Algebraic Geometry over C^inf Rings, Joyce

8/7/2019 Algebraic Geometry over C^inf Rings, Joyce

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arXiv:1001

.0023v1[math.AG

]31Dec2009

Algebraic Geometry over C-rings

Dominic Joyce

Abstract

If X is a manifold then the R-algebra C(X) of smooth functionsc : X R is a C-ring. That is, for each smooth function f : Rn Rthere is an n-fold operation f : C

(X)n C(X) acting by f :(c1, . . . , cn) f(c1, . . . , cn), and these operations f satisfy many naturalidentities. Thus, C(X) actually has a far richer structure than the

obvious R-algebra structure.We explain the foundations of a version of Algebraic Geometry in

which rings or algebras are replaced byC-rings. As schemes are the basicobjects in Algebraic Geometry, the new basic objects are C-schemes, acategory of geometric objects which generalize manifolds, and whose mor-phisms generalize smooth maps. We also study quasicoherent and coherentsheaves on C-schemes, and C-stacks, in particular DeligneMumfordC-stacks, a 2-category of geometric objects generalizing orbifolds.

Many of these ideas are not new: C-rings and C-schemes havelong been part of Synthetic Differential Geometry. But we develop themin new directions. In a sequel [20] the author will use these tools todefine d-manifolds and d-orbifolds, derived versions of manifolds andorbifolds related to Spivaks derived manifolds [37]. These in turn willhave applications in Symplectic Geometry, as the geometric structure onmoduli spaces ofJ-holomorphic curves.

Contents

1 Introduction 2

2 C-rings 52.1 Two definitions of C-ring . . . . . . . . . . . . . . . . . . . . . 52.2 C-rings as commutative R-algebras, and ideals . . . . . . . . . 72.3 Local C-rings, and localization . . . . . . . . . . . . . . . . . . 82.4 Fair C-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Good C-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.6 Pushouts ofC

-rings . . . . . . . . . . . . . . . . . . . . . . . . 13

3 The C-ring C(X) of a manifold X 15

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4 C-ringed spaces and C-schemes 184.1 C-ringed spaces and local C-ringed spaces . . . . . . . . . . . 18

4.2 Affine C-schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Locally finite sums in fair C-rings . . . . . . . . . . . . . . . . . 244.4 General C-schemes . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Modules over C-rings 275.1 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 Complete modules over fair C-rings . . . . . . . . . . . . . . . 285.3 Cotangent modules ofC-rings . . . . . . . . . . . . . . . . . . . 31

6 Sheaves of modules on C-schemes 376.1 Sheaves ofOX -modules on a C-ringed space (X, OX ) . . . . . 376.2 Sheaves on affine C-schemes, and MSpec . . . . . . . . . . . . . 396.3 Quasicoherent and coherent sheaves on C-schemes . . . . . . . 42

6.4 Cotangent sheaves of C

-schemes . . . . . . . . . . . . . . . . . 43

7 Background material on stacks 467.1 Grothendieck topologies, sites, prestacks, and stacks . . . . . . . 477.2 Commutative diagrams and fibre products . . . . . . . . . . . . . 487.3 Descent theory on a site . . . . . . . . . . . . . . . . . . . . . . . 507.4 Properties of 1-morphisms . . . . . . . . . . . . . . . . . . . . . . 517.5 Geometric stacks, and stacks associated to groupoids . . . . . . . 53

8 C-stacks and orbifolds 568.1 C- s t a c k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568.2 Properties of morphisms ofC-stacks . . . . . . . . . . . . . . . 588.3 Open C-substacks and open covers . . . . . . . . . . . . . . . . 608.4 DeligneMumford C-stacks . . . . . . . . . . . . . . . . . . . . 618.5 Characterizing DeligneMumford C- st acks . . . . . . . . . . . . 648.6 The underlying topological space of a C- st ack . . . . . . . . . . 678.7 Coarse moduli C-schemes of C-stacks . . . . . . . . . . . . . 708.8 Orbifolds as DeligneMumford C-stacks . . . . . . . . . . . . . 72

9 Sheaves on DeligneMumford C-stacks 759.1 OX-modules, quasicoherent and coherent sheaves . . . . . . . . . 769.2 Writing sheaves in terms of a groupoid presentation . . . . . . . 789.3 Pullback of sheaves as a pseudofunctor . . . . . . . . . . . . . . . 809.4 Cotangent sheaves of DeligneMumford C- st acks . . . . . . . . 83

References 86

1 Introduction

Let X be a smooth manifold, and write C(X) for the set of smooth functionsc : X R. Then C(X) is a commutative R-algebra, with operations ofaddition, multiplication, and scalar multiplication defined pointwise. However,

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exists at least one model with the properties you want, then as far as SyntheticDifferential Geometry is concerned the job is done. For this reason C-schemes

have not been developed very far in Synthetic Differential Geometry.Recently, C-rings and C-ringed spaces appeared in a very different con-

text, as part of David Spivaks definition of derived manifolds [36,37]. Spivakwas a student of Jacob Lurie, and his goal was to extend parts of Luries De-rived Algebraic Geometry programme [24] to Differential Geometry. Spivaksconstruction is very complex and technical, and his derived manifolds form asimplicial category, a kind of -category with n-morphisms for all n 1.

The authors interest in this area comes from Symplectic Geometry. Areas ofSymplectic Geometry such as open and closed GromovWitten invariants andLagrangian Floer cohomology involve the study of moduli spaces Mg,m(J, ) ofstable J-holomorphic curves in a symplectic manifold (M, ). Following Fukayaand Ono [12,13], such moduli spaces Mg,m(J, ) are supposed to be Kuranishispaces, topological spaces with an extra geometric structure. Unfortunately, all

definitions of Kuranishi spaces so far have been unsatisfactory, and there aresignificant problems with the foundations of this area.

If you compare the definitions of Kuranishi spaces in [12, 13] and derivedmanifolds in [36,37], it soon becomes clear that they should be related roughlyspeaking, a Kuranishi space should be a derived orbifold with corners. This bothsuggests a reason why the definitions of Kuranishi spaces were faulty to dothe job properly requires derived ideas which were not current in SymplecticGeometry at the time and suggests a way to fix the definition of Kuranishispace, and shore up the foundations. The author decided to attempt this.

However, upon studying the problem, the author became convinced thatthe right model for Kuranishi spaces are not Spivaks derived manifolds [36,37], which form an -category, but a truncated, simplified version of Spivaksconstruction we will call d-manifolds [20], which form a 2-category. In the

language of this paper, a d-manifold is a separated, locally good C-schemeX with an exact sequence of coherent sheaves E1 E2 TX 0 satisfyingcertain conditions. To get a d-orbifold we replace X by a separated, locallygood DeligneMumford C-stack X.

We will explain our theory of d-manifolds and d-orbifolds and their connec-tion to derived manifolds and Kuranishi spaces in [20]. But to set up the theoryrequires a lot of preliminary work on C-schemes and C-stacks, and coherentsheaves upon them. That is the purpose of this paper. We have tried to presenta complete, self-contained account which should be understandable to readerswith a reasonable background in Algebraic Geometry, and we assume no famil-iarity with Synthetic Differential Geometry. We expect this material may haveother applications quite different to those the author has in mind [20], which iswhy we have written it as a separate paper, and tried to give a general picture

rather than just those parts needed for [20].Section 2 explains C-rings. The archetypal examples ofC-rings, C(X)

for manifolds X, are discussed in 3. Section 4 studies C-schemes, 5 mod-ules over C-rings, and 6 sheaves of modules over C-schemes, quasicoherentsheaves, and coherent sheaves. Sections 79 generalize 4 and 6 to C-stacks.

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We are particularly interested in DeligneMumford C-stacks, which are C-stacks locally modelled on [U/G] for U an affine C-scheme and G a finite group

acting on U, since DeligneMumford C-stacks generalize orbifolds in the sameway that C-schemes generalize manifolds.

Almost all of24 is already understood in Synthetic Differential Geometry,such as Dubuc [10] and Moerdijk and Reyes [31], except for good C-rings,which appear to be a new idea, (locally) good C-schemes, and some factsabout the C-rings and C-schemes of manifolds with boundary or corners.But we believe it is worthwhile giving a detailed and self-contained exposition,from our own point of view. Section 7 summarizes material on stacks from[2,3, 14, 22,26, 32]. Sections 56 and 89 are new, so far as the author knows,though they are based on well known material in Algebraic Geometry.

Acknowledgements. I would like to thank Jacob Lurie for helpful conversations.

2 C-ringsWe begin by explaining the basic objects out of which our theories are built,C-rings, or smooth rings. The archetypal example of a C-ring is the vectorspace C(X) of smooth functions c : X R for a manifold X, and these willbe discussed at greater length in 3. The material on good C-rings in 2.52.6is new, as far as the author knows, but nearly everything else in this section willbe known to experts in Synthetic Differential Geometry, and much of it can befound in Moerdijk and Reyes [31, Ch. I], Dubuc [811] or Kock [21, III]. Weintroduce some new notation for brevity, in particular, our fair C-rings areknown in the literature as finitely generated and germ determined C-rings.

2.1 Two definitions ofC

-ringWe first define C-rings in the style of classical algebra.

Definition 2.1. A C-ring is a set C together with operations

f : Cn =

n copies

C C Cfor all n 0 and smooth maps f : Rn R, where by convention when n = 0 wedefine C0 to be the single point {}. These operations must satisfy the followingrelations: suppose m, n 0, and fi : R

n R for i = 1, . . . , m and g : Rm Rare smooth functions. Define a smooth function h : Rn R by

h(x1, . . . , xn) = g

f1(x1, . . . , xn), . . . , f m(x1 . . . , xn)

,

for all (x1, . . . , xn) Rn. Then for all (c1, . . . , cn) Cn we haveh(c1, . . . , cn) = g

f1(c1, . . . , cn), . . . , fm(c1, . . . , cn)

.

We also require that for all 1 j n, defining j : Rn R by j :

(x1, . . . , xn) xj , we have j (c1, . . . , cn) = cj for all (c1, . . . , cn) Cn.

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Usually we refer to C as the C-ring, leaving the operations f implicit.A morphism between C-rings

C, (f)f:RnR C

,D, (f)f:RnR C

is a map : C D such that f(c1), . . . , (cn) = f(c1, . . . , cn) forall smooth f : Rn R and c1, . . . , cn C. We will write CRings for thecategory of C-rings.

Here is the motivating example, which we will study at greater length in 3:Example 2.2. Let X be a manifold, which may be without boundary, or withboundary, or with corners. Write C(X) for the set of smooth functions c :X R. For n 0 and f : Rn R smooth, define f : C(X)n C(X) by

f(c1, . . . , cn)

(x) = f

c1(x), . . . , cn(x)

, (2)

for all c1, . . . , cn C(X) and x X. It is easy to see that C(X) and theoperations f form a C-ring.

