Cartier modules: Finiteness results - Heidelberg University · Cartier modules: Finiteness results By Manuel Blickle at Mainz and Gebhard Bo¨ckle at Heidelberg Abstract. On a locally

J. reine angew. Math., Ahead of Print

DOI 10.1515/CRELLE.2011.087

Journal fur die reine undangewandte Mathematik( Walter de Gruyter

Berlin � New York 2011

Cartier modules: Finiteness results

By Manuel Blickle at Mainz and Gebhard Bockle at Heidelberg

Abstract. On a locally Noetherian scheme X over a field of positive characteristic p,we study the category of coherent OX -modules M equipped with a p�e-linear map, i.e. anadditive map C : OX ! OX satisfying rCðmÞ ¼ Cðrpe

mÞ for all m A M, r A OX . The notionof nilpotence, meaning that some power of the map C is zero, is used to rigidify this cate-gory. The resulting quotient category, called Cartier crystals, satisfies some strong finitenessconditions. The main result in this paper states that, if the Frobenius morphism on X is afinite map, i.e. if X is F -finite, then all Cartier crystals have finite length. We further showhow this and related results can be used to recover and generalize other finiteness results ofHartshorne–Speiser [19], Lyubeznik [27], Sharp [30], Enescu–Hochster [14], and Hochster[20] about the structure of modules with a left action of the Frobenius. For example, weshow that over any regular F -finite scheme X Lyubeznik’s F -finite modules [27] havefinite length.

1. Introduction

The Frobenius morphism is the fundamental tool in algebra and geometry over a fieldof positive characteristic. For a scheme X over the finite field Fq of q ¼ pe elements, with p

a prime number, the Frobenius morphism is the unique Fq-linear map that sends sections ofthe structure sheaf OX to their q th power. In other words, there is a natural left action of theFrobenius F on the structure sheaf OX satisfying FðrsÞ ¼ rqFðsÞ. An additive map on anarbitrary OX -module satisfying such a relation is called a q-linear map. From the exampleof the structure sheaf OX with its natural Frobenius action, many quasi-coherent sheaves M

on X receive a natural left action of F , or equivalently are equipped with a map M ! F�M.Important examples are the local cohomology modules: The study of their natural Frobe-nius action has lead to deep structural results [21], [19], [27] which are important for thestudy of singularities, amongst other things [31], [32], [16], [3], [6].

In this article we consider a dual situation, namely that of a quasi-coherentOX -module M equipped with a right action of the Frobenius F . That is, M is equipped

The first author is fully supported by a Heisenberg grant of the DFG. Both authors are partially supported

by the DFG SFB/TR45 ‘‘Periods, moduli spaces and the arithmetic of algebraic varieties’’.

with an OX -linear map C ¼ CM : F�M !M. Such a map is called q�1-linear, or Cartier

linear. We call a pair ðM;CÞ consisting of a quasi-coherent OX -module M and a q�1-linearmap C on M a Cartier module. The prototypical example of a Cartier module is thedualizing sheaf oX of a smooth and finite type k-scheme X (k perfect) together with theclassical Cartier operator [10], [22]. The relation to the left actions described above is viaGrothendieck–Serre duality where the left action on OX corresponds to the Cartier mapon oX , cf. [7].

1.1. Finite length of Cartier modules up to nilpotence. In this article we develop thebasic theory of Cartier modules and derive a crucial structural result; namely, if X is anF -finite scheme, then

every coherent Cartier module has—up to nilpotence—finite length.

A Cartier module M ¼ ðM;CÞ is called nilpotent, if C e ¼ 0 for some ef 0. This notionof nilpotence is crucial in our treatment. In fact, we study a theory of coherent Cartiermodules up to nilpotence. To formalize this viewpoint one localizes the Abelian categoryof coherent Cartier modules at its Serre subcategory of nilpotent Cartier modules. Thelocalized category, which we call Cartier crystals, is an Abelian category. Its objects arecoherent Cartier modules, but the notion of isomorphism has changed: A morphismj : M ! N of Cartier modules induces an isomorphism of associated Cartier crystals if jis a nil-isomorphism, i.e. both, kernel and cokernel of j are nilpotent. Thus, more precisely,the main result of this paper states:

Main Theorem (Theorem 4.6). If X is a locally Noetherian scheme over Fq such

that the Frobenius F on X is a finite map, then every coherent Cartier module has—up to

nilpotence—finite length. More precisely, every coherent Cartier crystal has finite length.

Idea of the proof. Let us isolate the main points in the proof. The ascending chaincondition is clear since X is Noetherian and the underlying OX -module of the Cartier mod-ule is coherent. Hence one has to show that any descending chain of Cartier submodulesM MM1 MM2 MM3 M � � � stabilizes up to nilpotence. This is shown by induction on thedimension of X . Two main ideas enter into its proof.

(a) (Proposition 2.14) Any coherent Cartier module ðM;CÞ is nil-isomorphic to aCartier submodule ðM;CÞ with surjective structural map, i.e. with CðMÞ ¼M. It is easyto see that the support of M is a reduced1) subscheme of X .

This part is proved by showing that the descending chain of images

M MCðMÞMC2ðMÞM � � �

stabilizes and this stable image is then M (see Proposition 2.14). This step may be viewed asa global version of an important theorem in Hartshorne and Speiser [19] about co-finite

1) Throughout this article, when we speak of the support of a sheaf M of OX -modules we mean the sub-

scheme of X whose ideal sheaf is the annihilator of M.

2 Blickle and Bockle, Cartier modules: Finiteness results

modules (over a complete, local, and F -finite ring) with a left action of the Frobenius. Infact, the proof in [19] of the counterpart of Proposition 2.14 implicitly uses Cartier mod-ules, but is much less general than the proof given here which is due to Gabber [15], Lemma13.1 (see [23] for an application of this result to the theory of test ideals, and [4] for yet an-other version). In that same paper of Gabber one finds as well the other crucial ingredient;as part of the proof of his [15], Lemma 13.3:

(b) (Corollary 4.4) Let ðM;CÞ be a coherent Cartier module with CðMÞ ¼M. Thenthere is a dense open set U LX such that for all Cartier submodules N LM with the samegeneric rank as M, the quotient M=N is supported in the closed set Y ¼ XnU .

The critical point is that the closed subset Y only depends on ðM;CÞ, but not on thesubmodule N. With (a) and (b) at hand, the proof of the Main Theorem proceeds roughlyas follows: By (a) we may replace the above chain by a nil-isomorphic chain where allstructural maps Ci : Mi !Mi are surjective. The surjectivity is then automatically truefor all quotients M=Mi. Since the generic ranks of Mi can only drop finitely many times,we can assume that—after truncating—the generic ranks are constant. Part (b) thenimplies that all quotients M=Mi are supported on Y . In fact, the reducedness of the supportof M=Mi by (a) implies that if I is a sheaf of ideals cutting out Y , then I � ðM=MiÞ ¼ 0 andhence IM LMi for all i. So the stabilization of the chain fMigi>0 is equivalent to thestabilization of the chain fMi=M 0gi>0 where M 0 is the smallest Cartier submodule of M

containing IM. But the latter may be viewed as a chain of Cartier modules on the lowerdimensional scheme Y , hence by induction this chain stabilizes. r

There are reasons why one expects such a strong finiteness statement for Cartier mod-ules. The most apparent one, which also explains why Gabber’s [15] is so extremely helpfulin our proof, is as follows: Combining key results of Emerton and Kisin [13] or Bockle andPink [8] on a Riemann–Hilbert-type correspondence in positive characteristic with ongoingwork of the authors [7] (see also Section 5), one expects that the category of Cartier crystalson X is equivalent to the perverse constructible sheaves (for the middle perversity) ofFq-vector spaces on X�eet. One of the key results in [15] is that this latter category is Artinianand Noetherian.

A more explicit but entirely heuristic reason for the finite length of Cartier crystalsis the strong contracting property Cartier linear maps enjoy. This is quite elementary andnicely explained by Anderson [1] where he studies L-functions modulo p. He shows thatin a finitely generated R ¼ k½x1; . . . ; xn�-module M, equipped with a Cartier-liner map C,there is a finite dimensional k-subspace Mc such that some power of C maps any elementm A M into Mc. Very naively this indicates that—up to nilpotence (thought this is not quitecorrect!)—the finite dimensional k-vectorspace Mc with CjMc

contains all the information ofðM;CÞ. So we should expect that ðM;CÞ has finite length since Mc is finite dimensional.

1.2. Consequences, applications and relation to other finiteness results. There aresome immediate formal consequences of the finite length result for Cartier crystals.Namely, one has a theory of Jordan–Holder series, hence the notion of length of a Cartiercrystal, resp. of quasi-length (i.e. length up to nilpotence) of a Cartier-module. This is ex-plained in Section 4.1 where we further show a companion of a result of Hochster ([20],(5.1) Theorem) about the finiteness of homomorphisms of Lyubeznik’s F -finite modules.Our result in this context states:

3Blickle and Bockle, Cartier modules: Finiteness results

Theorem (Theorem 4.17). Let X be F-finite and let M, N be Cartier crystals. Then

the module HomCrysðM;NÞ is a finite-dimensional vector space over Fq.

The finiteness of the homomorphism set, together with the finite length, formally im-plies the finiteness of submodules of any Cartier module (up to nilpotence) (see also [20],(5.2) Corollary).

Theorem (Corollary 4.20). Let X be F-finite. Then a coherent Cartier module has, up

to nilpotence, only finitely many Cartier submodules.

As already mentioned, the theory of OX -modules equipped with a left action of theFrobenius is much more extensively studied as the right actions we investigate here, andthere are some deep known results which are quite similar to our Main Theorem. Examplesare the above mentioned result of Hartshorne and Speiser in [19] about co-finite moduleswith a left Frobenius action, and, most importantly, Lyubeznik’s result [27], Theorem 3.2about the finite length of objects in his category of F-finite modules over a regular ring(which is essentially of finite type over a regular local ring).

In the regular F -finite case, the connection between our and Lyubeznik’s result is ob-tained by tensoring with the inverse of the dualizing sheaf oX to obtain an equivalenceof Cartier modules and the category of g-sheaves which was recently introduced in [4].g-sheaves are OX -modules with a map g : M ! F �M, and it was shown that the associatedcategory of g-crystals is equivalent to the category of Lyubeznik’s F-finite modules [27] (inthe a‰ne F -finite case), resp. to Emerton and Kisin’s category of locally finitely generatedunit OX ½F �-modules [13] (finite type over an F -finite field case). Hence our main resultyields the following (partial) extension of the main result in [27], Theorem 3.2.

Theorem (Theorem 5.13). Let X be a regular F-finite scheme. Then every finitely

generated unit OX ½F �-module (resp. F-finite module in the terminology of [27]) has finite

length.

The connection to Hartshorne and Speiser’s theory of co-finite modules with a leftFrobenius action relies on Matlis duality and has been used, at least implicitly, many timesbefore, for an explicit use, see [15]. What it comes down to is that if F is finite, then theMatlis dual functor commutes with F�, and hence for a complete local and F -finite ring R

this functor yields an equivalence of categories between coherent Cartier modules overSpec R and co-finite left R½F �-modules. This equivalence preserves nilpotence and hencewe recover the analogous result of our main theorem for co-finite left R½F �-modules,cf. Theorem 5.3, [27], Theorem 4.7.

Besides these important consequences, we show how our theory implies global struc-tural results which generalize analogous results for modules with a left Frobenius actionover a local ring obtained recently by Enescu and Hochster [14], and Sharp [30]. Our globalversion states:

Theorem (Proposition 4.9). Let X be an F-finite scheme, and ðM;CÞ a coherent

Cartier module with surjective structural map C. Then the set

fSuppðM=NÞ jN LM a Cartier submoduleg


is a finite collection of reduced sub-schemes. It consists of the finite unions of the finitely many

irreducible closed sub-schemes contained in it.

In the same spirit we show the following result, which generalizes [14], Corollary 4.17:

Theorem (Proposition 5.6). Let R be a Noetherian and F-finite ring. Assume that R is

F-split, i.e. there is a map C : F�R! R splitting the Frobenius.

(a) The Cartier module ðR;CÞ has only finitely many Cartier submodules (Proposi-

tion 5.4).

(b) If R is moreover complete local, then the injective hull ER of the residue field of R

has some left R½F �-module structure such that ER has only finitely many R½F �-submodules.

(c) If ðR;mÞ is also quasi-Gorenstein, then the top local cohomology module H dmðRÞ

with its standard left R½F �-structure has only finitely many R½F �-submodules.

Note that in this result, we show the finiteness of the actual number of Cartier or ofR½F �-submodules, and not just the finiteness up to nilpotence which already follows byCorollary 4.20 as discussed above.

Along the way, in Remark 4.12, we point out how one may derive the finiteness re-sults in the theory of Frobenius splittings about the finiteness of compatibly split primesof [29], [24].

