Étale Cohomology of Rigid Analytic Spaces

Johan de Jong^{1} and Marius van der Put

Received: Oct. 16, 1995

Communicated by P. Schneider

Abstract. The paper serves as an introduction to étale cohomology of rigid analytic spaces. A number of basic results are proved, e.g. concerning cohomological dimension, base change, invariance for change of base fields, the homotopy axiom and comparison for étale cohomology of algebraic varieties. The methods are those of classical rigid analytic geometry and along the way a number of known results on rigid cohomology are re-established.

Key Phrases: ”Etale cohomology”, ”Rigid analytic spaces”, ”rigid cohomology”, ”overconvergent sheaves”

1991 Mathematics Subject Classification: Primary 26E30, secondary 14F20.

1 Introduction

The origin of this paper lies in the questions on étale cohomology for rigid analytic spaces posed in [S-S]. In that paper an étale site and a corresponding cohomology theory for analytic varieties are defined. We prove here that the axioms for an ‘abstract cohomology’ (as stated in [S-S]) hold for this cohomology theory. In addition, we prove a (quasi-compact) base change theorem for rigid étale cohomology and a comparison theorem comparing rigid and algebraic étale cohomology of algebraic varieties.

The main tools in this paper are analytic (resp. étale) points and rigid (resp. étale) overconvergent sheaves. The rigid overconvergent sheaves on affinoids were first introduced in [P82] and were called constructible in that paper. They were further studied in [S93] and were called conservative there. The term ‘overconvergent’, also used by P. Berthelot in recent work, seemed more appropriate this time.

In Section 2 we (re)introduce some basic notations concerning analytic points and rigid overconvergent sheaves, which are needed later on. We (re)prove a number of folklore results, most importantly: 1) Rigid cohomology agrees with Cech cohomology on quasi-compact spaces. 2) The cohomological dimension of a paracompact space is at most its dimension. 3) A base change theorem for rigid spaces which is more general than the results of [P82] or [S93].

1The research of Dr. A.J. de Jong has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.

Documenta Mathematica 1 (1996) 1–56

The rest of the paper deals with étale sites and étale cohomology. Étale points and étale overconvergent sheaves are introduced. A key point is the introduction of special étale morphisms of affinoids U → X, analogous to rational subdomains in the rigid case. Included in the paper is the proof by R. Huber that any étale morphism of affinoids is special étale. This simplifies the original exposition somewhat. A structure theorem for étale morphisms (3.1.2) allows us to give a proof of the étale base change theorem following closely the proof in the rigid case. We calculate the cohomology groups of one dimensional spaces in Section 4. This allows us to prove the basic results mentioned at the beginning of this introduction (Sections 5, 6 and 7).

We have tried to be complete in the proofs of various statements. We hope that this paper may serve as an introduction to rigid and étale cohomology of rigid analytic spaces.

Berkovich, in the paper [B93], developpes an étale cohomology theory for analytic spaces. The category of analytic spaces used there was introduced in [B90] and extended in [B93]. It is different from the category of rigid analytic spaces. For this reason we have not borrowed from his work. However, we have to mention that the approach taken here, in some sense, does not differ from his (allthough in this paper we have to deal with non-overconvergent sheaves also, which do not correspond to sheaves on the Berkovich analytic spaces). For example, Lemma 2.1.1, which controles the étale stalk functors, is more or less equivalent to Theorems 2.1.5 & 2.3.3 of [B93]. Furthermore, using the equality of Berkovich cohomology with ours in the case of paracompact varieties (see [Hu, Section 8.3]), all our results on cohomology of overconvergent sheaves are in principle deducible from the references [B93, B94a, B94b, B94c].

Étale cohomology theories for rigid analytic spaces were developped by O. Gabber (unpublished) and K. Fujiwara, who proved Delinge’s conjecture using his theory. As mentioned above R. Huber constructed an étale cohomology theory for his adic spaces, this specializes to give a theory for rigid analytic spaces also.

We thank P. Schneider for sending his informal notes [S91] to the authors for consultation.

1.1 Notations and conventions

• Unless stated otherwise k will be a complete non archimedean valued field.

• As general reference for the basic facts and definitions concerning rigid analytic varieties we take [BGR].

• All rigid analytic varieties occuring in this work will be quasi-separated analytic varieties. This means that the diagonal morphism X → X×X is quasi-compact, or equivalently that the intersection of any two affinoid subvarieties of X is a finite union of affinoid subvarieties of X. It is clear that fibre products of such are still quasi-separated.

• We work frequently with sites and associated topoi as in [SGA 4]. We recall that a morphism of sites f : S_{1} → S_{2} is a continuous functor u : S_{2} → S_{1} (remark that u goes in the opposite direction!), which induces a morphism of associated topoi S_{1}^{∼} → S_{2}^{∼} (see [SGA 4, IV 4.9]). We remark that if S_{2} allows finite projective limits then it suffices that u is continuous and preserves fibred products.

• A sheaf F on a site S is said to be flabby if for any object U in S we have H^{q}(U, F) = 0 for all q > 0. It is said to be flasque if for any morphism U → V the restriction map F(V) → F(U) is surjective. A flasque sheaf is flabby since Cech cohomology may be used to determine whether a sheaf is flabby ([M80, III 2.12]).

2 Analytic points and rigid overconvergent sheaves

In this section we will review the base change theorem for rigid analytic spaces (see [P82, S93]). We will introduce our basic notations and reprove the statements of [P82] (whose proofs are perhaps somewhat sketchy). We try to avoid using results from [B90] except for the basic fact that the space M(X) (see below) is Hausdorff and compact (this is not hard to prove). Finally, we prove a slightly stronger version of the base change theorem, namely that it holds for arbitrary sheaves.

2.1 Sites, sheaves and analytic points on affinoids

Let X be an affinoid space over some complete non archimedean valued field k. On X we consider the special Grothendieck topology given by the collection of finite unions of open affinoid subspaces and the admissible coverings. (See [FP, GP], this is a Gtopology slightly stronger than the weak G-topology of X in [BGR, 9.1.4].) We will write X_{rigid} for the following site:

1. The objects are the admissible open subsets of X. We choose here as admissible opens the finite unions of open affinoid subsets. These will also be called the special subsets of X.

2. A morphism between to objects is an inclusion between the admissible subsets.

3. For an object U the elements of Cov(U) are those set-theoretical coverings of U by admissible opens which can be refined to finite coverings.

We use the special G-topology rather than the strong G-topology since it behaves better with respect to base change and change of base field. We remark that this gives the same category of sheaves.

It is sometimes easier to work with a subcategory X_{rigid}^{rat} of X_{rigid}. The objects of X_{rigid}^{rat} are the rational subsets of X. A rational subset of X is a set of the form

{x ∈ X| |f_{1}(x)| ≥ |f_{i}(x)| for all i with 1 ≤ i ≤ n}

where f_{1}, …, f_{n} are elements of O(X) generating the unit ideal. We note that a small change of the f_{1}, …, f_{n} does not affect the subset above. It is known that every open affinoid subset of X is a finite union of rational subsets ([GG]). A rational covering of a rational U ⊂ X is a covering of the form U = ∪_{i=1}^{m}U_{i} given by elements f_{1}, ...f_{m} ∈ O(U) generating the unit ideal such that the U_{i} are the rational subsets (of U and also of X) U_{i} := {x ∈ U| |f_{i}(x)| ≥ |f_{j}(x)| for all j}. This defines for every object the collection of coverings. The morphism of sites X_{rigid} → X_{rigid}^{rat} (given by the inclusion functor X_{rigid}^{rat} → X_{rigid}, see our conventions) defines an isomorphism of associated topoi, this follows from the fact that any special subset of X is a finite union of rational subsets and any finite affinoid covering of an affinoid variety can be refined to a rational covering (see for example [BGR, 8.2.2/2]).

It is well known that the set of ordinary points of X is too small to ”separate” the sheaves on X_{rigid}. For this purpose one introduces new points, called analytic points. (See [P82, S93]). We will adopt here the terminology of [S93].

An analytic point a of X is a semi-norm | |_{a} : O(X) → R_{≥0} on the affinoid algebra O(X) of X satisfying:

1. |f + g|_{a} ≤ max(|f|_{a}, |g|_{a}) for all f, g ∈ O(X).

2. |fg|_{a} = |f|_{a}|g|_{a} for all f, g ∈ O(X).

3. For λ ∈ k the value |λ|_{a} is the absolute value of λ.

4. | |_{a} : O(X) → R_{≥0} is continuous with respect to the norm topology on O(X).

The filter of the analytic point a consist of the affinoid subdomains U of X for which there exists a rational covering given by f_{1}, ..., f_{n} and an i such that U ⊃ U_{i} and |f_{i}|_{a} ≥ |f_{j}|_{a} for all j. This is equivalent with the property that | |_{a} extends to a | |_{a} : O(U) → R_{≥0}, i.e., that a is also an analytic point of U. We write a ∈ U to denote that U belongs to the filter of a. We will also need the concept of a wide neighbourhood of an analytic point a of X (see [S93, p. 131]). An element U of the filter of a is a wide neighbourhood of a if there exists an affinoid generating system f_{1}, …, f_{n} of O(U) over O(X) such that |f_{i}|_{a} < 1 for all i.

Let M(X) denote the set of analytic points of X. We give M(X) the coarsest topology such that for every g ∈ O(X) the map M(Z) → R given by a ↦ |g|_{a} is continuous. For an analytic point a a fundamental system of neighbourhoods is given by the subsets M(U) where U runs through the (affinoid) wide neighbourhoods of a. The space M(Z) is Hausdorf and compact for this topology. These results are not hard to prove, they follow from 1.2.2 and 1.3.3 of [P82], but see [B90, §1], [S93, §1] for more details. We will repeatedly make use of the following corollary of the above: Suppose that {X_{i}}_{i∈I} are affinoid subdomains of X such that for any analytic point a of X some X_{i} is a wide neighbourhood of a, then the covering X = ⋃ X_{i} is admissible, i.e., finitely many of the X_{i} cover X.

The stalk of a sheaf S on X_{rigid} at an analytic point a is defined as S_{a} = lim_{→} S(U) where the direct limit is taken over all U in the filter of a. The modified stalk of S at a is S_{a}^{mod} = lim_{→} S(U) where the limit is over the wide open neighbourhoods of a in X.

For every U in the filter of a the seminorm | |_{a} extends to a seminorm on O(U). Hence we get a seminorm | |_{a} on O_{a} the stalk of O = O_{X} at a. A fundamental fact that we will use is (see [P82, 1.3.1]) that for f ∈ O(X):

|f|_{a} = inf{||f||_{U}}

where U runs through the filter of a. In fact it suffices to consider only wide open neighbourhoods of a (use that for U ⊂ X rational we have ||f||_{U} = inf_{r>1} ||f||_{U(r)} where U(r) is defined as in 2.3 below). It follows from these considerations that the ideal m_{a} of elements f ∈ O_{a} satisfying |f|_{a} = 0 is the unique maximal ideal of O_{a} (and similar for O_{a}^{mod}). The field O_{a}/m_{a} will be denoted by k_{a}. The semi-norm | |_{a} induces a valuation on k_{a}. This valuation extends the valuation of the subfield k of k_{a}. In general the field k_{a} is not complete and its completion is denoted by F_{a}. (The same constructions give k_{a}^{mod} and F_{a}^{mod}.)

Let ϕ : O(X) → F_{a} denote the continuous homomorphism of k-algebras obtained above from | |_{a}. Then one sees that |f|_{a} = |ϕ(f)|. This remark shows that our definition of analytic point coincides with the equivalence classes of analytic points as defined in [S93]. Every ordinary point of X is also an analytic point (with F_{a} = k_{a} a finite extension of k). The following lemma will be useful in our study of the étale site of X.

Lemma 2.1.1 Notations are as above.

1. O_{a} and O_{a}^{mod} are henselian local rings.

2. k_{a} and k_{a}^{mod} are henselian valued fields.

3. F_{a} is finite over a complete subfield K which has a dense subfield k(t_{1}, ..., t_{d}) with d ≤ the dimension of X.

4. The homomorphism O_{a}^{mod} → O_{a} is local, flat and induces an isomorphism F_{a}^{mod} ≅ F_{a}.

Proof. Let O_{a} ⊂ A be a finite free extension of rings. We claim the following: the ring A ˆ⊗F_{a} has a nontrivial idempotent if and only if A has one. (We also claim a similar result for O_{a}^{mod}.)

This immediately implies (1) (see [R70, I Proposition 5]). Statement (2) means that the valuation ring of k_{a} (resp. k_{a}^{mod}) is an henselian ring. Our claim implies that a finite separable ring extension k_{a} ⊂ k′ contains a copy of k_{a} if and only if the tensor product k′ ⊗ F_{a} contains a copy of F_{a} (use a lift O_{a} → A of the finite extension k_{a} → k′). This gives that any scheme étale over the valuation ring of k_{a} has a k_{a}-valued point if and only if it has a F_{a}-valued point. This assertion combined with the fact that the valuation ring of F_{a} is henselian implies that k_{a} is a henselian valued field (use the criterium of [R70, Proposition 3 page 76]).

To prove our claim, note that the ring extension O_{a} ⊂ A comes from a finite free ring extension O(U) ⊂ A_{U} for some U in the filter of a. Clearly, A_{U} is an affinoid algebra and hence determines a finite flat morphism ϕ : V = Spm(A_{U}) → U. The fact that A ˆ⊗F_{a} ≅ A_{U} ˆ⊗F_{a} has a nontrivial idempotent is equivalent to the fact that ϕ^{−1}(a) = b_{1}, …, b_{s} has at least two elements. Let us take disjoint wide neighbourhoods V_{i} of the b_{i} in V. There exists a smaller U′ in the filter of a such that ϕ^{−1}(U′) is contained in ∪V_{i} (see Lemma 3.1.6 below; the reader may check that this lemma is not used before that lemma). Therefore the algebra A_{U′} = A_{U} ⊗ O(U′) decomposes and hence so does A. The proof for O_{a}^{mod} is the same.

(3) After dividing O(X) by a prime ideal we may suppose that | |_{a} is a norm on O(X). The field of quotients of O(X) is a dense subfield of F_{a}. The algebra O(X) is finite over some A := k⟨T_{1}, …, T_{d}⟩ with d equal to the dimension of X. Let K ⊂ F_{a} denote the completion of the field of quotients of A with respect to | |_{a}. The field F_{a} is finite over K and K has k(T_{1}, …, T_{d}) as dense subfield with respect to | |_{a}.

(4) It is clear that the homomorphism O_{a}^{mod} → O_{a} is local and flat. Suppose that ℘ is the kernel of the seminorm | |_{a} on O(X). It is clear that the fraction field of O(X)/℘ is dense in both F_{a} and F_{a}^{mod}. The result follows. □

Remark 2.1.2 It follows from this lemma and its proof that there are equivalences between the following categories: the category of finite separable extensions of F_{a}, of finite separable extensions of k_{a}, of finite separable extensions of k_{a}^{mod}, of finite étale extensions of local rings O_{a} ⊂ A, and of finite étale extensions of local rings O_{a}^{mod} ⊂ A. Furthermore, any such extension comes from a finite étale (see paragraph 4) morphism V → U where U is a wide neighbourhood of a.

It is clear that the above constructions are functorial in the following sense. If f : Y → X is a morphism of affinoids over k, then we get a morphism of sites Y_{rigid} → X_{rigid} (resp. Y_{rigid}^{rat} → X_{rigid}^{rat}). Indeed, if U ⊂ X is an affinoid subdomain (resp. rational subset) then so is f^{−1}(U) ⊂ Y. Hence a functor X_{rigid} → Y_{rigid}, U ↦ f^{−1}(U), it is easy to see that this is continuous and compatible with fibre products (i.e., intersections). The associated adjoint functors on sheaves are denoted f^{∗}, f_{∗} as usual.

The morphism f also induces a continuous map: M(Y) → M(X). The seminorm O(Y) → R_{≥0} is mapped to the composition O(X) → O(Y) → R_{≥0}. We remark that if f identifies Y with an affinoid subdomain of X then 1) Y_{rigid} ≅ X_{rigid}/Y and 2) the analytic points of Y are identified with those analytic points a of X such that Y is in the filter of a, i.e., a ∈ Y.

2.2 Sites, sheaves and analytic points for general X

To the analytic variety X we associate the site X_{rigid} by exactly the same definition as for affinoid X’s. The objects are the finite unions of affinoid open subvarieties and the coverings are coverings which can be refined to finite coverings. (Since X is quasiseparated, the intersection of two affinoid open subvarieties is an object of the category X_{rigid}, so that X_{rigid} is indeed a site.) We remark that the the associated topos X_{rigid}^{∼} is again naturally isomorphic to the category of sheaves on X (as defined in [BGR, 9.2]). A morphism f : Y → X induces a morphism of topoi f_{rigid} : Y_{rigid}^{∼} → X_{rigid}^{∼} but not in general a morphism of sites Y_{rigid} → X_{rigid}. Indeed, this morphism of sites exists if and only if f is quasi-compact.

The space X has some admissible covering {X_{i}} by affinoids subsets. The analytic points of X are just the analytic points of the X_{i}, subject to the usual equivalence relation. (For a more precise definition see [S93, §2].) We remark that our f : Y → X induces a map on analytic points.

Finally, suppose f : Y → X is an open immersion (in the sense of [BGR, p. 354]). It is easy to prove (using the above) that: 1) f induces an injection between the sets of analytic points and 2) f induces an isomorphism Y_{rigid}^{∼} → X_{rigid}^{∼}/Y (where Y denotes the sheaf V ↦ Mor_{X}(V, Y) on X_{rigid}). However, it is not true that any f satisfying 1) and 2) is an open immersion.

2.3 Overconvergent sheaves on affinoids

Let X be an affinoid variety over k. The collection of analytic points of X is still not large enough to ”separate” the abelian sheaves on X_{rigid}. We can introduce a larger collection of points as in [P82] to remedy this fact. However, this larger collection of points seems not to be of much use for questions like base change theorems et cetera. We choose to work with a restricted collection of sheaves, namely the overconvergent sheaves on X_{rigid}.

Suppose that V ⊂ U are special subsets of X. We will say that V is inner in U (w.r.t. X), or that U is a wide neighbourhood of V in X, if for any analytic point a of V there is an affinoid wide neighbourhood U_{a} of a in X with U_{a} ⊂ U. Notation: V ⊂⊂_{X} U. It is proved in [S93, §1 Proposition 23] that this agrees with the notion V is relatively compact in U over X (see [BGR, 9.6.2]) if V and U are affinoid subdomains of X: V ⊂⊂_{X} U ⇔ there is an affinoid generating system f_{1}, …, f_{r} of O(U) over O(X) such that

V ⊂ {x ∈ U; |f_{1}(x)| < 1, …, |f_{r}(x)| < 1}.

Suppose V ⊂ X is rational in X given by the inequalities |g_{0}| ≥ |g_{1}|, ..., |g_{m}|. For r > 1 and r ∈ √|k^{∗}| we define the rational set V (r) by the inequalities r|g_{0}| ≥ |g_{1}|, ..., |g_{m}|. It is easy to see that V ⊂⊂_{X} V (r). (The notation V (r) will be used even if no explicit system g_{0}, …, g_{m} defining V and V (r) is indicated.)

Lemma 2.3.1 With notations as above.

1. The V (r) form a cofinal system of (special) wide neighbourhoods in X of the rational set V.

2. If V_{1}, …, V_{n} are rational in X then

V_{1} ∩ … ∩ V_{m} ⊂⊂_{X} V_{1}(r) ∩ … ∩ V_{m}(r)

(r > 1 and r ∈ √|k^{∗}|) and this forms a cofinal system of wide neighbourhoods of V_{1} ∩ … ∩ V_{m}. Similarly for V_{1} ∪ … ∪ V_{m} ⊂⊂_{X} V_{1}(r) ∪ … ∪ V_{m}(r).

Proof. Suppose that V ⊂⊂_{X} U (with U a special subset of X). We claim the covering X = U ∪ (X \ V) is admissible. This is proved in [P92, Lemma 1.1], but let us indicate another proof: For any analytic point a of X, a ∉ V choose an affinoid wide neighbourhood W_{a} of a with W_{a} ∩ V = ∅ (just define W_{a} by suitable inequalities). For an analytic point a ∈ V we choose the affinoid wide neighbourhood W_{a} of a in X which is contained in U. Since M(X) is compact the covering X = ⋃ W_{a} is admissible (see 3.1), hence so is X = U ∪(X \V). This proves our claim. In particular there is a special W ⊂ X \ V such that X = U ∪ W.

Next, put W_{i} = {w ∈ W; |g_{i}(x)| ≥ |g_{j}(x)| j = 0, …, n} for i = 1, …, n. Of course W = ⋃ W_{i} since W ∩ V = ∅. On W_{i} the function g_{i} is invertible hence we can put

ϵ_{i} = ||g_{0}/g_{i}||_{W_{i}} and ϵ = max_{i} ϵ_{i}.

By the maximum modulus principle on W_{i} and since W ∩ V = ∅ we get ϵ_{i} < 1 and ϵ < 1. It is now clear that for any r ∈ √|k^{∗}|, ϵ^{−1} > r > 1 we have V (r) ∩ W = ∅ and hence V (r) ⊂ U.

We prove 2) only in the case m = 2. Suppose that V_{1} is given by the inequalities |g_{0}| ≥ |g_{1}|, …, |g_{n}| and that V_{2} is given by the inequalities |f_{0}| ≥ |f_{1}|, …, |f_{n′}|. The intersection V_{1}(r)∩V_{2}(r) is given by the inequalities r^{2}|g_{0}f_{0}| ≥ |g_{i}f_{j}|, i = 0, …, n, j = 0, …, n′. The result follows. The statement for unions is trivial from 1). □

At this point we are able to define the rigid overconvergent sheaves on our affinoid variety X. A (pre)sheaf S (on X_{rigid}) is called (rigid) overconvergent if for every admissible open V ⊂ X we have

S(V) = lim_{\overrightarrow{V ⊂⊂_{X}U}} S(U).

It follows from the lemma above that if S is a sheaf then S is overconvergent if and only if S(V) = lim S(V (r)) for any rational V ⊂ X. These sheaves were called the constructible sheaves in [P82]; they agree with the conservative sheaves of [S93] by [S93, §1 Lemma 25]. In [S93, §1] it is shown that these overconvergent sheaves correspond to sheaves on the topological space M(X).

Lemma 2.3.2 (Properties of overconvergent sheaves.) In this lemma all (pre)sheaves are (pre)sheaves of abelian groups on the affinoid variety X.

1. The sheaf associated to a overconvergent presheaf is overconvergent.

2. For any overconvergent sheaf S the presheaves U ↦ H^{i}(U, S) are overconvergent.

3. The category of overconvergent sheaves is an exact subcategory of the category of all sheaves.

4. If f : Y → X is a morphism of affinoids then f^{∗} and f_{∗} preserve overconvergent sheaves. The same holds for R^{q}f_{∗}.

5. If X = ⋃ X_{i} is written as the finite union of affinoid subdomains then a sheaf S on X is overconvergent if and only if the restriction of S to any of the X_{i} is overconvergent.

