Open Access

Cohomological descent on the overconvergent site

Research in the Mathematical Sciences20141:8

DOI: 10.1186/s40687-014-0008-3

Received: 12 February 2014

Accepted: 12 June 2014

Published: 21 August 2014

Abstract

We prove that cohomological descent holds for finitely presented crystals on the overconvergent site with respect to proper or fppf hypercovers.

Background

Cohomological descent is a robust computational and theoretical tool, central to p-adic cohomology and its applications. On the one hand, it facilitates explicit calculations (analogous to the computation of coherent cohomology in scheme theory via Čech cohomology); on the other, it allows one to deduce results about singular schemes (e.g., finiteness of the cohomology of overconvergent isocrystals on singular schemes [1]) from results about smooth schemes, and, in a pinch, sometimes allows one to bootstrap global definitions from local ones (for example, for a scheme X which fails to embed into a formal scheme smooth near X, one actually defines rigid cohomology via cohomological descent; see [2], comment after Proposition 8.2.17).

The main result of the series of papers [3]–[5] is that cohomological descent for the rigid cohomology of overconvergent isocrystals holds with respect to both flat and proper hypercovers. The barrage of choices in the definition of rigid cohomology is burdensome and makes their proofs of cohomological descent very difficult, totaling to over 200 pages. Even after the main cohomological descent theorems [3] Theorems 7.3.1 and 7.4.1 are proved, one still has to work a bit to get a spectral sequence [3], Theorem 11.7.1. Actually, even to state what one means by cohomological descent (without a site) is subtle.

The situation is now more favorable. More than 25 years after Berthelot’s seminal papers [6]–[8], key foundational aspects have now been worked out. Le Stum’s recent advance [9] is the construction of an ‘overconvergent site’ [9] which gives an alternative, equivalent definition of rigid cohomology as the cohomology of the structure sheaf of a ringed site ( X AN , O X ) (and also of course an equivalence between the category of overconvergent isocrystals on X and the category of finitely presented O X -modules). This formalism is the correct setting for many problems; for instance, [10] uses the overconvergent site to develop a theory of constructible -modules on curves.

More applications are expected. And indeed, the main result of this paper is the application of the abstract machinery of [11], Exposé Vbis and VI to the overconvergent site to give a short proof of the following.

Theorem 1.1.

Cohomological descent for locally finitely presented modules on the overconvergent site holds with respect to

fppf hypercovers of varieties and

proper hypercovers of varieties.

Remark 1.2.

(New ideas and comparison to prior work) Classical results in rigid cohomology have no access to site theoretic techniques and instead one works directly with a choice of embedding into a formal scheme. This adds difficulties (again we highlight the length of [3]–[5]) and restricts the applicability of results (e.g., generalization to stacks; see Remark 1.1 (3)).

On the other hand, Theorem 1.1 is not merely a formal consequence of the techniques of [11], Exposé Vbis and VI. Cohomological descent for abelian sheaves on the étale site with respect to smooth hypercovers is simply the Čech theory (as in Theorem 3.1 (i)). In the overconvergent setting, an étale surjection is not a covering; and hence, the Čech theory does not apply. Another technical difficulty is that one cannot check triviality of an overconvergent sheaf F AN X by restricting to points of X, so that the template of the proof of proper cohomological descent for étale cohomology therefore does not apply to overconvergent cohomology, and a new argument is needed.

Moreover, we emphasize that, while similar-looking results appear in the literature (see e.g. section Zariski covers and Lemma 5.14), these results are not powerful enough to directly imply cohomological descent for the overconvergent site. New tools and ideas – the use of techniques from [11] and for instance, the use of Raynaud-Gruson’s theorem on ‘flattening stratifications’ [12], Théorèm 5.2.2, and le Stum’s main theorems (e.g. Proposition 4.17) - greatly simplify and extend the generality of our proof, and as an additional indication of this, we remark that (in contrast to cohomological descent for étale or rigid cohomology) the proofs of the étale and proper cases are intertwined.

Finally, in light of the central role that one expects le Stum’s work to play in the future development of rigid cohomology, we note that various ingredients of our proofs are useful lemmas which facilitate computations on the overconvergent site; see for instance Lemma 5.2.

Applications

We highlight a few direct applications of our main theorem.
  1. 1.

    (Spectral sequence). By le Stum’s comparison theorems between rigid and overconvergent cohomology [9], Corollary 3.5.9, we obtain a spectral sequence (see Remark 3.3) computing rigid cohomology; this gives a shorter proof of Theorem 11.7.1 of [3]. While this corollary of our work and [3],[4] are similar, the main results are independent and cannot be deduced from one another.

     
  2. 2.

    (Overconvergent de Rham-Witt cohomology). A recent result [13] proves directly that overconvergent de Rham-Witt cohomology agrees with classical rigid cohomology for smooth affine varieties, and a long argument with dagger spaces is needed to deduce agreement with general rigid cohomology. Use of the overconvergent site and Theorem 1.1 simplifies the globalization argument (this will appear in future work (Davis and Zureick-Brown: Overconvergent de Rham-Witt cohomology for stacks, in preparation); the main theorem is that the étale cohomology of each overconvergent de Rham-Witt sheaf vanishes on affines, and the globalization argument is then a direct application of Theorem 1.1 and the spectral sequence 3.3).

     
  3. 3.

    (Rigid cohomology for stacks.) Motivated by applications to geometric Langlands, Kedlaya proposed the problem of generalizing rigid cohomology to stacks. There are three approaches – le Stum’s site gives (via Definition 4.5) a direct approach realized in the author’s thesis [14]); the overconvergent de Rham-Witt complex of [13] gives an alternative, explicit and direct construction. Theorem 1.1 gives a third approach and a direct comparison of the first two approaches; moreover, Theorem 1.1 also gives a direct proof that the rigid cohomology of a stack is finite dimensional, and allows one to make various constructions (e.g., to define a Gysin map). Rigid cohomology for stacks is most naturally defined via the overconvergent site (one can take the same definition for a stack as for a scheme); cohomological descent for the overconvergent cohomology of a stack follows from the case of schemes by standard arguments, and both of these applications (finiteness and the ability to bootstrap constructions from the case of schemes) follow directly from the spectral sequence.

