′ and f to be such that there exists an open subset U ⊂ X for which U ′ = f −1(U) is dense in X ′,and for which the restriction of f to U ′ is an isomorphism U ′ ' U ; and

(iii) if X is reduced (resp. irreducible, integral), then we can assume that X ′ is reduced (resp. irreducible, integral).

Proof. The proof proceeds in multiple steps.

(A) We can first restrict to the case where X is irreducible. Indeed, in hypothesis (a), X is Noetherian, and so,in the two hypotheses, the irreducible components X i of X are finite in number. If the theorem is shownto be true for each of the reduced closed preschemes of X having the X i as their underlying spaces,and if X ′i and fi : X ′i → X i are the prescheme and the morphism corresponding to X i (respectively),then the prescheme X ′ given by the sum of the X ′i , and the morphism f : X ′→ X whose restriction toeach X ′i is ji fi (where ji is the canonical injection X i → X ) satisfy the conclusion of the theorem. It isimmediate that X ′ is reduced if all of the X ′i are; furthermore, we can satisfy (ii) by taking U to be

the union of the sets Ui ∩ û

⋃

j 6=i X j

. Finally, since the X ′i are quasi-projective over S, so is X ′ (5.3.6); II | 107similarly, the morphisms X ′i → X are projective by (5.5.5, i) and (5.5.5, ii), and so f is projective (5.5.6),and is clearly surjective, by definition.

(B) Now suppose that X is irreducible. Since the structure morphism r : X → S is of finite type, there existsa finite cover (Si) of S by ane open subsets, and for each i there is a finite cover (Ti j) of r−1(Si) byane open subsets, and the morphisms Ti j → Si are of finite type, and so quasi-projective (5.3.4, i);since in both hypotheses (a) and (b) the immersion Si → S is quasi-compact, it is also quasi-projective(5.3.4, i), and so the restriction of r to Ti j is a quasi-projective morphism (5.3.4, ii). Denote the Ti jby Uk (1 ¶ k ¶ n). There exists, for each index k, an open immersion φk : Uk → Pk, where Pk isprojective over S ((5.3.2) and (5.5.2)). Let U =

⋂

k Uk; since X is irreducible, and the Uk nonempty, Uis nonempty, and thus dense in X ; the restrictions of the φk to U define a morphism

φ : U −→ P = P1 ×S P2 ×S · · · ×S Pn

5. QUASI-AFFINE MORPHISMS; QUASI-PROJECTIVE MORPHISMS; PROPER MORPHISMS; PROJECTIVE MORPHISMS 194

such that the diagrams

(5.6.1.1) Uφ //

jk

P

pk

Uk

φk // Pk

commute, where jk is the canonical injection U → Uk, and pk the canonical projection P → Pk. If j isthe canonical injection U → X , then the morphism ψ = ( j,φ )S : U → X ×S P is an immersion (I, 5.3.14).In hypothesis (a), X ×S P is locally Noetherian ((3.4.1), (I, 6.3.7), and (I, 6.3.8)); in hypothesis (b),X ×S P is a quasi-compact scheme ((I, 5.5.1) and (I, 6.6.4)); in both cases, the closure X ′ in X ×S P ofthe subprescheme Z associated to ψ (and so with underlying space ψ(U)) exists, and ψ factors as

(5.6.1.2) ψ : Uψ′−→ X ′

h−→ X ×S P

where ψ′ is an open immersion and h a closed immersion (I, 9.5.10). Let q1 : X×S P → X and q2 : X×S P → Pbe the canonical projections; we set

(5.6.1.3) f : X ′h−→ X ×S P

q1−→ X ,

(5.6.1.4) g : X ′h−→ X ×S P

q2−→ P.

We will see that X ′ and f satisfy the conclusion of the theorem.(C) First we show that f is projective and surjective, and that the restriction of f to U ′ = f −1(U) is an

isomorphism from U ′ to U . Since the Pk are projective over S, so is P (5.5.5, iv), and so X ×S P isprojective over X (5.5.5, iii), and thus so is X ′, which is a closed subprescheme of X ×S P. Furthermore,we have f ψ′ = q1 (h ψ′) = q1 ψ = j, so f (X ′) contains the open everywhere-dense subset U of X ;but f is a closed morphism (5.5.3), so f (X ′) = X . Now note that q−1

1 (U) = U ×S P is induced on anopen subset of X ×S P, and, by definition, the prescheme U ′ = h−1(U ×S P) is induced by X ′ on theopen subset U ′; it is thus the closure relative to U ×S P of the prescheme Z (I, 9.5.8). But the immersion II | 108

ψ factors as UΓφ−→ U ×S P

j×1−−→ X ×S P, and since P is separated over S, the graph morphism Γφ is a

closed immersion (I, 5.4.3), and so Z is a closed subprescheme of U ×S P, whence U ′ = Z . Since ψ isan immersion, the restriction of f to U ′ is an isomorphism onto U , and the inverse of ψ′; finally, bythe definition of X ′, U ′ is dense in X ′.

(D) We now show that g is an immersion, which will imply that X ′ is quasi-projective over S, because P isprojective over S. Set

Vk =φk(Uk) (open subset of Pk)

Wk = p−1k (Vk) (open subset of P)

U ′k = f −1(Uk) (open subset of X ′)

U ′′k = g−1(Wk) (open subset of X ′).

It is clear that the U ′k form an open cover of X ′; we will first see that the U ′′k also form an open coverof X ′, by showing that U ′k ⊂ U ′′k . For this, it will suce to show that the diagram

(5.6.1.5) U ′kg|U ′k //

f |U ′k

P

pk

Uk

φk // Pk

commutes. But the prescheme U ′k = h−1(Uk ×S P) is induced by X ′ on the open subset U ′k, and is thusthe closure of Z = U ′ ⊂ U ′k relative to U ′k (I, 9.5.8). To show the commutativity of (5.6.1.5), it thussuces (since Pk is an S-scheme) to show that composing the diagram with the canonical injectionU ′→ U ′k (or, equivalently, thanks to the isomorphism from U ′ to U , with ψ) gives us a commutativediagram (I, 9.5.6). But, by definition, the diagram thus obtains is exactly (5.6.1.1), whence our claim.

The Wk thus form an open cover of g(X ′); to show that g is an immersion, it suces to showthat each of the restrictions g|U ′′k is an immersion into Wk (I, 4.2.4). For this, consider the morphism

uk : Wkpk−→ Vk

φ−1k−→ Uk → X ; since X is separated over S, the graph morphism Γuk

: Wk → X ×S Wk is a

5. QUASI-AFFINE MORPHISMS; QUASI-PROJECTIVE MORPHISMS; PROPER MORPHISMS; PROJECTIVE MORPHISMS 195

closed immersion (I, 5.4.3), and so the graph Tk = Γuk(Wk) is a closed subprescheme of X ×S W ; if

we show that U ′→ X ×S Wk factors through this subprescheme, then the map from the subpreschemeinduced by X ′ on the open subset X ′′k of X ′ to X ×S Wk will also factor through this graph, by (I, 9.5.8).Since the restriction of q2 to Tk is an isomorphism onto Wk, the restriction of g to X ′′k will be animmersion into Wk, and our claim will be proven. Let vk be the canonical injection U ′→ X ×S Wk; wehave to show that there exists a morphism wk : U ′→Wk such that vk = Γuk

wk. By the definition ofthe product, it suces to prove that q1 vk = uk q2 vk (I, 3.2.1), or, by composing on the right with II | 109the isomorphism ψ′ : U → U ′, that q1 ψ = uk q2 ψ. But since q1 ψ = j and q2 ψ =φ , our claimfollows from the commutativity of (5.6.1.1), taking into account the definition of uk.

(E) It is clear that since U , and thus U ′, is irreducible, so is the X ′ from the preceding construction, andthe morphism f is thus birational (I, 2.2.9). If in addition X is reduced, then so is U ′, and hence X ′ isalso reduced (I, 9.5.9). This finishes the proof.

Corollary (5.6.2). — Suppose that one of the hypotheses, (a) and (b), of (5.6.1) is satised. For X to be proper overS, it is necessary and sucient for there to exist a projective scheme X ′ over S, and a surjective S−morphism f : X ′→ X(which is thus projective, by (5.5.5, v)). Whenever this is the case, we can further choose f to be such that there exists adense open subset U of X for which the restriction of f to f −1(U) is an isomorphism f −1(U)' U , and for which f −1(U)is dense in X ′. If in addition X is irreducible (resp. reduced), then we can assume that X ′ is also irreducible (resp. reduced);when X and X ′ are irreducible, f is a birational morphism.

Proof. The condition is sucient, by (5.5.3) and (5.4.3, ii). It is necessary because, with the notation of(5.6.1), if X is proper over S, then X ′ is proper over S, because it is projective over X , and thus proper over X(5.5.3), and our claim follows from (5.4.2, ii); furthermore, since X ′ is quasi-projective over S, it is projectiveover S, by (5.5.3).

Corollary (5.6.3). — Let S be a locally Noetherian prescheme, and X an S-scheme of nite type over S, with structuremorphism f0 : X → S. For X to be proper over S, it is necessary and sucient that, for every morphism of finite typeS′→ S, ( f0)(S′) : X(S′)→ S′ be a closed morphism. It even suces for this condition to be veried only for every S-preschemeof the form S′ = S ⊗Z Z[T1, . . . , Tn] (where the Ti are indeterminates).

Proof. The condition being clearly necessary, we now show that it is sucient. Since the question is localon S and S′ (5.4.1), we can suppose that S and S′ are ane and Noetherian. By Chow’s lemma, there exists aprojective S-scheme P, an immersion j : X ′→ P, and a surjective projective morphism f : X ′→ X , such that thediagram

X

f0

X ′foo

j

S Proo

commutes. Since P is of finite type over S, the first hypothesis implies that the projection q2 : X ×S P → P is aclosed morphism. But the immersion j is the composition of q2 and the morphism f × 1 from X ′ ×S P to X ×S P;but f , being projective, is proper (5.5.3), and so f × 1 is closed. We thus conclude that j is a closed immersion,and thus proper (5.4.2, i). Furthermore, the structure morphism r : P → S is projective, and thus proper (5.5.3),so f0 f = r j is proper (5.4.2, ii); finally, since f is surjective, f0 is proper, by (5.4.3).

To prove the proposition using only the second, weaker hypothesis (where S′ is of the form S⊗ZZ[T1, . . . , Tn]),it suces to show that it implies the first. But, if S′ is ane and of finite type over S = Spec(A), then we have II | 110S′ = Spec(A[c1, . . . , cn]) (I, 6.3.3), and S′ is thus isomorphic to a closed subprescheme of S′′ = Spec(A[T1, . . . , Tn])(where the Ti are indeterminates). In the commutative diagram

X ×S S′1X× j //

( f0)(S′)

X ×S S′′

( f0)(S′′)

S′j // S′′

both j and 1X × j are closed immersions (I, 4.3.1), and ( f0)(S′) is closed by hypothesis; thus ( f0)(S′′) is alsoclosed.

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 196

§6. Integral morphisms and nite morphisms

6.1. Preschemes integral over another prescheme.

§7. Valuative criteria

7.1. Reminder on valuation rings.

§8. Blowup schemes; based cones; projective closure

8.1. Blowup preschemes.

(8.1.1). Let Y be a prescheme, and, for every integer n ¾ 0, let In be a quasi-coherent sheaf of ideals of OY ;suppose that the following conditions are satisfied:

(8.1.1.1) I0 = OY , In ⊂ Im for m¶ n,

(8.1.1.2) ImIn ⊂ Im+n for any m, n.II | 153

We note that these hypotheses imply

(8.1.1.3) I n1 ⊂ In.

Set

(8.1.1.4) S =⊕

n¾0

In.

It follows from (8.1.1.1) and (8.1.1.2) that S is a quasi-coherent graded OY -algebra, and thus defines aY -scheme X = Proj(S ). If J is an invertible sheaf of ideals of OY , then In ⊗OY

J ⊗n is canonically identified withInJ n. If we then replace the In by the InJ n, and, in doing so, replace S by a quasi-coherent OY -algebra S(J ),then X(J ) = Proj(S(J )) is canonically isomorphic to X (3.1.8).

(8.1.2). Suppose that Y is locally integral, so that the sheaf R(Y ) of rational functions is a quasi-coherentOY -algebra (1, 7.3.7). We say that a OY -submodule I of R(Y ) is a fractional ideal of R(Y ) if it is of nite type(0, 5.2.1). Suppose we have, for all n¾ 0, a quasi-coherent fractional ideal In of R(Y ), such that I0 = OY , andsuch that condition (8.1.1.2) (but not necessarily the second condition (8.1.1.1)) is satisfied; we can then againdefine a quasi-coherent graded OY -algebra by Equation (8.1.1.4), and the corresponding Y -scheme X = Proj(S );we will again have a canonical isomorphism from X to XJ for every invertible fractional ideal J of R(Y ).

Denition (8.1.3). — Let Y be a prescheme (resp. a locally integral prescheme), and I a quasi-coherentideal of OY (resp. a quasi-coherent fractional ideal of R(Y )). We say that the Y -scheme X = Proj(

⊕

n¾0I n) isobtained by blowing up the ideal I , or is the blow-up prescheme of Y relative to I . When I is a quasi-coherentideal of OY , and Y ′ is the closed subprescheme of Y defined by I , we also say that X is the Y -scheme obtainedby blowing up Y ′.

By definition, S =⊕

n¾0I n is then generated by S1 = I ; if I is an OY -module of nite type, then X isprojective over Y (5.5.2). Without any hypotheses on I , the OX -module OX (1) is invertible (3.2.5) and very ample,by (4.4.3) applied to the structure morphism X → Y .

We note that, if j : X → Y is the structure morphism, then the restriction of f to f −1(Y −Y ′) is an isomorphismto Y − Y ′ whenever I is an ideal of OY and Y ′ is the closed subprescheme that it defines: indeed, the questionbeing local on Y , it suces to assume that I = OY , and our claim then follows from (3.1.7).

If we replace I by I d (d > 0), then the blow-up Y -scheme X is replaced by a canonically isomorphicY -scheme X ′ (8.1.1); similarly, for every invertible ideal (resp. invertible fractional ideal) J , the blow-up preschemeX(J ) relative to the ideal IJ is canonically isomorphic to X (8.1.1).

In particular, whenever I is an invertible ideal (resp. invertible fractional ideal), the Y -scheme obtained byblowing up I is isomorphic to Y (3.1.7).

Proposition (8.1.3). — Let Y be an integral prescheme.

(i) For every sequence (In) of quasi-coherent fractional ideals ofR(Y ) that satises (8.1.1.2) and such that I0 = OY , II | 154the Y -scheme X = Proj(

⊕

n¾0I n) is integral, and the structure morphism f : X → Y is dominant.(ii) Let I be a quasi-coherent fractional ideal of R(Y ), and let X be the Y -scheme given by the blow up of Y relative

to I . If I 6= 0, then the structure morphism f : X → Y is then birational and surjective.

Proof.

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 197

(i) This follows from the fact that S =⊕

n¾0In is an integral OY -algebra ((3.1.12) and (3.1.14)), since,for all y ∈ Y , Oy is an integral ring (I, 5.1.4).

(ii) By (i), X is integral; if, furthermore, x and y are the generic points of X and Y (respectively), thenwe have f (x) = y, and it remains to show that k(x) is of rank 1 over k(y). But x is also the genericpoint of the fibre f −1(y); if ψ is the canonical morphism Z → Y , where Z = Spec(k(y)), then theprescheme f −1(y) can be identified with Proj(S ′), where S ′ = ψ∗(S ) (3.5.3). But it is clear thatS ′ =

⊕

n¾0(Iy)n, and, since I is a quasi-coherent fractional ideal of R(Y ) that is not zero, Iy 6= 0(I, 7.3.6), whence Iy = k(y); then Proj(S ′) can be identified with Spec(k(y)) (3.1.7), whence theconclusion.

We show a converse of (8.1.4) in (III, 2.3.8).

(8.1.5). We return to the setting and notation of (8.1.1). By definition, the injection homomorphisms In+1→In(8.1.1.1) define, for every k ∈ Z, an injective homomorphism of degree zero of graded S -modules

(8.1.5.1) uk : S+(k+ 1) −→S (k);

since S+(k+ 1) and S (k+ 1) are canonically (TN)-isomorphic, they give a canonical correspondence betweenuk and an injective homomorphism of OX -modules (3.4.2):

(8.1.5.2) euk : OX (k+ 1) −→ OX (k).

Recall as well (3.2.6) that we have defined canonical homomorphisms

(8.1.5.3) λ : OX (h)⊗OXOX (k) −→ OX (h+ k)

and, since the diagramS (h)⊗S S (k)⊗S S (l) //

S (h+ k)⊗S S (l)

S (h)⊗S S (k+ l) // S (h+ k+ l)

commutes, it follows from the functoriality of the λ (3.2.6) that the homomorphisms (8.1.5.3) define the structureof a quasi-coherent graded OX -algebra on

(8.1.5.4) SX =⊕

n∈Z

OX (n).

Furthermore, the diagramS (h)⊗S S (k+ 1) //

1⊗uk

S (h+ k+ 1)

uk+h

S (h)⊗S S (k) // S (h+ k)

commutes; the functoriality of the λ then implies that we have a commutative diagram

OX (h)⊗OXOX (k+ 1) λ //

1⊗euk

OX (h+ k+ 1)

euk+h

OX (h)⊗OX

OX (k)λ // OX (h+ k)

where the horizontal arrows are the canonical homomorphisms. We can thus say that the euk define an injectivehomomorphism (of degree zero) of graded SX -modules

(8.1.5.6) eu : SX (1) −→SX .

(8.1.6). Keeping the notation from (8.1.5), we now note that, for n ¾ 0, the composite homomorphism evn =eun−1 eun−2 . . . eu0 is an injective homomorphism OX (n)→ OX ; we denote by In,X its image, which is thus aquasi-coherent ideal of OX , isomorphic to OX (n). Furthermore, the diagram

OX (m)⊗OXOX (n)

λ //

evm⊗evn

OX (m+ n)

evm+n

OX id

// OX

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 198

commutes for m¾ 0, n¾ 0. We thus deduce the following inclusions:

(8.1.6.1) I0,X = OX , In,X ⊂ Im,X for 0¶ m¶ n;

(8.1.6.2) Im,XIn,X ⊂ Im+n,X for m¾ 0, n¾ 0.II | 156

Proposition (8.1.7). — Let Y be a prescheme, I a quasi-coherent ideal of OY , and X = Proj(⊕

n¾0I n) the Y -schemegiven by blowing up I . We then have, for all n> 0, a canonical isomorphism

(8.1.7.1) OX (n)∼−→I nOX = In,X

(cf. (0, 4.3.5)), and thus that I nOX is a very-ample invertible OX -module if n> 0.

Proof. The last claim is immediate, since OX (1) is invertible (3.2.5) and very ample for Y by definition((4.4.3) and (4.4.9)). Also by definition, the image of vn is exactly I nS , and (8.1.7.1) then follows from theexactness of the functor ÝM (3.2.4) and from Equation (3.2.4.1).

Corollary (8.1.8). — Under the hypotheses of (8.1.7), if f : X → Y is the structure morphism, and Y ′ the closedsubprescheme of Y dened by I , then the closed subprescheme X ′ = f −1(Y ′) of X is dened by IOX (which is canonicallyisomorphic to OX (1)), from which we obtain a canonical short exact sequence

(8.1.8.1) 0 −→ OX (1) −→ OX −→ OX ′ −→ 0.

Proof. This follows from (8.1.7.1) and from (I, 4.4.5).

(8.1.9). Under the hypotheses of (8.1.7), we can be more precise about the structure of the In,X . Note that thehomomorphism

eu−1 : OX −→ OX (−1)

canonically corresponds to a section s of OX (−1) over X , which we call the canonical section (relative to I )(0, 5.1.1). In the diagram in (8.1.5.5), the horizontal arrows are isomorphisms (3.2.7); by replacing h withk, and k with −1 in this diagram, we obtain that euk = 1k ⊗ eu−1 (where 1k denotes the identity on OX (h)), or,equivalently, that the homomorphism euk is given exactly by tensoring with the canonical section s (for all k ∈ Z).The homomorphism eu (8.1.5.6) can then be understood in the same way.

Thus, for all n ¾ 0, the homomorphism evn : OX (n)→ OX is given exactly by tensoring with s⊗n; we thusdeduce:

Corollary (8.1.10). — With the notation of (8.1.8), the underlying space of X ′ is the set of x ∈ X such that s(x) = 0,where s denotes the canonical section of OX (−1).

Proof. Indeed, if cx is a generator of the fibre (OX (1))x at a point x , then sx ⊗ cx is canonically identifiedwith a generator of the fibre of I1,X at the point x , and is thus invertible if and only if sx 6∈ mx(Ox(−1))x , or,equivalently, if and only if s(x) 6= 0.

Proposition. — Let Y be an integral prescheme, I a quasi-coherent fractional ideal of R(Y ), and X the Y -schemegiven by blowing up I . Then IOX is an invertible OX -module that is very ample for Y .