Example 2.3. Take X = {0} in Example 2.2. Then C({0}) = R, withoperations f : R

n R given by f(x1, . . . , xn) = f(x1, . . . , xn). This makesR into the simplest nonzero example of a C-ring.

Note that C-rings are far more general than those coming from manifolds.For example, if X is any topological space we could define a C-ring C0(X) tobe the set of continuous c : X R with operations f defined as in (2). ForX a manifold with dim X > 0, the C-rings C(X) and C0(X) are different.

There is a more succinct definition of C-rings using category theory:

Definition 2.4. Write Euc for the full subcategory of Man spanned by theEuclidean spaces Rn. That is, the objects of Euc are the manifolds Rn forn = 0, 1, 2, . . ., and the morphisms in Euc are smooth maps f : Rn Rm.Write Sets for the category of sets. In both Euc and Sets we have notions of(finite) products of objects (that is, Rn+m = Rn Rm, and products S T ofsets S, T), and products of morphisms. Define a (category-theoretic) C-ringto be a product-preserving functor F : Euc Sets.

Here is how this relates to Definition 2.1. Suppose F : Euc Sets is aproduct-preserving functor. Define C = F(R). Then C is an object in Sets,that is, a set. Suppose n 0 and f : Rn R is smooth. Then f is a morphismin Euc, so F(f) : F(Rn) F(R) = C is a morphism in Sets. Since F preservesproducts F(Rn) = F(R) F(R) = Cn, so F(f) maps Cn C. We definef : C

n C by f = F(f). The fact that F is a functor implies that the fsatisfy the relations in Definition 2.1, so

C, (f)f:RnR C

is a C ring.

Conversely, ifC, (f)f:RnR C

is a C-ring then we define F : Euc

Sets by F(Rn) = Cn, and if f : Rn

R

m is smooth then f = (f1, . . . , f m) for

fi : Rn R smooth, and we define F(f) : Cn Cm by F(f) : (c1, . . . , cn) f1(c1, . . . , cn), . . . , fm(c1, . . . , cn)

. Then F is a product-preserving functor.

This defines a 1-1 correspondence between C-rings in the sense of Definition2.1, and category-theoretic C-rings in the sense of Definition 2.4.

Since all small colimits exist in Sets [25, Ex. V.1.8], regarding C-rings asfunctors F : Euc Sets as in Definition 2.4, to take small colimits in the

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category of C-rings we can take colimits in Sets object-wise in Euc, so as inMoerdijk and Reyes [31, p. 21-22] we have:

Proposition 2.5. In the category CRings of C-rings, all small colimitsexist, and so in particular pushouts and all finite colimits exist.

2.2 C-rings as commutative R-algebras, and ideals

Every C-ring C has an underlying commutative R-algebra:

Definition 2.6. Let C be a C-ring. Then we may give C the structure ofa commutative R-algebra. Define addition + on C by c + c = f(c, c) for

c, c C, where f : R2 R is f(x, y) = x + y. Define multiplication on C by c c = g(c, c), where g : R2 R is f(x, y) = xy. Define scalarmultiplication by R by c = (c), where : R R is (x) = x. Defineelements 0 and 1 in C by 0 = 0(

) and 1 = 1(

), where 0 : R0

R and

1 : R0 R are the maps 0 : 0 and 1 : 1. One can then showusing the relations on the f that all the axioms of a commutative R-algebraare satisfied. In Example 2.2, this yields the obvious R-algebra structure onthe smooth functions c : X R. But a C-ring has far more structure andoperations than a commutative R-algebra.

Definition 2.7. An ideal I in C is an ideal I C in C regarded as a commu-tative R-algebra. Then we make the quotient C/I into a C-ring as follows. Iff : Rn R is smooth, define If : (C/I)n C/I by

If(c1 + I , . . . , cn + I)

(x) = f

c1(x), . . . , cn(x)

+ I.

To show this is well-defined, we must show it is independent of the choice of

representatives c1, . . . , cn in C for c1 + I , . . . , cn + I in C/I. By HadamardsLemma there exist smooth functions gi : R2n R for i = 1, . . . , n with

f(y1, . . . , yn) f(x1, . . . , xn) =n

i=1(yi xi)gi(x1, . . . , xn, y1, . . . , yn)for all x1, . . . , xn, y1, . . . , yn R. If c1, . . . , cn are alternative choices for c1, . . . ,cn, so that ci + I = ci + I for i = 1, . . . , n and c

i ci I, we have

f

c1(x), . . . , cn(x)

fc1(x), . . . , cn(x)

=n

i=1(ci ci)gi

c1(x), . . . , c

n(x), c1(x), . . . , cn(x)

.

The second line lies in I as ci ci I and I is an ideal, so If is well-defined,and clearly

C/I, (If)f:RnR C

is a C-ring.

If C is a C-ring, we will use the notation (fa : a

A) to denote theideal in C generated by a collection of elements fa, a A in C, in the sense ofcommutative R-algebras. That is,

(fa : a A) =n

i=1 fai ci : n 0, a1, . . . , an A, c1, . . . , cn C

.

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Definition 2.8. A C-ring C is called finitely generated if there exist c1, . . . , cnin C which generate C over all C-operations. That is, for each c

C there

exists smooth f : Rn R with c = f(c1, . . . , cn). (Note that this is a muchweaker condition than C being finitely generated as a commutative R-algebra).

By Kock [21, Prop. III.5.1], C(Rn) is the free C-ring with n generators.Given such C, c1, . . . , cn, define : C(R

n) C by (f) = f(c1, . . . , cn) forsmooth f : Rn R, where C(Rn) is as in Example 2.2 with X = Rn. Then is a surjective morphism of C-rings, so I = Ker is an ideal in C(Rn),and C = C(Rn)/I as a C-ring. Thus, C is finitely generated if and only ifC = C(Rn)/I for some n 0 and ideal I in C(Rn).

An ideal I in a C-ring C is called finitely generated ifI is a finitely generatedideal of the underlying commutative R-algebra of C in Definition 2.6, that is,I = (i1, . . . , ik) for some i1, . . . , ik C. A C-ring C is called finitely presented ifC = C(Rn)/I for some n 0, where I is a finitely generated ideal in C(Rn).

Given such C, n , I , choose generators i1, . . . , ik

C(Rn) for I. Define :

C(Rk) C(Rn) by (f)(x1, . . . , xn) = fi1(x1, . . . , xn), . . . , ik(x1, . . . , xn)for all smooth f : Rk R and x1, . . . , xn R. Then is a morphism ofC-rings, and

C(Rk)x1,...,xk 0

/G

C({0}) = R11

C(Rn) /G C

(3)

is a pushout square in CRings. Conversely, C is finitely presented if it fitsinto a pushout square (3).

A difference with conventional Algebraic Geometry is that C(Rn) is notnoetherian, so ideals in C(Rn) may not be finitely generated, and C finitelygenerated does not imply C finitely presented.

Write C

Ringsfg

and C

Ringsfp

for the full subcategories of finitelygenerated and finitely presented C-rings in CRings.

Example 2.9. A Weil algebra [8, Def. 1.4] is a finite-dimensional commutativeR-algebra W which has a maximal ideal m with W/m = R and mn = 0 forsome n > 0. Then by Dubuc [8, Prop. 1.5] or Kock [21, Th. III.5.3], there is aunique way to make W into a C-ring compatible with the given underlyingcommutative R-algebra. This C-ring is finitely presented [21, Prop. III.5.11].C-rings from Weil algebras are important in Synthetic Differential Geometry,in arguments involving infinitesimals.

2.3 Local C-rings, and localization

Definition 2.10. A C-ring C is called local if regarded as an R-algebra, as

in Definition 2.6, C is a local R-algebra with residue field R. That is, C has aunique maximal ideal mC with C/mC = R.

IfC,D are local C-rings with maximal ideals mC ,mD, and : C D isa morphism of C rings, then using the fact that C/mC = R = D/mD we seethat 1(mD) = mC , that is, is a local morphism of local C-rings. Thus,there is no difference between morphisms and local morphisms.

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Remark 2.11. Following Dubuc [10, Def. 4], we include the condition that Chas residue field R in the definition of local C-ring. Moerdijk and Reyes [2931]

omit this condition. They call local C-rings with residue field R pointed localC-rings in [31, I.3] and Archimedean local C-rings in [29, 3].

Localizations of C-rings are studied by Moerdijk et al. [29,30], [31, p. 23].

Definition 2.12. Let C be a C-ring and S a subset of C. A localizationC[s1 : s S] of C at S is a C-ring D = C[s1 : s S] and a morphism : C D such that (s) is invertible in D for all s S, with the universalproperty that ifE is a C-ring and : C E a morphism with (s) invertiblein E for all s S, then there is a unique morphism : D E with = .

A localization C[s1 : s S] always exists it can be constructed byadjoining an extra generator s1 and an extra relation s s1 1 = 0 for eachs S and is unique up to unique isomorphism. When S = {c} we have anexact sequence 0

I

C

R C(R)

C[c1]

0, where C

R C(R) is the

pushout in CRings of the unique C-ring morphisms R C, R C(R),with pushout morphisms 1 : C C R C(R), 2 : C(R) C R C(R),and I is the ideal in C R C(R) generated by 1(c) 2(x) 1, where x is thegenerator of C(R).

An R-point p of a C-ring C is a C-ring morphism p : C R, whereR is regarded as a C-ring as in Example 2.3. By [31, Prop. I.3.6], a mapp : C R is a morphism of C-rings if and only if it is a morphism of theunderlying R-algebras, as in Definition 2.7. Define Cp to be the localizationCp = C[s

1 : s C, p(s) = 0], with projection p : C Cp. Then Cp is alocal C-ring by [31, Lem. 1.1]. The R-points of C(Rn) are just evaluationat points p Rn.Example 2.13. For n 0 and p Rn, define Cp (Rn) to be the set of germsof smooth functions c : R

n

R at p Rn

, made into a C

-ring in the obviousway. Then Cp (R

n) is a local C-ring in the sense of Definition 2.10. Here arethree different ways to define Cp (R

n), which yield isomorphic C-rings:

(a) Defining Cp (Rn) as the germs of functions of smooth functions at p means

that points of Cp (Rn) are -equivalence classes [(U, c)] of pairs (U, c),

where U Rn is open with p U and c : U R is smooth, and(U, c) (U, c) if there exists p V U U open with c|V c|V.

(b) As the localization

C(Rn)

p = C(Rn)[g C(Rn) : g(p) = 0]. Then

points of

C(Rn)

p are equivalence classes [f /g] of fractions f /g forf, g C(Rn) with g(p) = 0, and fractions f /g, f/g are equivalent ifthere exists h C(Rn) with h(p) = 0 and h(f g fg) 0.