     

Organization of the paper

This paper is organized as follows: In section Notation and conventions, we recall notation. In section Simplicial methods, we review the machinery of cohomological descent. In section The overconvergent site, we recall the construction of the overconvergent site of [9]. In section Cohomological descent for overconvergent crystals, we prove Theorem 1.1, first in the case of Zariski hypercovers, then in the case of fppf hypercovers, and finally for proper hypercovers.

Notation and conventions

Throughout, K will denote a field of characteristic 0 that is complete with respect to a non-trivial non-archimedean valuation with valuation ring , whose maximal ideal and residue field we denote by 𝔪 and k. We denote the category of schemes over k by S c h k . We define an algebraic variety overk to be a scheme such that there exists a locally finite cover by schemes of finite type over k (recall that a collection of subsets of a topological space X is said to be locally finite if every point of X has a neighborhood which only intersects finitely many subsets X S ). Note that we do not require an algebraic variety to be reduced, quasi-compact, or separated.

Formal schemes: As in [9], 1.1 we define a formal -scheme to be a locally topologically finitely presented formal scheme P over , i.e., a formal scheme P with a locally finite covering by formal affine schemes Spf A, with A topologically of finite type (i.e., a quotient of the ring V { T 1 , , T n } of convergent power series by an ideal I + 𝔞 V { T 1 , , T n } , with I an ideal of V { T 1 , , T n } of finite type and 𝔞 an ideal of ). This finiteness property is necessary to define the ‘generic fiber’ of a formal scheme (see [15], Section 1).

We refer to [16], 1.10 for basic properties of formal schemes. The first section of [17] is another good reference; a short alternative is [9], Section 1, which contains everything we will need.

K-analytic spaces: We refer to [18] (as well as the brief discussion in [9], 4.2) for definitions regarding K-analytic spaces. As in [9], 4.2, we define an analytic variety over K to be a locally Hausdorff topological space V together with a maximal affinoid atlas τ which is locally defined by strictly affinoid algebras (i.e., an algebra A is strict if it is a quotient of a Tate algebra K{T1,,T n }) and denote by ( A ) the Gelfand spectrum of an affinoid algebra A. Moreover, recall that a K-analytic space is said to be good if every point has an open affinoid neighborhood.

Topoi: We follow the conventions of [19] (exposited in [9], 4.1) regarding sites, topologies, topoi, and localization. When there is no confusion, we will identify an object X of a category with its associated presheaf h X :YHom(Y,X). For an object X of category C, we denote by C/X the comma category - objects of C/X are morphisms YX, and morphisms are commutative diagrams - and by C / X ~ (resp. C / X ̂ ) the associated topos (resp. category of presheaves on C/X). Often (in this paper), a morphism ( f 1 , f ) : ( T , O T ) ( T , O T ) of ringed topoi will satisfy f 1 O T = O T , so that there is no distinction between the functors f−1 and f; in this case, we will write f for both. Finally, we note that the category Mod fp O T of O T -modules which locally admit a finite presentation i = 1 n O T i = 1 m O T M , is generally larger than Coh O T , since in general, the sheaf of rings O T is not itself coherent.

Simplicial methods

Here we recall the setup and main results of cohomological descent. The standard reference is [11], Exposé Vbis and VI; some alternatives are Deligne’s paper [20] and Conrad’s notes [21]; the latter has a lengthy introduction with a lot of motivation and gives more detailed proofs of some theorems of [11] and [20].

A morphism p0:XY of presheaves on a site C induces a morphism C / Y ~ C / X ~ of topoi. Setting X n to be the (n+1)-fold fiber product of p0 and p ij k : X i X j to be the k th map induced by composing projections, p0 also induces a morphism of topoi
p 1 , p : C / Y ~ C / X ~ ,
(1)
where C / X ~ is (equivalent to) the category of simplicial sheaves, i.e., collections of pairs F : = F n C / X n ~ , ι ij k : p ij k 1 F j F i (where the ι ij k ’s are maps of sheaves satisfying the usual cocycle compatibilities). The functors (p−1,p) are
p F : = eq p ( 0 ) F 0 p 1 F 1 , ( p 1 F ) n : = ( p ij k ) 1 F
(2)

with ι ij k the canonical isomorphisms; there is a clear natural map F p p F . Since C / X ~ is actually the topos of the simplicial site C / X , to F C / X ~ , we can associate the derived pushforward p F as well as the i th relative cohomology i p F and absolute cohomology H i ( C X , F ) .

We say that p0:XY is of cohomological descent (resp. universally of cohomological descent) with respect to F C / Y ~ if F p p F (resp. if for every f:YY, the map X× Y YY is of cohomological descent with respect to f F ). Similarly, we say that p0 is (universally) of cohomological descent with respect to a subcategory C Ab C / Y if p0 is (universally) of cohomological descent with respect to every F C .

Theorem 3.1.

  1. 1.

    Suppose that p 0:XY in C ̂ sheafifies to a covering in C ~ . Then p 0 is universally of cohomological descent. Moreover, for F Ab Y ~ , the map Fp p F is a quasi-isomorphism.

     
  2. 2.

    Any morphism in C ̂ which has a section locally (in C ~ ) is universally of cohomological descent.

     
  3. 3.

    Consider a Cartesian diagram

     
in C ̂ and let F C / S ~ be a sheaf of abelian groups. Suppose π0 is universally of cohomological descent with respect to . Then f0 is universally of cohomological descent with respect to if and only if f 0 is universally of cohomological descent with respect to π 0 F .
  1. 4.

    Let f:XY and g:YZ be maps in C and let F Z ~ be a sheaf of abelian groups. Suppose that the composition XZ is universally of cohomological descent with respect to . Then YZ is as well.

     
  2. 5.

    Let f:XY and g:YZ be maps in C and let F Z ~ be a sheaf of abelian groups. If g is universally of cohomological descent with respect to and f is universally of cohomological descent with respect to g F , then the composition gf is universally of cohomological descent with respect to .

     
  3. 6.

    Let f i :X i Y i be maps in C indexed by some arbitrary set I. For each iI let F i Y i ~ be a sheaf of abelian groups. Suppose that for each i, f i is of cohomological descent relative to F i . Then f i : X i Y i is of cohomological descent relative to F i (where disjoint unions are taken in C ̂ ).

     

Proof.