1.3. Structure of the paper. The theory of Cartier modules is presented in Section 2.We set up the notation and derive basic results needed in the rest of the article. In particu-lar, we deal with the all important notion of nilpotence for Cartier modules, and show thatnilpotent Cartier modules are a Serre-subcategory. This section also includes a treatment ofduality for a finite morphism (in a very basic way), a discussion of the main example of aCartier module (the dualizing sheaf), and a thorough discussion of the already interestingcase where X is the spectrum of a field.

In Section 3, the category of Cartier crystals is introduced, and the existence of aunique (i.e. minimal) representative of a Cartier crystal by a Cartier module is shown,cf. Theorem 3.10. This result is equivalent (see Theorem 5.2.1) to the existence of minimalg-sheaves recently proved by the first author in [4], but the proof in the present article ismuch simpler. We conclude this section with a Kashiwara type equivalence for Cartiercrystals, saying that if Y LX is a closed subscheme, then there is an equivalence of catego-ries of Cartier crystals on Y and Cartier crystals on X with support in Y .

The proof of the main result and some immediate consequences are given in Sec-tion 4. Since the arguments given in [15], Section 13 are rather brief, we decided to includethese proofs with many details added in this section. This also makes the present paperquite self contained and, as we hope, accessible to a reader with knowledge of algebraicgeometry at the level of [18].

The final Section 5 discusses the connections of the category of Cartier modules toother categories of modules with an action of the Frobenius, and it derives the conse-quences which we outlined above.


Acknowledgments. The main theorem of this article was worked out while the firstauthor was attending the Leopoldina-Symposium in Algebraic and Arithmetic Algebraic Ge-

ometry in Ascona in May 2009, and we thank the funding agencies who made this visit pos-sible. The basic theory of Cartier modules we present here is part of the more extensiveproject [7] of the authors, in particular the equivalences used in Section 5 were workedout in [7] as a by-product of showing an equivalence of the category of t-crystals of [8]and of finitely generated unit OX ½F �-modules of [13]. We are thankful to Karl Schwede fornumerous comments on an earlier draft of this manuscript.

Notation and conventions. We fix a prime number p and a power q ¼ pk. LetF ¼ Fq be the finite field with q elements. Our schemes X will always be locally Noetherianand separated over F. We denote by F the q th-power Frobenius, i.e. the morphism ofschemes F : X ! X which is the identity on the underlying topological space, and theq th-power map on sections of OX . For the most part we also assume that X is F -finite, i.e.that F is a finite morphism of schemes. It would be desirable to relax this hypothesis and toonly assume that X is excellent (to have an open regular locus) or possibly also assumingthat X has a dualizing complex. By [26] an F -finite ring is excellent, and in [15] it is statedthat an F -finite Noetherian scheme has a dualizing complex (with proof sketched in thea‰ne case).

2. Cartier modules

Definition 2.1. A Cartier module on X is a quasi-coherent sheaf M of OX -modulesequipped with a q�1-linear map C ¼ CM : M !M, that is an additive map satisfyingCðrqmÞ ¼ rCðMÞ for r A OX . Equivalently, C is an OX -linear map C : F�M !M and weuse both points of view interchangeably in this paper.

We refer to C ¼ CM as the structural map of M and often omit the subscript. IfX ¼ Spec R is a‰ne, denote by R½F � the ring

R½F � ¼ RfFgRhrqF � Fr j r A Ri

which is obtained from R by adjoining a non-commutative variable subject to the relationrqF ¼ Fr for all r A R. Then a Cartier module on X is nothing but a right module over R½F �.A general scheme X may be covered by finitely many a‰nes, gluing the respective ringsR½F � one obtains a sheaf of rings OX ½F �. A Cartier module on X is then a sheaf of right

OX ½F �-modules which is quasi-coherent as a sheaf of OX -modules.

A map of Cartier modules is a map j : M ! N of the underlying quasi-coherentsheaves such that the diagrams

M ��!j N

CM

???y???yCN

M ��!j N;

F�M ��!F�jF�N

CM

???y???yCN

M ��!j N


commute. Of course both diagrams are equivalent, the left one being a diagram of additivemaps, the right one of OX -linear maps. Kernels and cokernels of maps of Cartier modulesare the same as for (coherent) sheaves of OX -modules with the naturally induced Cartierlinear map. The category of (coherent) Cartier modules on X is an abelian category, sinceit is a category of right modules over the ring OX ½F �. We denote the category of Cartiermodules on X by CartðXÞ and the coherent ones by CohCartðXÞ.

2.1. Basic pushforward and pullback.

Lemma 2.2. Let f : Y ! X be a morphism of schemes, then the pushforward f� for

quasi-coherent OY -modules induces a functor f� : CartðYÞ ! CartðXÞ.

Proof. Clear since f� � F e� is naturally isomorphic to F e

� � f�. r

In the cases where f� preserves coherence for OY -modules, it also preserves coherencefor Cartier modules. In particular, if f is finite, then f� preserves coherence, hence induces afunctor from coherent Cartier modules on Y to coherent Cartier modules on X .

Now we discuss pullback for Cartier modules. For simplicity we only consider thespecial cases of open and closed immersions, leaving a more general discussion for later,see also [7].

Lemma 2.3. If S is a sheaf of multiplicative subsets of OX , then the localization func-

tor S�1 on quasi-coherent OX -modules restricts to an exact functor on Cartier modules which

preserves coherence. Furthermore, the localization map is a map of Cartier modules.

Proof. Note that for any quasi-coherent OX -module M, we have

F�ðS�1MÞ ¼ S�1F�ðMÞ

since S�qM ¼ S�1M (with the obvious meaning of S�q being the inverse of the multi-plicative system consisting of qth powers of S). This implies that for a Cartier module

M the structural map F�M !C M naturally equips S�1M with a Cartier structure

F�S�1M GS�1F�M ��!S�1C

M. Concretely on a fractionm

s¼ msq�1

sqthe structural map is

given by CS�1M

m

s

� �¼ CMðmsq�1Þ

s. Obviously the localization map m 7! m

1is Cartier

linear. r

Corollary 2.4. If i : U ,! X is an open immersion then the restriction i� for quasi-

coherent OX -modules induces a natural functor from Cartier modules on X to Cartier mod-

ules on U that preserves coherence.

Proof. Take local generators ð f1; . . . ; fkÞ of an ideal defining the complementX �U of U . Then MjU is the kernel of the first map in the Cech complex on theUi ¼ SpecOX ½ f �1

i �. Each MjUihas a natural structure of a Cartier module by Lemma 2.3,

hence so do the entries of the Cech complex. Since the maps in the Cech complex are(sums of) localization maps, which are Cartier linear by Lemma 2.3, we conclude that thekernel MjU ¼ i�M is naturally a Cartier module. r


Proposition 2.5. Let i : Y ,! X be a closed subscheme defined by the sheaf of ideals I .

Let ðM;CÞ be a Cartier module on X. Then

i[ðMÞ ¼defHomOX

ði�OY ;MÞ ¼M½I � ¼def fm A M j Im ¼ 0g ðviewed as an OY -moduleÞ

is naturally a Cartier module on Y. i[ defines a left exact functor from CartðY Þ to CartðX Þwhich preserves coherence.

Proof. Everything follows once we observe that CðM½I �ÞLM½I �: If Im ¼ 0, thenICðmÞ ¼ CðI ½q�mÞLCðImÞ ¼ 0. r

Proposition 2.6. Let i : Y ,! X be a closed subscheme defined by the sheaf of ideals I .

Let M (resp. N) a Cartier module on X (resp. Y ). Then

i�i[M GM½I � and i[i�N GN:

Proof. The first equality is clear from the definition, and the second follows sinceIi�N ¼ 0 and hence ði�NÞ½I � ¼ i�N. r

2.2. Nilpotence. In our treatment of Cartier modules the notion of nilpotence iscritical. It means that some power of the structural map acts as zero.

Definition 2.7. A Cartier module ðM;CÞ is called nilpotent if C eðMÞ ¼ 0 forsome M. The smallest e such that C eðMÞ ¼ 0 is called the order of nilpotence of M anddenoted by nilordðMÞ. A Cartier module is called locally nilpotent if it is the union of itsnilpotent Cartier submodules.

One might expect that local nilpotence for a Cartier module M is equivalent to theannihilation of every section of M by some power of C. As the example below shows,this is not the case. However, local nilpotence is equivalent to requesting that for everylocal section m A MðUÞ, there is e > 0 such that C e

�OX ðUÞ �m

�¼ 0, i.e. the whole

OX -submodule generated by m has to be nilpotent.2) The discrepancy between the annihila-tion of a section and of the sub-sheaf generated by this section is explained by the observa-tion that the kernel of C is generally not an OX -submodule of M; it is only an OX -submoduleof F�M. Hence, a Cartier module might have no nilpotent submodules even if C has a non-trivial kernel:

Example 2.8. Let C : F½x� ��!xi 7!xððiþ1Þ=qÞ�1ÞF½x� (where xððiþ1Þ=qÞ�1 ¼ 0 whenever the ex-

ponent is not integral). Then F½x� has no submodule on which C is zero and hence none onwhich C is nilpotent, even though every element of F½x� is annihilated by some power of C.

In this article we are mainly concerned with coherent Cartier modules. In this case thenotion of nilpotence and local-nilpotence agrees.

2) To check the equivalence, let Me be the subsheaf whose sections MeðUÞ are precisely the sections of

MðUÞ such that C e�OX ðUÞ �m

�¼ 0. One immediately verifies that each Me is a Cartier submodule of M, which

is nilpotent, and any nilpotent Cartier submodule N of M, say with nilordðMÞ ¼ e, is contained in Me; see also

the proof of Proposition 2.12.


Lemma 2.9. If ðM;CÞ is coherent then M is nilpotent if and only if M is locally

nilpotent.

Proof. If a Cartier module is nilpotent, it clearly is locally nilpotent. For theconverse, observe that if N1 and N2 are nilpotent Cartier subsheaves of M of nilpotencyorders e1 and e2, respectively, then N1 þN2 is nilpotent of order maxfe1; e2g. Using theNoetherianness of M, it now easily follows that locally nilpotent implies nilpotent. r

Lemma 2.10. A coherent Cartier module M is nilpotent if and only if for all (closed)points x A X the localization Mx is nilpotent.

Proof. If M is nilpotent then so are all of its localizations. Conversely, if for x A X

the map C e is zero, then C e is zero on some open neighborhood of x. Covering X by fi-nitely many such neighborhoods (for various (closed) points x A X ) and taking the maxi-mum of the appearing e’s it follows that C eðMÞ ¼ 0 as claimed. r

Lemma 2.11. The category of nilpotent Cartier modules on X is a Serre subcategory

of the category of coherent Cartier modules on X , i.e. it is an abelian subcategory which is

closed under extensions and subquotients.

Proof. Everything is clear except the closedness under extensions. For this we con-sider a short exact sequence 0! N 0 ! N !p N 00 ! 0 and assume that both N 0 and N 00 arenilpotent, i.e., there are e and f such that C e

N 0 ¼ Cf

N 00 ¼ 0. Given an arbitrary n A N,we have p

�C f ðnÞ

�¼ C f

�pðnÞ

�¼ 0 since N 00 is nilpotent of ordere f . Hence, C f ðnÞ is

in the kernel of p which is equal to N 0. Since N 0 is nilpotent of order e it follows thatC e

�C f ðnÞ

�¼ 0. Since e and f do not depend on n A N we have C eþf ðNÞ ¼ 0, so N is nil-

potent. r

The proof shows that

maxfnilordðN 0Þ; nilordðN 00Þge nilordðNÞe nilordðN 0Þ þ nilordðN 00Þ.

The following simple observation shows that each Cartier module has a largest (lo-cally) nilpotent submodule.

Proposition 2.12. Let ðM;CÞ be a coherent Cartier module. Then there exists a

unique Cartier submodule Mnil such that:

(a) Mnil is nilpotent.

(b) M ¼defM=Mnil contains no non-zero nilpotent Cartier submodule.

Proof. Define Mnil as the sum over all locally nilpotent Cartier sub-sheaves of M.By the proof of Lemma 2.9 this is a locally nilpotent Cartier subsheaf of M. Since M isNoetherian, Lemma 2.9 asserts that Mnil is nilpotent. Suppose now that N=Mnil is a nilpo-tent Cartier subsheaf of M=Mnil for some N IMnil. Then by Lemma 2.11, the Cartier sub-module N of M is nilpotent. By the definition of Mnil we have N HMnil. Hence Mnil sat-isfies both properties asserted in the proposition. r


Remark 2.13. If the scheme X is F -finite, it is shown in [7] that one obtains analogsof the above statements for locally nilpotent Cartier modules. Namely we show that locallynilpotent Cartier modules form a Serre subcategory, and that a quasi-coherent Cartiermodule M has a maximal locally nilpotent submodule Mnil such that the quotient M=Mnil

has no nilpotent submodules. One can construct an example which shows that both state-ments fail if X is not F -finite. These generalizations are extremely useful in the constructionof certain functors on the category of Cartier modules carried out in [7]. However, since wedo not need this added generality here, we only included the basic nilpotent case.