Proof. The question being local on Y (4.4.5), we can reduce to the case where Y = Spec(A), with A someintegral ring of ring of fractions K , and I = eI, with I some fractional ideal of K ; there then exists an elementa 6= 0 of A such that aI ⊂ A. Let S =

⊕

n¾0 In; the map x 7→ ax is an A-isomorphism from In+1 = (S(1))n to

aIn+1 = aISn ⊂ In = Sn, and thus defines a (TN)-isomorphism of degree zero of graded S-modules S+(1)→ aIS. II | 157On the other hand, x 7→ a−1 x is an isomorphism of degree zero of graded S-modules aIS

∼−→ IS. We thus obtain,

by composition (3.2.4), an isomorphism of OX -modules OX (1)∼−→IOX , and, since S is generated by S1 = I, OX (1)

is invertible (3.2.5) and very ample ((4.4.3) and (4.4.9)), whence our claim.

8.2. Preliminary results on the localisation of graded rings.

(8.2.1). Let S be a graded ring, but not assumed (for the moment) to be only in positive degree. We define

(8.2.1.1) S¾ =⊕

n¾0

Sn, S¶ =⊕

n¶0

Sn

which are both graded subrings of S, in only positive and negative degrees (respectively). If f is a homogeneouselements of degree d (positive or negative) of S, then the ring of fraction S f = S′ is again endowed with the

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 199

structure of a graded ring, by taking S′n (n ∈ Z) to be the set of the x/ f k for x ∈ Sn+kd (k ¾ 0); we defineS( f ) = S′0, and will write S¾f and S¶f for S

′¾ and S′¶ (respectively). If d > 0, then

(8.2.1.2) (S¾) f = S f

since, if x ∈ Sn+kd with n+ kd < 0, then we can write x/ f k = x f h/ f h+k, and we also have that n+ (h+ k)d > 0for h suciently large and > 0. We thus conclude, by definition, that

(8.2.1.3) (S¾)( f ) = (S¾f )0 = S( f ).

If M is a graded S-module, then we similarly define

(8.2.1.4) M¾ =⊕

n¾0

Mn, M¶ =⊕

n¶0

Mn

which are (respectively) a graded S¾-module and a graded S¶-module, and their intersection is the S0 moduleM0. If f ∈ Sd , then we define M f to be the graded S f -module whose elements of degree n are the z/ f k forz ∈ Mn+kd (k ¾ 0); we denote by M( f ) the set of elements of degree zero of M f , and this is an S( f )-module, andwe will write M¾f and M¶f to mean (M f )¾ and (M f )¶ (respectively). If d > 0, then we see, as above, that

(8.2.1.5) (M¾) f = M f

and

(8.2.1.6) (M¾)( f ) = (M¾f )0 = M( f ).

(8.2.2). Let z be an indeterminate, we we will call the homogenisation variable. If S is a graded ring (in positiveor negative degrees), then the polynomial algebra1

(8.2.2.1) bS = S[z]

is a graded S-algebra, where we define the degree of f zn (n¾ 0), with f homogeneous, as II | 158

(8.2.2.2) deg( f zn) = n+ deg f .

Lemma (8.2.3). — (i) There are canonical isomorphisms of (non-graded) rings

(8.2.3.1) bS(z)∼−→ bS/(z− 1)bS

∼−→ S.

(ii) There is a canonical isomorphism of (non-graded) rings

(8.2.3.2) bS( f )∼−→ S¶f

for all f ∈ Sd with d > 0.

Proof. The first of the isomorphisms in (8.2.3.1) was defined in (2.2.5), and the second is trivial; theisomorphism bS(z)

∼−→ S thus defined thus gives a correspondence between xzn/zn+k (where deg(x) = k for

k ¾ −n) and the element x . The homomorphism (8.2.3.2) gives a correspondence between xzn/ f k (wheredeg(x) = kd − n) and the element x/ f k of degree −n in S¶f , and it is again clear that this does indeed give anisomorphism.

(8.2.4). Let M be a graded S-module. It is clear that the S-module

(8.2.4.1) ÒM = M ⊗SbS = M ⊗S S[z]

is the direct sum of the S-modules M ⊗ Szn, and thus of the abelian groups Mk ⊗ Szn (k ∈ Z, n¾ 0); we defineon ÒM the structure of a graded bS-module by setting

(8.2.4.2) deg(x ⊗ zn) = n+ deg x

for all homogeneous x in M . We leave it to the reader to prove the analogue of (8.2.3):

Lemma (8.2.5). — (i) There is a canonical di-isomorphism of (non-graded) modules

(8.2.5.1) ÒM(z)∼−→ M .

(ii) For all f ∈ Sd (d > 0), there is a di-isomorphism of (non-graded) modules

(8.2.5.2) ÒM( f )∼−→ M¶f .

1This should not be confused with the use of the notation bS to denote the completed separation of a ring.

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 200

(8.2.6). Let S be a positively-graded ring, and consider the decreasing sequence of graded ideals of S

(8.2.6.1) S[n] =⊕

m¾n

Sm (n¾ 0)

(so, in particular, we have S[0] = S and S[1] = S+). Since it is evident that S[m]S[n] ⊂ S[m+n], we can define agraded ring S\ by setting

(8.2.6.2) S\ =⊕

n¾0

S\n with S\n = S[n].

S\0 is then the ring S considered as a non-graded ring, and S\ is thus an S\0-algebra. For every homogeneouselement f ∈ Sd (d > 0), we denote by f \ the element f considered as belonging to S[d] = S\d . With this notation:

II | 159

Lemma (8.2.7). — Let S be a positively-graded ring, and f a homogeneous element of Sd (d > 0). There are canonicalring isomorphisms

(8.2.7.1) S f∼−→⊕

n∈Z

S(n)( f )

(8.2.7.2) (S¾f ) f /1∼−→ S f

(8.2.7.3) S\( f \)∼−→ S¾f

where the rst two are isomorphisms of graded rings.

Proof. It is immediate, by definition, that we have (S f )n = (S(n) f )0, whence the isomorphism in (8.2.7.1),

which is exactly the identity. Next, since f /1 is invertible in S f , there is a canonical isomorphism S f∼−→ (S¾f ) f /1 =

(S f ) f /1, by (8.2.1.2) applied to S f ; the inverse isomorphism is, by definition, the isomorphism in (8.2.7.2).Finally, if x =

∑

m¾n ym is an element of S[n] with n= kd, then the element x/( f \)k corresponds to the element∑

m ym/ f k of S¾f , and we can quickly verify that this defines an isomorphism (8.2.7.3).

(8.2.8). If M is a graded S-module, then we similarly define, for all n ∈ Z,

(8.2.8.1) M[n] =⊕

m¾n

Mm

and, since S[m]M[n] ⊂ M[m+n] (m¾ 0), we can define a graded S\-module M \ by setting

(8.2.8.2) M \ =⊕

n∈Z

with M \n = M[n].

We leave to the reader the proof of:

Lemma (8.2.9). — With the notation of (8.2.7) and (8.2.8), there are canonical di-isomorphisms of modules

(8.2.9.1) M f∼−→⊕

n∈Z

M(n)( f )

(8.2.9.2) (M¾f ) f /1∼−→ M f

(8.2.9.3) M \

( f \)

∼−→ M¾f

where the rst two are di-isomorphisms of graded modules.

Lemma (8.2.10). — Let S be a positively-graded ring.

(i) For S\ to be an S\0-algebra of nite type (resp. a Noetherian S\0-algebra), it is necessary and sucient for S to bean S\0-algebra of nite type (resp. a Noetherian S\0-algebra).

(ii) For S\n+1 = S\1S\n (n¾ n0), it is necessary and sucient for Sn+1 = S1Sn (n¾ n0).

(iii) For S\n = S\1 (n¾ n0), it is necessary and sucient for Sn = Sn1 (n¾ n0).

(iv) If ( fα) is a set of homogeneous elements of S+ such that S+ is the radical in S+ of the ideal of S+ generated bythe fα, then S\+ is the radical in S\+ of the ideal of S\+ generated by the f \α .

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 201

Proof. (i) If S\ is an S\0-algebra of finite type, then S+ = S\1 is a module of finite type over S = S\0, by(2.1.6, i), and so S is an S0-algebra of finite type (2.1.4); if S\ is a Noetherian ring, then so too is S\0 = S(2.1.5). Conversely, if S is an S0-algebra of finite type, then we know (2.1.6, ii) that there exist h> 0 II | 160and m0 > 0 such that Sn+h = ShSn for n¾ m0; we can clearly assume that m0 ¾ h. Furthermore, the Sm

are S0-modules of finite type (2.1.6, i). So, if n¾ m0 + h, then S\n = ShS\n−h = S\hS\n−h; and if m< m0 + hthen, letting E = Sm0

+ . . .+ Sm0+h−1, we have that

S\m = Sm + . . .+ Sm0+h−1 + ShE + S2h E + . . . .

For 1 ¶ m ¶ m0, let Gm be the union of the finite systems of generators of the S0-modules Si form¶ i ¶ m0+h−1, thought of as a subset of S[m]. For m0+1¶ m¶ m0+h−1, let Gm be the union ofthe finite system of generators of the S0-modules Si for m¶ i ¶ m0 + h− 1 and of ShE, thought of as asubset of S[m]. It is clear that S\m = S\0Gm for 1¶ m¶ m0 + h− 1, and thus the union G of the Gm for

1¶ m¶ m0 + h− 1 is a system of generators of the S\0-algebra S\. We thus conclude that, if S = S\0 is aNoetherian ring, then so too is S\.

(ii) It is clear that, if Sn+1 = S1Sn for n ¾ n0, then S\n+1 = S1S\n, and a fortiori S\n+1 = S\1S\n for n ¾ n0.Conversely, this last equality can be written as

Sn+1 + Sn+2 + . . .= (S1 + S2 + . . .)(Sn + Sn+1 + . . .)

and comparing terms of degree n+ 1 (in S) on both sides gives that Sn+1 = S1Sn.(iii) If Sn = Sn

1 for n ¾ n0, then S\n = Sn1 + Sn+1

1 + . . .; since S\1 contains S1 + S21 + . . ., we have that S\n ⊂ S\n1 ,

and thus S\n = S\n1 for n¾ n0. Conversely, the only terms of S\n1 = (S1 + S2 + . . .)n that are of degree nin S are those of Sn

1 ; the equality S\n = S\n1 thus implies that Sn = Sn1 .

(iv) It suces to show that, if an element g ∈ Sk+h is considered as an element of S\k (k > 0, h¾ 0), then

there exists an integer n> 0 such that gn is a linear combination (in S\kn) of the f \α with coecients inS\. By hypothesis, there exists an integer m0 such that, for m¾ m0, we have, in S, that gm =

∑

α cαm fα,where the indices α here are independent of m; furthermore, we can clearly assume that the cαm arehomogeneous, with

deg(cαm) = m(k+ h)− deg fαin S. So take m0 suciently large enough to ensure that km0 > deg fα for all the fα that appear in gm0 ;for all α, let c′αm be the element cαm considered as having degree km− deg fα in S\; we then have, inS\, that gm =

∑

α c′αm f \α, which finishes the proof.

(8.2.11). Consider the graded S0-algebra

(8.2.11.1) S\ ⊗S S0 = S\/S+S\ =⊕

n¾0

S[n]/S+S[n].

Since Sn is a quotient S0-module of S[n]/S+S[n], there is a canonical homomorphism of graded S0-algebras

(8.2.11.2) S\ ⊗S S0 −→ S

which is clearly surjective, and thus corresponds (2.9.2) to a canonical closed immersion

(8.2.11.3) Proj(S) −→ Proj(S\ ⊗S S0).II | 161

Proposition (8.2.12). — The canonical morphism (8.2.11.3) is bijective. For the homomorphism (8.2.11.2) to be(TN)-bijective, it is necessary and sucient for there to exist some n0 such that Sn+1 = S1Sn for n ¾ n0. If this lattercondition is satised, then (8.2.11.3) is an isomorphism; the converse is true whenever S is Noetherian.

Proof. To prove the first claim, it suces (2.8.3) to show that the kernel I of the homomorphism (8.2.11.2)consists of nilpotent elements. But if f ∈ S[n] is an element whose class modulo S+S[n] belongs to this kernel, thenthis implies that f ∈ S[n+1]; then f n+1, considered as an element of S[n(n+1)], is also an element of S+S[n(n+1)],since it can be written as f · f n; so the class of f n+1 modulo S+S[n(n+1)] is zero, which proves our claim. Since

the hypothesis that Sn+1 = S1Sn for n¾ n0 is equivalent to S\n+1 = S\1S\n for n¾ n0 (8.2.10, ii), this hypothesis isequivalent, by definition, to the fact that (8.2.11.2) is (TN)-injective, and thus (TN)-bijective, and so (8.2.11.3)is an isomorphism, by (2.9.1). Conversely, if (8.2.11.3) is an isomorphism, then the sheaf eI on Proj(S\ ⊗S S0)is zero (2.9.2, i); since S\ ⊗S S0 is Noetherian, as a quotient of S\ (8.2.10, i), we conclude from (2.7.3) that Isatisfies condition (TN), and so S\n+1 = S\1S\n for n¾ n0, and this finishes the proof, by (8.2.10, ii).

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 202

(8.2.13). Consider now the canonical injections (S+)n→ S[n], which define an injective homomorphism of degreezero of graded rings

(8.2.13.1)⊕

n¾0

(S+)n −→ S\.

Proposition (8.2.14). — For the homomorphism (8.2.13.1) to be a (TN)-isomorphism, it is necessary and sucientfor there to exist some n0 such that Sn = Sn

1 for all n ¾ n0. Whenever this is the case, the morphisms corresponding to(8.2.13.1) is everywhere dened and also an isomorphism

Proj(S\)∼−→ Proj(

⊕

n¾0

(S+)n);

the converse is true whenever S is Noetherian.

Proof. The first two claims are evident, given (8.2.10, iii) and (2.9.1). The third will follow from (8.2.10, iand iii) and the following lemma:

Lemma (8.2.14.1). — Let T be a positively-graded ring that is also a T0-algebra of nite type. If the morphismcorresponding to the injective homomorphism

⊕

n¾0 T n1 → T is everywhere dened and also an isomorphism Proj(T )→

Proj(⊕

n¾0 T n1 ), then there exists some n0 such that Tn = T n

1 for n¾ n0.

Let gi (1 ¶ i ¶ r) be generators of the T0-module T1. The hypothesis implies first of all that the D+(gi)cover Proj(T ) (2.8.1). Let (h j)1¶ j¶s be a system of homogeneous elements of T+, with deg(h j) = n j , that form,with the gi , a system of generators of the ideal T+, or, equivalently (2.1.3), a system of generators of T as aT0-algebra; if we set T ′ =

⊕

n¾0 T n1 , then the element h j/g

n j

i of the ring T(gi) must, by hypothesis, belong to the

subring T ′(gi), and so there exists some integer k such that T k

1 h j ⊂ Tk+n j

1 for all j. We thus conclude, by induction

on r, that T k1 hr

j ⊂ T ′ for all r ¾ 1, and, by definition of the h j , we thus have that T k1 T ⊂ T ′. Also, there exists,

for all j, an integer m j such that hm j

j belongs to the ideal of T generated by the gi (2.3.14), so hm j

j ∈ T1T , and

hm j kj ∈ T k

1 T ⊂ T ′. There is thus an integer m0 ¾ k such that hmj ∈ T mn

1 for m¾ m0. So, if q is the largest of the II | 162integers n j , then n0 = qsm0 + k is the required number. Indeed, an element of Sn, for n ¾ n0, is the sum ofmonomials belonging to Tα1 u, where u is a product of powers of the h j ; if α ¾ k, then it follows from the above

that Tα1 u ⊂ T n1 ; in the other case, one of the exponents of the h j is ¾ m0, so u ∈ Tβ1 v, where β ¾ k and v is again

a product of powers of the h j ; we can then reduce to the previous case, and so we conclude that Tα1 u ⊂ T n1 in all

cases.

Remark (8.2.15). — The condition Sn = Sn1 for n ¾ n0 clearly implies that Sn+1 = S1Sn for n ¾ n0, but the

converse is not necessarily true, even if we assume that S is Noetherian. For example, let K be a field, A= K[x],and B = K[y]/y2K[y], where x and y are indeterminates, with x taken to have degree 1 and y to have degree 2,and let S = A⊗K B, so that S is a graded algebra over K that has a basis given by the elements 1, xn (n ¾ 1),and xny (n¾ 0). It is immediate that Sn+1 = S1Sn for n¾ 2, but Sn

1 = Kxn while Sn = Kxn + Kxny for n¾ 2.

8.3. Based cones.

(8.3.1). Let Y be a prescheme; in all of this section, we will consider only Y -preschemes and Y -morphisms. LetS be a quasi-coherent positively-graded OY -algebra; we further assume that S0 = OY . Following the notationintroduced in (8.2.2), we let

(8.3.1.1) cS = S [z] = S ⊗OYOY [z]

which we consider as a positively-graded OY -algebra by defining the degrees as in (8.2.2.2), so that, for everyane open subset U of Y , we have

Γ(U , cS ) = (Γ(U ,S ))[z].In what follows, we write

(8.3.1.2) X = Proj(S ), C = Spec(S ), bC = Proj(cS )

(where, in the definition of C , we consider S as a non-graded OY -algebra), and we say that C (resp. bC) is theane cone (resp. projective cone) defined by S ; we will sometimes say “cone” instead of “ane cone”. By anabuse of language, we also say that C (resp. bC) is the ane cone based at X (?) (resp. the projective cone based at X(?))2, with the implicit understanding that the prescheme X is given in the form Proj(S ); finally, we say that bC isthe projective closure of C (with the data of S being implicit in the structure of C).

2[Trans.] A more literal translation of the French ( cône projetant (ane/projectif)) would be the projecting (ane/projective) cone, but itseems that this terminology already exists to mean something else.

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 203

Proposition (8.3.2). — There exist canonical Y -morphisms

(8.3.2.1) Yε−→ C

i−→ bC

(8.3.2.2) Xj−→ bC

such that ε and j are closed immersions, and i is an ane morphism, which is a dominant open immersion, for which

(8.3.2.3) i(C) = bC \ j(X );

furthermore, bC is the smallest closed subprescheme of bC containing i(C).

Proof. To define i, consider the open subset of bC given by

(8.3.2.4) bCz = Spec(cS /(z− 1)cS )

(3.1.4), where z is canonically identified with a section of S over Y . The isomorphism i : C∼−→ bCz then

corresponds to the canonical isomorphism (8.2.3.1)

cS /(z− 1)cS∼−→S .

The morphism ε corresponds to the augmentation homomorphism S → S0 = OY , which has kernel S+(1.2.7), and, since the latter is surjective, ε is a closed immersion (1.4.10). Finally, j corresponds (3.5.1) to thesurjective homomorphism of degree zero cS →S , which restricts to the identity on S and is zero on z cS , whichis its kernel; j is everywhere defined, and is a closed immersion, by (3.6.2).

To prove the other claims of (8.3.2), we can clearly restrict to the case where Y = Spec(A) is ane, andS = eS, with S a graded A-algebra, whence cS = (bS)∼; the homogeneous elements f of S+ can then be identifiedwith sections of cS over Y , and the open subset of bC , denoted D+( f ) in (2.3.3), can then be written as bC f (3.1.4);similarly, the open subset of C denoted D( f ) in (I, 1.1.1) can be written as C f (0, 5.5.2). With this in mind, itfollows from (2.3.14) and from the definition of bS that, in this case, the open subsets bCz = i(C) and bC f (with fhomogeneous in S+) form a cover of bC . Furthermore, with this notation,

(8.3.2.5) i−1(bC f ) = C f ;

indeed, bC f ∩ i(C) = bC f ∩ bCz = bC f z = Spec(bS( f z)). But, if d = deg( f ), then bS( f z) is canonically isomorphic to(bS(z)) f /zd (2.2.2), and it follows from the definition of the isomorphism in (8.2.3.1) that the image of (bS(z)) f /zd

under the corresponding isomorphism of rings of fractions is exactly S f . Since C f = Spec(S f ), this proves(8.3.2.5) and shows, at the same time, that the morphism i is ane; furthermore, the restriction of i to C f ,thought of as a morphism to bC f , corresponds (I, 1.7.3) to the canonical homomorphism bS( f )→ bS( f z), and, bythe above and (8.2.3.2), we can claim the following result:

(8.3.2.6). If Y = Spec(A) is ane, and S = eS, then, for every homogeneous f in S+, bC f is canonically identifiedwith Spec(S¶f ), and the morphism C f → bC f given by restricting i then corresponds to the canonical injectionS¶f → S f .

Now note that (for Y ane) the complement of bCz in bC = Proj(bS) is, by definition, the set of graded prime II | 164ideals of bS containing z, which is exactly j(X ), by definition of j, which proves (8.3.2.3).