(c) As the quotient C(Rn)/I, where I is the ideal of f

C(Rn) with

f 0 near p Rn.One can show (a)(c) are isomorphic using the fact that if U is any open neigh-bourhood ofp in Rn then there exists smooth : Rn [0, 1] such that 0 onan open neighbourhood ofRn \ U in Rn and 1 on an open neighbourhoodof p in U. Any finitely generated local C-ring is a quotient of some Cp (R

n).

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2.4 Fair C-rings

Definition 2.14. An ideal I in C

(Rn

) is called fair if for each f C

(Rn

),f lies in I if and only if p(f) lies in p(I) Cp (Rn) for all p Rn, whereCp (R

n) is as in Example 2.13 and p : C(Rn) Cp (Rn) is the natural

projection p : c [(Rn, c)]. A C-ring C is called fair if it is isomorphicto C(Rn)/I, where I is a fair ideal. Equivalently, C is fair if it is finitelygenerated and whenever c C with p(c) = 0 in Cp for all R-points p : C Rthen c = 0, using the notation of Definition 2.12. Dubuc [9], [10, Def. 11] callsfair ideals ideals of local character, and Moerdijk and Reyes [31, I.4] call themgerm determined. By Dubuc [9, Prop. 1.8], [10, Prop. 12] any finitely generatedideal I is fair, so C finitely presented implies C fair. Write CRingsfa for thefull subcategory of fair C-rings in CRings.

Since C(Rn)/I is fair if and only if I is fair, we have:

Lemma 2.15. Let I C

(Rm

), J C

(Rn

) be ideals with C

(Rm

)/I =C(Rn)/J as C-rings. Then I is fair if and only if J is fair.Recall from category theory that if C is a subcategory of a category D, a

reflection R : D C is a left adjoint to the inclusion C D. That is, R : D Cis a functor with natural isomorphisms HomC(R(D), C) = HomD(D, C) for allC C and D D. We will define a reflection for CRingsfa CRingsfg,following Moerdijk and Reyes [31, p. 48-49] (see also Dubuc [10, Th. 13]).

Definition 2.16. Let C be a finitely generated C-ring. Let IC be the idealof all c C such that p(c) = 0 in Cp for all R-points p : C R then c = 0.Then C/IC is a finitely generated C-ring, with projection : C C/IC .It has the same R-points as C, that is, morphisms p : C/IC R are in 1-1correspondence with morphisms p : C

R by p = p

, and the local rings

(C/IC)p and Cp are naturally isomorphic. It follows that C/IC is fair. Definea functor Rfafg : C

Ringsfg CRingsfa by Rfafg(C) = C/IC on objects, andif : C D is a morphism in CRingsfg, then (IC) ID, so induces amorphism : C/IC D/ID, and we set Rfafg() = . It is easy to see Rfafg isa reflection.

Example 2.17. Let : R [0, ) be smooth with (x) > 0 for x (0, 1) and(x) = 0 for x / (0, 1). Define I C(R) by

I =

aA ga(x)(x a) : A Z is finite, ga C(R), a A

.

Then I is an ideal in C(R), so C = C(R)/I is a C-ring. The set off C(R) such that p(f) lies in p(I) Cp (R) for all p R is

I =

aZ ga(x)(x a) : ga C(R), a Z,where the sum

aZ ga(x)(x a) makes sense as at most one term is nonzero

at any point x R. Since I = I, we see that I is not fair, so C = C(R)/I isnot a fair C-ring. In fact I is the smallest fair ideal containing I. We haveIC(R)/I = I/I, and R

fafg

C(R)/I) = C(R)/I.

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2.5 Good C-rings

The following class of ideals in C

(Rn

) is defined by Moerdijk and Reyes [31,p. 47, p. 49] (see also Dubuc [9, 1.7(a)]), who call them flat ideals:Definition 2.18. Let X be a closed subset ofRn. Define mX to be the idealof all functions g C(Rn) such that kg|X 0 for all k 0, that is, g andall its derivatives vanish at each x X. If the interior X of X in Rn is densein X, that is (X) = X, then kg|X 0 for all k 0 if and only if g|X 0. Inthis case C(Rn)/mX

= C(X) :=

f|X : f C(Rn)

.

Here is an example from Moerdijk and Reyes [31, Th. I.1.3].

Example 2.19. Take X to be the point {0}. Iff, f C(Rn) then ff lies inm{0} if and only iff, f

have the same Taylor series at 0. Thus C(Rn)/m{0} is

the C-ring of Taylor series at 0 of f C(Rn). Since any formal power seriesin x1, . . . , xn is the Taylor series of some f C

(R

n

), we have C

(R

n

)/m

{0} =R[[x1, . . . , xn]]. Thus the R-algebra of formal power series R[[x1, . . . , xn]] canbe made into a C-ring.

Moerdijk and Reyes [31, Cor. I.4.12] prove a nontrivial result on such ideals:

Proposition 2.20. Let X Rm and Y Rn be closed. Then as ideals inC(Rm+n) we have (mX ,m

Y ) = m

XY.

The next definition is new, so far as the author knows.

Definition 2.21. An ideal I in C(Rn) is called good if it is of the formI = (f1, . . . , f k,mX ) for some f1, . . . , f k C(Rn) and closed X Rn. AC-ring C is called good ifC = C(Rn)/I for some n 0, where I is a goodideal. If I is finitely generated, then taking X =

shows that I is good. If I is

good then Moerdijk and Reyes [31, Cor. I.4.9] implies that I is fair.

We are interested in good C-rings as they have the following propertieswhich will be useful in applications later on and in [ 20]:

(a) If X is a manifold, possibly with boundary or with corners, then C(X)is a good C-ring, as in 3.

(b) Good C-rings are closed under finite colimits in CRings, as in 2.6.(c) The spectrum functor Spec from C-rings to affine C-schemes is full

and faithful on good C-rings, as in 4.(d) The cotangent module C of a good C-ring C is a finitely presented

C-module, as in 5.

Finitely presented C

-rings do not satisfy (a), fair C

-rings do not satisfy(b),(d), and finitely generated or arbitrary C-rings do not satisfy (c),(d). WriteCRingsgo for the full subcategory of good C-rings in CRings. Then

CRingsfp CRingsgo CRingsfa CRingsfg CRings.Here is an analogue of Lemma 2.15.

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Proposition 2.22. Suppose I C(Rm) and J C(Rn) are ideals withC(Rm)/I

= C(Rn)/J as C-rings. Then I is good, or finitely generated, if

and only if J is good, or finitely generated, respectively.

Proof. Write : C(Rm)/I C(Rn)/J for the isomorphism, and x1, . . . , xmfor the generators of C(Rm), and y1, . . . , yn for the generators of C

(Rn).Since is an isomorphism we can choose f1, . . . , f m C(Rn) with (xi + I) =fi + J for i = 1, . . . , m and g1, . . . , gn C(Rm) with (gi + I) = yi + J fori = 1, . . . , n. Write g = (g1, . . . , gn) : R

m Rn. Then(xi + I) = fi + J = fi(y1, . . . , yn) + J = fi(y1 + J , . . . , yn + J)

= fi

(g1 + I), . . . , (gn + I)

=

fi(g1, . . . , gn) + I

,

since and the projections C(Rm) C(Rm)/I, C(Rn) C(Rn)/J aremorphisms of C-rings. So as is injective we see that

xi fi(g1, . . . , gn) I for i = 1, . . . , m. (4)As I is the kernel of : C(Rm) C(Rn)/J we see that

I =

e C(Rm) : e(f1, . . . , f m) J

. (5)

Let K be the ideal in C(Rm) generated by xifi(g1, . . . , gn) for i = 1, . . . , m.Then K I by (4). Let e C(Rm). Then

e + K = e(x1, . . . , xm) + K = e(x1 + K , . . . , xm + K)

= e

f1(g1, . . . , gn) + K , . . . , fm(g1, . . . , gn) + K

= e

f1(g1, . . . , gn), . . . , fm(g1, . . . , gn)

+ K

= e(f1,...,fm)(g1, . . . , gn) + K.

So by (5), ife I then e + K = h(g1, . . . , gn) + K for some h J. Conversely,if h J then g(h) = h(g1, . . . , gn) I. Hence

I =

xi fi(g1, . . . , gn), i = 1, . . . , m , and g(h), h J

. (6)

Suppose J is good, so that J = (h1, . . . , hl,mY ) for some h1, . . . , hl C(Rn) and closed Y Rn. Define X = g1(Y), a closed subset in Rn. UsingProposition 2.20, one can show that the ideal

g(mY )

in C(Rn) generated

by g(mY ) is mX . Therefore (6) gives

I =

xi fi(g1, . . . , gn), i = 1, . . . , n, g(hj ), for j = 1, . . . , l, and mX

,

so I is good. Conversely, if I is good then J is good. If J is finitely generatedwe take Y = , and then I is finitely generated, and vice versa.Example 2.23. The local C-ring Cp (R

n) of Example 2.13 is the quotient ofC(Rn) by the ideal I of functions f with f 0 near p Rn. For n > 0 this Iis fair, but not good. So Cp (R

n) is fair, but not good, by Proposition 2.22.

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Moerdijk and Reyes [31, Cor. I.4.19] prove:

Proposition 2.24. Let X Rn

be closed with X = ,Rn

. Then the ideal mX

in C(Rn) is not countably generated.

Hence good ideals need not be finitely generated. With Proposition 2.22, thisimplies that good C-rings need not be finitely presented.

Proposition 2.25. Let C be a C-ring, c C, andC[c1] be the localizationof C at c. If C is finitely presented, or good, then C[c1] is finitely presented,or good, respectively.

Proof. There is an exact sequence 0 I C(Rn) C 0 as C is finitelygenerated in both cases. Write x1, . . . , xn for the generators of C(R

n), andlet c = (e) for e C(Rn). Then we have an exact sequence 0 J C(Rn+1)

C[c1]

0, where if we write the generators of C(Rn+1) as

x1, . . . , xn, y, then J =

y e(x1, . . . , xn) 1, f(x1, . . . , xn) : f I. IfC is goodthen I = (f1, . . . , f k, mX ) for some closed X Rn. Proposition 2.20 then yields

J =

y e(x1, . . . , xn) 1, f1(x1, . . . , xn), . . . , f k(x1, . . . , xn), and mXR

,

where X R Rn+1 is closed. Hence J,C[c1] are good. If C is finitelypresented we take X = , and then C[c1] is finitely presented.

However, localizations of fair C-rings need not be fair:

Example 2.26. Let C be the local C-ring C0 (R), as in Example 2.13. ThenC = C(R)/I, where I is the ideal of all f C(R) with f 0 near 0 in R.This I is fair, so C is fair. Let c = [(x,R)] C. Then the localization C[c1]is the C-ring of germs at 0 in R of smooth functions R \ {0} R. Takingy = x

1

as a generator of C[c1

], we see that C[c1

] = C

(R

)/J, where J isthe ideal of compactly supported functions in C(R). This J is not fair, so byLemma 2.15, C[c1] is not fair.

