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I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES by VLADIMIR G. BERKOVICH (1) CONTENTS ]NTI%ODUCTION .............................................................................. 7 w 1. Analytic spaces .......................................................................... 13 1.1. Underlying topological spaces ........................................................ 13 1.2. The category of analytic spaces ..................................................... 14 1.3. Analytic domains and G-topology on an analytic space ................................ 23 1.4. Fibre products and the ground field extension functor ................................. 1.5. Analytic spaces from [Ber] .......................................................... 31 1.6. Connection with rigid analytic geometry ............................................. 35 w 2. Local rings and residue fields of poh~ts of affnoid spaces .......................................... 2.1. The local rings 6x, x ............................................................... 2.2. Comparison of properties of ~x,x and 6~, x .......................................... 2.3. The residue fields ~:(x) ............................................................. 2.4. Q uasicomplete fields ............................................................... 45 2.5. The cohomological dimension of the fields It(x) ....................................... 49 2.6. GAGA over an affinoid space ....................................................... w 3. [~tale and smooth mcrpkisms ............................................................... Q uasifinite morphisms .............................................................. 54 3.1. Flat quasifinite morphisms .......................................................... 3.2. 3.3. t~tale morphisms ................................................................... Germs of analytic spaces ........................................................... 64 3.4. 3.5. Smooth morphisms ................................................................. 3.6. Smooth elementary curves ........................................................... 3.7. The local structure of a smooth morphism ........................................... w 4. [~tale cohomology . ........................................................................ 4.1. l~tale topology on an analytic space ................................................. Stalks of a sheaf ................................................................... 4.2. 4.3. Quasi-immersions of analytic spaces .................................................. 4.4. Q uasiconstructible sheaves .......................................................... (1) Incumbent of the Reiter Family Career Development Chair. Supported in part by NSF grant DMS-9100383. VLADIMIR G. BERKOVICH w 5. Cohomology with oompact support ............................................................ 5. I. Cohomology with support .......................................................... 5.2. Properties of cohomology with support ............................................... 5.3. The stalks of the sheaf R.q ~r F ...................................................... 5.4. The trace mapping for itat quasifinite morphisrns ...................................... 106 w 6. Calculation of cohomology for curves ......................................................... 109 6.1. The Comparison Theorem for projectives curves ...................................... 6.2. The trace mapping for curves ....................................................... 6.3. Tame ~tale coverings of curves ...................................................... I19 6.4. Cohomology of affmoid curves ...................................................... w 7. Main theorems .......................................................................... 7.1. The Comparison Theorem for Cohomology with Compact Support ..................... 7.2. The trace mapping ................................................................ 130 7.3. Poincarg Duality ................................................................... 7.4. Applications of Poincarg Duality .................................................... 7.5. The Comparison Theorem .......................................................... 7.6. The invariance of cohomology under extensions of the ground field ...................... 150 7.7. The Base Change Theorem for Cohomology with Compact Support ..................... 153 7.8. The Smooth Base Change Theorem ................................................. R~FEgSNeES ................................................................................. ISDEX OF NOTATIONS ........................................................................ 158 INDEX OF TERMINOLOGY ...................................................................... 160 INTRODUCTION The problem of constructing an &ale cohomology theory for non-Archimedean analytic spaces has arisen from Drinfeld's work on elliptic modules [Dr1]. In his work, Drinfeld defined (among other things) the first cohomology group Hi(X, ~,) as the set of pairs (L, ~), where L is an invertible sheaf on X, and ~ is an isomorphism ~x ~ L| He showed that for the one-dimensional p-adic upper half-plane ~ this group gives rise to a certain infinite dimensional representation of GL(k), where k is the ground local field. Afterwards, in [Dr2], Drinfeld constructed a certain family of equivariant coverings of the d-dimensional p-adic upper half-plane f~a+l, and suggested that all cuspidal representations of the group GLa+l(k) are realized in high dimensional &ale cohomology groups of this family of coverings. Since then, as far as I know, the only attempt to construct an &ale cohomology theory for non-Archimedean analytic spaces was undertaken by O. Gabber. We under- stand that O. Gabber has made progress in the subject, but, unfortunately, he has written nothing on it. Besides that, in [FrPu] and [ScSt], definitions of an &ale topology on a non-Archimedean analytic space were given, and in [ScSt] the cohomology of L2 a + 1 is calculated for arbitrary d under the hypotheses that this cohomology satisfies certain reasonable properties. Finally, in [Car], a conjecture, which is an explicit form of Drinfeld's suggestion, is proposed. The conjecture predicts the decomposition of the representations of GLd+l(k ) and the Galois group of/~ (k is a p-adic field) on the d-dimensional cohomology group of the equivariant system of coverings of fl TM in terms of the Langlands correspondence. The purpose of this work is to develop many basic results of &ale cohomology for non-Archimedean analytic spaces. We define the &ale cohomology and the &ale cohomology with compact support and calculate the cohomological dimension of an analytic space. We prove a Comparison Theorem for Cohomology with Compact Sup- port which states that, for a compactifiable morphism ~:~r _~ ~ between schemes of locally finite type over the spectrum of a k-affinoid algebra and a torsion sheaf ~" on ~/, there is a canonical isomorphism (R ~ ~I ~-)'~ ~ R ~ V~ ~-~, q/> 0. We also prove a Poincard Duality Theorem, the acyclicity of the canonical projection X � D ~ X, where D is an open polydisc in the affine space, a Cohomological Purity Theorem, the invariance of the cohomology under algebraically closed extensions of the ground field, a Base Change Theorem for Cohomology with Compact Support and a Smooth Base Change Theorem (all the results are proved for torsion sheaves with torsion orders prime to the characteristics of the residue field of k). In particular, all the properties of the " abstract" cohomology theory from [ScSt] hold. Our main result is a Compa- rison Theorem which states that, for a morphism of finite type ~:q/-~ 3~ between VLADIMIR G. BERKOVICH schemes of locally finite type over k and a constructible sheaf o~" on ~/ with torsion orders prime to the characteristics of the residue field of k, there is a canonical isomor- phism (R ~ 9. o~-) ~n -~ R" q~.n o~-~, q/> 0. We note that the only previous proof of the Comparison Theorem in the classical situation over C uses Hironaka's Theorem on resolution of singularities. Our proof of the Comparison Theorem works over l:l as well and does not use Hironaka's Theorem. Our approach to 6tale cohomology is completely based on the previous work [Ber]. In that work we introduced analytic spaces which are natural generalizations of the complex analytic spaces and have the advantage that they allow direct application of the geometrical intuition. One should say that although the analytic spaces from [Ber] were considered in a more general setting than that for rigid analytic geometry (for example, the valuation of the ground field is not assumed to be nontrivial), they don't give rise to all reasonable rigid spaces. And so our first purpose in this work is to extend the category of analytic spaces from [Ber] so that the new category gives rise to all reasonable rigid spaces, for example, to those that are associated with formal schemes of locally finite type over the ring of integers of k. We now give a summary of the material which follows. Let k be a non-Archimedean field, and let ~ denote its residue field. As in [Ber], we don't assume that the valuation of k is nontrivial. In w 1 we introduce a category of k-analytic spaces more general than those from [Ber]. These analytic spaces possess nice topological properties. For example, a basis of topology is formed by open locally compact paracompact arcwise connected sets. One does not need to use Grothendieck topology in the definition of the spaces, but they are naturally endowed with such a topology called the G-topology (w 1.3). The latter is formed by analytic domains of an analytic space. The spaces from [Ber] (they are said to be " good ") are exactly those in which every point has an affinoid neigh- borhood. The G-topology on an analytic space is a natural framework for working with coherent sheaves. (If the space is good, then it is enough to work with the usual topology as in [Ber].) In w 1.4 we show that the category of analytic spaces introduced admits fibre products and the ground field extension functor, and we associate with every point x of an analytic space a non-Archimedean field Yf(x) so that any mor- phism q0:Y-+.X induces, for a point y e Y, a canonical isometric embedding Jt(q~('y)) r In w 1.5 we define for a morphism q0:Y -+ X the relative interior Int(Y/X) (this is an open subset of Y), and we call the morphism closed if Int(Y/X) = Y. In w 1.6 we construct a fully faithful functor from the category of Hausdorff (strictly) analytic spaces to the category of quasiseparated rigid spaces and show that it induces an equivalence between the category of paracompact analytic spaces and the category of quasiseparated rigid spaces that have an admissible affinoid covering of finite type. In w 2 we establish properties of the local ring 0x, , and its residue field ~(x), where x is a point of a k-affinoid space X. (The completion of (x) is the field 34Z(x).) First, we establish those properties which are mentioned without proof in [Ber], w 2.3. I~.TAI.E COI,,IOMOLOGY FOR NON-ARCHIM'EDEAN ANALYTIC SPACES Furthermore, we prove that the ring Ox. x is Henselian, and the canonical valuation on ~(x) extends uniquely to any algebraic extension (fields with this property are said to be quasicomplete). The latter two facts are of crucial importance for the whole story. The quasicompleteness of K(x) implies, for example, an equivalence between the cate- gories of finite separable extensions of ~(x) and of .,~(x) and, in particular, an isomor- phism of their Galois groups G~<~)--% Ga~z). In w 2.4 we establish properties of quasi- complete fields (whose proofs are borrowed from [BGR] and [ZaSa]), and in w 2.5 we show that the g-cohomological dimension cdt(K(x)) of the field (x) (or, equivalently, of the field of(x)) is a most cdt(k ) + dim(X), where g is a prime integer. In w 2.6 to every scheme ~ of locally finite type over .%r - Spec(.ff), where d is an affinoid algebra, we associate an analytic space ~t,~ over X = Jt'(d). These objects are very important, in particular, for the proof of the Poincar6 Duality Theorem. We establish some basic facts on the correspondence ~r ~ ~/'~, which are necessary for this work. In w 3 we introduce and study the classes of ~tale and smooth morphisms. The first basic notion is that of a quasifinite morphism. A morphism ~ : Y ~ X is said to be quasifinite if for any pointy E Y there exist open neighborhoods r ofy and q/of ~(y) such that ~ induces a finite morphism r -7 ql. It turns out that a morphism is quasi- finite if and only if it has discrete fibres and is closed (in the sense ofw 1.5). Furthermore, we define 6tale morphisms. (By definition, they belong to the class of quasifinite mor- phisms.) For example, the canonical immersion of the closed unit disc in the affine line is not ~tale because it is not a closed morphism. In w 3.4 we introduce the notion of a germ of an analytic space and prove the very important fact that the category of germs finite and 6tale over the germ (X, x) of an analytic space X at a point x is equi- valent to the category of schemes finite and 6tale over the field Off(x). Furthermore, in w 3.5 we study smooth morphisms. A mosphism q~ : Y ~ X is said to be smooth if locally it is a composition of an 6tale morphism to the affine space A~ ~-- M � X and the canonical projection Aax ~ X. In particular, any smooth morphism is closed. The latter property of smooth morphisms is natural if we want to have for them Poincar6 Duality. In w 3.6 and w 3.7 we describe the local structure of a smooth morphism. This description is very important for the sequel and is actually an analog of the trivial fact that locally any smooth morphism of complex analytic spaces is isomorphic to the projection X � D ~ X, where D is an open polydisc in the affine space. In w 4 we define the 6tale topology on a k-analytic space X (the 6tale site X,~) and establish first basic properties of 6tale cohomology. In w 4.1 we verify that certain reasonable presheaves are actually sheaves and give an interpretation of the first coho- mology group with coefficients in a finite group. In w 4.2 we define the stalk Fz of a sheaf F at a point x ~ X. It is a discrete Gae~,)-set. It turns out that ff n is the canonical morphism of sites X,t ~ I X I, where I X [ is the site generated by the usual topology of X, then for any abelian sheaf F on X~t , there is an isomorphism (R ~ n, F)x -~ H~(G~e~,), F,). It follows that the sheaf F is flabby if and only if, for any point x e X, the fibre F, is a flabby Gae~,)-module and, for any 6tale morphism U ~ X, the restriction of F to the 2 VLADIMIR G. BERKOVICH I0 usual topology of U is flabby. This fact reduces the verification of many properties of the 6tale cohomology established in w 4 and w 5 to the verification of certain properties of the cohomology of profinite groups and the usual cohomology with coefficients in sheaves. As first applications of these considerations we prove that if X is good then the dtale cohomology of the sheaf induced by a coherent Ox-module coincides with its usual cohomology and that the l-cohomological dimension cdt(X) of a paracompact k-analytic space X is at most cdt(k) + 2 dim(X). The latter fact is easily obtained from the spectral sequence of the morphism of sites ~:Xa-+IX l, using the facts that the topological dimension of such a space is at most dim(X) and that cdt(gf~ ~< cdt(k) + dim(X). In w 4.3 we study quasi-immersions of analytic spaces. A morphism ~:Y-+ X of analytic spaces over k is said to be a quasi-immersion if it induces a homeomorphism of Y with its image ~0(Y) in X and, for any point y ~ Y, the field Jt~ is a purely inseparable extension of~(q~(y)). For example, the canonical embeddings of analytic domains in an analytic space and closed immersions are quasi-immersions. We prove that if ~ : Y ~ X is a quasi-immersion such that the set ~?(Y) has a basis of paracompact neighborhoods, then for any abelian sheaf F on X one has Hq(Y, F[y) = lira Ha(q/, F), where q/runs through open neighborhoods of the set ~(Y). Furthermore we construct a spectral sequence which relates the cohomology of a paracompact k-analytic space X to the cohomology of closed analytic domains from a locally finite coverings by such domains. We use it to show that the group Hx(X, ~t,) has the interpretation given to it by Drinfeld in [Drl]. In w 4.4 we introduce and study quasiconstructible sheaves which play the role of construcfible sheaves on schemes in the sense that any abefian torsion sheaf is a filtered inductive limit of quasiconstructible sheaves (the word " cons- tructible" is reserved for a future development). In w 5 we introduce and study the 6tale cohomology with compact support. All definitions and constructions are straightforward generalizations of the corresponding topological notions. In particular, the cohomology groups with compact support are defined as the right derived functors of the functor of sections with compact support. Theorem 5.3.1 gives, for an abelian sheaf F, a description of the stalks of the sheaves R q ~, F, where ~ is a Hausdorff morphism of k-analytic spaces, in terms of the coho- mology of the fibres of ~. As an application we show that if F is a torsion sheaf, then R a ~0~ F = 0 for all q > 2d, where d is the dimension of ?. In w 5.4 we construct for every separated flat quasifinite morphism ~ : Y -+ X and for every abelian sheaf F on X a trace mapping Tr~ : ~, ~'(F) ~ F. In w 6 we establish various facts on the cohomology of analytic curves. These facts are a basis for the induction used in the proof of the main theorems from w 7. In w 6.1 we prove the Comparison Theorem for Gohomology with Compact Support for curves. The proof of this theorem in the general case (9 7.1) may be read immediately after w 6.1. In the rest of w 6 we assume that the ground field k is algebraically closed. In w 6.2 we construct, for every smooth separated analytic curve X, a trace I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 11 mapping Trx: H~(X , [z~) -+ Z/nZ, where n is prime to char(k), and we show that it is an isomorphism if X is connected and n is prime to char(k). The central fact of w 6.3 (Theorem 6.3.2) states that any tame finite 6tale Galois covering of the one-dimensional closed disc is trivial (an 6tale morphism ~ : Y -+ X is said to be tame if for any point y ~ Y the degree [3~(y) :gff(~0(y))] is not divisible by char(~)). We deduce from this, in particular, a Riemann Existence Theorem which states that, for an algebraic curve of locally finite type over k, the functor ~/v-. ~n defines an equivalence between the category of finite 6tale Galois coverings of 5~" whose degree is prime to char(~) and the category of similar coverings of ~. We deduce also the Comparison Theorem for curves (the proof of this theorem in the general case (w 7.5) does not use the particular case). In w 6.4 we prove that, for a one-dimensional k-affinoid space X and a positive integer n which is prime to char(~), the group Hq(X, Z/nZ) is finite for q = 0, 1 and equal to zero for q I> 2, Furthermore, this group is preserved under algebraically closed extensions of the ground field. In w 7 we obtain our main results. In w 7.1 we prove the Comparison Theorem for Cohomology with Compact Support. This result implies that the cohomology groups with compact support of a scheme of locally finite type over k (recall that k may have trivial valuation) can be defined as the right derived functors of the functor of sections with compact support over the associated k-analytic space. In w 7.2 we construct a trace mapping Tr~ :R2a ~,(~,.x)~ -+ (Z/nZ)x for an arbitrary separated smooth mor- phism ~ : Y -+ X of pure dimension d and for n prime to char(k), and we show that it is an isomorphism if the geometric fibres of q~ are nonempty and connected and n is prime to char(k). In the rest of w 7 all sheaves considered are torsion with torsion orders prime to char(~). In w 7.3 we prove the Poincar6 Duality Theorem, which is actually a central result of this work. The main ingredients of the proof are our Theorem 3.7.2 and the Fundamental Lemma ([SGA4], Exp. XVlII, 2.14.2) from the proof of the Poincar6 Duality Theorem for schemes. In w 7.4 we give first applications of Poincar6 Duality. In particular, we prove the acyclicity of the canonical projections X � Ak a -+ X and X � D -+ X and the Cohomological Purity Theorem. In w 7.5 we prove the Comparison Theorem. The proof follows closely the proof of Deligne's " generic " theorem 1.9 from [SGA4{], Th. finitude, and uses it. (I am indebted to D. Kazhdan for suggesting that Deligne's " Th. finitude" could be useful for the proof of the Comparison Theorem.) The proof is actually a formal reasoning which works over the field of complex numbers (l as well. In w 7.6 we prove that the cohomology groups H~(X, F) and H~(X, F) are preserved under algebraically closed extensions of the ground field. In w 7.7 we deduce from this and from Theorem 5.3.1 the Base Change Theorem for Cohomology with Compact Support. It implies, in particular, a Kiinneth Formula. In w 7.8 we prove the Smooth Base Change Theorem. The proof uses Poincar6 Duality and the Base Change Theorem for Cohomology with Compact Support. VLADIMIR G. BERKOVICH Acknowledgments My interest in this subject was stimulated by D. Kazhdan who suggested I construct an 6tale cohomology theory for p-adic analytic spaces, and thereby show that the ana- lyric geometry of [Ber] is useful. I am very grateful to him for his interest, encouragement and valuable remarks. The first version of this paper was written at the end of 1990. It was also the topic of a series of talks I gave at the Institute for Advanced Study, Princeton, during the academic year 1991/1992. The listeners were P. Deligne and N. Katz. Their numerous critical remarks, explanations to me of the things I was talking about, and suggestions led me to a better understanding of the subject and, as a partial consequence of it, to many improvements of the original text. I am very grateful to them for all this. In the first two versions only the analytic spaces from [Ber] were considered. I am very grateful to P. Deligne for his further remarks and comments that stimulated me to introduce more general analytic spaces and to extend to them all the results obtained. I am very grateful to V. Hinich for many very useful discussions during all stages of this work. I greatly appreciate the hospitality and the financial support of the Institute for Advanced Study and Mathematisches Institut der Universit~t zu K61n where parts of this work were done. I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 13 w 1. Analytic spaces 1.1. Underlying topological spaces In this subsection we introduce some structures on topological spaces that will be used in the sequel. We also fix general topology terminology. All compact, locally compact and paracompact spaces are assumed to be Haus- dorff. (A Hausdorff topological space is called paracompact if any open covering of it has a locally finite refinement.) Recall that a locally compact space is paracompact if and only if it is a disjoint union of open and closed subspaces countable at infini W ([Bou], Ch. I, w 10, nO 12, [En], 5.1.27). Recall also that if a Hausdorff topological space has a locally finite covering by paracompact closed subsets, then the space is para- compact ([EnJ, 5.1.34). A topological space is said to be locally Hausdorff if each point of it has an open Hausdorff neighborhood. Let X be a topological space, and let v be a collection of subsets of X. (All subsets of X are provided with the induced topology.) For a subset Y C X we set "~1~ = {V e v [ V C Y }. We say that v is dense if, for any V e v, each point of V has a fundamental system of neighborhoods in V consisting of sets from v Iv" Furthermore, we say that -~ is a quasinet on X if, for each point x E X, there exist V1 .... , V, e v such that x e V1 n ... n V, and the set Va to ... w V, is a neighborhood of x. 1.1.1. Lemma. -- Let v be a quasinet on a topological space X. (i) a subset ql C X is open if and only if for each V ~ v the intersection rill n V is open inV. (ii) Suppose that 9 consists of compact sets. Then X is Hausdorff if and only if for any pair U, V ~ v the intersection U n V is compact. Proof. -- The direct implication in both statements is trivial. (i) Suppose that o//n V is open in V for all V ~ ~. For a point x ~ o// we take V1,...,V.e-~ such that xeVa n... nV, and V lu... uV, is a neighborhood of x in X. By hypothesis, there exist open sets ~CX with o//AV i= ~ nV~. Then the set r := ~r n... n Y/', is an open neighborhood of x in X. It follows that the set o//n (V 1 t3 ... t3 V,) is a neighborhood of x because it contains the inter- section 3r n (V 1 t3 ... u V,) which is a neighborhood of x. Therefore ~//is open in X. (ii) Suppose that U n V are compact for all pairs U, V e v. Since �215 Ve~} 14 VLADIMIR G. BERKOVICH is a quasinet on X � X, then, by (1), it suffices to verify that the intersection of the diagonal with any U � V for U, V e v is closed in U � V. But this intersection is homeomorphic to the compact set U n V, and therefore it is closed in U � V. 9 We remark that if X is Hausdorff, then to establish that a collection of compact subsets v is a quasinet, it suffices to verify that each point of X has a neighborhood of the form V1 u ... u V, with V s ~ v. We remark also that a Hausdorff space admitting a quasinet of compact subsets is locally compact. Furthermore, we say that a collection v of subsets of X is a net on X if it is a quasinet and, for any pair U,V ~, ~lvnv is a quasinet on U nV. 1.1.2. Lemma. ~ Let v be a net of compact sets on a topological space X. Then (i) for any pa# U, V e % the intersection U n V is locally closed in U and V; (ii) /fV C V1 u ... u V, for some V, V1, ..., V, 9 v, then there exist U1, ..., U,~ e .r such that V = U1 u ... u U,, and each U s is contained in some V s. Proof. -- (i) It suffices to verify that U n V is locally compact in the induced topology. But this is clear because X lvnv is a quasinet on U n V. (ii) For each point x E V and for each i with x ~ V~, we take a neighborhood of x in V n V s of the form V n u ... u Vs,,i , where Vii ~x. Then the union of such neighborhoods over all i with x EV s is a neighborhood of x in V of the form U 1 u ... u U,~ such that each U s belongs to .r and is contained in some V s. Since V is compact, we get the required fact. 9 The underlying topological spaces of analytic spaces will be, by Definition I. 2.3 below, locally Hausdorff and provided with a net of compact subsets. It will follow from the definition (Remark 1.2.4 (iii)) that they admit a basis of open locally compact paracompact arcwise connected subsets (see also Proposition 1.2.18), A continuous map of topological spaces q~ : Y -+ X is said to be Hausdorff if for any pair of different pointsy 1,y~ E Y with ~(Yl) = ~(Y2) there exist open neighborhoods r ofyl and ~e" 2 ofy~ with $/'x n ~e" 2 = O (i.e., the image of Y in Y � Y is closed). We remark that if q~ : Y ~ X is Hausdorff and X is Hausdorff, then Y is also Hausdorff. Furthermore, let X and Y be topological spaces and suppose that each point of X has a compact neighborhood. A continuous map q~ : Y --> X is said to be compact if the prei- mage of a compact subset of X is a compact subset of Y. It is clear that such a map is Hausdorff, it takes closed subsets of Y to closed subsets of X, and each point of Y has a compact neighborhood. 1.2. The category of analytic spaces Throughout the paper we fix a non-Archimedean field k. (We don't assume that the valuation on k is nontrivial.) The category of k-affinoid spaces is, by definition, the category dual to the category of k-affinoid algebras (see [Ber], w 2.1). The k-affinoid ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 15 space associated with a k-affinoid algebra off is denoted by X, where X = .It(off). (For properties of k-affinoid spaces and their affinoid domains, see loc. cit., w 2.) The notion of a k-analytic space we are going to introduce is based essentially on the following two fundamental facts. Let { V~ }i e i be a finite affinoid covering of a k-affinoid space X = ~t(off). Tate's Acyclicity Theorem. -- For any finite Banach off-module M, the ~ech complex 0 -+M ~ IIM| --> 1-IM| i ~... is exact and admissible. 