The following observation is at the heart of what is to follow. It roughly says that upto nilpotence one can replace a coherent Cartier module by one with a surjective structuralmap. For a Cartier module, to have a surjective structural map is equivalent to not havingnilpotent Cartier quotients. If X ¼ Spec R is a‰ne and R is local and F -finite, this state-ment is dual to a condition in [19] on the nilpotence of locally nilpotent co-finite moduleswith a left Frobenius action. A global version of it in the regular F -finite case can be foundin [4]. The following slick proof is due to Gabber [15]. Note that we do not assume X to beF -finite.

Proposition 2.14. Let ðM;CÞ be a coherent Cartier module on X. Then the descend-

ing sequence of images C iðMÞ

M MCðMÞMC2ðMÞM � � �

stabilizes.

Proof. We can and will assume that X ¼ Spec R is a‰ne. Note that C iðMÞ is notonly an R-submodule of M (since rC iðmÞ ¼ C iðrqmÞ), but even a coherent Cartier sub-module of M. Furthermore, CðS�1MÞ ¼ CðS�qMÞ ¼ S�1CðMÞ hence the formation ofthe image of the structural map C commutes with localization. Let

Yn ¼ Supp�C nðMÞ=C nþ1ðMÞ

�¼

�x A Spec R jC

�C n�1ðMÞ

�x3C n�1ðMxÞ

�be the closed subset of X where the n th step of the chain is not an equality. The secondequality in the displayed equation uses the fact that the formation of the image of C com-mutes with localization as explained above. If x B Yn, then C

�C n�1ðMxÞ

�¼ C n�1ðMxÞ and

applying C yields C�C nðMxÞ

�¼ C nðMxÞ such that x B Ynþ1. Hence fYngnf0 forms a de-

scending sequence of closed subsets of X . Since X is Noetherian, and hence compact,this sequence must stabilize. Hence there exists nf 0 such that for all mf n we haveYn ¼ Ymð¼def

YÞ. By replacing M by C nðMÞ, we may assume that for all n we haveY ¼ Supp

�C nðMÞ=C nþ1ðMÞ

�.

The statement that the chain C nðMÞ stabilizes means precisely that Y is empty. Solet us assume otherwise and let m be a generic point of Y (i.e. the generic point of anirreducible component Z of Y ). Localizing at m (and replacing X by Spec Rm and M

by Mm), we may hence assume that X ¼ Spec R, where ðR;mÞ is a local ring andSupp


�¼ fmg for all n. In particular, for n ¼ 0 we get that there is

k > 0 such that mkM LCðMÞ. Then for x A mk

x2M L xmkM L xCðMÞ ¼ CðxqMÞLCðx2MÞ


and by iterating we get for all x A mk and all a A N that x2M LC aðMÞ. Hence

mkðbþ1ÞM L ðmkÞ½2�M LC aðMÞ for all a A N

where b is the number of generators of mk and ðmkÞ½2� is the ideal generated by the squaresof the elements of mk (and it is easy to check that there is an inclusion mkðbþ1ÞL ðmkÞ½2�).Hence the chain C aðMÞ stabilizes if and only if the chain C aðMÞ=mkðbþ1ÞM does. But thisis a chain in M=mkðbþ1ÞM, which has finite length. This contradicts our assumption thatSupp


�3j for all n. r

Corollary 2.15. Let ðM;CÞ be a coherent Cartier module. Then there is a unique

Cartier submodule ðM;CÞ such that

(a) the quotient M=M is nilpotent, and

(b) the structural map C : M !M is surjective (equiv. M has no non-zero nilpotent

quotient).

Proof. The stable image M ¼defC eðMÞ for eg 0 which exists by Proposition 2.14

has all the desired properties. r

Remark 2.16. The canonically assigned modules M and M may also be character-ized by the following conditions, more resembling a universal property.

(a) M is the smallest Cartier submodule N of M such that the quotient M=N isnilpotent.

(b) M is the smallest Cartier quotient N of M such that the kernel of M !! N isnilpotent.

2.3. Duality for finite morphisms. In this section we briefly recall some parts of theduality theory for finite morphisms as explained in [17], Chapter III.6. We want to applythis in particular to the Frobenius morphism F , hence we shall assume now that F is a finitemap, i.e. we assume that X is an F -finite scheme. One consequence of this will be an inter-pretation of the Cartier module structure, i.e. the map C : F�M !M, via its adjoint underthe duality of the finite Frobenius map, i.e. a map k : M ! F [M GHomOX

ðF�OX ;MÞ.Here, and from here onward, we denote by HomOX

ðM;NÞ the sheaf of local homomor-phisms from M to N, i.e. the sheaf associated to the presheaf

U 7! HomOX ðUÞ�MðUÞ;NðUÞ

�:

Let f : Y ! X be a finite morphism of (locally Noetherian) schemes. Then thefunctor

f [ : OX -modules! OY -modules

is defined by f [M ¼ f �1 HomOXð f�OY ;MÞ viewed as an OY -module. Locally, for

Y ¼ Spec S and X ¼ Spec R, both a‰ne, it is just given by f [M ¼HomRðS;MÞ viewed


as an S-module via its action on the first entry of the Hom. As one easily checks, in thecase of a closed embedding i : Y ,! X this definition agrees with i[ given above inProposition 2.5. In the case that f : Y ! X is etale, one has that f [ G f �. Furthermoreð Þ[ is compatible with composition.

The duality for a finite morphism [17], Theorem III.6.7 states that the functor f [ is aright adjoint to the functor f�. More precisely

Theorem 2.17 (Duality for a finite morphism). Let f : Y ! X be a finite morphism

of locally Noetherian schemes. Then the trace map Trf : f� f [M !M (given by evaluation

at 1) induces an isomorphism

f�HomOYðM; f [NÞ !G HomOX

ð f�M;NÞ

for every quasi-coherent OY -module M, and OX -module N.

In the a‰ne case of a finite homomorphism of rings R! S, an R-module N, andS-module M (where everything can be reduced to) this comes down to nothing more thanthe well-known isomorphism

HomS

�M;HomRðS;NÞ

�!G HomRðM;NÞ

given by sending j to the map m 7!�jðmÞ

�ð1Þ (whose inverse is the map sending c to the

map m 7!�s 7! cðsmÞ

�. From this we obtain the following characterization of Cartier

modules:

Proposition 2.18. Let X be F-finite and M a quasi-coherent OX -module. Then a

Cartier module structure on M is equivalently given by one of the following:

(a) A right OX ½F �-module structure on M, compatible with the OX -structure.

(b) An OX -linear map C : F�M !M.

(c) An OX -linear map k : M ! F [M.

The equivalence of the first two items was already observed above, and the equiv-alence of the second and third item is the just discussed adjointness. One easily (but tedi-ously) verifies that the adjoint map to C i is the map k i defined inductively via k1 ¼ k andk i ¼ F [ðk i�1Þ � k ¼ F ði�1Þ[k � k i�1. As a corollary we get:

Corollary 2.19. Let X be F-finite, and ðM;C; kÞ a Cartier module on X. Then the

kernel of k i is the maximal nilpotent Cartier submodule of ordere i, and Mnil ¼Si

ker k i.

Proof. Clearly, the Ki ¼ ker k iM form an increasing sequence of OX -submodules.

Since kðKiÞLF [ðKi�1ÞLF [ðKiÞ, it follows that Ki is a Cartier submodule of M. So Ki isclearly the largest Cartier submodule of M on which k i acts as zero. But ki being zero isequivalent to C i being zero since k i and C i are adjoint morphisms. r


2.4. Dualizing complexes and the Cartier isomorphism. We recall some parts of Resi-dues and duality [17] on the existence and uniqueness of dualizing complexes. A dualizing

complex o�X is a bounded complex of finite injective dimension such that for all complexesof sheaves with coherent cohomology M � on X , the natural double-dualizing map

M � ! RHomOX

�RHomOX

ðM �;o�X Þ;o�X�

is an isomorphism. It is shown in [17], Proposition V.2.1, that this condition on o�X isequivalent to the validity of the same condition for all bounded complexes of sheaves withcoherent cohomology M �. We only consider Noetherian schemes of finite dimension. Thendualizing complexes exist in many circumstances, cf. [17], V.10:

(a) If X is regular, or more generally if X is Gorenstein, then OX itself is a dualizingcomplex.

(b) If X is essentially of finite type over a scheme that has a dualizing complex, thenX has a dualizing complex.

(c) In particular, if X is of finite type over a field, or over a Gorenstein local ring,then X has a dualizing complex.

As one readily verifies from the definition of a dualizing complex, if o�X is dualizing,then so is o�X nL½n� for any integer n and any invertible sheaf L on X . By [17], TheoremV.3.1, this is all that can happen, i.e. any two dualizing complexes on X di¤er only by ashift and tensorization with an invertible sheaf. If X is Cohen–Macaulay, then a dualizingcomplex has cohomology concentrated in a single degree. More generally, if X is normal,then we denote by the dualizing sheaf oX the unique reflexive sheaf on X which agrees withthe dualizing complex o�X on the Cohen–Macaulay locus. Hence, given two dualizingcomplexes o�X and o 0�X , on su‰ciently small Zariski open subsets U LX (trivializing L),the restrictions of o�X and o 0�X ½n� to U are isomorphic. Since in a normal scheme thenon-Cohen–Macaulay locus has codimensionf 3, this isomorphism extends to an isomor-phism of the corresponding dualizing sheaves. In particular, if X is the spectrum of a localring, then dualizing complexes are unique (up to shift), and so are dualizing sheaves.

One is interested in the behavior of dualizing complexes under pullback by a (finite)morphism f : Y ! X . More precisely, we consider the functor

f !M � ¼ RHomOXð f�OY ;M

�Þ

on the derived category of complexes of sheaves on X with bounded coherent cohomol-ogy.3) By [17], Proposition V.2.24, if o�X is a dualizing complex on X , then f !o�X is dualiz-ing on Y , hence if one has fixed dualizing complexes o�X on X , and o�Y on Y , then one hasZariski locally an isomorphism o�Y G f !o�X ½n�. One of the main points of duality theory in[17] is that in many cases this last statement also holds globally. More precisely, the

3) Generally, this functor is denoted by f [, however, we have reserved f [ for the zeroth cohomology of this

functor throughout the rest of this paper, see Section 2.3.


schemes X which are essentially of finite type over a base scheme S which has a dualizingcomplex o�S, can be equipped with a dualizing complex o�X (namely o�X ¼ h!o�X , whereh : X ! S is the morphism exhibiting the S-structure of X ), such that if f : Y ! X is amorphism of S-schemes, then f !ðo�X ÞGo�Y . Applying this to the Frobenius we obtain:

Proposition 2.20. Let X be an F-finite scheme over a local Gorenstein scheme

S ¼ Spec R. Then there is a dualizing complex o�X such that o�X GF !o�X . If X is also nor-

mal, one obtains an isomorphism oX GF [oX of dualizing sheaves.

Proof. Since X is an S-scheme we have a morphism h : X ! S. Since S is Goren-stein local, R ¼ OS itself is a dualizing sheaf on X . Let o�X ¼ h!R. Since R is local wehave an isomorphism F !

SRGR. Consider now the relative Frobenius diagram of X over S:

X

X 0 ��!F 0S

X???yh 0

???yh

S ��!FSS

�� FX¼F

FX=S

h

Applying h 0! and combining with the identities h! GF !X=S � h 0! and h 0! � F !

S GF 0! � h!, we get

o�X G h!RGF !X=Sh0!RGF !

X=Sh0!F !

SRGF !X=SF 0!S h

!RGF !Xo

�X :

Since the non-Cohen–Macaulay locus in a normal scheme has codimensionf 3, the in-duced isomorphism oX GF !oX on the Cohen–Macaulay locus extends to all of X . r

We point out that the proof in fact shows that Proposition 2.20 holds whenever thebase scheme S has a dualizing complex o�S such that F !o�S Go�S.

We need to extend this result slightly to a case that is not directly stated in [17].

Proposition 2.21. Let X be F-finite and a‰ne. Then X has a dualizing com-

plex o�X . Furthermore, for all su‰ciently small a‰ne open subsets U LX , one has

(quasi-)isomorphisms of dualizing complexes

o�X jU ¼ o�U GF !o�U

which, in the case that X is also normal, induce isomorphisms oU GF [oU of dualizing

sheaves.

Proof. The existence of a dualizing complex for any Noetherian, a‰ne, F -finitescheme is shown in [15], Remark 13.6. The local isomorphisms exist due to the uniquenessof dualizing complexes discussed above (the shift is irrelevant since F has relative dimen-sion zero). The final statement about the dualizing sheaves follows formally. r

Summarizing, we see that if X is normal and F -finite and either (a) X is essentiallyof finite type over a Gorenstein local ring, or (b) X is su‰ciently a‰ne, then there is a


dualizing sheaf oX such that oX GF [oX ; in other words oX carries the structure of aCartier module.