6. A overconvergent sheaf S is zero if and only if all of its stalks S_{a} at analytic points of X are zero.

Proof. Let S be a overconvergent presheaf. Suppose V ⊂ X is the union of rational subsets V_{1}, …, V_{m} of X. Denote by V = {V_{i}} the covering of V and by V(r) = {V_{i}(r)} the covering of V (r) := ⋃_{i} V_{i}(r). It is immediate from Lemma 2.3.1 that

C^{·}(V, S) = lim_{⟶}_{r>1}C^{·}(V(r), S).

(These symboles denote Cech complexes.) It is therefore clear that the map

lim_{\overrightarrow{V ⊂⊂_{X}U}} Ȟ^{p}(U, S) ⟶ Ȟ^{p}(V, S)

is surjective.

Let us prove that it is also injective. Take a special U ⊂ X with V ⊂⊂_{X} U, an admissible covering U = {U_{i}} of U, a cocycle ξ ∈ C^{p}(U, S) whose Cech cohomology class maps to zero in Ȟ^{p}(V, S). This means there is a covering V = {V_{j}} of V which refines U ∩ V, i.e., there is a function α such that V_{j} ⊂ U_{α(j)} ∩ V, and a chain η ∈ C^{p−1}(V, S) with α(ξ) − dη = 0 ∈ C^{p}(V, S). Here α(ξ) is the image of ξ under the map C^{p}(U, S) → C^{p}(V, S) determined by α. By refining U and V we may assume that U and V are finite and that all U_{i} and V_{j} are rational subdomains of X.

By the above, the cocycle ξ lifts to a cocycle ξ′ ∈ C^{p}(U(r), S) for some r > 1. Lemma 2.3.1 implies that there exists an r^{∗} > 1 such that V_{j}(r^{∗}) ⊂ U_{α(j)}(r)∀j. For an even smaller r^{∗}, we may also assume η lifts to a chain η′ ∈ C^{p−1}(V(r^{∗}), S). The cocycle α(ξ′) − dη′ ∈ C^{p}(V(r^{∗}), S) maps to zero as a chain in C^{p}(V, S), thus it is already zero in some C^{p}(V(r^{∗∗}), S), r^{∗} > r^{∗∗} > 1. We conclude that the cohomology class of ξ in Ȟ^{p}(V (r^{∗∗}), S) is zero, which was what we wanted to show.

The isomorphism of Cech cohomologies above proves that the presheaf Ȟ^{0}(S) is overconvergent if S is overconvergent. Hence also the sheaf associated to S is overconvergent. It proves (2) since Cech cohomology agrees with usual cohomology for any special U ⊂ X. (See [P82, 1.4.4] or our Proposition 2.5.4.)

The third statement of our lemma means that the kernels and cokernels of overconvergent sheaves are overconvergent and that if a short exact sequence of sheaves 0 → S_{1} → S_{2} → S_{3} → 0 is given, S_{1} and S_{3} are overconvergent then so is S_{2}. These statements follow easily from (1) and (2).

(4) If V ⊂ X is a rational subset, then f^{−1}(V) is a rational subdomain of Y and we have: f^{−1}(V (r)) = (f^{−1}(V))(r). Thus it is clear from Lemma 2.3.1 that for special V ⊂⊂_{X} U in X we have f^{−1}(V) ⊂⊂_{Y} f^{−1}(U) and that these f^{−1}(U) form a cofinal system of wide neighbourhoods of f^{−1}(V).

Take an overconvergent sheaf S on Y. The sheaf R^{q}f_{∗}S is the sheaf associated to the presheaf U ↦ H^{q}(f^{−1}(U), S). It is immediate from the remarks above and (2) that this presheaf is overconvergent.

If S is a sheaf on X then f^{∗}S is the sheaf associated to the presheaf P defined as follows on V ∈ X_{rigid}:

P (V) = lim_{\overrightarrow{U∈X_{rigid},f^{−1}(U)⊃V}} S(U)

Suppose S is overconvergent. If t ∈ P (V), i.e., t comes from s ∈ S(U) for some U ⊂ X as in the limit, then s comes from s′ ∈ S(U′) for some U′ ∈ X_{rigid} with U ⊂⊂_{X} U′. By the above we see that V ⊂⊂_{Y} f^{−1}(U′). We conclude that the map

lim_{\overrightarrow{V ⊂⊂_{Y} V′}} P (V′) → P (V)

is surjective. Let us prove that it is injective: Suppose t′ ∈ P (V′) comes from some s′ ∈ S(U′) with f^{−1}(U′) ⊃ V′ and maps to zero in some S(U) with U ⊂ U′ and f^{−1}(U) ⊃ V. There exists a wide neighbourhood U′′ of U′ and s′′ ∈ S(U′′) mapping to s′. Sinces S is overconvergent there is a special U′′′ with U′′′ ⊂ U′′, U′′′ ⊃⊃_{X} U such that s′′ maps to zero in U′′′. It is clear that V′′ := V′ ∩ f^{−1}(U′′′) is a wide neighbourhood of V in Y such that t′ maps to zero in P (V′′). We have proved that P, hence f^{∗}S, is overconvergent.

(5) This follows from (3) and (4) since any sheaf S on X fits into an exact sequence

0 ⟶ S ⟶ ⨁_{i} S|_{X_{i}} ⟶ ⨁_{i,j} S|_{X_{i}∩X_{j}}.

Here S|_{X_{i}} := j_{∗}j^{∗}S where j : X_{i} → X is the inclusion.

(6) Take a section s ∈ Γ(X, S). By assumption any analytic point a in X has an affinoid wide neighbourhood V_{a} ⊂ X such that s|_{V_{a}} = 0. By compactness of M(X) we get that the covering X = ⋃ V_{a} is admissible, hence s = 0. The same proof gives that Γ(V, S) = 0 for arbitrary special V ⊂ X. □

2.4 Overconvergent sheaves on general X

Let X be an arbitrary analytic variety over k. We will say that a sheaf S on X is overconvergent if for any affinoid open subvariety V ⊂ X, the restriction S|_{V} is overconvergent on V. Suppose X = ⋃ X_{i} is an admissible affinoid covering. It follows from Lemma 2.3.2 that S is overconvergent if and only if S|_{X_{i}} is overconvergent for all i.

Suppose f : Y → X is a morphism of rigid varieties. It is clear from Lemma 2.3.2 that f^{∗} preserves overconvergent sheaves. This is not true in general for f_{∗} or R^{q}f_{∗}. But it is true if f is quasicompact.

Proposition 2.4.1 If f : Y → X is a quasi-compact morphism then f_{∗} and R^{q}f_{∗} preserve overconvergent presheaves.

Proof. Take an overconvergent sheaf S on Y. The question is local on X, hence we may assume X affinoid. Thus Y is quasi-compact and hence by Lemma 2.5.3 we can find a finite admissible affinoid covering Y = ⋃ Y_{i} such that all intersections Y_{i_{0}...i_{q}} := Y_{i_{0}} ∩ … ∩ Y_{i_{q}} are affinoid. At this point we use the spectral sequence (deduced from the Cartan-Leray spectral sequence [SGA 4, V 3.3]) {E_{n}^{pq}} abutting to R^{n}f_{∗}S and with E_{2}-term:

E_{2}^{pq} = ⨁_{i_{0}...i_{q}} R^{p}(f|_{Y_{i_{0}...iq}} )_{∗}S|_{Y_{i_{0}...iq}}

By Lemma 2.3.2 all its terms are overconvergent sheaves. Hence by the same lemma we see that R^{n}f_{∗}S is overconvergent too. □

2.5 Cohomology and Cech cohomology

In this subsection we prove that cohomology agrees with Cech cohomology on quasicompact varieties. Further we prove that the cohomological dimension of such an analytic variety is at most its dimension.

Lemma 2.5.1 Let X be an affinoid variety, V ⊂ X special and a an analytic point of X. There exists a wide neighbourhood W = W_{a} of a such that W ∩ V is a finite union of Weierstrass domains, each defined by invertible functions.

Proof. Since V is a finite union of rational subsets of X we may assume that V is rational itself. Say it is defined by the inequalities |g_{0}| ≥ |g_{1}|, …, |g_{n}|, where the g_{i} generate the unit ideal of O(X). If a ∉ V, then we can find a wide neighbourhood W of a disjoint with V. If a ∈ V then |g_{0}|_{a} ≥ |g_{i}|_{a} and since the g_{i} generate the unit ideal we get |g_{0}|_{a} > 0. Thus we may replace X by a wide neighbourhood of a, so that g_{0} becomes invertible. In this situation V is defined by 1 ≥ |f_{i}| with f_{i} = g_{i}/g_{0}, i.e., V is a Weierstrass domain in X. For those i such that ϵ_{i} := |f_{i}|_{a} < 1, we may replace X by the wide neighbourhood of a defined by |f_{i}| ≤ 1/2(1 + ϵ_{1}) and drop f_{i}. At this point V ⊂ X is defined as 1 ≥ |f_{i}| with |f_{i}|_{a} = 1 for all i. Hence the subset |f_{i}| ≥ |π|, π ∈ k, 0 < |π| < 1 defines a wide neighbourhood of a such that f_{i} is invertible on it. □

Lemma 2.5.2 Suppose X is affinoid, V ⊂ X special. There exists a finite covering X = ⋃ X_{i} by affinoids of X such that X_{i} ∩ V is affinoid for all i.

Proof. By compactness of M(X) and the lemma above we may assume V ⊂ X is a finite union of Weierstrass domains, each given by invertible functions. Say V = ⋃_{i=1}^{n} V_{i} and V_{i} is defined by 1 ≥ |f_{1}^{i}|, …, |f_{n_{i}}^{i}| and each f_{j}^{i} invertible.

Consider combinatorial data of the form A = (i, (j_{1}, …, ĵ_{i}, …, j_{n})) where i ∈ {1, …, n} and j_{l} ∈ {1, …, n_{l}} for each l ≠ i, l ∈ {1, …, n}. We put

V_{A} = {x ∈ X; |f_{j}^{i}(x)| ≤ |f_{j_{l}}^{l}(x)|, l = 1, …, î, …, n, j = 1, …, n_{i}}

Remark that X = ⋃_{A} V_{A} since for any x ∈ X there is some i ∈ {1, …, n} such that max_{j} |f_{j}^{i}(x)| ≤ max_{j} |f_{j}^{l}(x)| for all l ≠ i. On the other hand, if A = (i, (j_{1}, …, ĵ_{i}, …, j_{n})) as above then

V_{A} ∩ V ⊂ V_{i},

and hence V_{A} ∩ V = V_{A} ∩ V_{i} is affinoid. This is immediate from the definitions. □

We remark that in proving the lemmata above we proved something slightly stronger: Suppose we had started with an admissible affinoid covering V = ⋃ V_{i}. This we can refine to a finite covering V = ⋃ V_{i} with V_{i} ⊂ X rational. The proof of Lemma 2.5.1 shows that we can cover X by finitely many affinoids X_{j} such that each X_{j} ∩ V_{i} is a Weierstrass domain in X_{j} defined by invertible functions. The proof of Lemma 2.5.2 shows that we can cover each X_{j} by finitely many X_{j,A}’s such that X_{j,A} ∩ (V ∩ X_{j}) = X_{j,A} ∩ V is contained in some V_{i}. Thus we have proved the first statement of the following lemma in the case that X is affinoid.

Lemma 2.5.3 Let X be a quasi-compact variety over k.

1. Given an admissible covering V : V = ⋃ V_{i} of the special subset V of X, there exists a finite affinoid covering U : X = ⋃ X_{j} such that the covering U ∩ V refines V. In addition we may assume X_{j} ∩ V_{i} affinoid for all j.

2. There exists a finite affinoid covering X = ⋃ X_{j} such that X_{i} ∩ X_{j} is affinoid for all i, j.

Proof. (1) This assertion follows immediately from the case X affinoid (proved above) by writing X as the finite admissible union of affinoids (use that X is quasi-separated by our conventions).

(2) Take first an arbitrary finite affinoid covering X = ⋃ X_{i}. By (1) we can find finite affinoid coverings U_{ij} : X_{i} = ⋃_{k} X_{ijk} such that X_{ijk} ∩ (X_{i} ∩ X_{j}) is affinoid for all k. Next we take a finite affinoid covering U_{i} : X_{i} = ⋃_{l} X_{il} refining U_{ij} for all j. It is clear that X_{il} ∩ X_{jm} = X_{il} ∩ (X_{i} ∩ X_{j}) ∩ X_{jm} is affinoid (all intersections are taken in X). Thus the covering X = ⋃ X_{il} works. □

Proposition 2.5.4 Suppose X is a quasi-compact (and quasi-separated) variety. Cech cohomology agrees with cohomology on X.

Proof. The Leray spectral sequence relating Cech cohomology with cohomology [SGA 4, V 3.4] shows that it suffices to prove: Ȟ^{p}(X, S) = 0 if S is a presheaf whose associated sheaf is zero. Suppose V is some finite admissible covering of X and ξ = ∏ ξ_{i_{o}...i_{p}} ∈ C^{p}(V, S) = ∏_{i_{o}...i_{p}} S(V_{i_{o}...i_{p}}). We can find a covering V_{i_{o}...i_{p}} of V_{i_{o}...i_{p}} such that ξ_{i_{o}...i_{p}} restricts to zero on each member of V_{i_{o}...i_{p}}. By Lemma 2.5.3 we can find a covering U_{i} : V_{i} = ⋃ U_{ij} of V_{i} such that U_{i} ∩ V_{i_{o}...i_{p}} (some i_{l} = i) refines V_{i_{o}...i_{p}} for all choices of the i_{l}. Put U = ⋃ U_{i}, it is an admissible covering of X and the map

α : C^{∗}(V, S) ⟶ C^{∗}(U, S)

is defined using U_{ij} ⊂ V_{i}. It is clear that the chain ξ maps to zero under α. □

Remark 2.5.5 By Lemma 2.5.3 this is a special case of [P82, 1.4.4]. The argument in the proof of [P82, 1.4.5] together with Lemma 2.5.3 shows that Cech cohomology agrees with cohomology on any (quasi-separated, see conventions) X which is of countable type (see Definition 2.5.6 below).

We introduce some convenient topological notions for the Grothendieck topology on our analytic varieties X.

Definition 2.5.6 Let X be an analytic variety over k.

1. We say that X is of countable type if there exists a countable admissible affinoid covering of X.

2. Suppose that X = ⋃ X_{i} is an admissible affinoid covering of X. We say that the covering is locally finite if each X_{i} meets finitely many X_{j}.

3. The variety X will be called paracompact if there exists an admissible locally finite affinoid covering.

Lemma 2.5.7 A paracompact space X is the admissible disjoint union of paracompact varieties of countable type. A connected paracompact variety X can be written as the admissible union X = ⋃_{n∈ℕ} X_{n}, with X_{n} quasi-compact and X_{i} ∩X_{j} = ∅ when |i − j| ≥ 2.

Proof. Since any rigid analytic space is the admissible disjoint union of its connected components, it suffices to prove the second statement. Therefore we assume that X is connected and has a locally finite affinoid admissible covering X = ⋃ X_{α}. Let us choose a fixed index α_{0}. For any α we define the distance d(α) of α to α_{0} to be the minimal length d of a sequence of indices α_{0}, α_{1}, …, α_{d} = α such that X_{α_{i}} ∩ X_{α_{i+1}} ≠ ∅ for all i = 0, …, d − 1. Since X is connected all distances are finite. We put X_{n} = ⋃_{d(α)=n} X_{α}. Since the covering was locally finite the spaces X_{n} are quasicompact. The last condition of the lemma follows immediately from our definition of distance. □

In the proof of the next proposition we need the relation of rigid analytic geometry with formal geometry (see [R70] and [BL]). We recall that if 𝔛 is a formal scheme of finite type and flat over Spf(k^{∘}) then there is canonically associated a quasi-compact rigid analytic variety X = 𝔛^{rig}. If 𝔘 ⊂ 𝔛 is a formal open subscheme then 𝔘^{rig} ⊂ 𝔛^{rig} is an open subvariety. If 𝔛 = ⋃ 𝔘_{i} then 𝔛^{rig} = ⋃ 𝔘_{i}^{rig} is an admissible covering (see [BL, §4]). Thus we get a morphism of sites X_{rigid} = 𝔛_{rigid}^{rig} → 𝔛_{Zar}.

It is also possible to perform the construction 𝔛 ↦ 𝔛^{rig} for formal schemes 𝔛 which are only locally of finite type over Spf(k^{∘}). It is not true that any rigid variety X comes from such a formal scheme. A counterexample can be constructed by gluing a countable number of closed discs to a fixed closed disc along mutually disjoint closed subdiscs. (This is also an example of a variety of countable type which is not paracompact.) It can be proved using the lemma above and [BL] that any paracompact X comes from a (paracompact) formal scheme 𝔛.

Proposition 2.5.8 (See [P82, 1.4.13]). If X is a quasi-compact rigid analytic variety of dimension d then H^{p}(X, S) = 0 for all p > d and all sheaves S on X.

Proof. Let us choose a formal scheme 𝔛 with 𝔛_{rig} ≅ X (see [R70] or [BL, Theorem 4.1]). Let us denote by {𝔛_{α}} the directed system of admissible blowing ups of 𝔛. These all satisfy 𝔛_{α}^{rig} ≅ X. Hence we get the morphism of sites π_{α} : X_{rigid} → 𝔛_{α,Zar}. Let us write S_{α} := π_{α,∗}S. There is a map H_{α}^{p} := H^{p}(𝔛_{α,Zar}, S_{α}) → H^{p}(X, S) deduced from the map π^{∗}π_{α,∗}S → S. It is proved in [BL, 4.4] that any finite covering of X comes from a covering of some 𝔛_{α}. Therefore, by our result that Cech cohomology agrees with cohomology on X, we see that any cohomology class in H^{p}(X, S) comes from some H_{α}^{p}. At this point we just remark that the underlying Zariski topological space associated to 𝔛_{α} is the underlying topological space of a scheme of finite type over the field k̄ of dimension at most n. The result follows. □

Remark 2.5.9 If we allow in X_{rigid} only finite coverings then it is true that lim 𝔛_{α,Zar} ≅ X_{rigid} as sites (see letter of Deligne to Raynaud of 23 august 1992). In this way it becomes clear that in fact lim H_{α}^{p} = H^{p}(X, S). This follows from the following general fact: Suppose the site S is the direct limit of a directed system of sites S_{α}. Then for any sheaf F on S there is a canonical isomorphism

lim_{\overrightarrow{α}} H^{q}(S_{α}, F|_{S_{α}}) ≅ H^{q}(S, F).

This isomorphism is in fact easy to prove by induction on q, using the Cartan-Leray spectral sequence and the fact that any cohomology class can be killed by some covering.

Corollary 2.5.10 If X is paracompact and of dimension ≤ d then cohomology of sheaves on X is zero in degrees ≥ d + 1.

Proof. It suffices to do the case where X is connected. Choose a covering X = ⋃ X_{n} as in Lemma 2.5.7. Put V_{1} = ⋃_{n odd} X_{n} and V_{2} = ⋃_{n even} X_{n}. The spaces V_{1}, V_{2} and V_{1} ∩ V_{2} are admissible disjoint unions of quasi-compact varieties. Note that for any sheaf S on X the maps H^{d}(X_{n}, S) ⊕ H^{d}(X_{n+1}, S) → H^{d}(X_{n} ∩ X_{n+1}, S) is surjective, otherwise the sheaf S on X_{n}∪X_{n+1} would have a nontrivial d+1^{th}-cohomology group, a contradiction with the proposition. With these remarks the result of the corollary follows from a consideration of the Cartan-Leray spectral sequence associated to the covering X = V_{1} ∪ V_{2}. □

Remark 2.5.11 Any separated variety of dimension 1 is paracompact. See [LP]. Similarly, the analytic space associated to a scheme of finite type over Spec(k) is paracompact.

2.6 General morphisms

Consider an extension of complete valued fields k ⊂ K. In [BGR, 9.3.6] there is constructed a base change functor X ↦ X ˆ⊗K of analytic varieties over k to analytic varieties over K. If X is affinoid then X ˆ⊗K is affinoid with algebra O(X) ˆ⊗_{k}K. In general, if X = ⋃ X_{i} is an admissible affinoid covering then X ˆ⊗K is defined as the glueing of the X_{i} ˆ⊗K. If V is an affinoid open subvariety of X then so is V ˆ⊗K ⊂ X ˆ⊗K. In this way (use [BGR, 9.3.6/1& 2]) we see that there is a morphism of sites

φ = φ_{K/k} : (X ˆ⊗K)_{rigid} → X_{rigid}.

Lemma 2.6.1 The functors φ^{∗}, φ_{∗} and R^{q}φ_{∗} preserve overconvergent sheaves.

Proof. There is a trivial reduction to the case that X is affinoid. Let V be a rational subdomain of X. It is clear that V (r) ˆ⊗K = (V ˆ⊗K)(r) for r > 1, r ∈ √|k^{∗}| (see [BGR, 9.3.6/1]). These form a cofinal system of wide neighbourhoods of V ˆ⊗K since √|k^{∗}| is dense in R_{≥0}. Thus it is clear from Lemma 2.3.1 that for special V ⊂⊂_{X} U in X we have V ˆ⊗K ⊂⊂_{X ˆ⊗K} U ˆ⊗K and that these U ˆ⊗K form a cofinal system of wide neighbourhoods of V ˆ⊗K. The rest of the proof is exactly the same as the proof of Lemma 2.3.2 part 4. □

Let k ⊂ K denote an extension of complete valued fields. Let X (resp. Y) denote an arbitrary analytic variety over the field k (resp. K). The most convenient way to define a general morphism f : Y → X is to say that f is a morphism of the K-analytic spaces Y → X ˆ⊗K. If both X and Y are affinoid then this is simply a continuous k-algebra homomorphism O(X) → O(Y), since any such factors as O(X) → O(X) ˆ⊗_{k}K → O(Y). By the above, a general morphism f : Y → X gives rise to a morphism of topoi f_{rigid} : Y_{rigid}^{∼} → X_{rigid}^{∼}. The pullback functor, written f^{∗}, preserves overconvergent sheaves. We say that the morphism f is quasicompact if Y → X ˆ⊗K is quasi-compact. In this case f induces a morphism of sites Y_{rigid} → X_{rigid} and R^{q}f_{∗} preserves overconvergent sheaves for all q. (Use the lemma above and Proposition 2.4.1.)

If, in addition, we are given a morphism Z → X of analytic varieties over k, then we can form the fibre product:

Y ×_{X} Z := Y ×_{X ˆ⊗K} X ˆ⊗K

It is an analytic variety over K which satisfies a certain universal property regarding general morphisms; we leave it to the reader to describe this property explicitly.

2.7 Base change

The aim of the base change theorem is to compare H^{q}(Y_{a}, S|_{Y_{a}}) with (R^{q}f_{∗}S)_{a} for sheaves S on Y. Here Y_{a} is the fibre of a morphism f over the analytic point a. Let us first define this fibre.