9 Kiehl' s Theorem. -- Suppose we are given, for each i e I, a finite off vi-m~ M~ and, for each pair i, j ~ I, an isomorphism ofsCvlnvj-modules % : M i | offvin vi -% Ms | i offvln vi such that ~a[w---- ~J,[w~ ~i~lw, W ---- V~ n V~ n V,, for all i,j,l eI. Then there exists a finite off-module M that gives rise to the dvfmodules M e and to the isomorphisms ~. 9 Both results are originally proved in the case when the valuation on k is nontrivial and all the spaces considered are strictly k-affinoid (see [BGR], 8.2.1[5 and 9.4.3/3). But the general case is reduced to this one by the standard argument from [Bet], w 2.1 (2.2.5 and 2.1.11). Tate's Acyclicity Theorem is sufficient to define the category of k-analytic spaces, and Kiehl's Theorem is used to establish their basic properties. 1.2.1. Remarks. -- (i) Let V be a subset of a k-affinoid space X = ~'(d) which is a finite union of affinoid domains ( V~ }i ~ i. From Tate's Acyclicity Theorem it follows that the commutative Banach k-algebra d v = Ker(H~Cv,-+.Hdvinvs) does not % J depend (up to a canonical isomorphism) on the covering. Furthermore, V is an affinoid domain if and only if the Banach algebra d v is k-affinoid and the canonical map V-+~r is bijective. (In [Ber], 2.2.6 (iii), the latter condition was missed.) (ii) From Tate's Acyclicity Theorem it follows that in the situation of KiehI's Theorem the d-module M is isomorphic to Ker(IIM,-+IIM~|162 dvinvi). (Recall that, by [Ber], 2.1.9, the category of finite Banach off-modules is equivalent to the category of finite domodules.) Our purpose is to introduce a category of *-analytic spaces associated with a system * of the following form. Suppose we are given for each non-Archimedean field K over k a class of K-affinoid spaces *~ so that the system * = { r } satisfies the following conditions: (1) .g(K) E,x; (2) *~ is stable under isomorpkisms and direct products; (3) ff ~0 : Y ~ X is a finite morphism of K-affinoid spaces and X ~ @~r, then Y e@x; 16 VLADIMIR G. BERKOVICH (4) if { V, },~i is a finite affinoid covering of a K-affinoid space X such that V~ e ~ for all i a I, then X s ~z; (5) if K ,-+ L is an isometric embedding of non-Archimedean fields over k, then for any X e ~x one has X (~r L e (I)r.. The class qbx is said to be dense if each point of each X e qb~r has a fundamental system of affinoid neighborhoods V e (I)~. The system 9 is said to be dense if all ~K are dense. The affinoid spaces from (I) K (resp. ~) and their algebras will be called ~K-(resp. ~-) affinoid. From (2) and (3) it follows that (I) K is stable under fibre products. In particular, if ? : Y -+ X is a morphism of qb~-affinoid spaces, then for any affinoid domain V C X with V e~K one has ?-I(V) eqb K. 1.9.. 2. Remark. -- In fact we shall consider in this paper only analytic spaces for the system of all affinoid spaces. The more general setting is necessary for establishing connection with rigid analytic geometry in w 1.6 (see also Remark 1.2.16). For this one takes for (I) K the class of strictly K-affinoid spaces. That this qb K satisfies (4) is shown as follows. Let X = ./g(d). By Tate's Acyclicity Theorem, the algebra ~r is a closed subalgebra of the direct product O~r It follows that for the spectral radius p(f) of an elementfe ~r one has p(f) = max 9vi(f)- Since Pv~(f) e V/[ k*] { 0 }, then p(f) ~%/[ k*l u{0}, and therefore ~' is strictly K-affinoid, by [Ber], 2.1.6. (Of course, in w 1.6 one assumes also that the valuation on k is nontriviah In this case the system (I) is dense.) Here is one more example of qb. Assume that the valuation on k is trivial. If the valuation on K is also trivial, then we take for @x the class of K-affinoid spaces X = J/(~r such that 9(f)~< 1 for all fe ~r Otherwise we take for ~K the class of all K-affinoid spaces. The system (I) = { (I)~r } satisfies the conditions (1)-(5) and it is dense. Let X be a locally Hausdorff topological space, and let v be a net of compact subsets on X. 1. ~.. 3. Definition. -- A rb~-affinoid atlas d on X with the net 9 is a map which assigns, to each V e % a (I)k-affinoid algebra ~r and a homeomorphism V ~%dc'(0~'v) and, to each pair U, V e v with U C V, a bounded homomorphism of k-affinoid algebras evm : ~r -+ ~r that identifies (U, ~r with an affinoid domain in (V, ~r 1. ~.. 4. Remarks. -- (i) It follows from the definition that, for any triple U, V, W e v with U C V C W, one has ~w/v = ~v/v o ~w/v. (ii) The family of qbk-affinoid atlascs with the samc net forms a catcgory. (iii) By [Ber], 2.2.8 and 3.2. I, each point ofa k-affinoid space has a fundamental system of opcn arcwisc connectcd subscts that arc countablc at infinity. It follows that a basis of topology of a ~k-analytic space is formcd by open locally compact paracompact arcwisc conncctcd subscts. I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 17 A triple (X, ~r ~) of the above form is said to be a ~.analytic space. To define morphisms between them, we need a preparatory work. First, we'll define a category. ~k-~Cn whose objects are the ~-analytic spaces and whose morphisms will be called strong morphisms. After that the category of Ok-analytic spaces ~k-dn will be constructed t~J as the category of fractions of ~k-~Cn with respect to a certain system of strong morphisms that admits calculus of right fractions. Let (X, d, ~) be a q~k-analytic space. 1.2.5. Lemma. --- If W is a @-affinoid domain in some U E % then it is a @-a~noid domain in any V ~ .r that contains W. Proof. -- Since vitro ~ v is a net and W is compact, we can find U1,..., U, ~ v[u ~ v with W C U 1 u ... u U,,. Furthermore, since W and Ui are ~-affinoid domains in U, then Wi :---- W n U i is a ~-affinoid domain in U~. It follows also that W~ and W i n W~ are q)-affinoid domains in V. By Tate's Acyclicity Theorem, applied to the affinoid covering { W~ } of W the Banach algebra ~r = Ker(H ~r ~ l-[ ~r wj) is k-aflinoid and W-%d/(~Cw). By Remark 1.2.1 (i), W is a ~-affinoid domain in V. 9 Let z denote the family of all W such that W is a (I)-affinoid domain in some V e ~. If ~k is dense, then Z is dense. 1.2.6. Proposition. -- The family ~ is a net on X, and there exists a unique (up to a canonical isomorphism) ~k-affinoid atlas ~ with the net u that extends e~'. Proof. -- Let U,V ~ and xcU nV. Take U',V'~v with UCU' and VCV'. We can find a neighborhood W 1 u... u W, of x in U'c~ V' with W~ ~'r and x E W 1 n ... n W,. Since U (resp. V) and Wi are (1)-affinoid domains in U' (resp. V'), then Ui := U n W~ (resp. V~:---- V n Wi) is a (1)-affinoid domain in W~, and therefore U~ n V~ is a ~-affinoid domain in W~, i.e., U~ c~ V~ e "~lv~v. Since U,(u, n v,) = (u v) (U, w,), then IJ~(U, c~ V~) is a neighborhood of x in U c~ V with x ~ I~,(U~ c~ V~). It follows that ~ is a net. Furthermore, for each Vev we fix V'ev with VCV' and assign to V the algebra ~r and the homeomorphism V ~Jc'(~tv) arising from (V', ~v'). We have to construct, for each pair U, V e v with U C V, a canonical bounded homomorphism a~r v -+ ~r that identifies (U, a~l~r ) with an affinoid domain in (V, ~r Consider first the case when V ev. Since z~,~v is a quasinet, we can find sets U~, ..., U,, that are 9 -affinoid domains in U' and V and such that U = U~ u ... u U,. By Tate's Acy- clicity Theorem, ~r ----Ker([I~Cv~ ~ .1-Iar and lherefore the homomorphisms ~r -+ ~1~i and eft v -~- ~r ~ tr i induce a bounded homomorphism ~'v ~ ~r that iden- tifies (U, ~r with an affinoid domain in (V, ~r In particular, the homomorphism 3 18 VLADIMIR G. BERKOVICH constructed does not depend on the choice of Ux, ..., U,. Assume now that V is arbi- trary. Then U C V' and, by the first case, there is a canonical bounded homomorphism ~r ~ ~1~ that identifies (U, ~qr with an affinoid domain in (V', ~r It follows that (U, ~r is a ~-affinoid domain in (V, ~'v). 9 1.9..7. Definition. -- A strong morphism of ~,-analytic spaces : (x, (x', is a pair which consists of a continuous map q~ : X -+ X', such that for each V e ~ there exists V' ~ ~' with ~(V) C V', and of a system of compatible morphisms of k-affinoid spaces ~v~v' : (V, ~r ~ (V', ~r for all pairs V ~ 9 and V' ~ ~' with ~(V) C V'. 1 .~..8. Proposition. -- Any strong morphism ~ : (X, ~1, ,) -+ (X', ~r ,') extends in a unique way to a strong morphisra ~ : (X, ~7, ~) ~ (X, aft', ~'). Proof. -- Let U and U' be ~-affinoid domains in V E z and V' c z', respectively, and suppose that ~(U) C U'. Take W' c z' with ~(V) C W'. Then q~(U) C Wx u ... u W. for some W1,..., W, e ~'lv'~w'. The morphism of k-affinoid spaces ~v/w' induces a morphism V~:----~v/~,(W~)-~W~ that induces, in its turn, a morphism Ui :-- U n V~ ~ U; :---- U" n W~ (the latter is a ~-affinoid domain in V'). Thus, we have a system of morphisms of k-affinoid spaces U~ ~ U; ~ U' that are compatible on intersections. It gives rise to a morphism ~gm' : (U, dr) ~ (U', d~,). It clear that the morphisms ~g/r' are compatible. 9 We now define the composition Z of two strong morphisms : (X, d, ~) ~ (X', ~1', ~') and + : (X', ~r #) ~ (X", ~1", ,"). The map X that is the composition of the maps ~ and + satisfies the necessary condition of the Definition 1.2.7. Furthermore, by Proposition 1.2.8, we may assume that and + are extended to the morphisms ~ and ~. Suppose now that we are given a pair V ~ ~ and V" E x" with z(V) C V". We have to define a morphism of k-affinoid spaces Xvjv" : (V, a~lv) -+ (V", ~r For this we take V' ~ ~' and U" ~ -d' with ~(V) C V' and +(V') C U". Since z(V) C U" n V" and V is compact, it follows that there exist IP II I It It I --I ~'r V~,...,V, ~"~,,~v,, with z(V) CV~ u... vV,. Then V~:=~v,m,,(V;) and --1 9 ,. V~ : ~ ?v~v' (V~) are r domains in V' and V, respectively, and V = Vx ~. U V.. The morphisms ~ and ~ induce morphisms of k-affinoid spaces V~ -+ V;', and since V~' are ~-affmoid domains in V", they induce a system of morphisms V~ -+ V" that are compatible on intersections. It gives rise to the required morphism of k-affinoid spaces Zv~v" : (V, ~r --~ (V", ~r It is easy to see that the morphisms ~(v~v,' are compatible. Hence we get a morphism ~( that is the composition of ~ and hb and is denoted by h~ o ? (or simply by @~). Thus, we get a category ~,-~r ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 19 1.2.9. Definition. -- A strong morphism ~ : (X, d, x) ~ (X', a~', x') is said to be a quasi-isomorphism if ~ induces a homeomorphism between X and X' and, for any pair V ~ 7 and V' ~ -r' with ~(V) C V', ~wv' identifies V with an affinoid domain in V'. It is easy to see that ff r is a quasi-isomorphism, then so is ~. t~J 1.2.10. Proposition. -- The system of quasi-isomorphisms in @k-dn admits calculus of right fractions. Proof. -- We have to verify (see [GaZi], Ch. I, w 2, 2.2) that the system satisfies the following properties: a) all identity morphisms are quasi-isomorphisms; b) the composition of two quasi-isomorphisms is a quasi-isomorphism; c) any diagram of the form (X, d, v) ~ (X', d', 7') ~- ('X', ~,7 ~' ~' ), where g is a quasi-isomorphism, can be complemented to a commutative square (X, ~, 7) ~, (X', ~', 7') T' T ~ (R, , (x', , . ) where f is a quasi-isomorphism; d) if for two strong morphisms ~, t~ : (X, d, v) & (X', d', 7') and for a quasi- isomorphism g : (X', d', '~') ~ (IK', .~', "~') one has g~ = gq4 then there exists a quasi- isomorphism f: (X, ~, '~) ~ (X, ~, 7) with q~f = ~f. (We'll show, in fact, that in this situation ~ = +.) The property a) is obviously valid. To verify b), it suffices to apply the construction of the composition and Remark 1.2.1 (i). To verify c), we need the following fact. 1.9.. 11. Lemma. -- Let ~ : (X, d, v) -+ (X', d', ~') be a strong morph#m. Then for any pair V ~ ~ and V' ~ § the intersection V n ,-I(V') is a finite union of (1)-affinoid domains in V. Proof -- Take U' ~u with ?(V)C U'. Then we can find U~, ..., U', ~'[w~v' t t --1 t with ~(V) C U a u ... w U,, and V t~ ~-I(V') = U~ ~v/v,(U~). 9 Suppose that we have a diagram as in c). We may assume that ,X' = X'. Then ~' C ~'. Let~' denote the family ofaU V e ~ for which there exists ~ff' ~" with ~(V) C ~'. From Lemma 1.2.11 it follows that 9 is a net. The (I),-affinoid atlas ~ defines a (I),-affinoid atlas ~Twith the net ~, and the strong morphism ~ induces a strong morphism : (X, .~, ~') -+ (X', .~', ~"). Then 9 and the canonical quasi-isomorphism f: (x, (x, g, 7) satisfy the required property c). 20 VLADIMIR G. BERKOVICH Finally, we claim that in the situation d) the morphisms ~? and + coincide. First of all, it is clear that they coincide as maps. Furthermore, let V e v and V' e v' be such that ~(V) C V'. Take V' e~' with g(V') C ~r Then we have two morphisms ofk-affinoid space ~v/v,, +v/v' : V -~ V' such that their compositions with gv'/~' coincide. Since V' is an affinoid domain in ~', it follows that ~v/v' = +v/v,. 9 The category of~-analytic spaces ~k-dn is, by definition, the category of fractions of ~k-~Cn with respect to the system of quasi-isomorphisms. By Proposition 1.2.10 morphisms in the category ~k-~Cn can be described as follows. Let (X, d, v) be a ~k-analytic space. If ~ is a net on X, we write a ~( v if ~ C ~. Then the ~k-affinoid atlas ~7 defines a ~-affinoid atlas ~r with the net ~, and there is a canonical quasi- isomorphism (X, do, ~) ~ (X, d, v). The system of nets { ~ } with ~ -< v is filtered and, for any ~k-analytic space (X', ~r v'), one has Hom((X, ~r v), (X', ~r v')) = lira Hom~((X, ~r g), (X', ~r v')). We remark that all the maps in the inductive system are injective. We now want to construct a maximal ~k-affinoid atlas on a ~k-analytic space and to describe the set of morphisms between two ~k-analytic spaces in terms of their maximal atlases. (Kiehl's Theorem will be used here for the first time.) Let (X, ~qr v) be a ~-analytic space. We say that a subset W C X is v-special if it is compact and there exists a covering W = W1 u... uW, such that W~, W~ c~Wj e v and ~r174162 is an admissible epimorphism. A covering of W of the above type will be said to be a ',:.-special covering of W. 1.9..12. Lemma. -- Let W be a d-special subset of X. If U, V eXlw, then U r3 V e ~: and dr| ~t v ---> dvn v is an admissible epimorphism. Proof. -- Since the sets U r3 W~ and V r3 W~ are compact, we can find finite coverings { U~ }~ of U r~ Wi and { V~}~ of V t~ W~ by sets from v. Furthermore, since W~ ~ W~ ~ W~ x W~ are closed immersions, is follows that Ue c~ V~ e § and U~ ~ V~ -+ U~ � V~ is a closed immersion. Consider now the finite affinoid covering { Ui~ x V~}~,~,k,~ of the k-affinoid space U � V. For each quadruplet i,j,k,l, '~V;kr, Vit is a finite ..~r215 and the system { ..~tvik~v~} satisfies the condi- tion of Kiehl's Theorem. It follows that this system is defined by a finite ~r � v'alg ebra isomorphic to ~r Ker(II~c~v~i->Haffv~v~s~,,.~v~,r), and therefore the latter algebra is r U c~ V -~ ~g(~Cv), and U r3 V ~ U � V is a closed immersion. By Remark 1.2.1 (i), U c~ V is a cI)-affinoid domain in U and V, i.e., U nV e~. 9 Let W be a d-special subset of X. From Lemma 1.2.12 it follows that any finite covering of W by sets from v is a T-special covering. Furthermore, if { W~ } is a v-special covering, then from Tate's Acyclicity Theorem it follows that the commutative Banach k-algebra ~r := Ker ([I ~r --> .H ~r wi) does not depend (up to a canonical iso- ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 21 morphism) on the covering, and a continuous map W -+ ~r162162 is well defined. Let denote the collection of all ~-special subsets W such that the algebra ~r is k-affinoid, W-%~'(Offw) and, for some ~-special covering { W,} of W, (W,, Offwi ) are affinoid domains in (W, dw). We remark that from the condition (4) for the class 9 k it follows that W belongs to ~k. Furthermore, the last property of W does not depend on the choice of the covering. 1.2.13. Proposition. -- (i) The collection .~ is a net, and for any net ~ -<.~ one has ~ = "~; (i.i) there exists a unique (up to a canonical isomorphism) aPk-analytic atlas ~ with the net ~ that extends the atlas ~r ; Proof. -- (i) Let U, V e ~. We take ~-special coverings { U~ } of U and { Vj } of V. Since U n V = U,,~.(U, n Vj) and ~ltrinv i are quasinets, it follows that ~lenv is a quasinet. Furthermore, let ~ is a net with a -< v. By Lemma 1.2.12, to verify the equality ~= ~, it suffices to show that for any V~u there exist U1, ...,U,~ with V=Ulu ... uU,. Since ~ is a net on X, we can find W1, ...,W,,~cr with VC W1 u ... u Wm. Since V, W~ E ~ and § is a net, then, by Lemma 1.1.2 (ii), we can find U1, ..., U, ~u such that V = U1 u ... u U, and each U~ is contained in some W~. Finally, since Wj E ~, it follows that U, ~ 5. (ii) For each V e ~ we fix a ~-special covering ( V~ } and assign to V the algebra ~r and the homeomorphism V ~ ~(~r arising from the covering. We have to construct for each pair U, V ~ ~ with U C V a canonical bounded homomorphism ~r -+ ~r that identifies (U, ~r with an affinoid domain in (V, ~r Consider first the case when U ev. By Lemma 1.2.12, U nV~ is an affinoid domain in V~ and therefore in V. It follows that U is an affinoid domain in V. If U is arbitrary, then by the first case each U, from some ~-special covering of U is an affinoid domain in V. It follows that U is an affinoid domain in V. (iii) From Lemma 1.2.12 it follows that ~ = ~. Let { V~ } be a ~-special covering of some V e ~. For each i we take a ~-special covering { V~ }~ of V~. Then { V,~ }~, is a ~-special covering of V, and therefore V ~ ~. 9 The sets from ~ are said to be ~-affinoid domains in X. The ~-special sets are said to be ~-special domains in X. They have a canonical ~-analytic space structure. The following statement follows from Lemma 1.2.11 and Proposition 1.2.13 (i). 1.9,.1~. Corollary. -- If r : (X, ,-r "r) --+ (X', 3]', "r') is a morphism of ebb-analytic spaces, then for any pair of dP-affinoid domains V CX and V' C X' the intersection V n q~-x(V') is a ~-special domain in X. 9 1.9,.15. Proposition. -- Let (X, ,~r v) and (X', ,~r "r') be aP~-analytic spaces. (i) There is a one.to-one correspondence between the set Hom((X, ,.qr ,r), (X', ,~1', ,r')) and the set of all pairs consisting of a continuous map ~ : X -+ X', such that for each point x ~ X VLADIMIR G. BERKOVICH there exist neighborhoods V1u ... w V, of x and V~ w ... u V', of 9(x) with x E Vx c~ ... c~ V, and 9(Vi) C V'~, where V, C X and V~ fi X' are O-affinoid domains, and of a system of compatible morphisms of k-affinoid spaces 9v/v' : (V, ~r -+ (V', ~r for all pairs of~-affinoid domains VC X and V'C X' with 9(V) C V'. (ii) A morphism 9 : (X, d, v) -+ (X', M', -d) is an #omorphism if and only if ~ induces a homeomorphism between X and X', ~('4) = "4' and, for any V ~ '4, (V, ~r ~ (V', ~r where V' = 9(V). Proof. -- (i) Let a be a net on X with a -~ v, and let ~0 : (X, ~r a) -+ (X', ~', v') be a strong morphism. It is easy to extend the system of compatible morphisms ofk-affinoid spaces ~v/v' : (V, ~v) --> (V', dv, ) for all pairs V ~ ~ and V' e '4 with ~(V) C V'. Since 8 ---- '4, we get a map (evidently injective) from the first set to the second one. Conversely, suppose that we have a pair of the above form. To verify that it comes from a morphism of ~k-analytic spaces, it suffices to show that the collection a of all V ~ ~ such that ~(V) C V' for some V' e ~' is a net. For this we take a point x e X and neighborhoods V x u ... u V, of x and V~ u ... V~ of ~(x) with x e V a r~ ... r~ V, and ~(V,) C V~, where V~ e ~ and V" e ~'. Then V~ e a, and we get the required fact. (ii) follows from (i). 9 In practice we don't make a difference between (X, ~1, ~) and the Ok-analytic spaces isomorphic to it. In particular, we shall denote it simply by X and assume that it is endowed with the maximal Ok-affinoid atlas. If it is necessary, we denote the under- lying topological space by [ X I" We remark that the functor that assigns to a Ok-affinoid space X = ~'(~1) the Ok-analytic space (X, ~r { X }) is fully faithful. A Ok-analytic space isomorphic to such a space is called a Ok-affinoid space. Furthermore, if 9 is the system of all affinoid spaces, then the category Ok-~ln is denoted by k-tin, and the corresponding spaces are called k-analytic spaces. In this case we withdraw the reference to 9 in the above and future definitions and notations. If 9 is the system of strictly affinoid spaces, then the category Ok-~ln is denoted by st-k-~n, and the corresponding spaces are called strictly k-analytic spaces. Similarly, instead of referring to O, we use the word " strictly" (strictly affinoid domains and so on). 1.2.16. Remark. -- For an arbitrary 9 there is an evident functor Ok-tin ~ k-tin. From Proposition I. 2.15 it follows that this functor is faithful. But we don't know whether it is fully faithful. (This was another reason for introducing the category of ~k-analytic spaces.) The only fact in this connection is Proposition 1.2.17. We say that a Ok-analytic space is good ff each point of it has a O-affinoid neigh- borhood. 1.2.17. Proposition. -- Let X and Y be Ok-analytic spaces, and assume that the class 9 k is dense and X is good. Then any morphism of k-analytic spaces ~ : Y ~ X is a morphism of Ok-analytic spaces. ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 23 Proof. -- It suffices to show that the family v of all ~-affinoid domains V C Y, for which there exists a *-affinoid domain U C X with ~(V) C U, is a net. For an arbi- trary pointy eY we take a r neighborhood U of ~(y). Since r is dense, we can find a neighborhood ofy in ~-I(U) of the form V1 u ... u V., where V, are *-affinoid domains in Y andy e V x n ... n V,. We get V~ e v, and therefore, is a net. 9 The dimension dim(X) of a Ck-analytic space X is the supremum of the dimensions of its r domains. (The dimension of a k-affinoid space is defined in [Ber], p. 34.) We remark that the supremum can be taken over ~-affinoid domains from some net, and, in particular, the dimension of X is the same whether the space is considered as an object of ~k-~Cn or of k-,~n. 1.9.. 18. Proposition. -- The topological dimension of a paracompact r space is at most the dimension of the space. If the space is strictly k-analytic, both numbers are equal. Proof. -- Suppose first that the space X ----Jt'(~r is k-affinoid. If X is strictly k-affinoid, the statement is proved in [Ber], 3.2.6. If X is arbitrary, we take a non- Archimedean field K of the form Krl ..... ,, (see [Ber], w 2.1) such that the algebra ~r = ~r | K is strictly k-affinoid, and consider the map a : X --> X' ---- dg(~r which takes a point x ~ X to the point x' ~ X' that corresponds to the multiplicative seminorm ]~a~ T ~ ~ max [ a,(x) [ r ~. The map cr induces a homeomorphism of X with a closed subset of X'. Therefore the topological dimension of X is at most dim(X') = dim(X). If X is an arbitrary paracompact k-analytic (resp. strictly k-analytic) space, then it has a locally finite covering by affinoid (resp. strictly affinoid) domains, and there- fore the statement follows from [En], 7.2.3. 9 1.3. Analytic domains and G-topology on an analytic space 1.3.1. Definition. -- A subset Y of a r space X is said to be a C-analytic domain if, for any point y e Y, there exist r domains V1,..., V, that are contained in Y and such thaty e V 1 n ... n V. and the set Vx u ... u V, is a neigh- borhood ofy in Y (i.e., the restriction of the net of r domains on Y is a net on Y). We remark that the intersection of two C-analytic domains is a C-analytic domain, and the preimage of a C-analytic domain with respect to a morphism of a r spaces is a C-analytic domain. Furthermore, the family of r domains that are contained in a C-analytic domain Y C X defines a r atlas on Y, and there is a canonical morphism of r spaces ~ : Y ~ X. For any morphism q~ : Z -+ X with ~(Z) C Y there exists a unique morphism + : Z -+ Y with q~ = v+. It is clear that a C-analytic domain that is isomorphic to a k-affinoid space is a r domain. A morphism ~:Y ~ X that induces an isomorphism of Y with an open C-analytic domain in X is said to be an open immersion. If the class Ck is dense, then all open subsets of X are C-analytic domains. 24 VLADIMIR G. BERKOVICH 1.3.9.. Proposition. -- Let { Y~ }~ e ~ be a covering of a (bk-analytic space X by (b-analytic domains such that each point of X has a neighborhood of the form Yq u ... r Y~, wltk y E Y~ ta ... n Y~, (i.e., { Y~ }, ~ i is a quasinet on X). Then for any (bk-analytic space X' the following sequence of sets is exact Horn(X, X') ~ H Hom(Y,, X') -~ IIHom(Y~ n Yj, X'). i i,j Proof. -- Let ~i:Yi-+ X' be a family of morphisms such that, for all pairs i,j eI, ~lyic~y i = q%]y/nyi. Then these q~i define a map N -+X' which is continuous, by Lemma 1.1.1 (i). Furthermore, let v be the collection of (b-affinoid domains V C X such that there exist i ~ I and a (b-affinoid domain V' C X' with V C Y~ and ~(V) C V'. It is easy to see that z is a net on X, and therefore there is a morphism q~ : X -+ X' that gives rise to all the morphisms ~. 9 We now consider a process of gluing of analytic spaces. Let { X~ }~ ~ x be a family of (b~-analytic spaces, and suppose that, for each pair i, j e I, we are given a (b-analytic domain X~C X~ and an isomorphism of (I)~-analytic spaces vis:X~-~ Xsi so that X u=X~, vi~(Xi~ ~X~) =X~I~X~ and v u=v~tov~ on X~ r We are looking for a ~-analytic space X with a family of morphisms ~ : X~ ~ X such that: (1) ~ is an isomorphism of X~ with a (b-analytic domain in X; (2) all ~(X~) cover X; (3) = n (4) ~=~ov. on Xi ~. If such X exists, we say that it is obtained by gluing of X~ along X~. 1.3.3. Proposition. -- The space X obtained by gluing of X~ along Xij exists and is unique (up to a canonical isomorphism) in each of the following cases: a) all Xij are open in Xi; b) for any i E I, all X~ are closed in Xi and the number of j ~ I with Xi~ 4= 0 # finite. Furthermore, in the case a) all ~(X~) are open in X. In the case b) all ~i(X~) are closed in X and, if all X~ are Itausdorff (resp. paracompact), then X is Hausdorff (resp. paracompact). Proof. -- Let ~ be the disjoint union Hi Xi. The system { vii } defines an equi- valence relation R on X. We denote by X the quotient space ,X/R and by N the induced maps X i. -+ X. In the case a), the equivalence relation R is open (see [Bou], Ch. I, w 9, n ~ 6), and therefore all tx~(X,) are open in X. In the case b), the equivalence rela- tion R is closed (see loc. cit., n ~ 7), and therefore all ~(Xi) are closed in X and tz~ induces a homeomorphism X~ -% ~(X~). Moreover, if all X~ are Hausdorff, then X is Hausdorff, by loc. cir., exerc. 6. If all X~. are paracompact, then X is paracompact because it has a locally finite covering by closed paracompact subsets ([En], 5.1.34). Furthermore, let z denote the collection of all subsets V C X for which there #,TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 2~ exists i eI such that VC Vt~(X~) and ~L~-I(V) is a r domain in X~ (in this case ~-I(V) is a r domain in Xj for any j with VC ~j(X~.)). It is easy to see that v is a net, and there is an evident r atlas ~r with the net X. In this way we get a r space (X, ~', ~) that satisfies the properties (1)-(4). That X is unique up to canonical isomorphism follows from Proposition 1.3.2. 