2.6 Pushouts ofC-rings

Proposition 2.5 shows that pushouts of C-rings exist. For finitely generatedC-rings, we can describe these pushouts explicitly.

Example 2.27. Suppose the following is a pushout diagram of C-rings:

C

/G

E

D

/G F,

so that F = D C E, with C,D,E finitely generated. Then we have exactsequences

0 I C(Rl) C 0, 0 J C(Rm) D 0,and 0 K C(Rn) E 0,

(7)

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where ,, are morphisms of C-rings, and I , J, K are ideals in C(Rl),C(Rm), C(Rn). Write x1, . . . , xl and y1, . . . , ym and z1, . . . , zn for the gen-

erators of C(Rl), C(Rm), C(Rn) respectively. Then (x1), . . . , (xl) gen-erate C, and (x1), . . . , (xl) lie in D, so we may write (xi) = (fi)for i = 1, . . . , l as is surjective, where fi : R

m R is smooth. Similarly (x1), . . . , (xl) lie in E, so we may write (xi) = (gi) for i = 1, . . . , l,where gi : R

n R is smooth.Then from the explicit construction of pushouts of C-rings we obtain an

exact sequence with a morphism of C-rings

0 L C(Rm+n) F 0, (8)where we write the generators of C(Rm+n) as y1, . . . , ym, z1, . . . , zn, and thenL is the ideal in C(Rq+r) generated by the elements d(y1, . . . , ym) for d J C(Rm), and e(z1, . . . , zn) for e K C(Rn), and fi(y1, . . . , ym) gi(z1, . . . , zn) for i = 1, . . . , l.Proposition 2.28. The subcategories CRingsfg, CRingsfp, CRingsgo

are closed under pushouts and all finite colimits in CRings.

Proof. First we show CRingsfg, CRingsfp, CRingsgo are closed underpushouts. Suppose C,D,E are finitely generated, and use the notation of Ex-ample 2.27. Then F is finitely generated with generators y1, . . . , ym, z1, . . . , zn,so CRingsfg is closed under pushouts. Suppose D,E are good. Then J =(d1, . . . , dj ,mY ), K = (e1, . . . , ek,m

Z ) for Y Rm, Z Rn closed. Proposition

2.20 gives (mY ,mZ ) = m

YZ , so by Example 2.27 we have

L =

dp(y1, . . . , ym), p = 1, . . . , j , ep(z1, . . . , zn), p = 1, . . . , k ,

fp(y1, . . . , ym) gp(z1, . . . , zn), p = 1, . . . , l , and mYZ

.(9)

Thus L is good, so F is good, and C

Ringsgo

is closed under pushouts. ForD,E finitely presented we take Y = Z = , and then L is finitely generated,and Ffinitely presented, and CRingsfp is closed under pushouts.

Finally, note that R is an initial object in CRings and lies in CRingsfg,CRingsfp and CRingsgo, and all finite colimits may be constructed byrepeated pushouts possibly involving the initial object. Hence CRingsfg,CRingsfp, CRingsgo are closed under finite colimits.

Here is an example from Moerdijk and Reyes [31, p. 49].

Example 2.29. Consider the pushout ofC-rings C(R)RC0 (R), using theunique morphisms R C(R), R C0 (R), where C0 (R) is the ring of germsof smooth functions at 0 in R as in Example 2.13. Then R, C(R), C0 (R) areall fair C-rings, but C0 (R) is neither finitely presented nor good.

By Example 2.27, C

(R) R C0 (R) = C

(R2

)/L, where L is the ideal inC(R2) generated by functions f(x, y) = g(y) for g C(R) with g 0 near0 R. This ideal L is not fair, since for example one can find f C(R2) withf(x, y) = 0 if and only if|xy| 1, and then f / L but p(f) p(L) Cp (R2)for all p R2. Hence C(R) R C0 (R) is not a fair C-ring, by Lemma 2.15,and pushouts of fair C-rings need not be fair.

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Our next result is referred to in the last part of Dubuc [10, Th. 13].

Proposition 2.30. C

Ringsfa

is not closed under pushouts in C

Rings.Nonetheless, pushouts and all finite colimits exist in CRingsfa, although theymay not coincide with pushouts and finite colimits inCRings.

Proof. Example 2.29 shows that CRingsfa is not closed under pushouts inCRings. To construct finite colimits in CRingsfa, we first take the colimitin CRingsfg, which exists by Propositions 2.5 and 2.28, and then apply thereflection functor Rfafg. By the universal properties of colimits and reflection

functors, the result is a colimit in CRingsfa.

3 The C-ring C(X) of a manifold X

We now study the C-rings C(X) of manifolds X defined in Example 2.2. We

are interested in manifolds without boundary (locally modelled on Rn), and inmanifolds with boundary (locally modelled on [0, ) Rn1), and in manifoldswith corners (locally modelled on [0, )k Rnk). Manifolds with corners wereconsidered by the author [19], and we use the conventions of that paper.

The C-rings of manifolds with boundary are discussed by Reyes [34] andKock [21, III.9], but Kock appears to have been unaware of Proposition 2.20,which makes C-rings of manifolds with boundary easier to understand.

If X, Y are manifolds with corners of dimensions m, n, then [19, 3] definedf : X Y to be weakly smooth if f is continuous and whenever (U, ), (V, )are charts on X, Y then 1 f : (f )1((V)) V is a smooth mapfrom (f )1((V)) Rm to V Rn. A smooth map is a weakly smoothmap f satisfying some complicated extra conditions over kX, lY in [19, 3].If Y =

these conditions are vacuous, so for manifolds without boundary,

weakly smooth maps and smooth maps coincide. Write Man, Manb, Mancfor the categories of manifolds without boundary, and with boundary, and withcorners, respectively, with morphisms smooth maps.

Example 3.1. Let 0 < k n, and consider the closed subset Rnk = [0, )k R

nk in Rn, the local model for manifolds with corners. Write C(Rnk ) for the

C-ring

f|Rnk : f C(Rn)

. Since the interior (Rnk ) = (0, )kRnk ofRnk

is dense in Rnk , as in Definition 2.18 we have C(Rnk ) = C

(Rn)/mRnk

. HenceC(Rnk ) is a good C

-ring, by Definition 2.21. Also, mRnk

is not countablygenerated by Proposition 2.24, so it is not finitely generated, and thus C(Rnk )is not a finitely presented C-ring by Proposition 2.22.

Consider the coproduct of C-rings C(Rmk ), C(Rnl ) in C

Rings, thatis, the pushout C(Rmk )

R C(R

nl ) over the trivial C

-ring R. By Example

2.29 and Proposition 2.20 we have

C(Rmk ) R C(Rnl ) = C(Rm+n)/(mRmk ,mRnl

) = C(Rm+n)/mRmk Rnl

= C(Rmk Rnl ) = C(Rm+nk+l ).This is an example of Theorem 3.6 below, with X=Rmk , Y =R

nl and Z={0}.

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Proposition 3.2. (a) If X is a manifold without boundary then the C-ringC(X) of Example 2.2 is finitely presented.

(b) If X is a manifold with boundary, or with corners, and X = , then theC-ring C(X) of Example 2.2 is good, but is not finitely presented.

Proof. Part (a) is proved in Dubuc [10, p. 687] and Moerdijk and Reyes [31,Th. I.2.3] following an observation of Lawvere, that if X is a manifold withoutboundary then we can choose a closed embedding i : X RN for N 0, andthen X is a retract of an open neighbourhood U of i(X), so we have an exactsequence 0 I C(RN) i

C(X) 0 in which the ideal I is finitelygenerated, and thus the C-ring C(X) is finitely presented.

For (b), if X is an n-manifold with boundary, or with corners, then we canembed X as a closed subset in an n-manifold X without boundary, such that theinclusion X X is locally modelled on the inclusion ofRnk = [0, )k Rnkin Rn for k n. We can take X diffeomorphic to the interior X ofX. Choose

a closed embedding i : X RN

for N 0 as above, giving 0 I C(RN)

i C(X) 0 with I generated by f1, . . . , f k C(RN). Theinterior X of X is open in X, so there exists an open subset U in RN withi(X) = U i(X). Therefore i(X) = U i(X).

Let I be the good ideal (f1, . . . , f k,mU ) in C(RN). Since U is open in RN

and dense in U, as in Definition 2.18 we have g mU

if and only if g|U 0.Therefore the isomorphism (i) : C(R

N)/I C(X) identifies the idealI/I in C(X) with the ideal of f C(X) such that f|X 0, since X =i1(U). Hence

C(RN)/I= C(X)/

fC(X) : f|X 0=

f|X : fC(X)

= C(X).As I is a good ideal, this implies that C(X) is a good C-ring. IfX = thenusing Proposition 2.24 we can show I is not countably generated, so C

(X) isnot finitely presented by Proposition 2.22.

Next we consider the transformation X C(X) as a functor.Definition 3.3. Write (CRings)op, (CRingsfp)op, (CRingsgo)op forthe opposite categories of CRings, CRingsfp, CRingsgo (i.e. directionsof morphisms are reversed). Define functors

FCRings

Man : Man (CRingsfp)op (CRings)op,FCRings

Manb: Manb (CRingsgo)op (CRings)op,

FCRings

Manc : Manc (CRingsgo)op (CRings)op

as follows. On objects the functors F

CRings

Man map X C

(X), where C

(X)is a C-ring as in Example 2.2. On morphisms, if f : X Y is a smooth mapof manifolds then f : C(Y) C(X) mapping c c f is a morphismof C-rings, so that f : C(Y) C(X) is a morphism in CRings, andf : C(X) C(Y) a morphism in (CRings)op, and FCRingsMan mapf f. Clearly FCRingsMan , FC

Rings

Manb, FC

RingsManc are functors.

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Iff : X Y is only weakly smooththen f : C(Y) C(X) in Definition3.3 is still a morphism of C-rings. From [31, Prop. I.1.5] we deduce:

Proposition 3.4. Let X, Y be manifolds with corners. Then the map f ffrom weakly smooth maps f : X Y to morphisms of C-rings : C(Y) C(X) is a 1-1 correspondence.

Using the conventions of [19], in the category Man of manifolds without

boundary, the morphisms are weakly smooth maps. So FCRings

Man is both injec-tive on morphisms (faithful), and surjective on morphisms (full), as in Moerdijkand Reyes [31, Th. I.2.8]. But in Manb, Manc the morphisms are smoothmaps, a proper subset of weakly smooth maps, so the functors are injective butnot surjective on morphisms. That is:

Corollary 3.5. The functor FCRings

Man : Man (CRingsfp)op is full and

faithful. However, the functors F

CRings

Manb : Manb

(C

Ringsgo

)op

andFCRings

Manc : Manc (CRingsgo)op are faithful, but not full.