Statement (1) is ([22], Lemma 1.4.24), (2) follows from (1) since any morphism with a section is a covering in the canonical topology. The proofs of (3) to (5) are identical to the proof of Theorem [21], Theorem 7.5, and (6) follows from the fact that, setting p 0 = f i , the induced morphism of simplicial topoi
( p 1 , p ) : X i ~ Y i ~
(3)
is also a morphism of topoi fibered over I, so that in particular, the natural map
F i p p F i
(4)
is an isomorphism if and only if, for all iI, the map F i f i f i F i is an isomorphism.

Corollary 3.2.

Let

be a commutative diagram in C ̂ and let F C / Y ~ . Suppose that is universally of cohomological descent with respect to Y i Y and that for each i, v i F is universally of cohomological descent with respect to X i Y i . Then is universally of cohomological descent with respect to XY.

Proof.

This follows immediately from Theorem 3.1 (4), (5), and (6).

Remark 3.3.

(Spectral sequence) If F p p 1 F , then composing derived functors gives
ℝΓ ( F ) ℝΓ p p 1 F = ( Γ p ) p 1 F = ℝΓ p 1 F ;
(5)
in particular, there is a spectral sequence
H j ( C / X i , F i ) H i + j ( C / Y , F ) .
(6)

Corollary 3.4.

Let p0:XY be an and suppose that F Y ~ is an abelian sheaf such that
  1. (1)

    for i≥0 and j>0, j p i p i F = 0 , and

     
  2. (2)

    F p p F is a quasi-isomorphism.

     

Then F p p F is a quasi-isomorphism.

Proof.

This follows from the spectral sequence.

Remark 3.5.

(Hypercoverings). We omit discussion of more general P-hypercovers except to note that by [21], Theorem 7.10, our results immediately extend to étale hypercovers and proper hypercovers.

The overconvergent site

In this section, we recall the setup and results of [9].

Definition 4.1.

([9], 1.2). Define an overconvergent variety over to be a pair ( X P , V λ P K ) , where XP is a locally closed immersion of an algebraic variety X over k into the special fiber P k of a formal scheme P (recall our convention that all formal schemes are topologically finitely presented over Spf V ), and V λ P K is a morphism of analytic varieties, where P K denotes the generic fiber of P, which is an analytic space. When there is no confusion, we will write (X,V) for ( X P , V λ P K ) and (X,P) for ( X P , P K id P K ) . Define a formal morphism (X,V)→(X,V) of overconvergent varieties to be a commutative diagram

where f is a morphism of algebraic varieties, v is a morphism of formal schemes, and u is a morphism of analytic varieties.

Finally, define AN ( V ) to be the category whose objects are overconvergent varieties and morphisms are formal morphisms. We endow AN ( V ) with the analytic topology, defined to be the topology generated by families {(X i ,V i )→(X,V)} such that for each i, the maps X i X and P i P are the identity maps, V i is an open subset of V, and V = V i is an open covering (recall that an open subset of an analytic space is admissible in the G-topology and thus also an analytic space – this can be checked locally in the G-topology, and for an affinoid this is clear because there is a basis for the topology of open affinoid subdomains).

Definition 4.2([9], Section 1.1).

The specialization map P K P k induces by composition a map VP k and we define the tube ]X[ V of X in V to be the preimage of X under this map. The tube ] X [ P K admits the structure of an analytic space and the inclusion i X : ] X [ P K P K is a locally closed inclusion of analytic spaces (and generally not open, in contrast to the rigid case). The tube ]X[ V is then the fiber product ] X [ P K × P K V (as analytic spaces) and in particular is also an analytic space. A formal morphism (f,u):(X,V)→(X,V) induces a morphism ] f [ u : ] X [ V ] X [ V of tubes. (Since ]f[ u is induced by u, when there is no confusion, we will sometimes denote it by u).

The fundamental topological object in rigid cohomology is the tube ]X[ V , in that most notions are defined only up to neighborhoods of ]X[ V . To make this precise, modify AN ( V ) .

Definition 4.3([9], Definition 1.3.3).

Define a formal morphism
( f , u ) : ( X , V ) ( X , V )
(7)
to be a strict neighborhood if f and ]f[ u are isomorphisms and u induces an isomorphism from V to a neighborhood W of ]X[ V in V.

Definition 4.4.

We define the category AN ( V ) of overconvergent varieties to be the localization of AN ( V ) by strict neighborhoods (which is possible by [9], Proposition 1.3.6): the objects of AN ( V ) are the same as those of AN ( V ) and a morphism (X,V)→(X,V) in AN ( V ) is a pair of formal morphisms
( X , V ) ( X , W ) ( X , V ) ,
(8)
where (X,W)→(X,V) is a strict neighborhood.

The functor AN ( V ) AN ( V ) induces the image topology on AN ( V ) (i.e., the largest topology on AN ( V ) such that the map from AN ( V ) is continuous). By [9], Proposition 1.4.1, the image topology on AN ( V ) is generated by the pretopology of collections {(X,V i )→(X,V)} with V i an open covering of a neighborhood of ]X[ V in V and ] X [ V = ] X [ V i .

Definition 4.5.

For any presheaf T AN ( V ) ̂ , we define AN (T) to be the localized category AN ( V ) / T whose objects are morphisms h(X,V)T (where h(X,V) is the presheaf associated to (X,V)) and morphisms are morphisms (X,V)→(X,V) which induce a commutative diagram

We may endow AN (T) with the induced topology (i.e., the smallest topology making continuous the projection functor AN ( T ) AN ( V ) ; see [9], Definition 1.4.7); concretely, the covering condition is the same as in 4.4. When T=h(X,V) we denote AN (T) by AN (X,V). Since the projection AN T AN V is a fibered category, the projection is also cocontinuous with respect to the induced topology. Finally, an algebraic space X over k defines a presheaf (X,V)Hom(X,X), and we denote the resulting site by AN (X).

There will be no confusion in writing (X,V) for an object of AN (T).

We use subscripts to denote topoi and continue the above naming conventions – i.e., we denote the category of sheaves of sets on AN (T) (resp. AN (X,V),AN (X)) by T AN (resp. ( X , V ) AN , X AN ). Any morphism f:TT of presheaves on AN ( V ) induces a morphism f AN : T AN T AN of topoi. For a morphism (f,u):(X,V)→(X,V) of overconvergent varieties, we denote the induced morphism of topoi by ( u AN , u AN ) .