9 Let X be a r space. The family of its aP-analytic domains can be considered as a category, and it gives rise to a Grothendieck topology generated by the pretopology for which the set of coverings of an analytic domain Y C X is formed by the families { Y~ }~ e ~ of analytic domains in Y that are quasinets on Y. For brevity, the above Grothendieck topology is called the G-topolog2 on X, and the corresponding site is denoted by X o. From Proposition 1.3.2 it follows that any representable presheaf on X o is a sheaf. The G-topology on X is a natural framework for working with coherent sheaves. Recall ([Ber], 1.5) that the n-dimensional affine space A ~ is the set of all mul- tiplicative semi-norm on the ring of polynomials k[T1, ..., T~] that extend the valuation on k endowed with the evident topology. The family of closed polydiscs with center at zero E(0; rl, ..., r,) = { x e A" [ [ T~(x)l < r,, 1 ~< i ~< n } defines a k-affinoid atlas on A% (We remark that A ~ is a good k-analytic space.) We remark that the affine line A 1 is a ring object of the category k-tin. If x -- vz(d) is a k-affinoid space, then Hom(X, A 1) ---- d. We return to aP-analytic spaces. Applying Proposition 1.3.2 to X' = A 1 and the category k-tin, we get a structural sheaf~xa on X G (this is a sheaf of rings). The category of d)xa-modules is denoted by Mod(XG). An dPx -module is said to be coherent if there exists a quasinet -r of ~-affinoid domains in X such that, for each V e % dYxo Ira is iso- morphic to the cokernel of a homomorphism of free d~vo-modules of finite rank. For example, suppose that X ----J4(~r is a qbk-affinoid space. Then a finite z~l-module M defines a coherent d)x~(M ) by V ~-~M| and Kiehl's Theorem tells that any coherent d~xcmodule is isomorphic to d~xG(M ) for some M. The latter fact enables one to define for a coherent d~x -module F the support Supp(F) of F. Namely, if X -=-- ~'(a') is k-affinoid and F = ~9xo(M), then Supp(F) is the support of the annihilator of M. If X is arbitrary, then Supp(F) is the set of point x e X such that for some (and therefore for any) affinoid domain V that contains x the support of F Iva contains x. Let Coh(Xo) denote the category of coherent d~xQ-modules, and let PiC(XG) denote the Picard group of invertible Ox -modules. (One has Pic(XG) ----HI(XG, d~xa). ) From Kiehl's Theorem it follows that Coh(Xo) and Pic(XG) are the same whether X is considered as an object of r or of k-~Cn. We now consider connection of the above objects with their analogs in the usual topology of X. For this we assume that the class a) k is dense. Then all open subsets of X are C-analytic domains, and there is a morphism of G-topological spaces = : X~ ---> X which induces a morphism of the corresponding topoi (~,, r:*) : X~ -~ X ~. The direct image functor =, is simply the restriction functor. In particular, we have the structural sheaf ~x :---- ~c, d~xo on X. The functor ~, is not fully faithful (see Remark 1.3.8). The inverse 4 VLADIMIR G. BERKOVIGH image functor n* is as follows. For a sheaf F on X and a ~-affinoid (or O-special) domain, one has r(v) = q/Dr where q/ runs through open neighborhoods of V. It is easy to see that F ~ 7:, rr* F. In particular, the functor z;* is fulIy faithful. Let Mod(X) denote the category of 0x-modules. The functor =, defines an evident functor Mod(XG) ~ Mod(X). The natural functor in the inverse direction is as follows: Mod(X) -+ Mod(Xo) : r ~ F o = ," F | OxQ. An 0x-module is said to be coherent if locally (in the usual topology of X) it is isomorphic to the cokernel of a homomorphism of free modules of finite rank. (For example, if X = ~(~r is O~-affinoid, then Ox(M):= ~, Ox~(M) is a coherent Ox-module. ) The Picard group Pic(X) is the group of invertible Ox-modules. One has Pic(X) = Ha(X, O;r). 1.3.4. Proposition. -- If X is a good ~b~-analytic space, then (i) for any Ox-module F one has F -% % FG; in particular, the funetor Mod(X) -+ Mod(XG) : F ~-~ F o is fully faithful; (ii) the functor F ~ F G induces an equivalence of categories Goh(X)--~ Goh(XG) ; (i)fi) a coherent Ox-module F is locally free if and only if F a is locally free. Proof. -- (i) It suffices to verify that for any point x e X there is an isomorphism of stalks F~ -% (=, FG) ~. But this easily follows from the definitions because x has an affinoid neighborhood. (ii) By (i), it suffices to verify that for a coherent 0x -module ~" the Ox-module F ---- =, ~" is coherent and FQ -% o~'. This also follows easily from the definitions. (iii) We may assume that X = Jg(~r is k-affinoid. It suffices to show that a finite d-module M is projective if and only if the dTx~-module d~x~(M ) is locally free. The direct implication is simple. Conversely, suppose that for some finite affinoid cove- ring { Vi }~ e i of X the finite dvi-modules M | ~qCvi are free. It suffices to verify that M is flat over d. For this we take an injecdve homomorphism of finite ~r P ~ O. Then the homomorphisms (M |162 P) |162 ~r -+ (M |162 Q) |162 ~r are also injective. Applying Tate's Acyclicity Theorem to the finite ~r M |162 P and M |162 Q, we obtain the injectivity of the homomorphism M | P-+ M |162 Q. 9 1.3.5. Corollary. -- If X is a good ~k-analytic space, then there is an isomorphism Pie(X) -% Pic(X~). 9 The structural sheaf 9 x will be used only for good spaces X. The group Pic(XQ) will appear in Corollary 4.3.8. We now compare the cohomology groups in both topologies. ~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 27 1.3.6. Proposition. -- (i) For any abelian sheaf F on X, one has H~(X, F) -~ H~(Xc, r~" F), q>~ O. (ii) If X is good, then Hq(X, F) ~ H~(X~, F~), q >>. O, for any coherent Ox-module F. (iii) IfX is paracompact, then fix(X, F) ~ ~II(Xo, ~" F) for any sheaf of groups F on X. Proof. -- (i) An open covering of X is a covering in the usual and the G-topology, and therefore it generates two Leray spectral sequences that are convergent to the groups Hq(X, F) and Ho(Xc, ~* F), respectively. Comparing them, we see that it suffices to verify the statement for sufficiently small X. In particular, we may assume that X is paracompact. It suffices to verify that ifF is injective, then H~(XG, ~* F) = 0 for q i> I. Since X is paracompact, it suffices to verify that the ~ech cohomology groups of ~" F with respect to a locally finite covering by compact analytic domains are trivial. But this is clear because they are also the Cech cohomology groups of F with respect to the same covering. (ii) The same reasoning reduces the situation to the case when X is an open paracompact subset of a k-affinoid space. (In particular, the intersection of two affinoid domains is an affinoid domain.) In this case H~(X, F) is an inductive limit of the q-th coho- mology groups of the ~ech complexes associated with locally finite open coverings { v//~ }~ e I of X. On the other hand, since the cohomology groups of a coherent sheaf on a G-ringed k-affinoid space are trivial, then H~(X~, FG) is the q-th cohomology group of the ~ech complex associated with an arbitrary locally finite affinoid covering { V~ }~ e a of X. It remains to remark that for any { q/~ }ieI we can find { V~ }5~ such that each Vj is contained in some q/~ and ~Jea Int(V~/X) ---- X. (Hi) is trivial. 9 We remark that a morphism of r spaces ~ : Y -+ X induces a morphism of G-ringed topological spaces ~Q : YG -+ Xo" If the spaces X and Y are good, then for any coherent Ox-module F there is a canonical isomorphism of coherent Oyo-modules (~" F) G -~ ~ Fo. We finish this subsection by introducing several classes of morphisms. 1.3.7. Lemma. -- The following properties of a morphism of dp~-analytic spaces : Y -+ X are equivalent: a) for any point x E X there exist ~-affinoid domains Vx, ..., V, C X such that x E V x n... n V,~ and ~ x(V~) ~ V~ are finite morphisms (resp. closed immersions) of k-affinoid spaces; b) for any O-affinoid domain VCX, ~-x(V) ->V /s a finite morphism (resp. a closed immersion) of k-affinoid spaces. Proof. -- Suppose that a) is true. Then the collection 9 of all r domains V C X such that ~-'(V) -* V is a finite morphlsm (resp. a closed immersion) ofk-affinoid spaces is a net. Let V be an arbitrary r domain. Then V C Vx u... u V, for some 28 VLADIMIR G. BERKOVICH V e,. By Lemma 1.1.2 (ii), we can find @-affinoid domains U1,...,U~,C X such that V = U1 u ... u U m and each Uj is contained in some V,. Then Uj ~ ,. It remains to apply Kiehl's Theorem. 9 A morphism ~:Y-+ X satisfying the equivalent properties of Lemma 1.3.7 is said to be finite (resp. a closed immersion). It is clear that this property of ~ is the same whether we consider it in the category qbk-dn or in k-tin. A finite morphism 9 : Y -+ X induces a compact map with finite fibres I Yl-+ I x I, and ~0~.(dYyQ) is a coherent d~x -module. If ~ is a closed immersion, then it induces a homeomorphism of I Y ] with its image in I X l, and the homomorphism 0xa -+ q~G.(0yQ) is surjective. Its kernel is a coherent sheaf of ideals in 0x. Furthermore, we say that a subset 2; C X is Zariski closed if, for any @-affinoid domain V C X, the intersection ~ n V is Zariski closed in V. The complement to a Zariski closed subset is called Zariski open. For example, the support of a coherent 0x -module is Zariski closed in X. If ~ : Y --~ X is a closed immersion, then the image of Y is Zariski closed in X. Conversely, if Y~ is Zariski closed in X, then there is a closed immersion Y -+ X that identifies I Y I with ~. Furthermore, a morphism of @k-analytic spaces ~0 : Y -+ X is said to be a G-locally (resp. locally) closed immersion if there exist a quasinet 9 ofqb-analydc (resp. open ~-analytic) domains in Y and, for each V E % a qb-analytic (resp. an open qb-analyfic) domain U C X such that 9 induces a closed immersion V --~ U. (It is clear that this property of q9 is the same whether we consider it in the category ~k-~Cn or in k-~Cn.) Of course, a locally closed immersion is a G-locally closed immersion. If the both spaces are good, then the converse is also true. Let now ~0 : Y ~ X be a G-locally closed immersion, and let V and U be as above. If J is the sheaf of ideals in d~vo that corresponds to V, then j/jz can be considered as an d~v -module. All these sheaves are compatible on intersections, and so they define a coherent d~r -module that is said to be the conormal sheaf of 9 and is denoted by dV'~rQ/Xo. If both spaces are good, then one can also define a similar dTx-module ~r and one has (~'Y/x)e, -% WYo/xQ- 1.3.8. Remark. -- Here is an example showing that the direct image functor r~. : Xo ~ -+ X ~ is not fully faithful. Let X be the closed unit disc E(0,1) ={x~A lllT(x)~< 1}, and let x 0 be the maximal point of X (it corresponds to the norm of the algebra k { T }). We construct two sheaves F and F' on X G as follows. Let Y be an analytic domain in X. Then F(Y)----Z if x 0 E Y, and F(Y)= 0 otherwise. Furthermore, F'(Y)= Z if { x ~ X ] r < I T(x)] < 1 } u { xo } C Y for some 0 < r < 1, and F'(Y) ---- 0 otherwise. The sheaves F and F' are not isomorphic, but ~. F = ~. F' = i. Z, where i is the embed- ding {Xo} ~X. ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 29 1.4,. Fibre products and the ground field extension funetor 1.4.1. Proposition. -- The category ~,-~r admits fibre products. Proof. -- First we shall show the existence of fibre products in the category k-dn, and after that we'll use this fact to show that the same is true for the category qbk-dn. Let q0:Y--~ X and f: X'--~ X be morphisms of k-analytic spaces. Consider first the case when all three spaces are paracompact. In this case we may assume that ~? and f are represented by strong morphisms (Y, ~, a) --~ (X, d, x) and (X', d', ~') --~ (X, d, ~), where % a and v' are locally finite nets. Let S denote the family of all triples (V, U, U'), where V e a, U e v, U' e v' and ~(V),f(U') C U. For a= (V,U,U') eS we denote by W~ the k-affinoid space V � and by E~ the topological space [ V [ � IuI [ U' l- The latter is a compact subset of the topological space E :-~ ] Y I � Ixl ] X' I, and the canonical map W~ -+ E~ induces a map ~ : W~ --~ E. We claim that, for any pair e, ~ 9 S, the set W~ :-- r~l(E~ c~ E0) is a special domain in W~, and there is a canonical isomorphism of k-analytic spaces v~:W~--~ W~. Indeed, let ~ = (V,U,U'). Then U nU =U 1 c~ ... c~U~ for U~ex. Furthermore, for each 1 ~< i<~ n, one has q~v/v(U~) - ~ c~ q>V/~(U~) -I = [.j~.i = 1 V~ and f~,/u(U,) f~,~(U,) = [.J~'~ U,1 for some Vi~. ca and U~t 9 One has W~ ~ (J~,j,~V~ � The right hand side of the latter equality can also be considered as a subset of W~. It follows that W~ and W~, are special domains in W~ and W~, respectively, and we get an isomorphism v~ : W~ ~ W~, that does not depend on the choice of the above coverings. It is clear that W~=W~, v~(W~nW~v ) =W~W~v and v~v=%vov~ on W~W w. By Proposition 1.3.3, we can glue all W~ along W ~ and get a k-analytic space (Y', ~', a') which is a fibre product of (Y, ~, o) and (X', ~qr v') over (X, ~r v). Consider now the case when only the space X is paracompact. In this case we take coverings { Y~ }~ 9 x of Y and { X'~ }~ 9 a of X' by open paracompact subsets, and we glue all the spaces Y~ � X~ along the open subspaces (Y, n Y,) � (X~ n X;). We get a locally Hausdorff space Y'. The collection ~' of sets of the forms V � U', where VC Y,, UC X and U'C X~ are affinoid domains, is a net of compact subsets on Y', and there is an evident k-affinoid atlas ~' with the net a'. The triple (Y', ~', a') is a fibre product of Y and X' over X. Finally, in the case when all three spaces are arbitrary we take a covering { X~ }~ 9 r of X by open paracompact subsets and construct a fibre product Y' by gluing the spaces -- -- 1 X Xj). a(X~.) � along the open subspaces ~-~(Xi c~X~) � ( in We remark that the above construction gives also a compact map :Y'- IYt � tX'I. VLADIMIR G. BERKOVICH In particular, if V C Y, U C X and U' C X' are affinoid domains with ~(V),f(U') C U, then the set ~-1(] V[ � U' I) is compact. It follows that the canonical morphlsm V' :---- V � v U' --> Y' identifies V' with an affinoid domain in Y'. Suppose now that T : Y -+ X and f: X' -+ X are morphisms of qbk-analytic spaces. Then the collection ~' of all affinoid domains of the form V � ~ U', where V C Y, U C X and U' C X' are qb-affinoid domains with ~(V), f(U') C U, is a net on Y', and there is an evident ek-affinoid atlas with the net ~'. It defines a r space structure on Y'. It is easy to see that the canonical projections Y' -+ Y and Y' --> X' are mor- phisms of r spaces and that Y' is a fibre product of Y and X' over X in the category ek-~cn. 9 Similarly, one constructs, for a non-Archimedean field K over k, a ground field extension functor ~k-dn -+ ~x-dn : X ~-~ X ~ K and a compact map X | K -+ X. A Oh-analytic space is a pair (K, X), where K is a non-Archimedean field K over k and X E ~K-z~n. A morphism (L, Y) ~ (K, X) is a pair consisting of an isometric embedding K '-+ L over k and a morphism of ~wanalytic spaces Y ~ X | L. The category of O-analytic spaces is denoted by ~-dn,. If 4) is the family of all affinoid spaces, then the category is denoted by dn k and its objects are called analytic spaces over k. For brevity we denote the analytic space (K, X) by X. We remark that with each point x e X e ~k-~Cn one can associate a non-Archi- medean field ~(x) over x so that, for any ~-affinoid domain V C X that contains x, there is a canonical bounded character ~r -+~(x) that identifies 3r with the cor- responding field of the point x with respect to V (see [Ber], 1.2.2 (i)). A morphism ~:Y ~ X induces, for each point y e Y, an isometric embedding 3r Furthermore, let x be a point of X. If V is a ~-affinoid domain that contains x, then the caracter d v (~W(x) ---~(x) :f| k ~-* ~f(x) defines an 3~(x)-point x v e V | It is clear that the image x' of x v in X | ~(x) does not depend on the choice of V. If now ~:Y-+ X is a morphism of ~k-analytic spaces, then the ~ff(x)-analytic space (Y | ~ff(x)) | ~ ~v(xl dt'(2g~(x)), where the morphism Jt'(~(x) ) -+ X | ~(x) corresponds to the point x', is denoted by Y, and is said to be thefibre of~ at the point x. The canonical morphism Yx -+ Y induces a homeomorphism Y, -~ ~-l(x). The dimension of ~, dim(~), is the supremum of the dimensions dim(Y,) over all x e X. Let ~:Y-+ X be a morphism of ~k-analyfic spaces, and consider the diagonal morphism Ay/x : Y ~ Y � x Y" The collection v of ~-affinoid domains V C Y for which there exists a ~-affinoid domain U C X with ~(V) C U is a net, and, for such V and U, V � v V is a ~-affinoid domain in Y and Ayt x induces a closed immersion V -+ V � v V. Thus, Ay/x is a G-locally closed immersion. The conormal sheaf of Ay/x is said to be the sheaf of differentials of ~ and is denoted by ~Yo/xo. If both spaces are good, then one can also define a similar coherent 0~-module ~Y/x, and one has (~Y/x)o --% ~/xo. The sheaves ~Yo~Xo and ~tx will be studied in w 3.3. A morphism of O,-analytic spaces ~:Y-+ X is said to be separated (resp. locally separated) if the diagonal morphism A~ x is a closed (resp. a locally closed) immersion. ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 31 If the canonical morphism X ~r is separated (resp. locally separated), then X is said to be separated (resp. local E separated). For example good *k-analytic spaces and morphisms between them are locally separated. If a morphism 9 9 Y -+ X is separated, then IYI is closed in I Y � I. Since the map n:[Y � ~[YI � is compact, then ] Y [ is closed also in [ Y [ � Y l, and therefore the map [ Y I -+ I X [ is Hausdorff. In particular, if Y is separated, then its underlying topological space [ Y [ is Hausdorff. 1.4.2. Proposition. -- A locally separated morphism of r spaces q~:Y-+ X is separated if and only if the induced map ]Y] ~ I XI is m~sdo~ Proof. -- Suppose that the map [ Y ] ~ [ X [ is Hausdorff. Then the complement of Y in [ Y [ � J xl I Y [ is open. Since the diagonal morphism A = Ay/x is a composition of a closed immersion with an open immersion, it suffices to show that A(Y) is closed in ] Y � x Y 1. For this we consider the compact map r~ : ] Y � x Y [ ~ [ Y [ � I xl [ Y 1" Let z ~ (Y � and let ~(z) = (Yl,Y~). If Y1 4=Y~, then 7:-1(W ") is an open neighborhood of z that does not meet A(Y). Ify 1 =Yz, then we take an open neighbor- hood 3r of y� = Y2 such that A~. : 3e" -+ 3r � x q/" is a closed immersion. Since 3r � x ~ is an open subset of Y � Y and z $A(Y), then we can find an open neighborhood of z that does not meet A(Y). The required fact follows. 9 It is clear that the classes of closed and locally closed immersions, finite, separated and locally separated morphisms are preserved under composition, under any base change functor and under extensions of the ground field. 1.4.3. Remarks. -- (i) The converse implication of Proposition 1.4.2 is not true in general. For example, the space obtained by gluing two copies of the unit one-dimen- sional disc along the closed annulus of radius one is Hausdorff but is not separated. (ii) We conjecture that every point of a separated k-analytic space has an open neighborhood which is isomorphic to an analytic domain in a k-affinoid space. !.5. Analytic spaces from [Berl In this subsection we recall the notion of a k-analytic space from [-Ber] (with the necessary details that were omitted in [Ber]), and we show that the category of k-analytic spaces from [Ber] is equivalent to the category of good k-analytic spaces from the previous subsection. First of all, recall that a k-quasiaffinoid space is a pair (q/, ~) consisting of a locally ringed space a//and an open immersion ,~ of q/in a k-affinoid space X. We remark that the immersion v induces a net v of all V C ql for which v(V) is an affinoid domain in X and a k-affinoid arias d with the net v for which My -~ d~cv~, and therefore we get a 32 VLADIMIR G. BERKOVICH k-analytic space (~, ~r -~) from k-dg'n. We remark also that if V is an affinoid domain in ~, then for any pair of open subsets ~r ~ C ~//with ~r C V C ~ there are canonical homomorphisms ~(~r ---> ~r ---> 0(r 9 Furthermore, a morphism of k-quasiaffinoid spaces (~, ,) -+ (~/', ~') is a morphism of locally ringed spaces ~ : o//_+ ~, such that for any pair of affinoid domains V Coy and V'C ~' with ~(V)C ~r = Int(V'/oY ') (the topological interior of V' in oy,), the induced homomorphism d v, ~0(r ~) -+~(~-~(~))-+~r is bounded. We remark that from the definition it follows that for any pair of affinoid domains U C V and U'C V' with ?(U)C Int(U'/~//') the homomorphisms ~,---> ~r and ~r ~v are compatible. 1 .ft. 1. Lemma. -- The system of homomorphism ~r J~'v extends canonically to the family of all pairs of affinoid domains V C ql and V' C ~ll' with ~(V) C V' so that one gets a well-defined morphism ( ~ s~', .~) -+ ( qg', d', v'). Proof. -- Let V, V' be such a pair. Assume first that ~(V)C Int(V'/~ We claim that the two maps from V to V' induced by q~ and by the homomorphism dr'-+ ~v coincide. Let + denote the second map, and let x e V. Take affinoid neighborhoods U of x in o-// and U' of q~(x) in ~//' such that ~(U) C Int(U'/~//'). Then ?(U n V) C Int(U' c~ V'/~ The homomorphisms ~'v' -+ ~r and db, nv, -+ dvn v are compatible, and therefore +(U n V)C U'A V'. Since U and U' can be taken sufficiently small, then ~?(x) = +(x), and our claim follows. It follows that one can construct in a canonical way bounded homomorphisms d~,-+ d v for every pair of affinoid domains U C V and U' C V' with ?(U) C U', and the two maps from U to U' induced by ~ and by the homomorphism ~r dv coincide. Assume now that V and V' are arbitrary. Then we can find affinoid domains t t t t V1, ...,V,C~andV1, ...,V~Cq/'suchthatVCV 1 u ... uV,,V'CV 1 t3 ... tAV, and ~0(V~)C Int(V~/~//'). By the first case, there are canonical bounded homomor- phisms ~r n v~ -+ dv n vi and d v, n vl c~ v~ -+ ~r n vi n v i that induce the maps t t t q~:V c~V,-+V' nV~ and V nV ic~V~-+V' t~V~ c~Vj. Applying Tate's Acyclicity Theorem to the coverings { V rh V~ } of V and { V' c~ V~ } of V', we get a bounded homomorphism d v, -+ d v that is compatible with the homo- morphisms dv, n vi -+ dv vi and such that the maps from V to V' induced by ~ and by the homomorphism d v, --~ ~v coincide. Thus, we get the required morphism (~, d, v) -- (~', ~', ~'). 9 We remark that any morphism (~', d, v) -+ (~', d', v') comes from a unique morphism (~, ~)-~ (~', ,'). Thus, k-quasiaffinoid spaces form a category which is equivalent to a full subcategory of k-dn. The latter consists of all k-analytic spaces that admit an open immersion in a k-affinoid space. ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN tMNALYTIC SPACES 33 1.5.2. Corollary. --- Let (q/, v) and (q/', ":.') be k-quasiaffinoid spaces, and let 9 : ql ---> at' be a morphism (resp. an isomorphism) of locally ringed spaces. Then the following are equivalent: a) ~ induces a morphism (resp. an isomorphism) of k-quasiaffinoid spaces (at, v) -+ (all', v'); b) there exist open coverings ~ ql~ }i ~ i of ql and { ql'~ )5 E 9 of all' such that, for each pair i, j, *? induces a morphism (resp. an isomorphism) of k-quasiaffinoid spaces (resp. (q/~ n ~-l(q/'j), ~) --% (q~(od~) n q/'j, ~')); e) property b) is true for arbitrary open coverings of ql and all'. 9 Let X be a locally tinged space. An (open) k-analytic atlas on X is a collection of k-quasiaffinoid spaces {(q/i, v~)}~ei called charts of the atlas such that { q/~}~r is an open coveting of X (each q/~ is provided with the locally ringed structure induced from X) and, for each pair i, j e I, the identity morphism induces an isomorphism of k-quasiaffinoid spaces (~ n ~, v~) --% (~/~ n q/~, ~). Furthermore, suppose that we are given an open subset ad C X and an open immersion ,J of q/in a k-affinoid space. Then (q/,~) is compatible with the atlas {(~,v~)}iEr if, for each i~I, the identity morphism induces an isomorphism of k-quasiaffinoid spaces (q/c~ q/j, ,~) -% (q/n o//i, v~). Two atlases are said to be compatible if every chart of one atlas is compatible with the other atlas. From Corollary 1.5.2 it follows that the compatibility of atlases is an equi- valence relation. A k-analytic space from [Ber] is a locally ringed space X provided with an equivalence class of k-analytic atlases. Let X, X' be two k-analytic spaces defined in the above way, and let 9 : X -+ X' be a morphism of locally ringed spaces. Then ~ is called a morphism of k-analytic spaces if there exists an atlas {(q/i, ~)}~ei of X and an atlas {(o//,, v,j)}~ea of X' such that, for each pair i, j, ~ induces a morphism of k-quasiaffinoid spaces (~ n q~-a(~//'~), v,) --> (~//], v'~). From Corollary 1.5.2 it follows that the same condition holds for any choice of adases on X and X' defining the same k-analytic structure, and that one can compose morphisms. Thus, one gets a category. This is the category intro- duced in [Ber] (and denoted there by k-ddn). We now construct a functor from the category of k-analytic spaces from [Ber] to k-s~tn. For each k-analytic space X from [Ber] we fix an open k-analytic atlas {(o7/~, v~)}~ex. Let 9 be the family of the subsets VC X for which there exists i e I such that V is an affinoid domain in q/, (in this case V is an affinoid domain in any q/~ that contains V). Then v is a net on X, and there is an evident k-affinoid atlas ~r with the net v. The k-analytic spaces (X, ~, v) obtained in this way is evidently good. Let now q~ : X -> X' be a morphism of k-analytic spaces from [Ber]. We denote by ~ the family of all V e v for which there exists V' e v' with q~(V) C V'. It is clear that a is a net with, < v, and the morphism 9 gives rise to a strong morphism (X, SIo, ~) ~ (X', SI', z'). Therefore we have the required functor, and it is easy to see that it is fully faithful. Let now X be a good k-analytic space from k-sin. For an affinoid domain V C X we denote 5 VLADIMIR G. BERKOVICH by q/v the topological interior of V in X and by Vv the canonical open immersion of locally ringed spaces q/v ~V. Then {(q/v, Vv)) is an open k-analytic atlas on X, and the k-analytic space from [Ber] obtained in this way gives rise to a k-analytic space from k-dn isomorphic to X. Thus, the correspondence X ~-~ (X, ~r ~) is an equivalence of the category of k-analytic spaces from [Bet] and the category of good k-analytic spaces. We now extend to the category k-.sC'n several classes of morphisms that were intro- duced in [Ber] for good k-analytic spaces. Let P be a class or morphisms of good k-analytic spaces which is preserved under compositions, under any base change and under exten- sions of the ground field. We say that a morphism ~ : Y ~ X in k-..~n is of class P if for any morphism X' ~ X from a good analytic space over k the space Y X x X' is good and the induced morphism Y x x X' ~ X' is of class P. It follows from the definition that the class P is also preserved under the same operations. Furthermore, if P contains locally closed immersions, then P processes the following property: if Y ~ X is a locally separated morphism, then any morphism Z ~ Y, for which the composition Z-+ X is of class P, is of class P. 1.5.3. Examples. -- (i) If P is the class of all morphisms of good analytic spaces, then the morphisms from P are said to be good. For example, finite morphisms and locally closed immersions are good morphisms. (ii) If P is the class of closed morphisms of good analytic spaces ([Ber], p. 49), then the morphisms from P are said to be closed. For example, finite morphisms and locally closed immersions are closed morphisms. (iii) If P is the class of proper morphisms of good analytic spaces ([Ber], p. 50), then the morphisms from P are said to be proper. It follows from the definitions that a morphism is proper if and only if it is compact and closed. For example, finite morphisms are closed. Conversely, if a proper morphism has discrete fibres, then it is finite ([Ber], 3.3.8). 1.5.4. Definition. -- The relative interior ofa morphism ~ : Y --~ X is the set Int(Y/X) of all points y ~ Y for which there exists an open neighborhood ~r ofy such that the induced morphism ~r -+ X is closed. The complement of Int(Y/X) is called the relative boundary of ? and is denoted by O(Y/X). If X = ~r these sets are denoted by Int(Y) and 0(Y) and are called the interior and the boundary of Y, respectively. It follows from the definition that 0(Y/X) = O if and only if the morphism ~ is closed. The following properties of the relative interior are easily deduced from the definition and [Ber], 3.1.3. 1.5.5. Proposition. -- (i) If Y is an analytic domain in X, then Int(Y/X) coincides with the topological interior of Y in X. (fi) For a sequence of morphisms Z L Y L X, one has Int(Z/Y) c~ +-'(Int(Y/X))C Int(Z/X). ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 35 If ? is locally separated (resp. and good) then Int(Z/X) C Int(Y/X) (resp. Int(Z/X) = Int(Z/Y) n OFI(Int(Y/X))). (ifi) For a morphism f: X' --> X, one has f'-l(Int(Y]X)) C Int(Y'/X'), where f' is y' = Y Xx x' -+Y. (iv) For a non-Arckimedean field K over k, one has =-l(Int(Y/X)) C Int(Y | K/X b K), where ~ is Y | K -+ Y. 9 1.5.6. Remark. -- The notion of a strictly k-analytic space introduced in [Ber], p. 48, is not consistent with that introduced in the previous subsection. First of all, if the valuation on k is trivial, the two notions are completely different. (For example, the affine line/k 1 is strictly k-analytic in the sense of [Ber] but is not such a space in the sense of w 2.2.) Assume now that the valuation on k is nontrivial. In this case the diffe- rence is that in [Ber] strictly k-analytic spaces were considered as objects of the whole category of k-analytic spaces, but here we consider them as objects of their own cate- gory st-k-~tn because we do not know whether the faithful functor st-k-~e~n -+ k-tin is fully faithful. 