Remark 2.22. An interesting question seems to be the existence of a dualizing com-plex for an arbitrary F -finite scheme X . Gabber states in [15], Remark 13.6, that this isthe case but only provides a proof in the case that X is a‰ne. Even more interestingly onemight ask if any F -finite scheme X has a dualizing complex such that o�X GF !o�X globally.This is not clear to us, even in the case that X is regular. This would of course imply, thaton an F -finite scheme, there is a dualizing sheaf equipped with a Cartier module structureF�oX ! oX such that its adjoint oX ! F [oX is an isomorphism.

We finish by pointing out an explicit example of the above general construction,namely the classical Cartier operator [10] on a smooth and finite type over a perfect fieldscheme.

Example 2.23. For simplicity let X ¼ Spec k½x1; . . . ; xn� and k ¼ Fp. Let

WX ¼ k½x1; . . . ; xn�hdx1; . . . ; dxni

be the module of Kahler di¤erentials and W�X the resulting de Rham complex. The inverse

Cartier operator is the unique map

C�1 : W1X ! H 1ðF�W�X Þ

which for a local section x A OX sends dx to the class of xp�1 dx (or, for easier memori-

zation it sends xi dx

x! xpi dx

x). The Cartier isomorphism states that the induced map

W iX ! H iðF�W�X Þ is an isomorphism, which is checked by an explicit calculation, see [11].

In particular, on top di¤erential forms oX ¼ WnX one obtains the Cartier operator

C : F�oX ! oX

as the composition of the natural surjection F�oX ! H nðF�W�X Þ and the inverse of the in-verse Cartier operator C�1 : oX !G H nðF�W�X Þ. Explicitly, this map is given by

xk1

1 � . . . � xknn

dx1

x15� � �5 dxn

xn

� �7! x

k1=p1 � . . . � xkn=p

n

dx1

x15� � �5 dxn

xn

� �

where we set xkj=p

j ¼ 0 if kj=p is not integral. By Theorem 2.17, the adjoint of the Cartieroperator C : F�oX ! oX is the OX -linear map

k : oX ! F [oX :

It can easily be checked by hand that k : oX ! F !oX is an isomorphism.

More generally, for any regular scheme X , essentially of finite type over a perfectfield k, one has, as above, a canonical isomorphism W i

X !H iðF�W�X Þ which induces theCartier operator C : F�oX 7! oX . If in addition X=k is normal, and denoting by oX the


unique reflexive sheaf on X which agrees with the top di¤erential forms on the regularlocus Xreg, then the Cartier operator on the smooth locus induces a Cartier linear mapC : F�oX ! oX .

2.5. Cartier modules over F-finite fields. In this section we investigate the case ofX ¼ Spec k where k is a field. That is we study vector spaces over k equipped with aq�1-linear endomorphism, or equivalently, with a right action of k½F �. The point we makehere is that if ½k : kq� < y then the duality Homkð ; kÞ can be used to reduce the study ofright k½F � vector-spaces to that of k-vectors-spaces with a left k½F �-action. This is slightlymore involved than one might suspect, owing to the just mentioned subtlety concerning theuniqueness of dualizing modules. The problem is, that there is no canonical identificationof a k-vector-space with its k-dual.

We assume that the field k is F -finite, that is, F�k is a finite dimensional k-vector-space. Then its k-dual ðF�kÞ4¼ HomkðF�k; kÞ is (non-canonically) isomorphic to F�k asa k-vector-space. But HomkðF�k; kÞ is also an F�k-vector-space via the action on thefirst factor of the Hom, and, for dimension reasons, necessarily one-dimensional as anF�k-vector-space. As fields, F�k is simply isomorphic to k. We write F [k for the one dimen-sional k GF�k vector-space ðF�kÞ4. The choice we make now, is the choice of a generatorof F [k as a k-vector-space, that is the choice of a k-isomorphism F [k G k. Clearly, two suchchoices will di¤er by multiplication with an element in k�. Since all that will follow dependson the choice of an isomorphism F [k G k, we will fix one from now on. Theorem 2.17shows that we have made the choice of a Cartier-module structure on F�k ! k such thatits adjoint is the isomorphism F [k G k. Also note, that in the case where k is perfect, thisproblem does not arise since then the Frobenius is an isomorphism k GF�k (canonically).

Lemma 2.24. Let k be an F-finite field, then there is an isomorphism of functors

F� � ð Þ4G ð Þ4� F� on k-vector-spaces.

Proof. We just compute for a k-vector-space V

HomkðF�V ; kÞ G2:17

F�HomkðV ;F [kÞGF�HomkðV ; kÞ

where the first isomorphism is the duality of the finite morphism (in the simple case of afield) in Theorem 2.17, and the second one is induced by our fixed isomorphism F [k G k.Note that a di¤erent choice of such isomorphism only changes everything by multiplicationof an element in k�. r

In the simple case of k-vectorspaces considered here, we had to fix an isomorphismbetween the dualizing module F [k and the dualizing module k. Once that has been fixedit induces for every finitely generated field extension i : k ,! K an isomorphism of theK-dualizing modules ði[kÞ (non-canonically isomorphic to K) with F [

Kði[kÞ via the functor-iality of ð Þ[. It is in this sense that the constructions that follow are compatible with fi-nitely generated field extensions. In particular for any finite extension field K of k we willfrom now on denote by ð Þ4K the functor HomKðV ; i[kÞ and hence get a natural isomor-phism FK�ð Þ4K G ðFK� Þ4K of functors on K-vector-spaces.

Proposition 2.25. Let K=k be a finitely generated extension of an F-finite field k.

Then the duality functor ð Þ4K induces an equivalence between the categories of coherent


Cartier modules over K (i.e. finite dimensional K-vector-spaces with a right action of the

Frobenius) and finite dimensional K-vector-spaces with a left action of the Frobenius. The

equivalence ð Þ4K furthermore preserves nilpotence.

Proof. Given a coherent Cartier module ðV ;CÞ over K , that is a finite dimensionalK-vector-space V with a K-linear map C : F�V ! V . Applying ð Þ4K ¼ ð Þ

4 we get

C4 : V4�!C4

ðF�VÞ4GF�V4

with the second isomorphism being the transformation of functors from the precedinglemma. Clearly, the map C4 gives V4 the structure of a K-vector-space with a left actionof the Frobenius, i.e. a left K½F �-module. Conversely, the same observation applies and it iseasy to verify that these functors induce inverse equivalences.

The fact that nilpotence is preserved follows immediately from the construction. r

Proposition 2.25 enables us to fall back on the much better studied theory of left k½F �modules concerning questions of right k½F �-modules. However, this appears to be only pos-sible in the case that k is F -finite, since otherwise F�k is infinite dimensional over k, andhence ðF�kÞ4 is not isomorphic to F�k, and in particular F [k (as defined here) cannot beisomorphic to k. This also shows that the assumption of F -finiteness we make on ourschemes is essential to most methods in this paper and quite likely to many of the resultsas well.

Since we took great care to set up everything compatibly with respect to finitely gen-erated field extensions, we get the following result.

Proposition 2.26. Let i : k ! K be a finitely generated extension of an F-finite field k.

Then there are natural functorial isomorphisms

ði�VÞ4k G i�ðV4K Þ; ði�WÞ4k G i�ðW4

K Þ;

ði[UÞ4K G i�ðU4k Þ; ði�HÞ4K G i[ðH4

k Þ

for all finite-dimensional right/left k½F �-vector-spaces V , W , and right/left K½F �-vector-spaces

U , H.

In other words, the duality functor ð Þ4K induces an equivalence of categories between

finite dimensional right and left k½F �-vector-spaces which commutes with i� and interchanges

i[ and i�.

Proof. It is su‰cient to show the isomorphisms in the first row as this will implythe ones in the second row since ði�; i !Þ and ði�; i�Þ are adjoint pairs and ð Þ4 is a duality.For this just observe that ði�VÞ4k ¼ Homkði�V ; kÞG i�HomkðV ; i[kÞ ¼ i�ðV4

K Þ where thelast equality is due to our convention concerning duality. The second isomorphism followsanalogously. r

With this at hand, we can restate some facts from the theory of left k½F �-modules interms of Cartier modules over k.


Proposition 2.27. Let k be F-finite, and V , W be Cartier modules. Then:

(a) There is a direct sum decomposition V ¼ Vnil lV where Vnil is nilpotent and V

does not have a nilpotent subspace.

(b) Let V ¼ V , then there is a finite separable extension K of k such that

VK ¼ HomkðK;VÞGK nk V is isomorphic toL

n

oK where oK denotes the Cartier module

i[k !G F [ði[kÞ corresponding to the standard Frobenius action on K via the duality ð Þ4K.

(c) The fixed points of CV form an Fq-vector-space of dimension less or equal to dimk V

with equality after a finite separable extension.

(d) If V ¼ V and W ¼W , then HomCartðkÞðV ;WÞ is a finite dimensional vector-space

over Fq.

(e) If V ¼ V , then there are only finitely many Cartier submodules of V.

Proof. Since dimk V ¼ dimk F [V , the structural map k : V ! F [V is injective ifand only if it is surjective, if and only if it is bijective, and each instance is equivalent to V

not having a nilpotent subspace. For a finite field extension, so in particular for F : k ! k,the functor ð Þ[ is exact, and it follows that the surjectivity of C is in fact equivalent to thesurjectivity of its adjoint k. Now let Vnil ¼ kerðk iÞ and V ¼ C iðVÞ for i su‰ciently large (ask-subspaces of a finite dimensional k-vectorspace these will stabilize, cf. Section 2.2). Thenit is easy to see by dimension reasons that Vnil and V have complementary dimension andzero intersection, which shows the first part. The second part follows from the well-knowncorresponding statement for left k½F �-modules using the preceding definition, see [12]. Thefinial three items follow again by the just proven duality ([20], (4.2) Theorem). r

Remark 2.28. The duality in the case of vector-spaces we work out here is a veryspecial case of the duality we describe in [7] where we show that Grothendieck–Serre dual-ity for quasi-coherent OX -modules can be extended to a duality between right and leftOX ½F �-modules. We decided to include the simple special case of fields here as an instructiveexample (already making apparent some of the di‰culties in duality theory), and because itwill be useful later in this paper, cf. Proposition 4.15.

3. Minimal Cartier modules and Cartier crystals

In this section we consider Cartier modules up to nil-isomorphism, leading to notion ofa Cartier crystal. The main result in this section is the existence of minimal Cartier mod-ules. This result allows one to pick in each nil-isomorphism class of Cartier modules (i.e.for each Cartier crystal) a canonical (minimal) representative. This minimal representativeis characterized by having neither nilpotent submodules nor nilpotent quotient modules. Inthis section we do not assume that our schemes are F -finite.

3.1. Cartier crystals.

Definition 3.1. A map of Cartier modules is called a nil-isomorphism if kernel andcokernel are nilpotent.


Lemma 3.2. The composition of nil-isomorphisms is again a nil-isomorphism.

Proof. Let M 0 !j M !c M 00 be a composition of nil-isomorphisms. The kernel of jand c are both nilpotent, say of order less or equal to e. Then it is easy to see thatkerðc � jÞ is nilpotent of order less or equal to 2e. Similarly for the cokernel of ðc � jÞ. r

Lemma 3.3. A map of coherent Cartier modules M !j N is a nil-isomorphism if and

only if for all x A X the induced map jx : Mx ! Nx is a nil-isomorphism.

Proof. The only if part is clear and the rest follows from Lemma 2.10. r

Definition 3.4. The category of Cartier crystals is the abelian category obtainedfrom the category of coherent Cartier modules by localization with respect to nil-isomorphisms.

This definition means, that the objects are just Cartier modules, but a homomorphismfrom M to N is a diagram (left fraction):

M (M 0 ! N

where M 0 )M is a nil-isomorphism (which we will denote for convenience by )), andM 0 ! N is a morphism of Cartier modules. The general theory of localization of an abe-lian category at a multiplicative system defined from a Serre subcategory yields that Cartiercrystals form also an abelian category, cf. [8] as a reference for standard facts about local-ization of categories. We will denote the Cartier crystals associated with a Cartier moduleM by the corresponding calligraphic letter M. Likewise, if M is a crystal, M denotes some

Cartier module giving rise to that crystal. Some immediate properties are:

Lemma 3.5. Let M, N be two coherent Cartier modules and M, N their associated

crystals.

(a) A map of Cartier modules M ! N is a nil-isomorphism if and only if the induced

map M!N of crystals is an isomorphism of Cartier crystals.

(b) MGN are isomorphic as crystals if and only if there is a Cartier module M 0 and

nil-isomorphisms M (M 0 ) N of Cartier modules.

(c) MG 0 as a crystal if and only if M is nilpotent.