For an analytic space V we denote by OpenV the category of open subsets of V and by Van the associated topos of sheaves of sets on OpenV. Recall that for an analytic variety (X,V), the topology on the tube ]X[ V is induced by the inclusion i X :]X[ V V.

Definition 4.6([9], Corollary 2.1.3).

Let (X,V) be an overconvergent variety. Then there is a morphism of sites
φ X , V : AN ( X , V ) Open ] X [ V .
(9)
The notation as usual is in the ‘direction’ of the induced morphism of topoi and in particular backward; it is associated to the functor Open ]X[ V →AN (X,V) given by U=W∩ ]X[ V (X,W) (and is independent of the choice of W up to strict neighborhoods). This induces a morphism of topoi
( φ X , V 1 , φ X , V ) : ( X , V ) AN ( ] X [ V ) an .
(10)

Definition 4.7([9], 2.1.7).

Let (X,V)AN (T) be an overconvergent variety over T and let F T AN be a sheaf on AN (T). We define the realizationFX,V of F on ]X[ V to be φ ( X , V ) ( F | ( X , V ) AN ) , where F | ( X , V ) AN is the restriction of F to AN (X,V).

We can describe the category T AN in terms of realizations in a manner similar to sheaves on the crystalline or lisse-étale sites.

Proposition 4.8 ([9], Proposition 2.1.8).

Let T be a presheaf on AN ( V ) . Then the category T AN is equivalent to the following category:
  1. (1)

    An object is a collection of sheaves F X,V on ]X[ V indexed by (X,V)AN (T) and, for each (f,u):(X ,V )→(X,V), a morphism ϕ f , u : ] f [ u 1 F X , V F X , V , such that as (f,u) varies, the maps ϕ f,u satisfy the usual compatibility condition.

     
  2. (2)

    A morphism is a collection of morphisms F X,VG X,V compatible with the morphisms ϕ f,u.

     

To obtain a richer theory, we endow our topoi with sheaves of rings and study the resulting theory of modules.

Definition 4.9([9], Definition 2.3.4).

Define the sheaf of overconvergent functions on AN ( V ) to be the presheaf of rings
O V : ( X , V ) Γ ( ] X [ V , i X 1 O V )
(11)
where i X is the inclusion of ]X[ V into V; this is a sheaf by [9], Corollary 2.3.3. For T AN ( V ) ̂ a presheaf on AN ( V ) , define O T to be the restriction of O V to AN (T). Following our naming conventions above, we denote by O ( X , V ) the restriction of O V to AN(X,V).

Remark 4.10.

By [9],Proposition 2.3.5, (i), the morphism of topoi of Definition 4.6 can be promoted to a morphism of ringed sites
( φ X , V , φ X , V ) : ( AN ( X , V ) , O ( X , V ) ) ( ] X [ V , i X 1 O V ) .
(12)
In particular, for (X,V)AN T and M O T , the realization MX,V is an i X 1 O V -module. For any morphism (f,u):(X,V)→(X,V) in AN (T), one has a map
( ] f [ u , ] f [ u ) : ( ] X [ V , i X , V 1 O V ) ( ] X [ V , i X , V 1 O V ) .
(13)
of ringed sites, and functoriality gives transition maps
ϕ f , u : ] f [ u M X , V M X , V
(14)
which satisfy the usual cocycle compatibilities.

Definition 4.11([9], Definition 2.3.7).

Define the category of overconvergent crystals onT, denoted Cris T, to be the full subcategory of Mod O T such that the transition maps ϕ f , u are isomorphisms.

Example 4.12.

The sheaf O T is a crystal, and in fact QCoh O T Cris T . More generally, the pair φ X , V and φX,V induces (see [9], Proposition 2.3.8) an equivalence of categories
Cris ( X , V ) Mod i X 1 O V .
(15)

One minor subtlety is the choice of an overconvergent variety as a base.

Definition 4.13.

Let ( C , O ) AN ( V ) be an overconvergent variety and let TC be a morphism from a presheaf on S c h k to C. Then T defines a presheaf on AN (C,O) which sends (X,V)→(C,O) to Hom C (X,T), which we denote by T/O. We denote the associated site by AN (T/O), and when (C,O)=(S k ,S) for some formal -scheme S, we write instead AN (T/S).

The minor subtlety is that there is no morphism Th(C,O) of presheaves on AN ( V ) . A key construction is the following.

Definition 4.14([9], Paragraph after Corollary 1.4.15).

Let ( X , V ) ( C , O ) AN ( V ) be a morphism of overconvergent varieties. We denote by X V /O the image presheaf of the morphism (X,V)→X/O, considered as a morphism of presheaves. Explicitly, a morphism (X,V)→X/O lifts to a morphism (X,V)→X V /O if and only if there exists a morphism (X,V)→(X,V) over X/O, and in particular different lifts (X,V)→(X,V) give rise to the same morphism (X,V)→X V /O. When ( C , O ) = ( Spec k , ( K ) ) , we may write X V instead X V / ( K ) .

Many theorems will require the following extra assumption of [9], Definition 1.5.10. Recall that a morphism of formal schemes PP is said to be proper at a subscheme X P k if, for every component Y of X ¯ , the map YP k is proper (see [9], Definition 1.1.5).

Definition 4.15.

Let ( C , O ) AN ( V ) be an overconvergent variety and let f:XC be a morphism of k-schemes. We say that a formal morphism (f,u):(X,V)→(C,O), written as

is a geometric realization of f if v is proper at X, v is smooth in a neighborhood of X, and V is a neighborhood of ] X [ P K × Q K O in P K × Q K O . We say that f is realizable if there exists a geometric realization of f.

We need a final refinement to AN ( V ) .

Definition 4.16.

We say that an overconvergent variety (X,V) is good if there is a good neighborhood V of ]X[ V in V (i.e., every point of ]X[ V has an affinoid neighborhood in V). We say that a formal scheme S is good if the overconvergent variety (S k ,S K ) is good. We define the good overconvergent site AN g ( T ) to be the full subcategory of AN (T) consisting of good overconvergent varieties. Given a presheaf T AN ( V ) , we denote by Tg the restriction of T to AN g ( V ) .