Of course, if we defined Manb, Manc to have morphisms weakly smooth

maps, then FCRings

Manb, FC

RingsManc would be full and faithful. But this is not

what we need for the applications in [20].Let X, Y, Z be manifolds and f : X Z, g : Y Z be smooth maps. If

X, Y, Zare without boundary then f, g are called transverse if whenever x Xand y Y with f(x) = g(y) = z Z we have TzZ = df(TxX) + dg(TyY). Iff, g are transverse then a fibre product XZ Y exists in Man.

For manifolds with boundary, or with corners, the situation is more com-plicated, as explained in [19, 6]. In the definition of smooth f : X Y weimpose extra conditions over j X, kY, and in the definition of transverse f, g

we impose extra conditions over

j

X,

k

Y,

l

Z. With these more restrictivedefinitions of smooth and transverse maps, transverse fibre products exist inManc by [19, Th. 6.3]. The nave definition of transversality is not a sufficientcondition for fibre products to exist. Note too that a fibre product of manifoldswith boundary may be a manifold with corners, so fibre products work best inMan or Manc rather than Manb.

Our next theorem is given in [10, Th. 16] and [31, Prop. I.2.6] for manifoldswithout boundary, and the special case of products Man Manb Manbfollows from Reyes [34, Th. 2.5], see also Kock [21, III.9]. It can be provedby combining the usual proof in the without boundary case, the proof of [19,Th. 6.3], and Proposition 2.20.

Theorem 3.6. The functors FCRings

Man , FCRingsManc preserve transverse fibre

products in Man, Manc, in the sense of [19,

6]. That is, if the following is aCartesian square of manifolds with g, h transverse

Wf

/G

e

Yh

Xg /G Z,

(10)

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so that W = Xg,Z,h Y, then we have a pushout square of C-rings

C(Z)h

/G

g

C(Y)f

C(X)e /G C(W),

(11)

so that C(W) = C(X) g,C(Z),h C(Y).

4 C-ringed spaces and C-schemesIn Algebraic Geometry, if A is an affine scheme and R the ring of regular func-tions on A, then we can recover A as the spectrum of the ring R, A = Spec R.One of the ideas of Synthetic Differential Geometry, as in [31, I], is to regard amanifold M as the spectrum of the C-ring C(M) in Example 2.2. So we can

try to develop analogues of the tools of Algebraic Geometry and scheme theoryfor smooth manifolds, replacing rings by C-rings throughout. This was doneby Dubuc [9,10]. The analogues of the Algebraic Geometry notions [16, II.2] ofringed spaces, locally ringed spaces, and schemes, are called C-ringed spaces,local C-ringed spaces and C-schemes. Almost nothing in this section isreally new, though we give more detail than our references in places.

4.1 C-ringed spaces and local C-ringed spaces

Definition 4.1. A C-ringed space X = (X, OX ) is a topological space Xwith a sheaf OX of C-rings on X. That is, for each open set U X we aregiven a C ring OX (U), and for each inclusion of open sets V U X we aregiven a morphism of C-rings UV :

OX (U)

OX (V), called the restriction

maps, and all this data satisfies the usual sheaf axioms [ 16, II.1], [15, 0.3.1].A morphism f = (f, f) : (X, OX ) (Y, OY) of C ringed spaces is a

continuous map f : X Y and a morphism f : OY f(OX ) of sheavesof C-rings on Y. That is, for each open U Y we are given a morphism ofC-rings f(U) : OY(U) OX (f1(U)) satisfying the obvious compatibilitieswith the restriction maps UV in OX and OY.

A local C-ringed space X = (X, OX ) is a C-ringed space for whichthe stalks OX,x of OX at x are local C-rings for all x X. As in Remark2.11, since morphisms of local C-rings are automatically local morphisms,morphisms of local C-ringed spaces (X, OX ), (Y, OY) are just morphisms ofC-ringed spaces, without any additional locality condition. Moerdijk et al. [29,3] call our local C-ringed spaces Archimedean C-spaces.

Write CRS for the category of C-ringed spaces, and LCRS for the

full subcategory of local C-ringed spaces.For brevity, we will use the notation that underlined upper case letters

X , Y , Z , . . . represent C-ringed spaces (X, OX ), (Y, OY), (Z, OZ ), . . ., and un-derlined lower case letters f , g , . . . represent morphisms of C-ringed spaces

(f, f), (g, g), . . .. When we write x X we mean that X = (X, OX ) and

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x X. When we write U is open in X we mean that U = (U, OU) andX = (X,

OX ) with U

X an open set and

OU =

OX

|U.

Example 4.2. Let X be a manifold, which may have boundary or corners.Define a C-ringed space X = (X, OX ) to have topological space X andOX (U) = C(U) for each open subset U X, where C(U) is the C-ring of smooth maps c : U R, and if V U X are open we defineU V : C

(U) C(V) by UV : c c|V.It is easy to verify that OX is a sheaf ofC-rings on X (not just a presheaf),

so X = (X, OX ) is a C-ringed space. For each x X, the stalk OX,x is thelocal C-ring of germs [(c, U)] of smooth functions c : X R at x X, as inExample 2.13, with unique maximal ideal mX,x =

[(c, U)] OX,x : c(x) = 0

and OX,x/mX,x = R. Hence X is a local C-ringed space.Let X, Y be manifolds and f : X Y a weakly smooth map. Define

(X, OX ), (Y, OY) as above. For all open U Y define f(U) : OY(U) =C

(U) OX (f1

(U)) = C

(f1

(U)) by f

(U) : c c f for all c C

(U).Then f(U) is a morphism of C-rings, and f : OY f(OX ) is a morphismof sheaves of C-rings on Y, so f = (f, f) : (X, OX ) (Y, OY) is a morphismof (local) C-ringed spaces.

As the category Top of topological spaces has all finite limits, and the con-struction ofCRS involves Top in a covariant way and the category CRingsin a contravariant way, using Proposition 2.5 one may prove:

Proposition 4.3. All finite limits exist in the category CRS.

Dubuc [10, Prop. 7] proves:

Proposition 4.4. The full subcategory LCRS of local C-ringed spaces inCRS is closed under finite limits in CRS.

4.2 Affine C-schemes

We define a functor Spec : CRingsop LCRS, following Hartshorne [16,p. 70], Dubuc [9, 10], and Moerdijk, Que and Reyes [29].

Definition 4.5. Let C be a C-ring, and use the notation of Definition 2.12.Write XC for the set of all R-points x ofC. Then each c C determines a mapc : XC R by c : x x(c). Let TC be the smallest topology on XC such thatc : XC R is continuous for all c C. That is, TC is generated by the opensets (c)

1(U) for all c C and open U R. Then XC is a topological space. Itis Hausdorff, since if x1 = x2 XC then there exists c C with x1(c) = x2(c),and then c : XC

R is continuous and c(x1)

= c(x2).

For each open U XC , define OXC (U) to be the set of functions s : U xUCx with s(x) Cx for all x U, and such that U may be covered by open

sets V for which there exist c, d C with x(d) = 0 for all x V, with s(x) =x(c)x(d)

1 Cx for all x V. Define operations f on OXC (U) pointwise inx U using the operations f on Cx. This makes OXC (U) into a C-ring. If

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V U XC are open, the restriction map U V : OXC (U) OXC (V) mappingU V : s

s

|V is a morphism of C

-rings.

It is then easy to see that OXC is a sheaf of C-rings on XC . The stalkOXC ,x at x X is isomorphic to Cx, which is a local C-ring. Hence (XC , OC)is a local C-ringed space, which we call the spectrum ofC, and write as Spec C.

Now let : C D be a morphism of C-rings. Define f : XD XCby f(x) = x . Then f is continuous. For U XC open define f(U) :OXC (U) OXD(f1 (U)) by f(U)s : x x(s(f(x))), where x : Cf(x) Dx is the induced morphism of local C

-rings. Then f : OXC (f)(OXD)is a morphism of sheaves of C-rings on XC , so f = (f, f

) : (XD, OD)

(XC , OC) is a morphism of local C-ringed spaces. Define Spec : SpecD SpecC by Spec = f. Then Spec is a functor CRings

op LCRS, whichpreserves limits by Dubuc [10, p. 687].

The global sections functor : LCRS

CRingsop acts on objects

(X, OX ) by : (X, OX ) OX (X) and on morphisms (f, f) : (X, OX ) (Y, OX ) b y : (f, f) f(Y). As in Dubuc [10, Th. 8] or Moerdijk etal. [29, Th. 3.2], is a left adjoint to Spec, that is, for all C CRings andX LCRS there are functorial isomorphisms

HomCRings(C, (X)) = HomLCRS(X, SpecC). (12)For any C-ring C there is a natural morphism ofC-rings C : C (SpecC)corresponding to idX in (12) with X = SpecC.

Remark 4.6. Our definition of the spectrum Spec C agrees with Dubuc [9,10],and with the Archimedean spectrum of [29, 3]. Moerdijk et al. [29, 1] give adifferent definition of the spectrum Spec C, in which the points are not R-points,but C-radical prime ideals.

Example 4.7. Let X be a manifold. It is easy to see that the local C-ringedspace X constructed in Example 4.2 is naturally isomorphic to Spec C(X).

Now suppose C is a finitely generated C-ring, with exact sequence 0 I C(Rn) C 0. Define a map : XC Rn by : z

z

(x1), . . . , z (xn)

, where x1, . . . , xn are the generators of C(Rn). Then

gives a homeomorphism

: XC= X

C =

(x1, . . . , xn) Rn : f(x1, . . . , xn) = 0 for all f I

, (13)

where the right hand side is a closed subset ofRn. So the topological spacesin SpecC for finitely generated C are homeomorphic to closed subsets of Rn.Comparing the definitions of Spec and the reflection Rfa

fg

, we can show:

Proposition 4.8. Let C be a fair C-ring. Then C : C (SpecC) is anisomorphism. More generally, if C is a finitely generated C-ring then SpecCis naturally isomorphic to Spec Rfafg(C), using the notation of Definition 2.16,and (SpecC) is naturally isomorphic to Rfafg(C), and C : C (SpecC) isidentified with the natural surjective projection C Rfafg(C).

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This is a contrast to conventional Algebraic Geometry, in which (Spec R) =R for arbitrary rings R, as in [16, Prop. II.2.2]. Proposition 4.8 shows that for

general C-rings C the functor Spec loses information about C, so Spec isneither full nor faithful, but for fair C-rings Spec loses no information up toisomorphism, so as in [10, Th. 13] we have:

Theorem 4.9. The functor Spec : (CRingsfa)op LCRS is full andfaithful. Hence Spec : (CRingsgo)op LCRS, Spec : (CRingsfp)op LCRS are also full and faithful, since (CRingsfp)op (CRingsgo)op (CRingsfa)op are full subcategories.

In the obvious way we define affine C-schemes.