Finally, to prove the last claim of (8.3.2), we can assume that Y is ane. With the above notation, note that,in the ring bS, z is not a zero divisor; since i(C) = bC , it suces to prove the following lemma:

Lemma (8.3.2.7). — Let T be a positively-graded ring, Z = Proj(T ), and g a homogeneous element of T of degreed > 0. If g is not a zero divisor in T , then Z is the smallest closed subprescheme of Z that contains Zg = D+(g).

By (I, 4.1.9), the question is local on Z ; for every homogeneous element h ∈ Te (e > 0), it thus suces toprove that Zh is the smallest closed subprescheme of Zh that contains Zgh; it follows from the definitions andfrom (I, 4.3.2) that this condition is equivalent to asking for the canonical homomorphism T(h)→ T(gh) to beinjective. But this homomorphism can be identified with the canonical homomorphism T(h)→ (T(h))g e/hd (2.2.3).But since g e is not a zero divisor in T , g e/hd is not a zero divisor in Th (nor a fortiori in T(h)), since the factthat (g e/hd)(t/hm) = 0 (for t ∈ T and m> 0) implies the existence of some n> 0 such that hn g e t = 0, whencehn t = 0, and thus t/hm = 0 in Th. This thus finishes the proof (0, 1.2.2).

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 204

(8.3.3). We will often identify the ane cone C with the subprescheme induced by the projective cone bC on theopen subset i(C) by means of the open immersion i. The closed subprescheme of C associated to the closedimmersion ε is called the vertex prescheme (?) of C ; we also say that ε, which is a Y -section of C , is the vertexsection (?), or the null section, or C ; we can identify Y with the vertex prescheme (?) of C by means of ε. Also,i ε is a Y -section of bC , and thus also a closed immersion (I, 5.4.6), corresponding to the canonical surjectivehomomorphism of degree zero cS = S [z]→OY [z] (3.1.7), whose kernel is S+[z] = S+ cS ; the subprescheme ofbC associated to this closed immersion is also called the vertex prescheme (?) of bC , and i ε the vertex section (?) ofbC ; it can be identified with Y by means of i ε. Finally, the closed subprescheme of bC associated to j is calledthe part at innity of bC , and can be identified with X by means of j.

(8.3.4). The subpreschemes of C (resp. bC) induced on the open subsets

(8.3.4.1) E = C − ε(Y ), bE = bC − i(ε(Y ))are called (by an abuse of language) the pointed ane cone and the pointed projective cone (respectively) definedby S ; we note that, despite this nomenclature, E is not necessarily ane over Y , nor bE projective over Y (8.4.3).When we identify C with i(C), we thus have the underlying spaces

(8.3.4.2) C ∪ bE = bC , C ∩ bE = E

so that bC can be considered as being obtained by gluing the open subpreschemes C and bE; furthermore, by(8.3.2.3),

(8.3.4.3) E = bE − j(X ).

If Y = Spec(A) is ane, then, with the notation of (8.3.2),

(8.3.4.4) E =⋃

C f , bE =⋃

bC f , C f = C ∩ bC f

where f runs over the set of homogeneous elements of S+ (or only a subset M of this set, with M generating anideal of S+ whose radical in S+ is S+ itself, or, equivalently, such that the X f for f ∈ M cover X (2.3.14)). Thegluing of C and bC f along C f is thus determined by the injection morphisms C f → C and C f → bC f , which, as wehave seen (8.3.2.6), correspond (respectively) to the canonical homomorphisms S→ S f and S¶f → S f .

Proposition (8.3.5). — With the notation of (8.3.1) and (8.3.4), the morphism associated (3.5.1) to the canonicalinjection ϕ : S → cS = S [z] is a surjective ane morphism (called the canonical retraction)

(8.3.5.1) p : bE −→ X

such that

(8.3.5.2) p j = 1X .

Proof. To prove the proposition, we can restrict to the case where Y is ane. Taking into account theexpression in (8.3.4.4) for bE, the fact that the domain of definition G(ϕ) of p is equal to bE will follow from thefirst of the following claims:

(8.3.5.3). If Y = Spec(A) is ane, and S = eS, then, for all homogeneous f ∈ S+,

(8.3.5.4) p−1(X f ) = bC f

and the restriction of p to bC f = Spec(S¶f ), thought of as a morphism from bC f to X f , corresponds to the canonical

injection S( f )→ S¶f . If, further, f ∈ S1, then bC f is isomorphic to X f ⊗Z Z[T] (where T is an indeterminate).

Indeed, Equation (8.3.5.4) is exactly a particular case of (2.8.1.1), and the second claim is exactly thedefinition of Proj(ϕ) whenever Y is ane (2.8.1). Then Equation (8.3.5.2) and the fact that p is surjective showthat the composition S → cS →S of the canonical homomorphisms is the identity on S . Finally, the last claimof (8.3.5.3) follows from the fact that S¶f is isomorphic to S( f )[T] whenever f ∈ S1 (2.2.1).

Corollary (8.3.6). — The restriction

(8.3.6.1) π : E −→ X

of p to E is a surjective ane morphism. If Y is ane and f homogeneous in S+, then

(8.3.6.2) π−1(X f ) = C f

and the restriction of π to C f corresponds to the canonical injection S( f )→ S f . If, further, f ∈ S1, then C f is isomorphic toX f ⊗Z Z[T, T−1] (where T is an indeterminate).

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 205

Proof. Equation (8.3.6.2) follows immediately from (8.3.5.3) and (8.3.2.5), and shows the surjectivity of π;we have already seen that the immersion i, restricted to C f , corresponds to the injection S¶f → S f (8.3.2). Finally, II | 166the last claim is a consequence of the fact that, for f ∈ S1, S f is isomorphic to S( f )[T, T−1] (2.2.1).

Remark (8.3.7). — Whenever Y is ane, the elements of the underlying space of E are the (not-necessarily-graded) prime ideals p of S not containing S+, by definition of the immersion ε (8.3.2). For such an ideal p, thep∩ Sn clearly satisfy the conditions of (2.1.9), and so there exists exactly one graded prime ideal q of S such thatq∩ Sn = p∩ Sn for all n; the map π : E→ X of underlying spaces can then be understood via the equation

(8.3.7.1) π(p) = q.

Indeed, to prove this equation, it suces to consider some homogeneous f in S+ such that p ∈ D( f ), and tonote that q( f ) is the inverse image of p f under the injection S( f )→ S f .

Corollary (8.3.8). — If S is generated by S1, then the morphisms p and π are of nite type; for all x ∈ X , the brep−1(x) is isomorphic to Spec(k(x)[T]), and the bre π−1 isomorphic to Spec(k(x)[T, T−1])

Proof. This follows immediately from (8.3.5) and (8.3.6) by noting that, whenever Y is ane and S isgenerated by S1, the X f , for f ∈ S1, form a cover of X (2.3.14).

Remark (8.3.9). — The pointed ane cone corresponding to the graded OY -algebra OY [T] (where T is anindeterminate) can be identified with Gm = Spec(OY [T, T−1]), since it is exactly CT , as we have seen in (8.3.2)(see (8.4.4) for a more general result). This prescheme is canonical endowed with the structure of a “Y -scheme incommutative groups”. This idea will be explained in detail later on, but, for now, can be quickly summarised asfollows. A Y -scheme in groups is a Y -scheme G endowed with two Y -morphisms, p : G ×Y G→ G and s : G→ G,that satisfy conditions formally analogous to the axioms of the composition law and the symmetry law of agroup: the diagram

G × G × Gp×1 //

1×p

G × G

p

G × G p

// G

should commute (“associativity”), and there should be a condition which corresponds to the fact that, for groups,the maps

(x , y) 7−→ (x , x−1, y) 7−→ (x , x−1 y) 7−→ x(x−1 y)and

(x , y) 7−→ (x , x−1, y) 7−→ (x , y x−1) 7−→ (y x−1)xshould both reduce to (x , y) 7→ y ; the sequence of morphisms corresponding, for example, to the first compositemap is

G × G(1,s)×1−−−−→ G × G × G

1×p−−→ G × G

p−→ G

and the reader should write down the second sequence. II | 167It is immediate (I, 3.4.3) that the data of a structure of a Y -scheme in groups on a Y -scheme G is equivalent

to the data, for every Y -prescheme Z , of a group structure on the set HomY (Z , G), where these structures shouldbe such that, for every Y -morphism Z → Z ′, the corresponding map HomY (Z ′, G)→ HomY (Z , G) is a grouphomomorphism. In the particular case of Gm that we consider here, HomY (Z , G) can be identified with the setof Z -sections of Z ×Y Gm (I, 3.3.14), and thus with the set of Z -sections of Spec(OZ[T, T−1]); finally, the samereasoning as in (I, 3.3.15) shows that this set is canonically identified with the set of invertible elements of thering Γ(Z ,OZ), and the group structure on this set is the structure coming from the multiplication in the ringΓ(Z ,OZ). The reader can verify that the morphisms p and s from above are obtained in the following way: theycorrespond, by (1.2.7) and (1.4.6), to the homomorphisms of OY -algebras

π :OY [T, T−1] −→ OY [T, T−1, T ′, T′−1]

σ :OY [T, T−1] −→ OY [T, T−1]

and are entirely defined by the data of π(T ) = T T ′ and σ(T ) = T−1.With this in mind, Gm can be considered as a “universal domain of operators” for every ane cone C = Spec(S ),

where S is a quasi-coherent positively-graded OY -algebra. This means that we can canonically define a Y -morphism Gm ×Y C → C which has the formal properties of an external law of a set endowed with a groupof operators; or, again, as above for schemes in groups, we can give, for every Y -prescheme Z , an externallaw on HomY (Z , C), having the group HomY (Z , Gm) as its set of operators, with the usual axioms of sets

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 206

endowed with a group of operators, and a compatibility condition with respect to the Y -morphisms Z → Z ′.In the current case, the morphism Gm ×Y C → C is defined by the data of a homomorphism of OY -algebrasS →S ⊗OY

OY [T, T−1] = S [T, T−1], which associates, to each section sn ∈ Γ(U ,Sn) (where U is an open subsetof Y ), the section snT n ∈ Γ(U ,S ⊗OY

OY [T, T−1]).Conversely, suppose that we are given a quasi-coherent, a priori non-graded, OY -algebra, and, on C = Spec(S ),

a structure of a “Y -scheme in sets endowed with a group of operators” that has the Y -scheme in groups Gm asits domain of operators; then we canonically obtain a grading of OY -algebras on S . Indeed, the data of aY -morphism Gm ×Y C → C is equivalent to that of a homomorphism of OY -algebras ψ : S →S [T, T−1], whichcan be written as ψ =

∑

n∈ZψnT n, where the ψn : S →S are homomorphisms of OY -modules (with ψn(s) = 0except for finitely many n for every section s ∈ Γ(U ,S ), for any open subset U of Y ). We can then prove thatthe axioms of sets endowed with a group of operators imply that the ψn(S ) = Sn define a grading (in positiveor negative degree) of OY -algebras on S , with the ψn being the corresponding projectors. We also have thenotation of a structure of an “ane cone” on every ane Y -scheme, defined in a “geometric” way without anyreference to any prior grading. We will not further develop this point of view here, and we leave the work of II | 168precisely formulating the definitions and results corresponding to the information given above to the reader.

8.4. Projective closure of a vector bundle.

(8.4.1). Let Y be a prescheme, and E a quasi-coherent OY -module. If we take S to be the graded OY -algebraSOY(E ), then Definition (8.3.1.1) shows that cS can be identified with SOY

(E ⊕OY ). With the ane cone Spec(S )defined by S being, by definition, V(E ), and Proj(S ) being, by definition, P(E ), we see that:

Proposition (8.4.2). — The projective closure of a vector bundle V(E ) on Y is canonically isomorphic to P(E ⊕OY ),and the part at innity of the latter is canonically isomorphic to P(E ).

Remark (8.4.3). — Take, for example, E = O rY with r ¾ 2; then the pointed cones E and bE defined by S are

nether ane nor projective on Y if Y 6=∅. The second claim is immediate, because bC = P(O r+1Y ) is projective on

Y , and the underlying spaces of E and bE are non-closed open subsets of bC , and so the canonical immersionsE→ bC and bE→ bC are not projective (5.5.3), and we conclude by appealing to (5.5.5, v). Now, supposing, forexample, that Y = Spec(A) is ane, and r = 2, then C = Spec(A[T1, T2]), and E is then the prescheme inducedby C on the open subset D(T1)∪ D(T2); but we have already seen that the latter is not ane (I, 5.5.11); a fortioribE cannot be ane, since E is the open subset where the section z over bE does not vanish (8.3.2).

However:

Proposition (8.4.4). — If L is an invertible OY -module, then there are canonical isomorphisms for both the pointedcones E and bE corresponding to C = V(L ):

(8.4.4.1) Spec

⊕

n∈Z

L ⊗n

∼−→ E

(8.4.4.2) V(L −1)∼−→ bE.

Furthermore, there exists a canonical isomorphism from the projective closure of V(L ) to the projective closure ofV(L −1) that sends the null section (resp. the part at innity) of the former to the part at innity (resp. the null section) ofthe second.

Proof. We have here that S =⊕

n¾0L ⊗n; the canonical injection

S −→⊕

n∈Z

L ⊗n

defines a canonical dominant morphism

(8.4.4.3) Spec

⊕

n∈Z

L ⊗n

−→ V(L ) = Spec

⊕

n¾0

L ⊗n

and it suces to prove that this morphism is an isomorphism from the scheme Spec(⊕

n∈ZL ⊗n) to E. Thequestion being local on Y , we can assume that Y = Spec(A) is ane and that L = OY , and so S = (A[T])∼ II | 169and

⊕

n∈ZL ×n = (A[T, T−1])∼. But A[T, T−1] is the ring of fractions A[T]T of A[T], and thus (8.4.4.3) identifies⊕

n∈ZL ⊗n (?) with the prescheme induced by C = V(L ) on the open subset D(T ); the complement V (T ) of thisopen subset in C is the underlying space of the closed subprescheme of C defined by the ideal TA[T], which isexactly the null section of C , and so E = D(T ).

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 207

The isomorphism in (8.4.4.2) will be a consequence of the last claim, since V(L −1) is the complementof the part at infinity of its projective closure, and bE is the complement of the null section of the projectiveclosure C = V(L ). But these projective closures are P(L −1⊕OY ) and P(L ⊕OY ) (respectively); but we can writeL ⊕OY =L ⊗ (L −1 ⊕OY ). The existence of the desired canonical isomorphism then follows from (4.1.4), andeverything reduces to showing that this isomorphism swaps the null sections and the parts at infinity. For this,we can reduce to the case where Y = Spec(A) is ane, L = Ac, and L−1 = Ac′, with the canonical isomorphismL ⊗ L−1→ A sending c ⊗ c′ to the element 1 of A. Then S(L ⊕ A) is the tensor product of A[z] with

⊕

n¾0 Ac⊗n,and S(L−1 ⊕ A) is the tensor product of A[z] with

⊕

n¾0 Ac′⊗n, and the isomorphism defined in (4.1.4) sends

zh ⊗ c′⊗(n−h) to the element zn−h ⊗ c⊗h. But, in P(L −1 ⊕OY ), the part at infinity is the set of points where the

section z vanishes, and the null section is the section of points where the section c′ vanishes; since we haveanalogous definitions for P(L ⊕OY ), the conclusion follows immediately from the above explanation.

8.5. Functorial behaviour.

(8.5.1). Let Y and Y ′ be prescheme, q : Y ′→ Y a morphism, and S (resp. S ′) a positively-graded quasi-coherentOY -algebra (resp. positively-graded quasi-coherent OY ′ -algebra). Consider a q-morphism of graded algebras

(8.5.1.1) ϕ : S −→S ′.

We know (1.5.6) that this corresponds, canonically, to a morphism

Φ = Spec(ϕ) : Spec(S ′) −→ Spec(S )

such that the diagram

(8.5.1.2) C ′ Φ //

C

Y ′ q

// Y

commutes, where we write C = Spec(S ) and C ′ = Spec(S ′). Suppose, further, that S0 = OY and S ′0 = OY ′ ; letε : Y → C and ε : Y ′→ C ′ be the canonical immersions (8.3.2); we then have a commutative diagram

(8.5.1.3) Y ′q //

ε ′

Y

ε

C ′Φ// C

which corresponds to the diagram II | 170

Sϕ //

S ′

OY

// OY ′

where the vertical arrows are the augmentation homomorphisms, and so the commutativity follows from thehypothesis that ϕ is assumed to be a homomorphism of graded algebras.

Proposition (8.5.2). — If E (resp. E′) is the pointed ane cone dened by S (resp. S ′), then Φ−1(E) ⊂ E′; if,further, Proj(ϕ) : G(ϕ)→ Proj(S ) is everywhere dened (or, equivalently, if G(ϕ) = Proj(S ′)), then Φ−1(E) = E′, andconversely.

Proof. The first claim follows from the commutativity of (8.5.1.3). To prove the second, we can restrict tothe case where Y = Spec(A) and Y ′ = Spec(A′) are ane, and S = eS and S ′ = eS′. For every homogeneous fin S+, writing f ′ = ϕ( f ), we have that Φ−1(C f ) = C ′f ′ (I, 2.2.4.1); saying that G(ϕ) = Proj(S′) implies that theradical (in S′+) of the ideal generated by the f ′ = ϕ( f ) is S′+ itself ((2.8.1) and (2.3.14)), and this is equivalent tosaying that the C ′f ′ cover E′ (8.3.4.4).

(8.5.3). The q-morphism ϕ canonically extends to a q-morphism of graded algebras

(8.5.3.1) bϕ : cS −→ÓS ′

by letting bϕ(z) = z. This induces a morphism

bΦ = Proj(bϕ) : G(bϕ) −→ bC = Proj(cS )

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 208

such that the diagram

G(bϕ)bΦ //

bC

Y ′ q

// Y

commutes (3.5.6). It follows immediately from the definitions that, if we write i : C → bC and i′ : C ′→ ÒC ′ to meanthe canonical open immersions (8.3.2), then i′(C ′) ⊂ G(bϕ), and the diagram

(8.5.3.2) C ′ Φ //

i

C

i′

G(bϕ)

bΦ//bC

commutes. Finally, if we let X = Proj(S ) and X ′ = Proj(S ′), and if j : X → bC and j′ : X ′→ ÒC ′ are the canonicalclosed immersions (8.3.2), then it follows from the definition of these immersions that j′(G(ϕ)) ⊂ G(bϕ), and thatthe diagram II | 171

(8.5.3.3) G(ϕ)Proj(ϕ) //

j′

X

j

G(bϕ)bΦ//bC

commutes.

Proposition (8.5.4). — If bE (resp. ÒE′) is the pointed projective cone dened by S (resp. by S ′), then bΦ−1(bE) ⊂ ÒE′;furthermore, if p : bE→ X and p′ : ÒE′→ X ′ are the canonical retractions, then p′(bΦ−1(bE)) ⊂ G(bϕ), and the diagram

(8.5.4.1) bΦ−1(bE)bΦ //

p′

bE

p

G(ϕ)

Proj(ϕ)// X

commutes. If Proj(ϕ) is everywhere dened, then so too is bΦ, and we have that bΦ−1(bE) = ÒE′

Proof. The first claim follows from the commutativity of Diagrams (8.5.1.3) and (8.5.3.2), and the twofollowing claims from the definition of the canonical retractions (8.3.5) and the definition of bϕ. To see that bΦ iseverywhere defined whenever Proj(ϕ) is, we can restrict to the case where Y = Spec(A) and Y ′ = Spec(A′) areane, and where S = eS and S ′ = eS′; the hypothesis is that, when f runs over the set of homogeneous elementsof S+, the radical in S′+ of the ideal generated in S′+ by the ϕ( f ) is S′+ itself; we thus immediately conclude thatthe radical in (S′[z])+ of the ideal generated by z and the ϕ( f ) is (S′[z])+ itself, whence our claim; this alsoshows that ÒE′ is the union of the ÒC ′ϕ( f ), and hence equal to bΦ−1(bE).

Corollary (8.5.5). — Whenever Proj(ϕ) is everywhere dened, the inverse image under bΦ of the underlying space ofthe part at innity (resp. of the vertex prescheme) of ÒC ′ is the underlying space of the part at innity (resp. of the vertexprescheme) of bC .

Proof. This follows immediately from (8.5.4) and (8.5.2), taking into account the equalities (8.3.4.1) and(8.3.4.2).

8.6. A canonical isomorphism for pointed cones.

(8.6.1). Let Y be a prescheme, S a quasi-coherent positively-graded OY -algebra such that S0 = OY , and let X bethe Y -scheme Proj(S ). We are going to apply the results of (8.5) to the case where Y ′ = X , and q : X → Y is thestructure morphism; let

(8.6.1.1) SX =⊕

n∈Z

OX (n)

which is a quasi-coherent graded OX -algebra, with multiplication defined by means of the canonical homomor- II | 172

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 209

phisms (3.2.6.1)OX (m)⊗OX

OX (n) −→ OX (m+ n)whose associativity is ensured by the commutative diagram in (2.5.11.4). Let S ′ be the quasi-coherent positively-graded OX -subalgebra S ¾X =

⊕

n¾0 OX (n) of SX .Finally, consider the canonical q-morphism

(8.6.1.2) α : S −→S ¾Xdefined in (3.3.2.3) as a homomorphism S → q∗(SX ), but which clearly sends S to q∗(S ¾X ). Write

(8.6.1.3) CX = Spec(S ¾X ), bCX = Proj(S ¾X [z]), X ′ = Proj(S ¾X )

and denote by EX and bEX the corresponding pointed ane and pointed projective cones (respectively); denotethe canonical morphisms defined in (8.3) by εX : X → CX , iX : C → bCX , jX : X ′ → bCX , pX : bEX → X ′, andπX : EX → X ′.