Proposition 3.6. The previously defined functors i[ ( for closed embeddings), j � (open

embeddings), and f� (in general) preserve nil-isomorphism, and hence induce well-defined

functors on Crys whenever they preserve coherence.4)

Proof. Since the functor i[ð Þ ¼ ½I � is left exact, it preserves nil-injections. Nowlet N ,!M be an injective map such that M=N is nilpotent. Since N½I � ¼M½I �XM, it

4) Coherence is clearly preserved under j � and i [, but generally not under f�.


follows that the cokernel of N½I � ,!M½I � is a submodule of the nilpotent M=N and henceit is also nilpotent. Hence i[ preserves nil-isomorphisms.

Since j � is exact for open embeddings j : U ! X , it preserves nil-isomorphisms sinceit preserves nilpotence. The case of f� follows also easily using that F e

� is exact, since F isa‰ne. r

3.2. Minimal Cartier modules. In this section we show that for a Cartier crystal Mone can choose a representing Cartier module in a functorial fashion.

Definition 3.7. A Cartier module M is called minimal if the following two conditionsare satisfied:

(a) M has no non-zero nilpotent Cartier submodules (equiv. (in the F -finite case) theadjoint structural map k : M ! F [M is injective).

(b) M has no non-zero nilpotent Cartier quotients (equiv. the structural mapC : F�M !M is surjective).

Lemma 3.8. A nil-isomorphism of minimal Cartier modules is an isomorphism.

Proof. Let j : M ! N be a nil-isomorphism of Cartier modules. If M satisfiesDefinition 3.7(a) then j cannot have a non-zero nilpotent kernel, hence j is injective. If N

satisfies Definition 3.7(b), then j cannot have non-zero nilpotent cokernel, so j must besurjective. r

Lemma 3.9. If M is minimal, then any localization of M is minimal.

Proof. Since localization is exact, the structural map on any localization is also sur-jective. Since any nilpotent submodule of a localization of M has non-zero intersection(pre-image under the localization map) with M, it follows that a localization cannot havenilpotent submodules, unless M did. r

Theorem 3.10. The assignment M 7!Mmin which maps a coherent Cartier module M

to the nil-isomorphic Cartier module Mmin defines a functor. The Cartier module Mmin can be

obtained as a sub-quotient of M, and any minimal Cartier module N which is nil-isomorphic

to M is isomorphic to Mmin.

Proof. Define

Mmin ¼def ðMÞ

where and are as in Proposition 2.12 and Proposition 2.14. Clearly Mmin is minimal, asub-quotient of M, and nil-isomorphic to M. Since and are both functorial, so is theircomposition. If N ( N 0 )M is another sequence of nil-isomorphisms with N minimal,the functoriality implies that we have nil-isomorphisms N ¼ Nmin ( N 0min )Mmin. Sincenil-isomorphisms of minimal Cartier modules are isomorphisms by Lemma 3.8, it followsthat N GMmin which implies uniqueness. r

Note that the uniqueness part of Theorem 3.10 shows that the order in which andare applied to arrive at a minimal Cartier module is irrelevant.


Lemma 3.11. If M, N are minimal Cartier modules and M, N denote their associ-

ated crystals, then HomCartðM;NÞ ¼ HomCrysðM;NÞ.

Proof. By Definition 3.7 of minimality for N, the map

HomCartðM;NÞ ! HomCrysðM;NÞ

induced by the functor M 7!M is injective. For surjectivity, let j A HomCrysðM;NÞ berepresented by a map of Cartier modules

M (M 0 !j0

N:

The functoriality of ð Þmin induces a map M GM 0min �!

j 0min

N which also represents j, but isa honest map of Cartier modules from M ! N. r

Theorem 3.12. The category of Cartier crystals is equivalent to the category of mini-

mal Cartier modules.

Proof. One has the natural functors

MinCartðX Þ ,! CartðXÞ ! CrysðXÞ:

Conversely, the map that assigns to each Cartier crystal M represented by some Cartiermodule M, the minimal Cartier module Mmin is a well defined functor from Cartier crystalsto minimal Cartier modules. Well definedness follows from Theorem 3.10. Now the preced-ing lemma shows that these two functors are inverse equivalences. r

3.3. Crystalline support and Kashiwara equivalence. A simple, however crucial,property of Cartier modules is captured by the following observation.

Lemma 3.13. Let ðM;CÞ be a Cartier module with surjective structural map, i.e.

CðMÞ ¼M. Then AnnOXM is a sheaf of radical ideals.

Proof. Let x A OX be a local section such that xkM ¼ 0. Then for qe f k we have0 ¼ xkM ¼ xqe

M ¼ C eðxqe

MÞ ¼ xC eðMÞ ¼ xM. r

Corollary 3.14. If X ¼ Spec R is a‰ne, p is a prime ideal, and M a Cartier module

with surjective structural map, then pkM ¼ 0 implies pM ¼ 0.

Definition 3.15. Let M be a coherent Cartier module on X . We define SuppcrysðMÞ,the crystalline support of M, as the set of all x A X such that Mx is not nilpotent.

Lemma 3.16. (a) If the structural map of M is surjective, then

SuppcrysðMÞ ¼ SuppðMÞ.

(b) If M and N are nil-isomorphic, then SuppcrysðMÞ ¼ SuppcrysðNÞ, In particular,using (a) we have SuppcrysðMÞ ¼ SuppðMÞ.

(c) SuppcrysðMÞ is a closed reduced subscheme.


Proof. Since surjectivity of the structural map passes to localization, every Mx forx A X has this property. But clearly, a Cartier module with surjective structural map is nil-potent if and only if it is zero, proving the first part. Similarly, if M and N are nil-isomor-phic, then so are Mx and Nx for all x A X . Hence Mx is nilpotent if and only if Nx is nilpo-tent. Part (a) shows that SuppcrysðMÞ is a closed set, equal to SuppðMÞ. But also the schemestructure given by AnnOX

M is reduced since by Lemma 3.13 AnnOXM is radical, since M

has surjective structural map. r

These observations guarantee the well definedness of the following notion of supportfor a Cartier crystal.

Definition 3.17. Let M be a Cartier crystal represented by the Cartier module M.Then the support of M is defined as SuppðMÞ ¼ SuppcrysðMÞ, which by (c) above is aclosed and reduced subscheme.

Let Y LX be a closed embedding, then the Kashiwara-type equivalence we will de-rive states that the category of Cartier crystals CrysðYÞ on Y is equivalent to the categoryof Cartier crystals on X which are supported in Y . This latter subcategory of CrysðX Þ wedenote by CrysY ðXÞ.

Lemma 3.18. Let Y LX be a closed embedding and let ðM;CÞ be a coherent Cartier

module on X with surjective structural map. Then M is supported in Y , if and only if

M ¼M½I � ¼ fm A M j Im ¼ 0g.

Proof. Let I be the sheaf of ideals defining Y LX . Then a coherent M is supportedin Y if and only if some power I n annihilates M, i.e. I nM ¼ 0. Since we assumed that thestructural map of M is surjective this is, by Lemma 3.13, equivalent to IM ¼ 0, and henceM½I � ¼M. r

Proposition 3.19. Let i : Y ! X be a closed embedding. Then the functors i� and i[

induce an equivalence of categories of Cartier crystals CrysðYÞ on Y and CrysY ðXÞ, the

Cartier crystals on X with support in Y.

Proof. In Proposition 2.6 we have seen that i[i� ¼ id and i�i[M ¼M½I � on Cartier

modules. But the two functors on Crystals are defined by evaluating them on representingCartier modules, cf. Proposition 3.6. We may hence choose the Cartier module M repre-senting a crystal M to have surjective structural map, by replacing M by M. Then the sup-port of M is equal to the support of M, and now the preceding lemma shows i�i

[ G id onCartier crystals. r

Remark 3.20. It is useful to point out that the property of minimality for coher-ent Cartier modules is preserved by the local cohomology functor H 0

I ð Þ. Furthermore,if M has surjective structural map, so in particular if M is minimal, Lemma 3.13shows that H 0

I ðMÞ ¼M½I �. To check minimality of H 0I ðMÞ we only need to show that

C�H 0

I ðMÞ�¼ H 0

I ðMÞ. But if I nm ¼ 0, and m ¼ C eðm 0Þ for some m 0 A M (since, by mini-mality C eðMÞ ¼M) then C e

�ðI nÞ½q

e�� ¼ I nC eðm 0Þ ¼ I nm ¼ 0. This means that the

OX -submodule ðI nÞ½qe�m 0 of M is killed by C e. Again by minimality of M this implies that

ðI nÞ½qe�m 0 ¼ 0, hence m 0 A H 0

I ðMÞ. This implies that for a closed immersion i : Y ! X thefunctor i[ preserves minimality. Furthermore, since i is a‰ne, i� is exact and it follows


that i� preserves minimality. Hence the preceding equivalence for Cartier crystals may alsobe thought of as an explicit equivalence on minimal coherent Cartier modules.

4. Finite length for Cartier crystals

In this section we prove our main result about coherent Cartier modules. Namely thatevery Cartier module has up to nilpotence finite length. From this result we will later derivea series of consequences, recovering and extending other finiteness-results of modules underthe presence of an action of the Frobenius.

The following result of Gabber is the key technical ingredient to showing that, up tonilpotence, coherent Cartier modules have finite length.

Proposition 4.1 ([15], Lemma 13.2). Let X be irreducible and F-finite, and let M be a

coherent Cartier module on X. Then there is an open subset U LX such that for all x A U

non generic5) we have:

All finite length Cartier quotients of Mx are nilpotent:ð�Þ

Proof. We may replace M by M and assume that the structural map of M is surjec-tive.6) The surjectivity of the structural map guarantees that the nil-radical I of OX annihi-lates M by Lemma 3.13, since I n ¼ 0 for ng 0. Hence we may replace X by Xred and as-sume that X is reduced, and hence integral.

We may further replace M by M ¼M=Mnil and hence assume that there are no nil-potent Cartier submodules.7) Since we assume that X is F -finite, this condition means thatthe adjoint to the structural map

M !k F [M ¼HomOXðF�OX ;MÞ

is injective, cf. Proposition 2.18 and Corollary 2.19. Let U ¼ Xreg be the dense open (sinceX is reduced and F -finite implies excellent) regular locus of M. By a result of Kunz [25],F is finite flat on U . Replacing X by U , F�OX is a locally free OX -module of finite rank.Hence, F [ ¼HomOX

ðF�OX ; Þ is an exact functor. Further shrinking U we may assumethat M is (locally) free of rank r, and hence so is F [M. Shrinking U once again we mayassume that k is surjective, hence an isomorphism (we already arranged that k is injectiveby killing Mnil). For x A U not the generic point of U , let Mx !! N be a Cartier modulequotient of finite length on SpecOX ;x. Since x is not the generic point, OX ;x ¼: ðR;mÞ is aregular local ring of dimensionf 1. The exactness of F [ implies that N ! F [N is still sur-

5) We call x non-generic in U if x is not the generic point of an irreducible component of U .

6) For this, one has to observe that ð�Þ for M at a point x implies ð�Þ for M at x. To ease notation we

temporarily replace M by Mx. If M=N is a finite length quotient, then the submodule M=ðN XMÞ has finite

length. By ð�Þ for M it is hence nilpotent. As a quotient of the nilpotent M=M, M=ðM þNÞ is nilpotent. But

then, since nilpotence is preserved under extensions by Lemma 2.11, it follows that M=N is nilpotent as desired.

7) For this, one has to check that for a point x A X condition ð�Þ for Mx implies ð�Þ for Mx. To simplify

notation we may temporarily replace M by Mx. Take N LM such that the quotient M=N has finite length. Then

M=ðMnil þNÞ ¼M=�N=ðMnil XNÞ

�has finite length, and is hence nilpotent by assumption that ð�Þ holds for M.

But clearly, Mnil XN is nilpotent as well, and hence so is M=N as an extension of nilpotents by Lemma 2.11.


jective, hence length N f lengthðF [NÞ. But since dim Rf 1, F [ ¼ HomRðF�R; Þ multi-plies the length of finite length modules by a power of q.8) But this implies that N ¼ 0, so,in particular, N is nilpotent as desired. r

Remark 4.2. The F -finiteness was crucially used in this proof when we employed theadjointness of F [ and F� for the finite Frobenius F . However, we expect that the result itselfholds much more generally. At least when R is local, or essentially of finite type over a reg-ular local ring. Faithfully flatness of completion can be used to reduce the local case to thecomplete local case. Then our Kashiwara equivalence Proposition 3.19 reduces to the reg-ular case.

Corollary 4.3. Let X be F-finite and ðM;CÞ a coherent Cartier module with surjective

structural map. Then there is a dense open subset U LX such that for each non-generic

x A U , the localization Mx does not have finite length Cartier quotients.

Proof. Let X1; . . . ;Xn be the irreducible components of X . On each irreducible opensubset X 0i ¼ X �

Sj3i

Xj of X consider the Cartier module MjX 0i

(Corollary 2.4). Let Ui LX 0i

be an open subset as in Proposition 4.1, and let U ¼S

Ui. Since C is surjective on M thesame is true for Mx for all x A X and also for all quotients Mx=N. By construction of U , forall x A U the finite length quotients of Mx are nilpotent and hence zero. r

Corollary 4.4. Let X be F-finite and ðM;CÞ a coherent Cartier module with surjec-

tive structural map. There is a closed subset Y ¼ VðIÞLX , not containing an irreducible

component of X , such that for all Cartier module quotients M=N of M whose support

Supp M=N does not contain an irreducible component of X , one has Supp M=N LY. In

fact AnnOXM=N M

ffiffiffiIp

M I .