Consider a morphism f : Y → X of analytic varieties over k and let an analytic point a of X be given. The fibre Y_{a} of f over a is defined as the fibre product of the general morphism Spm(F_{a}) → X with f. It can also be defined as the fibre of f ˆ⊗F_{a} : Y ˆ⊗F_{a} → X ˆ⊗F_{a} over the usual point a ∈ X ˆ⊗F_{a}. There results a general morphism α : Y_{a} → Y. We remark that α is quasi-compact; the morphism of sites (Y_{a})_{rigid} → Y_{rigid} comes from the functor V ↦ V_{a} on special subsets of Y. For a sheaf S on Y we write S|Y_{a} instead of α^{∗}(S). Finally, we remark that if both X and Y are affinoid then Y_{a} is affinoid with algebra O(Y) ˆ⊗_{O(X)}F_{a}.

Lemma 2.7.1 (Key lemma for the rigid case.) Let a morphism f : Y → X of affinoid spaces over k be given together with an analytic point a of X. Write α : Y_{a} → X for the resulting general morphism.

1. For every admissible open V ⊂ Y_{a} (i.e., V ∈ (Y_{a})_{rigid}) there is an admissible open W ⊂ Y such that V = W_{a}.

2. Suppose W, Z are admissible open in Y and W_{a} ⊂ Z_{a}. There is a U in the filter of a such that W ∩ f^{−1}(U) ⊂ Z.

Proof. (1) We may assume that V is a rational subset of Y_{a}. Thus V is given by inequalities |g_{1}| ≥ |g_{1}|, ..., |g_{m}| with elements g_{1}, ..., g_{m} ∈ O(Y_{a}) = O(Y) ˆ⊗_{O(X)}F_{a} generating the unit ideal. Say that f_{1}g_{1} + … + f_{m}g_{m} = 1. We may suppose that the g_{i} come from elements g_{i} ∈ O(Y) ⊗_{O(X)} k_{a}. So there is some U in the filter of a and elements G_{i} ∈ O(Y) ˆ⊗_{O(X)}O(U) = O(f^{−1}(U)) mapping to the g_{i}. If we take F_{i} ∈ O(Y) ˆ⊗_{O(X)}O(U) = O(f^{−1}(U)) mapping to elements close to the f_{i} then we see that F_{1}G_{1}+…+F_{m}G_{m} = 1+δ where δ maps to an element of O(Y_{a}) = O(Y) ˆ⊗_{O(X)}F_{a} with small norm, say with spectral norm < 1. By Lemma 2.7.2 this implies that δ gets spectral norm < 1 in O(Y) ˆ⊗_{O(X)}O(U) = O(f^{−1}(U)) for some smaller U in the filter of a. Hence we see that G_{1}, …, G_{m} generate the unit ideal in O(f^{−1}(U)). Thus W ⊂ f^{−1}U given by the inequalities |G_{1}| ≥ |G_{1}|, ..., |G_{m}| works.

(2) We may assume that W is a rational subdomain of Y. Next we write Z as a finite union Z = ⋃ Z_{i} of rational subdomains Z_{i} of Y. The finite covering W_{a} = ⋃_{i} W_{a} ∩ (Z_{i})_{a} can be refined by a rational covering W_{a} = ⋃_{j} V_{j} given by a number of elements g_{1}, …, g_{m} in O(W_{a}) generating the unit ideal. Arguing as above, we may suppose that the g_{i} come from G_{i} ∈ O(W) generating the unit ideal, after replacing X by some U in the filter of a. The rational subsets W_{j} of W defined by |G_{j}| ≥ |G_{1}|, ..., |G_{m}| cover W and each (W_{j})_{a} is contained in some (Z_{i})_{a}. If we solve the problem for all the pairs (W_{j}, Z_{i}) with (W_{j})_{a} ⊂ (Z_{i})_{a} then we solve the problem for (W, Z). Thus we have reduced to the case that both W and Z are rational subdomains of Y.

At this point we replace Z by Z ∩ W, then we are in the situation that Z ⊂ W is a rational subdomain, Z_{a} = W_{a} and we want to show that there is some U such that W ∩ f^{−1}(U) = Z ∩ f^{−1}(U) ⊂ Z. Suppose that Z is given by inequalities |h_{0}| ≥ |h_{1}|, ..., |h_{n}| where h_{0}, ..., h_{n} generate the unit ideal in O(W). In particular, h_{0} is an invertible function on Z, hence on Z_{a} = W_{a}. Arguing as in (1), we may shrink X and assume that h_{0} is invertible on W. Dividing by h_{0} we see that we may suppose that Z is given by the inequalities |h_{1}| ≤ 1, …, |h_{m}| ≤ 1. The h_{i} have norms ≤ 1 on W_{a}. Hence, by Lemma 2.7.2, we can find a U in the filter of a such that the h_{i} have norm ≤ 1 on W ∩ f^{−1}(U), i.e., such that W ∩ f^{−1}(U) = Z ∩ f^{−1}(U). □

Lemma 2.7.2 Let f : Y → X be a morphism of afinoid spaces over k, let a be an analytic point of X. Let g ∈ O(Y) whose image α(g) ∈ O(Y_{a}) has spectral norm ≤ 1 (resp. < 1). There is a U in the filter of a such that the spectral norm of g on f^{−1}(U) is ≤ 1 (resp. < 1).

Proof. Let us write O(Y) = O(X)⟨T_{1}, ..., T_{n}⟩/(G_{1}, ..., G_{m}). With obvious notations we have O(Y_{a}) = F_{a}⟨T_{1}, ..., T_{n}⟩/(G_{1}(a), ..., G_{m}(a)). If the spectral norm of α(g) is ≤ 1 it follows that α(g) is integral over the ring F_{a}⟨T_{1}, ...T_{n}⟩^{o}. Let such an equation be

α(g)^{e} + c_{e−1}α(g)^{e−1} + ... + c_{0} = 0

Write c_{i} = ∑ c_{i,β}T^{β} with all c_{i,β} ∈ F_{a} satisfying |c_{i,β}|_{a} ≤ 1.

Choose some π ∈ k with 0 < |π| < 1. For the c_{i,β} with |c_{i,β}|_{a} ≥ |π| (there are only finitely many of these!) we take a suitable U in the filter of a and elements C_{i,β} ∈ O(U) with images α(C_{i,β}) ∈ F_{a} such that |α(C_{i,β}) − c_{i,β}|_{a} < |π|. (This is possible, the image of O_{a} is dense in F_{a}.) It follows that |α(C_{i,β})|_{a} ≤ 1. Thus the inequalities |C_{i,β}| ≤ 1 define a smaller U in the filter of a where the elements C_{i,β} have spectral norm ≤ 1. For convenience we replace X by U and Y by f^{−1}U. The C_{i,β} ∈ O(X) are elements with spectral norm ≤ 1. We consider the expression

R := g^{e} + γ(∑ C_{e−1,β}T^{β})g^{e−1} + ... + γ(∑ C_{0,β}T^{β})

where γ denotes the map O(X)⟨T_{1}, ..., T_{n}⟩ → O(Y). This element R ∈ O(Y) has an image α(R) ∈ O(Y_{a}) with spectral norm < |π|. If we can find a U in the filter of a such that the spectral norm of R on f^{−1}U is < 1 then we replace again X by U and Y by f^{−1}U. After this is done the spectral norm of R on Y is < 1 and the spectral norms of the γ(∑ C_{i,β}T^{β}) are ≤ 1. It follows at once that the spectral norm of g on Y is ≤ 1.

In this way we have reduced the case ≤ 1 of the lemma to the case < 1. Let us therefore assume that the spectral norm of α(g) is < 1. For some N ≥ 1 the element α(g^{N}) ∈ O(Y_{a}) has a preimage g_{1} ∈ F_{a}⟨T_{1}, ..., T_{n}⟩ with norm < 1. Take also a g_{2} ∈ O(X)⟨T_{1}, ..., T_{n}⟩ with image g^{N} ∈ O(Y). Then α(g_{2}) − g_{1} ∈ F_{a}⟨T_{1}, ..., T_{n}⟩ lies in the ideal generated by the {G_{1}(a), ..., G_{m}(a)} and we can write

α(g_{2}) − g_{1} = ∑_{i} G_{i}(a)(∑_{β} a_{i,β}T^{β})

where the coefficients a_{i,β} ∈ F_{a} have limit 0. For the a_{i,β} with |a_{i,β}| ≥ |π| we choose a U in the filter of a and elements A_{i,β} ∈ O(U) such that the difference of the image of A_{i,β} and a_{i,β} in F_{a} has absolute value < |π|. We may suppose again that U = X. We suppose that π is choosen such that all coefficients of πG_{i}(a) have norm < 1 (in F_{a}). After changing g_{2} into g_{2} − ∑_{i} G_{i}(∑_{β} A_{i,β}T^{β}) ∈ O(X)⟨T_{1}, ..., T_{n}⟩ we have the situation that α(g_{2}) − g_{1} ∈ F_{a}⟨T_{1}, ..., T_{n}⟩ and α(g_{2}) ∈ F_{a}⟨T_{1}, ..., T_{n}⟩ are power series with coefficients having absolute values < 1. For the finitely many coefficients in O(X) of g_{2} with absolute value ≥ 1 we can find a U in the filter of a such that their spectral norms on f^{−1}U are < 1. After shrinking X to U and Y to f^{−1}U we arrive at the situation where all the coefficients of g_{2} are < 1. Hence the spectral norm of g^{N} on Y is < 1 and so the spectral norm of g on Y is < 1. □

Lemma 2.7.3 In the situation of the Key Lemma.

1. The functor S ↦ S|_{Y_{a}} preserves flasque sheaves.

2. For any sheaf S on Y the sheaf α_{rigid}^{∗}(S) = S|_{Y_{a}} can be described as follows:

Γ(W_{a}, S|_{Y_{a}}) = lim_{\overrightarrow{a∈U⊂X}} S(W ∩ f^{−1}(U)).

Here W is any special subset of Y.

Proof. (1) This is clear from the first assertion of our Lemma 2.7.1. (2) From Lemma 2.7.1 it follows that for any S we have

Γ(W_{a}, S|_{Y_{a}}) = lim_{\overrightarrow{Z⊂Y,Z_{a}=W_{a}}} S(Z).

The limit is over all admissible open Z ⊂ Y such that Z_{a} = W_{a}. ¿From Lemma 2.7.1 part 2 it follows that the Z = W ∩ f^{−1}(U) are cofinal in this system. □

Finally, we come to the base change theorem. To give a natural statement recall that a δ-functor between to abelian categories A and B is a sequence of functors {T_{n}}_{n≥0} (equipped with certain boundary operators) such that any short exact sequence in A is transformed into a long exact sequence in B. (For a more precise definition see for example [H77]).

Theorem 2.7.4 (Base change for rigid spaces.) Let f : Y → X be a quasi-compact morphism of rigid analytic varieties over k. Take any analytic point a of X and denote by Y_{a} the fibre of f over a. The functors S ↦ H^{n}(Y_{a}, S|_{Y_{a}}) (resp. S ↦ (R^{n}f_{∗}S)_{a}) on the category of abelian sheaves on Y_{rigid} form a δ-functor. These δ-functors are isomorphic: (R^{n}f_{∗}S)_{a} ≅ H^{n}(Y_{a}, S|_{Y_{a}}) for any abelian sheaf S on Y.

Proof. The functor S ↦ S|_{Y_{a}} is exact and so is the functor F ↦ F_{a} on sheaves on X. From this follows immediately that the functors under consideration form δ-functors.

Let us define the canonical morphisms:

(∗)  (R^{n}f_{∗}S)_{a} ⟶ H^{n}(Y_{a}, S|_{Y_{a}})

Since S|_{Y_{a}} = α^{∗}(S) there are canonical homomorphisms H^{n}(X, S) → H^{n}(Y_{a}, S|_{Y_{a}}) and these form a transformation of δ-functors. For any open subvariety U ⊂ X, with a ∈ U, we have f^{−1}(U)_{a} = Y_{a} and (S|_{f^{−1}(U)})|_{Y_{a}} = S|_{Y_{a}}. Hence the same argument gives

H^{n}(f^{−1}(U), S) ⟶ H^{n}(Y_{a}, S|_{Y_{a}}).

Since (R^{n}f_{∗}S)_{a} = lim H^{n}(f^{−1}(U), S) (the limit is taken over U as above) we get the desired map of δ-functors.

To prove that (*) is an isomorphism we may assume that X is affinoid. Let us first do the case that Y is affinoid. The result for n = 0 is Lemma 2.7.3 with W = Y. For a flasque sheaf on Y both sides of (*) are zero for n ≥ 1 (use 2.7.3), hence the standard argument gives the result for general n. (Inject S into a flasque sheaf and argue by induction on n.)

There are two ways to get the result for general quasi-compact Y.

(1) Choose a finite affinoid covering Y = ⋃ Y_{i} such that all Y_{i_{0}...i_{q}} = Y_{i_{0}} ∩ … ∩ Y_{i_{q}} are affinoid (see 2.5.3). The maps (*) for Y and all Y_{i_{0}...i_{q}} induce a morphism of spectral sequences {_{1}E_{n}^{pq}} → {_{2}E_{n}^{pq}} abutting to the maps (R^{p+q}f_{∗}S)_{a} ⟶ H^{p+q}(Y_{a}, S|_{Y_{a}}) and with as E_{2}-terms the maps (*):

⨁_{i_{0}...i_{q}} (R^{p}(f|_{Y_{i_{0}...iq}} )_{∗}S)_{a} ⟶ ⨁_{i_{0}...i_{q}} H^{p}((Y_{i_{0}...i_{q}})_{a}, S|_{Y_{i_{0}...iq}} )

These maps are isomorphisms by the above hence we get the result.

(2) Here we just remark that the Key Lemma holds for f : Y → X with X affinoid and Y quasi-compact. This follows immediately from the Key Lemma as it stands now. The base change theorem now follows from the same argument as for the case Y affinoid. □

Remark 2.7.5 (1) The result is of course most useful for overconvergent sheaves S since in that case the sheaves R^{n}f_{∗}S are overconvergent too and hence ”determined” by their stalks at analytic points.

(2) The proof given above is the one of [P82]. In [S93] the translation of rigid overconvergent sheaves on Z to sheaves on M(Z) is used to translate the statement into the topological base change theorem for the continuous map M(f) : M(Y) → M(X).

(3) One aim of this paper is to develop a theory of étale points and étale overconvergent sheaves such that the base change theorem and related theorems are valid.

3 Étale points and étale overconvergent sheaves

A morphism f : Y → X of analytic spaces over k is called étale if for every y ∈ Y the induced homomorphism of the local rings O_{X,f(y)} → O_{Y,y} is flat and unramified. The term unramified means that O_{Y,y}/mO_{Y,y} is a (finite) separabel field extension of the field O_{X,f(y)}/m where m denotes the maximal ideal of O_{X,f(y)}.

This notion of étale morphism is somewhat complicated. First of all the image of an étale morphism is in general not an admissible open subset. For affinoids Y, X however, it has been shown in [M81] that f(Y) is a finite union of affinoid subdomains of X. We will give a proof of this fact below (see Proposition 3.1.7).

We define the étale site in 3.2 (see [S-S]) and we compare the étale topology with the rigid topology. We define étale points and étale stalks in 3.3. In order to be able to work with étale overconvergent sheaves we construct étale wide neighbourhoods in the affinoid case. The proof of the étale base change theorem is then similar to the proof in the rigid case.

3.1 Étale morphisms of affinoids

Let an extension of rings A → B be given. Let d : B → Ω_{B/A}^{f} denote the uni-

versal finite differential module of B over A. By definition Ω_{B/A}^{f} is a finitely generated B-module and every derivation of B/A into a finitely generated B-module factors uniquely over d : B → Ω_{B/A}^{f}. This module exists in many cases where the usual universal differential module of B over A is not a finitely generated module. For affinoid algebras A, B over the same field k one can give B a presentation B = A⟨T_{1}, ..., T_{n}⟩/(G_{1}, ..., G_{m}). In this case Ω_{B/A}^{f} exists and is the quotient of the free B-module generated by dT_{1}, ..., dT_{n} by its submodule generated by dG_{1}, ..., dG_{m}. Let m be a maximal ideal of B and n the corresponding maximal ideal of A. The completions of the local rings are denoted by B̂_{m} and Â_{n}. One can show that Ω_{B̂_{m}/Â_{n}}^{f}

coincides with Ω_{B/A}^{f} ⊗ B̂_{m}. In the situation Y = Spm(B) and X = Spm(A) and y, x

corresponding to m, n one has that B̂_{m}, Â_{n} are the completions of the local rings O_{Y,y} and O_{X,f(y)}. The map between the last two local rings is unramified if and only if the map between the completed rings is unramified. The last statement is equivalent with Ω_{B̂_{m}/Â_{n}}^{f} = 0. From this and the fact that flatness is a local property one finds the following:

Observation 3.1.1 A morphism of affinoid spaces Y → X is étale if and only if O(X) → O(Y) is flat and Ω_{O(Y)/O(X)}^{f} = 0. Further Ω_{O(Y)/O(X)}^{f} = 0 if and only if the n × n minors of the matrix (\frac{∂G_{k}}{∂T_{l}}) generate the unit ideal in O(Y).

A special étale morphism of affinoid spaces f : Y → X is a morphism such that O(Y) has a presentation

O(Y) = O(X)⟨T_{1}, ..., T_{n}⟩/(G_{1}, ..., G_{n})

such that the functional determinant Δ := det (∂G_{i}/∂T_{j}) is an invertible function on Y. The morphism Y → X is indeed étale. We need only prove flatness. Let us check this in a point y ∈ Y where T_{i} = 0. The completion of the local ring O_{Y,y} is isomorphic to:

{ \atop O_{X,f(y)}}[[T_{1}, …, T_{n}]]/(G_{1}, …, G_{n})

Our assumption on Δ gives that (G_{1}, …, G_{n}) = (T_{1}, …, T_{n}) in this ring. Hence the map O_{X,f(y)} → O_{Y,y} is flat and unramified since it induces an isomorphism on completions (this is not true for general points y ∈ Y!).

We now present the proof by Huber of the fact that any étale morphism of affinoids is special étale.

Let Y → X be an étale morphism of affinoids. Choose a surjection O(X)⟨T_{1}, …, T_{n}⟩ → O(Y) with kernel I. Since the module of differentials of O(Y) over O(X) is zero, there is an isomorphism

I/I^{2} ⟶ O(Y)dT_{1} ⊕ … ⊕ O(Y)dT_{n}.

Thus we may choose G_{1}, …, G_{n} ∈ I whose classes mod I^{2} are a basis of I/I^{2}. It follows that Sp(O(X)⟨T_{1}, …, T_{n}⟩/(G_{1}, …, G_{n})) is equal to Y ⨿ Z for some affinoid Z. Let us choose an element G ∈ O(X)⟩T_{1}, …, T_{n}⟩ which is 1 on Y and 0 on Z. It follows that O(Y) has the presentation

O(Y) = O(X)⟨T, T_{1}, …, T_{n}⟩/(T G − 1, G_{1}, …, G_{n}).

It follows immediately that Y is special étale over X.

Observation 3.1.2 [Hu, 1.7.1] Any étale morphism Y → X of affinoids is a special étale morphism.

Let f : Y → X be an étale morphism of affinoids and choose a representation O(Y) = O(X)⟨T_{1}, …, T_{n}⟩/(G_{1}, …, G_{n}) such that {Δ, G_{1}, …, G_{n}} generate the unit ideal of the algebra O(X)⟨T_{1}, …, T_{n}⟩. If Z → X is a general morphism, where Z is an affinoid variety over K, then the fibre product Z ×_{X} Y is given by the affinoid algebra:

O(Z) ˆ⊗_{O(X)}O(Y) = O(Z)⟨T_{1}, ..., T_{n}⟩/(G_{1}, ..., G_{m})

Thus it is clear that Z ×_{X} Y → Z is again special étale.

Observation 3.1.3 Étale morphisms of affinoids are preserved by general base change. It follows that arbitrary étale morphisms f : Y → X are preserved by general base change. In particular, if b is an analytic point with image a in X then F_{b} is a finite separable extension of F_{a}.

The following proposition shows that any étale morphism of affinoids can locally be embedded in a finite étale morphism.

Proposition 3.1.4 Let f : Y → X be an étale morphism of affinoids. There exists a finite affinoid covering X = ⋃ X_{j}, finite étale morphisms g_{j} : Z_{j} → X_{j} and open immersions h_{j} : f^{−1}(X_{j}) → Z_{j} such that f|_{f^{−1}(X_{j})} = g_{j} ∘ h_{j}.

Proof. Let us take an analytic point a of X. By compactness of M(X) we need only to find a wide neighbourhood of a in X over which f can be factored as in the proposition. By Lemma 3.1.6 we may assume that f^{−1}({a}) = {b} for some analytic point b of Y. By the above the field extension F_{a} ⊂ F_{b} is finite separable. Hence we can find a wide U in the filter of a and a finite étale morphism ϕ : V → U such that ϕ^{−1}({a}) consists of one analytic point v with F_{v} ≅ F_{b} as F_{a} extensions. See Remark 2.1.2. Let us consider the fibre product Y ×_{X} V and its projections. The projection to the first factor is a finite étale map onto f^{−1}(U), the projection to the second factor is étale to V and there is an analytic point c =”b ×_{a} v” with F_{v} = F_{c} = F_{b}. Thus by the lemma below a wide neighbourhood W of b is isomorphic to a wide neighbourhood of c which is mapped isomorphically to an affinoid subdomain of a wide neighbourhood of v. Lemma 3.1.6 shows that replacing U by a smaller wide U we get that f^{−1}(U) ⊂ W is isomorphic to an affinoid open subdomain of V. □

Lemma 3.1.5 Let f : Y → X be an étale morphism of affinoids and b an analytic point of Y. Put a = f(b). If F_{a} ≅ F_{b} then there exists a wide U in the filter of a such that f^{−1}(U) = V ⨿ W where V is a wide neighbourhood of b in Y and the morphism V → U is an open immersion. If f is finite then V → U is an isomorphism.

Proof. In the case that f is finite we may assume that f^{−1}({a}) = {b} by replacing X by U as in Lemma 3.1.6 and Y by a connected component of f^{−1}(U). Now O(Y) is a finite locally free O(X)-module which we may assume to have constant rank by replacing X by one of its connected components. Our assumptions imply that F_{a} ≅ O(Y) ˆ⊗_{O(X)}F_{a} ≅ O(Y) ⊗_{O(X)} F_{a} hence this rank must be one. This proves the finite case.

The general case. We may replace X by a U as in the lemma below and Y by one of its connected components, hence we may assume that f^{−1}({a}) = {b}. Let us consider the fibre product Y ×_{X} Y. Since f is étale, the diagonal △(Y) is a union of connected components of Y ×_{X} Y: △ is a closed immersion and it is étale (look at local rings!), hence by the finite case above it is also open immersion. Put Z = Y ×_{X} Y \ △(Y), it is an affinoid variety. By assumption, b is not in the image of pr_{1}|_{Z} : Z → Y. Hence we can find a wide neighbourhood V of b in Y such that pr_{1}^{−1}(V) ∩ Z = ∅. (Use that the spaces M(Z) and M(Y) are Hausdorff and compact.) Next we replace X by a wide neighbourhood U of a such that f^{−1}(U) ⊂ V (see lemma below) and Y by f^{−1}(U). We see that Y ×_{X} Y ≅ Y. Thus Y → X is an open immersion ([BGR, 7.3.3], look at complete local rings in ordinary points of Y) and we have won. □

Lemma 3.1.6 Suppose f : Y → X is a morphism of affinoids and a is an analytic point of X.