Note that localization commutes with passage to good variants of our sites (e.g., there is an isomorphism AN g ( V ) / T g AN g ( T ) ). When making further definitions we will often omit the generalization to AN g when it is clear.

The following proposition will allow us to deduce facts about Mod fp O X g from results about (X,V) and X V .

Proposition 4.17.

Let ( C , O ) AN g ( V ) be a good overconvergent variety and let (X,V)→(C,O) be a geometric realization of a morphism XC of schemes. Then the following are true:

The map (X,V)g→(X/O)g is a covering in AN g ( V ) .

There is an equivalence of topoi ( X V / O ) AN g ( X / O ) AN g .

The natural pullback map Cris g X / O Cris g X V / O is an equivalence of categories.

Suppose that (X,V) is good. Then the natural map Cris X V / O Cris g X V / O is an equivalence of categories. In particular, the natural map Mod fp O X g Mod fp O ( X V ) g Mod fp O X V is an equivalence of categories.

Proof.

The first two claims are [9], 1.5.14, 1.5.15, the third follows from the second, and the last is clear.

Technical lemmas

We state here a few technical lemmas that will be useful in the proof of Theorem 1.1.

Lemma 4.19.

Let (Y,W)→(X,V) be a morphism of overconvergent varieties. Let Y=Y× X X and W=W× V X. Then (Y,W)(Y,W(X,V)(Y,V).

Proof.

This is the comment after [9], Proposition 1.3.10.

Lemma 4.20.

Let p:(X,V)→(X,V) be a morphism of overconvergent varieties such that the induced map on tubes is an inclusion of a closed subset. Then p is exact.

Proof.

It suffices to check that, for any Cartesian diagram

the map induced on tubes by p is exact; the lemma follows since for any base change of p, the induced map on tubes is also an inclusion of a closed subset and such maps are exact.

Cohomological descent for overconvergent crystals

In this section, we prove Theorem 1.1. The proof naturally breaks into cases: Zariski covers, modifications, finitely presented flat covers, and proper surjections. The full proof fails without the goodness assumption, but many special cases (e.g., cohomological descent with respect to Zariski hypercovers) hold without the goodness assumption.

Throughout, when discussing cohomological descent, we consider k-varieties, analytic varieties, etc., as presheaves on the overconvergent site.

Zariski covers

We begin with the case of a Zariski cover. One can restate the main result of [9], section 3.6 as the statement that a Zariski covering is universally of cohomological descent with respect to crystals. Throughout this subsection, we omit distinction between AN and AN g , but remark here that each result is true for either site.

Some care is needed to interpret le Stum’s results in the language of cohomological descent; to that end, we first prove a few lemmas that will be useful in later proofs as well.

Lemma 5.2.

Let p0=(f0,u0):(X0,V0)→(X−1,V−1) be a morphism of overconvergent varieties such that

the induced map ] f 0 [ : ] X 0 [ V 0 ] X 1 [ V 1 is an isomorphism, and

the natural map ] f 0 [ 1 i X 1 1 O V 1 i X 0 1 O V 0 is an isomorphism.

Then p0 is universally of cohomological descent with respect to crystals.

Remark 5.3.

Note that condition (ii) of Lemma 5.2 is satisfied if u0 is a finite quasi-immersion and thus in particular is satisfied if u0 is an isomorphism or if V0 is a neighborhood of ] X 1 [ V 1 . (Note that these are non-trivial conditions, since f0 may not be an isomorphism.) Moreover, condition (i) holds if f0 is surjective.

We also note that condition (ii) is necessary; in general, a morphism
( X , ] X [ V ) ( X , V )
(16)
is not universally of cohomological descent for crystals, since the Čech complex is not exact. For example, when X = 𝔸 1 and V = K 1 , condition (ii) fails, and indeed the Čech complex
0 K { t } K { t } 0 K { t }
(17)
is not exact.

Proof.

We check the hypotheses of Corollary 3.4. Denote by p i :(X i ,V i )→(X−1,V−1) the i+1-fold fiber product of the map p0:(X0,V0)→(X−1,V−1). Noting that formation of tubes commutes with base change (and in particular that the map on tubes induced by p i is an isomorphism), it follows from Lemma 4.20 that pi, is exact.

For each i≥0 and j>0, each projection p i j : ( X i , V i ) ( X i 1 , V i 1 ) also induces an isomorphism ] X i [ V i ] X i 1 [ V i 1 on tubes. Moreover, for a fixed i, the maps p i j are all equal. Finally, note that by condition (ii), the natural maps F p i , j p i j , F are all isomorphisms (contrast with Remark 5.3). It follows that the Čech complex F p p F is exact (since the maps alternate between an isomorphism and the zero map). The lemma follows.

Lemma 5.4.

Let {(X i ,V i )→(X,V)} be a collection of morphisms of overconvergent varieties such that each V i V is an open immersion and { ] X i [ V i } is an open covering of ]X[ V . Then the map
u 0 : ( X i , V i ) ( X , V )
(18)
is universally of cohomological descent with respect to crystals.

Proof.

Let W i be an open subset of V such that W i ] X [ V = ] X i [ V i (which exists since] X i [ V i is an open subset of ]X[ V ). Let W i be the preimage of W i′ under the map V i V. By Corollary 3.1 (4), it suffices to prove that the map u 0 : ( X i , W i ) ( X , V ) is universally of cohomological descent with respect to crystals.

Let v i denote the morphism (X i , W i )→(X, V). Then the morphism u 0′ factors as
( X i , W i ) v i ( X , W i ) w 0 ( X , V ) .
(19)

The map w0 is a covering in AN (X,V) and thus universally of cohomological descent by Theorem 3.1 (1). Since, by the construction of W i , the induced map ] X i [ W i ] X [ W i is an isomorphism, it follows from Lemma 5.2 that each map v i is universally of cohomological descent with respect to crystals. By Theorem 3.1 (6), v i is universally of cohomological descent with respect to crystals; by 3.1 (5), the composition is also universally of cohomological descent with respect to crystals and the lemma follows.

Definition 5.5.

We say that a collection {X i } of subspaces of a topological space X is a locally finite covering if X=X i and if each point x of X admits an open neighborhood U x on which {X i U x } admits a finite refinement which covers U x .

Lemma 5.6.