Proposition (8.6.2). — The structure morphism u : X ′ → X is an isomorphism, and the morphism Proj(α) iseverywhere dened and identical to u. The morphism Proj(bα) : bCX → bC is everywhere dened, and its restrictions to bEX

and EX are isomorphisms to bE and E (respectively). Finally, if we identify X ′ with X via u, then the morphisms pX andπX are identied with the structure morphisms of the X -preschemes bEX and EX .

Proof. We can clearly restrict to the case where Y = Spec(A) is ane, and S = eS; then X is the union ofane open subsets X f , where f runs over the set of homogeneous elements of S+, with the ring of each X f beingS( f ). It follows from (8.2.7.1) that

(8.6.2.1) Γ(X f ,S ¾X ) = S¾f .

So u−1(X f ) = Proj(S¾f ). But if f ∈ Sd (d > 0), then Proj(S¾f ) is canonically isomorphic to Proj((S¾f )(d)) (2.4.7),

and we also know that (S¾f )(d) = (S(d))¾f can be identified with S( f )[T] (2.2.1) by the map T 7→ f /1; we thus

conclude (3.1.7) that the structure morphism u−1(X f )→ X f is an isomorphism, whence the first claim. To provethe second, note that the restriction u−1(X f )∩ G(α)→ X = Proj(S) of Proj(α) corresponds to the canonical mapx 7→ x/1 from S to S¾f (2.6.2); we thus deduce, first of all, that G(α) = X ′, and then, taking into account the factthat u−1(X f ) = (u−1(X f )) f /1, that it follows from (2.8.1.1) that the image of u−1(X f ) under Proj(α) is containedin X f , and the restriction of Proj(α) to u−1(X f ), thought of as a morphism to X f = Spec(S( f )), is indeed identicalto that of u. Finally, applying (8.3.5.4) to pX instead of p, we see that p−1

X (u−1(X f )) = Spec((S¾f )

¶f /1), and this

open subset is, by (8.5.4.1), the inverse image under Proj(bα) of p−1(X f ) = Spec(S¶f ) (8.3.5.3). Taking (8.2.3.2)into account, the restriction of Proj(bα) to p−1

X (u−1(X f )) corresponds to the isomorphism inverse to (8.2.7.2),

restricted to S¶f , whence the third claim; the last claim is evident by definition. II | 173We note also that it follows from the commutative diagram in (8.5.3.2) that the restriction to CX of Proj(bα) is

exactly the morphism Spec(α).

Corollary (8.6.3). — Considered as X -schemes, bEX is canonically isomorphic to Spec(S ¶X ), and EX to Spec(SX ).

Proof. Since we know that the morphisms pX and πX are ane ((8.3.5) and (8.3.6)), it suces (given (1.3.1))to prove the corollary in the case where Y = Spec(A) is ane and S = eS. The first claim follows from theexistence of the canonical isomorphisms (8.2.7.2) (S¾f )

¶f /1

∼−→ S¶f and from the fact that these isomorphisms are

compatible with the map sending f to f g (where f and g are homogeneous in S+). Similarly, applying (8.3.6.2)to πX instead of π, we see that π−1

X (u−1(X f )) = Spec((S¾f ) f /1) for f homogeneous in S+, and the second claim

then follows from the existence of the canonical isomorphisms (8.2.7.2) (S¾f ) f /1∼−→ S f .

We can then say that bCX , thought of as an X -scheme, is given by gluing the ane X -schemes CX = Spec(S ¾X )and bEX = Spec(S ¶X ) over X , where the intersection of the two ane X -schemes is the open subset EX =Spec(SX ).

Corollary (8.6.4). — Assume that OX (1) is an invertible OX -module, and that SX is isomorphic to⊕

n∈Z(OX (1))⊗n

(which will be the case, in particular, whenever S is generated by S1 ((3.2.5) and (3.2.7))). Then the pointed projective conebE can be identied with the rank-1 vector bundle V(OX (−1)) on X , and the pointed ane cone E with the subpreschemeof this vector bundle induced on the complement of the null section. With this identication, the canonical retractionbE→ X is identied with the structure morphism of the X -scheme V(OX (−1)). Finally, there exists a canonical Y -morphismV(OX (1))→ C , whose restriction to the complement of the null section of V(OX (1)) is an isomorphism from this complementto the pointed ane cone E.

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 210

Proof. If we write L = OX (1), then S ¾X is identical to SOX(L ), and so bEX is canonically identified with

V(L −1), by (8.6.3), and CX with V(L ). The morphism V(L )→ C is the restriction of Proj(bα), and the claims ofthe corollary are then particular cases of (8.6.2).

We note that the inverse image under the morphism V(OX (1))→ C of the underlying space of the vertexprescheme of C is the underlying space of the null section of V(OX (1)) (8.5.5); but, in general, the correspondingsubpreschemes of C and of V(OX (1)) are not isomorphic. This problem will be studied below.

8.7. Blowing up based cones.

(8.7.1). Under the conditions of (8.6.1), we have, writing r = Proj(bα), a commutative diagram

(8.7.1.1) XiX εX //

q

bCX

r

Yiε //

bC

by (8.5.1.3) and (8.5.3.2); furthermore, the restriction of r to the complement bCX − iX (εX (X )) of the null section II | 174is an isomorphism to the complement bC − i(ε(Y )) of the null section, by (8.6.2). If we suppose, to simplify things,that Y is ane, that S is of finite type and generated by S1, and that X is projective over Y and bCX projectiveover X (5.5.1), then bCX is projective over Y (5.5.5, ii), and a fortiori over bC (5.5.5, v). We then have a projectiveY -morphism r : bCX → bC (whose restriction to CX is a projective Y -morphism CX → C) that contracts X to Y (?) andthat induces an isomorphism when we restrict to the complements of X and Y . We thus have a connection betweenCX and C , analogous to that which exists between a blow-up prescheme and the original prescheme (8.1.3). Wewill eectively show that CX can be identified with the homogeneous spectrum of a graded OC -algebra.

(8.7.2). Keeping the notation of (8.6.1), consider, for all n¾ 0, the quasi-coherent ideal

(8.7.2.1) S[n] =⊕

m¾n

Sm

of the graded OY -algebra S . It is clear that

(8.7.2.2) S[0] = S , S[n] ⊂ S[m] for m¶ n

(8.7.2.3) SnS[m] ⊂ S[m+n].

Consider the OC -module associated to S[n], which is a quasi-coherent ideal of OC = fS (1.4.4)

(8.7.2.4) In = (S[n])∼.

We thus deduce, from (8.7.2.2) and (8.7.2.3), using (1.4.4) and (1.4.8.1), the analogous formulas

(8.7.2.5) I[0] = OC , I[n] ⊂ I[m] for m¶ n

(8.7.2.6) InI[m] ⊂ I[m+n].

We are thus in the setting of (8.1.1), which leads us to introduce the quasi-coherent graded OC -algebra

(8.7.2.7) S \ =⊕

n¾0

In =

⊕

n¾0

S[n]

∼

.

Proposition (8.7.3). — There is a canonical C -isomorphism

(8.7.3.1) h : CX∼−→ Proj(S \).

Proof. Suppose first of all that Y = Spec(A) is ane, so that S = eS, with S a positively-graded A-algebra,and C = Spec(S). Definition (8.2.7.4) then shows, with the notation of (8.2.6), that S \ = (S\)∼. To define(8.7.3.1), consider a homogeneous element f ∈ Sd (d > 0) and the corresponding element f \ ∈ S\ (8.2.6); theS-isomorphism in (8.2.7.3) then defines a C -isomorphism

(8.7.3.2) Spec(S¾f )∼−→ Spec(S\( f \)).

II | 175

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 211

But with the notation of (8.6.2), if v : CX → X is the structure morphism, then it follows from (8.6.2.1) thatv−1(X f ) = Spec(S¾f ). We also have that Spec(S\( f \)) = D+( f \), which means that (8.7.3.2) defines an isomorphism

v−1(X f )→ D+( f \). Furthermore, if g ∈ Se (e > 0), then the diagram

v−1(X f g)∼ //

D+( f \g\)

v−1(X f )

∼ // D+( f \)

commutes, by definition of the isomorphism in (8.2.7.3). Finally, by definition, S+ is generated by the homogeneousf , and so it follows from (8.2.10, iv) and from (2.3.14) that the D+( f \) form a cover of Proj(S\), and that thev−1(X f ) form a cover of CX , since the X f form a cover of X ; in this case, we have thus defined the isomorphism(8.7.3.1).

To prove (8.7.3) in the general case, it suces to show that, if U and U ′ are ane open subsets of Y , givenby rings A and A′ (respectively), and such that U ′ ⊂ U , then, setting S |U = eS and S |U ′ = eS′, the diagram

(8.7.3.3) CU ′//

Proj(S′\)

CU

// Proj(S\)

commutes. But S is canonically identified with S ⊗A A′, and so S′\ is canonically identified with

S\ ⊗S S′ = S\ ⊗A A′;

thus Proj(S′\) = Proj(S\) ×U U ′ (2.8.10); similarly, if X = Proj(S) and X ′ = Proj(S′), then X ′ = X ×U U ′ and

SX ′ = SX ⊗OUU ′ (3.5.4), or, equivalently, SX ′ = j∗(SX ), where j is the projection X ′→ X . We then (1.5.2) have

that CU ′ = CU ×X X ′ = CU ×U U ′, and the commutativity of (8.7.3.3) is then immediate.

Remark (8.7.4). — (i) The end of the proof of (8.7.3) can be immediately generalised in the followingway. Let g : Y ′ → Y be a morphism, S ′ = g∗(S ), and X ′ = Proj(S ′); then we have a commutativediagram

(8.7.4.1) CX ′//

Proj(S′\)

CX

// Proj(S \)

Now let ϕ : S ′′ → S be a homomorphism of graded OY -algebras such that, if we write X ′′ =Proj(S ′′), then u = Proj(ϕ) : X → X ′′ is everywhere defined; we also have a Y -morphism v : C → C ′′ II | 176(with C ′′ = Spec(S ′′)) such that A (v) = ϕ, and, since ϕ is a homomorphism of graded algebras,ϕ induces a v-morphism of graded algebras ψ : S ′′\ → S \ (1.4.1). Furthermore, it follows from(8.2.10, iv) and from the hypothesis on ϕ that Proj(ψ) is everywhere defined. Finally, taking (3.5.6.1)into account, there is a canonical u-morphism SX ′′ →SX , whence (1.5.6) a morphism w : CX ′′ → CX .With this in mind, the diagram

(8.7.4.2) CX ′′∼ //

w

Proj(S ′′\)

Proj(ψ)

CX∼ // Proj(S \)

is commutative, as we can immediately verify by restricting to the case where Y is ane.(ii) Note that, by (8.7.2.5) and (8.7.2.6), we have I m

1 ⊂ Im ⊂ I1 for all m > 0. But, by definition,I1 = (S+)∼, and so I1 defines the closed subprescheme ε(Y ) in C ((1.4.10) and (8.3.2)); we thusconclude that, for all m> 0, the support of OC/Im is contained in the underlying space of the vertex preschemeε(Y ); on the inverse image of the pointed ane cone E, the structure morphism Proj(S \)→ C thusrestricts to an isomorphism (by (8.7.3) and (8.7.1)). Furthermore, by canonically identifying C with anopen subset of bC (8.3.3), we can clearly extend the ideals Im of OC to ideals Jm of O

bC , by asking for

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 212

it to agree with ObC on the open subset bE of bC . If we define T =

⊕

n¾0Jm, which is a quasi-coherentgraded O

bC -algebra, we can extend the isomorphism (8.7.3.1) to a bC -isomorphism

(8.7.4.3) bCX∼−→ Proj(T ).

Indeed, over bE, it follows from the above that Proj(T ) is canonically identified with bE, and wethus define the isomorphism (8.7.4.3) over bE by asking for it to agree with the canonical isomorphismbEX → bE (8.6.2); it is clear that this isomorphism and (8.7.3.1) then agree over bE.

Corollary (8.7.5). — Suppose that there exists some n0 > 0 such that

(8.7.5.1) Sn+1 = S1Sn for n¾ n0.

Then the vertex subprescheme (?) of CX (isomorphic to X ) is the inverse image under the canonical morphism r : CX → Cof the vertex subprescheme of C (isomorphic to Y ). Conversely, if this property is true, and if we further assume that Y isNoetherian and that S is of nite type, then there exists some n0 > 0 such that (8.7.5.1) holds true.

Proof. The first claim being local on Y , we can assume that Y = Spec(A) is ane, so that S = eS, withS a positively-graded A-algebra. The claim then follows from (8.2.12), since Proj(S\ ⊗S S0) = CX ×C ε(Y ) (bythe identification in (8.7.3.1)), or, in other words, since this prescheme is the inverse image of ε(Y ) in CX(I, 4.4.1). The converse also follows from (8.2.12) whenever Y is Noetherian ane and S is of finite type. If Y is II | 177Noetherian (but not necessarily ane) and S is of finite type, then there exists a finite cover of Y by Noetherianane open subsets Ui , and we then deduce from the above that, for all i, there exists an integer ni such thatSn+1|Ui = (S1|Ui)(Sn|Ui) for n¾ ni ; the largest of the ni then ensures that (8.7.5.1) holds true.

(8.7.6). Now consider the C -prescheme Z given by blowing up the vertex subprescheme ε(Y ) in the ane cone C ;by Definition (8.1.3), it is exactly the prescheme Proj(

⊕

n¾0S n+ ); the canonical injection

(8.7.6.1) ι :⊕

n¾0

S n+ −→S

\

defines (by the identification in (8.7.3)) a canonical dominant C -morphism

(8.7.6.2) G(ι) −→ Z

where G(ι) is an open subset of CX (3.5.1); note that it could be the case that G(ι) 6= CX , as shown by the examplewhere Y = Spec(K), with K a field, and S = eS, with S = K[y], where y is an indeterminate of degree 2; if Rn

denotes the set (S+)n, thought of as a subset of S[n] = S\n, then S\+ is not the radical in S\+ of the ideal generatedby the union of the Rn (cf. (2.3.14)).

Corollary (8.7.7). — Assume that there exists some n0 > 0 such that

(8.7.7.1) Sn = S n1 for n¾ n0.

Then the canonical morphism (8.7.6.2) is everywhere dened, and is an isomorphism CX∼−→ Z . Conversely, if this

property is true, and if we further assume that Y is Noetherian and that S is of nite type, then there exists some n0 suchthat (8.7.7.1) holds true.

Proof. The first claim is local on Y , and thus follows from (8.2.14); the converse follows similarly, arguingas in (8.7.5).

Remark (8.7.8). — Since condition (8.7.7.1) implies (8.7.5.1), we see that, whenever it holds true, not onlycan CX be identified with the prescheme given by blowing up the vertex (identified with Y ) of the ane cone C ,but also the vertex (identified with X ) of CX can be identified with the closed subprescheme given by the inverseimage of the vertex Y of C . Furthermore, hypothesis (8.7.7.1) implies that, on X = Proj(S ), the OX -modulesOX (n) are invertible ((3.2.5) and (3.2.9)), and that OX (n) = L ⊗n with L = OX (1) ((3.2.7) and (3.2.9)); byDefinition (8.6.1.1), CX is thus the vector bundle V(L ) on X , and its vertex is the null section of this vector bundle.

8.8. Ample sheaves and contractions.

(8.8.1). Let Y be a prescheme, f : X → Y a separated and quasi-compact morphism, andL an invertible OX -modulethat is ample relative to f . Consider the positively-graded OY -algebra

(8.8.1.1) S = OY ⊕⊕

n¾1

f∗(L ⊗n)

which is quasi-coherent (I, 9.2.2, a). There is a canonical homomorphisms of graded OX -algebras II | 178

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 213

(8.8.1.2) τ : f ∗(S ) −→⊕

n¾0

L ⊗n

which, in degrees ¾ 1, agrees with the canonical homomorphism σ : f ∗( f∗(L ⊗n))→L ⊗n (0, 4.4.3), and is theidentity in degree 0. The hypothesis that L is f -ample then implies ((4.6.3) and (3.6.1)) that the correspondingY -morphism

(8.8.1.3) r = rL ,τ : X −→ P = Proj(S )

is everywhere defined and is a dominant open immersion, and that

(8.8.1.4) r∗(OP(n)) =L ⊗n for all n ∈ Z.

Proposition (8.8.2). — Let C = Spec(S ) be the ane cone dened by S ; if L is f -ample, then there exists acanonical Y -morphism

(8.8.2.1) g : V = V(L ) −→ C

such that the diagram

(8.8.2.2) Xj //

f

V(L ) π //

g

X

f

Y

ε // Cψ // Y

commutes, where ψ and π are the structure morphisms, and j and ε the canonical immersions sending X and Y (respectively)to the null section of V(L ) and the vertex prescheme of C (respectively). Furthermore, the restriction of g to V(L )− j(X ) isan open immersion

(8.8.2.3) V(L )− j(X ) −→ E = C − ε(Y )into the pointed ane cone E corresponding to S .

Proof. With the notation of (8.8.1), let S ¾P =⊕

n¾0 OP(n) and Cp = Spec(S ¾P ). We know (8.6.2) that thereis a canonical morphism h= Spec(α) : Cp → C such that the diagram

(8.8.2.4) CP//

h

P

p

C

ψ // Y

commutes; furthermore, if εP : P → CP is the canonical immersion, then the diagram

(8.8.2.5) Pp //

εP

CP

h

Yε // C

commutes (8.7.1.1), and, finally, the restriction of H to the pointed ane cone EP is an isomorphism EP∼−→ E

(8.6.2). It follows from (8.8.1.4) thatr∗(S ¾P ) = SOX

(L )and so we have a canonical P -morphism q : V(L )→ CP , with the commutative diagram II | 179

(8.8.2.6) V(L ) π //

q

X

r

CP

// P

identifying V(L ) with the product CP×P X (1.5.2); since r is an open immersion, so too is q (I, 4.3.2). Furthermore,the restriction of q to V(L )− j(X ) sends this prescheme to Ep, by (8.5.2), and the diagram

(8.8.2.7) Xj //

r

V(L )

q

P

εP // CP

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 214

is commutative (since it is a particular case of (8.5.1.3)). The claims of (8.8.2) immediately follow from thesefacts, by taking g to be the composite morphism h q.

Remark (8.8.3). — Assume further that Y is a Noetherian prescheme, and that f is a proper morphism. Sincer is then proper(5.4.4), and thus closed, and since it is also a dominant open immersion, r is necessarily anisomorphism X

∼−→ P. Furthermore, we will see, in Chapter III (III, 2.3.5.1), that S is then necessarily an

OY -algebra of nite type. It then follows that S \ is an S \0 -algebra of nite type ((8.2.10, i) and (8.7.2.7)); sinceCP is C -isomorphic to Proj(S \) (8.7.3), we see that the morphism h : CP → C is projective; since the morphismr is an isomorphism, so too is q : V(L ) → CP , and we thus conclude that the morphism g : V(L ) → C isprojective. Furthermore, since the restriction of h to EP is an isomorphism to E, and since q is an isomorphism,the restriction (8.8.2.3) of g is an isomorphism V(L )− j(X )

∼−→ E.

If we further assume that L is very ample for f , then, as we will also see in Chapter III (III, 2.3.5.1), thereexists some integer n0 > 0 such that Sn = S n

1 for n ¾ n0. We then conclude, by (8.7.7), that V(L ) can beidentified with the prescheme Z given by blowing up the vertex prescheme (identified with Y ) in the ane cone C ,and that the null section of V(L ) (identified with Y ) is the inverse image of the vertex subprescheme Y of C .

Some of the above results can in fact be proven even without the Noetherian hypothesis:

Corollary (8.8.4). — Let Y be a prescheme (resp. a quasi-compact scheme), f : X → Y a proper morphism, and Lan invertible OX -module that is ample relative to f . Then the morphism in (8.8.2.1) is proper (resp. projective), and itsrestriction (8.8.2.3) is an isomorphism.