1.6. Connection with rigid analytic geometry We work here with the category of rigid k-analytic spaces which is defined in [BGR], w 9. Assume that the valuation on k is nontrivial, and let X be a Hausdorff strictly k-analytic space. The corresponding rigid k-analytic structure will be defined on the set X o = { x ~ X I [~st~ : k] < oo }. (We remark that from [Ber], 2.1.15, it follows that the set X 0 is everywhere dense in X.) First of all, ifX = ~(~1) is strictly k-affinoid, then the maximal spectrum X0 = Max(d) is endowed with a rigid k-analytic space structure as in [BGR], w 9.3.1. Suppose that X is arbitrary. We say that a subset q/C X 0 is admissible open if, for any strictly affinoid domain V C X, the intersection q/n V o is an admissible open set in the rigid k-affinoid space V 0. Furthermore, a covering { q/~ }i e I of an admissible open subset q/C X 0 by admissible open subsets is admissible if, for any strictly affinoid domain V C X, { o?/~ n V 0 }i e i is an admissible open coveting of ~ n V o. In this way we get a G-topology on the set X o. The sheaves of tings 0vo , where V runs through the strictly affinoid domains in X, are compatible on intersections, and to they glue together to form a sheaf of tings 0x~ on the G-topological space X 0. The locally G-ringed space (X0, 0x o) satisfies the conditions of Definition 9.3.1/4 from [BGR], and so we get a rigid k-analytic space. We remark that the rigid k-analytic space cons- tructed is quasiseparated. (A rigid k-analytic space is called quasiseparated if the inter- section of two open affinoid domains is a finite union of open affinoid domains.) 1.6.1. Theorem. n The correspondence X ~-~ X 0 is a fully faithful functor from the category of Hausdorff strictly k-analytic spaces to the category of quasiseparated rigid k-analytic 36 VLADIMIR G. BERKOVICH spaces. Furthermore, this functor induces an equivalence between the category of paracompact strictly k-analytic spaces and the category of quasiseparated rigid k-analytic spaces that have an admissible affinoid covering of finite type. A collection of subsets of a set is said to be of finite type if each subset of the col- lection meets only a finite number of other subsets of the collection. Proof. -- Let X be a Hausdorff strictly k-analytic space. First of all we establish the following fact. 1.6.2. /.emma. -- (i) Any open Crinoid domain in the rigid k-analytic space X o ~ of the form V0, where V is a strictly affinoid domain in X. (ii) Let { V~ }, el be a system of strictly affinoid domains in X. Then { V~. o }~ ~ i is an admissible covering of X o if and only if each point of X has a neighborhood of the form Vq u... uV~, (i.e., {V i}~I is a quasinet on X). Proof. -- (i) An open affinoid domain in X 0 is an open immersion of rigid k-analytic spaces f: U o -~ X0, where U is a strictly k-affinoid space. In particular, for any strictly affinoid domain VC X, f-l(V0) is a finite union of affinoid domains in U0, and {f-~(Vo)}, where V runs through strictly affinoid domains in X, is an affinoid covering. It follows that we can find strictly affinoid domains U1, ..., U, C U and Vx, ..., V, C X such that U ---- U1 u ... u U,, and f[v;,0 comes from a morphism of strictly affinoid spaces qh : Ui -~ V~ that identifies U~ with an affinoid domain in V~. Moreover, all ~ are compatible on intersections. Therefore, we get a morphism of strictly k-analytic spaces ? : U --> X. Since ~, as a map of topological spaces, is compact and induces an injection on the everywhere dense subset UoC U, it follows that ~ induces a homeo- morphism of U with its image in X. Finally, ~ identifies U~ with a strictly affinoid domain in V~, and therefore ? identifies U with a strictly analytic domain in X. It is clear that this is a strictly affinoid domain. (ii) Suppose first that (V~. 0 }~cx is an admissible covering of X o. This means that, for any strictly affinoid domain V C X, { Vi, o r3 V o }~ e x is an admissible covering of V o. It follows that V is contained in a finite union Vq u ... u V~,, and therefore each point of X has a neighborhood of the required form. Conversely, assume that the latter property is true. Then any strictly affinoid domain is contained in a finite union Vq u ... u V~,, and therefore { Vi, o c~ V 0 }i e i is an admissible covering of V 0. 9 Let ? : Y ~ X be a morphism of strictly k-analytic spaces. First of all we claim that the induced map 9o : Y0 ~ Xo is continuous with respect to the G-topologies on X o and Y0. Let ~/C X 0 be an admissible open subset, and let V C Y be a strictly affinoid domain. By [BGR], 9.1,4/2, and Lemma 1.6.2 (i), the set q/has an admissible cove- ring {U~,0}~ex, where U~ are strictly affinoid domains in X. By Corollary 1.2.14, for each i ~ I one has ~-I(U~) c~ V = Ujeji V~j, where Vl~ are strictly affinoid domains in V andJ~ is finite. We get a covering {Vii.0 } of the set ~ol(ad) c~ V 0. To verify that ~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 37 the latter set is admissible open in V0, it suffices to show that for any morphism of strictly k-affinoid spaces ~b : W -+ V with ~bo(Wo) C ~o 1(o//) n V0, the covering { 90 l(Vij. 0)} has a finite affinoid covering that refines it. But this follows from the fact that the latter condition is satisfied by the covering {(9~b)ol(Ui,0 },~i. Thus, the set 9o1(~ is admis- sible open in Yo. In the same way one shows that the preimage of an admissible covering of an admissible open set is an admissible open covering. Hence the map of G-topological spaces 90:Yo ~ X o is continuous. That 9 induces a morphism of locally G-ringed spaces easily follows from this. Let now f: Yo -+ Xo be a morphism between the above rigid k-analytic spaces. We have to show that it comes from a unique morphism of strictly k-analytic spaces 9 : Y -+ X. First of all, from Proposition 1.3.2 it follows that it suffices to verify the required fact only in the case when Y is strictly k-affinoid. For this we remark that the system { U o }, where U runs through strictly affinoid domains in X, is an admissible covering of X o. Therefore {f-l(U0) } is an admissible covering of Y0 by admissible open subsets. By [BGR], 9.1.4/2, the latter covering has a finite affinoid covering that refines it. In this way we get strictly affinoid domains V~, ..., V, C Y and Ut, ..., U, C X such that Y0 = V1,0 u ... u V,, o (and therefore Y = V t u . .. u V~) andf(V~,0) C U~, 0. The induced morphisms of strictly affinoid spaces V, -+ U, are obviously compatible on intersections, and therefore we get a morphism of strictly k-analytic spaces 9 : Y -+ X. It is easy to see that % =f and that 9 is a unique morphism satisfying this property. If X is a paracompact strictly k-analytic space, then it has a strictly k-affinoid atlas with a locally finite net, and therefore the rigid k-analytic space X 0 has an admis- sible affinoid covering of finite type. It is also evident that it is quasiseparated. Conversely, let 5F be a quasiseparated rigid k-analytic space that has an admissible affinoid cove- ring { ~ }i~i of finite type. First of all, let q/~-~ U~,0, where the U, are strictly k-affinoid spaces. Since ~r is quasiseparated, for any pair i, j ~ I the intersection q/, n q/~. is a finite union of open affinoid domains in 5~. Thus, there are strictly special domains U,~ C U~ and U~, C U~ that correspond to q/i n o//j under the identifications q/~ = U,, 0 and q/~ = Uj. 0- Let ~ denote the induced isomorphism U~--% Uj~. It is clear that U~I=U~, 'Jij(U~jnUil) =U~inUj~ and v~z=~lov~ on U~nU~. By Proposi- tion 1.3.3, we can glue all U~ along U~ and get a paracompact strictly k-analytic space X. It is easy to see that X o is isomorphic to 5F. 9 Let X be a Hausdorff strictly k-analytic space. From Lemma 1.6.2 it follows that there is an isomorphism of topoi Xo ~ ~ X o. In particular, there is a morphism of topoi (~., n*):X o -~X ~ such that the functor n* is fully faithful (and n. is not). Furthermore, from Proposition 1.3.6 it follows that if F is an abelian sheaf on X, then H~(X, F) --% H~(Xo, ~* F), q >1 0, and if X is good and F is a coherent Ox module, then H~(X, F) -% H~(X0, F0) , q/> 0, where F 0 = n* F |162 ~)x0" Finally, there are equi- valences of categories Mod(X~) -% Mod(Xo) and Coh(Xo) --% Coh(Xo) and an isomor- phism of groups PiC(XG) --% Pic(X0). $8 VLADIMIR G. BERKOVICH w 2. Local rings and residue fields os points os afllnoid spaces 2.1. The local rings Ox,. Throughout the section we consider a k-affinoid space X = Jt'(d). The stalk 0x, , of the structural sheaf 9 x at a point x ~ X is a local ring. Its maximal ideal is denoted by m,, and its residue field 0x.,/m, is denoted by !r The field ~:(x) has a canonical valuation. The completion of ~:(x) is the field oCg(x). Furthermore, let ~ denote the affine scheme Spec(d). There is a morphism of locally ringed spaces ~ : X ~ ~r. For a point x E X we denote by x its image in Y', by ~0~ the corresponding prime ideal of d (it is the kernel of the seminorm on d which corresponds to the point x) and by k(x) the fraction field of d/ga x . 9.. t.1. Proposition. -- The map z~:X---> s is surjective. Proof. -- Suppose that the algebra is strictly k-affinoid. It suffices to show that if d has no zero divisors, then there exists a point x ~ X with go x = 0. By Noether Normaliza- tion Lemma, there exists a finite injective homomorphism ~ = k (TI, ..., T, }-~ d. By [Ber], 2.1.16, the map r ~.,r is surjective. So it suffices to consider the algebra k{ T1,..., T, }. In this case gox = 0 for the point x corresponding to the norm of k { Tx, ..., T, }. 2.1.2. Lemma. -- For a k-offinoid algebra d and a non-Archimedean field K over k the algebra aft" = d ~ K is faithfully flat over d. Proof. -- First of all we recall that for Banach spaces B and M over k the canonical map M | ~ M | B is injective and, if 0 -+ M ---> N ---> P ---> 0 is an exact admissible sequence of Banach spaces over k, then the sequence 0 --> M | B -+ N | B ~ P | B -+ 0 is also exact and admissible (see [Gru]). (One assumed in [Gru] that the valuation on k is nontrivial, but in the case of trivial valuation one obtains the same fact by tensoring with the field K, for some 0 < r < I.) Let now M be a finite d-module. It can be regarded as a finite Banach d-module and, in particular, as a Banach space over k (see [Ber], 2.1.9). We have M |162 d' = M | d' = M | K. By the above fact, if Mae 0, then M |162 d' 4= 0. Furthermore, ifa homomorphism M ~ N of finite d-modules is injective, the homomorphism of Banach K-spaces M | K ~ N | K is injective. This implies that the homomorphism M | ---> N | d' is injecfive. Hence, ~r is a faith- fully flat d-algebra. 9 2.1.3. Corollary. -- In the situation of Lemma 2.1.2 for any pair of points x E X x' E X' = ,~(d') with ~(x') = x, where ~? is the canonical map X' ~ X, Ox, ' ~, is a faithfully flat Ox,,-algebra. I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 39 Proof. -- It suffices to show that for any pair of affinoid subdomains V C X' and U C X with ~(V) C U, a/v is a flat a/u-algebra. By Lemma 2.1.2, a/',-~cv~ = a/u | K I t is a fiat a/u-algebra. Since a/v is a flat a/,_xlul-algebra ([Ber], 2.2.4 (ii)), then a/v is a flat a/u-algebra. 9 We now consider an arbitrary k-affinoid algebra a/. We can take a non-Archi- medean field K over k such that the algebra a/' = at | K is strictly K-affinoid. By the previous case, the map X' =~/t'(a/')~'= Spec(a/') is surjective, and, by Lemma 2.1.2, the map s _+ s is surjective. It follows that the map rc is surjective. 9 2.1.4. Theorem. -- The ring Ox, , is a Noetherian ring faithfully flat over 0~,: = a/~ . Proof. -- Suppose first that the valuation ofk is nontrivial, the algebra a/is strictly k-affinoid and x E Max(a/). In this case Ox, , coincides with the algebra of germs ofaffinoid functions on Max(a/) considered in [BGR], 7.3.2. By [BGR], 7.3.2/7, the ring Ox,, is Noetherian, and, by [BGR], 7.3.2]3, there is an isomorphism ~ --% a/~x --% Ox, 9 between go., go x a/~. ,.__. ~-adic_~_ completions of the rings a/, a/~., Ox, ,, respectively. By [Mat], 8.14, the ring a/~a x --% ~x,~ is faithfully flat over a/~: and Ox, ,. We now consider the general case. We can find a non-Archimedean field K over k with nontrivial valuation such that the algebra a/'= a/| K is strictly K-affinoid, and there exists a point x'e Max(a/') which goes to x under the canonical map X' ---= ~/t'(a/') -+ X. By Lemma 2.1.2 (resp. Corollay 2.1.3), the algebra a/~,, (resp. Ox,,,, ) is faithfully flat over a/~. (resp. Ox,,). Since Ox,.,, is faithfully flat over a/~.,, then Ox, , is faithfully flat over a/~. Furthermore, Corollary 2.1.3 implies that a = aOx,,,, r30x, , for any finitely generated ideal a E Ox, ~. From this it follows easily that Ox., is a Noe- therian ring. 9 2.1.5. Theorem. -- The ring Ox., is Igenselian. Proof. -- We use the following criterion for a local ring A to be Henselian (see [Ray], I. 1.5). A is Henselian if and only if any finite free A-algebra B is a direct product of local rings. Let B be a finite free Ox,,-algebra. We claim that there exist an affinoid neigh- borhood U of the point x and a finite free a/v-algebra ~ such that B = & |162 Ox,," Indeed, let bl, ..., b, be free generators of the 0x,,-module B and set 1 = Y'~'-i ai b~ and b~bj = 5]l"_la~bz, where a~, a~ l ~Ox, ,. The fact that B is an associative and commutative ring with identity is equivalent to certain identities between the coeffi- cients a~ and a~ I. Take a sufficiently small affinoid neighborhood U of the point x such that all the a~, a~j~ come from d v and all the identities are true in d v. Consider the free a/v-module ~ = a/v bl A- ... q- a/v b, and endow it with the multiplication b~ b~ = Y~]'_~ a~j I b~. Then ~ is a finite free a/v-algebra, and, by construction, B = ~|162 Ox, ,. 40 VLADIMIR G. BERKOVICH Furthermore, we may assume that U =-X and consider ~ as a finite Banach ag-algebra (see [Ber], 2.1.12). Then we have a finite morphism of k-affinoid spaces : Y = ~r ~ X, and the theorem follows from the following lemma. 2.1.6. Lemma. -- Let ~ : Y = .~r ~ X = ..K(d) be a finite morphism of k-affinoid spaces. Then for any point x ~ X there is an isomorphism of rings ~ |162 ~x, 9 = lle~-I @Y, vi where = {yx, ...,yd }. Proof. -- Since ? is a map of compact spaces, one has ~-I(U) = [Jd x V,, for any sufficiently small affinoid neighborhood U of x, where V i are affinoid neighborhoods of the points y, such that Vr n V~ = O for i 4=j. Moreover, the domains V, form a basis of affinoid neighborhoods of y,. We have i-1 Here we used the equality ~-l(v> = ~ |162 ~r = ~ |162 ~qlu which follows from the fact that ~ is a finite Banach aft-module. Therefore, ~ | Ox,, H i_ , Oy.,i. 9 2.2. Comparison of properties of Ox.,~ and O.r,x Let P be a property of local rings which is preserved under localizations with respect to the complements to prime ideals. A commutative ring A is said to possess the property P (or A is a P-ring) if all of the local rings A~, where go runs through prime ideals of A, possess the property P. More generally, let Y be a locally ringed space. The set of pointsy ~ Y such that OY, v is a P-ring is denoted by P(Y). If P(Y) ---- Y, then Y is said to possess the property P. 2.2.1. Theorem. -- Let P be the property of being Red (reduced), Nor (normal), Reg (regular), CI (complete intersection), Gor (Gorenstein), CM (Cohen-Macauley). Then P(X) is Zariski open in X and P(X) = r~-l(P(~)). For the definition of these properties and the verification of the fact that P is pre- served under localizations see Matsumura's book [Mat]. We shall deduce Theorem 2.2.1 from known results which are formulated in the following lemmas. 2.2.2. Lemma. -- Let (A, m) ~ (B, n) be a faithfully flat homomorphism of local Noetherian rings. (i) If B is a P-ring, then so is A. (ii) If A is a P-ring, where P :~ Red, Nor, and n = mB, then B is a P-ring. 9 2.2.3. Lemma. --- Strictly k-affinoid algebras are excellent rings. 9 ]~TALE GOHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 41 2.2.4. Lemma. -- Let A be an excellent ring. (i) If go is a prime ideal of A such that A~, is a P-ring, where P = Red or Nor, then the completion d~ is a P,ring. (ii) The set of prime ideals go C A such that A~ is a P-ring is open in Spec(A), 9 For Lemma 2.2.2 and the assertion (ii) of Lemma 2.2.4 in the cases P = OI or Gor see [Mat], w 23-24. Lemma 2.2.3 is proved in [Kie]. The assertions (i) and (ii) (for P ae CI, Gor) of Lemma 2.2.4 are proved in [EGAIV], 7.8.3. Proof of Theorem 2.2.1. -- From Theorem 2.1.4 and Lemma 2.2.2 (i) it follows that P(X) C ~-l(p(:y)). Let x be a point of X such that d~x is a P-ring. We have to show that 0x, . is a P-ring. Suppose that the valuation of k is nontrivial and the algebra d is strictly k-affinoid. From Lemmas 2.2.3 and 2.2.4 it follows that the set P(f) is open in Y'. Furthermore, ifx ~ Max(d), then d~x = g)x,,, and therefore Ox, ~ is a P-ring. Let x be an arbitrary point. Since P(W) is open in ~, then V C ~-l(p(~)) for any sufficiently small strictly affinoid neighborhood V of x. Ify ~ Max(dv) C Max(~), then Oar,=(u ~ is a P-ring, and therefore 0x, ~ is a P-ring. Since Ox. u = g)v, u, then 0~../~ / is a P-ring, where r -~ Spec(~'V). It follows that P(r = r because P(r is open in r Thus, the algebras d v are P-rings for all sufficiently small strictly affinoid neighborhoods V of x. Since 0x ~ = lim ~'v, then 0x,~ is a P-ring. Indeed, this is evident if P ----- Red or Nor. If P 4= Red, Nor, we remark that rn~ ---- ~ax, v 0x, ~ for a sufficiently small strictly affinoid neighborhood V of x, where gox, v is the prime ideal of d v corresponding to the point x. From Lemma 2.2.2 (ii) it follows that Ox,~ is a P-ring. We remark that (under the same assumptions) the fact already verified implies that the subsheaf of ideals Jx COx consisting of nilpotent elements is coherent and, in fact, is generated by the nilradical rad(~) of ~'. Furthermore, if X is reduced, then the subsheaf ~)~m of the sheaf ~g//~ of meromorphic functions consisting of elements, whose images in all stalks ~'x., are integral over Ox,~, is coherent, and there exists a e ~', which is not a zero divisor, such that aO~ ~ C (9 x. We now consider the general case. 9,.9,.5. Lemma. -- Let K be a field of the form K~ ....... (see [Ber], w 2.1) and ~r -~ ~/| K. Consider the map ~ : X -+ X' = .#{(~") which sends a point x ~ X to the point x' e X' corresponding to the multiplicative semi-norm Y~, a, T ~ ~ max I a,(x) [ r ~. Then go~, = go~ s]'. Furthermore, if Y'is a Zariski closed subset of X', then ~r-l(Y ') is a Zariski closed subset of X. Proof. -- Let fl,...,f, be generators of g0~. Since the canonical epimorphism d" ~ ~ : (al, ..., a,) ~ ~E~= 1 air is admissible (see [Ber], 2.1.9), there exists a constant C > 0 such that any element a e ~0~ can be represented in the form ~=~ a~f with l[a,[[~< C[[a[[, 1~< i~< n. Let a'----Y%a.,T ~go~,. By construction, all the a, 6 42 VLADIMIK G. BERKOVICH belong to fo x. For every v we take a representation a v = ~-1 av,~f as above. Then b~ = ~ a~,~ T ~ are well defined elements of d', and we have a' = ~'_1 b~f~ ~ go x d'. Thus ~x' = ~~ d'. Let Y' be defined by an ideal a'C ~1'. Denote by a the ideal of off generated by all of the coefficients a, from the representations a' = ~, av T ~ of elements a' ~ a'. We claim that ~-l(y,) is the closed k-analytic subset of X defined by the ideal a. Indeed, let x~X and x' =a(x). Then x~o-l(Y ') ~x'~Y'~a'C~o x, ~aC~o,. 9 Take a field K of the form K n ..... ,,, n/> 1, such that the algebra off' = off ~ K is strictly K-affinoid (the valuation on K is nontrivial since n >t I). First we consider the case when P 4= Red, Nor. Since ~0~, = ~o~ off', where x'= ,(x), it follows from Lemma 2.2.2 (ii) that off~x' is a P-ring. By the strictly affinoid case, Ox,,~, is a P-ring. Therefore Ox, ~ is a P-ring, by Lemma 2.2.2 (i). We have P(X) = = From Lemma 2.2.5 it follows that P(X) is Zariski open in X. Let P = Red. It suffices to verify that the subsheaf of ideals Jr x C d~ x consisting of nilpotent elements is generated by rad(off). We may assume that rad(off)= 0. Then rad(off') = 0. Hence Jx' = 0. Since Sx,, is embedded to Ox',~', where x' = ,(x), we have Jx = 0. Let P = Nor. By the previous case, we may assume that X is reduced. We want to verify that there exists a ~ off which is not a zero-divisor such that ad~x r= C ~x. Since this is true for the subsheaf of.lr x generated by the normalization of off (see [Ber], 2.1.14 (i)), we may assume that off is a (normal) integral domain. By the strictly affinoid case, there exists a non-zero element a' = ~ a~ T ~ ~ off' with a' d~ ~ C Ox'. It suffices to show that a~ &x ~ C #x for any v. Let ad be an open subset of X, and letf be an element from the full ring of fractions of Ox(q/). Then a'fr #x,(q~-l(ad)), where r denotes the canonical map X'--~X. But Ox,(Cp-l(q/)) consists of the series ~f~T ~ such that f, E ex(~,'), and, for any affinoid subdomain V Cad, IIf~ ]Iv r~ ~ 0 as v ~ oo. It follows that a~f~ #x(ad) for any v. 9 B. B. 6. CoroUary. -- Let P be one of the properties in Theorem 2.2.1. Then for any good k-analytic space Y the set P(Y) is Zariski open in Y. Furthermore, if Y is reduced, then the comple- ment to Reg(Y) /s nowhere dense in Y. 9 2.2.7. Corollary. -- Let P be one of the properties in Theorem 2.2.1. Let q/be a scheme of locally finite type over k, and let ~ be the canonical map ~/'~ ~ q/. Then P(~/") = ~-t(p(q/)). Proof. -- We may assume that ~ ---- Spec(B) is an affine scheme, and B is a finitely generated k-algebra. Suppose first that the valuation on k is trivial. Iff~, ...,f, gene- rate B over k, then q/= is a union of affinoid subdomains of the type v ={y Ilf,(x)I r, 1~< i~< n). But if r >i 1, then ~v = B. Therefore the required statement follows from Theorem 2.2.1. ~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 43 Suppose now that the valuation on k is nontrivial, and lety e q/'~. By [Ber], 3.4.1, tP~. ~ is a faithfully flat O~,~-algebra, and ify E Max(B), then 0~, = tg~, T. Since A is an excellent ring, we get the required statement, by the reasoning from the proof of Theorem 2.2.1. 9 2.8.8. Corollary. --For any affinoid subdomain V C X one has Reg(V) = Reg(X) c~ V. Proofs. -- If X and V are strictly k-affinoid, this is clear because both sets are determined by their intersections with X 0. The general case is reduced to the strictly affinoid one by the reasoning from the proof of Theorem 2.2.1. 9 8.2.9. Remark. ~ The maximal ideal m~ of the local ring Ox, ~ can be strictly larger than go x Ox. ,. This is related to the fact that the Zariski topology on an affinoid subdomain V C X can be strictly stronger than that induced by the Zariski topology on X. Here is an example. Let J~T)= ~'-1 ~ T~ be a formal power series in one variable over the residue field k, which is algebraically independent of T, and set f(T)= ~~ ta~T ~, where a~ are representatives of ~ in k ~ We claim that for any non-zero g(T1, T2) Ek{ TI, T2} one has g(T,f(T)) . 0. Indeed, let g(Tx, T,.) = Y~,~.j_oa,.,T~xT~. Multiplying g by a constant, we may assume that []g [l = max [a~.j [ = 1. Let S ={(i,j) lla,,jl = 1} (it is a finite set) and set P(Tx, T,) = ~g,,, j, e s a~, t "P1 T~ and h(Tx, T~) = g(Tx, T~) -- P(Tx, T~). Since II h II < l, we have II h(T,f(T))II < X. Ifg(T,f(T)) = 0, then II P(T,f(T))II < I. It follows that ]~(T,J~T)) = 0. This is impossible because a~T) is algebraically inde- pendent of T over ~. Thus, the Zariski closed subset of the two-dimensional disc V of radius r < I, which is defined by the equation T,. --f(Tx) = 0, does not extend to a Zariski closed subset of the two-dimensional unit disc X. If now x is the point of X, which corresponds to the multiplicative seminorm on k ( Tx, T 2 } : g(Tx, Tz) ~-* [ g(T,f(T))[, then x~ * 0 because T, --f(Tx) ~ m~, but go x = 0. 2.3. The residue fields ~:(x) 2.3. t. Definition. -- A field K with valuation is said to be quasicomplete if the valuation extends uniquely to any algebraic extension of K. For example, if K is complete with respect to its valuation (i.e. K is a valuation field in the terminology of [Bet], w 1.1), then it is quasicomplete. The following lemma easily follows from [BGR], w 3.2. 44 VLADIMIR G. BERKOVICH 2.3.2. Lemma. ~ The following properties of a field K with valuation are equivalent: (i) K is quasicomplete; (ii) for any irreductible polynomial T" + al T"-I + ... + a, ~ KIT], one has [a~ Ira<. [a, I ~/", 1 <~ i<~ n; (iii) the spectral norm of any finite extension of K is a valuation. 9 2.3.3. Theorem. -- The residue field r,(x) of a point x e X is quasicomplete. Proof. -- Note that the field K(x) does not change if we replace X by a smaller affinoid neighborhood of x or by a closed k-analytic subset which contains x. Since the ring Ox.x is Noetherian, mx= ~ox, v Ox,, for some affinoid neighborhood V of x, where fox, v is the prime ideal of ~v which corresponds to x. So we can replace X by ~r and assume that m, = 0. Furthermore, since X is regular at x, we can decrease X and assume that the algebra ~ is regular and has no zero divisors. Let L be a finite extension of K----K(x). It suffices to show that the spectral norm I [.p of L is a valuation. Recall (see [BGR], 3.2.1/1) that for an element g e L one has [g [,0 = max If ] m, where T" +fl T"-I + -.- +f, is the minimal poly- nomial of g and I ] is the valuation of K (iff comes from ~, then Ill = If(x)[). We may assume that L is separable over K and, in particular, that L is generated by one element a. Let T" + a x T m- i + ... + a,, be the minimal polynomial of a over K. Decreasing X, we may assume that all the a~ andfj belong to d. Let 3U be the fraction field of ~. Consider the finite extension .~o of ~l which corresponds to the minimal polynomial of a. We may assume that ~, g e .~o and L = K.oq ~. Let ~ be the integral closure of ~ in .o~P, and let ~ denote the morphism ofk-affinoid spaces Y = ~r -+ X. By construction, 0-i(x) = {y} and ~(y) = L. It suffices to show that ]g [~ = [g(y)[. One has I g(Y)] = invf Pv(g), where V runs through a basis of affinoid neighborhoods of y, and 0v(g) is the spectral norm of g in the Banach algebra ~r Since ~-l(x) = {y }, we have I g(Y)[ = inf P~-~,v, (g), where U runs through a basis of affinoid neighborhoods of x. 2.3.4. Lemma. -- Let s/~ ~ be a finite injective homomorphism of regular k-affinoid algebras, and suppose that ~ has no zero divisors. Then for any element g e g~ one has p(g) = max O(f) m, where T" + fl T"-I + 9 .. + f, is the minimal polynomial of g over ,~. I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 45 Proof. -- Suppose that the valuation of k is nontrivial and the algebra M is stHctly k-affinoid. Then for an elementfe d (resp. g e ~) the spectral norm P(f) (resp. P(g)) is equal to the supremum semi-norm If],,p (resp. I g I,,p) on the maximal spectrum Max(M) (resp. Max(~)). Hence the required fact follows from [BGR], 3.8.1[7. In the general case we take a field k' of the form K,~ ..... ,,, n/> 1, such that the algebra M'= M~ k' is strictly k'-affinoid; then the rings M' and ~'= ~| are regular. From Lemma 2.2.5 it follows that ~' has no zero divisors. Since the canonical homomorphisms M -+ M' and M ~ ~' are isometric, it suffices to show that the minimal polynomial of g over M remains to be irreducible over M'. But this is clear because ~' has no zero divisors. 9 Using Lemma 2.3.4, we have I g(Y)[ = info~-x,c,(g) = inf max p~(f)~/~ U l~<i~<n = max i f Pu = max If(x)I a/' = I g I.~. Theorem 2.3.3 is proved. 9 2.4. Quasicomplete fields In this subsection we establish properties of quasicomplete fields which will be very useful in the sequel. The Galois group of a normal extension L/K will be denoted by G(L/K). The Galois group G(K'/K) of the separable closure K' of K will be denoted by G x. If K is a quasicomplete field, then the valuation on K uniquely extends to its algebraical closure K a. The same, of course, is true for the completion N: of K. 2.4.1. Proposition. - Let K be a quasicomplete field. Then for any finite separable extension L/K one has I.--% L| x K, and the correspondence ]_, ~ I. induces an equivalence between the categories of finite separable extensions of K and of ft. In particular, there is an iso- morphism G~ -~ GK. Proof. -- Since the valuation of L coincides with the spectral norm, L is weakly K-cartesian ([BGR], 3.5.1/3). By [BGR], 2.3.3/6, one has [L: ~] = [L : K], and therefore L-% L| K K. Our assertion now follows from Krasner's Lemraa (see [BGR], 3.4.2). 