Let {(f i ,u i ):(X i ,V i )→(X,V)} be a collection of morphisms of overconvergent varieties such that

the maps ] u i [ : ] X i [ V i ] X [ V are closed inclusions of topological spaces,

{ ] X i [ V i } is a locally finite covering of ]X[ V , and

for each i, the natural map ] f i [ 1 i X 1 O V i X i 1 O V i is an isomorphism.Then the map
p : ( X i , V i ) ( X , V )
(20)
is universally of cohomological descent with respect to crystals.

Proof.

Let FCris (X,V) be a crystal. By Corollary 3.4, it suffices to prove that (i) q p , p F = 0 for q>0, and (ii) the Čech complex F p , p F is exact. Let p j be the j-fold fiber product of p. By the spectral sequence (Remark 3.3), it suffices to prove that q p j p j F = 0 for j≥0 and q>0; since for each j, p j is a disjoint union of maps which induce closed inclusions on tubes, this follows from Lemma 4.20.

For (ii), it suffices to check that, for every map π:(X,V)→(X,V), the realization with respect to (X,V) of the Čech complex of π−1F with respect to
p : ( X i , V i ) ( X , V )
(21)
(where X i = X i × X X and V i = V i × V V ) is exact, which (noting that our hypotheses are stable under base change) since the tubes form a locally finite closed covering, follows from condition (c) and the proof of [9], Proposition 3.1.4.

Corollary 5.7.

Let (XPV) be an overconvergent variety and let {P i }iI be a collection of Zariski open formal subschemes of P. Let (X i ,V)=(X× P P i PV) and let ( X i , V i ) = ( X × P P i P i V × P K ( P i ) K ) . Suppose that {X i } forms a locally finite Zariski open cover of X. Then the following are true:

The map ( X i , V ) ( X , V ) is universally of cohomological descent with respect to crystals.

Suppose (X,V) is good and that the tubes { ] X i [ V i } cover a neighborhood of ]X[ V in V.

Then ( X i , V i ) ( X , V ) is universally of cohomological descent with respect to finitely presented crystals.

Remark 5.8.

By Remark 5.3, the extra hypothesis on the tubes in claim (2) is necessary.

Proof.

Since specialization is anti-continuous, the tubes form a locally finite closed covering and claim (1) thus follows from Lemma 5.6. For claim (2), since (X,V) is good, we may assume that V is affinoid. The claim then follows by Tate’s Acylicity Theorem [23], 8.2, Corollary 5.

We say that a morphism of schemes XY over k is universally of cohomological descent (resp., with respect to a sheaf F Ab ( Y AN ) ) if the associated morphism X AN Y AN is universally of cohomological descent (resp., with respect to ).

Theorem 5.9.

Let (C,O) be an overconvergent variety and let XC be a morphism of algebraic varieties. Let {U i }iI be a locally finite covering of X by open subschemes (resp., a covering of X by closed subschemes) and denote by α 0 : U = i I U i X the induced morphism of schemes. Denote by α:UX the 0-coskeleton of α0. Then the morphism of topoi U / O AN X / O AN is universally of cohomological descent with respect to .

Proof.

The proof for α AN g is identical to the proof for α AN . We note that the map ( X , V ) X , where the coproduct is taken over AN (X/O), is a covering in the canonical topology on AN (X/O) and thus universally of cohomological descent. Setting U i = X × X U i , the diagram (of sheaves on AN O)

commutes. By Lemma 5.6 (resp., Lemma 5.4) the maps ( U i , V ) ( X , V ) are universally of cohomological descent with respect to crystals; the theorem thus follows from Corollary 3.2.

Remark 5.10.

Let {X i } be a collection of schemes. Then the presheaf on AN V represented by the disjoint union X i (as schemes) is not equal to the disjoint union (as presheaves) of the presheaves represented by each X i . Nonetheless, Theorem 3.1 (6) also holds for the map in AN V represented by a disjoint union Y i X i of morphisms of schemes (taken as a disjoint union of schemes instead of as presheaves on AN V ); indeed, the sheafification of X i is the same in each case, and in general for a site C and a presheaf F C ̂ with sheafification F a , there is a natural equivalence
C / F ~ C ~ / F a
(22)
of topoi.

Corollary 5.11.

Let (C,O) be an overconvergent variety and let XY be a morphism of algebraic varieties over C. Let {Y i } be a locally finite open cover of Y and denote by X i the fiber product X× Y Y i . Then X / O AN Y / O AN is universally of cohomological descent with respect to crystals if and only if for each i, the map X i / O AN Y i / O AN is universally of cohomological descent with respect to crystals.

Proof.

This follows from Theorems 5.9 and 3.1 (3) and (6) applied to the diagram of sheaves on AN (Y/O) induced by the Cartesian diagram of schemes

The following direct corollary to Theorem 5.9 allows us to reduce to the integral case.

Corollary 5.12.

Let Y be an algebraic variety. Let { Y i } be the set of irreducible components of Y and let Y i : = ( Y i ) red be the reduction of Y i . Then the morphism Y i Y is universally of cohomological descent with respect to crystals.

Modifications

In order to apply Raynaud-Gruson’s theorem on ‘flattening stratifications’, we now address cohomological descent for modifications. The following lemma is an adaptation of [5], Lemma 3.4.5 to the overconvergent site, with a minor variation in that we work with non-archimedean analytic spaces. Note also that some care (e.g., the use of Lemma 5.2) is necessary to apply his argument to the overconvergent site.

Lemma 5.14.

Let Y be a scheme and let Z be a closed subscheme whose sheaf of ideals I is generated by two elements f and g. Then the blow up XY of Y with respect to I is universally of cohomological descent with respect to crystals.

Proof.

By Corollary 5.11, we may assume that Y is affine and thus admits an embedding Y P : = V n ̂ . Let { P i : = 𝔸 n ̂ } i = 0 n (where we invert the i th coordinate on n ) and let Y i =Y× P P i . Let Y i ¯ be the closure of Y i in P i .

Let f i (resp. g i ) denote the restriction of f (resp. g) to Y i and denote by Z i the subscheme defined by f i and g i . We claim that f and g lift to sections f and g of P. Indeed, the immersion YP factors through a closed immersion Y 𝔸 V n ; since this is a closed immersion the sections lift to 𝔸 n , and we can lift to n by homogenizing.