1. If Y_{a} = ⋃ Y_{i} is the decomposition of the fibre of f into connected components, then there is a wide U in the filter of a such that f^{−1}(U) = ∐ V_{i} with (V_{i})_{a} = Y_{i}.

2. If Y_{a} = {b_{1}, …, b_{s}} and we are given wide neighbourhoods W_{i} ⊂ Y of b_{i} then we may choose U such that f^{−1}(U) ⊂ ⋃ W_{i}.

Proof. The first assertion is a direct consequence of the base change theorem combined with the fact that f_{∗}ℤ is overconvergent. For 2) take neighbourhoods W′_{i} of b_{i} in Y such that W′_{i} ⊂⊂_{Y} W_{i}. By our Key Lemma we can find a neighbourhood U′ of a such that f^{−1}(U′) ⊂ ⋃ W′_{i}. For some U ⊂ X with U′ ⊂⊂_{X} U we get f^{−1}(U) ⊂ ⋃ W_{i}. (Since ⋃ W′_{i} ⊂⊂_{Y} ⋃ W_{i}, compare with proof of Lemma 2.3.2 part 4.) □

Proposition 3.1.7 Let f : Y → X be an étale morphism with Y quasi-compact.

1. The image f(Y) of f is a special subset of X, i.e., it is a finite union of open affinoid subvarieties of X.

2. An analytic point a of X comes from an analytic point of f(Y) if and only if there exists an analytic point of Y mapping to a.

3. The formation of the image of f commutes with general base change: if X′ → X is a general morphism then f(Y ×_{X} X′) = f(Y) ×_{X} X′.

Proof. We remark that the last assertion follows from the other two.

Let us take an admissible affinoid covering X = ⋃ X_{j}. The admissible covering Y = ⋃ f^{−1}(X_{j}) has a finite affinoid refinement Y = ⋃_{i=1}^{n} Y_{i}. It suffices to prove the proposition for the maps Y_{i} → X_{α(i)}. Thus we may assume that both X and Y are affinoid. At this point let us prove the assertion on analytic points assuming proven the result on the image. Take an analytic point a of X. If a is not an analytic point of f(Y) then there exists a neighbourhood U ⊂ X of a such that U ∩ f(Y) = ∅. Hence f^{−1}(U) = ∅ and so Y_{a} = ∅. On the other hand, if a = f(b) for some analytic point b of Y then for any U in the filter of a, f^{−1}(U) ≠ ∅. Hence U ∩ f(Y) ≠ ∅, hence a is an analytic point of f(Y).

Let us prove the first assertion. Using our preceding proposition we may assume that f factors as Y → Z → X where Y → Z is an open immersion and Z → X is finite étale. We may also assume that Y is a rational subdomain of Z. We have a morphism

φ : Z ⟶ (ℙ_{n})^{an}

with Y = φ^{−1}(R) where R = {(x_{0}, …, x_{n}); |x_{0}| ≥ |x_{i}|}.

Suppose the degree of Z → X is constant and equal to d. Consider the d-fold fibre product

Z^{d} := Z ×_{X} Z × … ×_{X} Z

The diagonals △_{ij} = {(z_{1}, …, z_{d}) ∈ Z^{d}|z_{i} = z_{j}} are unions of connected components of Z^{d} since Z → X is étale. We put

W := Z^{d} \ ⋃_{i,j} △_{ij}

It is an affinoid variety, finite étale over X endowed with an action of S_{d} (the symmetric group on d letters). The quotient of W under this action is X in the sense that Γ(W, O_{W})^{S_{d}} = Γ(X, O_{X}). (Since Z → X is finite we are doing just algebraic geometry here.) There is a S_{d}-equivariant map

φ × … × φ : W ⟶ ℙ_{n}^{an} × … × ℙ_{n}^{an}

which descends to a morphism

S_{d}(φ) : X → ((ℙ_{n})^{d}/S_{d})^{an}.

It is clear that f(Y) = S_{d}(φ)^{−1}(R(d)/S_{d}) with

R(d) = ⋃_{i} ℙ_{n}^{an} × … × R × … ℙ_{n}^{an}.

There is an obvious formal scheme (ℙ_{n,k^{∘}}^{d})^{∧} giving rise to (ℙ_{n}^{d})^{an} and R(d) corresponds to a S_{d}-stable formal open subscheme of it, namely:

U := ⋃_{i}(ℙ_{n,k^{∘}})^{∧} × … × (𝔸_{n,k^{∘}})^{∧} × … × (ℙ_{n,k^{∘}})^{∧}

It follows that R(d)/S_{d} corresponds to the formal open subsecheme U/S_{d} of (ℙ_{n,k^{∘}}^{d})^{∧}. Thus R(d)/S_{d} is a special subset of ((ℙ_{n})^{d}/S_{d})^{an} and hence so is S_{d}(φ)^{−1} of it. □

3.2 The étale site

Let X be an analytic variety over k. In this subsection we recall the definition of the étale site of X (see [S-S, p. 58]). We give a criterium for a presheaf to be a sheaf and we give some examples of étale sheaves. Finally, we prove Hilbert 90 in our situation and we prove that étale cohomology of coherent O-modules agrees with rigid cohomology.

The underlying category of the site X_{étale} will be the category of étale morphisms f of analytic varieties f : Y → X. A morphism of f into f′ is a morphism g : Y → Y′ such that f′ ∘ g = f; the morphism g is automatically étale.

We say that a familly of étale morphisms {g_{i} : Z_{i} → Y }_{i∈I} is an étale covering if it has the following property:

For any (some) choice of admissible affinoid coverings Z_{i} = ⋃_{j} Z_{i,j} we have Y = ⋃_{i,j} g_{i}(Z_{i,j}) and this is an admissible covering in the G-topology of Y.

This makes sense since the subsets g_{i}(Z_{i,j}) are admissible (special) subsets (see Proposition 3.1.7). We remark that the property is local on Y in the following sense: if Y = ⋃ Y_{l} is an admissible affinoid covering then {g_{i} : Z_{i} → Y } is an étale covering if and only if {g_{i} : g_{i}^{−1}(Y_{l}) → Y_{l}} is an étale covering for all l. This is so since both assertions are equivalent to the following assertion:

For each l there are finitely many (i_{α}, j_{α}), α = 1, …, n
such that Y_{l} ⊂ ⋃_{α=1}^{n} g_{i_{α}}(Z_{i_{α},j_{α}}).

¿From this it also immediately follows that if {Z_{i} → Y } is an étale covering and {X_{i,j} → Z_{i}} are étale coverings then {X_{i,j} → Y } is an étale covering.

Lemma 3.2.1 Suppose {Y_{i} → X} is an étale covering and Z → X is a general morphism. The fibre product {Z ×_{X} Y_{i} → Z} is an étale covering.

Proof. This follows immediately from the definition, the remarks above and Proposition 3.1.7. □

It follows from the above that the category X_{étale}, equipped with the familly of étale coverings as defined above is a site. It is also clear from the lemma that any (general) morphism f : Z → X defines a morphism of sites Z_{étale} → X_{étale} (given by the functor (Y → X) ↦ (Z ×_{X} Y → Z)). The functors on étale sheaves will be denoted by f_{∗} and f^{∗} as usual.

For any object Y → X of X_{étale} we get a mophism of sites

r_{Y/X} : X_{étale} ⟶ Y_{rigid},

comparing rigid and étale topologies. It is defined by the inclusion of categories Y_{rigid} ⊂ X_{étale}, if S is a sheaf on X_{étale} then Γ(V, (r_{Y/X})_{∗}S) = Γ(V, S). Sometimes we will use the notation S|_{Y_{rigid}} in stead of (r_{Y/X})_{∗}S; we will also use this notation for presheaves S on X_{étale}. If Y = X the morphism r_{X/X} will be denoted r : X_{étale} → X_{rigid}. If a is an analytic point of X then we put S_{a} := (S|_{X_{rigid}})_{a} = r_{∗}(S)_{a}.

Proposition 3.2.2 The presheaf S on X_{étale} is a sheaf if and only if the following two conditions hold:

1. For any Y in X_{étale} the presheaf S|_{Y_{rigid}} is a sheaf.

2. For any surjective finite étale morphism Y′ → Y of affinoids in X_{étale} the sequence ∅ → S(Y) → S(Y′)\overrightarrow{→}S(Y′ ×_{Y} Y′) is exact.

Proof. Suppose S satisfies 1) and 2). We claim that S also satisfies 2) for any finite étale morphism Y′ → Y in X_{étale} with Y quasi-compact. Just cover Y by affinoids as in Lemma 2.5.3 and use 1) to show that it suffices to know 2) for all the resulting affinoid finite étale coverings.

Let us take a morphism φ : Z → U in X_{étale} such that

1. φ is surjective,

2. Z and U are quasi-compact,

3. φ factors as Z → V → U with V → U finite étale and Z → V an open immersion.

We claim that for any such φ the sequence

∅ → S(U) → S(Z)\overrightarrow{→}S(Z ×_{U} Z)

is exact. We prove this by induction on the degree of the morphism V → U. (If it is 1 then φ is an isomorphism and our claim trivial.) Suppose therefore that the degree of V → U is d and that we have proved our claim in the cases where the corresponding degree is less than d.

Since V → U is finite étale we have that the diagonal △(V) ⊂ V ×_{U} V is a union of connected components of V ×_{U} V. Its complement W ⊂ V ×_{U} V is thus a quasi-compact variety and the morphism pr_{2} : W → V is finite étale of degree < d. Put Z′ = Z ×_{U} V ∩ W and U′ = pr_{2}(Z′), both are quasi-compact (see Propositon 3.1.7). The surjective étale morphism φ′ = pr_{2} : Z′ → U′ factors through V′ := W ∩ pr_{2}^{−1}(U′) → U′ which is finite étale of degree < d. Furthermore, it is clear that V = U′ ∩ Z.

We have the following commutative diagram:

∅ ⟶ S(U) ⟶ S(Z) \overrightarrow{⟶} S(Z ×_{U} Z)
↓ ↓ ↓
∅ ⟶ S(U′) ⟶ S(Z′) \overrightarrow{⟶} S(Z′ ×′_{U} Z′)

The diagram shows that any element s ∈ S(Z) such that p_{1}^{∗}(s) = p_{2}^{∗}(s) gives a unique element (by induction) s′ ∈ S(U′) such that s′|_{Z′} = s|_{Z′}. It is also true that s′|_{U′∩Z} = s|_{U′∩Z} (use induction hypothesis for the morphism (φ′)^{−1}(U′ ∩ Z) → U′ ∩ Z). Hence by 1) for the covering V = Z ∪ U′ we get a unique section s_{V} ∈ S(V) with s_{V} |_{Z} = s and s_{V} |_{U′} = s′. We want to show that p_{1}^{∗}(s_{V}) = p_{2}^{∗}(s_{V}) on V ×_{U} V. Remark that V ×_{U} V has the following admissible special covering

V ×_{U} V = Z ×_{U} Z ∪ U′ ×_{U} Z ∪ Z ×_{U} U′ ∪ U′ ×_{U} U′.

Hence by 1) we need only to prove p_{1}^{∗}(s_{V}) = p_{2}^{∗}(s_{V}) on each of these. For the most difficult case, namely U′ ×_{U} U′, we remark that the morphism Z′ ×_{U} Z′ → U′ ×_{U} U′ is the composition Z′ ×_{U} Z′ → U′ ×_{U} Z′ → U′ ×_{U} U′ of morphisms to which our induction hypothesis applies. Hence the map S(U′ ×_{U} U′) → S(Z′ ×_{U} Z′) is injective. At this point the commutative diagram

S(Z) \overrightarrow{⟶} S(Z ×_{U} Z)
↓  ↓
S(Z′) \overrightarrow{⟶} S(Z′ ×′_{U} Z′)

gives the desired result.

To prove that the presheaf S is a sheaf we have to show that any étale covering {g_{i} : Z_{i} → Y } in X_{étale} gives an exact sequence

∅ ⟶ S(Y) ⟶ ∏ S(Z_{i})\overrightarrow{⟶} ∏ S(Z_{i} ×_{Y} Z_{j}).

By choosing an admissible affinoid covering Y = ⋃ Y_{j} and using 1) it is easy to reduce to the case Y affinoid. Similarly we may reduce to the case all Z_{i} affinoid also. Using propositions 3.1.4 and 3.1.7 we may assume that each Z_{i} → Y factors as Z_{i} → V_{i} → U_{i} ⊂ Y as above. It is now easy to deduce the result from our claim above. Compare also with [M80, II 1.5]. □

Examples of sheaves on the étale site. It follows easily from the criterium given above that the following presheaves are sheaves. A general object of X_{étale} will be denoted by f : Y → X.

1. The structure sheaf 𝔾_{a} defined by Y ↦ Γ(Y, O_{Y}).

2. The sheaf 𝔾_{m} defined by Y ↦ Γ(Y, O_{Y}^{∗}).

3. For any real number r we can look at the subsheaf of 𝔾_{a} given by Y ↦ {f ∈ Γ(Y, O_{Y}) : |f(y)| ≤ r∀y ∈ Y }. We can also replace the ≤-sign by the < sign. If r ≤ 1 we can define a subsheaf of 𝔾_{m} by inequalities of the form |1 − f(y)| ≤ r.

4. Any representable sheaf Y ↦ Mor_{X}(Y, Z) given by some variety Z over X.

5. For any abelian group A we have the constant sheaf A_{X} with stalks A defined by: Y ↦ the set of maps Y → A constant on connected components of Y. (This is in fact a representable sheaf, namely represented by ∐_{a∈A} X.)

6. If n is prime to the characteristic of k then we define μ_{n} as the kernel of the homomorphism 𝔾_{m} → 𝔾_{m} given by multiplication by n. If k contains a primitive n^{th} root of unity ζ then μ_{n} ≅ ℤ/nℤ_{X} · ζ. There is a Kummer exact sequence

1 ⟶ μ_{n} ⟶ 𝔾_{m} ⟶ 𝔾_{m} ⟶ 1.

7. Suppose that F is a coherent sheaf of O_{X}-modules on X. We define a sheaf W (F) of 𝔾_{a}-modules on X_{étale} as follows: Y ↦ Γ(Y, f^{∗}F), here f^{∗} denotes pullback of coherent O-modules: f^{∗}F := f^{∗}(F) ⊗_{f^{∗}O_{X}} O_{Y}. It is clear that W (O_{X}) = 𝔾_{a}.

Suppose that we are given an étale covering {Y_{i} → X}. We claim that this coverings allows effective descent of coherent O-modules. This means the following: suppose we are given for each i a coherent O_{Y_{i}}-module F_{i} and descent data. This means isomorphisms of coherent sheaves

φ_{ij} : pr_{1}^{∗}F_{i} ⟶ pr_{2}^{∗}F_{j}

on Y_{i} ×_{X} Y_{j} satisfying the cocycle relation φ_{ij} ∘ φ_{jk} = φ_{ik} on Y_{i} ×_{X} Y_{j} ×_{X} Y_{k}. In this situation there exists a unique coherent sheaf of O_{X}-modules F giving rise to F_{i} on each Y_{i} and inducing the isomorphisms φ_{ij}. In addition, homomorphisms of systems (F_{i}, φ_{ij}) as above should correspond to homomorphisms of the corresponding sheaves F. The proof of this is gotten by paraphrasing the proof of Proposition 3.2.2 in this case. Indeed, the question we are considering is whether the association Y ↦ the category of coherent O_{X}-modules defines a sheaf of categories. It is clear that rigid coverings and finite étale coverings allow effective descent for coherent Omodules and hence the reasoning of the Proposition applies.

Corollary 3.2.3 Any descent datum for coherent sheaves over an étale covering of X is effective.

Corollary 3.2.4 For any analytic variety X we have the following isomorphisms:

H^{1}(X, 𝔾_{m}) ≅ H^{1}(X, O_{X}^{∗}) ≅ P ic(X)

Proof. Of course the group P ic(X) is the group of isomorphism classes of line bundles on X. Since H^{1} = Ȟ^{1} any element in H^{1}(X, 𝔾_{m}) can be considered as descent data for invertible O-modules. By the above these are effective and hence come from an element of H^{1}(X, O_{X}^{∗}). □

Proposition 3.2.5 Suppose F is a coherent sheaf of O_{X}-modules. The natural maps H^{i}(X_{rigid}, F) → H^{i}(X, W (F)) are isomorphisms.

Proof. The maps arise from the identification r_{∗}W (F) ≅ F and the adjunction map r^{∗}F = r^{∗}r_{∗}W (F) → W (F). Hence the result for i = 0. We are going to prove the proposition by induction on i. Take n and suppose the proposition is proven for all X, F and i ≤ n − 1.

For any f : Y → X in X_{étale} consider the map

H^{n}(Y, f^{∗}F) = H^{n}(Y, (r_{Y/X})_{∗}W (F)) ⟶ H^{n}(Y, W (F)).

This map is injective: by induction hypothesis the sheaves R^{i}(r_{Y/X})_{∗}W (F) on Y_{rigid} are zero for i = 1, …, n − 1 (they are the sheaves associated to the presheaves U ↦ H^{i}(U, W (F))). Thus the spectral sequence H^{j}(Y, R^{i}(r_{Y/X})_{∗}W (F)) ⇒ H^{i+j}(Y, W (F)) gives the result. Consider the presheaf H^{n} on X_{étale} defined by

Y ⟼ H^{n} := Coker(H^{n}(Y, f^{∗}F) → H^{n}(Y, W (F))).

The sheaf associated to this presheaf is zero since any cohomology class in H^{n}(Y, W (F)) can be killed by an étale covering. Therefore, if we show that H^{n} is a sheaf then we are done. To do this we use the criterium from Proposition 3.2.2.

Take any admissible covering U : Y = ⋃ Y_{j} of some étale f : Y → X. We have the morphism of spectral sequences

Ȟ^{i}(U, \underline{H}^{j}(f^{∗}F)) ⇒ H^{i+j}(Y, f^{∗}F)
↓  ↓
Ȟ^{i}(U, \underline{H}^{j}(W (F))) ⇒ H^{i+j}(Y, W (F))

(see for example [M80, III Proposition 2.7]). We leave it to the reader to verify that this and our induction hypothesis immediately imply that

0 ⟶ H^{n}(Y) ⟶ ∏ H^{n}(Y_{j}) ⟶ ∏ H^{n}(Y_{i} ∩ Y_{j})

is exact.

Finally, let Z → Y be a finite étale morphism of affinoids in X_{étale}. Put A = O(Y), B = O(Z) and M = Γ(Y, W (F)). We use the notation Z^{n} = Z×_{X} Z×…×_{X} Z. It is an affinoid variety. Thus we have that H^{i}(Z_{rigid}^{n}, F ⊗ O_{Z}^{n}) = 0 for all i, n. Furthermore, the complex

0 ⟶ M ⟶ M ˆ⊗_{A}B ⟶ M ˆ⊗_{A}B ˆ⊗_{A}B ⟶ …

is exact. (As the ring extension A ⊂ B is finite we may replace the completed tensor products by usual ones and then the result is classical.) Thus the spectral sequence Ȟ^{i}(U, \underline{H}^{j}(W (F))) ⇒ H^{i+j}(Y, W (F)) for the covering U = \{Z → Y \} and induction hypothesis gives that

0 ⟶ H^{n}(Y, W (F)) ⟶ H^{n}(Z, W (F)) ⟶ H^{n}(Z ×_{Y} Z, W (F))

is exact. We have won. □

Corollary 3.2.6 Suppose the homomorphism A → B of affinoid algebras defines a surjective étale morphism of affinoids. For any finite A-module M the complex

0 ⟶ M ⟶ M ˆ⊗_{A}B ⟶ M ˆ⊗_{A}B ˆ⊗_{A}B ⟶ …

is exact.

3.3 Étale points and stalks

Let us define an étale point of the analytic variety X. An étale point e above the analytic point a of X is a separable closure F_{a} ⊂ H_{e} of F_{a}. We will always denote by F_{e} the completion of H_{e}. Note that the field F_{e} is algebraically closed (see [BGR, 3.4.1/6]). Therefore an étale point e over a also corresponds to an algebraically closed complete extension F_{a} ⊂ F_{e} such that the algebraic closure of F_{a} lies dense in F_{e}. The group Gal(H_{e}/F_{a}) is equal to the group of continuous F_{a}-isomorphisms F_{e} → F_{e}; this profinite group will be denoted G_{e}.

An étale neighbourhood of e is a triple (Y, b, ϕ), where Y is a variety étale over X, the analytic point b of Y maps to a and ϕ : F_{b} → F_{e} is an F_{a}-embedding. A morphism (Y, b, ϕ) → (Y′, b′, ϕ′) is a morphism g : Y → Y′ over X such that g(b) = b′ and ϕ′ = ϕ ∘ g^{∗}. Two étale neighbourhoods (Y_{1}, b_{1}, ϕ_{1}) and (Y_{2}, b_{2}, ϕ_{2}) are dominated by a third one: take Y = Y_{1} ×_{X} Y_{2}, take the point b in Y corresponding to some factor of F_{b_{1}} ⊗_{F_{a}} F_{b_{2}} and ϕ = ϕ_{1} ⊗ ϕ_{2}. In this way we see that the category of all étale neighbourhoods of e give a filtered system.

The stalk S_{e} of a sheaf S on X_{étale} at the étale point e is defined by the formula:

S_{e} := lim_{\overrightarrow{(Y,b,ϕ)}} S(Y)

The limit is take over the category of étale neighbourhoods of e. If e′ (given by F_{a} ⊂ F_{e′}) is another étale point lying over a, we get by choosing a continuous isomorphism ψ : F_{e} → F_{e′} a functor (Y, b, ϕ) ↦ (Y, b, ψ ∘ϕ) of the category of étale neighbourhoods of e to the category of étale neighbourhoods of e′. This gives an isomorphism of stalks

S_{e} := lim_{\overrightarrow{(Y,b,ϕ)}} S(Y)  ⟶^{ψ^{∗}}  S_{e′} := lim_{\overrightarrow{(Y,b,ϕ)}} S(Y)
s ∈ S(Y) w.r.t. (Y, b, ϕ) ⟼ s ∈ S(Y) w.r.t. (Y, b, ψ ∘ ϕ)

In particular we get an action of G_{e} on the stalk functor S ↦ S_{e}. It is clear that this action is continuous (with the discrete topology on S_{e}), since any ϕ is stabilized by an open subgroup of G_{e}.

Let us construct some étale neighbourhoods of e. Take a finite Galois extension F_{a} ⊂ L, say with group G, contained in F_{e}. By Remark 2.1.2 we can find an affinoid neighbourhood U of a in X and a finite étale morphism g : V → U, such that G acts on V over U, g^{−1}({a}) = {v} and F_{v} ≅ L (G-equivariant). We may also assume that V and U are connected. It is clear that (V, v, F_{v} → L ⊂ F_{e}) is an étale neighbourhood of e. We claim that these étale neighbourhoods are cofinal in the system of all étale neighbourhoods of e.