Definition 4.10. A local C-ringed space X is called an affine C-schemeif it is isomorphic in LCRS to SpecC for some C-ring C. We call X afinitely presented, or good, or fair, affine C-scheme if X

= SpecC for C that

kind of C-ring. Write ACSch, ACSchfp, ACSchgo, ACSchfa forthe full subcategories of affine C-schemes and of finitely presented, good, andfair, affine C-schemes in LCRS respectively. Then Theorem 4.9 impliesthat Spec : (CRingsfa)op ACSchfa is an equivalence of categories, andsimilarly for ACSchfp, ACSchgo.

We did not define finitely generated affine C-schemes, because they coin-cide with fair affine C-schemes, as Proposition 4.8 implies.

Corollary 4.11. Suppose C is a finitely generated C-ring. Then SpecC is afair affine C-scheme.

Theorem 4.12. The full subcategories ACSchfp, ACSchgo, ACSchfa,ACSch are closed under all finite limits in LCRS. Hence, fibre products

and all finite limits exist in each of these subcategories.

Proof. ACSch is closed under small limits in LCRS as small limits existin (CRings)op by Proposition 2.20 and Spec preserves limits by [10, p. 687].Fibre products and all finite limits exist in (CRingsfa)op by Proposition2.30, although they may not coincide with fibre products and finite limits in(CRings)op. The subcategories (CRingsfp)op, (CRingsgo)op are closedunder fibre products and finite limits in (CRings)op by Proposition 2.28, andhence under fibre products and finite limits in (CRingsfa)op. By Dubuc [10,Th. 13] the functor Spec : (CRingsfa)op LCRS preserves limits. (Herewe mean limits in (CRingsfa)op, rather than limits in (CRings)op.)

Let X , Y , Z be finitely presented, or good, or fair, affine C-schemes, andf : X Z, g : Y Z be morphisms in LCRS. Then we have isomorphisms

X = SpecC, Y= SpecD, Z = SpecE in LCRS, (14)

where C,D,E are finitely presented, or good, or fair, C-rings, respectively.Since Spec : (CRingsfa)op LCRS is full and faithful by Theorem 4.9,

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there exist unique morphisms of C-rings : E C, : E D such that (14)identifies f with Spec and g with Spec . Then [10, Th. 13] implies that

Xf,Z,g Y= SpecC Spec ,SpecE,Spec SpecD = SpecC ,E, D

,

where C E D is the pushout in CRingsfa rather than in CRings. ThenCED is finitely presented, or good, or fair, respectively, since (CRingsfp)op,(CRingsgo)op are closed under fibre products in (CRingsfa)op. HenceX Z Y is finitely presented, or good, or fair, respectively, and ACSchfp,ACSchgo, ACSchfa are closed under fibre products in LCRS. SinceSpecR is a terminal object, we see that ACSchfp, ACSchgo, ACSchfa

are also closed under finite limits in LCRS.

Definition 4.13. Define functors

F

CSch

Man : Man AC

Sch

fp

AC

Sch,FCSch

Manb : Manb ACSchgo ACSch,

FCSch

Manc : Manc ACSchgo ACSch,

by FCSch

Man = Spec FCRings

Man , in the notation of Definitions 3.3 and 4.5.

By Example 4.7, if X is a manifold with corners then FCSch

Manc (X) is natu-rally isomorphic to the local C-ringed space X in Example 4.2.

If X , Y , . . . are manifolds, or f , g , . . . are (weakly) smooth maps, we may useX , Y , . . . , f , g , . . . to denote FC

SchManc (X , Y , . . . , f , g , . . .). So for instance we will

write Rn and [0, ) for FCSchMan (Rn) and FCSch

Manb

[0, ).

By Corollary 3.5, Theorems 3.6 and 4.9 and Spec : (CRingsfa)op LC

RS preserving fibre products, we find as in Dubuc [10, Th. 16]:Corollary 4.14. FC

SchMan : Man ACSchfp ACSch is a full and

faithful functor, and FCSch

Manb: Manb ACSchgo ACSch, FCSchManc :

Manc ACSchgo ACSch are both faithful functors, but are not full.Also these functors take transverse fibre products in Man, Manc to fibre prod-ucts in ACSchfp, ACSchgo.

In Definition 2.8 we saw that a C-ring C is finitely presented if and only ifit fits into a pushout square (3) in CRings. Applying Spec, which preservesfibre products, implies:

Lemma 4.15. A C-ringed space X is a finitely presented affine C-schemeif and only if it may be written as a fibre product inCRS :

X /G

{0}0

Rn

/GR

k,

where : Rn Rk is a smooth map, and 0 : {0} Rk is the zero map.

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Our next two results show that ACSchfp, ACSchgo and ACSchfa

are closed under taking open subsets.

Definition 4.16. Let (X, OX ) be a C-ringed space, and U X an opensubset. A characteristic function for U is a morphism of C-ringed spacesf = (f, f) : (X, OX ) R = (R, OR) such that U = {x X : f(x) = 0}.

Let C be a C-ring, and c C. Define c : C(R) C by c(f) = f(c)for all smooth f : R R. Then c is a morphism of C-rings, so Spec c :SpecC Spec C(R) = (R, OR) is a morphism of affine C-schemes. Hence ifthe C-ringed space (X, OX ) is SpecC then elements c C generate morphismsofC-ringed spaces fc = Spec(c) : (X, OX ) R. The characteristic functionswe consider will always be of this form for c C.Proposition 4.17. Let (X, OX ) be a fair affine C-scheme. Then every openU X admits a characteristic function.

Proof. By definition (X, OX ) = SpecC = (XC , OXC ) for some finitely generatedC-ring C, which fits into an exact sequence 0 I C(Rn) C 0.Thus X is homeomorphic to XC , which is homeomorphic to the closed subsetXC

in Rn given in (13). Let U X be open, and U be the open subset of XC

identified with U by these homeomorphisms. As XC has the subspace topology,

there exists an open V Rn with U = V XC .Every open subset in Rn has a characteristic function, [31, Lem. I.1.4]. Hence

there exists f C(Rn) with V = x Rn : f(x) = 0. Definition 4.16 givesa morphism Spec (f) : SpecC R. Let (f, f) : (X, OX ) (R, OR) be themorphism identified with Spec (f) by the isomorphism (X, OX ) = SpecC.Then f : X R is identified with f|X

C

: XC

R by the homeomorphismsX

= XC

= X

C

. As U is identified with U = x X

C

: f(x)= 0, it follows

that U = {x X : f(x) = 0}, so (f, f) is a characteristic function for U.Proposition 4.18. Let (X, OX ) be a finitely presented, or good, or fair, affineC-scheme, and U X be an open subset. Then (U, OX |U) is also a finitelypresented, or good, or fair, affine C-scheme, respectively.

Proof. As (X, OX ) is fair, there exists a characteristic function f : (X, OX ) Rfor U by Proposition 4.17. Consider the fibre product

(X, OX ) f,R,i R \ {0}, (15)

in LCRS, where i : R \ {0} R is the inclusion, and i : R \ {0} R isthe image morphism of affine C-schemes under FC

SchMan . Since R

\ {0

}=

(R \ {0}, OR|R\{0}), it follows on general grounds that (15) is isomorphic to(U, OX |U). But R,R \ {0} are manifolds without boundary, so R,R \ {0} liein ACSchfp, ACSchgo, ACSchfa, which are closed under fibre productsby Theorem 4.12. Thus if (X, OX ) is a finitely presented, or good, or fair, affineC-scheme then so is (15), and hence so is (U, OX |U).

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Note that this is better than the situation in conventional Algebraic Geom-etry, where for instance C2 is an affine C-scheme, but its open subset C2

\{0

}is

not an affine C-scheme. This is because characteristic functions need not existfor open subsets of affine C-schemes.

For general affine C-schemes (X, OX ) = SpecC, open subsets (U, OX |U)need not be affine C-schemes, but we can say the following. A principalopen subset is one of the form Uc = {x X : x(c) = 0} for some c C.They are closed under finite intersections, since Uc1 Ucn = Uc1cn . Also(Uc, OX |Uc) = SpecC[c1], so principal open subsets of affine C-schemes areaffine. Since principal open subsets generate the topology on X, every opensubset in X is a union of principal open subsets. Thus we deduce:

Lemma 4.19. Let (X, OX ) be an affine C-scheme, and U X an opensubset. Then U can be covered by open subsets V U such that (V, OX |V) isan affine C-scheme.

Our next result describes the sheaf of C-rings OX in SpecC for C afinitely generated C-ring. It is a version of [16, Prop. I.2.2(b)] in conven-tional Algebraic Geometry, and reduces to Moerdijk and Reyes [31, Prop. I.1.6]when C = C(Rn).

Proposition 4.20. Let C be a finitely generated C-ring, write (X, OX ) =SpecC, and let U X be open. By Proposition 4.17 we may choose a char-acteristic function f : (X, OX ) R for U of the form f = Spec c for somec C. Then there is a canonical isomorphism OX (U) = Rfafg(C[c1]), in thenotation of Definitions 2.12 and 2.16. If C is finitely presented, or good,then OX (U) = C[c1].Proof. We have morphisms of C-rings c : C(R)

C and i : C(R)

C(R\{0}), and C(R), C(R\{0}) are finitely presented C-rings by Propo-sition 3.2(a). So as Spec preserves limits in (CRingsfg)op we have

SpecC c,C(R),i C(R \ {0})

= SpecC f,R,i R \ {0}= (U, OX |U).

But C C(R) C(R \ {0}) = C[c1] for formal reasons. Thus Proposition 4.8gives OX (U) =

(U, OX |U)

= Rfafg(C[c1]). IfC is finitely presented, or good,then C[c1] is too by Proposition 2.25, so C[c1] is fair and Rfafg

C[c1]

= C[c1],

and therefore OX (U) = C[c1].

4.3 Locally finite sums in fair C-rings

We discuss infinite sums in fair C-rings, broadly following Dubuc [11].

Definition 4.21. Let C be a C-ring, and write (X, OX ) = SpecC. Considera formal expression

aA ca, where A is a (usually infinite) indexing set and

ca C. We say that

aA ca is a locally finite sum ifX can be covered by opensets U X such that for all but finitely many a A we have x(ca) = 0 in Cxfor all x U, or equivalently, XU C(ca) = 0 in OX (U).

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If

aA ca is a locally finite sum, we say that c C is a limit of

aA ca,written c = aA ca, ifx(c) = aA x(ca) for all x

X, where aA x(ca)makes sense as there are only finitely many nonzero terms. For general C limitsneed neither exist, nor be unique. In C(Rn), every locally finite sum

iI ci

has a unique limit, defined pointwise.Suppose the topological space X is locally compact. (This is automatic ifC

is finitely generated, since X is homeomorphic to a closed subset ofRn.) Thenwe can express locally finite sums in terms of a topology on C. For each c Cand each compact subset S X, define Uc,S to be the set of c C such thatx(c) = x(c) in Cx for all x S. We think ofUc,S as an open neighbourhoodofc in C. Let C have the topology with basis Uc,S for all c, S. Then c =

a=1 ca

is equivalent to c = limN(N

a=1 ca) in this topology on C.Let U X be open. We say that c C is supported on U if x(c) = 0 in Cx

for all x X\ U. We do not define supported on U for non-open U.An open cover

{Ua : a

A

}ofX is called locally finite ifX can be covered by

open sets W which intersect only finitely many Ua. An open cover {Vb : b B}of X is called a refinement of another open cover {Ua : a A} of X if Vb Uabfor all b B and some ab A. If X is paracompact (this is automatic if C isfinitely generated) then every open cover has a locally finite refinement.