Denote by f i ¯ and g i ¯ restrictions of the lifts f and g to Γ ( O Y i ¯ ) and define Z i ¯ be the subscheme of Y i ¯ defined by f i ¯ and g i ¯ . Let X i ¯ be the blow up of Y i ¯ along Z i ¯ . Then Z i ¯ × Y i ¯ Y i = Z i and X i ¯ × Y i ¯ Y i = X i .

Let Ui,1 be the tube ] Z i ¯ [ ( P i ) K of Z i ¯ in (P i ) K . Fix a rational number λ in (0,1), let f i ̂ and g i ̂ be lifts of f i ¯ and g i ¯ to g amma ( O P i ) , and define
U i , 2 = { x ] X i ¯ [ ( P i ) K : | f i ̂ | > λ or | g i ̂ | > λ } ;
(23)
by construction Ui,1Ui,2 is a cover of ] X i ¯ [ ( P i ) K .

The scheme X i ¯ is a subscheme of P i 1 (indeed, if s and t are coordinates for 1 , then X i ¯ is defined by the equation f i ¯ t g i ¯ s ). Set V i , 1 : = U i , 1 × ( P i ) K ( P i 1 ) K U i , 1 1 . The map (X i ,Vi,1)→(Y i ,Ui,1) of overconvergent varieties factors as (X i ,Vi,1)→(Y i ,Vi,1)→(Y i ,Ui,1). The second map has a section and is thus universally of cohomological descent by Theorem 3.1 (2), and the first map is universally of cohomological descent with respect to crystals by Lemma 5.2 (for the first map, note that since X i Y i is surjective, the map on tubes is an isomorphism); we conclude that (X i ,Vi,1)→(Y i ,Ui,1) universally of cohomological descent with respect to crystals by Theorem 3.1 (5).

Let R i be the closed formal subscheme of P i 1 defined by the equation f ̂ i t ĝ i s . Then (R i ) K →(P i ) K is an isomorphism away from the vanishing locus of f i ̂ and g i ̂ in (P i ) K . Denote by Vi,2 the pre-image of Ui,2 under the map (R i ) K →(P i ) K . Then the map (X i ,Vi,2)→(Y i ,Ui,2) of overconvergent varieties factors as (X i ,Vi,2)→(Y i ,Vi,2)→(Y i ,Ui,2); the second map is an isomorphism, and the first map is universally of cohomological descent with respect to crystals by Lemma 5.2 (again, since X i Y i is surjective, the map on tubes is an isomorphism); we conclude that (X i ,Vi,2)→(Y i ,Ui,2) is universally of cohomological descent with respect to crystals by Theorem 3.1 (5).

We get a diagram

The middle horizontal map is universally of cohomological descent with respect to crystals by Lemma 5.7 and the left and right horizontal maps are universally of cohomological descent by Theorems 4.17 and 3.1 (1); thus the composition is universally of cohomological descent with respect to crystals by Theorem 3.1 (5). By the previous two paragraphs and Theorem 3.1 (6), the left vertical map is universally of cohomological descent with respect to crystals; the lemma thus follows from Theorem 3.1 (4).

The next lemma lets us reduce the case of a general blow up to the situation of Lemma 5.14.

Lemma 5.15.

Let Y be a Noetherian integral scheme, let I O Y be a sheaf of ideals globally generated by r≥2 many elements and let XY be the blow up of Y along I. Then there exists a map XX such that the composition XY factors as
X = X r X r 1 X i X i 1 X 0 = Y ,
(24)
where each map X i Xi−1 is a blow up centered at an ideal which is globally generated by two elements.

Proof.

This is a special case of [5], Lemma 3.4.4.

Recall that a morphism p:XY is a modification if it is proper and an isomorphism over a dense open subscheme of Y. The next proposition shows that modifications are universally of cohomological descent with respect to crystals.

Proposition 5.16.

Let p:XY be a modification. Then p is universally of cohomological descent with respect to crystals.

Proof.

By Chow’s lemma [24], Theorem 5.6.1 and 3.1 (4), we may assume that p is projective. By Corollary 5.12 and Theorem 3.1 (4) and (5), we may assume that Y is integral, and then by [25], Section 8, Theorem 1.24, there exists an affine open cover {Y i } of Y such that for each i, Y i × Y XY i is a blow up of Y i along a closed subscheme; by Corollary 5.11, we may thus assume that p is a blow up. By the structure lemma for blow ups (Lemma 5.15), we may reduce to the case of a codimension one blow up which is Lemma 5.14.

Flat covers

In this section, we prove Theorem 1.1 (i) – that finitely presented crystals are universally cohomologically descendable with respect to fppf (faithfully flat locally finitely presented) morphisms of schemes.

Definition 5.18.

A map (f,u):(X,V)→(X,V) of overconvergent varieties is said to be finite (see [9], Definition 3.2.3) if, up to strict neighborhoods, u is finite (see [18], paragraph after Lemma 1.3.7) and u 1 ( ] X [ V ) = ] X [ V . Moreover, u is said to be universally flat if u is quasi-finite and, locally for Grothendieck topology on V and V, u is of the form ( A ) ( A ) with AA flat (see [18], Definition 3.2.5).

Proposition 5.19.

Let (f,u):(X,V)→(X,V) be a finite map of overconvergent varieties and suppose that, after possibly shrinking V and V, u is universally flat and surjective. Then (f,u) is universally of cohomological descent with respect to finitely presented overconvergent crystals.

Proof.

To ease notation we set p:=(f,u). Let FModfp(X,V). By Corollary 3.4, it suffices to prove that (i) q p , p F = 0 for q>0 and (ii) the Čech complex F p , p F is exact. Let p i :=(f i ,u i ):(X i ,V i )→(X,V) be the i-fold fiber product of p; p i also satisfies the hypotheses of this proposition. By the spectral sequence (Remark 3.3), it suffices to prove that q p i p i F = 0 for i≥0 and q>0.

Shrink V and V i such that u i is finite and such that FX,V is isomorphic to i X 1 G for some G Coh O V (which is possible by [9], Proposition 2.2.10). To prove (i), one can work with realizations as in [9], Proof of Proposition 3.2.4; it thus suffices to prove that q ] u i [ ] u i [ F X , V = 0 for q>0. Then q ] u i [ ] u i [ F X , V = i X 1 q u i u i G ; by [18], Corollary 4.3.2 q u i u i G = 0 and (i) follows.