Proof. To prove that g is proper, we can restrict to the case where Y is ane, and it then suces to considerthe case where Y is a quasi-compact scheme. The same arguments as in (8.8.3) first of all show that r is anisomorphism X

∼−→ P; then q is also an isomorphism, and, since the restriction of h to EP is an isomorphism

EP∼−→ E, we have already seen that (8.8.2.3) is an isomorphism. It remains only to prove that g is projective.Since f is of finite type, by hypothesis, we can apply (3.8.5) to the homomorphism τ from (8.8.1.2): there is II | 180

an integer d > 0 and a quasi-coherent OY -submodule E of finite type of Sd such that, if S ′ is the OY -subalgebraof S generated by E , and τ′ = τ q∗(ϕ) (where ϕ is the canonical injection S ′ → S ), then r ′ = rL ,τ′ is animmersion

X −→ P ′ = Proj(S ′).Furthermore, since ϕ is injective, r ′ is also a dominant immersion (3.7.6); the same argument as for r then shows

that r ′ is a surjective closed immersion; since r ′ factors as Xr−→ Proj(S )

Φ−→ Proj(S ′), where Φ = Proj(ϕ), we thus

conclude that Φ is also a surjective closed immersion. But this implies that Φ is an isomorphism; we can restrictto the case where Y = Spec(A) is ane, and S = eS and S ′ = eS′, with S a graded A-algebra and S′ a gradedsubalgebra of S. For every homogeneous element t ∈ S′, we have that S′(t) is a subring of S(t); if we return tothe definition of Proj(ϕ) (2.8.1), we see that it suces to prove that, if B′ is a subring of a ring B, and if themorphism Spec(B)→ Spec(B′) corresponding to the canonical injection B′ → B is a closed immersion, thenthis morphism is necessarily an isomorphism; but this follows from (I, 4.2.3). Furthermore, Φ∗(OP ′(n)) = OP(n)((3.5.2, ii) and (3.5.4)), and so r

′∗(OP ′(n)) is isomorphic to L ⊗n (4.6.3). Let S ′′ = S′(d), so that (3.1.8, i) X is

canonically identified with P ′′ = Proj(S ′′), and L ′′ =L ⊗d with OP ′′(1) (3.2.9, ii).Now, if C ′′ = Spec(S ′′), then S ¾P ′′ =

⊕

n¾0 OP ′′(n) can be identified with⊕

n¾0L ′′⊗n, and thus CP ′′ =Spec(S ¾P ′′) with V(L ′′); we also know (8.7.3) that CP ′′ is C ′′-isomorphic to Proj(S ′′\); by the definition of S ′′,we know that S ′′\ is generated by S ′′\1 , and that S ′′\1 is of finite type over S ′′\0 = S ′′ ((8.2.10, i and iii)), and soProj(S ′′\) is projective over C ′′ (5.5.1). Consider the diagram

(8.8.4.1) V(L )g //

u

Spec(S ) = C

v

V(L ′′)g ′′ // Spec(S ′′) = C ′′

where g and g ′′ correspond, by (1.5.6), to the canonical j-morphisms

S −→⊕

n¾0

L ⊗n and S ′′ −→⊕

n¾0

L ′′⊗n

(3.3.2.3) (see (8.8.5) below), and v and u to the inclusion morphisms S ′′→S and⊕

n¾0L ⊗nd →⊕

n¾0L ⊗n

(respectively); it is immediate (3.3.2) that this diagram is commutative. We have just seen that g ′′ is a projectivemorphism; we also know that u is a nite morphism. Since the question is local on X , we can assume that X is

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 215

ane of ring A, and that L = OX ; everything then reduces to noting that the ring A[T] is a module of finite typeover its subring A[T d] (with T an indeterminate). Since Y is a quasi-compact scheme, and since C ′′ is aneover Y , we know that C ′′ is also a quasi-compact scheme, and so g ′′ u is a projective morphism (5.5.5, ii); by II | 181commutativity of (8.8.4.1), v g is also projective, and, since v is ane, thus separated, we finally conclude thatg is projective (5.5.5, v).

(8.8.5). Consider again the situation in (8.8.1). We will see that the morphism g : V(L )→ C can be also bedefined in a way that works for any invertible (but not necessarily ample) OX -module L . For this, consider thef -morphism

(8.8.5.1) τ[ : S −→⊕

n¾0

L ⊗n

corresponding to the morphism τ from (8.8.1.2). This induces (1.5.6) a morphism g ′ : V → C such that, ifπ : V → X and ψ : C → Y are the structure morphisms, the diagrams

(8.8.5.2) X

f

Vπoo

g ′

Y C

ψoo

Xj //

f

V

g ′

Y

ε // C

commute ((8.5.1.2) and (8.5.1.3)). We will show that (if we assume that L is f -ample) the morphisms g and g ′ areidentical.

The question being local on Y , we can assume that Y = Spec(A) is ane, and (by (8.8.1.3)) identifyX with an open subset of P = Proj(S), where S = A⊕

⊕

n¾0 Γ(X ,L ⊗n); we then deduce, by (8.8.1.4), thatΓ(X ,OP(n)) = Γ(X ,L ⊗n) for all n ∈ Z. Taking into account the definition of h = Spec(α), where α is thecanonical p-morphism eS→S ¾P (8.6.1.2), we have to show that the restriction to X of α] : p∗(eS)→S ¾P is identicalto τ. Taking (0, 4.4.3) into account, it suces to show that, if we compose the canonical homomorphismαn : Sn → Γ(P,OP(n)) with the restriction homomorphism Γ(P,OP(n)) → Γ(X ,OP(n)) = Γ(X ,L ⊗n), then weobtain the identity, for all n > 0; but this follows immediately from the definition of the algebra S and of αn(2.6.2).

Proposition (8.8.6). — Assume (with the notation of (8.8.5)) that, if we write f = ( f0,λ), then the homomorphismλ : OY → j∗(OX ) is bijective; then:

(i) if we write g = (g0,µ), then µ : OC → g∗(OV ) is an isomorphism; and(ii) if X is integral (resp. locally integral and normal), then C is integral (resp. normal).

Proof. Indeed, the f -morphism τ[ is then an isomorphism

τ[ : S = ψ∗(OC) −→ f∗(π∗(OV )) = ψ∗(g∗(OV ))

and the Y -morphism g can be considered as that for which the homomorphism A (g) (1.1.2) is equal to τ[. Tosee that µ is an isomorphism of OC -modules, it suces (1.4.2) to see that A (µ) : ψ∗(OC)→ ψ∗(g∗(OV )) is anisomorphism. But, by Definition (1.1.2), we have that A (µ) =A (g), whence the conclusion of (i).

To prove (ii), we can restrict to the case where Y is ane, and so S = eS, with S =⊕

n¾0 Γ(X ,L ⊗n); the II | 182hypothesis that X is integral implies that the ring S is integral (I, 7.4.4), and thus so too is C (I, 5.1.4). To showthat C is normal, we will use the following lemma:

Lemma (8.8.6.1). — Let Z be a normal integral prescheme. Then the ring Γ(Z ,OZ) is integral and integrally closed.

Proof. It follows from (I, 8.2.1.1) that Γ(Z ,OZ) is the intersection, in the field of rational functions R(Z), ofthe integrally closed rings Oz over all z ∈ Z .

With this in mind, we first show that V is locally integral and normal; for this, we can restrict to the casewhere X = Spec(A) is ane, with ring A integral and integrally closed (6.3.8), and where L = OX . Since thenV = Spec(A[T]), and A[T] is integral and integrally closed [Jaf60, p. 99], this proves our claim. For every aneopen subset U of C , g−1(U) is quasi-compact, since the morphism g is quasi-compact; since V is locally integral,the connected components of g−1(U) are open integral preschemes in g−1(U), and thus finite in number, and,since V is normal, these preschemes are also normal (6.3.8). Then Γ(U ,OC), which is equal to Γ(g−1(U),OV ),by (i), is the direct sum (?) of finitely-many integral and integrally closed rings (8.8.6.1), which proves that C isnormal (6.3.4).

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 216

8.9. Grauert’s ampleness criterion: statement. We intend to show that the properties proven in (8.8.2)characterise f -ample OX -modules, and, more precisely, to prove the following criterion:

Theorem (8.9.1). — (Grauert’s criterion). Let Y be a prescheme, p : X → Y a separated and quasi-compact morphism,andL an invertible OX -module. ForL to be ample relative to p, it is necessary and sucient for there to exist a Y -preschemeC , a Y -section ε : Y → C of C , and a Y -morphism q : V(L )→ C , satisfying the following properties:

(i) the diagram

(8.9.1.1) Xj //

p

V(L )

q

Y

ε // C

commutes, where j is the null section of the vector bundle V(L ); and(ii) the restriction of q to V(L )− j(X ) is a quasi-compact open immersion

V(L )− j(X ) −→ X

whose image does not intersect ε(Y ).

Note that, if C is separated over Y , we can, in condition (ii), remove the hypothesis that the open immersionis quasi-compact; to see that this property (of quasi-compactness) is in fact a consequence of the other conditions,we can restrict to the case where Y is ane, and the claim then follows from (I, 5.5.1)i and (I, 5.5.10). We canalso remove the same hypothesis if we assume that X is Noetherian, since then V is also Noetherian, and the II | 183claim follows from (I, 6.3.5).

Corollary (8.9.2). — If the morphism p : X → Y is proper, then we can, in the statement of Theorem (8.9.1), assumethat q is proper, and replace “open immersion” by “isomorphism”.

In a more suggestive manner, we can say (whenever p : X → Y is proper) that L is ample relative to p if andonly if we can “ contract” the null section of the vector bundle V(L ) to the base prescheme Y . An important particularcase is that where Y is the spectrum of a field, and where the operation of “contraction” consists of contract thenull section V(L ) to a single point.

(8.9.3). The necessity of the conditions in Theorem (8.9.1) and Corollary (8.9.2) follow immediately from (8.8.2)and (8.8.4).

To show that the conditions of (8.9.1) suces, consider a slightly more general situation. For this, let (withthe notation of (8.8.2))

S ′ =⊕

n¾0

L ⊗n

andV = V(L ) = Spec(S ′).

The closed subprescheme j(X ), null section of V(L ), is defined by the quasi-coherent sheaf of idealsJ = (S ′+)

∼ of OV (1.4.10). This OV -module is invertible, since this property is local on X , and this reduces toremarking that the ideal TA[T] in a ring of polynomials A[T] is a free cyclic A[T]-module. Furthermore, it isimmediate (again, because the question is local on X ) that

L = j∗(J )

andj∗(L ) = J /J 2.

Now, ifπ : V(L ) −→ X

is the structure morphism, then π∗(J ) = S ′+ and π∗(J /J 2) =L ; there are thus canonical homomorphismsL → π∗(J )→ L , the first being the canonical injection L → S ′+, and the second the canonical projectionfrom S ′+ to S ′1 =L , and their composition being the identity. We can also canonically embed π∗(J ) = S ′+ =⊕

n¾1L ⊗n into the product∏

n¾1L⊗n = lim←−n

π∗(J /J n+1) (since π∗(J /J n+1) =L ⊕L ⊗2⊕ . . .⊕L ⊗n), and wethus have canonical homomorphisms

(8.9.3.1) L −→ lim←−π∗(J /Jn+1) −→L

whose composition is the identity.With this in mind, the generalisation of (8.9.1) that we are going to prove is the following:

8. BLOWUP SCHEMES; BASED CONES; PROJECTIVE CLOSURE 217

Proposition (8.9.4). — Let Y be a prescheme, V a Y -prescheme, and X a closed subprescheme of V dened by an idealJ of OV , which is an invertible OV -module; if j : X → V is the canonical injection, then let L = j∗(J ) = J ⊗OV

OX , II | 184so that j∗(L ) = J /J 2. Assume that the structure morphism p : X → Y is separated and quasi-compact, and that thefollowing conditions are satised:

(i) there exists a Y -morphism π : V → X of nite type such that π j = 1X , and so π∗(J /J 2) =L ;(ii) there exists a homomorphism of OX -modules ϕ :L → lim←−π∗(J /J

n+1) such that the composition

Lϕ−→ lim←−π∗(J /J

n+1)α−→ π∗(J /J 2) =L

(where α is the canonical homomorphism) is the identity;(iii) there exists a Y -prescheme C , a Y -section ε of C , and a Y -morphism q : V → C such that the diagram

(8.9.4.1) Xj //

p

V

q

Y ε

// C

commutes; and(iv) the restriction of q to W = V − j(X ) is a quasi-compact open immersion into C , whose image does not intersect

ε(Y ).Then L is ample relative to p.

8.10. Grauert’s ampleness criterion: proof.

Lemma (8.10.1). —

8.11. Uniqueness of contractions.

8.12. Quasi-coherent sheaves on based cones.

(8.12.1). Let us take the hypotheses and notation of (8.3.1). LetM be a quasi-coherent graded S -module; to avoidany confusion, we denote by ÝM the quasi-coherent OC -module associated toM (1.4.3) whenM is considered as II | 192a nongraded S -module, and by Proj0(M ) the quasi-coherent OX -module associated toM ,M being consideredthis time as a graded S -module (in other words, the OX -module denoted by ÝM in (3.2.2)). In addition, we set

(8.12.1.1) MX =Proj0(M ) =⊕

n∈Z

Proj0(M (n));

the quasi-coherent graded OX -algebra SX being defined by (8.6.1.1), Proj(M ) is equipped with a structure of a(quasi-coherent) graded SX -module, by means of the canonical homomorphisms (3.2.6.1)

(8.12.1.2) OX (m)⊗OXProj0(M (n)) −→Proj0(S (m)⊗S M (n)) −→Proj0(M (m+ n)),

the verification of the axioms of sheaves of modules being done using the commutative diagram in (2.5.11.4).If Y = Spec(A) is ane, S = eS, andM = eM , where S is a graded A-algebra and M is a graded S-module,

then, for every homogeneous element f ∈ S+, we have

(8.12.1.3) Γ(X f ,Proj( eM)) = M f

by the definitions and (8.2.9.1).Now consider the quasi-coherent graded cS -module

(8.12.1.4) ÓM =M ⊗S cS

(cS defined by (8.3.1.1)); we deduce a quasi-coherent graded ObC -module Proj0( ÓM ), which we will also denote by

(8.12.1.5) M =Proj0( ÓM ).It is clear (3.2.4) thatM is an additive functor which is exact inM , commuting with direct sums and with

inductive limits.

CHAPTER III

Cohomological study of coherent sheaves (EGA III)

build hack [CC]

Summary

§1. Cohomology of ane schemes.§2. Cohomological study of projective morphisms.§3. Finiteness theorem for proper morphisms.§4. The fundamental theorem of proper morphisms. Applications.§5. An existence theorem for coherent algebraic sheaves.§6. Local and global Tor functors; Künneth formula.§7. Base change for homological functors of sheaves of modules.§8. The duality theorem for projective bundles§9. Relative cohomology and local cohomology; local duality§10. Relations between projective cohomology and local cohomology. Formal completion technique along a divisor§11. Global and local Picard groups1

This chapter gives the fundamental theorems concerning the cohomology of coherent algebraic sheaves,with the exception of theorems explaining the theory of residues (duality theorems), which will be the subject ofa later chapter. Amongst all those included here, there are essentially six fundamental theorems, and each one isthe subject of one of the first six chapters. These results will prove to be essential tools in all that follows, even inquestions which are not truly cohomological in their nature; the reader will see the first such examples startingfrom §4. §7 gives some more technical results, but ones which are constantly used in applications. Finally, in§§8–11, we will develop certain results, related to the duality of coherent sheaves, that are particularly importantfor applications, and which can be explained even before the introduction of the full general theory of residues.

The content of §§1 and 2 is due to J.-P. Serre, and the reader will observe that we have had only to follow(FAC). §8 and 9 are equally inspired by (FAC) (the changes necessitated by the dierent contexts, however,being less evident). Finally, as we said in the Introduction, §4 should be considered as the formalisation, inmodern language, of the fundamental “invariance theorem” of Zariski’s “theory of holomorphic functions”.

We draw attention to the fact that the results of no3.4 (and the preliminary propositions of (0, 13.4 to 13.7))will not be used in what follows Chapter III, and can thus be skipped in a first reading.

§1. Cohomology of ane schemes

1.1. Review of the exterior algebra complex.

(1.1.1). Let A be a ring, f= ( fi)1¶i¶r a system of r elements of A. The exterior algebra complex K•(f) correspondingto f is a chain complex (G, I, 2.2) defined in the following way: the graded A-module K•(f) is equal to the exterioralgebra ∧(Ar), graded in the usual way, and the boundary map is the interior multiplication if by f considered asan element of the dual (Ar)∨; we recall that if is an antiderivation of degree −1 of ∧(Ar), and if (ei)1¶i¶r is thecanonical basis of Ar , then we have if(ei) = fi ; the verification of the condition if if = 0 is immediate.

An equivalent definition is the following: for each i, we consider a chain complex K•( fi) defined as follows:K0( fi) = K1( fi) = A, Kn( fi) = 0 for n 6= 0,1: the boundary map is defined by the condition that d1 : A→ A ismultiplication by fi . We then take K•(f) to be the tensor product K•( f1)⊗ K•( f2)⊗ · · · ⊗ K•( fr) (G, I, 2.7) with itstotal degree; the verification of the isomorphism from this complex to the complex defined above is immediate.

1EGA IV does not depend on §§8–11, and will probably be published before these chapters. [Trans.] These last four chapters were neverpublished.

218

1. COHOMOLOGY OF AFFINE SCHEMES 219

(1.1.2). For every A-module M , we define the chain complex

(1.1.2.1) K•(f, M) = K•(f)⊗A M

and the cochain complex (G, I, 2.2)

(1.1.2.2) K•(f, M) = HomA(K•(f, M).

If g is a k-cochain of this latter complex, and if we set

g(i1, . . . , ik) = g(ei1 ∧ · · · ∧ eik),

then g identifies with an alternating map from [1, r]k to M , and it follows from the above definitions that we have

(1.1.2.3) dk g(i1, i2, . . . , ik+1) =k+1∑

h=1

(−1)h−1 fih g(i1, . . . , bih, . . . , ik+1).

III | 83(1.1.3). From the above complexes, we deduce as usual the homology and cohomology A-modules (G, I, 2.2)

(1.1.3.1) H•(f, M) = H•(K•(f, M)),

(1.1.3.2) H•(f, M) = H•(K•(f, M)).

We define an A-isomorphism K•(f, M) ' K•(f, M) by sending each chain z =∑

(ei1 ∧ · · · ∧ eik) ⊗ zi1,...,ikto the cochain gz such that gz( j1, . . . , jr−k) = εzi1,...,ik , where ( jh)1¶h¶r−k is the strictly increasing sequencecomplementary to the strictly increasing sequence (ih)1¶h¶k in [1, r] and ε = (−1)ν , where ν is the numberof inversions of the permutation i1, . . . , ik, j1, . . . , jr−k of [1, r]. We verify that gdz = d(gz), which gives anisomorphism

(1.1.3.3) Hi(f, M)' Hr−i(f, M) for 0¶ i ¶ r.

In this chapter, we will especially consider the cohomology modules H•(f, M).For a given f, it is immediate (G, I, 2.1) that M 7→ H•(f, M) is a cohomological functor (T, II, 2.1) from the

category of A-modules to the category of graded A-modules, zero in degrees < 0 and > r. In addition, we have

(1.1.3.4) H0(f, M) = HomA(A/(f), M),

denoting by (f) the ideal of A generated by f1, . . . , fr ; this follows immediately from (1.1.2.3), and it is clear thatH0(f, M) identifies with the submodule of M killed by (f). Similarly, we have by (1.1.2.3) that

(1.1.3.5) Hr(f, M) = M/ r∑

i=1

fi M

= (A/(f))⊗A M .

We will use the following known result, which we will recall a proof of to be complete:

Proposition (1.1.4). — Let A be a ring, f = ( fi)1¶i¶r a nite family of elements of A, and M an A-module. If, for1¶ i ¶ r , the scaling z 7→ fi · z on Mi−1 = M/( f1M + · · ·+ fi−1M) is injective, then we have Hi(f, M) = 0 for i 6= r .

It suces to prove that Hi(f, M) = 0 for all i > 0 according to (1.1.3.3). We argue by induction on r, thecase r = 0 being trivial. Set f′ = ( fi)1¶i¶r−1; this family satisfies the conditions in the statement, so if we setL• = K•(f′, M), then we have Hi(L•) = 0 for i > 0 by hypothesis, and H0(L•) = Mr−1 by virtue of (1.1.3.3) and(1.1.3.5). To abbreviate, set K• = K•( fr) = K0 ⊕ K1, with K0 = K1 = A, d1 : K1→ K0 multiplication by fr ; we haveby definition (1.1.1) that K•(f, M) = K• ⊗A L•. We have the following lemma:

Lemma (1.1.4.1). — Let K• be a chain complex of free A-modules, zero except in dimensions 0 and 1. For every chaincomplex L• of A-modules, we have an exact sequence

0 −→ H0(K• ⊗Hp(L•)) −→ Hp(K• ⊗ L•) −→ H1(K• ⊗Hp−1(L•)) −→ 0

for every index p.III | 84

This is a particular case of an exact sequece of low-order terms of the Künneth spectral sequence (M, XVII,5.2 (a) and G, I, 5.5.2); it can be proved directly as follows. Consider K0 and K1 as chain complexes (zero indimensions 6= 0 and 6= 1 respectively); we then have an exact sequence of complexes

0 −→ K0 ⊗ L• −→ K• ⊗ L• −→ K1 ⊗ L• −→ 0,

to which we can apply the exact sequence in homology

· · · −→ Hp+1(K1 ⊗ L•)∂−→ Hp(K0 ⊗ L•) −→ Hp(K• ⊗ L•) −→ Hp(K1 ⊗ L•)

∂−→ Hp−1(K0 ⊗ L•) −→ · · · .