Indeed, given an arbitrary (Y, b, ϕ) take L such that it contains ϕ(F_{b}). The fibre product Y ×_{X} V contains a point c with pr_{1}(c) = b, pr_{2}(c) = v and F_{b} → F_{c} ≅ F_{v} ≅ L ⊂ F_{e} equals ϕ. Using Lemma 3.1.5 we see that there is a commutative diagram

V′ ⟶  Y
↓  ↑
V ⟵ Y ×_{X} V

where V′ ⊂ V is an affinoid subdomain containing v. By the Key Lemma we can find a smaller affinoid neighbourhood U′ of a in X such that g^{−1}(U′) ⊂ V′. It is clear that (g^{−1}(U′), v, F_{v} → L ⊂ F_{e}) is of the form described above and dominates (Y, b, ϕ).

Lemma 3.3.1 In the situation above.

1. The association S ↦ S_{e} is an exact functor of the category of étale sheaves on X to the category of continuous G_{e}-sets.

2. For any étale neighbourhood (Y, b, ϕ) we have

(S|_{Y_{rigid}})_{b} = ((r_{Y/X})_{∗}S)_{b} = H^{0}(Gal(H_{e}/ϕ(F_{b})), S_{e}).

In particular S_{a} = (r_{∗}S)_{a} = H^{0}(G_{e}, S_{e}).

3. The cohomological dimension cd(G_{e}) of the pro-finite group G_{e} = Gal(H_{e}/F_{a}) is less than or equal to dim X + cd(Gal(k^{sep}/k)).

Proof. (1) We have to show that a surjection of étale sheaves S → T induces a surjection S_{e} → T_{e}. Take an element t ∈ T (Y) for some étale neighbourhood (Y, b, ϕ). There is some étale covering {Z_{i} → Y } of Y and elements s_{i} ∈ S(Z_{i}) such that s_{i} maps to the element t|_{Z_{i}} in T (Z_{i}). By Lemma 3.2.1 the familly {(Z_{i})_{b} → b} is a covering, hence there is some i_{0} and analytic point b_{i_{0}} ∈ Z_{i_{0}} mapping to b. Thus (Z_{i_{0}}, b_{i_{0}}, χ) is a neighbourhood of e (here χ is some extension of ϕ) and s_{i_{0}} gives an element of S_{e} lifting t. (Another proof follows from the result S_{e} ↔ i_{a}^{∗}S below.)

(2) We only do the case X = Y, a = b. Take an element s ∈ S_{e}, which is fixed by the group G_{e}. By our results above we may assume that s ∈ S(V) for some special neighbourhood (V, v, F_{v} → L ⊂ F_{e}) constructed above. Our assumption is that s is stable under the action of G acting on S(V) via its action on V. If we show that V ×_{U} V = V ×_{X} V is isomorphic to V × G then the sheaf property of S will imply that s comes from a unique section of S over U and hence we will be done. However, this again is a consequence of Remark 2.1.2 at least after shrinking U a bit.

(3) We use the notations of Lemma 2.1.1. By [S64, II 4.1] we may replace F_{a} by K, since this can at most increase the cohomological dimension. The field K is the completion of k(\underline{t}) = k(t_{1}, …, t_{d}) for some valuation and some d ≤ dim X. But then the group Gal(K^{sep}/K) is a closed subgroup of the group Gal(k(\underline{t})^{sep}/k(\underline{t})). We conclude by [S64, I Proposition 14, II Proposition 11]. □

There is a more canonical way to understand the étale stalks S_{e}. Consider the general morphism

i_{a} : a = Spm(F_{a}) ⟶ X.

It is clear that the category of sheaves on a_{étale} is equivalent to the category of discrete G_{e}-sets. (Compare [M80, II 1.9].) Therefore i_{a}^{∗} is a functor of sheaves on X_{étale} to the category of discrete Gal − cont(F_{e}/F_{a})-sets. This functor is precisely our functor S ↦ S_{e}. The functor (i_{a})_{∗} has the following description: if M is a set with a continuous G_{e}-action, then

Γ(Y, (i_{a})_{∗}M) = ∏_{b∈Y_{a}} (Hom_{F_{a}}(F_{b}, F_{e}) × M)^{G_{e}}

We leave it to the reader as a nice exercise that this functor is exact. It follows from the yoga of adjoint functors that S ↦ (i_{a})^{∗}S = S_{e} transforms injective sheaves into injective G_{e}-modules.

Corollary 3.3.2 There are canonical isomorphisms

(R^{q}r_{∗}S)_{a} ≅ H^{q}(G_{e}, S_{e}).

Proof. For q = 0 this is the lemma above. It follows for general q by the usual argument using that if S is injective then both sides are zero. (See above.) □

As in the rigid case we do not have enough étale points to separate étale sheaves. To overcome this difficulty we introduce the étale overconvergent sheaves: A sheaf S on X_{étale} is said to be (étale) overconvergent if S|_{Y_{rigid}} is overconvergent for all Y in X_{étale}. Before we can prove interesting properties of these sheaves we need some technical preparations; these will be done in the next section.

3.4 Étale overconvergent sheaves on affinoids

In this section X will be an affinoid variety. Let f : Y → X be a morphism with Y affinoid and let b be an analytic point of Y. We will say that Y is a wide neighbourhood of b over X if there exists an affinoid generating system f_{1}, …, f_{n} of O(Y) over O(X) such that |f_{i}|_{b} < 1 for all i = 1, …, n. Note that this agrees with our definition in §3.1 in the case that f is an open immersion.

Next we define the notion of relative compactness over X. Let us take a quasicompact analytic variety Z over X and a quasi-compact open subvariety Y ⊂ Z. We say that Y is relatively compact in Z over X, or that Z is a wide neighbourhood of Y over X, if for any analytic point b of Y there is an affinoid neighbourhood V ⊂ Z of b which is a wide neighbourhood of b over X. Notation: Y ⊂⊂_{X} Z. Remark that if Y ⊂⊂_{Z} Y′ in this situation then also Y ⊂⊂_{X} Y′. We note that if both Y and Z are affinoid then this agrees with the definition of [BGR, p. 394] (proof same as proof of [S93, Proposition 23], see also [B90, §2.5]).

Suppose that f : Y → X is an étale morphism with Y quasi-compact. We want to construct wide neighbourhoods of f. We only do this in the case that f is an étale morphism of affinoids. Thus Y is affinoid and O(Y) has a presentation:

O(Y) = O(X)⟨T_{1}, …, T_{n}⟩/(G_{1}, …, G_{n})

such that Δ = det(∂G_{i}/∂T_{j}) generates the unit ideal of O(Y). A fundamental property of special étale morphisms is that we may always choose this presentation such that G_{1}, …, G_{n} ∈ O(X)[T_{1}, …, T_{n}]. This follows immediately from the proposition below; in it we use |R| for the supremum norm of an element R ∈ O(X)⟨T_{1}, …, T_{n}⟩.

Lemma 3.4.1 In the situation above there exists an ϵ > 0 such that if we take any R_{1}, …, R_{n} ∈ O(X)⟨T_{1}, …, T_{n}⟩ with |R_{i}| < ϵ then we have:

1. The affinoid algebra O(X)⟨T_{1}, …, T_{n}⟩/(G_{1} +R_{1}, …, G_{n} +R_{n}) defines a special étale morphism f′ : Y′ → X.

2. There exists an isomorphism Y ≅ Y′ of analytic varieties over X.

Proof. Let us write Δ + R for the determinant of the matrix (∂(G_{i} + R_{i})/∂T_{j}). It is clear that if the R_{i} have small norm then R has small norm. Since Δ, G_{i} generate the unit ideal of O(X)⟨T_{1}, …, T_{n}⟩ it follows that Δ + R, G_{i} + R_{i} also generate the unit ideal if |R_{i}| is small enough. This proves (1).

We claim there exists for any positive δ < 1 an ϵ > 0 such that for any affinoid O(X)-algebra A the following holds: If there are a_{1}, ..., a_{n} ∈ A with all |a_{i}| ≤ 1 and all |G_{i}(a_{1}, ..., a_{n})| < ϵ, then there are b_{1}, …, b_{n} ∈ A such that all |a_{i} − b_{i}| < δ and all G_{i}(b_{1}, ..., b_{n}) = 0.

We suppose given a_{1}, ..., a_{n} ∈ A with all |a_{i}| ≤ 1 and all |G_{i}(a_{1}, ..., a_{n})| < ϵ, the size of ϵ will be determined later. For an element b = (b_{1}, ..., b_{n}) ∈ A^{n} we write ∥b∥ = max |b_{i}|. Further G = (G_{1}, ..., G_{n}) is seen as a map from {b ∈ A^{n}; ∥b∥ ≤ 1} to A^{n}. Let ∂G/∂T denote the Jacobian matrix of G. Note that |(∂G/∂T)| is bounded from below away from zero on Y, hence is bounded from below by η > 0 in a neighbourhood of the form |G_{i}| ≤ ϵ_{0}, some ϵ_{0} > 0. We apply Newton’s method; consider the map Z : b ↦ b − (∂G/∂T (b))^{−1}G(b). By the remark above, and by considering a power series expansion of the map G, we see that for δ small enough (so that the quadratic and higher oreder terms of the power series are neglectible) and ϵ < ηδ (for the constant terms) this defines a selfmap of the set S := {b ∈ A^{n}; ∥b−a∥ ≤ δ}. Moreover it is then clear that the map Z : S → S is a contraction. The fixed point b of the contraction satisfies G(b) = 0.

We apply this claim to A = O(X)⟨T_{1}, ..., T_{n}⟩/(G_{1} + R_{1}, ..., G_{n} + R_{n}) with the |R_{i}| < ϵ and where a_{i}, i = 1, ..., n is the image of T_{i} in A. There results a morphism of affinoid O(X)-algebras α : O(X)⟨T_{1}, ..., T_{n}⟩/(G_{1}, ..., G_{n}) → O(X)⟨T_{1}, ..., T_{n}⟩/(G_{1} + R_{1}, ..., G_{n} + R_{n}), with α(T_{i}) close to T_{i}. We can do the same in the other direction to get β : O(X)⟨T_{1}, ..., T_{n}⟩/(G_{1} + R_{1}, ..., G_{n} + R_{n}) → O(X)⟨T_{1}, ..., T_{n}⟩/(G_{1}, ..., G_{n}), with β(T_{i}) close to T_{i}. The composition is an endomorphism of O(Y) = O(X)⟨T_{1}, ..., T_{n}⟩/(G_{1}, ..., G_{n}) as an O(X)-algebra which is close to the identity. It follows that this must be the identity by looking at the graph of it in the fibre product Y ×_{X} Y, where the diagonal is a union of connected components. □

Let us take an étale morphism of affinoids f : Y → X and take a presentation O(Y) = O(X)⟨T_{1}, …, T_{n}⟩/(G_{1}, …, G_{n}) with G_{i} ∈ O(X)[T_{1}, …, T_{n}]. The functional determinant of this presentation Δ = det(∂G_{i}/∂T_{j}) is viewed as a function on X × 𝔸^{N,an}. We define a morphism f(r) : Y (r) → X for r ∈ √|k^{∗}|, r > 1 as follows:

Y (r) = {(x, t_{1}, …, t_{n}) ∈ X × 𝔸^{n,an}; |t_{i}| ≤ r and G_{i}(x, t_{1}, …, t_{n}) = 0}

We claim that if our r is close to 1 then f(r) will again be special étale. To see this we note that there is a presentation:

O(Y (r)) = O(X)⟨S_{1}, …, S_{n}, T′_{1}, …, T′_{n}⟩/((T′_{i})^{m} − π^{−m+1}S_{i}, G_{i}(πT′_{1}, …, πT′_{n}))

Here π is an element of k with r^{m} = |π| and the relation of the coordinates is that S_{i} = π^{−1}T_{i}^{m} and T′_{i} = π^{−1}T_{i}. The functional determinant of this presentation is π^{−mn}Δ|_{Y (r)}. It is therefore clear that Y (r) → X is special étale as soon as Δ ∈ Γ(Y (r), O_{Y (r)}) is invertible; this will be the case for r sufficiently close to 1 (the zero locus of Δ lies a positive distance away from Y!). Finally it is clear that Y ⊂⊂_{X} Y (r). We will use the notation Y (r) even if no explicit presentation of O(Y) is given, the number r will always denote an element of √|k^{∗}| bigger than 1 and small enough.

At this point we want to prove the analog of Lemma 2.3.1 in this situation. However, we need to be carefull since any étale U → X has many nonseparated wide neighbourhoods, so the wide neighbourhoods Y ⊂ Y (r) can only be cofinal in the system of separated wide neighbourhoods. Allthough this is in fact true, we restrict ourselves to the case of affinoid varieties.

Lemma 3.4.2 With notations as above.

1. Let W, U ⊂⊂_{X} V be affinoid varieties étale over X and f : V → W a morphism over X. If V′ is a wide neighbourhood of U in V, then f(V′) is a wide neighbourhood of f(U) in W. For varying V′ these give a cofinal system of wide neighbourhoods of f(U). If f|_{U} is an isomorphism U → f(U) then for some U ⊂⊂_{V} V′ f induces an isomorphism V′ → f(V′).

2. If U ⊂⊂_{X} V are affinoid varieties étale over X and φ : Y → U is a morphism then for some r > 1 there exists an extension φ(r) : Y (r) → V of φ. This extension is unique if r is sufficiently close to 1. In particular the Y (r), r > 1 form a cofinal system of affinoid wide neighbourhoods of Y over X.

3. Suppose that Y_{i} → X, i = 1, …, n is étale and Y_{i} affinoid. We have:

Y_{1}(r) ×_{X} … ×_{X} Y_{n}(r) = (Y_{1} ×_{X} … ×_{X} Y_{m})(r)

Proof. Suppose we show that if b is an analytic point of U (with image a in W), then there is a wide neighbourhood V_{b} of b in V′ such that f(V_{b}) is a wide neighbourhood of a in W. This immediately implies the first statement of (1). The statement on cofinality then follows immediately by letting V′ run through the inverse images of such a system of wide neighbourhoods of f(U). Let Z denote the complement of the diagonal in V ×_{W} V; it is a union of connected components and hence affinoid. Under the last assumption of (1) we have pr_{1}(Z)∩U = ∅. Thus for some wide neighbourhood V′ of V we have V′ ×_{W} V′ ≅ V′ and hence it will map isomorphically onto f(V′) (compare with Lemma 3.1.5).

Let us construct the neighbourhood V_{b}. By assumption there exists an affinoid generating system f_{1}, …, f_{r} of O(V) over O(X) such that |f_{i}|_{b} < 1. Take a wide neighbourhood V_{b} of b such that ||f_{i}||_{V_{b}} < 1. By Lemma 3.1.6 we can find a wide neighbourhood W_{a} of a in W such that f^{−1}(W_{a}) = ⋃ V_{i} as in 3.1.6.1 and V_{1} ⊂ V_{b} is a wide neighbourhood of b. Thus we may replace W by W_{a} and V by V_{1} and assume that ||f_{i}||_{V} < 1 for a generating system f_{1}, …, f_{r} of O(V) over O(W). But then V is finite over W [BGR, 9.6.3/6], so we get the existence of V_{b} by Lemma 3.1.5.

To prove (2) we apply (1) to the projection Y (r) ×_{X} V → Y (r). We see that there exists a wide neighbourhood of the graph Γ_{φ} ⊂ Y ×_{X} U ⊂ Y (r) ×_{X} V which maps isomorphically onto a wide neighbourhood Y′ of Y in Y (r). Hence we can find some r′, 1 < r′ < r such that Y (r′) ⊂ Y′ (see Lemma 2.3.1). This r′ works.

The proof of (3) is formal. □

The lemma above allows us to work with étale morphisms of affinoids only. Therefore we introduce the special étale site of X. (Recall that X is affinoid.) It is denoted X_{étale}^{sp} and is defined as follows:

1. Objects are étale morphisms Y → X with Y affinoid, i.e. special étale ones.

2. Morphisms are morphisms of analytic spaces over X.

3. Coverings are those finite families of morphisms {f_{i} : Y_{i} → Y } such that ⋃ f_{i}(Y_{i}) = Y.

It follows from the remarks made after the definition of special étale morphisms and Lemma 3.2.1 that this is indeed a site. It is functorial with respect to (general) morphisms of affinoids: Z → X induces a morphism of sites Z_{étale}^{sp} → X_{étale}^{sp}.

The morphism of site X_{étale} → X_{étale}^{sp}, given by the inclusion functor, induces an equivalence of associated topoi. (Use 3.1.2.)

Lemma 3.4.3 The topos of sheaves on X_{étale}^{sp} is coherent (see [SGA 4, Exposé VI]). In particular, étale cohomology of étale abelian sheaves on X commutes with filtered direct limits, see [Ibid, 5.2].

Proof. In the site X_{étale}^{sp} finite fibred products are representable and any object is quasi-compact (see [Ibid, Definition 1.1]). Since it also has a final object we are done, see [Ibid, 2.4.1]. □

The following lemma characterizes overconvergent étale sheaves on X in terms of the site X_{étale}^{sp}.

Lemma 3.4.4 A sheaf S on X_{étale}^{sp} corresponds to an overconvergent sheaf on X_{étale} if and only if the natural map

lim_{\overrightarrow{r>1}} S(Y (r)) ⟶ S(Y)

is an isomorphism for all Y → X affinoid étale.

Proof. Suppose that S is overconvergent. In this case S|_{Y (r_{0})} is overconvergent for some r_{0} > 1. Hence 3.4.4 is an isomorphism since Y ⊂⊂_{Y (r_{0})} Y (r), 1 < r < r_{0} forms a cofinal system of wide neighbourhoods of Y in Y (r_{0}).

Conversely suppose 3.4.4 is an isomorphism always. Let Y → X be an étale morphism of affinoids. We have to show that S|_{Y} is rigid overconvergent. Let U ⊂ Y be a rational subset of Y. Choose some r_{0} > 1 such that Y ⊂⊂_{X} Y (r_{0}) and Y (r_{0}) is étale over X. Denote for r > 1 by U(r) the wide neighbourhood of U in Y (r_{0}) defined in §3.3. It follows easily from Lemma 3.4.2 that these wide neighbourhoods U ⊂⊂_{X} U(r) form a cofinal system of affinoid étale wide neighbourhoods of U over X. Hence our assumption gives the isomorphism lim S(U(r)) = S(U).

However, we want to show that the map lim S(U(r) ∩ Y) → S(U) is an isomorphism. It is clear from the above that this is a surjection. Using for all the rational subdomains U(r)∩Y of Y the bijectivity of the map S((U(r)∩Y)(r′)) → S(U(r)∩Y), it also follows that the map is injective. This proves our lemma. □

We will say that a presheaf on X_{étale}^{sp} is overconvergent if the map 3.4.4 is always an isomorphism. At this point we introduce a useful method to produce overconvergent (pre)-sheaves on X_{étale}^{sp}. Let S be a presheaf on X_{étale}^{sp}. We define the presheaf cS on X_{étale}^{sp} as follows:

Γ(Y, cS) = cS(Y) := lim_{\overrightarrow{r>1}} Γ(Y (r), S)

for any Y in X_{étale}^{sp}. Note that Lemma 3.4.2 implies that this is independent of the chosen representation of O(Y) over O(X) and that cS is indeed a presheaf. The construction c is a functor, there is an obvious functorial arrow cS → S and the map ccS → cS is an isomorphism. Hence the presheaf cS is overconvergent. It is therefore clear that the functor c is a right adjoint of the inclusion functor: overconvergent presheaves on X_{étale}^{sp} → presheaves on X_{étale}^{sp}.

Lemma 3.4.5 With notations as above.

1. If S is a sheaf then cS is a (overconvergent) sheaf. The functor S ↦ cS is a right adjoint of the inclusion functor: overconvergent sheaves on X → sheaves on X. The functor S ↦ cS is left exact on the category of sheaves on X_{étale}^{sp}.

2. If J is an injective sheaf on X_{étale}^{sp} and U = {Y_{i} → Y } is a covering in X_{étale}^{sp}

then

Ȟ^{i}(U, cJ) = 0 ∀i > 0.

It follows that cJ is a flabby sheaf on X_{étale}^{sp}.

3. Any overconvergent sheaf can be embedded into a sheaf of the form cJ with J injective.

Proof. Let U = {g_{i} : Y_{i} → Y } be a covering in X_{étale}^{sp}. We want to show the following: there exists a set of coverings U_{α} such that for any (pre)sheaf S we have a canonical isomorphism:

(∗)  lim_{\overrightarrow{α}} Č(U_{α}, S) ≅ Č(U, cS)

(The symboles Č denote Cech-complexes.) It is clear that this will prove that cS is a sheaf if S is a sheaf and it will prove the second assertion of the lemma. We leave the adjointness property to the reader, as well as the third part of the lemma.

We will only prove the above in the case that the covering U = {g : Z → Y } is given by one map. Since for an arbitrary (and hence finite) covering in X_{étale}^{sp} there exists a covering consisting of a single morphism giving an isomorphic Cech-complex there is no loss of generality. To do this we fix r_{0} > 1 small enough such that Y (r_{0}) is étale over X and a r_{1} > 1 small enough such that g extends to g̃ : Z(r_{1}) → Y (r_{0}). Next, for any r_{2}, 1 < r_{2} < r_{1}, we choose a r_{3}(r_{2}), 1 < r_{3}(r_{2}) < r_{0} such that Y (r_{3}(r_{2})) ⊂ g̃(Z(r_{2})). This is possible by Lemma 3.4.2, which also implies that we may choose r_{3}(r_{2}) to be a decreasing function of r_{2}, decreasing to 1 in fact.

We put Z_{r_{2}} = Z(r_{2}) ∩ g̃^{−1}(Y (r_{3}(r_{2}))). The coverings we are looking for are U_{r_{2}} = {Z_{r_{2}} → Y (r_{3}(r_{2}))}. Note that there are commutative diagrams for 1 < r′_{2} < r_{2}:

Z ⟶  Z_{r′_{2}}  ⟶  Z_{r_{2}}
↓  ↓  ↓
Y ⟶ Y (r_{3}(r′_{2})) ⟶ Y (r_{3}(r_{2}))

Hence we get the map (*). To show that (*) is an isomorphism we only need to prove that

Z ×_{Y} … ×_{Y} Z ⊂⊂_{X} Z_{r_{2}} ×_{Y (r_{3}(r_{2}))} … ×_{Y (r_{3}(r_{2}))} Z_{r_{2}}

forms a cofinal system of wide neighbourhoods of Z ×_{Y} … ×_{Y} Z as r_{2} decreases to 1. This is clear from the following three facts: 1) Z_{r_{2}} ×_{X} … ×_{X} Z_{r_{2}} forms a cofinal system of wide neighbourhoods of Z ×_{X} …×_{X} Z (see Lemma 3.4.2), 2) Z ×_{Y} …×_{Y} Z is a union of connected components of Z ×_{X} … ×_{X} Z and 3) the intersection of Z_{r_{2}} ×_{Y (r_{3}(r_{2}))} … ×_{Y (r_{3}(r_{2}))} Z_{r_{2}} with Z ×_{X} … ×_{X} Z is Z ×_{Y} … ×_{Y} Z. □

Lemma 3.4.6 (Properties of overconvergent sheaves on X_{étale}^{sp}.) In this lemma all (pre-)sheaves are (pre-)sheaves of abelian groups.