Proof. Let U be as in the preceding corollary and let Y ¼ X �U . Let Z ¼ VðpÞ bean irreducible component of Supp M=N. By assumption p is not the generic point of anirreducible component of U (¼ generic point of an irreducible component of X ). The local-ization ðM=NÞp ¼Mp=Np is a non-zero and finite length module over OX ;p. Since we as-sumed that the structural map of M, and hence of Mp=Np is surjective, it cannot be nil-potent, hence p B U (since by assumption p is not the generic point of an irreduciblecomponent of U which would be the only other option). It follows that Z LY , henceSupp M=N LY . Again, since M=N has surjective structural map, the ideal Ann M=N isradical by Lemma 3.13, hence I L

ffiffiffiIp

LAnn M=N. r

As a corollary of this result, one derives by induction the following.

Proposition 4.5 ([15]). Let X be F-finite and M a coherent Cartier module. There is a

finite subset S LX such that for all x A XnS

the finite length quotients of Mx are nilpotent:ð�Þ

8) Since F [ is exact it is enough to show that length�F [ðR=mÞ

�¼ lengthðR=m½q�Þ. For this we note

that HomRðF�R;R=mÞ ¼ HomR=m

�F�ðR=m ½q�Þ;R=m

�since for x A m we have 0 ¼ xjðrÞ ¼ jðxqrÞ for all

j A HomRðF�R;R=mÞ. But R=m½q� is free of dimension equal to lengthðR=m½q�Þ over R=m, hence so is

HomR=m

�F�ðR=m ½q�Þ;R=m

�over F�ðR=mÞ. This shows the claim.


Proof. As before we may replace M by M and assume that C is surjective. Wefirst observe that if Y LX is a closed irreducible subset given by a sheaf of ideals I ,then for all x A Y the Cartier module M on X satisfies ð�Þ at x if the Cartier module~MM ¼M=

Pif0

C iðIMÞ on Y satisfies ð�Þ: To see this, let N LMx be such that Mx=N has

finite length. Since C is surjective, this implies that mxMx LN by Corollary 3.14. Sincex A Y , and hence Ix Lmx we have IxMx LN. It follows that

Pif0

C iðIxMxÞLN since the

former is the smallest Cartier submodule of Mx that contains IxMx. If we denote by ~NN theimage of N in ~MMx, it can be checked (using the snake lemma, for example) that the naturalmap Mx=N ! ~MMx= ~NN is an isomorphism. But ð�Þ for ~MM on Y implies that the latter is nil-potent.

To conclude the argument we can now use induction on the dimension. By consider-ing each irreducible component of X separately we may reduce to X irreducible. ThenProposition 4.1 yields an open set U (minus the generic point) where ð�Þ holds for M. Byinduction we know that the conclusion of the theorem holds for ~MM on each of the finitelymany irreducible components of the complement of U . This finishes the argument. r

Gabber uses the preceding proposition in [15] as a crucial ingredient in his proof thatthe category of perverse constructible sheaves of Fq-vectorspaces

PervcðXet; FqÞLDbc ðXet; FqÞ

is Noetherian and Artinian. Via Grothendieck–Serre duality and using [8] or [13], and [7]the category of Cartier crystals is expected to be equivalent to PervcðXet; FqÞ. This, in turnwould imply the finite length for Cartier crystals. The purpose of this section is to give adirect argument for this expected finite length of Cartier crystals:

Theorem 4.6. Let X be a scheme satisfying Proposition 4.1 (e.g. X is F-finite) and M

a coherent Cartier module. Then any descending chain

M MM1 MM2 MM3 M � � �

of Cartier submodules of M stabilizes up to nilpotence. This means that for i g 0 the quo-

tients Mi=Miþ1 are nilpotent.

Proof. We may replace the given chain by the nil-isomorphic one

M MM1 MM2 MM3 M � � �

by Proposition 2.14—since clearly—Mi=Mj is nilpotent if and only if Mi ¼Mj. Hence wemay assume that for all i we have CðMiÞ ¼Mi. One proceeds by induction on the dimen-sion of X , the case of dimension zero being obvious. There are only finitely many steps inthis chain where the generic rank on some irreducible component of X can drop. By trun-cating we may hence assume that on each irreducible component the generic rank is con-stant in the descending chain. This implies, that for each i the support Supp M=Mi does notcontain any irreducible component of X . Applying Corollary 4.4 we have Y ¼ VðIÞLX

of strictly smaller dimension than X such that for all i the support Supp M=Mi LY . In factwe even have IM LMi for all i. This implies that M 0 ¼def P

if0

C iðIMÞ (which is the smallest


Cartier submodule of M containing IM) is also contained in each Mi, since each Mi is aCartier submodule containing IM. But this implies that the original chain stabilizes if andonly if the chain

M=M 0MM1=M 0MM2=M 0M � � �

stabilizes. The latter, however, is a chain of Cartier submodules of M=M 0 on Y (sinceIM LM 0) with the property that CM=M 0 ðMi=M 0Þ ¼Mi=M 0 for all i. By induction weknow this chain stabilizes. r

As an immediate corollary we obtain, in the language of Cartier crystal, our mainresult.

Corollary 4.7. Let X be a scheme satisfying Proposition 4.1 (e.g. X is F-finite), then

every coherent Cartier crystal is Noetherian and Artinian, i.e. ascending, as well as descend-

ing chains of Cartier sub-crystals are eventually constant.

Proof. That ascending chains stabilize already follows from the Noetheriannessof X . The stabilization of descending chains is precisely the statement of Theorem 4.6. r

As a first application we derive global versions of analogs of results of Enescu andHochster [14] and Sharp [30], Corollary 3.11. The following (and also its proof) is analo-gous to [14], Proposition 3.5.

Lemma 4.8. Let M be a Cartier module on X. Then

S :¼ fY ¼ Suppcrys M=N jN a Cartier submodule of Mg

is a collection of reduced subschemes, closed under taking irreducible components and finite

unions.

Proof. By Lemma 3.13 the elements of S are reduced subschemes. That S is closedunder taking finite unions is a consequence of the equality

AnnOXM=ðN XN 0Þ ¼ AnnOX

M=N XAnnOXM=N 0.

It remains to show that the irreducible components of every Y ¼ Suppcrys M=N A S lieagain in S. For this, we may replace ðX ;MÞ by ðY ;M=NÞ. Then we need to show thatthe irreducible components of X lie in S provided that X ¼ SuppðMÞ ¼ SuppcrysðMÞ.Let Z be such a component, let Z 0 be the union of those components of X di¤erentfrom Z and let I and I 0 denote the corresponding ideal sheaves. Since X is reduced, I

and I 0 are radical. Then I X I 0 ¼ 0 by their definition, and so AnnOXðI 0Þ ¼ I . Proposition

2.6 shows that M½I 0� is a Cartier submodule of M. Using AnnOXðMÞ ¼ 0, we compute

AnnOXðM=M½I 0�Þ ¼ f f A OX j fM HM½I 0�g ¼ f f A OX j fI 0M ¼ 0g

¼ f f A OX j fI 0 ¼ 0g ¼ AnnOXðI 0Þ ¼ I :

Since I is the ideal sheaf of Z, the proof is complete. r


Proposition 4.9. Let X be F-finite and M a coherent Cartier module. Then the collec-

tion

fY ¼ Suppcrys M=N jN a Cartier submodule of Mg

is a finite collection of reduced subschemes of X. In fact, it consists of the finite unions of the

finitely many irreducible subschemes in the collection.

Proof. By the preceding lemma it is enough to show that there are only finitelymany irreducible subschemes in this collection. As before, we may replace M by M sothat the structural map is surjective. If V ¼ x for x A X is an irreducible component ofSuppðM=NÞ, then ðM=NÞx is a non-zero finite length quotient of Mx (over OX ;x). Sincethe structural map of Mx, and hence of ðM=NÞx, is surjective, ðM=NÞx cannot be nilpotent.It follows that x is an element of the finite set S of Proposition 4.5. r

Remark 4.10. Enescu and Hochster show in [14], Theorem 3.6 (see also [30], Corol-lary 3.11) that if R is local and W is an Artinian left R½F �-module, then the collection

fAnnR V jV LW an R½F �-submoduleg

is a finite set of radical ideals, consisting of all intersections of the finitely many primes in it.We will explain in Section 5 that the precise connection to our result is via Matlis duality.In fact, in the local case, one may give an alternative proof of Proposition 4.9 using [14],Theorem 3.1.

Remark 4.11. Note that the collection in Proposition 4.9 is not closed under scheme-theoretic intersection in general. See, however, the following remark.

Remark 4.12 (Compatibly split subvarieties). From the viewpoint of Frobeniussplittings (see [9]) Proposition 4.9 may also be interpreted as a generalization of the finite-ness of compatibly split subvarieties of an F -split variety X obtained by Schwede [29](see also [24] for a very short proof): A scheme X is called F -split if there is a mapC : F e

�OX ! OX splitting the Frobenius. Due to the splitting property, the coherent Cartiermodule ðOX ;CÞ has surjective structural map C. A subvariety Y cut out by a sheaf ofideals I is called compatibly split, if CðIÞL I , i.e. if and only if the defining ideal I of Y

is a Cartier submodule of OX , or equivalently, the quotient OY is a Cartier module quotientof OX . Since OY is a Cartier module quotient, it has also surjective structural map. In par-ticular, by Lemma 3.13, Y is reduced. Since Y ¼ SuppOX

OY the finiteness of compatiblysplit subschemes Y , follows from Proposition 4.9. Note that in this case, the collection is,in fact, closed under scheme-theoretic intersection, since AnnOX

�OX=ðI þ JÞ

�¼ I þ J.

4.1. Finiteness of homomorphisms of Cartier crystals. We show that the Hom-setsin the category of Cartier crystals are finite sets. An analog of this result for Lyubeznik’sF-finite modules has been obtained by Hochster [20].

Proposition 4.13. A non-zero coherent Cartier module is simple if either

(a) M is minimal and simple, or

(b) M has zero structural map and is simple as an OX -module, i.e. isomorphic to OX=mfor some maximal ideal m.


Proof. Let M be simple. If Mmin is non-zero, then M GMmin since Mmin is a sub-quotient of M. Otherwise Mmin ¼ 0, hence M is nilpotent which is the same as C nðMÞ ¼ 0for some n. This implies that C cannot be surjective, implying that CðMÞ is a proper Cartiersubmodule of M. Hence CðMÞ ¼ 0 by simplicity of M. But if C acts as zero on M, then M

is simple as a Cartier module if and only if M is simple as an OX -module. r

Proposition 4.14. A simple Cartier module M has a unique associated prime.

Proof. Let p be an associated prime of M. Then M½p�LM is clearly a non-zeroCartier submodule. Since M is simple we must have M½p� ¼M which shows our assertion.

r

Proposition 4.15. Suppose X is F-finite and M is a simple coherent Cartier module

on X.

(a) If M is minimal and simple, then EndCartðMÞ is a finite field containing Fq.

(b) Otherwise AnnOXðMÞ ¼: m is a maximal ideal sheaf, CM ¼ 0 and EndCartðMÞ is

isomorphic to the residue field at m.

Proof. In view of Proposition 4.13, the proof of (b) is obvious. To prove (a), let pdenote the unique associated prime of M. Since M HMp and EndCartðMÞHEndCartðMpÞ,we may localize at p and thus assume that X ¼ Spec R for R a local ring with maximalideal p ¼ AnnRðMÞ (the condition CðMÞ ¼M persists under localization). Therefore M

may be regarded as a finite dimensional vector space over the residue field k :¼ R=p, andso we are reduced to R ¼ k. Since X is F -finite, the same holds for k. From Proposition2.27 we deduce that EndCartðMÞ is a finite dimensional vector space over Fq and in particu-lar it is finite. As we also assume that M is simple, the lemma of Schur implies thatEndCartðMÞ is a skew field. Now (a) follows since every finite skew field is a field. r

Corollary 4.16. If M is a simple (non-zero) Cartier crystal on X , then EndCrysðMÞ is

a finite field containing Fq.

Combining Proposition 4.15 with the finite length of a Cartier crystal proved inCorollary 4.7, we obtain the following analog of the main result in [20]—and in fact, wefollow its arguments closely.

Theorem 4.17. Let X be an F-finite scheme, and suppose that M and N are coherent

Cartier sheaves with CðMÞ ¼M and Nnil ¼ 0. Then HomCartðM;NÞ is finite dimensional

over Fq, hence a finite set.