Let {Ua : a A} be a locally finite open cover of X. A partition of unity inC subordinate to {Ua : a A} is {a : a A} with a C supported on Ua fora A, such that aA a = 1 in C. Note that

aA a is a locally finite sum

in C, since {Ua : a A} is locally finite and a is supported on U.If we just say {a : a A} is a partition of unity in C, we mean it is a

partition of unity subordinate to some locally finite open cover of X.

Following Dubuc [11], it is now easy to prove:

Proposition 4.22. (a) An ideal I in C

(R

n

) is fair if and only if it is closedunder locally finite sums.

(b) Let C be a fair C-ring. Then every locally finite sum

aA ca inC hasa unique limit.

(c) Let C be a fair C-ring, (X, OX ) = SpecC, and {Ua : a A} be a locallyfinite open cover of X. Then there exists a partition of unity{a : a A} inC subordinate to {Ua : a A}.

4.4 General C-schemes

As in conventional Algebraic Geometry [16, II.2], we define a C-scheme tobe a local C-ringed space covered by affine C-schemes.

Definition 4.23. Let X = (X, OX ) be a local C

-ringed space. We call X aC-scheme if X can be covered by open sets U X such that (U, OX |U) isan affine C-scheme. We call a C-scheme X locally fair, or locally good, orlocally finitely presented, if X can be covered by open U X with (U, OX |U) afair, or good, or finitely presented, affine C-scheme, respectively.

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We call a C-scheme X separated if the underlying topological space X isHausdorff. Affine C-schemes are always separated.

Write CSchlf, CSchlg, CSchlfp, CSch for the full subcategoriesof locally fair, and locally good, and locally finitely presented, and all, C-schemes, respectively. Our categories of spaces so far are related as follows:

Man

FCSch

Man

/G Manb

FCSch

Manb

/G Manc

FCSch

Manc

vvmmmmmm

mmmm

ACSchfp /G

ACSchgo

/G

ACSchfa /G

ACSch

'9yyyyy

yyyy C

RS

CSchlfp /G CSchlg

/G CSchlf /G CSch

/G LCRS.

Oy

Ordinary schemes are much more general than ordinary affine schemes, and

central examples such as CPn are affine schemes. However, affine C-schemesare already general enough for many purposes, and constructions involving affineC-schemes often yield affine C-schemes. For example:

All manifolds are affine C-schemes. If a C-scheme X is Hausdorff and can be covered by finitely many fair

affine C-schemes, one can show X is a fair affine C-scheme.

Let X be a locally fair C-scheme with X Hausdorff and paracompact.Then one can prove X is an affine C-scheme.

From Proposition 4.18 and Lemma 4.19 we immediately deduce:

Proposition 4.24. Let (X, OX ) be a locally finitely presented, locally good,locally fair, or general, C

-scheme, and U X be open. Then (U, OX |U)is also a locally finitely presented, or locally good, or locally fair, or general,C-scheme, respectively.

Here is the analogue of Theorem 4.12.

Theorem 4.25. The full subcategories CSchlfp, CSchlg, CSchlf andCSch are closed under all finite limits in LCRS. Hence, fibre productsand all finite limits exist in each of these subcategories.

Proof. We first show CSchlfp, . . . , CSch are closed under fibre products.Let f : X Z, g : Y Z be morphisms in one of these categories andW = Xf,Z,gY be the fibre product in LC

RS, with projections X : W X,Y : W Y. Write W = (W,OW), f = (f, f), and so on. Let w W, andset x = X (w) X, y = Y(w) Y and z = f(x) = g(y) Z. Choose V Zwith z V and (V, OZ |V) in ACSchfp, . . . , ACSch respectively. Thenf1(V) is open in X so

f1(V), OX |f1(V)

lies in CSchlfp, . . . , CSch

by Proposition 4.24. Thus we may choose T f1(V) open with x T and(T, OX |T) in ACSchfp, . . . , ACSch, and similarly we choose U g1(V)open with y U and (U, OY|U) in ACSchfp, . . . , ACSch.

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Let S = 1X (T)1Y (U). Then S is an open neighbourhood ofw in W, and(S,

OW

|S )

= (T,

OX

|T)

(V,OZ |V) (U,

OY

|U). But AC

Schfp, . . . , ACSch are

closed under fibre products in LCRS by Theorem 4.12, so (S, OW|S ) lies inACSchfp, ACSchgo, ACSchfa or ACSch respectively. As W can becovered by S, W lies in CSchlfp, CSchlg, CSchlf or CSch. HenceCSchlfp, . . . , CSch are closed under fibre products in LCRS. They arealso closed under finite limits, as in the proof of Theorem 4.12.

5 Modules over C-ringsNext we discuss modules over C-rings. The author knows of no previous workon these, so all this section may be new, although much of it is a straightforwardgeneralization of well known facts about modules over commutative rings.

5.1 Modules

Definition 5.1. Let C be a C-ring. A module (M, ) over C, or C-module, isa module over C regarded as a commutative R-algebra as in Definition 2.7. Thatis, M is a vector space over R equipped with a bilinear map : C M M,satisfying (c1 c2, m) =

c1, (c2, m)

and (1, m) = m for all c1, c2 C and

m M. A morphism : (M, ) (M, ) ofC-modules (M, ), (M, ) is alinear map : M M such that = (idC ) : C M M. ThenC-modules form an abelian category, which we write as C-mod. Often we writeM for the C-module, leaving implicit, and write c m rather than (c, m).

Let W be a real vector space. Then we define a C-module (CR W, W) byW(c1, c2 w) = (c1 c2) w for c1, c2 C and w W. A C-module (M, )is called free if it is isomorphic to (C R W, W) in C-mod for some W. Noteas in Example 5.5 below that if W is infinite-dimensional then free C-modulesC R W may not be well-behaved, and not a useful idea in some problems.

A C-module (M, ) is called finitely generated if there is an exact sequence(C R Rn, Rn) (M, ) 0 in C-mod for some n 0. A C-module (M, )is called finitely presented if there is an exact sequence (C R Rm, Rm) (C R Rn, Rn) (M, ) 0 in C-mod for some m, n 0. We write C-modfpfor the full subcategory of finitely presented C-modules in C-mod.

If E F G 0 is an exact sequence in C-mod with E, F finitelypresented (or more generally E finitely generated and F finitely presented) thenG is finitely presented. This is because ifC R Rl E 0 and C R Rm C R Rn F 0 are exact, we can make an exact sequence C R Rl+m C R Rn G 0. Similarly, if E, G are finitely presented, then F is finitelypresented. Hence C-modfp is closed under cokernels and extensions in C-mod.

But it may not be closed under kernels, since C may not be noetherian as acommutative R-algebra.Now let : C D be a morphism of C-rings. If (M, ) is a D-module

then (M, ) =

M, ( idM)

is a C-module, and this defines a functor : D-mod C-mod. However, is not very well-behaved, for instance itneed not take finitely generated D-modules to finitely generated C-modules, and

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we will not use it. If (M, ) is a C-module then (M, ) =

MCD, D

is a D-module, where D = C

idD : D

M

CD

= C

CD

M

CD

M

CD, and

this induces a functor : C-mod D-mod which does take finitely generatedor presented C-modules to finitely generated or presented D-modules.

Vector bundles E over manifolds X give examples of modules over C(X).

Example 5.2. Let X be a manifold, which may have boundary or corners. LetE X be a vector bundle, and C(E) the vector space of smooth sectionse of E. Define E : C

(X) C(E) C(E) by E (c, e) = c e. ThenC(E), E

is a C(X)-module. If E is a trivial rank k vector bundle, E =

X Rk, then C(E), E =

C(X) R Rk, Rk

, so

C(E), E

is a free

C(X)-module.Let E, F X be vector bundles over X and : E F a morphism of

vector bundles. Then : C(E) C(F) defined by : e e is amorphism of C

(X)-modules.Now let X, Y be manifolds and f : X Y a (weakly) smooth map. Thenf : C(Y) C(X) is a morphism of C-rings. If E Y is a vectorbundle over Y, then f(E) is a vector bundle over X. Under the functor (f) :C(Y)-mod C(X)-mod of Definition 5.1, we see that (f)

C(E)

=

C(E) C(Y) C(X) is isomorphic as a C(X)-module to C

f(E)

.

If E X is any vector bundle over a manifold then by choosing sectionse1, . . . , en C(E) for n 0 such that e1|x, . . . , en|x span E|x for all x Xwe obtain a surjective morphism of vector bundles : X Rn E, whosekernel is another vector bundle F. By choosing another surjective morphism

: XRm F we obtain an exact sequence of vector bundles XRm XR

n

E

0, which induces an exact sequence of C(X)-modules C(X)

R

Rm C(X) R Rn C(E) 0. Thus we deduce:Lemma 5.3. Let X be a manifold, which may have boundary or corners, andE X be a vector bundle. Then the C(X)-module C(E) in Example 5.2is finitely presented.

5.2 Complete modules over fair C-rings

We now extend the ideas in 4.3 on infinite sums in C-rings to modules.Definition 5.4. Let C be a fair C-ring, and M a module over C. Write(X, OX ) = SpecC. Consider a formal expression

aA ma, where A is a (usu-

ally infinite) indexing set and ma M. We say that

aA aa is a locally finitesum if X can be covered by open sets U

X such that for all but finitely

many a A we have (idM x)(ma) = 0 in M C Cx for all x U, whereidM x : M = MC C MC Cx is induced by the projection x : C Cx.

If

aA ma is a locally finite sum, we say that m M is a limit of

aA ma,written m =

aA ma, if (idM x)(m) =

aA(idM x)(ma) for all x X,

where the sum makes sense as there are only finitely many nonzero terms.

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We say two locally finite sums

aA ma,

aA ma are equivalent if for all

x

X we have aA(idM

x)(ma) = aA(idM x)(m

a) in Cx. Equiva-

lent locally finite sums have the same limits.We call M a complete C-module if every locally finite sum in M has a unique

limit. Write C-modco for the full subcategory of complete modules in C-mod.

Example 5.5. Let C be a fair C-ring. Consider C as a module over itself.Then Proposition 4.22(b) implies that C is complete. More generally, C R Rnis a complete C-module for all n 0. However, if W is an infinite-dimensionalvector space then CR W is in general not a complete C-module. The problemis with the notion of tensor product: by definition, elements ofCRW are finitesums

na=1 cawa, whereas we want to consider infinite, but locally finite, sums

aA ca wa. So, to obtain a complete module we need to pass to some kindof completed tensor productCRW, using the topology on C in Definition 4.21.