9 2.4.2. Corollary. -- Let K be a quasicomplete field, and let K' be a bigger quasicomplete feld whose valuation extends the valuation of K. Suppose that the maximal purely inseparable extension of ~ in ~' is dense in K'. Then the correspondence L ~-~ L | K' induces an equivalence between the categories o.ffinite separable extemions of K and of K', and there is an isomorphism G K, ~ G~. 9 46 VLADIMIR G. BERKOVICH 2.4.3. Proposition. -- The following properties of a field K with valuation are equivalent: a) K is quasicomplete; b) the local ring K ~ = { ~ e K II ~ ] ~< 1 ) is ttenselian. Proof. -- a) =, b) (see [BGR], 3.3.4). Let F be a polynomial in K~ with [ F [ = 1 (the Gauss norm). Suppose that F is a product of two coprime polynomials g, h E KIT]. Take a decomposition F = Fa, ..., F, of F into irreducible polynomials. We may assume that [F~I= 1 for all i and that the polynomials F1, ...,F~ are monic and the leading coefficients of F,~+ 1, ..., F, have the norm < 1. Then F~ =f~ for some irreducible polynomials f e KIT], 1 ~< i~< m (this follows from Proposition 2.4.4 (ii)). From Lemma 2.3.2 it follows that F,,+x, -.-, ~, are elements of g.~ Since g and h are coprime in K.[T], we may assume that g = ~f111, ...,fr ~r for some a e K with [ a [ = 1 and r~< m. Then for the polynomials G = aF1, . .., F r and H=a-IFr+I,...,F, one has G=g, ~----h and F-----GH. b) :~ a). Let L be a finite extension of K, and let B the integral closure of A = K ~ in L. We claim that B is a local ring. Indeed, it suffices to show that the set b ---- B\B" is an ideal in B. Suppose that for some elements f g e b one has f + g r b. Consider the A-subalgebra C orB generated by the elementsf g and (f+ g)-L Then C is a finite A-algebra. But any finite algebra over a local Henselian ring is a product of local rings. Since C is an integral domain, it follows that C is a local ring. We get that the elementsf and g belong to the maximal ideal of C but their sum f + g is invertible in C. Thus, B is a local ring. Furthermore, from the definition of the spectral norm it follows that B =(feLl [f[,~<, 1}. Let ] {x, ..., [ [, be the valuations on L which extend the valuation on K. One has Ill| = max If I, (see [BGR], w 3.3). Suppose that n > 1. By the Artin-Waples l~i~<tt Lemma, one can find for each 1~< i~ n an elementf~eL such that Ill,---- 1 and ]f~ 1~ < 1 forj 4= i. The elements fl, ...,f, are not invertible in B, and therefore belong to the maximal ideal of B. But for the element f=fx + ... +f~ one has Ifh = 1, 1~< i~< n. It follows that If-ll, = l, l,< i,< n, and therefore If-~ 1,~ = l, i.e., f is invertible in B. Hence, n = 1, i.e., the spectral norm on L is a valuation. 9 Let K be a quasicomplete field, let L be a Galois extension of K (finite or infinite). We set I(L/K) = { a ~ G(L/K) [ a acts trivially on L } (~ is the residue field of L) and W(L/K) ---- { ~ e G(L/K) [ ]~ -- ~ [ < J ~ ] for all ct e L* }. We set p = char(~). 9..4.4. Proposition. -- (i) I(L/K) and W(L/K) are normal divisors of G(L/K) and W(L/K) c I(L/K); (il) the extension ~/K is normal and G(L/K)/I(LIK) -% G("L/K); (iii) there is a canonical isomorphism I(L/K)/W(L/K) -% Hom(] L" J/] K' 1, ]~'); (iv) W(L/K) is a pro-p-group. #.TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 47 Proof. -- (i) is trivial. (ii) Let ~ e ~. We take a representative ~ of ~ in L ~ and the minimal polynomial P(T) =T"+a 1T "-a+ ... +a, of ~ over K. Since the valuation of L coincides with the spectral norm, [ ~ I -- max I ~, I~'- < 1, and therefore all a~ belong to K ~ The clement ~ is a root of the polynomial P(T) = T" + ~ T "-1 + ... + ~", e K,[T]. Hence all the elements of (g,)a conjugated to ~ are also roots of P(T). Since the group G(L/K) acts transitively on the set of roots of P(T), its image in the automorphism group of~ over K acts transitively on the set of roots ofP(T). It follows that ~ is normal over g,, and the canonical map G(L/K) -~ G(t/g,) is surjective. (iii) For ~ ~ I(L]K) and ~ e L* we denote by ~b(~, ~) the image of the element "Gt/0t in ~*. The map ~b : I(L/K) � L" -+ ~" is bilinear because for a, v e I(L/K) one has ..... <1. 0t 0~ 0t 0t 0t Furthermore, if I ~ [ = I ~ I, then ;)l i.e., ~b(~, ,r depends only on I ~ [. If ~ e K*, then d~(,, ~c) = 1. Therefore + induces an embedding M(L/K) = I(L/K)IW(LIK ) ~ Hom([ L* ]/] K" [, %*). 2.4.5. Lemma. -- If K' is a Galois extension of K with K C K'C L, then there are exact sequences 0 -+ I(L/K') -+ I(L/K) -+ I(K'/K) -~ 0, 0 -+ W(L/K') -+ W(L/K) -+ W(K'/K) -+ 0. Proof. -- The only nontrivial fact is the surjectivity of the maps I(L/K) -+ I(K'/K) and W(L/K) -> W(K'/K). The surjcctivity of the first map is equivalent to the sur- jectivity of the map G(L/K) -> G(~/Y-.) which is proved in (ii). The surjcctivity of the second map is equivalent to the injectivity of the map M(L/K') -+ M(L/K). The latter fact follows from the part of (iii) which is already verified. 9 Suppose that L is finite over K and set K'= L w~r'm). By Lemma 2.4.5, W(K']K) = 0. We claim that g.' is the maximal subfield ~0 C ~ separable over g,, and [I(K'/K) : 1] = [Hom(] K'* ][[ K* ], g.'*) : 1] = [] K'*]: ] K* []. Indeed, since G(R'/R) = G(~/R)=G(~./R), 48 VLADIMIR G. BERKOVICH we have ~, C ~.'. Furthermore, [K': K] = [G(f{'/K) : 1] [I(K'/K) : 1] < [~' : ~] [Hom(I K'* {/I K* l, K'*) : 1] ~< [K': K] [I K'*[: I K* l] ~< [K': K]. Therefore all the inequalities are actually equalities, and our claim follows. The case of an arbitrary L is deduced from this, using Lemma 2.4.5. (iv) As above, it suffices to assume that L is finite over K. Suppose that W contains an element a of order l which is prime to p. Let K' be the subfield of L, which consists of elements fixed by a. Then L is a cyclic extension of K' of degree l. Take an element ~ L with L = K'(~). If a = TrT,/x,(~), then replacing ~ by ~ -- a/l, we may assume that TrT./x,(~) = 0. On the other hand, since ~ e W then ~ = ~ + ~, where ] ~ 1< { ~ 1. We have l--1 0=~+%~+ ... +~ =l~+ Y, ~. i=0 This is impossible because l Y/-1 ,:o~,l<l~l =ll~l . 9 The group I(L/K) is said to be the inertia group, and the group M(L/K) = I(L/K)/W(L/K) (resp. W(L/K)) is said to be the moderate (resp. wild) ramification group of the Galois extension LIK. Furthermore, applying Proposition 2.4.3 to the separable closure K s of K, one gets the maximal unramified (resp. moderately ramified) extension K nr (resp. K mr) of K. We set err= G(Kmr/K), 6~ = a(KnrlK), Mx = G(Kmr/K "r) and W x = G(KS/K~). 2.4.6. Corollary. -- (i) K "~ is the separable closure K" of K and [K"rl = l K I; (ii) G~ = G~; (iii) M x -% Hom (~/IK~[/I K* l, (~s),) ; (iv) W x is a pro-p-group. 9 We say that an algebraic extension L/K is unramified (resp. moderately ramified) if L C K ~ (resp. E C Kmr). 2.4.7. Proposition. -- A finite separable extension L/K is unramified (resp. moderately ramified) if and only if it satisfies the following conditions: a) L is K-cartesian, i.e., [L:K] = [~:K] []L*[:IK*I]; b) ~ is separable over K; c) J L* I = [ K* [ (resp. p X[I L*]: I K* 1]). I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 49 Proof. -- The direct implication follows from Proposition 2.4.4 and the fact that a subfield of a K-cartesian field is K-cartesian. 9..4.8. Lemma. -- Any finite extension L/K with p ,~'[L:K] is moderately ramified. Proof. -- The assertion follows from the fact that the wild ramification group W(L']K) is an invariant p-subgroup of the Galois group G(L'/K), where L' is the minimal Galois extension of K which contains L. 9 Suppose that L satisfies the conditions a)-c). Let L' be a finite Galois extension of K which contains L, and let K 0 be the maximal subfield of L' which is unramified over K. We take the field K' C K 0 for which K' = ~ and claim that K' C L. Indeed, let = be an element of K '~ with ~' = K(~). Since [K': K] = [K': K], we have K' = K(a). Let P(T) be the (monic) minimal polynomial of ~ over K. It is clear that P(T) e K~ The polynomial P(T) is separable and has a root in ~. Since L ~ is Hen- selian, there exists a root ~ of P(T) in L with ~ = ~. From this it follows that ~ = because that polynomial P(T) is separable. We have [L:K'] = [[L*I:IK*I]. If IL*I =]K*[, then L=K'. Ifp does not divide [1 L* I :l K* I], then p X[L: K'], and, by Lemma 2.4.8, L is moderately ramified over K'. Since K' is unramified over K, L is moderately ramified over K. 9 2.5. The cohomological dimension of the fields K(x) Recall that the /-cohomological dimension cd~(G) of a profinite group G is the minimal integer n (or m) such that HI(G, A) = 0 for all i > n and all /-torsion G-mo- dules A (l is a prime integer). The/-cohomological dimension cd,(K) of a field K is, by definition, the /-cohomological dimension cdz(G). Recall also that if l = char(K), then cd,(K) ~< 1 ([Ser], Ch. II, w 2.2). 2.5.1. Theorem. -- For a point x eX, one has cdz(~:(x)) < cdt(k ) + dim(X). Proof. -- First of all we remark that the statement is evidently true if dim(X) = 0 and that, by Proposition 2.4.1, one has cd~(n(X)) = cd,(g'(x)). Consider first the case when X is a closed disc in A 1. If [~(x) : k] < oo, then cd~(K(x)) ~< cdz(k). Assume there- fore that [n(x) : k] = oo. Then the field of the rational functions in one variable k(T) is embedded in n(x) and everywhere dense in it. Fix an embedding k(T) 8 ~-~ n(x)* over the canonical embedding k(T) ,-~ ~(x). Since the field K(x) is quasicomplete, it follows that K(x) 8= k(T) 8 K(x). In particular, the Galois group G=) can be identified with a closed subgroup of G~T ), and therefore one has cd~(K(x)) ~< cdz(k(T)) (lot. cit., Ch. I, w 3.3). By Tsen's Theorem (loc. cit., Oh. II, w 4.2), one has cd~(k(T)) ~< cdz(k ) + 1, and hence cdz(~(x)) ~< cd~(k) + 1. Suppose now that dim(X) t> 1 and that the theorem is true for affinoid spaces whose dimension is at most dim(X) -- 1. Take an analytic function f on W which is 7 50 VLADIMIR G. BERKOVICH nonconstant at any irreducible component of X and consider the induced morphism f: X-~ A 1. The morphism f can be considered as a morphism to a closed disc of a sufficiently big radius Y. Lety =f(x). The point x is also a point of the Yf(y)-affinoid space X,, whose dimension is at most dim(X) -- 1. By induction, cd~(~(x)) ~< cdt(gff(y)) q- dim(X) -- 1, and, by the first case, cd~(W(y))~< cdz(k ) q-1. The required inequality follows, 9 If l is not equal to the characteristic of the residue field of k, then one can get as follows a more strong inequality for cd,(K(x)) (the result will not be used in the sequel). For this we recall some definitions and facts from [Ber], w 9.1. Let K be an extension of k with a valuation which extends the valuation of k. We denote by s(K/k) the transcendence degree of K over k" and by t(K/k) the dimension of the Q-vector space [~/~-//~/[ k* ], and we set d(K/k)= s(K/k)+ t(K/k) (the dimension of K over k). It is clear that d(K/k) = d(K/k). 2.5.2. Lemma. -- For a point x e X, on~ has d(,:(x)/k) <<. dim(X). Moreover, the equality is achieved for some point of X. Proof. ~ If X is strictly k-affinoid, the assertion is proved in [Ber], 9.1.3. The proof also shows that dOc(x)[k ) = dim(X) for some point x from the Shilov boundary of off (see [Ber], w 2.4). In the general case we take a field K of the form K,1 ..... r, for which the algebra d'= d~ K is strictly K-affinoid. Let r denote the canonical map X' = J/t'(off') -+ X, and let ~ denote the map X -+ X' from Lemma 2.2.5. Since the fiber of q~ at x coincides with Jl(SC~(x) | K), it follows that dOc(o(x))/lc(x)) = n. By the strictly affinoid case, dOc((r(x))/K ) <~ dim(X). Let now x' be a point from the Shilov boundary of off' for which dOc(x')/K ) = dim(X). It is easy to see that x' = ~(x) where x = ~(x'). Therefore dOc(x)[k ) = d(K(x')/K) = dim(X). 9 ~..5.3. Theorem. -- Suppose that l ~: char(T). Then for a point x ~X, one has cdt(~(x)) ~< cdt(k ) + d(K(x)/k). ~..5. 4. Lemma. -- Let K be a quasicomplete field, and let g be a prime integer different from char(K). Suppose that the numbers cdt(K ) and st(K ) = dimr~(]K*]/[ K* it) are finite. Then cd,(K) ~< cd~(K) -[-st(K ). Proof. -- Since W,r is a p-group, then cdt(K ) = cdl(G~r). Furthermore, since the group M x is abelian, cdt(Mx) coincides with the l-cohomological dimension of the /-component of M x. The latter group is isomorphic to Z~tor), and therefore cdt(Mx) = s:(K). Since G~ ~ G~, the required fact follows from the spectral sequence H~(G~, Hq(MK, A)) =~ H~+q(G~, A). 9 Proof of Theorem 2.5.3. -- As in the proof of Theorem 2.5.1 consider first the case when X is a closed disc in AL Let x' be a point of X' ---- X | k" over x. Since the ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 51 field K(x)k ~ is everywhere dense in K(x') and the both fields are quasicomplete, from Corollary 2.4.2 it follows that G~,~--~ G~x~k,. Therefore there is an exact sequence 0 ~ G~,,,} ~ G~(~, ~ G(*:(x) k*/r~(x)) ~ O. The latter group is a closed subgroup of G,, and hence its cohomological dimension is at most cd,(k). It follows that cd,(~:(x)) ~< cd,(k) q- cd,(~:(x')). Since d(K(x')/k ~) = d(K(x)[k), Lemma 2.5.4 implies that cd,(~:(x')) = d(r~(x)/k). Suppose now that dim(X)>/ 1 and that the theorem is true for affinoid spaces whose dimension is at most dim(X) -- 1. As in the proof of Theorem 2.5.1 we can find a morphism f: X -+ Y, where Y is a closed disc in A 1, such that all of the fibres of X has dimension at most dim(X) -- 1. Lety =f(x). By induction, cd,(~f(x)) <~ cd~(~(y)) q- d(~(x)[~(y)), and, by the first case, cd~(J(f(y)) ~< cd~(k) q- d(JF(y)/k). Since + d(ae(y)/k) = the required inequality follows. 9 2.6. GAGA over an affinoid space Let ~/be a scheme of locally finite type over 5f, and let F be the functor from the category of morphisms Z ~ X, where Z is a good analytic space over k, to the cate- gory of sets which associates with Z ~ X the set of morphisms of locally ringed spaces over W, Hom~(Z, ~r 2.6.1. Proposition. -- The functor F is representable by a closed morphism of k-analytic spaces ~t~___> X and a morphism of locally ringed spaces 7~ : ~r ~. The correspondence ~t ~ ~t~ is a functor which commutes with extensions of the ground field and with fibred products. Proof. -- The k-analytic space ~t~n is constructed in the same way as in the case when X = ~r (see [Ber], 3.4.1). Namely, one shows that if Y/ is the affine space over s A~, then ~n = Aax = X � A a. After that one shows that if Y/~ exists for ~, then ~e~n exists for any subscheme ~" C ~/. In particular, q/~ exists for any affine scheme of finite type over 5~ and for any its open subscheme. Finally, if ~/is an arbitrary scheme of locally finite type over 5F, then one takes an open covering { Y/~ }i ~ x of ~/by affine subschemes of finite type over s One glues together all of the q/~'s and obtains the k-analytic space qC'n associated with ~/. That the correspondence q/~ q/~ is a functor possessing the necessary properties follows from the universal property of ~t~. 9 2.6.2. Proposition. -- The map 7~ : Y = ~/~n _+ q/ is surjective, and for any point y E Y the ring Oy.u is flat over O~,y, where y = re(y) (i.e., 7~ is a faithfully flat morphism). 52 VLADIMIR G. BERKOVICH Proof. -- By Proposition 2.1.1, the map X ~ W is surjective. Therefore to show that the map 7: : Y ~ ~/is surjective, it suffices to verify that for any point x r X the map Y~-+ q/x is surjective. One has Y, = (~/x| Since the morphism of schemes ~t x | --~ ~r is faithfully flat, the situation is reduced to the case when X = ,//(k). In this case it suffices to verify that if ~/ is an irreducible affine scheme of finite type over k and y is its generic point, then there exists a point y ~ Y whose image in q/is y. The Noether Normalization Lemma reduces the problem to the case when ~/is the affine space A a. In this case, for any point y ~ Y = A a associated with a closed polydisc, n(y) is the generic point of ~t = A a. 9,. 6.8. Lemma. -- The map r: : Y = Y/~ ~ Y/ induces a bijection Yo -% ~o ---- { Y ~ ~/] [k(y) : k] < oo }. If y ~ Yo, then there is an isomorphism of completions O~,y-% Oy,~. Proof. -- Let y ~Y/o. For n >/ 1 we set 2Y = Spec(O,,~/m~). The scheme ~e consists of one point z and is finite over k. Therefore Z = ~" consists of one point z, and one has Oa-, ---- 0,, y[n~ ---% Oz, .. Furthermore, there is a canonical closed immersion 2~ e -~ q/ which takes z to y. Therefore Z-+ Y is also a closed immersion, and the point y, which is the image of z in Y, is the only preimage of y in Y. (In particular, Yo --% q/o-) Moreover, one has = r r If n = 1, we get m v = my dgy, v and k(y) = K(y). Hence O~,y --% Or. v. 9 From Lemma 2.6.3 it follows that Or, v is flat over ~, y at least in the case when Y ~ Y0. In the general case we take a sufficiently big non-Archimedean field K over k such that there exists a K-point y' e Y' = Y ~ K over y. We set X' = Jt'(~r | K) and ~"= Spec(z~r174 K). One has Y'= ~/,~n, where @t,= ~t � W,. Let y' be the image of the point y' in ~'. We know that Oy,,r is flat over dT,, y,. Since ~' is faithfully flat over ~ (Lemma 2.1.2), it follows that O~,,y, is flat over 0~,y, and therefore 0y,r is flat over dT,,~. Finally, from Corollary 2.1.3 it follows that Oy,r is faithfully flat over 0y, v. Hence ~r,~ is flat over 0,,y. 9 2.6.4. Proposition. -- Let T be a constructible subset of ~. Then n-~(T) = 7~-1(T). 2.6.5. Lemma. -- Suppose that ~/ is affine of finite type over 3Y. Let y, z be points of with z ~ ~, and let z be a point of ~ t~ with ~:( z) = z. Then any open neighborhood of z contains a pointy with n(y) = y. Proof. -- For a non-Archimedean field K over k, we set ~r = A | K, X' = ~/(~r ~' = Spec(M'), ~' = q/ � f'. Since ~t' is faithfully flat over ~, there exists points ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 53 y', z' ~ ~' over y and z, respectively, with z' e ~'. Since the tensor product ~(z) | k(z') is nontrivial, the valuation on r(z) extends to a multiplicadve seminorm on it. It follows that there exists a point z' e q/"~ whose image in ~/' is z' and in ~/'~ is z. Thus, we can extend the field k, and, in particular, we may assume that the valuation on k is nontrivial and ~f is strictly k-affinoid. Let q/= Spec(B). Replacing B by B/ga~, we may assume that ~/is reduced and irreducible and that y is its generic point. Since ~t~ is everywhere dense in ~t~, we may assume that z e ~/~. Furthermore, since the ring B is excellent (it is finitely generated over the excellent ring ~), Reg(~/) is an everywhere dense Zariski open subset of ~/. Shrinking ~t, we may assume that ~t is regular. In this case the homomorphism B -+ ~,, is injective. From Lemma 2.6.3 it follows that the homo- morphism B -+ ~,,, is injective. Take a sufficiently small connected strictly affinoid neighborhood V = r162 of the point z. The latter homomorphism goes through ~v, therefore the homomorphism B -+ ~v is injective. Since ff/'~ is regular and V is connected, the ring ~v is an integral domain. By Proposition 2.1. I, there exists a point y ~ V for which the character ~v ~-'~(Y) is injective. It follows that the character ~ -~ag'(y) is injective, and therefore n(y) is the generic point of ~/. 9 Proof of Proposition 2.6.4. -- It is clear that ~-I(T) C 7~-I(T). To verify the inverse inequality, we may assume that T = ~/. Since T is constructible, it contains an every- where dense Zariski open subset of ~/and, in particular, all of the generic points of ~t. From Lemma 2.6.5 it follows that 7~-1(T) = ~r 9 2.6.6. Corollary. -- A constructible set T is open (resp. cloud, resp. everywhere dense) in ~,r if and only if 7~-I(T) is open (resp. closed, resp. everywhere dense) in ~r 9 2.6.7. CoroUary. -- A morphism ~ : .~ ~ ~/ between schemes of locally finite type over YE is separated if and only if ~" : ~.~" -~ ~t,~ is separated. Proof. -- Let A : .~ ~ ~ /~ ~ be the diagonal morphism. If ~ is separated, then A is a closed immersion, and it follows that so is An. Assume that ~ is separated. It suffices to verify that the set A(~) is closed in ~e x~ .~. This follows from Corol- lary 2.6.6 because A(~ e) is a constructible set and its preimage in ~e~ � ~.~ is closed. 9 2.6.8. Proposition. -- For a morphism ~ : .~ -+ ~t between schemes of locally finite type over ~F, one has 7r-I(~(.~))= ~,,(.~e~). Proof. -- From the construction of the analytification it follows that the statement is true when ~e is an open subscheme of ~t and, if { ~t i }i e i is [an open covering of ~t, then { ~t~ }~e x is an open covering of ~t~. Hence the situation is reduced to the case when ~/---- Spec(B) and ~e ___ Spec(C) are affine schemes of finite type over ~. The inclusion ~'~(.o~ ~) C 7~-1(~(~e)) is evident. To verify the converse inclusion, it suffices 54 VLADIMIR G. BERKOVICH to show that if q is a prime ideal of C, then any multiplicative norm on B/p, where p is the preimage of q in B, extends to a mulfiplicative norm on C/q. But this follows from the well known fact that a valuation on a field can be extended to a valuation on any bigger field. 9 The following statement is proved in the same way as its particular case when = .//(k) (see [Ber], 3.4.7). 8.6.9. Proposition. -- Let ~ : .o~ -+ ~ be a morphism of finite type between schemes of locally finite type over Y~. Then ~ is proper (resp. finite, resp. a closed immersion) if and only if 9~n possesses the same property. 9 2.6.10. Proposition. -- Let q~ : .~e = Spec(C) ~r = Spec(B) be a finite morphism between affine schemes of finite type over ~. Let z ~ Y~" and y e ~1'~ be points with ~(z) = re(y) = y, and let ~'~-l(y)={zl,...,z,} and ~"-~(y) nT:-x(z)={zx,...,z,,}, m~n. Then there is an isomorphism of rings t~1 i=ra+l where (~,,~)~ is the localization with respect to the complement of the prime ideal of C cor- responding to the point z. Proof. -- If V is an affinoid domain in ~/~, then for its preimage W in .o~ e~ one has cg w -~ 6~v| B C. Since q~'~ is finite, from Lemma 2.1.6 it follows that 0~r t/| B C-~ 1-I~= 10~van, z/. The ring dT~,.,~ | r is the localization of 0~,,,| C with respect to the complement of ga,. IfT~(z~) --~ z, then the ring ~,. ,i does not change under this localization. The required statement follows. 9 w 3. Etale and smooth morphisms 3.1. O uas|~-~te morphisms 3.1.1. Definition. -- A morphism of k-analytic spaces ~ : Y ~ X is said to be finite at a pointy e Y if there exist open neighborhoods r ofy and q/of ~(y) such that ~ induces a finite morphism r ~ o~; ~ is said to be quasifinite if it is finite at any point y e Y. It follows from the definition that quasifinite morphisms are locally separated and closed. 3.1.8. Lemma. -- If a morphism q~ : Y -+ X is finite at a point y e Y, then the neigh- borhoods "f" and ~ from the Definition 3.1.1 can be found arbitrary small. Proof. -- We may assume that the morphism ~ is finite. Let x = q~(y) and let ~-1(x) = {Yl =Y, Y2, .- .,Y, }. Since the map I Y I ~ [ X is I compact, we can find a ]~TALE COHOMOLOGY FOR. NON-ARCHIMEDEAN ANALYTIC SPACES 55 sufficiently small open neighborhood q/ of x such that q~-l(~)= He,=~ $/], where y~e~ and ~n~=O for i#j. It follows that all of the morphisms r are finite, and the neighborhood ~ ofy is sufficiently small. 9 3.1.3. Corollary. -- Quasifinite morphisms are preserved under compositions, under any base change functor, and under any ground field extension functor. 9 We remark that if ~ : Y -+ X is a quasifinite morphism and the space X is good, then Y is also good. Q uasifinite morphisms between good k-analytic spaces can be characterized as follows. 3.1.4. Proposition. -- Let ~ : Y -+ X be a morphism of good k-analytic spaces, and let y e Y and x -~ ~(y). The following are equivalent: a) ~ is finite at y; b) there exist sufficiently small affinoid neighborhoods V of y and U of x such that ~ induces a finite morphism V ~ U; c) the point y is isolated in the fibre r and y Elnt(Y/X). Proof. -- The implications a) :~ c) and b) :~ c) are trivial. The implication b) :~ a) follows from the following Lemma. 3.1.5. Lemma. -- Let ~ : Y -+ X be a morphism, and let V C Y and U C X be affinoid subdomains such that ~ induces a finite morphism + : V ~ U. Then Int(V/Y) = +-l(Int(U/X)). In particular, q~ induces a finite morphism Int(V]Y) -+ Int(U]X). Proof. -- From [Ber], 3.1.3, it follows that Int(V/X) = Int(V]Y) n Int(Y/X) = Int(V/Y). On the other hand, Int(V/X) ---- Int(V/U) n hb-l(Int(U/X)) = hb-l(Int(U/X)). There- fore Int(V/Y) = ~b-l(Int(U/X)). 9 To verify the implication c) ~ b), it suffices to assume that X = ~r and Y = ~(~) are k-affinoid. Furthermore, since X and Y are compact, we can decrease them and assume that q~-l(x) -- {y }. Since y e Int(Y/X), there exists an admissible epimorphism : off {r~-lT1, ..., r~-lT,} ~ such that I ~(T~) (y) [ < r~, 1 ~< i ~< n. For any affinoid neighborhood U of x, n induces an admissible epimorphism '~v: ~r r~-I T1, " ", r:l T, } ~ ~,-1~. If U is sufficiently small, then ?-I(U) is a sufficiently small affinoid neighborhood of the point 2. Therefore we can find sufficiently small U such that [ n(T~) (y')[ < r 56 VLADIMIR G. BERKOVICH for all points y' e ~-~(U). This means that the induced morphism of k-affinoid spaces ~-I(U) -~ U is closed. By [-Ber], 2.5.13, the latter morphism is finite. 9 3.1.6. Corollary. -- Ira morphism of good k-analytic spaces ~ : Y -+ X is finite at a point y e Y, then Oy,, is a finite ~x.,~,~-algebra. 9 Recall that a morphism of schemes ~? : ~t _+ ~ is called quasifinite if any point y e~ is isolated in the fibre ~-~(x) = Y/x, where x = ~(y) (see [SGA1], 1.2). 3.1.7. Corollary. -- A morphism ~ : ~ -~. ~ between schemes of locally finite type over = Spec(z~r where ~ is a k-affinoid algebra, is quasifinite if and only if the corresponding morphism ~ :.o~ ~r is quasifinite. Proof. -- For any point y e y/M, there is an isomorphism of ~t~(y)-analytic spaces Therefore our assertion follows from the fact that the morphism ~ is closed (Pro- position 2.6.1). 9 3.1.8. Proposition. -- Let ~ : Y -+ X be a morphism of k-analytic spaces, and let y e Y and x = ~(y). Then the following are equivalent: a) ~? is finite at y; b) there exist analytic domains X1,...,X, CX such that x~Xl ~... c~X,, X 1 u ... w X, is a neighborhood ofx and the morphisms ?-l(X~) -+X~ arefinite aty; c) there exist affinoid domains VI,...,V, CY and U1,...,U,C X such that yeV lr~.., nV,, VlW... uV, and Uitg... wU, are neighborhoods ofy and x, respectively, ~(V~) C U~, and the induced morphisms ~ : Vi -+ U~ and ~ : V~ n V~ -+ U~ n Uj are finite at y. We remark that if the spaces X and Y are separated, then the finiteness at y of all the morphisms ~ from c) implies the same property for the morphisms r Indeed, in this case Ui r~ Uj and V~ rn Vj are affinoid domains and, by Proposition 3.1.4, it suffices to verify that y e Int(Vi r~ Vj/Ui n U~). If V~ := 9~-l(U~ n Uj), then y e Int(VJU~ r~ U~), and therefore y e Int(V~i n VJU~ n Uj). It remains to note that V~ n V~.~ = V~ c~ Vj. Proof. -- The implication a) :~ b) is trivial. Furthermore, we remark that all three properties remain true if we replace X and Y by sufficiently small open neigh- borhoods of the point x andy, respectively. In particular, we may assume that X and Y are Hausdorff. b) :* c). We may assume that X~ = U~ are affinoid domains. Then we can find affinoid neighborhoods V~ of y in ~-I(U~) such that the induced morphisms I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 57 ~ : V~ -+ U~ are finite at y. Furthermore, shrinking X and Y, we may assume that the morphisms q~ are finite and q~-~(x) ={y}. Thus we get a morphism +:V -+ U, where V----V 1 ~... u V, and U = U~ u... ~ U, are compact analytic neigh- borhoods ofy and x, respectively. It follows that we can find on open neighborhood q/ of x in X with q/C U such that ~e" := +-1(~//) is an open neighborhood ofy in Y and, for every 1 ~< i~< n, +-~(U~ n ~//)C V~. If now Ui is an affinoid neighborhood of x in U~ c~ q/, then V, := +- ~(Ui) is an affinoid neighborhood of y in V; n ~ and the induced morphism ~?, : V~ ~ U~ is finite. Since V, ~ V~ = q~-~(U, n U~), the induced morphisms ?i~ : V~ c~ Vj -+ U i n U~. are finite. Hence, q~ satisfies c). 3.1.9. Lemma. -- If Y is an analytic domain in X and the canonical morphism Y ~ X is quasifinite, then Y is open in X. Proof. -- Since quasifinite morphisms are closed, Int(Y/X)= Y. By Proposi- tion 1.5.5 (ii), Int(Y/X) coincides with the topological interior of Y in X. It follows that Y is open in X. 9 c) ~ a). We can shrink all the affinoid domains V~ and assume that q~-l(x) = {y }. Then for sufficiently small affinoid neighborhoods U~ of x in U~, q~-~(U~) and q~l(U~ c~ U;) = q~-l(U~) n q~/l(U'j) are sufficiently small neighborhoods of y in V and V~ A V~, respectively. Thus, we can shrink all the affinoid domains U~ and assume that all of the morphisms q~ and qhj are finite. Furthermore, the morphism qh induces finite morphisms q~ : V~ A Vj -+ U~ c~ Uj and V~j:=-q0~-a(U~ nUj)~U~ n Uj, and therefore the canonical embedding of special domains V~ n Vj ~ V~j is finite. From Lemma 3.1.9 it follows that V~ A Vj is open in V~j, and therefore V~; = (V~ c~ Vj) 11 W~j for some special domain W~j C V~. We get q0-~(U~) = V~I_I W~, where W~ = U~.