1. COHOMOLOGY OF AFFINE SCHEMES 220

But it is evident that Hp(K0⊗ L•) = K0⊗Hp(L•) and Hp(K1⊗ L•) = K1⊗Hp−1(L•) for all p; in addition, we verifyimmediately that the operator ∂ : K1 ⊗Hp(L•)→ K0 ⊗Hp(L•) is none other than d1 ⊗ 1; the lemma thus followsfrom the above exact sequence and the definition of H0(K• ⊗Hp(L•)) and H1(K• ⊗Hp−1(L•)).

The lemma having been established, the end of the proof of Proposition (1.1.4) is immediate: the inductionhypothesis of Lemma (1.1.4.1) gives Hp(K• ⊗ L•) = 0 for p ¾ 2; in addition if we show that H1(K•,H0(L•)) = 0,then we also deduce from Lemma (1.1.4.1) that H1(K• ⊗ L•) = 0; but by definition, H1(K•, H0(L•)) is none otherthan the kernel of the scaling z 7→ fr · z on Mr−1, and as by hypothesis this kernel is zero, this finishes the proof.

(1.1.5). Let g = (gi)1¶i¶r be a second sequence of r elements of A, and set fg = ( fi gi)1¶i¶r . We can define acanonical homomorphism of complexes

(1.1.5.1) φg : K•(fg) −→ K•(f)

as the canonical extension to the exterior algebra ∧(Ar) of the A-linear map (x1, . . . , x r) 7→ (g1 x1, . . . , gr x r) fromAr to itself. To see that we have a homomorphism of complexes, it suces to note, in general, that if u : E→ Fis an A-linear map, and if x ∈ F∨ and y= tu(x) ∈ E∨, then we have the formula

(1.1.5.2) (∧u) iy = ix (∧u);

indeed, the two elements are antiderivations of ∧F , and it sues to check that they coincide on F , which followsimmediately from the definitions.

When we identify K•(f) with the tensor product of the K•( fi) (1.1.1), φg is the tensor product of the φgi,

where φgiis the identity in degree 0 and multiplication by gi in degree 1.

(1.1.6). In particular, for every pair of integers m and n such that 0 ¶ n ¶ m, we have homomorphisms ofcomplexes

(1.1.6.1) φfm−n : K•(fm) −→ K•(f

n)

and as a result, homomorphisms

(1.1.6.2) φfm−n : K•(fn, M) −→ K•(fm, M),

(1.1.6.3) φfm−n : H•(fn, M) −→ H•(fm, M).III | 85

The latter homomorphisms evidently satisfy the transitivity condition φfm−p = φfm−n φfn−p for p ¶ n ¶ m;they therefore define two inductive systems of A-modules; we set

(1.1.6.4) C•((f), M) = lim−→n

K•(fn, M),

(1.1.6.5) H•((f), M) = H•(C•((f), M)) = lim−→n

H•(fn, M),

the last equality following from the fact that passing to the inductive limit commutes with the functor H• (G,I, 2.1). We will later see (1.4.3) that H•((f), M) does not depend on the ideal (f) of A (and similarly on the(f)-pre-adic topology on A), which justifies the notations.

It is clear that M 7→ C•((f), M) is an exact A-linear functor, and M 7→ H•((f), M) is a cohomological functor.

(1.1.7). Set f = ( fi) ∈ Ar and g = (gi) ∈ Ar ; denote by eg the left multiplication by the vector g ∈ Ar on theexterior algebra ∧(Ar); we know that we have the homotopy formula

(1.1.7.1) ifeg + egif = ⟨g, f⟩1

in the A-module Ar (1 denotes the identity automorphism of Ar); this relation also implies that in the complexK•(f) we have

(1.1.7.2) deg + egd = ⟨g, f⟩1.

If the ideal (f) is equal to A, then there exists a g ∈ Ar such that ⟨g, f⟩=∑r

i=1 gi fi = 1. As a result (G, I, 2.4):

Proposition (1.1.8). — Suppose that the ideal (f) generated by the fi is equal to A. Then the complex K•(f) ishomotopically trivial, and so are the complexes K•(f, M) and K•(f, M) for every A-module M .

Corollary (1.1.9). — If (f) = A, then we have H•(f, M) = 0 and H•((f), M) = 0 for every A-module M .

Proof. Indeed, we then have (fn) = A for all n.

1. COHOMOLOGY OF AFFINE SCHEMES 221

Remark (1.1.10). — With the same notations as above, set X = Spec(A) and Y the closed subpreschemeof X defined by the ideal (f). We will prove in §9 that H•((f), M) is isomorphic to the cohomology H•Y (X , eM)corresponding to the antifilter Φ of closed subsets of Y (T, 3.2). We will also show that Proposition (1.2.3)applied to X and to F = eM is a particular case of an exact sequence in cohomology

· · · −→ HpY (X ,F ) −→ Hp(X ,F ) −→ Hp(X − Y,F ) −→ Hp+1

Y (X ,F ) −→ · · · .

1.2. Čech cohomology of an open cover.

Notation (1.2.1). — In this section, we denote:(1) X a prescheme;(2) F a quasi-coherent OX -module; III | 86(3) A= Γ(X ,OX ), M = Γ(X ,F );(4) f= ( fi)1¶i¶r a finite system of elements of A;(5) Ui = X fi

, the open set (0, 5.5.2) of the x ∈ X such that fi(x) 6= 0;(6) U =

⋃ri=1 Ui ;

(7) U the cover (Ui)1¶i¶r of U .

(1.2.2). Suppose that X is either a prescheme whose underlying space is Noetherian or a scheme whose underlyingspace is quasi-compact. We then know (I, 9.3.3) that we have Γ(Ui ,F ) = M fi

. We set

Ui0 i1···ip=

p⋂

k=0

Uik = X fi0fi1··· fip

(0, 5.5.3); so we also have

(1.2.2.1) Γ(Ui0 i1···ip,F ) = M fi0

fi1··· fip

.

We have (0, 1.6.1) that M fi0fi1··· fip

identifies with the inductive limit lim−→nM (n)

i0 i1···ip, where the inductive system is

formed by the M (n)i0 i1···ip

= M , the homomorphisms φnm : M (m)i0 i1···ip

→ M (n)i0 i1···ip

being multiplication by ( fi0 fi1 · · · fip)n−m

for m ¶ n. We denote by C pn (M) the set of alternating maps from [1, r]p+1 to M (for all n); these A-modules

also form an inductive system with respect to the φnm. If C p(U,F ) is the group of alternating Čech p-cochainsrelative to the cover U, with coecients in F (G, II, 5.1), then it follows from the above that we can write

(1.2.2.2) C p(U,F ) = lim−→n

C pn (M).

With the notations of (1.1.2), C pn (M) identifies with K p+1(fn, M), and the map φnm identifies with the map

φfn−m defined in (1.1.6). We thus have, for every p ¾ 0, a canonical functorial isomorphism

(1.2.2.3) C p(U,F )' C p+1((f), M).

In addition, the formula (1.1.2.3) and the definition of the cohomology of a cover (G, II, 5.1) shows that theisomorphisms (1.2.2.3) are compatible with the coboundary maps.

Proposition (1.2.3). — If X is a prescheme whose underlying space in Noetherian or a scheme whose underlying space isquasi-compact, then there exists a canonical functorial isomorphism in F

(1.2.3.1) Hp(U,F )' Hp+1((f), M) for p ¾ 1.

In addition, we have a functorial exact sequence in F

(1.2.3.2) 0 −→ H0((f), M) −→ M −→ H0(U,F ) −→ H1((f), M) −→ 0.III | 87

Proof. The isomorphisms (1.2.3.1) are immediate consequences of what we saw in (1.2.2). On the other hand,we have C0(U,F ) = C1((f), M); as a result, H0(U,F ) identifies with the subgroup of 1-cocycles of C1((f), M); asM = C0((f), M), the exact sequence (1.2.3.2) is none other than the one given by the definition of the cohomologygroups H0((f), M) and H1((f), M).

Corollary (1.2.4). — Suppose that the X fiare quasi-compact and that there exists gi ∈ Γ(U ,F ) such that

∑

i gi( fi |U) =1|U . Then for every quasi-coherent (OX |U)-module G , we have Hp(U,G ) = 0 for p > 0; if in addition U = X , then thecanonical homomorphism (1.2.3.2) M → H0(U,F ) is bijective.

Proof. As by hypothesis the Ui = X fiare quasi-compact, so is U , and we can reduce to the case where U = X ;

the hypothesis then implies that Hp((f), M) = 0 for all p ¾ 0 (1.1.9). The corollary then follows immediatelyfrom (1.2.3.1) and (1.2.3.2).

1. COHOMOLOGY OF AFFINE SCHEMES 222

We note that since H0(U,F ) = H0(U ,F ) (G, II, 5.2.2), we have again proved (I, 1.3.7) as a special case.

Remark (1.2.5). — Suppose that X is an ane scheme; then the Ui = X fi= D( fi) are ane open sets, as well as

the Ui0 i1···ip(but U is not necessarily ane). In this case, the functors Γ(X ,F ) and Γ(Ui0 i1···ip

,F ) are exact inF (I, 1.3.11). If we have an exact sequence 0→F ′→F →F ′′→ 0 of quasi-coherent OX -modules, then thesequence of complexes

0 −→ C•(U,F ′) −→ C•(U,F ) −→ C•(U,F ′′) −→ 0is exact, and thus gives an exact sequence in cohomolology

· · · −→ Hp(U,F ) −→ Hp(U,F ) −→ Hp(U,F ′′)∂−→ Hp+1(U,F ′) −→ · · · .

On the other hand, if we set M ′ = Γ(X ,F ′) and M ′′ = Γ(X ,F ′′), then the sequence 0→ M ′→ M → M ′′→ 0 isexact; as C•((f), M) is an exact functor in M , we also have the exact sequence in cohomology

· · · −→ Hp((f), M ′) −→ Hp((f), M) −→ Hp((f), M ′′)∂−→ Hp+1((f), M ′) −→ · · · .

This being so, as the diagram

0 // C•(U,F ′) //

C•(U,F ) //

C•(U,F ′′) //

0

0 // C•((f), M ′) // C•((f), M) // C•((f), M ′′) // 0

is commutative, we conclude that the diagrams III | 88

(1.2.5.1) Hp(U,F ′′) ∂ //

Hp+1(U,F ′)

Hp+1((f), M ′′) ∂ // Hp+2((f), M ′)

are commutative for all p (G, I, 2.1.1).

1.3. Cohomology of an ane scheme.

Theorem (1.3.1). — Let X be an ane scheme. For every quasi-coherent OX -module F , we have Hp(X ,F ) = 0 for allp > 0.

Proof. Let U be a finite cover of X by the ane open sets X fi= D( fi) (1 ¶ i ¶ r); we then know that the

ideal of A= Γ(X ,OX ) generated by the fi is equal to A. We thus conclude from Corollary (1.2.4) that we haveHp(U,F ) = 0 for p > 0. As there are finite covers of X by ane open sets which are arbitrarily fine (I, 1.1.10),the definition of Čech cohomology (G, II, 5.8) shows that we also have Hp(X ,F ) = 0 for p > 0. But this alsoapplies to every prescheme X f for f ∈ A (I, 1.3.6), hence Hp(X f ,F ) = 0 for p > 0. As we have X f ∩ X g = X f g ,we deduce that we also have Hp(X ,F ) = 0 for all p > 0, by virtue of (G, II, 5.9.2).

Corollary (1.3.2). — Let Y be a prescheme, f : X → Y an ane morphism (II, 1.6.1). For every quasi-coherentOX -module F , we have Rq f∗(F ) = 0 for q > 0.

Proof. By definition Rq f∗(F ) is the OY -module associated to the presheaf U 7→ Hq( f −1(U),F ), where Uvaries over the open subsets of Y . But the ane open sets form a basis for Y , and for such an open set U , f −1(U)is ane (II, 1.3.2), hence Hq( f −1(U),F ) = 0 by Theorem (1.3.1), which proves the corollary.

Corollary (1.3.3). — Let Y be a prescheme, f : X → Y an ane morphism. For every quasi-coherent OX -module F ,the canonical homomorphism Hp(Y, f∗(F ))→ Hp(X ,F ) (0, 12.1.3.1) is bijective for all p.

Proof. It suces (by (0, 12.1.7)) to show that the edge homomorphisms ′′Ep02 = Hp(Y, f∗(F ))→ Hp(X ,F )

of the second spectral sequence of the composite functor Γ f∗ are bijective. But the E2-term of this spectralsequence is given by ′′Epq

2 = Hp(Y, Rq f∗(F )) (G, II, 4.17.1), so it follows from Corollary (1.3.2) that ′′Epq2 = 0 for

q > 0, and the spectral sequence degenerates; hence our assertion (0, 11.1.6).

Corollary (1.3.4). — Let f : X → Y be an ane morphism, g : Y → Z a morphism. For every quasi-coherent III | 89OX -module F , the canonical homomorphism Rp g∗( f∗(F ))→ Rp(g f )∗(F ) (0, 12.2.5.1) is bijective for all p.

Proof. It suces to note that, according to Corollary (1.3.3), for every ane open subset W of Z , the canonicalhomomorphism Hp(g−1(W ), f∗(F ))→ Hp( f −1(g−1(W )),F ) is bijective; this proves that the homomorphism ofpresheaves defining the canonical homomorphism Rp g∗( f∗(F ))→ Rp(g f )∗(F ) is bijective (0, 12.2.5).

1. COHOMOLOGY OF AFFINE SCHEMES 223

1.4. Application to the cohomology of arbitrary preschemes.

Proposition (1.4.1). — Let X be a scheme, U = (Uα) be a cover of X by ane open sets. For every quasi-coherentOX -module F , the cohomology modules H•(X ,F ) and H•(U,F ) (over Γ(X ,OX )) are canonically isomorphic.

Proof. As X is a scheme, every finite intersection V of open sets in the cover U is ane (I, 5.5.6), soHq(V,F ) = 0 for g ¾ 1 by Theorem (1.3.1). The proposition then follows from a theorem of Leray (G, II,5.4.1).

Remark (1.4.2). — We note that the result of Proposition (1.4.1) is still true when the finite intersections ofthe sets Uα are ane, even when we do not necessarily assume that X is a scheme.

Corollary (1.4.3). — Let X be a scheme with quasi-compact underlying space, A= Γ(X ,OX ), and f= ( fi)1¶i¶r a nitesequence of elements of A such that the X fi

(notation of (1.2.1)) are ane. Then (with the notations of (1.2.1)), for everyquasi-coherent OX -module F , we have a canonical isomorphism which is functorial in F

(1.4.3.1) Hq(U ,F )' Hq+1((f), M) for q ¾ 1,

and an exact sequence which is functorial in F

(1.4.3.2) 0 −→ H0((f), M) −→ M −→ H0(U ,F ) −→ H1((f), M) −→ 0.

Proof. This follows immediately from Propositions (1.4.1) and (1.2.3).

(1.4.4). If X is an ane scheme, then it follows from Remark (1.2.5) and Proposition (1.4.1) that for all q ¾ 0, thediagrams

(1.4.4.1) Hq(U ,F ′′) ∂ //

Hq+1(U ,F ′)

Hq+1((f), M ′′) ∂ // Hq+2((f), M ′)

corresponding to an exact sequence 0→F ′→F →F ′′→ 0 of quasi-coherent OX -modules (with the notationsof Remark (1.2.5)) are commutative.

III | 90Proposition (1.4.5). — Let X be a quasi-compact scheme, L an invertible OX -module, and consider the graded ringA∗ = Γ∗(L ) (0, 5.4.6); then H•(F ,L ) =

⊕

n∈Z H•(X ,F ⊗L ⊗n) is a graded A∗-module, and for all f ∈ An, we have acanonical isomorphism

(1.4.5.1) H•(X f ,F )' (H•(F ,L ))( f )of (A∗)( f )-modules.

Proof. As X is a quasi-compact scheme, we can calculate the cohomology of all the OX -modules F ⊗L ⊗n

using the same finite cover U= (Ui) consisting of the ane open sets such that the restriction L|Ui is isomorphicto OX |Ui for each i (1.4.1). It is then immediate that the Ui ∩ X f are ane open sets (I, 1.3.6), and we canthus calculate the cohomology H•(X f ,F ⊗L ⊗n) using the cover U|X f = (Ui ∩ X f ) (1.4.1). It is immediate thatfor all f ∈ An, multiplication by f defines a homomorphism C•(U,F ⊗L m)→ C•(U,F ⊗L ⊗(m+n)), hence ahomomorphism H•(U,F ⊗L ⊗m)→ H•(U,F ⊗L ⊗(m+n)), which establishes the first assertion. On the other hand,for a given f ∈ An, it follows from (I, 9.3.2) that we have an isomorphism of complexes of (A∗)( f )-modules

C•(U|X f ,F )'

C•

U,⊕

n∈Z

F ⊗L ⊗n

( f ),

taking into account (I, 1.3.9, ii). Passing to the cohomology of these complexes, we induce the isomorphism(1.4.5.1), recalling that the functor M 7→ M( f ) is exact on the category of graded A∗-modules.

Corollary (1.4.6). — Suppose that the hypotheses of Proposition (1.4.5) are satised, and in addition suppose thatL = OX . If we set A= Γ(X ,OX ), then for all f ∈ A, we have a canonical isomorphism H•(X f ,F ) ' (H•(X ,F )) f ofA f -modules.

Corollary (1.4.7). — Let X be a quasi-compact scheme, f an element of Γ(X ,OX ).(i) Suppose that the open set X f is ane. Then for every quasi-coherent OX -module F , every i > 0, and every

ξ ∈ Hi(X ,F ), there exists an integer n> 0 such that f nξ = 0.(ii) Conversely, suppose that X f is quasi-compact and that for every quasi-coherent sheaf of ideals J of OX and every

ζ ∈ H1(X ,J ), there exists an n> 0 such that f nζ = 0. Then X f is ane.

1. COHOMOLOGY OF AFFINE SCHEMES 224

Proof.(i) If X f is ane, then we have Hi(X f ,F ) = 0 for all i > 0 (1.3.1), so the assertion follows directly from

Corollary (1.4.6).(ii) By virtue of Serre’s criterion (II, 5.2.1), it suces to prove that for every quasi-coherent sheaf of idealsK of OX |X f , we have H1(X f ,K ) = 0. As X f is a quasi-compact open set in a quasi-compact schemeX , there exists a quasi-coherent sheaf of ideals J of OX such that K = J |X f (I, 9.4.2). According toCorollary (1.4.6), we have H1(X f ,K ) = (H1(X ,J )) f , and the hypothesis implies that the right handside is zero, hence the assertion.

Remark (1.4.8). — We note that Corollary (1.4.7, i) gives a simpler proof of the relation (II, 4.5.13.2).III | 91

Lemma (1.4.9). — Let X be a quasi-comact scheme, U = (Ui)1¶i¶n a nite cover of X by ane open sets, and F aquasi-coherent OX -module. The complex of sheaves C •(U,F ) dened by the cover U (G, II, 5.2) is then a quasi-coherentOX -module.

Proof. It follows from the definitions (G, II, 5.2) that C p(U,F ) is the direct sum of the direct image sheavesof the F|Ui0···ip

under the canonical injection Ui0···ip→ X . The hypothesis that X is a scheme implies that these

injections are ane morphisms (I, 5.5.6), hence the C p(U,F ) are quasi-coherent (II, 1.2.6).

Proposition (1.4.10). — Let u : X → Y be a separated and quasi-compact morphism. For every quasi-coherentOX -module F , the Rqu∗(F ) are quasi-coherent OY -modules.

Proof. The question is local on Y , so we can suppose that Y is ane. Then X is a finite union of ane opensets Ui (1¶ i ¶ n); let U be the cover (Ui). In addition, as Y is a scheme, it follows from (I, 5.5.10) that for everyane open V ⊂ Y , the canonical injection u−1(V )→ X is an ane morphism; we conclude (Proposition (1.4.1)and (G, II, 5.2)) that we have a canonical isomorphism

(1.4.10.1) H•(u−1(V ),F )' H•(Γ(V,K •)),

where we set K • = u∗(C •(U,F )). According to Lemma (1.4.1) and (I, 9.2.2), K • is a quasi-coherent OY -module;moreover, it constitutes a complex of sheaves since so is C •(U,F ). It then follows from the definition of thecohomology H •(K •) (G, II, 4.1) that the latter consists of quasi-coherent OY -modules (I, 4.1.1). As (for V anein Y ) the functor Γ(V,G ) is exact in G on the category of quasi-coherent OY -modules, we have (G, II, 4.1)

(1.4.10.2) H•(Γ(V,K •)) = Γ(V,H •(K •)).