1. The sheaf associated to an overconvergent presheaf is overconvergent.

2. For any overconvergent sheaf S the presheaves Y ↦ H^{q}(Y, S) are overconvergent; for any q the rigid sheaf R^{q}(r_{X/Y})_{∗}S is overconvergent on Y, in particular the sheaves R^{q}r_{∗}S are overconvergent on X_{rigid}.

3. The category of overconvergent sheaves is an exact subcategory of the category of all sheaves on X_{étale}^{sp}.

4. If f : Z → X is a general morphism of affinoids then f^{∗} and f_{∗} preserve overconvergent sheaves. The same holds for R^{q}f_{∗} for any q.

5. If {f_{i} : X_{i} → X} is a special étale covering of X then a sheaf on X_{étale}^{sp} is overconvergent if and only if each f_{i}^{∗}(S) is overconvergent.

6. An overconvergent sheaf S is zero if and only if its étale stalks S_{e} are zero for all étale points e of X.

Proof. 1) Let a(S) denote the sheaf associated to S. The map S → a(S) factors as S → ca(S) → a(S) since S is overconvergent. By the universal property of aS we get a section a(S) → ca(S) (as ca(S) is a sheaf). It follows that a(S) is a direct summand of the overconvergent sheaf ca(S) and hence overconvergent.

2) Embed S in an overconvergent flabby sheaf as in the preceding lemma: 0 → S → cJ. The quotient presheaf is overconvergent hence so is the quotient sheaf Q by 1). For any affinoid Y étale over X we get the exact sequence

0 ⟶ H^{0}(Y, S) ⟶ H^{0}(Y, cJ) ⟶ H^{0}(Y, Q) ⟶ H^{1}(Y, S) ⟶ 0

and isomorphisms H^{q}(Y, S) ≅ H^{q−1}(Y, Q) for q > 1. It follows immediately that the presheaf Y ↦ H^{1}(Y, S) is overconvergent and the usual induction on q does the rest.

3) Follows from 1) and 2) and the results on rigid overconvergent sheaves.

4) Remark that if Y → X is affinoid étale then Y (r) ×_{X} Z ≅ (Y ×_{X} Z)(r). The rest of the argument is completely analogous to the proof of Lemma 2.3.2 part 4.

5) Same argument as in the rigid case.

6) This is immediate from Lemma 3.3.1 combined with the result for rigid overconvergent sheaves. □

3.5 Étale overconvergent sheaves on general X

Let X be an arbitrary analytic variety over k. Recall that a sheaf S on X_{étale} is overconvergent if S|_{Y} is rigid overconvergent for any Y étale over X. It is clear from Lemma 3.4.6 that this condition is local in the étale topology on X.

There are now a number of easy consequences of the above which we list here:

1. If f : Z → X is an arbitrary (general) morphism then f^{∗} preserves overconvergent sheaves.

2. If f : Z → X is quasi-compact then R^{q}f_{∗} preserves overconvergent sheaves. (Compare proof of Proposition 2.4.1.)

3. For any overconvergent sheaf S on X the rigid sheaves R^{q}(r_{Y/X})_{∗}S (in particular R^{q}r_{∗}S) are overconvergent.

Finally, we have the following result.

Proposition 3.5.1 If X is paracompact and S is an overconvergent torsion sheaf on X_{étale} then H^{q}(X, S) = 0 for all q > 2 dim X + cd(k), where cd(k) denotes the cohomological dimension of k.

Proof. Consider the spectral sequence with E_{2}-term H^{i}(X_{rigid}, R^{j}r_{∗}S) converging to H^{i+j}(X, S). By Corollary 3.3.2 and Lemma 3.4.6 we get that the sheaves R^{j}r_{∗}S are zero for j > dim X + cd(k). Hence we get the result from Corollary 2.5.10. □

3.6 Galois action on cohomology

Let us take a separable closure k^{sep} of k and let us denote by K the completion of k^{sep} with respect to the absolute value | |. Note that K is algebraically closed (see e.g. [BGR, 3.4]). We remark that the group G := Gal(k^{sep}/k) can be identified with the group of continuous automorphisms of K over k.

Take an analytic variety X over k and an étale sheaf S on it. Consider the variety X ˆ⊗K over K and the general morphism α : X ˆ⊗K → X. For any σ ∈ G there is an obvious general morphism φ_{σ} : X ˆ⊗K → X ˆ⊗K. This is not a morphism of analytic varieties over K unless σ = id_{K}; it lies over the continuous field homomorphism σ : K → K. Since it is clear that α = α ∘ φ_{σ}, we get an isomorphism α^{∗}(S) ≅ (α ∘ φ_{σ})^{∗}(S) ≅ (φ_{σ})^{∗}α^{∗}(S). Thus we get

φ_{σ}^{∗} : H^{i}(X ˆ⊗K, α^{∗}S) ⟶ H^{i}(X ˆ⊗K, α^{∗}S).

This defines an action of G on H^{i}(X ˆ⊗K, α^{∗}S).

Another way to get a G-module is to consider the morphism

p : X → Sp(k)

As was noted above the sheaves R^{i}p_{∗}S correspond to G-modules (R^{i}p_{∗}S)_{e}. It will be shown below that these two Galois modules agree in the case that X is quasi-compact.

3.7 Étale base change

Let f : Y → X be a quasi-compact morphism of analytic varieties over k and S an étale sheaf on Y. The étale base change theorem compares the cohomology of S on the étale fibre Y_{e} with the étale stalks at e of the sheaves R^{q}f_{∗}S. The étale fibre is just defined as Y_{a} ˆ⊗F_{e}, or as the fibre product of the general morphism Sp(F_{e}) → X with the morphism Y → X. The result will be an isomorphism of G_{e}-modules. As in the rigid case the theorem will follow formally from a lemma describing the étale site of the fibre Y_{e} in the affinoid case.

Therefore we suppose that f : Y → X is a morphism of affinoids over k and we fix an étale point e lying over the analytic point a of X. For any étale neighbourhood (U, b, ϕ) of e with U affinoid we can consider the special étale site of Y_{U} := Y ×_{X} U. Using ϕ we can see e as an étale point of U lying over b and then it is clear that (Y_{U})_{e} = Sp(F_{e}) ×_{U} Y_{U} ≅ Sp(F_{e}) ×_{X} Y = Y_{e}. Thus a general morphism Y_{e} → Y_{U} which gives rise to the functor

\begin{matrix}(Y_{U})_{étale}^{sp} & ⟶ & (Y_{e})_{étale}^{sp} \\ V → Y_{U} & ⟼ & V_{e} → Y_{e}. \end{matrix}

On the other hand, if the affinoid étale neighbourhood (U′, b′, ϕ′) dominates (U, b, ϕ), there is clearly a functor (Y_{U})_{étale}^{sp} → (Y_{U′})_{étale}^{sp} compatible with the functor described above.

Lemma 3.7.1 The functors above define an equivalence of sites:

lim_{\overrightarrow{(U,b,ϕ)}} (Y_{U})_{étale}^{sp} ≅ (Y_{e})_{étale}^{sp}.

Proof. Note that all functors defined above underly morphisms of sites in the reverse directions. The statement follows from the following three assertions:

1. For any étale morphism V → Y_{e} with V affinoid there exists an affinoid étale neighbourhood (U, b, ϕ) and an étale W → Y_{U} morphism of affinoids such that W_{e} ≅ V as varieties over Y_{e}.

2. Given two étale morphims W_{i} → Y_{U}, W_{i} affinoid (i = 1, 2 and U as above) and a morphism ψ_{e} : W_{1,e} → W_{2,e} there exists an affinoid étale neighbourhood (U′, b′, ϕ′) dominating (U, b, ϕ) and a morphism ψ_{U′} : W_{1,U′} → W_{2,U′} such that ψ_{U′,e} = ψ_{e}. This ψ_{U′} is unique if U′ is small enough.

3. If {g_{i} : W_{i} → W } is a finite set of morphisms in (Y_{U})_{étale}^{sp} and {W_{i,e} → W_{e}} is an étale covering then {W_{i,U′} → W_{U′}} is an étale covering if U′ is small enough.

Let us prove 1). By definition O(V) has a presentation

O(V) ≅ O(Y_{e})⟨T_{1}, …, T_{n}⟩/(G_{1}, …, G_{n})

such that Δ := det(∂G_{i}/∂T_{j}) is invertible. By Lemma 3.4.1 we may suppose that the G_{i} are polynomials. Since O(Y_{e}) = O(Y) ˆ⊗_{O(X)}F_{e} we can approximate the G_{i} by polynomials with coefficients in O(Y)⊗_{O(X)} L for some finite separable field extension F_{a} ⊂ L ⊂ F_{e}. By Lemma 3.4.1 we may assume G_{i} ∈ O(Y) ⊗_{O(X)} L[T_{1}, …, T_{n}]. We can construct (U, b, ϕ) such that ϕ(F_{b}) ⊃ L (see 3.3); for this U we can find polynomials P_{i} ∈ O(Y) ˆ⊗_{O(X)}O(U)[T_{1}, …, T_{n}] mapping to the G_{i}. The function Δ(P) := det(∂P_{i}/∂T_{j}) on

W := Sp(O(Y) ˆ⊗_{O(X)}O(U)⟨T_{1}, …, T_{n}⟩/(P_{1}, …, P_{n})

is such that its restriction to W_{e} is invertible. Hence, Δ(P) is invertible on W_{b}, hence by Lemma 2.7.2 or 2.7.1 we get that Δ(P) is invertible on W after shrinking U. This gives that W → Y_{U} is special étale and W_{e} ≅ V by construction.

Next we do 2). Note that the morphism ψ_{e} : W_{1,e} → W_{2,e} gives rise to a graph morphism Γ_{e} : W_{1,e} → (W_{1} ×_{Y_{U}} W_{2})_{e} and that this morphism identifies W_{1,e} with a union of connected components of (W_{1} ×_{Y_{U}} W_{2})_{e}. Hence Γ_{e} is an étale morphism of affinoids. By 1) there exists a smaller étale neighbourhood (U′, b′, ϕ′) and an étale morphism of affinoids Γ_{U′} : W → (W_{1} ×_{Y_{U}} W_{2}) ×_{U} U′ ≅ W_{1,U′} ×_{Y_{U′}} W_{2,U′} with W_{e} ≅ W_{1,e} and Γ_{U′,e} = Γ_{e}. We replace (U, b, ϕ) by (U′, b′, ϕ′) and hence we have Γ : W → W_{1} ×_{Y_{U}} W_{2}. Consider p_{2} = pr_{2} ∘ Γ. By the above we have that (p_{2})_{e} is an isomorphism. It follows that (p_{2})_{b} is a bijective (on analytic points) étale morphism of affinoids and hence an isomorphism. Thus for any analytic point c ∈ W_{1,b} we have that p_{2}^{−1}(c) consists of one analytic point c′ ∈ W with F_{c} ≅ F_{c′}. Lemma 3.1.5 implies that p_{2} is an open immersion in a wide open neighbourhood W_{1}(c) of c in W_{1}. Finitely many W_{1}(c)_{b}’s cover W_{1,b} and W_{b} = W_{1,b} hence by the key lemma for the rigid case we may shrink U and get that p_{2} is an isomorphism (apply the key lemma to both W and W_{1}). Clearly, the morphism pr_{1} ∘ Γ ∘ (p_{2})^{−1} : W_{1} → W_{2} does the job.

The uniqueness follows easily from the rigd key lemma by looking at graphs as above.

Finally, if the assumptions are as in 3) then W_{e} = ⋃_{i} g_{i,e}(W_{i,e}) implies W_{b} = ⋃ g_{i,b}(W_{i,b}), since formation of image commutes with arbitrary base change, see Lemma 3.1.7. Thus the statement follows from the rigid case, i.e., if U small enough then W = ⋃ g_{i}(W_{i}). □

This was the hard part of the proof of the base change theorem in the étale case. We can deduce the following analog of 2.7.3.

Corollary 3.7.2 Consider the general morphism α : Y_{e} → Y.

1. The functor α^{∗} preserves flabby sheaves.

2. For any sheaf S on Y_{étale}^{sp}, any (U, b, ϕ) and any W → Y_{U} as above we have:

Γ(W_{e}, α^{∗}S) = lim_{\overrightarrow{(U′,b′,ϕ′)≥(U,b,ϕ)}} Γ(W ×_{U} U′, S)

Proof. For any S on Y_{étale}^{sp} we have H^{q}(Y_{e}, α^{∗}S) = lim H^{q}(Y_{U}, S), by Remark 2.5.9 and the previous lemma. The same argument gives H^{q}(W_{e}, α^{∗}S) = lim H^{q}(W_{U}, S) for W as in 2). The results of the corollary follow directly from this. □

Theorem 3.7.3 Let f : Y → X be a quasi-compact morphism of analytic spaces over k. Take an étale point e of X and denote by Y_{e} the (étale) fibre of f at e. The functors

S ↦ H^{q}(Y_{e}, α^{∗}S) resp. S ↦ (R^{q}f_{∗}S)_{e}

are δ-functors of the category of abelian sheaves on Y_{étale} to the category of continuous G_{e}-modules. These δ-functors are isomorphic.

Proof. Remark that Y_{e} = Y_{a} ˆ⊗F_{e} and that α^{∗}S is the pullback of S|_{Y_{a,étale}} via the general morphism Y_{e} → Y_{a}. Thus we see by 3.6 that the groups H^{q}(Y_{e}, α^{∗}S) indeed have a Galois module structure. In the same way as in the proof of the rigid base change theorem it is proved that the functors under consideration form δ-functors. The maps

(R^{q}f_{∗}S)_{e} ⟶ H^{q}(Y_{e}, α^{∗}S)

are defined similarly as in the proof of Theorem 2.7.4. These maps commute with Galois action since the action on both sides is defined through the action of G_{e} on F_{e}.

Let us prove that these maps are isomorphisms only in the case that both X and Y are affinoid. The general case then follows as it did in the rigid case. The result for q = 0 is just Corollary 3.7.2 part 2) with W = Y. The general result follows by induction on q and the fact that α^{∗} preserves flabby sheaves. □

Corollary 3.7.4 If f : Y → X is quasi-compact and has finite fibres then R^{q}f_{∗}S is zero for q ≥ 1 and any overconvergent sheaf S on Y_{étale}. In particular the cohomology of S on Y is equal to the cohomology of f_{∗}S on X.

Corollary 3.7.5 (Hochschild-Serre spectral sequence.) Let K be a completion of a separable closure k^{sep} of k. Let G = Gal(k^{sep}/k) denote the continuous Galois group of K over k. For any quasi-compact variety X over k and any abelian sheaf S on X_{étale} there is a spectral sequence

H^{i}(G, H^{j}(X ˆ⊗K, α^{∗}S)) ⇒ H^{i+j}(X, S).

Here α : X ˆ⊗K → X is as in 3.6 and H^{q}(G, −) denotes continuous cohomology.

Proof. Let us write p : X → Sp(k) as in 3.6. Let a denote the unique analytic point of Sp(k) and let e be an étale point lying over a. First we note that (if S is overconvergent)

H^{q}(X ˆ⊗K, α^{∗}S) ≅ (R^{q}p_{∗}S)_{e}

as G-modules by the theorem above. This shows that the G-module on the left has a continuous G-action if it is given the discrete topology. It also proves that

H^{0}(G, H^{0}(X ˆ⊗K, α^{∗}S)) = H^{0}(G, (p_{∗}S)_{e}) = (p_{∗}S)_{a} = H^{0}(Sp(k), p_{∗}S) = H^{0}(X, S).

Here we used Lemma 3.3.2. Hence we only need to show that the functor which maps S to the Galois module H^{0}(X ˆ⊗K, α^{∗}S) transforms an injective sheaf S on X into an acyclic G-module. Since we are taking continuous cohomology we have:

H^{q}(G, H^{0}(X ˆ⊗K, α^{∗}S)) = lim_{\overrightarrow{k⊂k′}} H^{q}(Gal(k′/k), H^{0}(X ˆ⊗K, α^{∗}S)^{G′}),

where the limit runs over all finite Galois extensions k ⊂ k′ contained in K. By an argument as above this is the limit over the groups

H^{q}(Gal(k′/k), H^{0}(X ⊗ k′, S)).

But since S is injective these groups compute the cohomology groups H^{q}(X, S) (compare [M80, Theorem 2.20]) and these are zero for q ≥ 1. □

4 Cohomology of varieties of dimension at most 1

In this section we suppose that the field k is algebraically closed. Let p ≥ 1 denote the characteristic of the residue field of k. We put p = 1 if the residue field of k contains the field of rational numbers. Further X will denote an analytic space over k of dimension ≤ 1.

4.1 Some general results

For n > 1 which is not divisible by the characteristic of k, we consider the exact sequence

0 ⟶ μ_{n} ⟶ 𝔾_{m} ⟶^{n} 𝔾_{m} ⟶ 0

of sheaves on X_{étale}. This sequence induces the following distinguished triangle of complexes on X_{rigid}:

⟶ Rr_{∗}μ_{n} ⟶ Rr_{∗}𝔾_{m} ⟶ Rr_{∗}𝔾_{m} ⟶ Rr_{∗}μ_{n}[1]

We already know quite a lot about the homology sheaves of these complexes: We know that R^{q}r_{∗}μ_{n} = 0 for all q ≥ 2 by Lemma 3.3.1 and Corollary 3.3.2 combined with the fact that R^{q}r_{∗}μ_{n} are overconvergent (Lemma 3.4.6). Further we know that R^{1}r_{∗}𝔾_{m} = 0 by Corollary 3.2.4. The following result is a formal consequence of this.

Lemma 4.1.1 In the derived category of abelian sheaves on X_{rigid} we have the following isomorphism

Rr_{∗}μ_{n} ≅ (O_{X}^{∗} ⟶^{n} O_{X}^{∗}),

where the first term on the right is placed in degree 0.

Lemma 4.1.2 Let X be connected paracompact (and still have dimension ≤ 1). We denote by ℤ/nℤ_{X} the constant sheaf with fibre ℤ/nℤ on X_{étale}, where n is prime to the characteristic of k. We have H^{0}(X, ℤ/nℤ_{X}) = ℤ/nℤ and H^{q}(X, ℤ/nℤ_{X}) = 0 for q ≥ 3.

1. There is an exact sequence

0 ⟶ H^{1}(X_{rigid}, ℤ/nℤ) ⟶ H^{1}(X, ℤ/nℤ_{X}) ⟶ H^{0}(X_{rigid}, R^{1}r_{∗}ℤ/nℤ_{X}) → 0

and we have H^{2}(X, ℤ/nℤ_{X}) = H^{1}(X_{rigid}, R^{1}r_{∗}ℤ/nℤ_{X}).

2. A choice of a primitive n^{th}-root of unity determines an exact sequence

0 ⟶ O(X)^{∗}/O(X)^{∗ n} ⟶ H^{1}(X, ℤ/nℤ_{X}) ⟶ ker(n, P ic(X)) ⟶ 0

and an isomorphism H^{2}(X, ℤ/nℤ_{X}) = P ic(X)/nP ic(X).

Proof. The statement on H^{0} is trivial. Consider the spectral sequence with E_{2}- terms H^{p}(X_{rigid}, R^{q}r_{∗}ℤ/nℤ_{X}) abutting to H^{p+q}(X, ℤ/nℤ_{X}). Clearly 1) follows since R^{q}r_{∗}ℤ/nℤ_{X} = (0) for q ≥ 2 and cohomology of rigid sheaves is zero on X in dimensions ≥ 2 by Corollary 2.5.10.

A choice of a primitive n^{th}-root of unity determines an isomorphism of sheaves ℤ/nℤ_{X} ≅ μ_{n}. Thus statement 2) follows from the lemma above and the vanishing of rigid cohomology in degrees ≥ 2 on X. □

4.2 The cohomology of ℤ/nℤ with (n, p) = 1

In this subsection we will determine the cohomology of X in certain cases where X is smooth and irreducible. We will use the word curve to denote a separated analytic variety of pure dimension 1. Recall that we are working over an algebraically closed field.

Proposition 4.2.1 Let C be a nonsingular projective curve of genus g. We compute the cohomology with values in ℤ/nℤ for (n, p) = 1 of an open subvariety X of C as follows.

1. If X = C, then H^{1}(X, ℤ/nℤ_{X}) ≅ (ℤ/nℤ)^{2g} and H^{2}(X, ℤ/nℤ_{X}) ≅ ℤ/nℤ.

2. If X is the complement of finitely many points c_{1}, …, c_{a} (a > 0) in C, then H^{1}(X, ℤ/nℤ_{X}) ≅ (ℤ/nℤ)^{2g+a−1} and H^{2}(X, ℤ/nℤ_{X}) = 0.

3. Suppose X is C \ (D_{1} ∪ … ∪ D_{a}) where the D_{i} are disjoint open discs in C. In this case H^{1}(X, ℤ/nℤ_{X}) ≅ (ℤ/nℤ)^{2g+a−1} and H^{2}(X, ℤ/nℤ_{X}) = 0.

In all of these cases, for any extension of algebraically closed complete valued fields k ⊂ K, the natural map H^{q}(X, ℤ/nℤ_{X}) → H^{q}(X_{K}, ℤ/nℤ_{X_{K}}) is an isomorphism.

Proof. Part 1) follows from Lemma 4.1.2 part 2) and the fact that P ic(C) corresponds to the algebraic Picard group of C by GAGA. Note that the isomorphism for H^{2} is given by the isomorphism P ic(C)/nP ic(C) ≅ ℤ/nℤ induced by taking degrees of line bundles on C.

Note that in case 2) the space X is the admissible increasing union X = ⋃ X_{n} of affinoid spaces X_{n} as in 3). Just take D_{i,n} to be smaller and smaller open discs in C with center c_{i}. Thus if we prove 3) then 2) will follow by considering the Cartan-Leray spectral sequence associated to the covering X = ⋃ X_{n}. (Here we also need that the maps H^{i}(X_{n+1}) → H^{i}(X_{n}) are isomorphisms; this follows from the proof of 3) below.)

Let us assume X is as in 3). Any line bundle on X is the restriction of a line bundle of degree zero on C. In other terms, P ic^{0}(C) → P ic(X) is surjective. In particular P ic(X) is a divisible group and by Lemma 4.1.2 we get H^{2}(X, ℤ/nℤ_{X}) = 0. For the calculation of the group H^{1}(X, ℤ/nℤ_{X}) we start with the case where X is the closed unit disk 𝔹 := {z ∈ k; |z| ≤ 1}.

Now H^{1}(𝔹, 𝔾_{m}) = P ic(𝔹) = 0 and by 4.1.2 one has H^{1}(𝔹, ℤ/nℤ_{𝔹}) = O(𝔹)^{∗}/O(𝔹)^{∗n}. The invertible functions on 𝔹 are of the form λ(1 + f) with λ ∈ k^{∗} and f ∈ O(𝔹) has a norm < 1. The condition on n implies that such a function has an n-th root. Hence H^{1}(𝔹, ℤ/nℤ_{𝔹}) = 0.