~W~. Since y sVt n ... nV,, we have x r ~(W~), and therefore we can find, for each 1 ~< i ~< n, an affinoid neighborhood U~ ofx in U~ such that U~ c~ qg(W~) = O. The latter implies that q~-l(Us = V~ := q~-l(U;). I t I t Hence, the morphism V~ u... u V, ~U1 w... w U, is finite, and the required statement follows. 9 3.1.10. Corollary. -- A morphism of k-analytic spaces ~:Y ~ X is finite at a point y ~Y if and only if the pointy is isolated in the fibre q0-1(~(y)) andy ~ Int(Y/X). Proof. -- The direct implication is clear. Suppose that y is isolated in q~-l(q0(y)) and y e Int(Y/X). We can shrink Y and assume that q0 is closed. Let U1, ..., U, be affinoid domains in X such that q~(y) e U1 n ... n U, and U1 u ... u U, is a neigh- borhood of q0(y). From Proposition 3. 1.4 is follows that ~?-~(U~) -+ U i are closed morphisms of good k-analytic spaces. In particular, the property b) of Proposition 3.1.8 holds. It follows that ? is finite at y. 9 58 VLADIMIR G. BERKOVICH 3.2. Flat quasitln;te morphisms A morphism of good k-analytic spaces ~ : Y -+ X is said to be flat at a point y e Y if #y, ~ is a flat d~x. ~l~)-algebra. 9 is said to be fiat if it is flat at all points y ~ Y. 3.2.1. Proposition. -- A finite morphism ofk-affinoid spaces ~ : Y = ~g ( ~) ~ X = .~r ( sY) is flat at a point y e Y if and only if ~y is a flat d~x-algebra , where x = ~?(y), In particular, r is flat if and only if ~ is a flat d-algebra. Proof. -- If ~y,~ is flat over Ox,,, then it is also flat over ~ex' by Theorem 2.1.4. Thus, the ring 0y,~ is faithfully flat over ~ey and is flat over alex. It follows that ~Oy is flat over de. Conversely, suppose that ~ey is a flat alex-algebra. Then the ring Ox, x| ~ ~y is flat over Ox, x. If ~-~(x)= {y~ =Y, Y2,...,Y,}, then d~ x, 9 | ~ = II",=~ ~Y, ~i (Lemma 2.1 .6). But the ring d~x, ~ | ~ey is the localization of 0x, ~ | ~ with respect to the complement of ~ay, and therefore, is a direct product of the same localizations of the rings ~y,,~. Since the ring 0y,, does not change under this localization, it is a direct summand of the flat (9x,,-module d~x,,| ~y. It follows that 0v, ~ is a flat #x, ,-algebra. 9 3.2.2. Corollary. ~ A morphism of good k-analytic spaces ~ : Y -+ X is flat quasifinite if and only if for any point y e Y there exist affinoid neighborhoods V of y and U of x = ?(y) such that ~(V) C U and g~v is a flat finite Mc-algebra. Proof. -- The converse implication follows from Proposition 3.2.1. Suppose that ~ is flat quasifinite. By Corollary 3.1.6, O~,u is a finite 0x,,-algebra. Therefore there is an isomorphism ~,, --% Oy, u. It is clear that it comes from a homomorphism M~ -+ ~v for some affinoid neighborhoods V ofy and U of x such that ~0 induces a finite morphism +:V-+ U. The homomorphism considered is related to a homomorphism of sheaves 0~ -+ +.(Ov). The supports of the kernel and cokernel of the latter homo- morphism are Zariski closed in U and do not contain the point x. It follows that one can decrease U and V such that 0~--% +.(Ov) , i.e., ~4~-% ~v. Hence +:V--~U is a flat finite morphism. 9 3.2.3. Proposition. -- Let ~ : Y ~ X be a finite morphism of k-analytic spaces, and let y e Y and x = ~(y). Then the following are equivalent: a) there exist affinoid domains V~,..., V,C X such that x e Vx c~... c~ V,, V~ ~ ... ~ V, is a neighborhood of x and ~-a(V~) -+V~ are flat at y; b) for any affinoid domain x e V C X, ?-I(V) -+ V isflat aty. Proof. -- The implication b) :~ a) is trivial. Suppose that a) is true. Then b) is true for any V that is contained in some V,. Assume that V is arbitrary. Replacing V by a small affinoid neighborhood of x in V, we may assume that V C V x tJ ... u V.. By I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES Lemma 1.1.2 (ii), there exist affinoid domains U1, ..., U~ such that V = U1 w ... u U,, and each U# is contained in some V,. By the first remark, if x e U~, then the finite morphism cp-l(U~) ~ Uj is flat at y. Therefore the required fact follows from the fol- lowing lemma. 3.2.4. Lemma. -- A finite morphism of k-affinoid spaces c? : Y ~ X is flat at a point y E Y if and only if there exists a finite affinoid covering { V~ }1 <<.~ <,, of X suck that, for each i with ,?(y) e V,, the induced finite morpkism 9-1(V,) -->V, is flat at y. Proof. -- Shrinking X, we may assume that q)(y) e V 1 n ... n V,. Furthermore, from Proposition 3.2.1 it follows that we can shrink X and Y and assume that all of the morphisms q)- I(V,) --> V, are flat. It suffices to show that ~ is flat after an extension of the ground field. Therefore, we may assume that the valuation on k is nontrivial, and all V, are strictly k-affinoid. (Then X and Y are also strictly k-affinoid.) If x ~ X0, then Ox, x = Ov~.x for any V, that contains x. It follows that ~ is flat at all points of Y, i.e., ~? is flat. 9 Let r : Y -+ X be a quasifinite morphism of k-analytic spaces. 3.2.5. Definition. -- The morphism q0 is said to beflat at a pointy e Y if there exist open neighborhoods ~r ofy and q/of ~0(y) such that 9 induces a finite morphism ~ -+ q/ that possesses the equivalent properties of Proposition 3.2.3; 9 is said to be flat if it is flat at all points of Y. From Proposition 3.2.3 it follows that if a quasifinite morphism 9 : Y ~ X is flat at a point y z Y, then the neighborhoods q/" and q/can be found sufficiently small. 8.2.6. Corollary. -- Flat quasifinite morphisms are preserved under compositions, under any base change functor, and under any ground field extension functor. 9 3.2.7. Proposition. -- A flat quasifinite morphism ~?:Y-+ X is an open map. Proof. -- We may assume that X and Y are k-affinoid. Lety ~Y and x = q)(y). By the proof of Corollary 3.2.2, there exist affinoid neighborhoods V ofy and U of x for which there is an isomorphism ~v --% d~. In particular, the canonical homo- morphism dv ~ ~v is injective and finite. By [Ber], 2.1.16, the map + : v = u = is surjective. By Lemma 3.1.5, Int(V/Y)= +-l(Int(U/X)), and therefore q(Int(V/Y)) = Int(U/X). 9 3.2.8. Proposition. -- Let 9 : Y -> X be a quasifinite morphism. Then the set of points y ~ Y such that ~ is not flat at y is Zariski closed. 60 VLADIMIR G. BERKOVICH Proof. -- We may assume that ~ is a finite morphism of k-affinoid spaces. In this case Proposition 3.2.1 reduces the statement to the corresponding fact for morphisms of affine schemes. 9 3.~..9. Proposition. -- Let ~:Y-+ X be a closed morphism of pure one-dimensional good k-analytic spaces, and suppose that X is regular and Y is reduced. Then ~ is flat quasifinite if and only if q~ is nonconstant at any irreducible component of Y. Proof. -- The direct implication is trivial. Suppose that ~? is nonconstant at any irreducible component of Y. Then q0 has discrete fibres, and therefore is quasifinite, by Proposition 3.1.4. It follows that for any point y e Y we can find connected affinoid neighborhoods V ofy and U of ?(y) such that ? induces a finite morphism of k-affinoid spaces V-+ U. Then ~r is a one-dimensionai regular integral domain, the ring ~v is reduced, and the canonical homomorphism d v -+ ~v is injective. By [Ha2], III.9.7, ~v is a flat ~v-algebra. 9 3.2.10. Proposition. -- A morphism ~ : ~ __~ ~t between schemes of locally finite type over ~ = Spec(~r where dg is a k-affinoid algebra, is flat quasifinite if and only if the corres- ponding morphism ~ : ~ -+ ~ is flat quasifinite. Proof. -- Since ~tan and ~.~a" are faithfully flat over ~ and ~e, respectively, the converse implication follows. Assume that ~0 is fiat quasifinite. Then ?~n is quasifinite, by Corollary 3.1.7. Let z ~ ~", y = ~0~(z), z = ~(z), and y = z~(y). We may replace ~t and ~ by open affine subschemes of finite type over 5~. By Zariski's Main Theorem, there is an open immersion of ~3~ in an affine scheme ~ finite over Y/. Then ~L ~n -+ ~ean is also an open immersion, and ~e~,, is finite over ~". By hypothesis, ~. ~ is flat over ~/, y. Therefore 0~,,~|162 O~r,, is flat over O~,,,y. By Proposition 2.6.10, O~r~,~ is a direct factor of the above tensor product. It follows that ~a',-,, is fiat over 0~,, ~. 9 3.3. l~tale morphlsms We start this subsection with establishing basic properties of the sheaves of dif- ferentials that were introduced in w 1.4. Of course, the essential case is that of k-affinoid spaces. Let q~:Y = ~g(~)--~ X = ~g(d) be a morphism of k-affinoid spaces. In this case the sheaf ~Y/x is associated with the finite Banach ~-module f~e/~ = j/j2, where J is the kernel of the multiplication ~z : ~ | ~ --~ ~. Furthermore, let M be a Banach ~-module. An d-derivation from ~ to M is a bounded map D:N--~ M such that D(x -J-y) = Dx 4: Dy, D(xy) = x Dy -t- y Dx and D(d) = 0. The set of all d-derivations from ~ to M is a Banach ~-module with respect to the evident norm. It is denoted by Der~(~, M). For example, the mapping ~ ~J :x ~ 1 | x - x N 1 induces an d-deri- vation d : ~ ~ g]~/~. t~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 3.3.1. Proposition. -- (i) The finite M-module ~)~l.e is generated by the elements dx, x e d. (ii) For any Banach d-module M there is a canonical isomorphism of Banach d-module~ Hom~(fl~/~,, M) -% Der~,(d, M) (the left hand side is the set of all bounded d-homomorphisms). Proof. -- (i) Let T be the d-submodule off~a~/~ generated by the elements dx, x e d. Recall that the ring d is Noetherian, and all its ideals are closed. Since f~s~/d is a finite Banach d-module, T is closed in it. We claim that for any w e ~a/d and any r > 0 there exists an element t ~ T with I1 w -- t }1 < e" Indeed, let v be an inverse image of w in J. There exists an element ]~=lx~ y, ed| such that [Iv Y~=ax~| < e. Since F(o) = 0, II I:L x~y, ]] < ~. We have xi| , = ~ (x,| 1)(1 |174 1) + ,~x,y,| 1. ~=I i~l Therefore I1 v - (x,| 1) (l | --Y~| 1) qI .< max(I v- ll, ll The required claim follows. (ii) It is clear that the homomorphism considered is bounded. From (i) it follows that it is injective. Therefore it suffices to construct an inverse bounded homomorphism. Consider the following Banach d-algebra d * M. As a Banach d-module it is the direct sum of d and M. Its multiplication is defined as follows: (x, m) (y, n) = (xy, xn -+-ym). Let now D : d ~ M be an d-derivation. The bounded ~-bilinear mapping d � d --~d* M: (x,y) ~--~(xy, xDy) induces a bounded homomorphism of Banach ~r ?:d | d--+ d * M. The reasoning from (i) shows that q~(J) c M. Since M s : 0, the homomorphism ~ induces a bounded homomorphism of Banach d-modules f: f~/~, :J/J~-+ M. We have Dx :f(dx) for all x e d. That the correspondence D ~f is bounded follows from the construction. 9 3.3.2. Proposition. -- Suppose we are given a commutative diagram of morpkisms of k-analytic spaces Y ~> X S 62 VLADIMIR G. BERKOVICH (i) There is an exact sequence (i.i) If ~ is a closed immersion and d~o is the subsheaf of ideals in Ox. that corresponds to Y, then there is an exact sequence Proof. -- We may assume that X = ~'(M), Y = Jr'(&) and S---- J t'(~s are k-affinoid. (i) We have to show that the sequence of finite Banach ~-modules is exact (note that D~r | ~ = ~)a/~e | ~). For this it suffices to show that for any finite Banach ~-module M the induced sequence 0 -+ Hom~(D~/~r M) -+ Hom~(D~/~e , M) --~ Hom,(D~c/~e, M) is exact. But, by Proposition 3.3.1, the latter sequence coincides with the sequence 0--~ Der=r M) -+ Der,(~, M) -+ Der,(~, M) which is exact for trivial reasons. (ii) Let J be the ideal of ~r corresponding to J. We have to show that the sequence of finite Banach ~-modules j/j2 _~n D~c/~e |162 ~ -+ Da/~e -+ 0 is exact, where 8(x) = dx | 1. As above, it suffices to show that for any finite Banach ~-module M the sequence 0 -+ Der,(~, M) ~ Der,(d, M) -+ Hom~(J/J 2, M) is exact, but this is evident. 9 3.3.3. Proposition. -- Let ~ : Y -+ X be a morphism of k-analytic spaces. Then: (i) for any morphism of k-analytic spaces f: X'-+X, one has ~Y~lX'o =fd*(Dvo/xo), where f' is the induced morphism Y' = Y � x X' -+ Y; (ii) for any non-Archimedean field K over k, one has Dy,~/x, o =f~*(~o/xo), where f' is the induced morphism Y' = Y | K ~ Y. Proof. -- We may assume that Y = Jt'(~), X = d((~r and X' = d((~") from (i) are k-affinoid. In this case D~/x is defined by the finite Banach ~-module j/jz, where J is the kernel of the multiplication cg = ~ | 3~ _+ ~. Note that the exact admissible sequence 0 -+ J -+ (g -+ ~ -+ 0 is split. (i) If ~' = ~ | a", then Dy,/x, is defined by the finite Banach ~'-module J'[J'~, where J' is the kernel of the multiplication cg, = ~, | ~, _+ ~,. We have to show that J/JZ | ~' --%J'/J'~ (since J/J~ is a finite Banach ~-module, J/J= | ~' = J/J~ ~a~ ~'). The exact sequence 0-+J'-+ ~'-+ ~'-+ 0 is obtained from the above exact sequence by tensoring with ~r over ~. It follows that J' = J |162 ~r = J ~, cg, = jog, because J is a finite Banach Cg-module. Tensoring the exact sequence o o ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 63 with ~r over 5, we get an exact sequence j2@r 5'-+J' _+j/j2|162 ~,,_+ 0. Since j/j2 @r (#, = j/ja | 2@, and the image ofJ ~ @~ ~" in J' --- J~ coincides with j,2 _ jz 5, we get the required isomorphism. (ii) If = d | K and 2@' = 2@ | K, then ~Y'lx' is defined by j,[j,2, where J' is the kernel of the multiplication 5'= 2@' |162 2@'-+ 2@'. As above we get that J' = J | K ---- J~' and j,2 = j2 | K = j2 ~r Therefore j/j2 | K ---- j,/j,2. It remains to note that J/J~ ~ K = j/j2 | 2@'. 9 Let q~ : Y ---> X be a quasifinite morphism. 3.3.4. Definition. -- The morphism 9 is said to be unramified if flYo/xo = 0. It is said to be aale if it is unramified and flat. It is said to be unramified (resp. dtale) at a pointy e Y if there exists an open neighborhood Y/" ofy such that the induced morphism Y/"--> X is unramified (resp. 6tale). For example, if q~ is a local isomorphism at a pointy e Y (i.e., there exist open neigh- borhoods $: ofy and q/of r such that r induces an isomorphism ~ --% q/), then 9 is 6tale aty. Therefore if ~ is a local isomorphism (i.e., q~ is a local isomorphism at every point y e Y), then q~ is dtale. 3.3.5. Lemma. -- If ~ : Y ---> X is a quasifinite morphism of good k-analytic spaces, then the stalk of ~Yix at a point y ~ Y coincides with the module of differentials ~lx, where B = Oy.v, A ---- Ox, , and x = ~(y). Proof. -- We may assume that ~ is a finite morphism of k-affinoid spaces Y = ~r162 -+ X = ~r In this case 2@ |162 2@ -~ 2@ | 2@, and therefore Layt x is defined by the module of differentials ~2a/~, (regarded as a finite Banach 2@-module). The required statement easily follows from this. 9 3.3.6. Corollary. -- A quasifinite morphism of good k-analyt# spaces r X is unramified (resp. gtale) at a point y ~ Y if and only if Oy, ~/m, Oy. v is a finite separable extension of the field ~(x) (resp. and Oy, v #flat over Ox,,), where x = ~(y). 9 The following statement is straightforward from the definitions. 3.3.7. Proposition. -- Let ~:Y ~ X be a quasifinite morphism and y E Y. Then the following are equivalent: a) 9 is unramified at y; b) the support of ~/x does not contain y; c) the diagonal morphism A : Y ---> Y � x Y is a local isomorphism at y. 9 3.3.8. Corollary. -- Unramified (resp. gtale) morphisms are preserved under compositions, under any base change functor, and under any ground field extension functor. 9 3.3.9. Corollary. -- Let + : Z ~ Y and ~ : Y -> X be quasifinite morphisms and suppose that ,?~b is gtale and ~ is unraraified. Then ~b is ~tale. 64 VLADIMIR G. BERKOVICH Proof. -- The morphism + is a composition of the graph morphism F, : Z -+ Z � Y with the projection P2 : Z � x Y ~ Y- The first morphism is an open immersion because it is a base change of the open immersion Y --~ Y x x Y with respect to the evident mor- phism Z � x Y -+ Y x x Y, and the second one is 6tale because it is a base change of the 6tale morphism ~+:Z ~ X. 9 Let ~t(X) denote the category of 6tale morphisms U -+ X. Corollary 3.3.9 implies that any morphism in the category gt(X) is 6tale. 3.3.10. Proposition. --- Suppose that ~ : Y ~ X is a quasifinite morphism or is a closed morphism of good k-analytic spaces. Then the set of points y ~ Y such that q~ is not unramified (resp. gtale) at y is Zariski closed. Proof. -- We can replace Y by the complement to the support of the coherent 0y -module ~Y~/Xo and assume that ~YotX, = 0. In the second case the latter equality implies that ~ has discrete fibres, and therefore q~ is quasifinite, by Proposition 3.1.4. Hence our statement follows from Propositions 3.3.7 and 3.2.8. 9 3.3.11. Proposition. -- A morphism q~ : ~ -+ ~r between schemes of locally finite type over ~ = Spec(d), where off is a k-affinoid algebra, is unramified (resp. 6tale) if and only if the corresponding morphism ~: ~.~ -+ ~/~ is unramified (resp. gtale). Proof. ~ The unramifiedness statement follows from Corollary 3.3.7 and the simple fact that ~a, an/~ .... (~/~)'~. The 6taleness statement now follows from Pro- position 3.2.10. 9 3.4. Germs of analytic spaces A germ of k-analytic space (or simply a k-germ) is a pair (X, S), where X is a k-analytic space, and S is a subset of the underlying topological spaoe I X I" (S is said to be the under- l~ng topological space of the k-germ (X, S).) If S = { x }, then (X, S) is denoted by (X, x). The k-germs form a category in which morphisms from (Y, T) to (X, S) are the morphisms q~:Y-+X with q~(T)C S. The category k-fgerms we are going to work with is the category of fractions of the latter category with respect to the system of morphisms ~:(Y, T) ~ (X, S) such that ~ induces an isomorphism of Y with an open neigh- borhood of S in X. This system obviously admits a calculus of right fractions, and so the set of morphisms Hom((Y, T), (X, S)) in k fCerms is the inductive limit of the set of morphisms ~0 : 3r p ~ X with ~(T) C S, where ~r runs through a fundamental system of open neighborhoods of T in Y. (Such a morphism q~ : 3 r ~ X is said to be a repre- sentative ofa morphism (Y, T) ~ (X, S).) It is easy to see that a morphism (Y, T) -+ (X, S) is an isomorphism in k-fgerms if and only if it induces an isomorphism between some open neighborhoods of T and S. We remark that the correspondence X ~ (X, ] X [) induces a fully faithful functor k-offn -+ k-f~erms. I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 65 The category k-~'mns admits fibre products. Indeed, let (Y, T) ~ (X, S) and (X', S') -+ (X, S) be two morphisms. If ? : ~e- _~ X and f: q/' ~ X are their repre- sentatives, then (~e-� q/', 7~-1( T XS S')) is a fibre product of (Y, T) and (X', S') over (X, S), where rc is the canonical map [ ~e" � q/' [---~ 1~"[ � [ q/' [. Thus, for any morphism cp : (Y, T) -~ (X, S) and a point x e S one can define the fibre of ? at x in the category k-fferms as the fibre product (Y, T) � ~x. s~ (X, x) which is actually iso- morphic to the k-germ (Y, ?-l(x)), where q~-l(x) is the inverse image of x in T. In par- ticular, for a morphism of k-analytic spaces r : Y --* X and a point x ~ X, one has the fibre (Y, 9-1(x)) of ? at x in the category k-~erms. (Recall that in w 1.4 we defined the fiber Y~ of ? at x in the category .~r k of analytic spaces over k.) Furthermore, for a non-Archimedean field K over k there is a ground field extension functor k-~erms --* K-fferms : (X, S) ~ (X| K, ~-~(S)), where rc is the canonical map X ~ K ~ X. Similarly to offn~ one can define the category fferms~ of germs of analytic spaces over k (or simply germs over k). Its objects are pairs (K, (X, S)), where K is a non- Archimedean field over k and (X, S) is a K-germ. A morphism (L, (Y, T)) ~ (K, (X, S)) is a pair consisting of an isometric embedding K r L and a morphism of L-germs (Y, T) ~ (X, S) ~= L. As above, there is a fully faithful functor ~r ~ ~erms k : (K, X) ~* (K, (X, I X D)" For a k-germ (X,S), let Et(X,S) denote the category of the morphism (Y, T) ~ (X, S) that have an 6tale representative ? : ~e" ~ X with T = ?-I(S). It is clear that for X e k-..rCn there is an equivalence of categories ]~t(X)--% ~t(X, [X D. For a point x e X, let Ftt(X, x) denote the full subcategory of ~t(X, x) consisting of the morphisms (Y, T)--~ (X, x) that have an 6tale representative 9 : ~r X such that the morphism ~e-~ ?(~e-) is finite. (Equivalently, Ftt(X, x) consists of the mor- phisms (Y, T) ~ (X, x) with finite set T that have an 6tale separated representative ? : q/" --0- X.) For a field K, let Ftt(K) denote the category of schemes finite and 6tale over the spectrum of K. 3.4.1. Theorem. -- Let X be a k-analytic space. Then for any point x ~ X there is an equivalence of categories Ftt(X, x) -% F&(~(x)). Proof. m Consider first the case when the point x has an affinoid neighborhood. Let F~t(~(x)) denote the category of schemes finite and ~tale over the affine scheme ~(x) = Spec(Ox,,). Then the functor considered is a composition of the three evident functors F&(X, x) -+ Ftt(~(x)) ~ F~t(~:(x)) ~ Fft(~(x)). The third functor is an equivalence of categories because the field r(x) is quasicomplete. The second one is an equivalence because the ring Ox,~ is Henselian. We now verify that the first functor is faithful. Let ?, h~ : (Y,)) -+ (X, x) be two morphisms that induce 9 66 VJ..ADIMIR G. BERKOVICH the same homomorphism Oy,~ -~ ~x,= (we do not need here the dtaleness of ~ and 4/). We may assume that X = ~(d) and Y = Jg(~) are k-affinoid, and q~ and 4/ are induced by two homomorphisms of k-affinoid algebras ~, ~:d-~ ~. Consider an admissible epimorphism T : k{ r~ -1 Tx, ..., r~ -1 T, } ~ d and set f = T(T~). Since the images off(f) and ~(f) in Oy, v coincide, we can find an affinoid neighborhood V ofy such that ~(~)Iv = ~(f)Iv- By [Ber], 2.1.5, the induced morphisms ~[v, ~lv: V --~ X coincide, and therefore the first functor is faithful. Furthermore, we claim that a morphism in Fft(X, x) that becomes an isomorphism in Fdt(SF(x)) is an isomorphism. Indeed, let : Y ~ X be an dtale morphism with x = q~(y) such that Oy,, ~ $x., is an isomorphism. We can shrink X and Y and assume that X = .~r and Y ----~r are k-affinoid, ~-l(x) ={y} and ~ is a free d-module. From Lemma 2.1.6 it follows that ~| ~Vx, = -% ~Py,~, and therefore the rank of 5g over d is one, i.e., d--% ~. Finally, that any finite dtale morphism over /~'(x) comes from an 6tale morphism over (X, x) is obtained by the construction from the proof of Theorem 2.1.5. That the first functor is fully faithful now follows from the fact that any morphism in the categories Fdt(X, x) and Fdt(aq(x)) is dtale. Suppose now that the point x is arbitrary. We may assume that the space X Hansdorff, and we take aflinoid domains U1, 9 9 U, such that x ~ U1 r3 ... t3 U, and Ux u... u U, is a neighborhood of x. First we verify that the functor considered is faithful. Indeed, let q~ : Y -~ X and 4/: Z --~ X be two 6tale morphisms with ~- ~(x) = {y } and q2--1(X)={Z}, and suppose that f,,g: Z--*Y are two morphisms over X with f(z) = g(z) =y that give rise to the same embedding of fields ~(y) ,--~W(z). By the first case, we can find for each 1 <~ i<~ n an affinoid neighborhood Wi of z in ~-~(U~) such thatf[wi = g Iwi" Then the analytic domain W = W x u ... u W, is a neighborhood of the point z and f[ w = g[w. It follows that the morphism from (Z, z) to (Y,y) induced by f and g coincide. In the same way one shows that a morphism in Fdt(X, x) that becomes an isomorphism in F6t(o~(x)) is an isomorphism. Since any morphism in the category Fdt(X, x) is dtale, to prove the theorem it remains to show that the functor considered is essentially surjective. Let K be a finite separable extension of the field ~r176 By the first case, we can shrink all U, and find tinite dtale morphism q~:V,--~U~ such that q~-X(x) -={y~} and there are isomorphisms of fields K ~ o.~(y~) over o~(x). (We fix such isomorphisms.) Suppose first that X is separated at x. Then we may assume that X is separated, and therefore U~ ~ U~. are affinoid domains. Setting V~ = q~-~(U~ c~ U~), we have two finite dtale morphisms V~-§ U~ n Ui and V~ ~ U~ n Ui and, for the points yi~e V~j and y~ ~ Vii , an isomorphism of fields ~O(y~) __%~,,(y~) over ~(x) induced by the isomorphisms K-~.,Y'(y~) and K--%o~(y~). By the first case, we can shrink all Ui and assume that there exist isomorphisms v~: V~t ~ Vi~ over X that give rise to the above isomorphisms of fields. By construction, V, = V~ and v~(V~ c~ V~) = V~ c~ V~. We now can shrink all U~ and assume that v, = ~ o ,~ on Vi~ c~ V,. By Proposi- tion 1.3.3, we can glue all Vi along Vii and get a k-analytic space Y with a morphism I~TALE COHOMOLOGY FOR NON-ARGHIMEDEAN ANALYTIC SPACES 67 q~ : Y-+ X. By Proposition 3.1.8, the morphism r is finite at the pointy (that corres- ponds to the points y~). It is clear that the point y is not contained in the support of f~Yorx,, i.e., q~ is unramified at y. It is flat at y, by Proposition 3.2.3. Suppose now that X is arbitrary (and Hausdorff). The intersections U~ n Uj are not affinoid now, but they are separated compact k-analytic spaces, and therefore we can apply the above construction using the fact that everything is already verified for separated spaces. The theorem is proved. 9 3.4.2. Corollary. -- Let ~:Y-+ X be a morphism of analytic spaces over k, and let y ~ Y, x : ~(y). Suppose that the maximal purely inseparable extension of Jr(x) in ~(y) is dense in .r Then the correspondence U ~-* U x x Y induces an equivalence of categories F~t(X, x) -~ F~t(Y,y). 9 3.5. Smooth morphlsms For a k-analytic spaces X we set A~- = Ad X X (the d-dimensional affine spaces over X). 3.5.1. Definition. -- A morphism of k-analytic spaces 9:Y-+ X is said to be smooth at a pointy ~ Y if there exists an open neighborhood ~e" ofy such that the induced morphJsm ~r ~ X can be represented as a composition of an ~tale morphism r --* A a with the canonical morphism A~x ~ X; r is said to be smooth if it is smooth at all points y ~ Y. If the canonical morphism X -~.a(t'(k) is smooth, then X is said to be smooth. We remark that the number d is equal to the dimension of ~ at the pointy, i.e., to the dimension of the fibre Y,, where x = q~(y), at the point y. If this number is inde- pendent of y, we say that r is of pure dimension d. For example, smooth morphisms of pure dimension 0 are exactly ~tale morphisms. We remark that smooth morphisms are locally separated and closed. The following proposition follows easily from the definition of smooth morphisms and Corollary 3.3.8. 3.5.2. Proposition. -- Smooth morphisms are preserved under compositions, under any base change functor, and under extensions of the ground field. 9 3.5.3. Proposition. -- Suppose we are given a commutative diagram of morphisms of k-analytic spaces Y ~> X (i) If ? is (tale, then e?o (xolso) --~ ~Y~/so. (ii) If f and g are smooth and 9~(f~xo/so) ~ f2~o/Sa, then q~ is gtale. Since ~s~dx ~ is a free ~)s~ -module of rank d and, if X is good, the canonical mor- phism Aax -+ X is flat, then the statement (i) implies 68 VLADIMIR G. BERKOVICH 3.5.4. Corollary. -- Let e? : Y -+ X be a smooth morphism between good k-analytic spaces. Then ~ is flat, and flY/X is a locally free Or-module whose rank at a point y ~ Y is equal to the dimension of ~ at y. 9 8.5.5. Lemma. -- Suppose we are given a cartesian diagram Y *>X T, T, Y' r X' where ~ is a G-locally closed immersion and f is flat quasifinite. Then f"(,A/'ro/xo) = ~"ratx~" Proof. -- We may assume that all the spaces are k-affinoid, ~ is a closed immersion, andfis a finite morphism. Let X = Jt'(~r Y = ,/t'(~), X' = ~t'(a") and Y' = ~r where &' = & | ~r One has exact admissible sequences 0 ~J ~ ,W ~ ~ ~ 0 and 0 ~J' -+ ,~r -+ ~' ~ 0. Since ~r is a flat finite ~r then the second sequence is obtained by tensoring of the first one with d' over ~r In particular, J' = J ~ a~r = J~r It follows that J'/J'* : J/J~ |162 ~' : J/J* | ~'. 