Proof. The condition on M ensures that its image under any homomorphism to N

lies in N, the condition on N that Mnil is in the kernel of any such homomorphism. Hencewe may replace M by M ¼M min and N by N ¼ N min, so that

HomCartðM;NÞ ¼ HomCrysðM;NÞ

if M and N denote the associated crystals by Lemma 3.11. Since by Corollary 4.7, both Mand N have finite length (equal to the quasi-length of M, N), we can now proceed byinduction on lengthðMÞ þ lengthðNÞ, the base case of M and N being simple was just


treated in Proposition 4.15. Now take any 03N 0kN and consider the long exact se-quence for HomCrys:

0! HomCrysðM;N 0Þ ! HomCrysðM;NÞ ! HomCrysðM;N=N 0Þ ! � � � :

By induction hypothesis, both ends are finite, so the middle term is finite as well. Proceed-ing similarly for M, the result follows. r

Corollary 4.18. Let X be an F-finite scheme. Then any Cartier crystal M on X con-

tains only finitely many subcrystals.

Proof. By Corollary 4.7 and Corollary 4.16, the category of Cartier crystals is anArtinian and Noetherian Abelian category in which the endomorphism ring of any simpleobject is finite. For any such it is well known that any object has up to isomorphism onlyfinitely many subobjects. Lacking a suitable reference, we include a short proof: Supposethere is a counterexample M. Since every object has finite length, we take M to be thecounterexample of smallest length. By the Jordan–Holder theorem and Theorem 4.17, M

contains only finitely many simple subobjects. Since any subobject of M contains a simpleone, there must be a simple subobject S of M for which there are infintely many distinctsubobjects of M containing S. But then M=S contains infinitely many distinct subobjects.Since the length of M=S is smaller than that of M, we have reached a contradiction. r

The following result allows one to obtain an analogous result for Cartier modules:

Proposition 4.19. Let X be an F-finite scheme and M a coherent Cartier module on X

with associated crystal M. Then the assignment N 7!N from the submodules N of M which

satisfy CðNÞ ¼ N to the subcrystals of M, sending N to its associated crystal, is a bijection.

Moreover this assignment is inclusion preserving, i.e., if N and N 0 are submodules of M with

CðNÞ ¼ N and CðN 0Þ ¼ N 0 representing the subcrystals N and N 0 of M, respectively, then

N HN 0 if and only if N is a subcrystal of N 0.

Proof. If N HM is any Cartier submodule representing the subcrystal N of M,then N HN will also represent N. Hence the assignment in the proposition is surjec-tive. Suppose now that two Cartier submodules N and N 0 of M with CðNÞ ¼ N andCðN 0Þ ¼ N 0 both represent N. Then N þN 0 will also be a Cartier submodule of M withCðN þN 0Þ ¼ N þN 0 which represents N. Thus we may assume that N HN 0. But thenthe injective homomorphism N ,! N 0 is a nil-isomorphism, and in particular N 0 will havethe nilpotent quotient N 0=N. Since CðN 0Þ ¼ N 0 we deduce N 0=N ¼ 0. Hence the assign-ment in the proposition is also injective. The argument employed in the proof of the injec-tivity, also easily proves that the assignment is inclusion preserving. r

The following result is now immediate:

Corollary 4.20. Let X be an F-finite scheme. A coherent Cartier module M on X has

only finitely many submodules N such that CðNÞ ¼ N.

Remark 4.21. If we call two submodules N;N 0LM equal up to nilpotence ifN ¼ N 0 as submodules of M, then Corollary 4.20 says that a coherent Cartier module hasup to nilpotence, only finitely many Cartier submodules.


4.2. Nil-decomposition series. Our main result Theorem 4.6 and its Corollary 4.7state that in the category of Cartier crystals every object has finite length. Via standardarguments one derives from this a Jordan–Holder theory for Cartier crystals. For Cartiermodules, this yields in turn a theory of Jordan–Holder series up to nilpotence. Let us spellout what this means. Let M be a coherent Cartier module. Then a nil-decomposition series

is a sequence

M ¼M0 MM1 MM2 M � � �MMn�1 MMn ¼ 0

of submodules of M such that for each i the quotients Mi=Miþ1 are not nilpotent (equiv-alently ðMi=Miþ1Þmin 3 0). A nil-decomposition series is called maximal, if and only if itcannot be refined to a longer nil-decomposition series. Clearly, a nil-decomposition seriesis maximal if and only if all quotients ðMi=Miþ1Þmin are simple. We define the quasi-length

of M, denoted qlðMÞ, as the length of the shortest maximal nil-decomposition series of M.Clearly qlðMÞ is equal to the length of the associated Cartier crystal M. As an immediateconsequence of Proposition 4.19, we deduce:

Proposition 4.22. Let M be a coherent Cartier module on the F-finite scheme X with

M ¼ CðMÞ. Then there is a bijection between nil-decomposition series of M satisfying

CðMiÞ ¼Mi for all i and decomposition series of the crystal represented by M.

This yields the following type of nil-decomposition series for M:

Proposition 4.23. Let M be a coherent Cartier module on the F-finite scheme X. Then

there exists a finite sequence

M ¼M0 MM0 MM1 MM1 M � � �MMn�1 MMn�1 MMn MMn ¼ 0

such that Mi=Mi is nilpotent (and possibly zero) and Mi=Miþ1 is not nilpotent and simple

(equiv. minimal, simple, and non-zero).

Proof. Let M ¼M0 MM1 M � � �MMn�1 MMn ¼ 0 be a decomposition series of M

corresponding by the previous proposition to a Jordan–Holder series of the crystals repre-sented by M. Defining Mi as the pullback under Miþ1 !Miþ1=Mi of ðMiþ1=MiÞnil, the re-sult follows. r

The following theorem follows now formally from the finite length up to nilpotence

via standard arguments in Jordan–Holder theory.

Theorem 4.24. Let M be a coherent Cartier module on the F-finite scheme X.

(a) Every nil-decomposition series of M can be refined to a maximal nil-decomposition

series of M.

(b) Every maximal nil-decomposition series of M has the same length.

(c) The minimal simple sub-quotients ðMi=Miþ1Þmin are unique (up to re-ordering).


5. Co-finite left R[F ]-modules and Lyubeznik’s F-finite modules

The category of Cartier modules can be viewed as central axis which relates othercategories of OX -modules with other types of Frobenius related actions on them. The pic-ture that one has is roughly as follows:

co-finite left R½F �-modules

ðX¼Spec R; localÞ

��!½27� f :g: unit OX ½F �-modules

ðregularÞ

Matlis duality

ðlocal; F -finiteÞ

x??y Gen

x???½4�coherent Cartier modules

ðNoetherianÞ

��! ��ðregular F -finiteÞ

��noX ��!no�1X

coherent g-sheaves

ðregular F -finiteÞ

Matlis duality

ðregular localÞ

��

��

The precise categories and the arrows between them will be explained in this section. Theparenthesized assumptions in the diagram denote the generality in which the respective cat-egory or functor exists. It turns out that all arrows, except the ones to the top right corner,are equivalences of categories; these remaining two arrows become equivalences when thesource is replaced by the respective category of crystals, i.e. after killing nilpotence. Thearrows in the bottom left triangle all preserve nil-isomorphisms and hence induce an equiv-alence on the level of the associated crystals.

In addition to this—and this is a main point in the manuscript [7]—Grothendieck–Serre duality induces an equivalence between Cartier modules and so called t-sheavesof [8], i.e. left OX -modules. However this equivalence is on the level of an appropriate de-rived category and hence quite more technical than the current paper. In particular, thepreservation of certain functors, which we only hinted at here, is a non-trivial matter. Wewill not discuss this (related) viewpoint here but refer the reader to the upcoming [7], orto [5], Chapter 6, for an overview of these results.

5.1. Co-finite modules with Frobenius action. In this section, we explain the relation-ship of our theory with the theory of co-finite left R½F �-modules for a local ring ðR;mÞ asstudied before by many authors, mainly in connections to questions about local cohomol-ogy, see for example [19], [30], [31], [28], [16], [27]. The relation to our theory is via Matlisduality, quite similar to the duality in the case of a field we described above.

5.1.1. Matlis duality. Let ðR;mÞ be complete, local and F -finite. Denote by E ¼ ER

an injective hull of the residue field R=m of R. Since R is F -finite one has thatF [ER ¼ HomRðF�R;ERÞGEF�R, and since R and F�R are isomorphic as rings we mayidentify EF�R with ER, and hence get an isomorphism ER GF [ER. As in the case of fieldstreated in Section 2.5, we fix from now on an isomorphism F [ER GER; note that any otherchoice of isomorphism F [ER GER will di¤er from our fixed one by an automorphismof ER, i.e. by multiplication with a unit in R. We denote by ð Þ4R ¼ HomRð ;ERÞ theMatlis duality functor.

Lemma 5.1. Let ðR;mÞ be local and F-finite. Then there is a functorial isomorphism

F�ð Þ4R G ðF� Þ4R.


Proof. Using the duality for finite morphisms we get for any R-module M

HomRðF�M;ERÞGF�HomRðM;F [ERÞGF�HomRðM;ERÞ

where the final isomorphism is induced by the fixed isomorphism above. r

Proposition 5.2. Let ðR;mÞ be complete, local and F-finite. Then Matlis duality

induces an equivalence of categories between left R½F �-modules which are co-finite9) as

R-modules and coherent Cartier modules, i.e. R-finitely generated right R½F �-modules. The

equivalence preserves nilpotence, and hence, since duality is exact, it also preserves nil-

isomorphism.

Proof. If M is a finitely generated Cartier module, then the dual of the structuralmap, together with the functorial isomorphism of the preceding lemma gives a map

M4! ðF�MÞ4GF�ðM4Þ

which is nothing but a left action of R½F � on the co-finite R-module M4. Conversely, thesame construction works and it is easy to check that this indeed induces an equivalence ofcategories. r

This correspondence precisely explains the relationship between our Proposition 4.9and the result of Enescu and Hochster [14], Theorem 3.6, discussed above in Remark 4.10.Further consequences are:

Theorem 5.3. Let ðR;mÞ be local and F-finite. Let N be a co-finite R-module with a

left R½F �-action. Then:

(a) Nnil ¼ fn A N jF eðnÞ ¼ 0 for some eg is nilpotent, i.e. there is e > 0 such that

F eðNnilÞ ¼ 0, see [19], Proposition 1.11.

(b) Up to nilpotence, N has finite length, i.e. every increasing chain of submodules

of N eventually becomes nil-constant (i.e. successive quotients are nilpotent). N has a nil-

decomposition series, see [27], Theorem 4.7.

(c) Up to nilpotence, N has only finitely many R½F �-submodules. More precisely, N has

only finitely many submodules for which the action of F on the quotient is injective.

Proof. All the statements follow via the just described duality from the correspond-ing statements for coherent Cartier modules (Proposition 2.14, Theorem 4.6, Corollary 4.20and Remark 4.21). r

In [14], Enescu and Hochster derive under certain purity and Gorenstein conditionsthe finiteness of the actual number (and not just their number up to nilpotence) of R½F �-submodules of the top local cohomology module H d

mðRÞ. We will give a simple version ofthis for Cartier modules, and show how this implies a slight generalization of results in [14],

9) By co-finite we mean here Artinian as an R-module.


see [14], Discussion 4.5. This is directly related to Remark 4.12 on compatibly split sub-varieties.

Proposition 5.4. Let R be F-finite and F-split, i.e. there is a map C : F e�R! R split-

ting the Frobenius F : R! F e�R. Then the Cartier module ðR;CÞ has only finitely many

Cartier submodules.

Proof. Let I LR be a Cartier submodule, i.e. an ideal such that CðIÞL I . Thensince C splits F we have I ¼ C

�FðIÞ

�¼ CðI ½qe�ÞLCðIÞL I . This shows that any Cartier

submodule has the property that CðIÞ ¼ I , hence the claim follows by Corollary 4.20. r

In analogy with a definition in [14], we say that a Cartier module ðM;CÞ is anti-

nilpotent if for all Cartier submodules N LM, we have CðNÞ ¼ N. For such Corollary4.20 yields:

Proposition 5.5. An anti-nilpotent Cartier module has only finitely many Cartier sub-

modules.10)

It is not di‰cult to check that anti-nilpotent is equivalent to each of the followingconditions.11) (a) M does not have non-trivial nilpotent Cartier sub-quotients. (b) M doesnot have non-trivial Cartier sub-quotients with zero structural map. (c) M and all itsCartier submodules and quotients are minimal. (d) M and all its Cartier submodules andquotients are anti-nilpotent.

Specializing to the case that X ¼ Spec R with R a complete local ring, we obtain fromProposition 5.4 via Matlis duality in Proposition 5.2 that a coherent Cartier module M isanti-nilpotent if its Matlis dual M4 is an anti-nilpotent left R½F �-module, meaning that M4

has no nilpotent R½F �-sub-quotients, cf. [14], Definition 4.7. We obtain the following exten-sion of [14], Corollary 4.17.

Proposition 5.6. Let ðR;mÞ be complete, local, F-finite, and F-split.