Finally, we note that it follows from the definition of the canonical homomorphism

H•(U,F ) −→ H•(X ,F ),

given in (G, II, 5.2), that if V ′ ⊂ V is a second ane open subset of Y , then the diagram

H•(u−1(V ),F ) ∼ //

H•(Γ(V,K •))

H•(u−1(V ′),F ) ∼ // H•(Γ(V ′,K •))

is commutative. We thus conclude from the above that the isomorphisms (1.4.10.1) define an isomorphism ofOY -modules

(1.4.10.3) R•u∗(F )'H •(K •),

and as a result, R•u∗(F ) is quasi-coherent. III | 92

In addition, it follows from (1.4.10.3), (1.4.10.2), and (1.4.10.1) that:

Corollary (1.4.11). — Under the hypotheses of Proposition (1.4.10), for every ane open set V of Y , the canonicalhomomorphism

(1.4.11.1) Hq(u−1(V ),F ) −→ Γ(V, R1u∗(F ))

is an isomorphism for all q ¾ 0.

Corollary (1.4.12). — Suppose that the hypotheses of Proposition (1.4.10) are satised, and in addition suppose that Yis quasi-compact. Then there exists an integer r > 0 such that for every quasi-coherent OX -module F and every integerq > r , we have Rqu∗(F ) = 0. If Y is ane, then we can take for r an integer such that there exists a cover of X consistingof r open ane sets.

1. COHOMOLOGY OF AFFINE SCHEMES 225

Proof. As we can cover Y by a finite number of ane open sets, we can reduce to proving the secondassertion, by virtue of Corollary (1.4.11). If U is a cover of X by r ane open sets, then we have Hq(U,F ) = 0for q > r, since the cochains of Cq(U,F ) are alternating; the assertion thus follows from Proposition (1.4.1).

Corollary (1.4.13). — Suppose that the hypotheses of Proposition (1.4.10) are satsied, and in addition suppose thatY = Spec(A) is ane. Then for every quasi-coherent OX -module F and every f ∈ A, we have

Γ(Yf , Rqu∗(F )) = (Γ(Y, Rqu∗(F )) fup to canonical isomorphism.

Proof. This follows from the fact that Rqu∗(F ) is a quasi-coherent OY -module (I, 1.3.7).

Proposition (1.4.14). — Let f : X → Y be a separated and quasi-compact morphism, g : Y → Z an ane morphism.For every quasi-coherent OX -module F , the canonical homomorphism Rp(g f )∗(F ) → g∗(Rp f∗(F )) (0, 12.2.5.2) isbijective for all p.

Proof. For every ane open subset W of Z , g−1(W ) is an ane open subset of Y . The homomorphism ofpresheaves defining the canonical homomorphism

Rp(g f )∗(F ) −→ g∗(Rp f∗(F ))

(0, 12.2.5) is thus bijective by Corollary (1.4.11).

Proposition (1.4.15). — Let u : X → Y be a separated morphism of nite type, v : Y ′ → Y a at morphism ofpreschemes (0, 6.7.1); let u′ = u(Y ′), such that we have the commutative diagram

(1.4.15.1) X

u

X ′ = X(Y ′)v′oo

u′

Y Y ′.voo

Then for every quasi-coherent OX -module F , Rqu′∗(F′) is canonically isomorphic to Rqu∗(F )⊗OY

OY ′ = v∗(Rqu∗(F )) forall q ¾ 0, where F ′ = v′∗(F ) =F ⊗OY

OY ′ .III | 93

Proof. The canonical homomorphism ρ :F → v′∗(v′∗(F )) (0, 4.4.3.2) defines by functoriality a homomor-

phism

(1.4.15.2) Rqu∗(F ) −→ Rqu∗(v′∗(F

′)).

On the other hand, we have, by setting w= u v′ = v u′, the canonical homomorphisms (0 ,12.2.5.1 and12.2.5.2)

(1.4.15.3) Rqu∗(v′∗(F

′)) −→ Rqw∗(F ′) −→ v∗(Rqu′∗(F

′)).

Composing (1.4.15.3) and (1.4.15.2), we have a homomorphism

ψ : Rqu∗(F ) −→ v∗(Rqu′∗(F

′)),

and finally we obtain a canonical homomorphism (whose definition does not make any assumptions on v)

(1.4.15.4) ψ] : v∗(Rqu∗(F )) −→ Rqu′∗(F′),

and it is necessary to prove that it is an isomorphism when v is at. It is clear that the question is local on Yand Y ′, and we can therefore suppose that Y = Spec(A) and Y ′ = Spec(B); we will also use the following lemma:

Lemma (1.4.15.5). — Let φ : A→ B be a ring homomorphism, Y = Spec(A), X = Spec(B), f : X → Y the morphismcorresponding to φ , and M a B-module. For the OX -module eM to be f -at (0, 6.7.1), it is necessary and sucient for M tobe a at A-module. In particular, for the morphism f to be at, it is necessary and sucient for B to be a at A-module.

This follows from the definition (0, 6.7.1) and from (0, 6.3.3), taking into account (I, 1.3.4).This being so, it follows from (1.4.11.1) and the definitions of the homomorphisms (1.4.15.3) (cf. (0, 12.2.5))

that ψ then corresponds to the composite morphism

Hq(X ,F )ρq−→ Hq(X , v′∗(v

′∗(F )))θq−→ Hq(X ′, v′∗(v′∗(v

′∗(F ))))σq−→ Hq(X ′, v′∗(F )),

1. COHOMOLOGY OF AFFINE SCHEMES 226

where ρq and σq are the homomorphisms in cohomology corresponding to the canonical morphisms ρ andσ : v′∗(v′∗(G

′))→G ′, and θq is the φ -morphism (0, 12.1.3.1) relative to the OX -module v′∗(v′∗(F )). But by the

functoriality of θq, we have the commutative diagram

Hq(X ,F )ρq //

θq

Hq(X , v′∗(v′∗(F )))

θq

Hq(X ′, v′∗(F ))

v′∗(ρq) // Hq(X ′, v′∗(v′∗(v′∗(F )))),

and as by definition (0, 4.4.3) v′∗(ρ) is the inverse of σ, we see that the composite morphism considered above III | 94is finally none other than θq; as a result, ψ] is the associated B-homomorphism Hq(X ,F )⊗A B→ Hq(X ′,F ′). Asu is of finite type, X is a finite union of ane open sets Ui (1¶ i ¶ r); let U be the cover (Ui). As v is an anemorphism, so is v′ (II, 1.6.5, iii), and as a result the U ′i = v′−1(Ui) form an ane open cover U′ of X ′. We thenknow (0, 12.1.4.2) that the diagram

Hq(U,F )θq //

Hq(U′,F ′)

Hq(X ,F )

θq // Hq(X ′,F ′)is commutative, and the vertical arrows are isomorphisms since X and X ′ are schemes (I, 1.4.1). As a result,it suces to prove that the canonical φ -morphism θq : Hq(U,F ) → Hq(U′,F ′) is such that the associatedB-homomorphism

Hq(U,F )⊗A B −→ Hq(U′,F ′)is an isomorphism. For every sequence s= (ik)0¶k¶p of p+1 indices of [1, r], set Us =

⋂pk=0 Uik , U ′s =

⋂pk=0 U ′ik =

v′−1(Us), Ms = Γ(Us,F ), and M ′s = Γ(U ′s,F′). The canonical map Ms ⊗A B→ M ′s is an isomorphism (I, 1.6.5),

hence the canonical map C p(U,F )⊗A B → C p(U′,F ′) is an isomorphism, by which d ⊗ 1 identifies with thecoboundary map C p(U′,F ′) → C p+1(U′,F ′). As B is a at A-module, it follows from the definition of thecohomology modules that the canonical map Hq(U,F )⊗A B → Hq(U′,F ′) is an isomorhism (0, 6.1.1). Thisresult will later be generalized in §6.

Corollary (1.4.16). — Let A be a ring, X an A-scheme of nite type, and B an A-algebra which is faithfully at over A.For X to be ane, it is necessary and sucient for X ⊗A B to be.

Proof. The condition is evidently necessary (I, 3.2.2); we show that it is sucient. As X is separated over Aand the morphism Spec(B)→ Spec(A) is flat, it follows from Proposition (1.4.1) that we have

(1.4.16.1) Hi(X ⊗A B,F ⊗A B) = Hi(X ,F )⊗A B

for every i ¾ 0 and every quasi-coherent OX -module F . If X ⊗A B is ane, the left hand side of (1.4.16.1) is zerofor i = 1, hence so is H1(X ,F ) since B is a faithfully flat A-module. As X is a quasi-compact scheme, we finishthe proof by Serre’s criterion (II, 5.2.1).

Proposition (1.4.17). — Let X be a prescheme, 0→Fu−→G

v−→H → 0 an exact sequence of OX -modules. If F and

H are quasi-coherent, then so is G .III | 95

Proof. The question is local on X , so we can suppose that X = Spec(A) is ane, and it then suces to provethat G satisfies the conditions (d1) and (d2) of (I, 1.4.1) (with V = X ). The verification of (d2) is immediate,because if t ∈ Γ(X ,G ) is zero when restricted to D( f ), then so is its image v(t) ∈ Γ(X ,H ); therefore there existsan m> 0 such that f mv(t) = v( f m t) = 0 (I, 1.4.1), and as Γ is left exact, f m t = u(s), where s ∈ Γ(X ,F ); as u isinjective, the restriction of s to D( f ) is zero, hence (I, 1.4.1) there exists an integer n> 0 such that f ns = 0; wefinally deduce that f m+n t = u( f ns) = 0.

We now check (d1); let t ′ ∈ Γ(D( f ),G ); as H is quasi-coherent, there exists an integer m such thatf mv(t ′) = v( f m t ′) extends to a section z ∈ Γ(X ,H ) (I, 1.4.1). But in virtue of Theorem (1.3.1) (or (I, 5.1.9.2))applied to the quasi-coherent OX -module F , the sequence Γ(X ,G ) → Γ(X ,H ) → 0 is exact, so there existst ∈ Γ(X ,G ) such that z = v(t); we thus see that v( f m t ′ − t ′′) = 0, denoting by t ′′ the restriction of t to D( f );thus we have f m t ′ − t ′′ = u(s′), where s′ ∈ Γ(D( f ),F ). But as F is quasi-coherent, there exists an integer n> 0

3. FINITENESS THEOREM FOR PROPER MORPHISMS 227

such that f ns′ extends to a section s ∈ Γ(X ,F ); as f m+n t ′ − f n t ′′ = u( f ns′), we see that f m+n t ′ is the restrictionto D( f ) of a section f n t + u( f ns) ∈ Γ(X ,G ), which finishes the proof.

§2. Cohomological study of projective morphisms

2.1. Explicit calculations of certain cohomology groups.

§3. Finiteness theorem for proper morphisms

3.1. The dévissage lemma.

Denition (3.1.1). — Let C be an abelian category. We say that a subset C′ of the set of objects of C is exact if0 ∈ C′ and if, for every exact sequence 0→ A′→ A→ A′′→ 0 in C such that two of the objects A, A′, A′′ are in C′,then the third is also in C′.

Theorem (3.1.2). — Let X be a Noetherian prescheme; we denote by C the abelian category of coherent OX -modules. LetC′ be an exact subset of C, X ′ a closed subset of the underlying space of X . Suppose that for every closed irreducible subset Yof X ′, with generic point y , there exists an OX -module G ∈ C′ such that Gy is a k(y)-vector space of dimension 1. Thenevery coherent OX -module with support contained in X ′ is in C′ (and in particular, if X ′ = X , then we have C′ = C).

Proof. Consider the following property P(Y ) of a closed subset Y of X ′: every coherent OX -module withsupport contained in Y is in C′. By virtue of the principle of Noetherian induction (0, 2.2.2), we see that we canreduce to showing that if Y is a closed subset of X ′ such that the property P(Y ′) is true for every closed subset Y ′ of Y ,distinct from Y , then P(Y ) is true.

Therefore, let F ∈ C have support contained in Y , and we show that F ∈ C′. Denote also by Y the reducedclosed subprescheme of X having Y for its underlying space (I, 5.2.1); it is defined by a coherent sheaf of idealsJ of OX . We know (I, 9.3.4) that there exists an integer n> 0 such that J nF = 0; for 1¶ k ¶ n, we thus havean exact sequence

0 −→J k−1F/J kF −→F/J kF −→F/J k−1F −→ 0

of coherent OX -modules ((I, 5.3.6) and (I, 5.3.3)); as C′ is exact, we see, by induction on k, that it suces toshow that each of the Fk = J k−1F/J kF is in C′. We thus reduce to proving that F ∈ C′ under the additionalhypothesis that JF = 0; it is equivalent to say that F = j∗( j∗(F )), where j is the canonical injection Y → X .Let us now consider two cases:

(a) Y is reducible. Let Y = Y ′ ∩ Y ′′, where Y ′ and Y ′′ are closed subsets of Y , distinct from Y ; denote alsoby Y ′ and Y ′′ the reduced closed subpreschemes of X having Y and Y ′′ for their respective underlyingspaces, which are defined respectively by sheaves of ideals J ′ and J ′′ of OX . Set F ′ =F ⊗OX

(OX/J ′)and F ′′ = F ⊗OX

(OX/J ′′). The canonical homomorphisms F → F ′ and F → F ′′ thus definea homomorphism u : F → F ′ ⊕ F ′′. We show that for every z 6∈ Y ′ ∩ Y ′′, the homomorphismuz :Fz →F ′z ⊕F

′′z is bijective. Indeed, we have J ′ ∩J ′′ = J , since the question is local and the above III | 116

equality follows from ((I, 5.2.1) and (I, 1.1.5)); if z 6∈ Y ′′, then we have J ′z = Jz , hence F ′z =Fz andF ′′z = 0, which establishes our assertion in this case; we reason similarly for z 6∈ Y ′. As a result, thekernel and cokernel of u, which are in C (0, 5.3.4), have their support in Y ′ ∩ Y ′′, and thus is in C′by hypothesis; for the same reason, F ′ and F ′′ are in C′, hence also F ′ ⊕F ′′, as C′ is exact. Theconclusion then follows from the consideration of the two exact sequences

0 −→ Im u −→F ′ ⊕F ′′ −→ Coker u −→ 0,

0 −→ Ker u −→F −→ Im u −→ 0,

and the hypothesis that C′ is exact.(b) Y is irreducible, and as a result, the subprescheme Y of X is integral. If y is its generic point, then

we have (OY )y = k(y), and as j∗(F ) is a coherent OY -module, Fy = ( j∗(F ))y is a k(y)-vector spaceof finite dimension m. By hypothesis, there is a coherent OX -module G ∈ C′ (necessarily of supportY ) such that Gy is a k(y)-vector space of dimension 1. As a result, there is a k(y)-isomorphism(Gy)m 'Fy , which is also an OY -isomorphism, and as Gm and F are coherent, there exists an openneighborhood W of y in X and an isomorphism Gm|W ' F|W (0, 5.2.7). Let H be the graph ofthis isomorphism, which is a coherent (OX |W )-submodule of (Gm ⊕F )|W , canonically isomorphic toGm|W and to F|W ; there thus exists a coherent OX -submodule H0 of Gm ⊕F , inducing H on W and0 on X − Y , since Gm and F have Y for their support (I, 9.4.7). The restrictions v :H0 → Gm andw :H0→F of the canonical projections of Gm⊕F are then homomorphisms of coherent OX -modules,which, on W and on X − Y , reduce to isomorphisms; in other words, the kernels and cokernels of v

3. FINITENESS THEOREM FOR PROPER MORPHISMS 228

and w have their suppoer in the closed set Y − (Y ∩W ), distinct from Y . They are in C′; on the otherhand, we have Gm ∈ C′ since G ∈ C′ and since C′ is exact. We conclude successively, by the exactnessof C′, that H0 ∈ C′, then F ∈ C′. Q.E.D.

Corollary (3.1.3). — Suppose that the exact subset C′ of C has in addition the property that any coherent direct factor ofa coherent OX -moduleM ∈ C′ is also in C′. In this case, the conclusion of Theorem (3.1.2) is still valid when the condition“Gy is a k(y)-vector space of dimension 1” is replaced by Gy 6= 0 (this is equivalent to Supp(G ) = Y ).

Proof. The reasoning of Theorem (3.1.2) must be modified only in the case (b); now Gy is a k(y)-vectorspace of dimension q > 0, and as a result, we have an OY -isomorphism (Gy)m ' (Fy)q; the end of the reasoningin Theorem (3.1.2) then proves that F q ∈ C′, and the additional hypothesis on C′ implies that F ∈ C′.

3.2. The niteness theorem: the case of usual schemes.

Theorem (3.2.1). — Let Y be a locally Noetherian prescheme, f : X → Y a proper morphism. For every coherentOX -module F , the OY -modules Rq f∗(F ) are coherent for q ¾ 0.

Proof. The question being local on Y , we can suppose Y Noetherian, thus X Noetherian (I, 6.3.7). Thecoherent OX -modules F for which the conclusion of Theorem (3.2.1) is true forms an exact subset C′ of thecategory C of coherent OX -modules. Indeed, let 0 → F ′ → F → F ′′ → 0 is an exact sequence of coherent III | 117OX -modules; suppose for example that F ′ and F ′′ belong to C′; we have the long exact sequence in cohomology

Rq−1 f∗(F ′′)∂−→ Rq f∗(F ′) −→ Rq f∗(F ) −→ Rq f∗(F ′′)

∂−→ Rq+1 f∗(F ′),

in which by hypothesis the outer four terms are coherent; it is the same for the middle term Rq f∗(F ) by((0, 5.3.4) and (0, 5.3.3)). We show in the same way that when F and F ′ (resp. F and F ′′) are in C′, then so isF ′′ (resp.F ′). In addition, every coherent direct factor F ′ of an OX -moduleF ∈ C′ belongs to C′: indeed, Rq f∗(F ′)is then a direct factor of Rq f∗(F ) (G, II, 4.4.4), therefore it is of finite type, and as it is quasi-coherent (1.4.10), itis coherent, as Y is Noetherian. By virtue of Corollary (3.1.3), we reduce to proving that when X is irreduciblewith generic point x , there exists one coherent OX -module F belonging to C′, such that Fx 6= 0: indeed, if thispoint is established, then it can be applied to any irreducible closed subprescheme Y of X , since if j : Y → Xis the canonical injection, then f j is proper (II, 5.4.2), and if G is a coherent OY -module with support Y ,then j∗(G ) is a coherent OX -module such that Rq( f j)∗(G ) = Rq f∗( j∗(G )) (G, II, 4.9.1), therefore we can applyCOrollary (3.1.3).

By virtue of Chow’s lemma (II, 5.6.2), there exists an irreducible prescheme X ′ an a projective and surjectivemorphism g : X ′→ X such that f g : X ′→ Y is projective. There exists an ample OX -module L for g (II, 5.3.1);we apply the fundamental theorem of projective morphisms (2.2.1) to g : X ′→ X and with L : there thus exixtsan integer n such that F = g∗(OX ′(n)) is a coherent OX -module and Rq g∗(OX ′(n)) = 0 for all q > 0; in addition,as g∗(g∗(OX ′(n)))→OX ′(n) is surjective for n large enough (2.2.1), we see that we can suppose, at the genericpoint x of X , that we have Fx 6= 0 (II, 3.4.7). On the other hand, as f g is projective as Y is Noetherian,the Rq( f g)∗(OX ′(n)) are coherent (2.2.1). This being so, R•( f g)∗(OX ′(n)) is the abutment of a Leray spectralsequence, whose E2-term is given by Epq

2 = Rp f∗(Rq g∗(OX ′(n))); the above shows that this spectral sequencedegenerates, and we then know (0, 11.1.6) that Ep0

2 = Rp f∗(F ) is isomorphic to Rp( f g)∗(OX ′(n)), which finishesthe proof.

Corollary (3.2.2). — Let Y be a locally Noetherian prescheme. For every proper morphism f : X → Y , the direct imageunder f of any coherent OX -module is a coherent OY -module.

Corollary (3.2.3). — Let A be a Noetherian ring, X a proper scheme over A; for every coherent OX -module F , theHp(X ,F ) are A-modules of nite type, and there exists an integer r > 0 such that for every coherent OX -module F and allp > r , Hp(X ,F ) = 0.

Proof. The second assertion has already been proved (1.4.12); the first follows from the finiteness theo-rem (3.2.1), taking into account Corollary (1.4.11).

In particular, if X is a proper algebraic scheme over a field k, then, for every coherent OX -module F , theHp(X ,F ) are nite-dimensional k-vector spaces.

Corollary (3.2.4). — Let Y be a locally Noetherian prescheme, f : X → Y a morphism of nite type. For every coherentOX -module F whose support in proper over Y (II, 5.4.10), the OY -modules Rq f∗(F ) are coherent.

III | 118

3. FINITENESS THEOREM FOR PROPER MORPHISMS 229

Proof. The question being local on Y , we can suppose Y Noetherian, and it is the same for X (I, 6.3.7). Byhypothesis, every closed subprescheme Z of X whose underlying space is Supp(F ) is proper over Y , in otherwords, if j : Z → X is the canonical injection, then f j : Z → Y is proper. We can suppose that Z is suchthat F = j∗(G ), where G = j∗(F ) is a coherent OZ -module (I, 9.3.5); as we have Rq f∗(F ) = Rq( f j)∗(G ) byCorollary (1.3.4), the conclusion follows immediately from Theorem (3.2.1).