9 Proof of Proposition 3.5.3. -- (i) Consider the diagram X ~z/~ > X X s X y aY~ y � Y > Y � Y where Y � x Y is identified with the fibre product of X and Y x s Y over X � s X. The morphism d~ is fiat quasifinite. By Lemma 3.5.5, the conormal sheaf of the G-locally closed immersion Y � x Y ~ Y � s Y coincides with +"(f~xo/s~). Since At/x is an open immersion, then ~ro/so = ~'(~xo/so)" (ii) Consider first the case when the space S is good. (Then X and Y are also good.) From the exact sequence 3.3.2 (i) it follows that f~r/x = 0, and therefore the mor- phism q~ has discrete fibres. By Proposition 3.1.4, q~ is quasifinite. Since flr/x = 0, it is unramified. Let y E Y, x = q~(y) and s = g(y). We have to verify that Or, , is a flat Ox, z-algebra. Suppose first that [o~~ : k] < oo. Since Or. ~ and Ox, = are flat Os, ,-algebras (Corollary 3.5.4), then, by Corollary 5.9 from [SGA1], Exp. IV, it suffices to verify that Or,,/m , 0~, v is a flat 0x, Jm . 0x,=-algebra. Since [~(y) : k) < oo, we have 0r, Jm, 0r, v = 0y,,~ and 0x,,/m, 0x,= = Ox, ,,. Therefore we may assume that S = .~r In this case the ring 0x, = is regular and, in particular, normal. Since 0y, v is a finite unramified 0x,=-algebra, it suffices by Theorem 9.5 (ii) from [SGAI], Exp. I, to verify that the canonical homomorphism 0x, = ~ 0r, y is injective or, equivalently, that 0x, ~ and 0r, u have the same dimensions. But these dimensions are equal to the I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 69 ranks of ~x~ and D~ at the points x and y, respectively. Since ~'(~x~) ~ ~, they are equal. Suppose now that the pointy is arbitrary. Let K be a big enough non-Archimedean field such that there exists a pointy' ~ Y' = Y| K with [~f~(y') : K] < oo and ~(y') =y, where ~ is the canonical mapping Y' -+ Y. From Proposition 3.3.3 it follows that ~"(D.x,/s,)-~,/s, , where ?' is the induced morphism Y'-+X' =X| By the previous case, Or,,r is a flat 0x,.,-algebra, where x'= ~'(y'). By Corollary 2.1.3, Ox',.' and 0~,r are faithfully flat over Ox, , and 0r,~, respectively. It follows that 0~,. is flat over Ox,.. Consider now the general case. If U is an affinoid domain in S, then, by the first case, the induced morphism g-z(U) ~f-~(U) is dtale and, in particular, it is quasifinite. From Proposition 3.1.8 it follows that the morphism ~ is quasifinite. This implies immediately that it is dtale. 9 3.5.6. Corollary. -- In the situation of Proposition 3.3.2 (i) suppose that ~ is smooth. Then there is an exact sequence 3.5.7. Corollary. -- Let q~ : Y ~ X be a smooth morpkism, and let f: Y ~ A~x be an X-morphism defined by some functions fx, ... ,f~ ~ O(Y). Then f is gtale at a point y ~ Y if and only if for some affinoid domain U C X that contains the point x = ~(y) the elements dfl, ..., dfa form a base of ~,_~u~m at y. Proof. -- The direct implication is trivial. Suppose that dfa, ..., dfa form a base of D~-x~u~tv at y. By Proposition 3.5.3, the induced morphism ~-a(U) -~A~ is dtale at y. In particular, the point y is isolated in the fibre f- ~ (f(y)). From Proposition 3.1.4 it follows that for any affinoid domain V C X that contains the point x the induced morphism ?-~(V) ~ A~ is finite aty, and therefore, by Proposition 3.1.8, the morpbismf is finite aty. It is dtale because the elements dfx, ..., dfa form a base of ~-a~v)/v at y for any V as above. 9 3.5.8. Proposition. -- A morphism ? : ~ ~ ~ between schemes of locally finite type over Y" = Spec(d), where d is a k-affinoid algebra, is smooth if and only if the corresponding morphism ~*~ : ~,n ~ q/,~ is smooth. Proof. -- The direct implication follows from Proposition 3.3.11. Suppose that ~'~ is smooth. By Corollary 3.5.4, ~.n is flat, and X2~/~an is a locally free 0~..n-module. Since the morphisms ~/'~ ~ q/and ~r.~ -+ Z are faithfully flat, it follows that ~ is flat. Since ~l~.a~/~.~ -- (~./~) , it follows that ~2~./. is a locally free 0~.-module. Therefore is smooth. 9 70 VLADIMIR G. BERKOVICH 3.5.9. Proposition. -- Suppose we are given a commutative diagram of morphisms of good k-analytic spaces Y ~> X where ~ is a closed immersion and f is smooth. Then the following are equivalent: a) g is smooth; b) for any point y ~ Y there exist an open neighborhood X 1 of y in X and an ~tale morphism h : X x -+ A~ over S such that Y1 = Y n X 1 is the inverse image of the closed k-analytic subset of A~ defined by the equations T1 ..... T, = 0 (T1, ..., T a are the coordinate functions on M). Proof. -- The implication b) => a) is trivial. Suppose that g is smooth. Then the Or-modules q~*(g)x/s)= f~x/s| 0y and f~Y/s from the exact sequence 3.3.2 (ii) are locally free. We may decrease X and assume that they are free. Since the ele- ments dh, h ~O(X), generate Dx/s over Ox, then we can find he+l, ...,h a ~0(X) such that the restrictions ofdh~+l, . .., dh a to Y form a base of/)y/s. After that we can decrease Y and find hi,..., he ~J(X) such that dhl, ...,d ha form a base of f~xls, where is the subsheaf of ideals in 9 x that corresponds to Y. By Corollary 3.5.7, the induced morphism h:X-+ A~ is Etale. Let Z be the closed k-analytic subset of A~ defined by the equations TI ..... T~ = 0, and let Y' be the inverse image of Z in X. By construction, Y is a closed k-analytic subset of Y'. Corollary 3.5.7 implies that the induced morphism Y -+ Z is Etale. Therefore the closed immersion i : Y -+ Y' is dtale (Corollary 3.3.9). Since it is an open map (Proposition 3.2.7), it follows that we can decrease X and assume that i is a homeomorphism. Finally, since the sheaf i.(0y) is a locally free ~y,-module and the homomorphism Oy, ~ i.(Oy) is surjective, we have 0y,--% i.(0y). It follows that i is an isomorphism. 9 3.5.10. Corollary. -- In the situation of Proposition 3.3.2 (ii) suppose that f and g are smooth. Then there is an exact sequence o -+ o. In particular, JG/,r is a locally flee Oyo,module whose rank is the codimension of Y in X. 9 3.6. Smooth elementary curves In this subsection we recall some results from [Ber] on the structure of the k-analytic curve X----~ associated with a smooth geometrically connected projective curve over k of genus g/> 0, and we recall the Stable Reduction Theorem of Bosch and Liitkehbomert from [BL] which is actually the most important ingredient in the study ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 71 of X. Furthermore, we introduce the notion of an elementary triple (X, Y, x) where Y is an open neighborhood of a point x e X. (A k-analytic curve Y and pairs (X, Y) and (Y, x) for which such a triple exists will be called elementary.) And we show that any point of a smooth k-analytic curve has, after a finite separable extension of k, an elementary open neighborhood. First we consider the case of trivial valuation on k because it is very simple. But we remark that everything considered in the case of nontrivial valuation has the same meaning in the trivial valuation case. Thus, suppose that the valuation on k is trivial. Then points of X are of the following three types. First, there is a canonical embedding of the set ~0 of closed points of in X, ~0 -%x0={x~xl[~ :k]< oo}. For x~X 0 one has Ox,~=O~r,x and !r ---- Y~(x) = k(x) (x is the image of x in 5~). Furthermore, there is a generic point which corresponds to the trivial valuation on the field of rational functions k(3V). For this point x one has Ox, . = K(x) = 3r ----k(x) -~ k(s (The one-element set { x } is denoted by A(X) and called the skeleton of X.) Finally, for any closed point a there is an interval which connects a with the generic point and which is parametrized by the unit interval [0, 1]. Namely, the point x associated with a number 0 < r < corres- 1 ponds to the valuation on k(5~') which takes the value r E~la) :kj on a local paremeterfat a, i.e., [f(x) [ = t~(a' r :kl. This point x is denoted by p(E(a, r)). One has Ox, . ~- 1r = 5Of(x), and this field coincides with the fraction field of the ring Oa.a. We also set E(a, r) = {y e X [If(Y)[ ~< rt*'~':kJ }, D(a, r) = {y e X [if(Y)[ < rr*"):k~} and B(a; r, R) = {y e X I rE*(~' :k, < [f(Y) l < Rt*(""k' }. The topology of X induces the usual topology on each of the intervals, and a basis of open neighborhoods of the generic point is formed by sets of the form X\U~ 1 E(a~, r~), where ai~X 0 and 0< ri< 1. Let Y be an open neighborhood of a point x ~ X. We say that the triple (X, Y, x) is elementary if one of the following is true: a) g=O, x-=aeX(k) and Y=D(a,r), where.0<r< 1; b) g-=O, x =- p(E(a, r) ), where aeX(k) and 0<r< 1, and Y=B(a;r',r"), where O<r'<r<r"< 1; c) x is the generic point of X and Y = X\II~=I E(ai, q), m>_. 1, where a~ eX(k) and 0< r~< 1. It is clear that for any open neighborhood Y of an arbitrary point x e X one can find a finite separable extension K of k and an open subset Y" C Y' = Y | K such that the point x has a unique preimage x' in Y" and the triple (X', Y", x') is elementary where X' = X | K. Suppose that the valuation on k is nontrivial. As above, one has Y'0 ~ X0 and K(x) =:~(x) ---- k(x) for x ~ X 0. But in this case the set X 0 is everywhere dense in X. For a point x e X\X 0 one has Ox, ~ ---- !r 72 VLADIMIR G. BERKOVICH Consider first the case ~ = p1. Let a E A~ and r > 0. Then the real valued function on k[T] f ~ max I O,f(a) l r', i %+~ 0~ is the operator ~ ~ , is a multiplicative norm, and therefore it defines a point x s A 1 which is denoted by p(E(a, r)). The notation tells that the point depends only on the closed disc E(a, r) ={y A11 I/(y)l ,< : l, wheref = T" + ~1 T~- 1 + ... + ~. is the monic generator of the maximal ideal ofk[T] which corresponds to a. (We also define the open disc D(a, r) = (yeAXllf(Y)l < r ~} and the open annulus B(a; r, R) = {y e/l, 1 I r" < If(y)l < R" }. We remark that the radius r of the disc E(a, r) does not depend on the choice of the center a. If r e %/I k* ] (such a point is said to be of type (2)), then the extension M'(x)/k is finitely generated of transcendence degree one, and the group [ o~f'(x)* I/1 k" I is finite. If r ~ A/I k* I (such a point is said to be of type (3)), then the extension o~(x)[k is finite and the group [ M'(x)* I is generated by E ~(a)" I and r. A more general construction of points of A 1 is as follows. Let 8-----{ E } be a decreasing family of closed discs in 11,1. Then the real valued function on k[T] f ~-* inf If(P(E)) I is a multiplicative seminorm, and therefore it defines a point x=p(g)eA 1. By [Ber], 1.4.4, any point of Al, 1 is obtained in this way. We set , -= f'lz~t(E c~/1~) and r -- inf,. e t r(E). Suppose first that cr 4: O. Then one does not obtain a new point. Namely, if r = 0, then x e Ao ~, and if r > 0, then x -----p(E(a, r)) for any a e ,. Suppose therefore that , = 121. Then one obtains a new point. If r = 0, then x is the image of an element from k'\k ~ under the mapping/kk k, ~ A 1. Points with r > 0 (they are said to be of type (4)) exist if and only if the field k~ is not maximally complete. Points with r = 0 (for arbitrary a) are said to be of type (1). For a point x of type (1) or (4) the extension .~(x)/k is algebraic and the group [~(x)* I/I k* I is torsion. A basis of topology on A 1 is formed by open sets of the form D(a, r)\l.J~= 1 E(a~, rl). (We recall that any affinoid domain in /k I is a disjoint union of the standard affinoid domains which have the form E(a, r)\l.J~= I D(ai, r,).) Let Y be an open neighborhood of a point x e A 1. We say that the triple (pt, y, x) is elementary if one of the following is true: a) x is of type (1) or (4) and Y = D(a, r), where a ek and r> 0; b) x=p(E(a,r)), where aek and re%/Ik*l, and Y=B(a;r',r"), where 0<r'<r<r"; c) x =p(E(a, r)), where a ek and r elk" l, and Y = D(a, r')\ll~= t E(a,, r,), m1> 0, where 0<r~<r<r', a, ek, [a,--a[~< r and [a,--ajl =r for i4:j. I~TALE COHOMOLOGY FOR. NON-AR.CHIMEDEAN ANALYTIC SPACES 73 It is clear that if Y" is of genus zero, then for any open neighborhood Y of a point x ~ X one can find a finite separable extension K ofk and an open subset Y" C Y' = Y | K such that ~| K-%P~, the point x has a unique preimage x' in Y", and the triple (P~, Y", x') is elementary. Consider now the case of an arbitrary smooth geometrically connected projective curve ~g'. Assume that the k-analytic curve X admits a distinguished formal covering q/by strictly affinoid domains with k~-split semistable reduction X = X~, and let n : X -+ X be the reduction map (see [Ber], w 4.3). Furthermore, let { ~L~, }~ ~ i be the irreducible compo- nents of X. One has g = b + ~]~eig(~), where b is the Betti number of the incidence graph A(X) of X. For a point ~ E X one of the following possibilities holds ([Ber], 4.3.1). (i) If ~ is the generic point of :~e, then there exists a unique point x~ ~ X with n(x~) = ~. One has ~~ ~(~ei) ([Ber], 2.4.4 (ii)). (ii) If 7 is a smooth closed point belonging to ~r then ~-i(~) is a connected open set which becomes isomorphic, after a finite separable extension of k, to a dis- joint union of a finite number of copies of the open unit disc with center at zero, and 7~-1(~) = 7~-1(~) k.J{xi}. If YeX(~), then 7~-1(~)-~D(0, 1). (iii) If ~is a double point belonging to components ~e i and ~e (which may coin- cide), then ~-1(~) --% B(0; r, 1), where r e k* I ] andr< 1, and n-l(Y) = ~-l(y) u {xi, xj}. One constructs as follows a closed subset A~,(X)C X which has the structure of a finite graph and is isomorphic to the incidence graph A(X) (it is called the skeleton of X with respect to the covering ~). The vertices of Ae(X ) are the points xl, i e I. The edges of A,(X) correspond to the double points of X as follows. If Yis a double point belonging 1 '~ to components ~ and ~ej, then the subset t~ C ~- (x), which is the preimage of the set {p(E(0, t))lr < t < 1 under } the isomorphism ~-~(~) -% B(0; r, 1), does not depend on the choice of the isomorphism, and the set [~ = tz u { xi, xj } is an edge of Ae(X). There is a canonical (deformational) retraction v:X---> Ae(X). The reduction is said to be good if :K is smooth. In this case Aez(X) consists of one point x (the generic point) for which ~v~~ = ~(.X). The reduction is said to be stable if .X is a stable curve. In this case, if g >i 2 or if g = 1 and the reduction is good, then any other distinguished formal covering of X with stable reduction is equivalent to ql, and therefore the reduction map X --> .X and the finite graph A(X) = Ae(X) do not depend on ql. If g = 1 and the reduction is bad, then the set A(X) = A~,(X) (without the structure of a graph) does not depend on oy. The graph A(X) is called the skeleton of X. The complement X\A(X) is the set of points x e X which have an open neighborhood such that it becomes isomorphic, after a finite separable extension of k, to a disjoint union of a finite number of copies of the open unit disc with center at zero. Assume that g/> 1, and let Y be an open neighborhood of a point x e X. We say that the triple (X, Y, x) is elementary if X has good reduction, x is the generic point of X, and Y X\ ~=IE~, m>~ l, whcrc E~ is an affinoid domain in ~ (x~), ~eX(~), isomorphic to a closed disc E(O, r~), and the points x~,..., ~ are pairwisc different. I0 74 VLADIMIR G. BERKOVICH The Stable Reduction Theorem asserts that, for every smooth geometrically connected projective curve ~ over k of genus g >i 1, there exists a finite separable exten- sion K of k such that the K-analytic space (5~ | K) ~ has stable reduction. Finally, we drop any assumptions. A triple (X, Y, x) is said to be elementary if it is iso- morphic to one of the elementary triples which were defined above. The following Propo- sition 3.6.1 is a particular case of the main result of the next subsection, Theorem 3.7.2, on the local structure of a smooth morphism of pure dimension one. The proof of Theorem 3.7.2 is essentially a generalization of the proof of Proposition 3.6.1. 3.6.1. Proposition. -- Let Y be a smooth k-analytic curve. Then for any point y ~ Y there exist a finite separable extension K of k and an open subset Y" C Y' = Y | K such that the point y has a unique preimagey' in Y" and the pair (Y",y') is elementary. The following statement will be used also in w 7.3. 3.6.2. Lemma. -- Let q~:Y ~ X be a smooth morphism of pure dimension one with k-affinoid X = all(d). Then for any point y ~ Y there exist an open neighborhood Y' of y and a commutative diagram y' c J> q#~ where d/: ~ ~ ~ = Spec(d) is a smooth affine curve of finite type over ~, and j is an open immersion. Proof. ~ We can shrink Y and find an &ale morphism g : Y ~ &~. Let z denote the image of the point z = g(y) in A}. The field k(z) is everywhere dense in ~(z), and ~(y) is a finite separable extension of K(z). By Proposition 2.4.1, there exists a finite separable extension K of k(z) which embeds in K(y) and is everywhere dense in it. Take an arbitrary &ale morphism of finite type between affine schemes h:~ ~ A~- for which there exists a point y e ~ h(y) = z and k(y) = K. Y g> &~ , A~ < ~r X > ~ The embedding of K in 1r (y) defines a point y' e q/~. Since K is everywhere dense in ~ (y), we have lc(y) = K(y'). We get two &ale morphisms of k-germs (Y,y) ~ (A~, z) and (Y/~,y') --~ (A~, z) such that ~r = !r From Theorem 3.4.1 it follows that the k-germs (Y,y) and (q/~,y') are isomorphic. 9 Proof of Proposition 8.6.1. -- By Lemma 3.6.2, we can shrink Y and find an open embedding of Y in the analytificadon ~r,~ of a smooth affine curve ~' of finite type I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 75 over k. Let ~ be the smooth projectivization of 5F'. Increasing the field k, we may assume that ~ is geometrically connected. If the valuation on k is trivial or if the genus g of is zero, then the statement is clear. Thus, assume that the valuation on k is nontrivial and g/> 1. By the Stable Reduction Theorem, we can increase the field k and assume that X = Y'~ has ~-split stable reduction. If the pointy is not a vertex of the skeleton A(X), then, after increasing of k, y has an open neighborhood in X (and therefore in Y) iso- morphic to an open subset of px. Assume therefore that y is a vertex of A(X). First, we want to reduce the situation to the case when A(X)-~ {y }. Let L be a connected open neighborhood of the point y in A(X) which does not contain loops and other vertices of A(X). One has L {y } t3 [J~=m ~, where gi is homeo- morphic to an open interval. Furthermore, for each 1 ~< i ~< m there is an isomor- phism v-~(g~) -% B(0; r~, 1), where 0 < r~ < 1. We fix such an isomorphism so that the point p(E(0, t)) tends to the pointy for t -+ 1. We now glue the open set "~-I(L) with m copies of D (0, 1) via the isomorphisms v- ~(g~) -% B(0; r~, 1) C D(0, 1). We get a new proper smooth k-analytic curve X which is the analytification of a smooth geometrically connected projective curve s of (new) genus g 1> 0. Suppose that g/> 1. Increasing the field k, we may assume that X has stable reduction. Since any point x 4= y has an open neighborhood such that, after a finite separable extension of k, it becomes iso- morphic to a disjoint union of open discs, it follows that the reduction of X is good andy is the generic point of X. Consider the reduction map r~ : X ---> X. Sincey is a unique preimage of the generic point of.~, there exist m/> 1 closed points ~x, ..., ~',~ ~ :~ with rc-'(X\( ~1, 9 9 ~,, }) C Y. Increasing the field k, we may assume that ~ .~(~), and therefore r~-l(~) -% D(0, 1). It follows that we can replace Y by a smaller open neighborhood ofy of the form X\[.J~'= 1 Ei, where ~C z~-a(~) and E~ --% E(0, r~). 9 3.6.3. Remark. -- (i) Let (Y,y) be an elementary pair. Then Y contains a disjoint union of m >/ 1 open annuli B(ri, Ri) such that the set Y\[.J~= 1 B(ri, R~) is a connected compact neighborhood of the point y (it is actually an affinoid domain). (ii) Suppose that the field k is algebraically closed. If (Y,y) is an elementary pair such that Y is not isomorphic to an open disc, then the open set Y\{y } is isomorphic to a disjoint union of a finite number of open annuli and an infinite number of open discs. From Proposition 3.6.1 it follows that any smooth k-analytic curve has a covering by elementary open subsets. (iii) A broader class of smooth k-analytic curves is that of standard curves which are isomorphic to an open subset of the analytification of a smooth geometrically connected projective curve such that its complement is a disjoint union of m 1> 1 closed discs with center at zero. We remark that a standard curve is connected. We remark also that an elementary (resp. standard) curve remains elementary (resp. standard) after any extension of the ground field. 76 VLADIMIR G. BERKOVICH 3.7. The local structure of a smooth morphlsm 3.7.1. Definitions. -- (i) A morphism 9 : Y ~ X is said to be an elementary fibration of pure dimension one (resp. at a pointy ~ Y) if it can be included to a commutative diagram y C 9 Z < V = II (X � E3 i=1 such that a) d~ : Z ~ X is a smooth proper morphism whose geometric fibres are irreducible curves of genus g/> O; b) Y is an open subset of Z, and V = Z~Y; c) V is an analytic domain in Z isomorphic to a disjoint union ll'~=l (X x Ei), m/> 1, where E~ are closed discs in A1 with center at zero, andpr is the canonical projection; d) there exists an analytic domain V C V' such that the isomorphism V-~ H'~= l(X x E~) c) extends to an isomorphism V'-~ 117= I(X x E~), where El is a closed disc from in A 1 which contains E~ and has a bigger radius; e) the pair (Zx, Yx) (resp. the triple (Z~, Y~,y)), where x = q~(y), is elementary. (ii) A morphism is said to be an elementary fibration if it is a composition of ele- mentary fibrations of pure dimension one. (iii) A morphism q~ : Y ~ X is said to be standard of pure dimension one if everything from (i), except the property e), is true for it. A composition of standard morphisms of pure dimension one is said to be standard. We remark that the geometric fibers of a standard morphism are nonempty and connected. Furthermore, elementary fibrations (resp. standard morphisms) are preserved under any base change functor and under any ground field extension functor. 3.7.2. Theorem. -- Let ep : Y ~ X be a smooth morphism of pure dimension one, and suppose that X (and therefore Y) is good. Then for any point y e Y there exist an ~tale morphism f: X' ~ X and an open subset Y" C Y' = Y � X' Y ~ X T T, Y' ~' > X' g,, such that y has a unique preimage y' in Y" and ~" : Y" ~ X' is an elementary fibration of pure dimension one at the point y'. I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 77 8.7.8. Corollary. -- Let ~ : Y ~ X be a smooth morphism of good k-analytic spaces. Then for any point y e Y there exist ~tale morphisms f : X' ~ X and g : Y" ~ Y' = Y Xx X' Y ~> X Y' ~' ~ X' Tooy yt~ such that y ef'(g(Y")) and ~?" is an elementary fibration. 9 8.7.4. Corollary. -- A smooth morphism is an open map. 9 Proof of Theorem 8.7.2. -- All the analytic spaces considered in the proof are assumed to be good. For numbers 0 < r' ~< r" (resp. 0 < r' < r") we denote by A(r', r") (resp. B(r', r")) the closed (resp. open) annulus E(0, r")\D(0, r') (resp. D(0, r")\E(0, r')) with center at zero. 8.7.5. Proposition. -- Let ~ : Y ~ X be a separated smooth morphism of pure dimension one, and let x ~ X. Suppose that the fibre Y~ is isomorphic to the open annulus B(r, R)ae~l. Then there exist numbers r < r' < R' < R, an open neighborhood ~ of x and an open subset 3r C ~-l(~ such that a) ~r coincides with B(r', R')~e,x, under the identification of Yx with B(r, R)ae,,,; b) ~'~ is isomorphic to the direct product q/ x B(r', R') over ~ 8.7.6. Lemma. -- Let Y be an open subset of A~, and suppose that Y~ = D(0, r)ar,~, \ U E(a i, r,)~(~,, m >t O, i=l where r i < r and [ T(ai)] < r (T is the coordinate function on A1). Then (i) /f m = 0, then for any number 0 < r' < r there exists an open neighborhood all of x such that Y contains the direct product o// � D(0, r'). (ii) /fm 1> 1, then for any numbers max{[ T(ai)[, ri) < r' < r" < r, there exists an open neighborhood all of x such that Y contains the direct product q/ x B(r', r"). Proof. -- In the case (i) (resp. (ii)) the intersection of all compact sets of the form U x E(0, r') (resp. U � A(r', r")), where U runs through compact neighborhoods of the point x, coincides with E(0, r')ae/x~ (resp. A(r', r")~cx~), and therefore it is contained in the open set Y. It follows that there exists a compact neighborhood U of x with U � E(0, r')CY (resp. U � A(r',r")CY). 9 78 VLADIMIR G. BERKOVICH 3.7.7. Lemma. --Suppose we are given a commutative diagram Y S>Z with smooth ~ and + of the same pure dimension, and let y ~ Y and x = ?(y). Suppose that the induced morphism f~ : Y~ --+ Z, is a local isomorphism at the point y. Then f is a local isomorphism at y. Proof. -- Since Y~ is the fibre product of Y and Z~ over Z, the inverse image of the sheaf of differentials off on Y, coincides with that off~. But the latter sheaf is equal to zero at the point y. Therefore we may decrease Y and assume that f is unramified. By Proposition 3.3.2 (i), the canonical homomorphism f*(f~z/x) --+fLc/x is surjective. Since both 0y-modules are locally free and of the same rank, we have f*(~z/x) ~ f~Y~x. From Proposition 3.5.3 (ii) it follows thatfis ~tale at the pointy. By hypothesis, f induces an isomorphism of fields ~(z) -~(y). By Proposition 2.4.1, (z) ~ ~(y), and there- fore f is a local isomorphism at y, by Theorem 3.4.1. 9 Proof of Proposition 3.7.5. -- We may assume that X ----Jr is k-affinoid. Take a number r < t < R and set y = p(E(0, t)) ~ Y,. Let V be an affinoid neighborhood ofy in Y. Shrinking X and Y, we may assume that the fibre V, is connected. Then V~ = E(O, R')g,,)\ U D(a~, i=l where t < R' < R, r~ < R' and [ T(a~) [ < R' (T is the coordinate function on B(r, R)~vl,I ). Let V ----,/g(~). Since the image of M | ,:(x) in ~ ~g~(x) is everywhere dense, we can shrink X and assume that there exists an elementf e ~ such that the norm off-- T in ~ bg~,~ is less than rnin(r~, t). It follows that the morphism f: V -+ A~ induces an isomorphism V,-~ V, and f(y)=p(E(0, t)). Take an open neighborhood ~r of the point y such that r C V. Then the induced morphism f: r A~ satisfies the hypothesis of Lemma 3.7.7, and therefore f is a local isomorphism at the point y. The required statement now follows from Lemma 3.7.6. 9 3.7.8. Proposition. -- Let ? : Y -+ X be a smooth morphism of pure dimension one, and let x e x. Suppose that thefibre Y, is isomorphic to the open disc D(0, R)g~ I and that the morphism is a composition of an open immersion Y '-+ Z with a compact morphism + : Z --+ X. Then for any 0 < r< R there exist an open neighborhood ~ of x and an open subset "/PC ~-l(~ such that a) ~ coincides with D(0, r)ae~,~ under the identification of ~, with D(0, R)ael,l; b) r is isomorphic to ql x D(0, r) over ql. Let P and Q be Hausdorff topological spaces, and let PI C P and QI C Q be their open subsets for which there is a homeomorphism f: P1 --% Q1- The topological space, r,)ae,,), I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 79 which is obtained by gluing P and Q vial, will be denoted by P u I Q. The following statement is trivial. 3.7.9. Lemma. -- Suppose that there exists an open subset P' C P such that P1 C P' and the space P' u y Q is Ilausdorff. Then the space P u y Q is Hausdorff. Furthermore, suppose in addition that P\P1 is compact and Q-~ PI u y Q is relatively compact in P' u I Q. Then the space P uy Q is compact. 9 Proof of Proposition 3.7.8. -- We may assume that X is k-affinoid. In particular, Z is Hausdorff and compact. From Proposition 3.7.5 it follows that we can shrink X and find an open subset r Y such that ~",-= B(r', R')~ec,~ and ~e'__% X x B(r', R') for some r < r'< R'< R. Take numbers r'< r"< t < R"< R' and denote by ~r the open subset of r which corresponds to X � B(r", R"). The subset Y, C ~, which corresponds to X x A(t, t), is compact because X is compact, and therefore Z\Z is an open subset of Z. We apply the above gluing procedure to the spaces P = Z\Z, P' = zCr\Z, Px = r Q = X � (PX\E(0, r")) I1 X � D(0, R"), Q~ = X x B(r", t)IJ X � B(t, R"), and to the canonical isomorphism f: P1 --~ Q1. One has P' uf Q--% X � (PX\E(0, r')) 11 X � D(0, R'). From Lemma 3.7.9 it follows that the space Z' = P uy Q is Hausdorff and compact. In particular, the morphism 4' : Z' -+ X is compact. Furthermore, the above construction restricted to the fibre at x gives a compact ~ff(x)-analytic space Z: such that the connected component of D(0, R)av~,> is isomorphic to the projective line P~vI,>. Moreover, the morphism 4' is smooth at all points of this connected component. Since +' is compact, we can shrink X and replace Z' by the connected component of D (0, R)ae(,~ in Z' so that the morphism 4' becomes proper smooth of pure dimension one. It has a section ~ : X -+ Z' defined by the point infinity of the disc PX\E(0, r'). One has dim~el~> H~ d)z;) = 1 and HI(Z~, d)z; ) = 0. By the Semi- continuity Theorem ([Ber], 3.3.11), we can shrink X and assume that the same is true for all x' e X. It follows that all the fibres of ~b' are isomorphic to the projective line. Finally, let L be the invertible sheaf on Z' which corresponds to a(X), and let L,, denote the inverse image of L on the fibre Z~,, x' s X. Then HI(Z~,, L,,) = 0 and dim H~ L,,) = 2. It follows that ~V(L) is a locally free 0x-module of rank two. Shrinking X, we may assume that ~'.