For (ii), since one can check exactness of a complex of abelian sheaves on the collection of all good realizations and since our hypotheses are stable under base change, it suffices to prove that the Čech complex of FX,V with respect to ]u[ is exact. Since i X 1 is exact, it suffices to prove that the Čech complex of G with respect to u is exact.

By [18], Proposition 4.1.2, G is a sheaf in the flat quasi-finite topology, so by Theorem 3.1 (1), G u , u G is exact in the flat quasi-finite topology; since G is coherent and (X,V) is good, this is exact in the usual topology.

Recall that a monogenic map of rings is a map of the form AA[t]/f(t), where fA[t] is a monic polynomial, and a map of affine formal schemes is said to be monogenic if the associated map on rings is monogenic.

Proof of Theorem 1.1 (i). By Theorem 5.9 and Corollary 3.2, we may assume that everything is affine.

Step 0: (Reduction to the finite and locally free case). Let p:XY be an fppf cover. By [26], Lemma (http://math.columbia.edu/algebraic_geometry/stacks-git/locate.php?tag=05WN), there exists a map XX such that the composition XY is a composition of surjective finite locally free morphisms and Zariski coverings; by Theorem 5.9 and Corollary 3.2, we may assume that XY is finite and locally free.

Step 1: (Monogenic case). Suppose that XY is monogenic and choose a closed embedding Y 𝔸 V n (which exists since Y is affine and of finite type) and then an open immersion 𝔸 V n P : = V n . The polynomial defining XY lifts to a monic polynomial with coefficients, giving a monogenic (and thus finite and flat) map X 0 𝔸 V n , and then homogenizing this polynomial gives a map π:PP of schemes over Spec V and an embedding XP which is compatible with the embedding YP. The map π may not be finite or flat (see Remark 5.20 below), but (noting that π is projective) by [12], Théorèm 5.2.2, there exists a modification P ~ P , centered away from X, such that the strict transform P ~ P ~ of PP is flat and (since it is generically finite, flat, and proper) finite.

Replacing PP with the formal completion of P ~ P ~ , we thus have a finite flat map PP of formal schemes and an embedding XP which is compatible with the embedding YP. Consider the diagram

By Theorems 4.17 and 3.1 (1), (Y,P K )→Y is universally of cohomological descent with respect to crystals, so by Corollary 3.2 it suffices to prove that ( X , P K ) ( Y , P K ) is universally of cohomological descent with respect to finitely presented crystals. Since X=Y× P P, ( X , P K ) ( Y , P K ) satisfies the hypotheses of Proposition 5.19 and step 1 follows.

Step 2: (Base extension). Now let kk be a finite field extension of the residue field. We claim that it suffices to check that X k Y k is universally of cohomological descent with respect to finitely presented crystals. Indeed, let k=k0k1kn−1k n =k be a sequence of field extensions such that k i =ki−1(α i ) for some α i k i (note that one may not be able to choose n=1 since kk may not be separable). Consider the diagram

Each map Y k i Y k i 1 is monogenic and thus universally of cohomological descent with respect to finitely presented crystals, the claim thus follows from Theorem 3.1 (4), (5), and (6).

Step 3: (Reduction to the monogenic case). Let k ¯ be the algebraic closure of k and let p k ¯ : X k ¯ Y k ¯ be the base change of p to k ¯ . Let x X k ¯ be a closed point and set y = p k ¯ ( x ) . Let k ¯ ( x ) and k ¯ ( y ) denote the residue fields of x and y; since k ¯ is algebraically closed, k ¯ ( x ) = k ¯ ( y ) . In particular, k ¯ ( y ) is a separable extension of k ¯ ( x ) , and thus, by the argument of [27], 2.3, Proposition 3, there exists a (generally non-cartesian) commutative diagram

where X x (resp. Y y ) is an affine open neighborhood of x (resp. y) and X x Y y is monogenic. By quasi-compactness of Y k ¯ , there thus exists a (generally non-cartesian) commutative diagram

such that {Y i } is a finite cover of Y k ¯ by affine open subschemes of finite type over k ¯ , X i is an affine open subscheme of X k ¯ , and each map f i :X i Y i is monogenic. Since the covering is finite, there exists a finite field extension kk and a (generally non-Cartesian) commutative diagram

with the same properties. By step 2, it suffices to check that X k Y k is universally of cohomological descent with respect to finitely presented crystals. By Corollary 3.2, it suffices to prove this for each i, the map X i Y i , which follows from step 1.

Remark 5.20.

The modification in step 1 of the proof is necessary. Indeed, the monogenic map X 𝔸 2 given by t2+x1x2t+x1+x2 homogenizes to the map X 2 given by x 0 2 t 2 + x 1 x 2 ts + ( x 1 + x 2 ) x 0 s 2 which is not flat, since it is generically quasi-finite but not quasi-finite (since the fiber over x0=x1=0 is 1 ).

Proper surjections

The proper case of the main theorem will now follow from Chow’s lemma and the Raynaud-Gruson theorem on ‘Flattening Blow Ups’.

Proof of Theorem 1.1 (ii). Let p:XY be a proper surjection. By Chow’s lemma [24], Theorem 5.6.1 and Theorem 3.1 (4), we may assume that p is projective. By Corollary 5.11 we may assume that Y is affine and thus by [12], Théorèm 5.2.2, there exists a modification YY such that the strict transform XY is flat. By Theorem 1.1 (i) (resp. 5.16) XY (resp. YY) is universally of cohomological descent with respect to finitely presented crystals. By 3.1 (5), the composition XY is universally of cohomological descent, and the proper case of the main theorem follows from 3.1 (4).

Declarations

Acknowledgements

We would like to thank Bjorn Poonen, Brian Conrad, Arthur Ogus, Bernard le Stum, Bruno Chiarellotto, and Anton Geraschenko for many useful conversations and encouragement. We also remark that the étale case was part of the author’s thesis [14].

This work was partially supported by a National Defense Science and Engineering Graduate Fellowship and by a National Security Agency Young Investigator grant.

Authors’ Affiliations

(1)
Department of Mathematics and Computer science, Emory University

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