Next, we want to investigate a ring domain (or annulus) ∂𝔹 := {z ∈ k; |z| = 1}. Again P ic(∂𝔹) = 0. Further every invertible function on ∂𝔹 has uniquely the form λz^{s}(∑_{m} a_{m}z^{m}) where λ ∈ k^{∗}, s ∈ ℤ and where the Laurent series satisfies a_{0} = 1, |a_{m}| < 1 for all m ≠ 0 and lim |a_{m}| = 0. It follows that H^{1}(∂𝔹, ℤ/nℤ_{∂𝔹}) = O(∂𝔹)^{∗}/O(∂𝔹)^{∗n} = ℤ/nℤ, a generator is given by the class of z. Clearly this is independent of the base field k.

Now we start proving the general statement. The pre-sheaves U ↦ H^{i}(U_{etale}, ℤ/nℤ_{C}) are overconvergent on C_{rigid}. Hence it suffices to prove the statement for all wide neighbourhoods X′ of X in C. For such an X′ we can find closed unit discs B_{i} ⊂ D_{i} such that X′ ∩ B_{i} is isomorphic to a ring domain ∂𝔹. If we have this then the covering

C = X′ ∪ ⋃ B_{i}

will be admisssible. In particular it is also an étale covering of C. Therefore, we have the Mayer-Vietoris sequence exact sequence [M80, p. 110]

0 ⟶ H^{1}(C, ℤ/nℤ_{C}) ⟶ H^{1}(X′, ℤ/nℤ_{C}) ⊕ ⨁ H^{1}(B_{i}, ℤ/nℤ_{C})
⟶ ⨁ H^{1}(X′ ∩ B_{i}, ℤ/nℤ_{C}) ⟶ H^{2}(C, ℤ/nℤ_{C}) ⟶ 0.

The zero on the right follows by the vanishing of H^{2} on affinoid curves proved above and the zero on the left is trivial to establish. The result follows by the computation of cohomology of 𝔹 and ∂𝔹 given above.

For a precise definition of the map H^{q}(X, ℤ/nℤ_{X}) → H^{q}(X_{K}, ℤ/nℤ_{X_{K}}) see 5.1 below. The invariance of cohomology under extension of base field for X follows from the invariance of cohomology for the algebraic curve C and the spaces 𝔹, resp. ∂𝔹. □

Remark 4.2.2 Let k_{0} ⊂ k denote a complete subfield of k such that k is the completion of the separable algebraic closure of k_{0}. Suppose that the g.c.d.(n, p) = 1. Let ∂𝔹 be the ring domain {z ∈ k_{0}; |z| = 1} over k_{0}. The galois action (see 3.6) on H^{1}(((∂𝔹) ˆ⊗_{k_{0}}k), μ_{n}) = ℤ/nℤ is trivial. By the proof above this cohomology group is canonically isomorphic to O(∂𝔹 ˆ⊗_{k_{0}}k)^{∗}/O(∂𝔹 ˆ⊗_{k_{0}}k)^{∗ n}. The generator of this group is the class of the invertible function z. This is clearly invariant under the Galois group.

Remark 4.2.3 The open unit disc D is the increasing union of closed discs. Thus we see, by the argument that proved part 2 of the proposition, that H^{q}(D, ℤ/nℤ_{D}) = 0 for q ≥ 1. This result is partially generalized in the corollary below.

Corollary 4.2.4 Let L be a compact subset of ℙ_{k}^{1} and put X = ℙ_{k}^{1} \ L. Then H^{1}(X, ℤ/nℤ_{X}) coincides with the group of ℤ/nℤ-valued currents on the tree of X (or the tree of L). More generally, for any connected open subspace X of ℙ_{k}^{1}, the group H^{1}(X, ℤ/nℤ_{X}) is equal to O(X)^{∗}/O(X)^{∗n}.

Proof. The line bundles on any open subspace X of ℙ_{k}^{1} are trivial (see [FP]) and hence H^{1}(X, ℤ/nℤ_{X}) ≅ O(X)^{∗}/O(X)^{∗n}. (Use Lemma 4.1.2.) The structure of the group O(X)^{∗} is well known if X = ℙ_{k}^{1} − L. Namely, there is an exact sequence

0 ⟶ k^{∗} ⟶ O(X)^{∗} ⟶ C(T) → 0,

where T denotes the tree of L and where C(T) denotes the group of currents with values in ℤ on T. (See [FP].) It follows that O(X)^{∗}/O(X)^{∗n} = C(T)/nC(T) is the group of currents on T with values in ℤ/nℤ. □

Proposition 4.2.5 If X is a connected smooth affinoid curve then there is an embedding X ⊂ C as in Proposition 4.2.1 part 3) above. We deduce from this the following results. Let A be an abelian torsion group of exponent n, with (n, p) = 1.

1. There are natural isomorphisms H^{q}(X, ℤ/nℤ_{X}) ⊗ A ≅ H^{q}(X, A_{X}).

2. The cohomology groups H^{q}(X, A_{X}) are invariant under algebraically closed extensions of base fields.

Proof. The existence of such an embedding X → C is proved in [P80]. The group A is the direct limit of its finite subgroups. Taking cohomology commutes with direct limits (3.4.3), hence it suffices to do the case A is finite. Writing A as the direct sum of cyclic subgroups it follows that we may assume A ≅ ℤ/n′ℤ where n′|n. In this case both 1) and 2) follow easily from Proposition 4.2.1. □

Remark 4.2.6 Other constant sheaves.

1. The cohomology of ℚ_{X}. Since in this case the rigid sheaves R^{q}r_{∗}ℚ_{X} for q ≥ 1 are both torsion (by 3.3.2 and 3.4.6) and sheaves of ℚ-vectorspaces, they are zero. Hence we have

H^{q}(X, ℚ_{X}) ≅ H^{q}(X_{rigid}, ℚ)

for all q. If X is a separated quasi-compact smooth curve then we have

H^{1}(X, ℚ_{X}) = H^{1}(X_{rigid}, ℚ) = ℚ^{b},

where b is the Betti number of the graph of a semi-stable reduction of X. A semi-stable reduction of X is defined as follows: take a separated formal scheme 𝔛 of finite type over Spf(k^{∘}), whose associated rigid space 𝔛^{rig} is isomorphic to X. (See [R74] or [BL].) By blowing up 𝔛 a bit we may assume that the singularities of the special fibre are ordinary double points. This special fibre is a semi-stable reduction of X. Since any other such formal scheme 𝔛′ may be compared with 𝔛 by a sequence of blow ups and blow downs in points it follows that the associated graphs have the same homotopy type. The result now follows from Remark 2.5.9 and a computation of the Zariski cohomology of a constant sheaf on an algebraic semistable curve.

2. The constant sheaf ℤ/pℤ_{X}.

(a) Let the characteristic of k be p > 1. We will give a calculation of H^{1}(𝔹, ℤ/pℤ_{X}) where 𝔹 is the closed unit disk. Consider the Artin-Schreier exact sequence

0 ⟶ ℤ/pℤ_{𝔹} ⟶ 𝔾_{a} ⟶^{ϕ} 𝔾_{a} ⟶ 0

on X_{étale}. Here of course ϕ(f) = f^{p} − f. On cohomology we get an exact sequence

0 ⟶ ℤ/pℤ ⟶ O(𝔹) ⟶^{ϕ} O(𝔹) ⟶ H^{1}(𝔹, ℤ/pℤ_{𝔹}) ⟶ 0,

since H^{1}(𝔹, 𝔾_{a}) = (0) by 3.2.5. The cokernel of ϕ : O(𝔹) → O(𝔹) is a rather large group and is not invariant under algebraically closed extensions of base fields. This reflects the fact that the closed disk has many p-cyclic unramified coverings.

(b) Here the characteristic of k is zero, but the characteristic of the residue field k̃ is p > 1. With the methods above it follows that H^{1}(𝔹, ℤ/pℤ_{𝔹}) = O(𝔹)^{∗}/O(𝔹)^{∗p}. This is again a very large group not invariant under base field extensions. It can be shown that every (algebraic) finite étale covering of the affine line over k̃ lifts to a finite étale covering of 𝔹. The conjecture of S.S.Abhyankar on the coverings of the affine line in characteristic p (proved by M.Raynaud) implies that the totality of nontrivial finite étale coverings of 𝔹 is very large.

5 Base change revisited

In this section we prove a general base change theorem for quasi-compact morphisms and overconvergent étale sheaves. In order to be able to apply Theorem 3.7.3 we have to prove invariance of cohomology under extensions of algebraically closed base fields. This is done below for rigid and étale cohomologies and overconvergent sheaves.

5.1 A change of fields, étale case.

Let k ⊂ K be an extension of complete and algebraically closed fields. For any analytic space X over k we denote by p_{K} : X_{K} = X ˆ⊗_{k}K → X the general morphism associated to the change of fields. For an étale sheaf S on X we have the étale sheaf p_{K}^{∗}S on X_{K} and comparison maps

H^{q}(X, S) ⟶ H^{q}(X_{K}, p_{K}^{∗}S).

We would like to know when these are isomorphisms. As before we put p ≥ 1 equal to the characteristic of the residue field of k (and p = 1 if char(k̃) = 0).

Theorem 5.1.1 The canonical maps H^{q}(X, S) ⟶ H^{q}(X_{K}, p_{K}^{∗}S) are isomorphisms if S satisfies the following conditions:

1. The sheaf S is overconvergent.

2. All étale stalks S_{e} of S are torsion groups, with torsion prime to p.

Proof. By taking an admissible affinoid covering of X we see that it suffices to do the case that X is affinoid. Let us consider S as a sheaf on the site X_{étale}^{sp}. For any n ∈ ℕ with (n, p) = 1 let S_{n} ⊂ S be the subsheaf of S consisting of sections annihilated by n, i.e., S_{n} := Ker(S →^{n} S) is also overconvergent. By our two conditions on S and Lemma 3.3.1 we see that any section s ∈ S(Y) is torsion (Y → X is special étale, hence Y affinoid, hence quasi-compact). Thus we see that S = ⋃ S_{n}. By looking at stalks we see that (p_{K}^{∗}S)_{n} = p_{K}^{∗}S_{n}, hence also p_{K}^{∗}S = ⋃ p_{K}^{∗}S_{n}. Since cohomology commutes with direct limits (3.4.3) it suffices to do the case that S is a sheaf of ℤ/nℤ-modules.

Consider fields L with k ⊂ L ⊂ K, which are complete and algebraically closed. We say that L has topological transcendence degree ≤ r over k if there exist elements t_{1}, …, t_{r} ∈ L such that L is the completion of the algebraic closure of k(t_{1}, …, t_{r}). The reasoning of Lemma 3.7.1 shows that the site (X_{K})_{étale}^{sp} is the direct limit of the sites (X_{L})_{étale}^{sp}, taken over all L of finite topological transcendence degree over k. Therefore it suffices to prove H^{q}(X, S) = H^{q}(X_{L}, p_{L}^{∗}S) for k ⊂ L of topological transcence degree ≤ r. By induction on r it suffices to do the case: k ⊂ K of topological transcendence degree 1.

Take an element t ∈ K such that K is the completion of the algebraic closure of k(t). We may assume that |t| ≤ 1. Consider the continuous k-algebra homomorphism k⟨T ⟩ → K mapping T to t. This determines an étale point e of the closed unit disc 𝔹 over k with F_{e} = K.

The problem we are studying may now be formulated with the help of the following diagram of analytic spaces and general morphisms.

X_{K} ⟶ X × 𝔹 ⟶^{p_{1}} X
↓  ↓^{p_{2}}  ↓^{q_{2}}
Sp(K) ⟶^{e}  𝔹  ⟶^{q_{1}} Sp(k).

There is a general base change morphism (see [SGA 4, Exp. XVII 4.1.5])

q_{1}^{∗}R^{q}(q_{2})_{∗}S ⟶ R^{q}(p_{2})_{∗}p_{1}^{∗}S  (1).

The comparison map from the theorem is the base change map for the big rectangle of the diagram. Our étale base change theorem 3.7.3 asserts that the base change map for the left square with p_{1}^{∗}S is an isomorphism. By the functoriality properties of the base change morphism it will suffice to prove that (1) is an isomorphism.

Let V → 𝔹 be étale and V affinoid. Put p : X × V → X equal to the projection. We write H^{m}(V) = H^{m}(V, ℤ/nℤ_{V}) and H^{m}(V)_{X} is the constant sheaf with fibre H^{m}(V) on X_{étale}. There is a natural map

H^{m}(V)_{X} ⊗_{ℤ/nℤ} S ⟶ R^{m}p_{∗}p^{∗}S.  (2)

It is the composition

H^{m}(V)_{X} ⊗ S ⟶ R^{m}p_{∗}(ℤ/nℤ_{X×V}) ⊗ S ⟶ R^{m}p_{∗}p^{∗}S,

the first map given by base change, the second deduced from S → p_{∗}p^{∗}S by the cupproduct R^{m}p_{∗}(ℤ/nℤ_{X×V}) ⊗ R^{0}p_{∗}p^{∗}S → R^{m}p_{∗}p^{∗}S associated to ℤ/nℤ_{X×V} ⊗ p^{∗}S → p^{∗}S. By étale base change 3.7.3 the stalk of R^{m}p_{∗}p^{∗}S in the étale point f of X is H^{m}(V ˆ⊗F_{f}, (S_{f})_{V ˆ⊗F_{f}}). Hence, by Proposition 4.2.5 (2) is an isomorphism on étale stalks for all f. Since both sides of (2) are overconvergent we get that (2) is an isomorphism. Thus we get that R^{m}p_{∗}p^{∗}S = (0) for n ≥ 2. Finally, since p : X × V → X has a section, the maps H^{m}(X, S) → H^{m}(X × V, p^{∗}S) have sections. We conclude that the spectral sequence H^{i}(X, R^{j}p_{∗}p^{∗}S) ⇒ H^{i+j}(X × V, S) degenerates and gives:

\begin{aligned}H^{m}(X × V, S) & ≅ & H^{m}(X, H^{0}(V)_{X} ⊗ S) & ⊕ & H^{n−1}(X, H^{1}(V)_{X} ⊗ S) \\ & ≅ & H^{0}(V) ⊗ H^{m}(X, S) & ⊕ & H^{1}(V) ⊗ H^{n−1}(X, S) \end{aligned}

Therefore, the sheaf associated to the presheaf V ↦ H^{m}(X × V, S) (on 𝔹_{étale}^{sp}) is the constant sheaf with fibre H^{m}(X, S). Clearly this means that the right side of (1) is constant and hence that (1) is an isomorphism (look at fibres over 0 ∈ 𝔹). □

5.2 A change of fields, rigid case.

We think it is quite amusing that a similar theorem also holds for the rigid case. Notations are as in 5.1.

Theorem 5.2.1 Let S be an overconvergent sheaf on X_{rigid}. The canonical maps

H^{q}(X_{rigid}, S) ⟶ H^{q}((X_{K})_{rigid}, p_{K}^{∗}S)

are isomorphisms.

Proof. As in the proof of the étale case we may assume that X is affinoid and k ⊂ K of topological transcendence degree 1 (using X_{rigid}^{rat} in stead of X_{étale}^{sp}). We consider subfields L ⊂ K, which are complete and are the completion of a function field of transcendence degree 1 over k. In this case we remark that (X_{K})_{rigid}^{rat} is the direct limit of the sites (X_{L})_{rigid}^{rat} for such fields L. Again it suffices to do the case K = L. (The field K is no longer algebraically closed!)

Suppose Z is a nonsingular projective irreducible curve over k, whose function field k(Z) is a dense subfield of K. The embedding k(Z) → K defines an analytic point a of Z with F_{a} = K.

Observation 5.2.2 There is an affinoid subdomain U ⊂ Z in the filter of a with the following property: For every affinoid V ⊂ U and any constant sheaf T on U the cohomology groups H^{n}(V, T) are zero for n ≥ 1.

This follows quite easily from the stable reduction of Z. As was noted in Remark 4.2.6 part 1) the cohomology of a rigid constant sheaf on an affinoid smooth curve depends only on the Betti number of the graph of its stable reduction. Thus we take for U the preimage of a Zariski open W of the stable reduction of Z, such that W contains no cycles. The assertion of the observation then holds for V = U. But also for any such V ⊂ U it holds, since this corresponds to a Zariski open part in a blow up of the stable model of Z. Blow ups do not introduce extra cycles.

The rest of the proof of the theorem is similar to the proof of Theorem 5.1.1: just replace 𝔹 by U and étale by rigid cohomology. □

Remark 5.2.3 Both theorems are false when k is not algebraically closed. Just take X = Sp(k′) where k ⊂ k′ is a finite Galois extension and S = ℤ/nℤ_{X}. In this case H^{0}(X, S) = ℤ/nℤ and H^{0}(X_{K}, p_{K}^{∗}S) = (ℤ/nℤ)^{[k′:k]}. Even if X is a geometrically connected smooth projective curve and S is a constant sheaf the result is false in general. (Both rigid and étale case.)

Corollary 5.2.4 Suppose that S is an overconvergent sheaf of ℤ[1/p]-modules on X_{étale}. The canonical comparison maps H^{q}(X, S) → H^{q}(X_{K}, p_{K}^{∗}S) are isomorphisms.

Proof. There is an exact sequence

0 ⟶ S_{tors} ⟶ S ⟶ S ⊗ ℚ ⟶ Q ⟶ 0

By Theorem 5.1.1 the result is true for both S_{tors} and Q. Since H^{q}(X, S ⊗ ℚ) agrees with H^{q}(X_{rigid}, r_{∗}S ⊗ ℚ) (compare Remark 4.2.6) we see the result is true for S ⊗ ℚ also by the theorem above. The snake lemma gives the result for the sheaf S. □

5.3 Quasi-compact base change.

By a combination of our previous results we can now prove a general base change theorem for quasi-compact morphisms.

Theorem 5.3.1 (Quasi-compact base change.) Consider a diagram

Z ×_{X} Y ⟶^{g′} Y
↓^{f′}  ↓^{f}
Z  ⟶^{g} X

and an overconvergent sheaf of ℤ[1/p]-modules S on Y_{étale}. Here f is a quasi-compact morphism of analytic varieties over k and g is a (arbitrary) general morphism of analytic varieties. The base change morphism [SGA 4, Exposé XVII]

g^{∗}Rf_{∗}S ⟶ Rf′_{∗}(g′)^{∗}S

is a quasi-isomorphism.

Proof. We only have to show that this morphism induces an isomorphism on the étale stalks of the overconvergent sheaves g^{∗}R^{q}f_{∗}S and R^{q}f′_{∗}(g′)^{∗}S. Any étale point e′ of Z lies over an étale point e of X, i.e., such that F_{e} ⊂ F_{e′}. By the étale base change theorem 3.7.3 the map on the stalks is the map 5.1 between

(g^{∗}R^{q}f_{∗}S)_{e′} = (R^{q}f_{∗}S)_{e} = H^{q}(Y_{e}, S|_{Y_{e}})

and

(R^{q}f′_{∗}(g′)^{∗}S)_{e′} = H^{q}(Y_{e′}, (g′)^{∗}S|_{Y_{e′}}) = H^{q}(Y_{e} ˆ⊗F_{e′}, p_{F_{e′}}^{∗} (S|_{Y_{e}})).

The statement follows from Corollary 5.2.4. □

6 The axioms for cohomology

Let k be a complete valued field. An ‘abstract’ cohomology theory for rigid analytic spaces over k is defined in [S-S, section 2] to be a cohomology theory X ↦ H^{∗}(X) satisfying four axioms. There is also given a candidate for such a cohomology theory. Let A be a finite ring of order prime to the residue field of k. Let K be the completion of the algebraic closure of k. We put

H^{∗}(X) := H^{∗}(X ˆ⊗K, A_{X ˆ⊗K}).

As remarked in [S-S, p. 58], the nontrivial axioms to check in this case are the ‘homotopy axiom’ and the axiom concerning the cohomology of the projective space. In this section we will prove those axioms.

The homotopy axiom states that H^{∗}(X × D) ≅ H^{∗}(X) for an open disc D. This follows immediately from the following theorem.

Theorem 6.0.2 (The homotopy axiom.) Let X be an analytic space over k. Let S be an overconvergent sheaf of ℤ[1/p]-modules on X_{étale}. Suppose D is an open or closed disc over k; let p : X × D → X denote the projection. The canonical maps H^{q}(X, S) → H^{q}(X × D, p^{∗}S) are isomorphisms.

Proof. If the disc D is open then it is the admissible union D = ⋃ B_{n} of closed discs B_{n} of radius ρ_{n} ∈ √|k^{∗}|. The covering X × D = ⋃ X × B_{n} is the also admissible. Therefore, it suffices to prove the theorem for a closed disc B.

In this case we prove that p_{∗}p^{∗}S ≅ S and that R^{q}p_{∗}p^{∗}S = (0) for q ≥ 1. By the étale base change theorem the étale stalk at e of these sheaves are equal to H^{q}(B ˆ⊗F_{e}, (S_{e})_{B ˆ⊗F_{e}}). Note that B ˆ⊗F_{e} ≅ 𝔹, the closed unit disc of radius 1 over F_{e}. If we prove that H^{q}(𝔹, A_{𝔹}) = (0) for q ≥ 1 for any ℤ[1/p]-module A then we are done. A standard argument, compare with 4.2.5, reduces to the cases A = ℚ or A = ℤ/nℤ. These cases where done in Remark 4.2.6 and Proposition 4.2.1. □

For the formulation of the following theorem, we need to be more precise about the Galois action on the cohomology groups. Let K denote the completion of the separable closure k^{sep} of k. The symbole G = Gal(k^{sep}/k) denotes the continuous Galois group of K over k. See 3.6. The finite ring A is given the trivial G action. For any i ∈ ℤ we define A(i) := A ⊗ (μ_{n}(K))^{⊗i} as a G-module, where n = #G. The following result also follows from the comparison theorem in the following section.

Theorem 6.0.3 (Cohomology of ℙ^{d}.) Let ℙ^{d} denote the d-dimensional projective space over k. We have as Galois modules

H^{q}(ℙ^{d}) = H^{q}(ℙ_{K}^{d}, A_{ℙ_{K}^{d}}) = { A(−q/2) for q even, 0 ≤ q ≤ 2d, (0) otherwise.

Proof. The calculation of the cohomology is done by applying the Mayer-Vietoris sequence to the covering {U_{0}, U_{1}} of ℙ^{d} given by U_{0} = {(z_{0}; ...; z_{d})| |z_{j}| ≤ |z_{0}| for all j} and U_{1} = {(z_{0}; ...; z_{d})| |z_{0}| ≤ max(|z_{1}|, ..., |z_{d}|)}. The space U_{0} is a product of disks and has therefore trivial cohomology. The spaces U_{1} and U_{0} ∩ U_{1} respectively, admit a surjective morphism to ℙ^{d−1} given by (z_{0}; ...; z_{d}) ↦ (z_{1}; ...; z_{d}). The fibres are disks or ring domains respectively and the fiberings are locally trivial. Base change, with respect to the first map, yields H^{i}(U_{1}) →^{∼} H^{i}(ℙ^{d−1}). Base change applied to the second map gives rise to an exact sequence

0 ⟶ H^{i}(U_{1}) ⟶ H^{i}(U_{0} ∩ U_{1}) ⟶ H^{i−1}(ℙ^{d−1})(−1) ⟶ 0.