When C = C(Rn) for n > 0 and W is an infinite-dimensional vector space,

consider the following three sets of maps Rn

W:(i) M1 =

smooth maps w : Rn W with w(Rn) contained in a finite-

dimensional vector subspace W of W

;

(ii) M2 =

smooth w : Rn W such that Rn is covered by open U Rnwith w(U) contained in a finite-dimensional subspace W of W

; and

(iii) M3 =

all smooth maps w : Rn W.Then M1, M2, M3 are C(R

n)-modules with M1 M2 M3, where M1 is thetensor product C(Rn) R W, and M2 is the correct completed tensor productC(Rn)RW, a complete C(Rn)-module. For our purposes M3 is too big.To see this, note when we pass to germs at x Rn we have

M1 C(Rn) Cx (Rn) = M2 C(Rn) Cx (Rn) = Cx (Rn) R W,but M3 C(Rn) Cx (Rn) is much larger than Cx (Rn) R W.

As for Rfafg in Definition 2.16, one can show:

Proposition 5.6. Let C be a fair C-ring. Then there is a reflection functorRcoall : C-modC-modco, left adjoint to the inclusionC-modco C-mod.

This can be proved by defining Rcoall(M) to be the set of equivalence classes[

aA ma] of locally finite sums

aA ma, with C-action

c, [

aA ma]

=aA (c, ma)

for c C, and checking Rcoall(M) has the required properties.

Alternatively, we can define Rcoall to be the functor MSpec in 6 below, andverify it is a reflection. The correct notion of completed tensor product C

RW

in Example 5.5, up to isomorphism, is CRW = Rcoall(C R W).To test whether a C-module M is complete, it is enough to consider only

locally finite sums of the form

bB b mb where {b : b B} is a partition ofunity in C and mb M. The proof requires C to be fair.

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Lemma 5.7. LetC be a fair C-ring, and M aC-module. Then every locallyfinite sumaA ma is equivalent to one of the formbB b

mb, where

{b :

b B} is a partition of unity inC and mb M for all b B. Conversely, allsuch

bB b mb are locally finite sums.

Proof. Let

aA ma be a locally finite sum. Then X has an open cover ofU such that ma|U 0 for all but finitely many a A. Since C is fair X isparacompact, so we can choose a locally finite refinement {Vb : b B} of thisopen cover, and Proposition 4.22(c) gives a partition of unity {b : b B} inC subordinate to {Vb : b B}. For each b B, define mb =

aAb

ma whereAb A is the finite set of a A with ma|Vb 0. It is then easy to see that

bB b mb is a locally finite sum equivalent to

aA ma. The last part isimmediate as {b : b B} is locally finite.Proposition 5.8. Let C be a fair C-ring. Then C-modco is closed underkernels, cokernels and extensions in C-mod, that is, C-modco is an abelian sub-category of C-mod.

Proof. Let 0 M1 M2 M3 0 be an exact sequence in C-mod. Firstsuppose M2, M3 C-modco, and

aA ma is a locally finite sum in M1. Then

aA (ma) is locally finite in M2 which is complete, so m =

aA (ma)

for some unique m M2. As morphisms of modules preserve limits we have(m) =

aA (ma) =

aA 0 = 0,

so (m) = 0 as limits in M3 are unique as M3 is complete. Hence m = (m)for some unique m M1 by exactness. This m is the unique limit of

aA ma,

so M1 is complete, and C-modco is closed under kernels.

Now suppose M1, M2

C-modco. Let

aA ma be a locally finite sum

in M3. By Lemma 5.7 we can choose an equivalent sum

bB b mb with{b : b B} a partition of unity. By exactness mb = (mb) for some mb M2,all b B. Then bB b mb is a locally finite sum in M2, so

bB b mb = m

for some unique m M2. As morphisms preserve limits we have

bB b mb =(m) =

aA ma, so limits always exist in M3.

To show limits are unique in M3, it is enough to consider the zero sequence0. Suppose m M3 is a limit of

0. Then m = (m) for some m M2,

and X has an open cover of U such that m|U 0. Choose a locally finiterefinement {Vb : b B} and a subordinate partition of unity {b : b B}; wecan also arrange that b =

2b for b C supported on Vb. Then b m = 0 in

M3, so (b m) = 0, and thus b m = (mb) for mb M1. Hence

bB bmbis a locally finite sum in M1, so

bB bmb = m for some unique m M1 as

M1 is complete. Therefore

(m) =

bB b(mb) =

bB 2b m =

bB b m = m,

and (m) = m by uniqueness of limits in M2, so m = (m) = 0. Thuslimits in M3 are unique, M3 is complete, and C-mod

co is closed under cokernels.Closedness under extensions follows by a similar argument.

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As Rcoall is a reflection, it is an exact functor between abelian categories.Since every finitely presented C-module is the cokernel of a morphism between

C R Rm,C R Rn, which are complete as in Example 5.5, we have:Corollary 5.9. Let C be a fair C-ring. Then every finitely presented C-module is complete, that is, C-modfp C-modco.

5.3 Cotangent modules ofC-rings

Given a C-ring C, we will define the cotangent module (C , C) ofC. Althoughour definition ofC-module only used the commutative R-algebra underlying theC-ring C, our definition of the particular C-module (C , C) does use theC-ring structure in a nontrivial way. It is a C-ring version of the module ofrelative differential forms in Hartshorne [16, p. 172].

Definition 5.10. Suppose C is a C-ring, and (M, ) a C-module. A C-derivation is an R-linear map d : C M such that whenever f : Rn R is asmooth map and c1, . . . , cn C, we have

df(c1, . . . , cn) =n

i=1

fxi

(c1, . . . , cn), dci

. (16)

Note that d is not a morphism of C-modules. We call such a pair (M, ), d acotangent module for C if it has the universal property that for any C-module(M, ) and C-derivation d : C M, there exists a unique morphism ofC-modules : (M, ) (M, ) with d = d.

There is a natural construction for a cotangent module: we take (M, )to be the quotient of the free C-module with basis of symbols dc for c Cby the C-submodule spanned by all expressions of the form df(c1, . . . , cn)

n

i=1

fxi

(c1, . . . , cn), dci for f : Rn

R smooth and c1, . . . , cn

C. Thus

cotangent modules exist, and are unique up to unique isomorphism. When wespeak of the cotangent module, we mean that constructed above. We maywrite (C , C), dC : C C for the (or a choice of) cotangent module for C.

Let C,D be C-rings with cotangent modules (C , C), dC , (D, D), dD,and : C D be a morphism of C-rings. Then the action C ( idD)makes D into a C-module, and dD : C D is a C-derivation. Thusby the universal property of C , there exists a unique morphism ofC-modules : C D with dD = dC . This then induces a morphism ofD-modules () : C CD D with () (dC idD) = dD as a compositionD = C C D C C D D. If : C D, : D E are morphisms ofC-rings then = : C E.Example 5.11. Let X be a manifold. Then the cotangent bundle TX is a vec-tor bundle over X, so as in Example 5.2 it yields a C(X)-module C(TX).The exterior derivative d : C(X) C(TX), d : c dc is then a C-derivation, since equation (16) follows from

d

f(c1, . . . , cn)

=n

i=1f

xi(c1, . . . , cn) dcn

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for f : Rn R smooth and c1, . . . , cn C(X), which holds by the chainrule. It is easy to show that

C(TX), TX

, d have the universal property

in Definition 5.10, and so form a cotangent module for C(X).Now let X, Y be manifolds, and f : X Y a (weakly) smooth map. Then

f(T Y), T X are vector bundles over X, and the derivative of f is a vectorbundle morphism df : T X f(T Y). The dual of this morphism is (df) :f(TY) TX. This induces a morphism of C(X)-modules ((df)) :C

f(TY)

C(TX). This ((df)) is identified with (f) under thenatural isomorphism C

f(TY)

= C(TY) C(Y) C(X), where weidentify C(Y), C(X), f with C,D, in Definition 5.10.

The importance of Definition 5.10 is that it abstracts the notion of cotangentbundle of a manifold in a way that makes sense for any C-ring.

Remark 5.12. (a) There is a second way to define a cotangent-type module for

a C

-ringC

, namely the module KdC of Kahler differentials of the underlyingR-algebra ofC. This is defined as for C , but requiring (16) to hold only whenf : Rn R is a polynomial. Since we impose many fewer relations, KdC isgenerally much larger than C , so that KdC(Rn) is not a finitely generatedC(Rn)-module for n > 0, for instance.

(b) Cotangent modules should be part of a more complicated story about cotan-gent complexes. Iff : A B is a morphism of rings then the cotangent complexLf, constructed by Illusie [18], is an object in the derived category of B-modulesDb(B-mod). As discussed by Spivak [37, 7], presumably the same constructionworks for C-rings, so that if : C D is a morphism of C-rings then oneshould obtain a C-cotangent complex LC

. One can also form the cotangentcomplex L of regarded just as a ring morphism. Then L and LC

both liein Db(D-mod), but will in general be different, since L ignores the C-ring

structures ofC,D but LC

does not.IfC is a C-ring then there is a unique C-ring morphism : R C, sinceR is an initial object in CRings. Thus we can form cotangent complexes LC

and L in Db(C-mod). We expect that our cotangent module C is canonicallyisomorphic to H0(LC

), whereas the module of Kahler differentials KdC in (a)is canonically isomorphic to H0(L).

Theorem 5.13. If C is a finitely generated C-ring then C is a finitelygenerated C-module. If C is a fair C-ring then C is complete. If C is afinitely presented, or good, C-ring, thenC is finitely presented.

Proof. IfC is finitely generated we have an exact sequence 0 I C(Rn)C 0. Write x1, . . . , xn for the generators of C(Rn). Then any c C

may be written as (f) for some f

C(Rn), and (16) implies that

dc = df

(x1), . . . , (xn)

=n

i=1

fxi

((x1), . . . , (xn)), d (xi)

.

Hence the generators dc of C for c C are C-linear combinations of d (xi),i = 1, . . . , n, so C is generated by the d (xi), and is finitely generated.

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Now suppose C is fair. From the first part we have an exact sequence 0 M

C

R R

n

C

0, where C

R R

n is the free C-module with basis

e1, . . . , en, and (ei) = d (xi), i = 1, . . . , n, and M is the C-submodule ofC R Rn generated by elements

fx1

e1 + +

fxn

en for f I. (17)

By Example 5.5 CRRn is complete. We will show M is complete. Then C iscomplete, as C-modco is closed under cokernels by Proposition 5.8. Since locallyfinite sums have unique limits in C R Rn M, it is enough to show that M isclosed under locally finite sums.

Let

aA ma be a locally finite sum in M. Then by Lemma 5.7 and (17),

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Algebraic Geometry over C^inf Rings, Joyce