3.3. Generalization of the niteness theorem (usual schemes).

Proposition (3.3.1). — Let Y be a Noetherian prescheme, S a quasi-coherent OY -algebra of nite type, graded in positivedegrees, Y ′ = Proj(S ), and g : Y ′ → Y the structure morphism. Let f : X → Y be a proper morphism, S ′ = f ∗(S ),M =

⊕

k∈ZMk a quasi-coherent graded S ′-module of nite type. Then the Rp f∗(M ) =⊕

k∈Z Rp f∗(Mk) are gradedS -modules of nite type for all p. Suppose in addition that the S are generated by S1; then, for every p ∈ Z, there existsan integer kp such that for all k ¾ kp and all r > 0, we have

(3.3.1.1) Rp f∗(Mk+r) = SrRp f∗(Mk).

Proof. The first assertion is identical to the statement of Theorem (2.4.1, i), where we have simply replaced“projective morphism” by “proper morphism”. In the proof of Theorem (2.4.1, i), the hypothesis on f was onlyused to show (with the notation of this proof) that Rp f ′∗ ( ÝM ) is a coherent OY ′ -module. With the hypothesis ofProposition (3.3.1), f ′ is proper (II, 5.4.2, iii), so we can resume without change in the proof of Theorem (2.4.1, i),thanks to the finiteness theorem (3.2.1).

As for the second assertion, it suces to remark that there is a finite ane open cover (Ui) of Y such thatthe restrictions to the Ui of the two sides of (3.3.1.1) are equal for all k ¾ kp,i (II, 2.1.6, ii); it suces to take forkp the largest of the kp,i .

Corollary (3.3.2). — Let A be a Notherian ring, m an ideal of A, X a proper A-scheme, and F a coherent OX -module.Then, for all p ¾ 0, the direct sum

⊕

k¾0 Hp(X ,mkF ) is a module of nite type over the ring S =⊕

k¾0 mk; in particular,

there exists an integer kp ¾ 0 such that for all k ¾ kp and all r > 0, we have

(3.3.2.1) Hp(X ,mk+rF ) = mrHp(X ,mkF ).

Proof. It suces to apply Proposition (3.3.1) with Y = Spec(A), S = eS,Mk = mkF , taking into accountCorollary (1.4.11).

It should be remembered that the S-module structure on⊕

k¾0 Hp(X ,mkF ) is obtained by considering,for every a ∈ mr , the map Hp(X ,mkF )→ Hp(X ,mk+rF ), which comes from the passage to cohomology of themultiplication map mrF → mk+rF defined by a (2.4.1).

III | 1193.4. Finiteness theorem: the case of formal schemes. The results of this section (except the defini-

tion (3.4.1)) will not be used in the rest of this chapter.

(3.4.1). Let X and S be two locally Noetherian formal preschemes (I, 10.4.2), f : X→S a morphism of formalpreschemes. We say that f is a proper morphism if it satisfies the following conditions:

1st. f is a morphism of nite type (I, 10.13.3).2nd. IfK is a sheaf of ideals of denition for S and if we set J = f ∗(K )OX, X0 = (X,OX/J ), S0 = (S,OS/K ),

then the morphism f0 : X0→ S0 induced by f (I, 10.5.6) is proper.It is immediate that this definition does not depend on the sheaf of ideals of definition K for S considered;indeed, if K ′ is a second sheaf of ideals of definition such that K ′ ⊂ K , and if we set J ′ = f ∗(K ′)OX,X ′0 = (X,OX/J ′), S′0 = (S ,OS/K ′), then the morphism f ′0 : X ′0→ S′0 induced by f is such that the diagram

X0f0 //

i

S0

j

X ′0

f ′0 // S′0

is commutative, i and j being surjective immersions; it is equivalent to say that f0 or f ′0 is proper, by virtueof (II, 5.4.5).

We note that, for all n¾ 0, if we set Xn = (X,OX/J n+1), Sn = (S,OS/K n+1), then the morphism fn : Xn→ Sninduced by f (I, 10.5.6) is proper for all n whenever it is for n= 0 (II, 5.4.6).

If g : Y → Z is a proper morphism of locally Noetherian usual preschemes, Z ′ a closed subset of Z , Y ′ aclosed subset of Y such that g(Y ′) ⊂ Z ′, then the extension bg : Y/Y ′ → Z/Z ′ of g to the completions (I, 10.9.1) isa proper morphism of formal preschemes, as it follows from the definition and from (II, 5.4.5).

3. FINITENESS THEOREM FOR PROPER MORPHISMS 230

Let X and S be two locally Noetherian formal preschemes, f : X→S a morphism of nite type (I, 10.13.3);the notation being the same as above, we say that a subset Z of the underlying space of X is proper over S (orproper for f ) if, considered as a subset of X0, Z is proper over S0 (II, 5.4.10). All the properties of proper subsetsof usual preschemes stated in (II, 5.4.10) are still true for the proper subsets of formal preschemes, as it followsimmediately from the definitions.

Theorem (3.4.2). — Let X and Y be locally Noetherian formal preschemes, f : X→Y a proper morphism. For everycoherent OX-module F , the OY-modules Rq f∗(F ) are coherent for all q ¾ 0.

Let J be a sheaf of ideals of definition for Y, K = f ∗(J )OX, and consider the OX-modules

(3.4.2.1) Fk =F ⊗OY (OY/Jk+1) =F/K k+1F (k ¾ 0)

which evidently form a projective system of topological OX-modules, such that F = lim←−kFk (I, 10.11.3). On the III | 120

other hand, it follows from Theorem (3.4.2) that each of the Rq f∗(F ), being coherent, is naturally equippedwith a topological OY-module structure (I, 10.11.6), and so are the Rq f∗(Fk). The canonical homomorphismsF →Fk =F/K k+1F canonically correspond to homomorphisms

Rq f∗(F ) −→ Rq f∗(Fk),

which are necessarily continuous for the topological OY-module structures above (I, 10.11.6), and form aprojective system, giving the limit a canonical functorial homomorphism

(3.4.2.2) Rq f∗(F ) −→ lim←−k

Rq f∗(Fk),

which will be a continuous homomorphism of topological OY-modules. We will prove along with Theorem (3.4.2)the

Corollary (3.4.3). — Each of the homomorphisms (3.4.2.2) is a topological isomorphism. In addition, if Y is Noetherian,then the projective system (Rq f∗(F/K k+1F ))k¾0 satises the (ML)-condition (0, 13.1.1).

We will begin by establishing Theorem (3.4.2) and Corollary (3.4.3) when Y is a Noetherian formal anescheme (I, 10.4.1):

Corollary (3.4.4). — Under the hypotheses of Theorem (3.4.2), suppose in addition that Y = Spf(A), where A is anadic Noetherian ring. Let J be an ideal of denition for A, and set Fk =F/Jk+1F for k ¾ 0. Then the Hn(X,F ) areA-modules of nite type; the projective system (Hn(X,Fk))k¾0 satises the (ML)-condition for all n; if we set

(3.4.4.1) Nn,k = Ker

Hn(X,F ) −→ Hn(X,Fk)

(also equal to Im(Hn(X,Jk+1F )→ Hn(X,F )) by the exact sequence in cohomology), then the Nn,k dene on Hn(X,F ) aJ-good ltration (0, 13.7.7); nally, the canonical homomorphism

(3.4.4.2) Hn(X,F ) −→ lim←−k

Hn(X,Fk)

is a topological isomorphism for all n (the left hand side being equipped with the J-adic topology, the Hn(X,Fk) with thediscrete topology).

SetS = gr(A) =

⊕

k¾0

Jk/Jk+1, M = gr(F ) =⊕

k¾0

JkF/Jk+1F .

We know that J∆ is a sheaf of ideals of definition for Y (I, 10.3.1); let K = f ∗(J∆)OX, X0 = (X,OX/K ),Y0 = (Y,OY/J∆) = Spec(A0), with A0 = A/J. It is clear that the Mk = JkF/Jk+1F are coherent OX0

-modules (I, 10.11.3). Consider on the other hand the quasi-coherent graded OX0

-algebra

(3.4.4.4) S = OX0⊗A0

S = gr(OX) =⊕

k¾0

K k/K k+1.

The hypothesis that F is a OX-module of finite type implies first thatM is a graded S -module of nite type. III | 121Indeed, the question is local on X, and we can thus suppose that X = Spf(B), where B is an adic Noetherianring, and F = N∆, where N is a B-module of finite type (I, 10.10.5); we have in addition X0 = Spec(B0),where B0 = B/JB, and the quasi-coherent OX0

-modules S andM are respectively equal to eS′ and ÝM ′, whereS′ =

⊕

k¾0((Jk/Jk+1)⊗A0

B0) and M ′ =⊕

k¾0((Jk/Jk+1)⊗A0

N0), with N0 = N/JN ; we then evidently have M ′ =S′⊗B0

N0, and as N0 is a B0-module of finite type, M ′ is a S′-module of finite type, hence our assertion (I, 1.3.13).As the morphism f0 : X0 → Y0 is proper by hypothesis, we can apply Corollary (3.3.2) to S ,M , and the

morphism f0: taking into account Corollary (1.4.11), we conclude that for all n ¾ 0,⊕

k¾0 Hn(X0,Mk) is a

4. THE FUNDAMENTAL THEOREM OF PROPER MORPHISMS. APPLICATIONS 231

graded S-module of nite type. This proves that the condition (Fn) of (0, 13.7.7) is satisfied for all n¾ 0, whenwe consider the strictly projective system (F/JkF )k¾0 of sheaves of abelian groups on X0, each equipped withits natural “filtered A-module” structure. We can thus apply (0, 13.7.7), which proves that:

1st. The projective system (Hn(X,Fk))k¾0 satisfies the (ML)-condition.2nd. If H′n = lim←−k

Hn(X,Fk), then H′n is an A-module of finite type.3rd. The filtration defined on H′n by the kernels of the canonical homomorphisms H′n → Hn(X,Fk) is

J-good.

Note that on the other hand, if we set Xk = (X,OX/K k+1), then Fk is a coherent OXk-module (I, 10.11.3), and

if U is an ane open set in X0, then U is also an ane open set in each of the Xk (I, 5.1.9), so Hn(U ,Fk) = 0 for alln> 0 and all k (1.3.1) and H0(U ,Fk)→ H0(U ,Fh) is surjective for h¶ k (I, 1.3.9). We are thus in the conditionsof (0, 13.3.2) and applying (0, 13.3.1) proves that H′n canonically identifies with Hn(X, lim←−k

Fk) = Hn(X,F ); thisfinishes the proof of Corollary (3.4.4).

(3.4.5). We return to the proof of (3.4.2) and (3.4.3). We first prove the propositions for the case Y = Spf(A)envissaged in (3.4.4); for this, for all g ∈ A, apply (3.4.4) to the Noetherian ane formal scheme inducedon the open set Yg = D(g) of Y, which is equal to Spf(Ag), and to the formal prescheme induced by X onf −1(Yg); note that Yg is also an ane open set in the prescheme Yk = (Y,OY/(J∆)k+1), and as Fk is a coherentOXk

-module, we haveHn( f −1(Yg),Fk) = Γ(Yg , Rn f∗(Fk))

for all k ¾ 0 by virtue of Corollary (1.4.11). The canonical homomorphism

Hn( f −1(Yg),F ) −→ lim←−k

Γ(Yg , Rn f∗(Fk))

is an isomorphism; but we have (0, 3.2.6)

lim←−k

Γ(Yg , Rn f∗(Fk)) = Γ(Yg , lim←−k

Rn f∗(Fk)),

and as the sheaf Rn f∗(F ) is the sheaf associated to the presheaf Yg 7→ Hn( f −1(Yg),F ) on the Yg (0, 3.2.1), III | 122we have shown that the homormorphism (3.4.2.2) is bijective. Let us now prove that Rn f∗(F ) is a coherentOY-module, and more precisely that we have

(3.4.5.1) Rn f∗(F ) =

Hn(X,F )∆

.

With the above notation, we have, since Fk is a coherent OXk-module (1.4.13),

Γ(Yg , Rn f∗(Fk)) = (Γ(Y, Rn f∗(Fk)))g = (Hn(X,Fk))g .

Now the Hn(X,Fk) form a projective system satisfying (ML), and their projective limit Hn(X,F ) is anA-module of finite type. We conclude (0, 13.7.8) that we have

lim←−k

(Hn(X,Fk))g

= Hn(X,F )⊗A Ag = Γ(Yg , (Hn(X,F ))∆),

taking into account (I, 10.10.8) applied to A and Ag; this proves (3.4.5.1) since Γ(Yg , Rn f∗(F )) = lim←−kΓ(Yg , Rn f∗(Fk)).

As (3.4.2.2) is then an isomorphism of coherent OY-modules, it is necessarily a topological isomorphism (I, 10.11.6).Finally, it follows from the relations Rn f∗(Fk) = (Hn(X,Fk))∆ that the projective system (Rn f∗(Fk))k¾0 satisfies(ML) (I, 10.10.2).

Once (3.4.2) and (3.4.3) are proved in the case where the formal prescheme Y is ane Noetherian, it isimmediate to pass to the general case for (3.4.2) and the first assertion of (3.4.3), which are local on Y. As forthe second assertion of (3.4.3), it suces, Y being Noetherian, to cover it by a finite number of Noetherian aneopen sets Ui and to note that the restricyions of the projective system (Rq f∗(Fk)) to each of the Ui satisfies (ML).

Along the way, we have in addition proved:

Corollary (3.4.6). — Under the hypotheses of Corollary (3.4.4), the canonical homomorphism

(3.4.6.1) Hq(X,F ) −→ Γ(Y, Rq f∗(F ))

is bijective.

§4. The fundamental theorem of proper morphisms. Applications

4.1. The fundamental theorem.

7. BASE CHANGE FOR HOMOLOGICAL FUNCTORS OF SHEAVES OF MODULES 232

§5. An existence theorem for coherent algebraic sheaves

5.1. Statement of the theorem.

§6. Local and global Tor functors; Künneth formula

6.1. Introduction.

§7. Base change for homological functors of sheaves of modules

7.1. Functors of A-modules.

CHAPTER IV

Local study of schemes and their morphisms (EGA IV)

build hack [CC]

SummaryIV-1 | 222

§1. Relative finiteness conditions. Constructible sets of preschemes.§2. Base change and flatness.§3. Associated prime cycles and primary decomposition.§4. Change of base field for algebraic preschemes.§5. Dimension and depth for preschemes.§6. Flat morphisms of locally Noetherian preschemes.§7. Application to the relations between a local Noetherian ring and its completion. Excellent rings.§8. Projective limits of preschemes.§9. Constructible properties.§10. Jacobson preschemes.§11.1 Topological properties of finitely presented flat morphisms. Local flatness criteria.§12. Study of fibres of finitely presented flat morphisms.§13. Equidimensional morphisms.§14. Universally open morphisms.§15. Study of fibres of a universally open morphism.§16. Dierential invariants. Dierentially smooth morphisms.§17. Smooth morphisms, unramified morphisms, and étale morphisms.§18. Supplement on étale morphisms. Henselian local rings and strictly local rings.§19. Regular immersions and transversely regular immersions.§20. Hyperplane sections; generic projections.§21. Infinitesimal extensions.

IV-1 | 223The subjects discussed in the chapter call for the following remarks.

(a) The common property of all the subjects discussed is that they all related to local properties ofpreschemes or morphisms, i.e. considered at a point, or the points of a fibre, or on a (non-specified)neighbourhood of a point or of a fibre. These properties are generally of a topological, dierential, ordimensional nature (i.e. bringing the ideas of dimension and depth into play), and are linked to theproperties of the local rings at the points considered. One type of problem is the relating, for a givenmorphisms f : X → Y and point x ∈ X , of the properties of X at x with those of Y at y = f (x) andthose of the fibre X y = f −1(y) at x . Another is the determining of the topological nature (for example,the constructibility, or the fact of being open or closed) of the set of points x ∈ X at which X has acertain property, or for which the fibre X f (x) passing through x has a certain property at x . Similarly,we are interested in the topological nature of the set of points y ∈ Y such that X has a certain propertyat all the points of the fibre X y , or those such that this fibre itself has a certain property.

(b) The most important idea for the following chapters is that of at morphisms of nite presentation, as wellas the particular cases of smooth morphisms and étale morphisms. Their detailed study (as well as that ofconnected questions) really starts in §11.

(c) Sections §§1–10 can be considered as being preliminary in nature, and as developing three typesof techniques, used, not only in the other sections of the chapter, but also, of course, in the followchapters:

1The order and content of §§11–21 are given only as an indication of what the titles will be, and will possibly be modified before theirpublication. [Trans.] This was indeed the case: many of §§11–21 ended up having entirely dierent titles.

233

1. RELATIVE FINITENESS CONDITIONS. CONSTRUCTIBLE SETS OF PRESCHEMES 234

(c1) Sections §§1–4 are envisaged as treating the diverse aspects of the idea of change of base, above allin relation with the conditions of niteness or atness; we there initiate the technique of descent,with its most elementary aspects (the questions of “eectiveness” linked to this technique will bestudied in Chapter V).

(c2) Sections §§5–7 are focused on what we may call Noetherian techniques, since the preschemesconsidered are always locally Noetherian, whereas, on the contrary, there is generally no finitenesscondition imposed on the morphisms; this is essentially due to the fact that the ideas of dimensionand depth are hardly manageable except in the case of Noetherian local rings. Recall that §7constitutes a “delicate (?)” theory of Noetherian local rings, not much used in what follows in thechapter.

(c3) Sections §§8–10 describe, amongst other things, the means of eliminating the Noetherian hypotheseson the preschemes considered, by substituting such hypotheses for suitable ones of niteness(“finite presentation”) on the morphisms considered: the advantage of this substitution is thatthe latter such hypotheses (those of finiteness on the morphisms) are stable under base change,which is not the case for the Noetherian hypotheses on the preschemes. The technique permittingthis substitution relies, in some part, on the use of the idea of the projective limit of preschemes,thanks to which we can reduce a question to the same question with Noetherian hypotheses; onthe other hand, it relies on the systematic use of constructible sets, which have the double interestof being preserved under taking inverse images (of arbitrary morphisms) and by direct images IV-1 | 224(of morphisms of finite presentation), and having manageable topological properties in locallyNoetherian preschemes. The same techniques often even allow to restrict to the case of morespecific Noetherian rings, for example the Z-algebras of nite type, and it is here that the propertiesof “excellent” rings (studied in §7) intervene in a decisive manner. Independently of the questionof elimination of Noetherian hypotheses, the techniques of §§8–10, elementary in nature, findconstant use in nearly all applications.

§1. Relative niteness conditions. Constructible sets of preschemes

In this section. we will resume the exposé of “finiteness conditions” for a morphism of preschemes f : X → Ygiven in (I, 6.3 and 6.6). There are essentially two notions of “finiteness” of a global nature on X , that ofquasi-compact morphism (defined in (I, 6.6.1)) and that of a quasi-separated morphism; on the other hand, thereare two notions of “finiteness” of a local nature on X , that of a morphism locally of nite type (defined in (I, 6.6.2))and that of a morphism locally of nite presentation. By combining these local notions with the precedingglobal notions, we obtain the notion of a morphism of nite type (defined in (I, 6.3.1)) and of a morphism ofnite presentation. For the convenience of the reader, we will give again in this section the properties statedin (I, 6.3 and 6.6), referring to their labels in Chapter I for their proofs.

In nos1.8 and 1.9, we complete, in the context of preschemes, and making use of the previous notions offiniteness, the results on constructible sets given in (0III, §9).

1.1. Quasi-compact morphisms.

Denition (1.1.1). — We say that a morphism of preschemes f : X → Y is quasi-compact if the continous mapf from the topological space X to the topological space Y is quasi-compact (0, 9.1.1), in other words, if theinverse image f −1(U) of every quasi-compact open subset U of Y is quasi-compact (cf. (I, 6.6.1)).

If B is a basis for the topology of Y consisting of ane open sets, then for f to be quasi-compact, it isnecessary and sucient that for all V ∈B, f −1(V ) is a nite union of ane open sets. For example, if Y is aneand X is quasi-compact, every morphism f : X → Y is quasi-compact (I, 6.6.1).

If f : X → Y is a quasi-compact morphism, then it is clear that for every open subset V of Y , the restrictionof f to f −1(V ) is a quasi-compact morphism f −1(V ) → V . Conversely, if (Uα) is an open cover of Y andf : X → Y is a morphism such that the restrictions f −1(Uα)→ Uα are quasi-compact, then f is quasi-compact.As a result, if f : X → Y is an S-morphism of S-preschemes, and if there exists an open cover (Sλ) of S such that IV-1 | 225the restrictions g−1(Sλ)→ h−1(Sλ) of f (where g and h are the structure morphisms) are quasi-compact, then fis quasi-compact.

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