The (−1) in the last cohomology group is a consequence of the Galois action on the cohomology of a ring domain. See 4.2.2. Induction on d and the Mayer-Vietoris sequence imply that H^{i}(ℙ^{d}) ≅ H^{i−2}(ℙ^{d−1})(−1) for i ≥ 2 and the expected values of H^{0} and H^{1}. □

7 Purity and comparison

Let X be a scheme of finite type over the complete valued field k. We write X_{ét} for the small étale site of the scheme X. Further, X^{an} denotes the rigid analytic variety associated to X. There is a morphism of sites

ϵ : X_{étale}^{an} ⟶ X_{ét}

comparing the algebraic and rigid étale sites. It is given by the functor that associates to the scheme Y étale over X the analytic space Y^{an} étale over X^{an}. We want to compare sheaves on both sides and their cohomology. It will turn out that if the characteristic of k is zero then we get results as proved in [SGA 4] comparing étale cohomology and classical cohomology over ℂ. However, if the characteristic is p > 1, only a weaker version holds. We will give counterexamples for the full statement.

In order to prove the statements above we use a purity result for rigid étale cohomology. It tells us what the cohomology of the complement of a smooth divisor in a smooth rigid analytic variety is.

7.1 A preliminary result

We start by proving that sheaves of the form ϵ^{∗}S are overconvergent.

Lemma 7.1.1 For any sheaf S on X_{ét} the sheaf ϵ^{∗}S is overconvergent.

Proof. Let Y → X be an algebraic étale morphism, with Y affine. We also denote by Y the sheaf on X_{ét} it defines. We only need to show that the sheaf ϵ^{∗}(Y) is overconvergent. (The sheaves Y generate the category of sheaves on X_{ét} and the direct limit of overconvergent sheaves is overconvergent.) By Zariski’s main theorem we can embed Y as a Zariski open set in a scheme \overline{Y} finite over X. Suppose U ⊂ X^{an} is an affinoid subdomain and V → U is an étale morphism of affinoids. The notation V (r) → U is as in 3.4. We have to show that any morphism φ : V → Y^{an} over X^{an} extends (uniquely) to some V (r) → Y^{an}. See 3.4.4. By Lemma 3.4.2 it suffices to show that φ(V) ⊂⊂_{U} Y^{an} ×_{X^{an}} U. Clearly, we have that φ(V) ⊂⊂_{U} \overline{Y}^{an} ×_{X^{an}} U, since the last space is finite over U. The result follows since Y^{an} ×_{X^{an}} U is Zariski open in \overline{Y}^{an} ×_{X^{an}} U. See for example [S93, §3 Proposition 3]. □

7.2 Purity for rigid étale cohomology

Let i : Z → X be a closed immersion of analytic varieties over k. Let U = X \ Z denote the admissible open subvariety of X which is the complement of Z. As usual j : U → X denotes the open immersion of U into X. We want to prove that sheaves on Z_{étale} correspond to sheaves S on X_{étale} such that j^{∗}S is a final object in the category of sheaves on U, i.e., a sheaf which has exactly one section over each object of U_{étale}. This means that the category of sheaves on Z can be viewed as the closed sub-topos of X_{étale}^{∼} complementary to the open sub-topos U_{étale}^{∼}. Compare [SGA 4, Exposé IV 9.3.5]. Although this follows easily for overconvergent sheaves, we need the result in general for the proof of purity below. It implies in particular that Ri_{∗}F ≅ i_{∗}F for any abelian sheaf F on Z_{étale}.

Lemma 7.2.1 The functor i_{∗} identifies the category of sheaves (of sets) on Z with the category of sheaves S on X such that j^{∗}S is a final object of U_{étale}^{∼}.

Proof. Let us take an admissible affinoid covering X = ⋃ X_{i} of X and admissible affinoid coverings X_{i} ∩ X_{j} = ⋃ X_{ijk}. Any sheaf on X is given by sheaves on X_{i} glued on the X_{ijk}, whereas a sheaf on Z (resp. U) is given by sheaves on Z ∩ X_{i} (resp. U ∩ X_{i}) glued on the Z ∩ X_{ijk} (resp. U ∩ X_{ijk}). In this way one reduces to the case that X is affinoid.

In this case we work with the sites X_{étale}^{sp} and Z_{étale}^{sp}. The functor X_{étale}^{sp} → Z_{étale}^{sp} is denoted W ↦ W_{Z} = Z ×_{X} W. Consider the following statements:

1. For any étale W_{0} → Z, W_{0} affinoid, there exists an étale W → X morphism of affinoids such that W_{0} ≅ W_{Z}.

2. If V, W ∈ X_{étale}^{sp} and ϕ_{0} : W_{Z} → V_{Z} is a morphism over Z then there is a Weierstrass domain W′ ⊂ W with W′_{Z} = W_{Z} and a morphism ϕ : W′ → V lifting ϕ_{0}.

3. If W ∈ X_{étale}^{sp} then any special étale covering {W_{i,0} → W_{Z}} may be lifted to a special étale covering of W.

Let us first prove that these imply the lemma.

We denote by e a final object of (U_{étale})^{∼}. Further for any sheaf S on X we denote by P (S) the presheaf on Z_{étale}^{sp} defined by the formula:

P (S)(W_{0}) = lim_{\overrightarrow{V,ϕ_{0}:W_{0}→V_{Z}}} Γ(V, S)

By definition, i^{∗}S is the sheaf associated to the presheaf P (S). It is clear from 1) and 2) above that P (S) may be described as follows

P (S)(W_{Z}) = lim_{\overrightarrow{W′⊂W as in 2)}} Γ(W′, S).

Such a subdomain W′ is automatically a wide neighbourhood of W_{Z} in W, since W_{Z} is closed in W. Therefore there exists a special subset V ⊂ W, disjoint with W_{Z} such that W = W′ ∪ V. This means that if S has the property that j^{∗}S ≅ e then Γ(W, S) = Γ(W′, S) since both Γ(V, S) and Γ(V ∩ W′, S) consist of one element. In particular we see that for such S we have P (S)(W_{Z}) = Γ(W, S). Property 3) above implies that P (S) is a sheaf in this case and hence i^{∗}(S) = P (S). It follows immediately that i_{∗}i^{∗}S ≅ S for such sheaves S.

Conversely, if F is a sheaf on Z, it is immediate that j^{∗}i_{∗}F ≅ e. Hence by the above we have that i^{∗}i_{∗}F = P (i_{∗}F) and

Γ(W_{Z}, i^{∗}i_{∗}F) = Γ(W, i_{∗}F) = Γ(W_{Z}, F).

We have proved that i_{∗} and i^{∗} are mutually inverse funtors defining the desired equivalence of categories.

Let us prove 1). Let Y_{0} → Z be an étale morphism of affinoids. We can choose a presentation (see Lemma 3.4.1)

O(Y_{0}) = O(Z)⟨T_{1}, …, T_{n}⟩/(G_{1}, …, G_{n})

with G_{1}, …, G_{n} ∈ O(Z)[T_{1}, …, T_{n}] such that the determinant Δ_{0} = det(∂G_{j}/∂T_{i}) is invertible in O(Z). Let us lift the polynomials G_{i} to polynomials F_{i} ∈ O(X)[T_{1}, …, T_{n}]. Put Δ = det(∂F_{j}/∂T_{i}). Take π ∈ k^{∗} such that |π| is smaller than the infinum of |Δ_{0}| on Z. We consider the algebra

O(X)⟨T_{0}, T_{1}, …, T_{n}⟩/(F_{1}, …, F_{n}, ΔT_{0} − π).

This defines a special étale morphism Y → X since the corresponding functional determinant is Δ^{2}, which is invertible. The isomorphism Y_{0} ≅ Z ×_{X} Y follows by construction.

The proof of 2) is similar to the proof of 2) in Lemma 3.7.1. We consider the product V ×_{X} W and the graph morphism Γ_{0} : W_{Z} → (V ×_{X} W)_{Z}. This morphism is étale. By 1) (with W ×_{X} V in stead of X) we can find Γ : Y → V ×_{X} W such that Y_{Z} ≅ W_{Z} and Γ_{Z} = Γ_{0}. Next argue as in the proof of 3.7.1 to see that there is some wide neighbourhood W′ ⊂ W of W_{Z} such that pr_{1} ∘ Γ : Γ^{−1}(V ×_{X} W′) → W′ is an isomorphism. Thus we get W′ → V. Finally, the Weierstrass domains in W are cofinal in the set of neighbourhoods of W_{Z} in W. To see this apply the rigid key lemma to a morphism f : W → 𝔹^{n} with W_{Z} = f^{−1}(0).

For 3) we first note that by 1) we may lift each of the special étale W_{i,0} → W_{Z} to special étale f_{i} : W_{i} → W. The special subset ⋃ f_{i}(W_{i}) is a neighbourhood of W_{Z} in W, hence a wide neighbourhood, hence there exists some special V ⊂ W such that V ∩ W_{Z} = ∅ and W = V ∪ ⋃ f_{i}(W_{i}). Write V = ⋃ V_{j} as a finite union of rational subdomains of W, then the special étale covering of W we are looking for is the covering {W_{i} → W, V_{j} → W }. □

Next, we prove some kind of purity in the rigid étale case. Let X be a smooth rigid variety over k. let i : H → X be a closed immersion, with H smooth over k and everywhere in X of codimension 1. Thus it is a smooth divisor in X. Let U denote the admissible open subset X \ H of X and let j denote the open immersion j : U → X.

Theorem 7.2.2 (Purity.) With the notations as above and with n prime to the characteristic of k we have

R^{q}j_{∗}ℤ/nℤ_{U} = \{ \begin{matrix}ℤ/nℤ_{X} & q = 0 \\ i_{∗}(μ_{n}^{⊗−1}) & q = 1 \\ (0) & q ≥ 2. \end{matrix}

Proof. The statement is local on X, hence we may assume X affinoid. Locally on X (in the Zariski topology) the ideal of H is generated by a single function, hence we may assume that H is given as f = 0 for some f ∈ O(X). By [K68] we can find an affinoid neighbourhood of H in X which has an admissible covering by affinoids of the form H_{i} × 𝔹. Here 𝔹 is the closed unit ball over k with coordinate z. Thus we may assume that X = H × 𝔹 and U = H × 𝔹^{∗} where 𝔹^{∗} is the punctured unit disc. Let us write f̄: X → H for the projection and f = f̄|_{U} so that we have the following commutative diagram:

U ⟶^{j} X ⟵^{i} H
f ↘ ↓f̄ ↙ id
H

We note that the sheaves R^{q}j_{∗}ℤ/nℤ_{U} for q ≥ 1 have are zero restricted to U, hence are of the form i_{∗}F_{q} for certain sheaves F_{q} on H (use lemma above). Further, it is clear that j_{∗}ℤ/nℤ_{U} = ℤ/nℤ_{X} on X. We study the spectral sequence associated to the isomorphism Rf_{∗} ≅ Rf̄_{∗} ∘ Rj_{∗}. For the sheaf ℤ/nℤ_{U} its E_{2}-terms are E_{2}^{ab} = R^{a}f̄_{∗}R^{b}j_{∗}ℤ/nℤ_{U} and it abuts to R^{a+b}f_{∗}ℤ/nℤ_{U}. In view of the fact that R^{b}j_{∗}ℤ/nℤ_{U} = i_{∗}F_{b} ≅ Ri_{∗}F_{b} for b ≥ 1 (by 7.2.1), we see that E_{2}^{ab} = 0 for a, b ≥ 1 and E_{2}^{0b} = F_{b} for b ≥ 1. Also we have E_{2}^{a0} = R^{a}f̄_{∗}ℤ/nℤ_{X}. This is an overconvergent sheaf, whose étale stalks are H^{a}(𝔹, ℤ/nℤ_{𝔹}), over various algebraically closed base fields. Hence, by Lemma 4.1.2 and since P ic(𝔹) = (0), we see that R^{a}f̄_{∗}ℤ/nℤ_{X} = (0) for a ≥ 2. The upshot of all of this is: 1) we have R^{0}f_{∗}ℤ/nℤ_{U} = ℤ/nℤ_{H}, 2) there is an exact sequence

0 ⟶ R^{1}f̄_{∗}ℤ/nℤ_{X} ⟶ R^{1}f_{∗}ℤ/nℤ_{U} ⟶ F_{1} ⟶ 0

and 3) there are isomorphisms R^{q}f_{∗}ℤ/nℤ_{U} ≅ F_{q} for q ≥ 2.

We have already used the morphism

Rf̄_{∗}ℤ/nℤ_{X} ⟶ Rf_{∗}ℤ/nℤ_{U}.

In addition, there is a map

ℤ/nℤ_{H}[−1] ⟶ Rf_{∗}μ_{n},

which associates to 1 ∈ ℤ/nℤ the section of R^{1}f_{∗}μ_{n} corresponding to the μ_{n}-torsor of U = H × 𝔹^{∗} given by the equation y^{n} = z. We claim that together these induce a quasi-isomorphism

Rf̄_{∗}ℤ/nℤ_{X} ⊕ μ_{n}^{⊗−1}[−1] ⟶ Rf_{∗}ℤ/nℤ_{U}.  (1)

¿From the considerations above it follows that this implies the theorem.

To prove the claim we may assume that n is a prime power n = p^{r}. Let us treat the case that p equals the characteristic of k. In this case k is a p-adic field. The other cases are easier and similar arguments work.

Let us take c ∈ ℕ large enough. For m ∈ ℕ we put R_{m} = {x ∈ 𝔹; |x| ≥ |p^{cm}|}, a ring domain. Put U_{m} = H × R_{m}, note that the covering U = ⋃ U_{m} is admissible. Let us write f_{m} : U_{m} → H for the projection. We will study the overconvergent sheaf R^{q}f_{m,∗}μ_{n}. Its étale fibres are H^{q}(R_{m} ˆ⊗F_{e}, μ_{n}). These are zero for q ≥ 2 and equal to μ_{n} for q = 0. Note that

ℙ^{1} = {|z| ≥ |p^{cm}|} ∪ {|z| ≤ 1},

hence that we have an exact sequence

0 ⟶ F_{e}^{∗} ⟶ O^{∗}(|z| ≥ |p^{cm}|) ⊕ O^{∗}(|z| ≤ 1) ⟶ O^{∗}(R_{m}) ⟶ ℤ ⟶ 0.

This follows by the computation of cohomology of ℙ^{1}, use for example Lemma 4.1.2. We see immediately that

H^{1}(R_{m} ˆ⊗F_{e}) = ℤ/nℤ ⊕ H^{1}(|z| ≤ 1) ⊕ H^{1}(|z| ≥ |p^{cm}|).

This implies a corresponding decomposition of the overconvergent sheaf

R^{1}f_{m,∗}μ_{n} = ℤ/nℤ_{H} ⊕ R^{1}f̄_{∗}μ_{n} ⊕ Rest_{m}.

The maps Rest_{m+1} → Rest_{m} are zero, since by [L93, Theorem 2.1] the maps on the étale fibres H^{1}(|z| ≥ |p^{c(m+1)}|) → H^{1}(|z| ≥ |p^{cm}) are zero if c is large enough.

This means that for any V → H affinoid étale we have a decomposition

H^{q}(V × R_{m}) = H^{q−1}(V, μ_{n}^{⊗−1}) ⊕ H^{q}(V × 𝔹) ⊕ Rest_{m},

the transition maps H^{q}(V × R_{m+1}) → H^{q}(V × R_{m}) are the identity on the first two summands, zero on the last one. This proves that

lim_{⟵} H^{q}(V × R_{m}) = H^{q−1}(V, μ_{n}^{⊗−1}) ⊕ H^{q}(V × 𝔹)

and the derived limit

lim_{⟵}^{(1)}H^{q}(V × R_{m}) = (0).

We conclude that H^{q}(V × 𝔹^{∗}) = H^{q−1}(V, μ_{n}^{⊗−1}) ⊕ H^{q}(V × 𝔹), hence (1) is an isomorphism. □

7.3 Comparison

In this section X will denote a variety of finite type over over the complete valued field k. We state the results corresponding to [SGA 4, Exposé XI Theorem 4.4] in our case. Further, we will indicate the nessecary changes in the proof given there so that it will work in our case also.

Theorem 7.3.1 Suppose the charateristic of k is zero. There is an equivalence between the category of locally constant sheaves on X_{ét} with finite stalks and the category of locally constant sheaves on X_{étale}^{an} with finite stalks. The equivalence is given by the functors ϵ^{∗} and ϵ_{∗}.

Proof. Since sheaves of this kind are representable by finite étale coverings we see that it suffices to prove the following statement: If Y → X^{an} is finite étale then there exists a (unique) finite étale morphism of schemes Z → X such that Y ≅ Z^{an}. This was recently proved by Lütkebohmert, see [L93]. □

In the next theorem k is no longer of characteristic zero.

Theorem 7.3.2 Let X be smooth over Spec(k) and let k be algebraically closed. Suppose S is an abelian locally constant sheaf on X_{ét} with finite stalks where all orders of torsion are prime to the characteristic of k. In this case we have R^{q}ϵ_{∗}ϵ^{∗}S = (0) for q ≥ 1. The canonical morphisms H^{q}(X_{ét}, S) → H^{q}(X_{étale}^{an}, ϵ^{∗}S) are isomorphisms. In particular we have

H^{q}(X_{ét}, ℤ/nℤ) ≅ H^{q}(X^{an}, ℤ/nℤ_{X^{an}}).

Proof. With the results proved above, we can use the proof of [SGA 4, Exposé XI Theorem 4.4 part (ii)]. In stead of the ‘calcul direct’ of line 1 on page 13 we use Theorem 7.2.2. The only other fact used in the proof which is not immediately clear is the following: Suppose f̄ : X̄ → S is a familly of smooth projective curves over the scheme S, which is of finite type over k, suppose n is relatively prime to the characteristic of k. In this case R^{1}f̄_{∗}^{an}ℤ/nℤX̄^{an} is a locally constant sheaf on S_{étale}^{an}. However, this is immediately clear from: 1) The corresponding fact in the algebraic case. 2) The base change map ϵ^{∗}R^{1}f_{∗}ℤ/nℤ → R^{1}f̄_{∗}^{an}ℤ/nℤX̄^{an} is an isomorphism (look at étale fibres). □

Remark 7.3.3 The more general results proved in [SGA 4, Exposé XVI §4] should hold true for the rigid analytic case also. At least if the characteristic of k is zero then it should be possible with some effort to follow the reasoning of locus citatus in this case.

7.4 Counterexamples in characteristic p > 0.

Take k algebraically closed of characteristic p > 0. Riemann’s existence theorem is no longer valid in this case. We give an example of this.

Lemma 7.4.1 Consider the covering ψ : Y → 𝔸^{1 an}, given by the equation T^{p} − T = F := ∑_{i≥0} a_{i}z^{p^{i}}, where the series F converges on 𝔸^{1 an}. We suppose that there are infinitely many non zero a_{i} and that for every k ≥ 0 one has

|a_{k} + a_{k−1}^{p} + a_{k−2}^{p^{2}} + ... + a_{0}^{p^{k}}| = max _{0≤i≤k}(|a_{i}|^{p^{k−i}}).

Then Y is not isomorphic to Z^{an} for any covering Z → 𝔸^{1}.

Proof. Any p-cyclic (unramified) covering of the unit disk D is given by an equation T^{p} − T = f with f ∈ O(D). Two functions f_{1}, f_{2} ∈ O(D) define isomorphic coverings if and only if λ_{1}f_{1} + λ_{2}f_{2} = h^{p} − h holds with λ_{1}, λ_{2} ∈ 𝔽_{p}^{∗} and h ∈ O(D). Using that the structure sheaf O on the analytic space 𝔸^{1,an} has trivial cohomology, one finds for 𝔸^{1,an} similar results. Namely: Any p-cyclic analytic covering of 𝔸^{1 an} is given by an equation T^{p} − T = f with f a holomorphic function on 𝔸^{1 an}. Two holomorphic functions f_{1}, f_{2} define the same p-cyclic extension if and only if there is a λ ∈ 𝔽_{p} such that the equation T^{p} − T = λ_{1}f_{1} + λ_{2}f_{2} has a holomorphic solution.

If Y = Z^{an} then Z is a p-cyclic covering of 𝔸^{1} given by an equation of the form U^{p} − U = g with g ∈ zk[z]. Let the equation T^{p} − T = −G := F − λg have a holomorphic solution h = ∑_{i≥1} h_{i}z^{i}. In some disk around 0 the spectral norm of G is less than 1. Therefore ∑_{i≥0} G^{p^{i}} converges and is on this disk a solution of T^{p} − T = −G. So h coincides with ∑_{i≥0} G^{p^{i}} on this disk and the power series expansion of h is equal to the power series expansion of ∑_{i≥0} G^{p^{i}}. One takes a disc D(0, R) around 0 such that 1 < B := ∥G∥_{R} = ∥F ∥_{R} > ∥g∥_{R}. After replacing z by zλ for a suitable λ ∈ k^{∗}, we may suppose that R = 1. First we look at ∑_{i≥0} F^{p^{i}} = ∑_{k≥0} A_{k}z^{p^{k}} with A_{k} = (a_{k} + a_{k−1}^{p} + a_{k−2}^{p^{2}} + ... + a_{0}^{p^{k}}). A calculation shows that for N >> 0 one has |A_{N}| = |A_{N−1}|^{p} and |A_{N}| ≥ B. Then we look at ∑_{i≥0} g^{p^{i}} = ∑_{k≥1} b_{k}z^{k}. One can calculate that the absolute value of b_{p^{k}} grows less fast than |A_{k}|. This implies that the power series representing h is not convergent on D(0, 1). This contradiction ends the proof. □

Corollary 7.4.2 The map

H^{1}(𝔸_{et}^{1}, ℤ/pℤ) ⟶ H^{1}(𝔸_{etale}^{1 an}, ℤ/pℤ)

is injective but not surjective. In particular Theorem 7.3.2 is not valid for sheaves consisting of p-torsion.

Corollary 7.4.3 Let n > 1 with p|ϕ(n) and with n not divisible by p. There is a locally constant sheaf S on 𝔸_{étale}^{1 an} with stalk ℤ/nℤ, which is not of the form ϵ_{∗}T.

Proof. We consider the p-cyclic analytic covering ψ : Y → 𝔸^{1 an} of Lemma 7.4.1. Let σ denote the generator of the Galois group G of this extension. Let M denote the constant étale sheaf on Y with stalk ℤ/nℤ. Let a ∈ ℤ/nℤ^{∗} be an element of order p. We define an action G on ℤ/nℤ by σ(i) = ai. This induces an action of G on ℤ/nℤ × Y by σ((i, y)) = (ai, σ(y)). The quotient by this group action is a sheaf S on 𝔸_{étale}^{1 an} which is locally the constant sheaf with stalk ℤ/nℤ. (And of course ψ^{∗}S is the constant sheaf on Y). However, there is no étale covering {Y_{i}} of 𝔸^{1} which trivializes S. Indeed, such an étale covering would give a trivialization of Y → 𝔸^{1 an}. □

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Johan de Jong

dejong@math.harvard.edu Marius van der Put