(a) The injective hull of the residue field ER has some left R½F �-structure for which ER

is anti-nilpotent.

(b) If R is also quasi-Gorenstein (i.e. H dmðRÞGER), then the top local cohomology

module H dmðRÞ with its canonical left R½F �-module structure is anti-nilpotent and hence

by Proposition 5.5 has only finitely many R½F �-submodules, cf. [14], Theorem 3.7, Corollary

4.17.

Proof. For part (a), let C : F�R! R be a splitting of the Frobenius. In the proofof Proposition 5.4 it was observed that the Cartier module ðR;CÞ is anti-nilpotent. The

10) The converse does not hold, as the example of a simple OX -module with zero structural map shows.

11) Anti-nilpotent means that all Cartier submodules have surjective structural map. The surjectivity of

the structural map, however, always passes to Cartier module quotients, hence if M is anti-nilpotent, then all

sub-quotients have surjective structural map. In particular there are no nilpotent sub-quotients, cf. [14], Prop-

osition 4.6.


duality of Proposition 5.2 induces an R½F �-module structure C4 : ER ! F�ER and the anti-nilpotence of ðR;CÞ immediately translates into the anti-nilpotence of ðER;C

4Þ.

Let us show (b). Abbreviating H ¼ H dmðRÞ, we have H ¼ H d

mðRÞGER since R isquasi-Gorenstein. Part (a) shows that ðH;C4Þ is an anti-nilpotent left R½F �-module. Atthe same time, by the functoriality of local cohomology the Frobenius on R inducesanother (natural) Frobenius action FH : H ! F�H. We claim that C4 factors through FH ,i.e., that there is an r A R such that C4¼ r � FH . Suppose that this claim holds. Then everyR½F �-submodule N LH under the action coming from FH is also an R½F �-submodule of H

under the action coming from C4, because C4ðNÞ ¼ rFHðNÞL rN LN. Hence the sub-quotients of H for the FR-action are a subset of those for the C4-action. By (a) none ofthe non-zero subquotients in the latter set is nilpotent for the action of C4. The factoriza-tion C4¼ rFH implies the same for the subquotients in the former set for the action of FH ,proving the anti-nilpotence of ðH;FHÞ.

It remains to show the factorization C4¼ rFH . Since the top local cohomology isthe cokernel of an appropriate Cech complex one easily observes that the adjoint (underadjointness for F � and F�) of FH is an isomorphism, see [4], Example 2.7. If y 0 : F �H ! H

denotes the adjoint of C4, the composition y 0 � y�1 is an R-linear endomorphism of H.Since R is complete, and H GER we have that y 0 � y�1 is given by multiplication byan element r A R. Hence y 0 ¼ ry which implies for their adjoints the claimed equalityC4¼ rFH . r

5.2. Lyubeznik’s F-finite modules. The connection of our theory of Cartier moduleswith Lyubeznik’s theory of F-finite modules is via his notion of roots, or generators.This has been worked out in quite some detail in [4] where the first author introducesa category of g-sheaves (corresponding to Lyubeznik’s generators) and shows that thecategory of g-crystals (g-sheaves modulo nilpotence) is equivalent to Lyubeznik’s F-finitemodules. In this section we show that under a reasonable hypothesis (X is regular,F -finite and su‰ciently a‰ne) the category of Cartier modules is equivalent to the cate-gory of g-sheaves and the equivalence preserves nilpotence. This is a slight variant of workin [7] where the case X regular and essentially of finite type over an F -finite field istreated.

5.2.1. Cartier modules and g-sheaves. In [4] the first author introduced—motivatedby Lyubeznik’s concept of a root in [27]—the category of g-sheaves. For a regular andF -finite scheme X , this is the category consisting of quasi-coherent OX -modules N

equipped with an OX -linear map g : N ! F �N. The theory develops quite analogouslyas in the case of Cartier modules, taking the viewpoint that a Cartier module is given bya map k : M ! F [M and replacing F [ by F �. In particular, there is a notion of nil-potence, meaning that g i ¼ 0 for some i, where g i ¼ F �g i�1 � g. Nilpotent g-sheaves forma Serre sub-category of the abelian category of g-shaves, there is an abelian category ofg-crystals and so forth. A reason why one has to assume regularity for X in the context ofg-sheaves is because one needs the exactness of F � to guarantee that the kernel of a mapof g-sheaves naturally is also a g-sheaf. In the case of Cartier modules, the exactness ofF� holds in general since F is an a‰ne morphism. The passage from g-sheaves to Cartiermodules is achieved by tensoring with the dualizing sheaf oX . We recall the necessaryfacts below.


Lemma 5.7. Let f : Y ! X be a finite flat morphism and M a quasi-coherent

OX -module, then there is a functorial isomorphism

f [OX nOYf �M !G f [M:

Proof. One has an OY -linear map given by sending a section jnOYsnOX

m to themap sending a section r A OY to jðsrÞ �m. To verify that it is an isomorphism we mayassume that Y ¼ Spec S, X ¼ Spec R are a‰ne and S is a finitely generated and freeR-module. Then everything comes down to checking that the homomorphism

HomRðS;RÞnR Mjnn 7! ðr 7!jðrÞnÞ��! HomRðS;MÞ

is bijective, which is easily verified since S is finite and free over R. r

Applying this lemma to M ¼ oX , a dualizing sheaf, we obtain as an immediate cor-ollary:

Corollary 5.8. Let f : Y ! X be finite and flat, and suppose that oX is invertible.

Then

f [OX G f [oX n f �o�1X ;

f [ðoX nMÞG f [oX n f �M;

f �ðo�1X nMÞG ð f [oX Þ�1 n f [M:

If we assume in addition that f [oX GoY , then

f [OX ¼ oY=X the relative dualizing sheaf ;

f [ðoX nMÞGoY n f �M;

f �ðo�1X nMÞGo�1

Y n f [M:

Note that the additional assumption that f [oX GoY is satisfied if X is normal andeither X is essentially of finite type over a local Gorenstein ring, or X is su‰ciently a‰ne,as explained in Section 2.4.

Theorem 5.9. Let X be regular and F-finite, and assume that there is a dualizing sheaf

oX such that F [oX GoX . Then the category of Cartier modules on X is equivalent to the

category of g-sheaves on X.

The equivalence is given by tensoring with o�1X , its inverse by tensoring with oX . This

preserves coherence, nilpotence, and nil-isomorphism, and hence induces an equivalence be-

tween Cartier crystals and g-crystals.

Proof. Since, if X is regular, the Frobenius F is flat by [25], we may apply the pre-ceding corollary to obtain isomorphisms

o�1X nF [M GF �ðo�1

X nMÞ and oX nF �N GF [ðoX nNÞ:


This shows that if M !k F [M is a Cartier module, then

o�1X nM ��!o�1

Xnk

o�1X nF [M GF �ðo�1

X nMÞ

gives o�1X nM a natural structure of a g-sheaf. Conversely, if N !g F �N is a g-sheaf, then

oX nN ��!oXngoX nF �N GF [ðoX nNÞ

equips oX nN with the structure of a Cartier module. It is immediate that these two oper-ations are inverse to one another. By functoriality, nilpotence is clearly preserved (nil-potence is k i ¼ 0, resp. g i ¼ 0, which is preserved by a functor) and since tensoring with alocally free module is exact, nil-isomorphisms are also preserved. Hence one gets an in-duced equivalence on the level of crystals. r

As a corollary of this equivalence our Theorem 3.10 yields the main result of [4] onthe existence of minimal g-sheaves.

Theorem 5.10 ([4], Theorem 2.24). Let X be regular and F-finite. For each coherent

g-sheaf M there is a unique ( functorial) minimal g-sheaf Mmin which is nil-isomorphic to M.

Proof. Minimality for g-sheaves is defined in the same way as for Cartier modules,namely, a g-sheaf is called minimal if it has neither nilpotent submodules or quotients. Sincethe equivalence in Theorem 5.9 is exact and preserves nilpotence it follows immediatelythat minimality is also preserved. It was observed in [4], Lemma 2.17, that minimality forg-sheaves localizes, and that minimal g-sheaves are unique in their nil-isomorphism class[4], Proposition 2.25. It follows that one can reduce the proof of the existence to the mem-bers of a finite a‰ne cover of X . Hence we may assume that X is a‰ne and furthermorethat F [oX GoX . In this situation we may apply Theorem 5.9 to Theorem 3.10 to derivethe result. r

As a translation of our finite length result Theorem 4.6 for coherent Cartier crystalsand of Corollary 4.16 we get the following statement for coherent g-sheaves.

Theorem 5.11. Let X be regular and F-finite. Then every coherent g-sheaf has, up to

nilpotence, finite length in the category of g-sheaves. In other words, the category of g-crystals

is Artinian (DCC) and Noetherian (ACC).

Moreover, the endomorphism ring of any simple non-zero g-crystal is a finite field con-

taining Fq.

Proof. The Noetherianness is clear, since already as OX -modules a coherent g-sheafsatisfies the ascending chain condition on submodules (we always assume that X is locallyNoetherian). Hence we have to show that any descending chain of g-sheaves stabilizes(up to nilpotence). But this may be checked on a finite a‰ne cover. Hence we may assumethat X is a‰ne and furthermore one has an isomorphism F [oX GoX . This enables us toemploy Theorem 5.9 which reduces the statement to the finite length result for coherentCartier modules (up to nilpotence) shown in Theorem 4.6.

For the last assertion observe first that, by the lemma of Schur, the endomorphismring of a simple non-nilpotent g-crystal is a skew field. Next note that any endomorphism


on X induces endomorphisms on the restrictions to any open of a fixed finite a‰ne cover.On su‰ciently small a‰ne open sets we can apply Theorem 5.9, and so there the endomor-phism rings are finite. But the map sending global endomorphisms to its restrictions on ana‰ne cover is clearly injective. r

5.2.2. Finitely generated unit R½F �-modules. There is a functor from (coherent)g-sheaves to the category of (locally finitely generated) unit OX ½F �-modules. This latter cat-egory was introduced in [27], under the name of F-finite modules, in the regular a‰ne case,and in [13] for regular schemes of finite type over a field. It consist of OX -quasi-coherent leftOX ½F �-modules M which are locally finitely generated over OX ½F �, and such that the adjointmap to the F -action

y : F �M!M

is an isomorphism. Already in [27] it was observed that there is a functor from coherentg-sheaves to finitely generated unit OX ½F �-modules. This functor, denoted Gen, sends ag-sheaf M !g F �M to the limit GenðMÞ of the directed system

M �!g F �M �!F�g

F 2�M �! � � � :

It is shown in [4] that this functor induces an equivalence of categories from g-crystals tofinitely generated unit OX ½F �-modules.

Proposition 5.12 ([4], Theorem 2.7). Let X be regular and F-finite. Then the functor

Gen from coherent g-sheaves on X to finitely generated unit OX ½F �-modules induces an equiv-

alence

fg-crystalsg !F f f:g: unit OX ½F �-modulesg:

As an immediate application of Proposition 5.12 and Theorem 5.11 we obtain the fol-lowing generalization of the main result of [27], Theorem 3.2:

Theorem 5.13. Let X be regular and F-finite, then every finitely generated unit OX ½F �-module has finite length in the category of unit OX ½F �-modules. Moreover the endomorphism

ring of any simple finitely generated unit OX ½F �-module is a finite field containing Fq.

Remark 5.14. Lyubeznik shows in [27], Theorem 3.2, that the finite length forfinitely generated unit OX ½F �-modules holds in the case that X is regular and essentiallyof finite type over a regular local ring. So the result just given extends this to all regularF -finite schemes, but does not completely recover Lyubeznik’s result. Important caseswhich are covered by [27], Theorem 3.2, but not our result is that of a finite type schemeX over a field with ½k : kp� ¼y, or X ¼ Spec R with R local but not F -finite. We suspectthat the main result in this paper, the finite length of Cartier crystals, Theorem 4.6, will alsohold in these cases. However, our proof, as well as the transition from Cartier modules tofinitely generated unit OX ½F �-modules, is closely tied to the F -finiteness. There seems to besome di¤erent techniques necessary to obtain these results.

Note also that in combining Theorem 5.9 with the above quoted [4], Theorem 2.7, weobtain


Theorem 5.15. Let X be regular and F-finite, and assume that there is a dualizing

sheaf oX such that F [oX GoX . Then the category of Cartier modules on X is equivalent to

the category of finitely generated unit OX ½F �-modules.

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Institut fur Mathematik, FB 08, Johannes Gutenberg-Universitat Mainz, 55099 Mainz, Germany

e-mail: [email protected]

Interdisziplinares Zentrum fur wissenschaftliches Rechnen, Universitat Heidelberg, 69120 Heidelberg, Germany

e-mail: [email protected]

Eingegangen 17. September 2009, in revidierter Fassung 15. Juli 2010


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Cartier modules: Finiteness results - Heidelberg University · Cartier modules: Finiteness results By Manuel Blickle at Mainz and Gebhard Bo¨ckle at Heidelberg Abstract. On a locally