(L) is free. In this case it defines a morphism Z' ~ I~ over X which is evidently an isomorphism. The required statement now follows from Lemma 3.7.6 (i). 9 We are now ready to prove the theorem. First of all we remark that if K is a finite separable extension of the field ~:(x), then there exists an 6tale morphismf: X' ~ X such that f-l(x) -=-- { x' } and ~:(x') = K. The preimage of the pointy under the morphism 80 VLADIMIR G. BERKOVICH f' : Y' -= Y � x X' --~ Y is a nonempty finite set of points. Fixing one of them, say y', we can replace q~ by a morphism Y" --~ X', where Y" is an open neighborhood ofy' which does not contain other points from f'-~(y). If we make the above procedure, we say for brevity that we increase the field K(x). Furthermore, we may assume that the space X is k-affinoid. Step 1. -- One can increase the field K(x) so that ~ is included in a commutative diagram yc~>Z where a) + is a smooth proper morphism of pure dimension one whose geometric fibres are irreducible curves of genus g >>. O, and j is an open immersion; b) if g >1 1, then Z, has good reduction and y is the generic point of Z,. Let Z be an affinoid neighborhood of the point y. Replacing Y by a small open neighborhood ofy which is contained in Z, we may assume that the morphism q~ : Y -+ X is a composition of an open immersion Y ~ Z with a compact morphism + : Z -+ X. By Proposition 3.6.1, we can increase the field K(x) and assume that the pair (Y,,y) is elementary. In particular, Y, contains a disjoint union of m i> 1 open annuli B(r~, R,)ae(~l such that the set Y,\[IT=I B(r~, R,)ae~, I is a connected compact neighborhood of the point y (Remark 3.6.3 (i)). By Proposition 3.7.5, we can shrink X, Y and the annuli and assume that there are pairwise disjoint open subsets ~ C Y, 1 ~< i <~ m, such that ~e]~,~ = B(r~, R~)ae,~ , and ~ ~ X X B(r,, R~). We take numbers r~ < r[ < t, < R~ < R, and denote by ~ the open subset of ~i which corresponds to X x B(r', R~). The set Z~ C r162 which corresponds to X � A(t,, t~), is compact because X is compact, and therefore Z\lJ~"=l E, is an open subset of Z. We apply the gluing procedure to the spaces P = Z\OL1 Z~, P' = [J~=l(~i\Z~), Pa = O~=l(Yg~\Z~), Q = II~m= l(X X (P'\E(O, rf)) I] X � D(O, R~)), Q1 = Ll'~=a(X x B(r~, t,) H X � B(t~, R~)), and to the canonical isomorphism f: P1 --% Q1- One has V' ucQ~ II (x x (P~\E(0, r,)) II X x D(0, R,)). i=l From Lemma 3.7.9 it follows that the space Z' = P t31 Q is Hausdorff and compact. In particular, the morphism qb' : Z' ~ X is compact. Furthermore, the above construction restricted to the fibre at x gives a compact 3f'(x)-analytic space Z~ such that the connected component of the point y is a smooth ]~TALE COHOMOLOGY FOR NON-AKCHIMEDEAN ANALYTIC SPACES 81 geometrically connected proper .~(x)-analytic curve. Moreover, q,' is smooth at all points of this connected component. Since +' is compact, we can shrink X and replace Z' by the connected component of the point y so that the morphism +' becomes proper smooth of pure dimension one and the fibre Z~ becomes a smooth geometrically connected proper.Cd(x)-analytic curve. The point infinity ofPl\E(0, r;) or the point zero of E(0, R') defines a section r : X ~ Z'. From the construction it follows that we can increase the field (x) and assume that the curve Z~ has good reduction. Moreover, if the genus g of Z~ is positive, then y is the generic point of Z' x. Finally, one has dimae(~ ~ H~ dTz,)= 1 and dim~e~ ~ HI(Z'~, Oz~ ) --~ g. By the Semicontinuity Theorem, we can shrink X and assume that the above equalities hold for all points x' e X. In particular, the fibres of '~' are connected. They are geometrically connected because qb' is smooth and has a section. Step 2. - .... One can increase the field !r and shrink Y so that the morphism ~ : Y ~ X is an elementary fibration at y. We can increase the field ~:(x) and assume that the triple (Z~, Y,,y) is elementary. In particular, Y,=Z,\IJ~__IE~, where Ei-%E(0, R~)aec~j, R~< I, and there are bigger open subsets E~C D~C Z~ with D~ -% D(0, R~)~(z), R~ < R~ < 1. From Propo- sition 3.7.8 it follows that we can shrink X and find pairwise disjoint open subsets CZ such that ~e]~,,=D(0, t;),,e,,, and ~--%X � D(0, t~), where R,<t;< R I. Take numbers R~< t~< t[ and set ~ = X X D(0, ti). Since (z\U~"=aW~),cY,, we can shrink X and assume that z\U~'= ~ ~ C Y. Finally, we take numbers R~ < ri < t~ and set V = ll]" a(X � E(0, r~)). By construction, Z\V C Y. Therefore we can replace Y by Z\V so that all the conditions of Definition 3.7.1 (i) hold. (The condition d) holds for V'= I_I~=t(X � E(0, r~)), where r~< r" < t~.) The theorem is proved. 9 The following remarks will not be used in the sequel. 3.7.10. Remarks. -- (i) One can add to the definition of an elementary fibra- tion 3.7.1 (i) the condition that all the fibres of the morphism qb have stable reduction. Indeed, taking an integer n/> 3 which is prime to char (~), one can increase the field ~:(x) and assume that all the points of order n on the Jacobian of the curve Z, are rational. One can then shrink X so that, for any x' e X, all the points of order n on the Jacobian of Zx, are rational, and therefore Z~, has stable reduction. (ii) One can show that if X = ..r162162 is k-affinoid, then the space Z from Definition 3.7.1 (i) is the analytification :Lr~- of a smooth projective curve ~ over y" = Spec(~r (iii) One can give a similar local description of a flat morphism of good k-analytic spaces q~ : Y -~ X at a pointy ~ Y such that the fibre Y~, x = 9(y), is a curve smooth outside the point y. (iv) We think that the theorem 3.7.2 is true without the assumption that the space X (and therefore Y) is good. At least this is so if each point of X has 11 82 VLADIMIK G. BERKOVICH an open neighborhood which is isomorphic to an analytic domain in a k-affinoid space (see Remark 1.4.3 (ii)). Indeed, shrinking X and Y, we may assume that X is a special domain in a k-affinoid space X' and that ~0 factorizes through an dtale morphism Y ~ A~. Since A~ is an analytic domain in A~:,, then, by Corollary 3.4.2, we can shrink Y and assume that the 6tale morphism Y -~ A~ is a base change of an 6tale morphism Y' --* A~r i.e., the morphism q0 is the base change of a smooth morphism q~' : Y' -+ X' with respect to X ~ X'. Then the validity of the theorem 3.7.2 for q~' implies its validity for ~. w 4. l~tale cohomology 4.1. i~tale topology on an analytic space The gtale topology on a k-analytic space X is the Grothendieck topology on the category ]~t(X) generated by the pretopology for which the set of coverings of (U ~ X) ~ ]~t(X) is formed by the families { U~ ~ U },~x such that U ---- [J~xf(U,). We denote by X~t the site obtained in this way (the gtale site of X) and by X~ the cate- gory of sheaves of sets on X~t (the (tale topos of X). Furthermore, we denote by S(X) (resp. S(X, A)) the category of abelian sheaves (resp. sheaves of A-modules) on X~t and by D(X) (resp. D(X, A)) the corresponding derived category. For a sheaf F on X~t we often say that F is a sheaf on X. The cohomology groups of an abelian sheaf F e S(X) will be denoted by Hq(X, F). Any morphism ~0 : Y ~ X of analytic spaces over k induces a morphism of sites Y~t ~ X~t. If~ is a scheme of locally finite type over Spec(d), where d is a k-affinoid algebra, then Propositions 2.6.8 and 3.3.11 imply that there is a morphism of sites (f~n)~t -+ Y'~t, where ~Y,t is the dtale site of the scheme f. The inverse image of a sheaf ~ e ~ on :~O n will be denoted by ~'~. Our first purpose is to show that certain reasonable presheaves on X,t are actually sheaves. For this we introduce a (big) flat quasifinite site Xrq f of X. This is the site with the underlying category k-dn/x of k-analytic spaces over X and with the Grothendieck topo- logy generated by the pretopology for which the set of coverings of (Y ~ X) e k-~n/x is formed by the families { Y~-~ Y}~ei such that the f are flat quasifinite and Y = U~exf(Y,). There is an evident morphism of sites Xf~f-+X~t. It is clear that if a presheaf on Xfqf is a sheaf, then its restriction on X,t is also a sheaf. 4.1.1. Lemma. -- The following conditions are sufficient for a presheaf F on Xfqf to be a sheaf: (1) for any k-analytic space Y over X the restriction of F to the G-topolog~ on Y is a sheaf; (2) for any finite faithfully flat morphism of k-affinoid spaces Z ~ Y over X the sequence F(Y) -+ F(Z) ~ F(Z � y Z) is exact. I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 83 Proof. -- Let o//=_ (U i ~J> Y)iai be a covering in Xfqf. We have to show that the sequence (.) F(Y) ~ II F(U,) -% 11 F(U, � y Uj) i i,j is exact. For every i e I we take an open covering { V~ }J~Ji of Ui such that the induced morphisms V 5 --> q0,(V~) are finite. From (1) it follows that if (,) is exact for the cove- ring {V~}~ea, where J = UiEiJi, then (,) is exact for ~//. So we may assume that all the morphisms U i -+ q~i(Ui) are finite. Furthermore, since (,) is exact for the open covering { ~0,(Ui)}iei of Y, it suffices to show that the sequence F(Y) -+ F(Z) -% F(Z � y Z) is exact for any finite faithfully flat morphism ~ : Z ~ Y over X. Finally, if r = { V~ }~ e i is a quasinet of affinoid domains on Y, then zcf = { ~-I(V~)}~ e i is a quasinet of affinoid domains on Z and, by (1), the sequences (.) for the coverings r and ~ of Y and Z, respectively, are exact. Thus, the situation is reduced to the case of k-affinoid spaces. But the required fact in this case is guaranteed by (2). 9 4.1.2. Corollary. -- Let F be a coherent sheaf on X G. Then the presheaf F, which assigns to a k-analytic space Y over X the group of global sections of the inverse image of F on YG, is a sheaf on X~qf and therefore on X~t. 9 4.1.3. Proposition. -- A presheaf representable by a k-analytic space good over X is a sheaf on Xrq f and therefore on X~t. Proof. -- Let F be representable by a k-analytic space X' over X, i.e., F(Y) = Homx(Y,X' ). The condition (1) holds for F, by Proposition 1.3.2. Let Y = J/f (N) and Z = ~/(~). Then ~ is a faithfully flat N-algebra. It is well known that in this situation the sequence 0 -+ N --> ~f -% c~ | W is exact. Since ~ is a finite Banach N-algebra, we have ~'| ~ = ~| ~, and the above sequence is admissible. It follows that the condition (2) holds, at least, for k-affinoid X'. In the general case we have to show that for any g : Z ~ X' over X with g o p~ = g o P2, where p, are the canonical projections Z � y Z, there exists a unique f: Y -+ X' over X with g =fo ~ (~ is Z ~ Y). Uniqueness off. (Here the assumption that X' is good over X is not used.) Assume that we are given fl,f2 : Y -+ X' with fl o qo =f2 o qo. Since ~? is surjective, fl and ./2 coincide as maps of topological spaces. Let U be an affinoid domain in X'. Then ~-I(U) = f2-1(U) = [J~=~ Vi for some affinoid domains V~ C Y. Applying the particular case to the morphisms q0-~(V~) -+ V~ and the k-affinoid space U, we get f~lvi =f2lvi. Since Y is covered by a finite number of such V~, it follows that f~ ----f~. Existence off. We can replace X by Y and X' by Y x x X' and assume that X' is good. By the uniqueness, it suffices to construct f locally. Let y ~ Y, z e 9-a(y), U an affinoid neighborhood of g(z) in X'. Since ~ is an open map, ~(g-~(U)) contains an 84 VLADIMIR G. BERKOVICH affinoid neighborhood V of the point y. We claim that ~-~(V)Cg-a(U). Indeed, if ~(zl) = q~(z~), then there exists z' ~Z � with pl(z')= zl and p,(z'~ = z2. If z 1 zg-~(U), then g(zl) = gpl(z') = gp2(z') = g(z,) z U. And so the situation is reduced to the morphism ~-~(V) ~V and the k-affinoid space U. 9 4.1.4. Corollary. -- (i) There is a fullyfaithfulfunctor l~t(X) ~ X~. In particular, the dtale topology on ]~t(X) is weaker than the canonical one. (ii) If a sheaf F is representable by an X-space T dtale over X, then for any morphism : Y -+ X the sheaf ~* F on Y is representable by the space T � x Y. 9 4.1.5. Remark. -- The Proposition 4. I. 3 certainly should be true without the assumption that the space X' is good over X. For this it would be enough to know that for a finite 6tale morphism of k-affinoid spaces 9 : Y ~ X and for any affinoid domain V C Y the image 9(V) is a finite union of affinoid domains in X. If everything is strictly k-affinoid, this is a particular case of a result of Raynaud that gives the same fact for an arbitrary flat morphism. 4.1.6. Example. -- (i) Let A be a set. Then the k-analytic space ~Jx~A X over X represents the constant sheaf A x. (ii) The sheaf of abelian groups G~, x is defined by G~,x(Y ) = 0(Y). (iii) The sheaf of multiplicative groups Gin, x is defined by Gm, x(Y ) = 0(Y)*. (iv) The sheaf of n-th roots of unity ~,x is defined by ~,x(Y) -=- {f~ 0(Y)* If" = 1 }. If the field k contains all n-th roots of unity and n is prime to char(k), then the sheaf ~.x is isomorphic to the constant sheaf (Z/nZ)x. 4.1.7. Proposition. -- (i) The Kummer sequence IH'f 0 --+ ~,x ----~ Gin, x ~ Gin, x-+ 0 is exact in S(Xfqf). If n is prime to char(k), then it is exact also in S(X). (ii) Ifp = char(k) > 0, then the Artin-Schreier sequence (z/p" G ,x o is exact in S(X) (and therefore in S(Xfqf)). Proof. -- (i) It suffices to show that for a compact k-analytic space X and an element f E O(Y)* there exists a flat quasifinite morphism Y ~X such that the image off in #(Y)* is in 0(Y)*". Let {Ui),~ I be a finite affinoid covering of X. Then ~ := dgi[T]/(T"--1) is a finite Banach du/-algebra , and the mor- phism ~,:Vi = .~/{(g~) -+ U~ is finite flat (6tale if n is prime to char(k)). Further- more, for any pair i,j ~I, there is a canonical isomorphism of special domains 'qi~ : Vii := q~i-l(U~ ('~ Uj) -~ Vii-'= ~gj- I(U$ ('~ Uj), and one has Vi~----Vi, n = n ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 85 and '~,i = vii o v,~ on V,~ r3 V,. Therefore we can glue all V, along Vii , and we get a compact k-analytic space Y and a flat (6tale if n is prime to char(k)) finite morphism Y ~ X. The image of T in each g, defines an element g ~ #(Y)* with g" =f. (ii) is proved in the same way. 9 The cohomology groups of A x (if A is an abelian group), G,, x, Gin. x and ~,, x will be denoted by Hq(X, A), Hq(X, G~), Hq(X, Gin) and Hq(X, ~t,), respectively. The first (~ech cohomology set I~I(X, F) can be defined for any sheaf of groups F on X. This set contains a marked element that corresponds to the trivial cocycle. (If F is abelian, then ~tl(X, F)= Hi(X, F).) The set ~I~(X, F) has the usual interpretation as the set of sheaves on X that are principal homogeneous spaces of F over X. On the other hand, if F is representable by an X-group G (a group object in k-dn/x), then one has the set PHS(G/X) of isomorphism classes of principal homogeneous spaces of G in the category k-~C/x , and there is an evident mapping PHS(G/X) -> ~I~(X, G). 4.1.8. Proposition. -- If G is an X-group itale over X, then PHS(G/X) -~ tYtl(X, G). Proof -- It suffices to verify that a sheaf F, which is a principal homogeneous space of G over X, is representable by an ~tale X-space. Furthermore, by Corollary 4.1.4, it suffices to show that F is representable locally in the usual topology of X. In particular, we may assume that X is paracompact and there is a finite dtale surjective morphism U ---> X for which FIu is representable. If X is k-affinoid, then U is also k-affinoid and the representability of F follows from the descent theory of schemes. In the general case we take a locally finite net v of affinoid domains of X. For V e z, let fv : Yv -~ V be an dtale morphism that represents the inverse image of F on V. For a pair V, W ~ ~, there is a canonical isomorphism ~v.w: Yv, w :-~-fvl( v ~ W) -+ Yw, v :=fwl( V c~ W), and the system of isomorphisms ~v,w satisfies the necessary conditions for gluing of Yv along Yv, w. In this way we get an X-space Y. It is easy to see that the canonical mor- phism Y ~ X is 6tale and that it represents the sheaf F. 9 If G is an abstract group, then the principal homogeneous spaces of the constant X-group G x are called gtale Galois coverings of X with the group G. 4.1.9. Corollary. -- Let G be an abstract group. Then there is a bijection between tF-II(X, G) and the set of isomorphism classes of gtale Galois coverings of X with the Galois group G and with a given action of G. 9 We remark that if ~ : Y ---> X is an 6tale Galois covering with the Galois group G, then for any abelian sheaf F on X there is a spectral sequence H~(G, Hq(Y, r)) ~ H~+q(X, r). The group H:(X, G~) can be interpreted as the group of invertible Ox~(modules , where ~x+t is the sheaf of rings on X~t associated with the structural sheaf tYxo (see Corollary 4.1.2). In particular, there is an inj ective homomorphism Pie(X) -~ Hi(X, Gin). 86 VLADIMIR G. BERKOVICH 4.1.10. Proposition. -- (Hilbert Theorem 90). If X is good, then Pie(X) ~ H~(X, %). Proof. -- Let ~a be an invertible 0x~t-module. Since the homomorphism considered is injective (for an arbitrary X), it suffices to show that any point x ~ X has an open neighborhood q/ such that ~cPI~ comes from Pic(~//). Therefore we can shrink X and assume that there is a finite 6tale morphism Y ~ X such that ~tY comes from Pie(Y). If V is an affinoid neighborhood of x, then ~ defines an invertible d~r on the affine scheme r ~-Spec(~Cv). By the descent theory for schemes, the latter module comes from an invertible 0,--module. Thus, if q/is an open neighborhood of x that is contained in V, then ~[~ comes from an invertible d~-module. 9 For an arbitrary k-analytic spaces X we can prove only the following 4.1.11. Corollary. -- There is a canonical homomorphism H~(X,G~)--->Pic(X~) such that its composition with Pie(X) -+ Ha(X, Gin) coincides with the canonical homomorphism Pic(X) -+ Pic(Xo). Proof. -- Let &o be an invertible 0x~t-module. Then it defines for any affinoid domain V C X an invertible 0v-module Lv, and these modules are glued together to an invertible 0xQ-module L. The correspondence .oq ~ ~ L gives the required homo- morphism. 9 The group Hi(X, ~n) (n is prime to char(k)) can be interpreted as the group of isomorphism classes of the pairs (.LP, ~?), where .oCt is an invertible 0x~t-module and ~? is an isomorphism d~x~ t ~ .La| From Proposition 4.1. I0 it follows that if X is good, then the latter group coincides with the group of isomorphism classes of the pairs (L, q~), where L ~ Pie(X) and q~ is an isomorphism 0 x --% L | In the general case Corol- lary 4.1.11 gives a canonical homomorphism from H~(X, ~,) to the group of isomorphism classes of the pairs (L, q~), where L e Pic(Xa) and q0 is an isomorphism d~xo--% L | (The latter is the group Hi(X, P-n) introduced by Drinfeld in [Drl].) In w 4.3 we'll show that this is an isomorphism. 4.2. Stalks of a sheaf For technical reason we consider the gtale topology on a k-germ (X, S). It is the Grothendieck topology on the category I~t(X, S) (see w 3.4) generated by the preto- pology for which the set of coverings of ((U, T) -+ (X, S)) ~ s S) is formed by the families {(U~, T~)~ (U,T)}~e I such that T = I.J~if(T~). We denote by (X, S)~ the corresponding site (the gtale site of (X, S)) and by (X, S)~ the category of sheaves of sets on (X, S)~ t (the dtale topos of (X, S)). The category of abelian sheaves (resp. presheaves) on (X, S)~ will be denoted by S(X, S) (resp. P(X, S)). There is an evident rnorphism of sites i~x,s I : (X, S)~ t ~ X~t. (If S = [ X [, it is an isomorphism.) For a sheaf F on X we set Fcx, s I = i*~x,s~ F and F(X, S) = F~x,s~(X, S). I~TALE COHOMOLOGY FOR NON-ARGt-IIMEDEAN ANALYTIC SPACES 87 For a field K we denote by K,t the 6tale site of the spectrum of K and by K~ the category of sheaves of sets on K a. (The latter is equivalent to the category of dis- crete G,~-sets.) The following statement follows immediately from Theorem 3.4.1. 4.9.. 1. Proposition. -- For any point x of a k-analytic space X there is an equivalence of categories ~(x)~ -% (X, x)~. 9 Let (X, S) be a k-germ. For a point x ~ S let i x denote the canonical morphism of sites o~(x)~t-+ (X, S)~. The inverse image of a sheaf F with respect to i x is denoted by F, and is called the stalk of F at x. (We identify F, with the corresponding discrete G~el,l-set.) The image of an elementf ~ F(X, S) in F, is denoted by f,. If F is an abelian sheaf, then the support of an elementf~ F(X, S) is the set Supp(f) = { x e Ifx S ~e 0 } (from the following Proposition 4.2.2 it follows that this is a closed subset of S.) Further- more, for a subextension~(x) C K C o~r s we denote by F,(K) the subset of G(~(x)S/K)- invariant elements. (For example, Fx(~~ = F~%e(x~ and Fx(~~ s) = Fx. ) 4.9..2. Proposition. -- For a sheaf F on (X, S) and a point x ~ X, one has F~(o~(x)) = lim F(ql, S n o//), #g~z where ql runs through open neighborhoods of x in X. Proof. -- The set F,(~(x)) is the inductive limit of the sets F(Y, T) over all ((Y, T) ~ (X, S)) s l~t(X, S) with a fixed pointy ~ T over x such that ~(x) ~tt~ By Theorem 3.4.1, the latter implies that the morphismf induces an isomorphism of k-germs (Y,y) -~ (X, x), and the required statement follows. 9 4.2.8. Corollary. -- A morphism of sheaves F ~ G on (X, S) is a mono/epi/isomorphism if and only if for all x ~ X the induced maps F, ~ G, possess the same properties. 9 Let (X, S) be a k-germ. For each open subset U C S we fix an open subset Y/C X with ~/n S = U. Then the correspondence U ~-~ q/defines a functor from the category of open subsets of S to the category l~t(X, S), and this functor does not depend (up to a canonical isomorphism) on the choice of the sets ~/. In this way we get a morphism of sites = : (X, S)~ t ~ S, where S is the site induced by the usual topology of S. 4.2.4. Proposition. -- For an abelian sheaf F on (X, S) and a point x e S, one has (R q re, F), -% H"(Gae(,,, F,), q >/ 0. Proof. -- The case q = 0 follows from Proposition 4.2.2. Therefore in the general case it suffices to verify that ifF is a flabby sheaf on (X, S), then F(x,, ~ is a flabby sheaf on (X, x). For this it suffices to show that ~Iq(~cK, F~x.,)) = 0, q/> 1, for any covering #~ of (X, x) of the form ((U, u) ~ (X, x)) where f is finite. By Proposition 4.2.2, the ~ech complex 88 VLADIMIR G. BERKOVICH of the covering ~" is an inductive limit of the ~ech complexes of the coverings ((~e', T n r (q/, S nag)), where ag runs through sufficiently small open neigh- borhoods of the point x, T =f-l(S) and r =f-l(O//). Since F is flabby, those ~ech complexes are acyclic. 9 In the previous proof and in the sequel, an abelian sheaf F on a site C is called flabby if its q-dimensional cohomology groups on any object of C are trivial for all q 1> 1. In [SGA4], Exp. V, 4.1, such a sheaf is called C-acyclic. Recall (loc. cit., 4.3) that F is flabby if and only if the ~ech cohomology groups ff-I a of any object of C (resp. for all coverings of any object of C) are trivial for q/> 1. We remark that if C is the site of a topological space, then the notion of a flabby sheaf is not related to the notion of a flasque sheaf from [God]. If G is a profinite group, then a discrete G-module M is flabby if Ha(H, M) = 0 for all open subgroups H C G and all q >i 1. 4.2.5. Corollary. -- An abelian sheaf F on (X, S) is flabby if and only if (1) for any point x ~ S, F, is a flabby G~e,,)-module; (2) for any ((Y, T) ~ (X, S)) E ]~t(X, S), the restriction of F to the usual topology of T is a flabby sheaf. 9 We now obtain first applications of the above results. They are obtained using the spectral sequence (,) E$ ,a ---- H~(] X l, Ra re, F) => H~+a(X, F) of the morphism of sites ~ : X~t ~ ] X [. Let l be a prime integer. The l-cohomological dimension cdz(X ) of a k-analytic space X is the minimal integer n (or oo) such that Ha(X, F) = 0 for all q > n and for all abelian/-torsion sheaves F on X. (F is said to be/-torsion if all its stalks are/-torsion.) For example, if X = d/(k), then cdz(X ) = cdl(k ). 4.2.6. Theorem. -- Let X be a paracompact k-analytic space, and let l be a prime integer. Then cdz(X ) ~< cdz(k ) -F 2 dim(X). If l -= char(k) then cd~(X) ~< 1 q- dim(X). Proof. -- Let F be an abelian/-torsion sheaf. From Proposition 1.2.18 it follows that the member E~ 'a of the spectral sequence (,) is zero for p > dim(X). Since (R a z~, F), ---- Hq(G~,~, F,), then Theorem 2.5.1 implies that E~ 'q = 0 for q > cd~(k) q- dim(X) (resp. q > 1 if l = char(k)). The required fact now follows from the spectral sequence. 9 4.2.7. Theorem. -- If X is a good k-analytic space, then for any coherent Ox-module F there is a canonical isomorphism H~(IXt, F)-~Ha(X,Y), q~> 0. I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 89 4.2.8. Lemma. -- Let x E X. Then (G,,x) , = ~hx,,, where ~x, ~ is the strict Hense- lization of the local Henselian ring Ox. ~. Proof. -- The assertion straightforwardly follows from Proposition 4.2.2. 9 Proof of Theorem. -- It is clear that ~,(F) = F. It suffices to show that R ~ ~,(~) = 0 for all q/> 1. For this it suffices to verify that H"(G~,I, (F),) --0 for all x e X and q/> 1. We set A = Ox,,. From Lemma 4.2.8 it follows that (F), = F,| A A ~, where F~ is the stalk of the coherent module F at the point x (this is a finitely generated A-module). We claim that Hq(G(K/~(x)), F,| ) = 0, q/> 1, for any finite Galois extension K of ~:(x), where B is the finite extension of A which corresponds to the extension K/~:(x). For this we remark that the cohomology groups considered coincide with the coho- mology groups of the ~ech complex F~|174174 -+F~| | Since B is a faithfully flat A-algebra, this complex is exact. 9 4.3. Quasi-immersions of analytic spaces Let q~ : (Y, T) ~ (X, S) be a morphism of germs over k. Then for any pair of points yeT and x E S with x = q~(y) there is an isometric embedding of fields o~ff(x) ,---~f'(y). We always fix for such a pair an extension of the above embedding to an embedding of separable closures o~ff(x) 8 '-+o~'~ '. It induces a homomorphism of Galois groups G~er ~ Ggr The following statement follows straightforwardly from the definitions and Proposition 4.2.2. 4.3.1. Proposition. -- (i) For any sheaf F on (X, S) and any pair of points y ~ T and x e S with x = (p(y), there is a canonical bijection F, ~ (q~* F)~ that is compatible with the action of the groups G,,e,~ and Ggc~. (ii) For any sheaf F on (Y, T) and any point x e S, one has (% F), (af(x)) = li_~m F(q~-l(#/), T r3 ?-1(0//)), where all runs through open neighborhoods of x in X. 9 4.3.2. Corollary. -- Let r X be a finite morphism of k-analytic spaces. Then the functor cO, :S(Y) -+ g(x) is exact. In particular, for any abelian sheaf F on Y one has Hq(X, % F) -% Hq(Y, F), q 1> 0. Proof. -- From Proposition 4.3.1 (ii) it follows easily that for any point x e X the stalk (% F)x is isomorphic to the direct sum over all y e q~-l(x) of the induced Gar(~ _modules Indav(x~(F~). ~o(~ It follows that the functor ~, is exact. 9 12 90 VLADIMIR G. BERKOVIGtt 4.3.3. Definition. -- A morphism 9 : (Y, T) -+ (X, S) of germs over k is said to be a quasi-immersion if it induces a homeomorphism of T with its image 9(T) in S and, for every pair of points y e T and x e S with x = 9(Y), the maximal purely inse- parable extension of o~(x) in oct~ is everywhere dense in o~(y). For example, analytic domains, closed immersions and the morphisms of the form X | K -+ X, where K is a purely inseparable extension of k, are quasi-immersions. Furthermore, if 9 : Y ~ X is morphism a of k-analytic spaces, then for any point x e X the canonical morphism Y, -~ Y is a quasMmmersion. We remark that quasi-immersions are preserved under compositions and under any base change in the category ~erm~ (when it is well defined). 4.8.4. Proposition (Rigidity Theorem). -- Let q~ : (Y, T) ~ (X, S) be a quasi-immersion of germs over k. Then (i) ~ induces an equivalence of categories (Y, T)~ ~ (X, 9(T))~; (i.i) /f 9(T) is closed in S, then e? induces an equivalence between the category S(Y, T) and the full subcategory 0f S(X, S) which consists of such F that F~ = O for all x ~ S\q~(T). Proof. -- (i) We may assume that S = ~(T). Let y e T and x = 9(Y). By hypo- thesis, there is an isomorphism G;e~, -~ G;e~,~. We claim that for any sheaf F on (Y, T) there is a bijection of Ga.~,~ = Gaec,)-sets (q~. F), -% F~. It suffices to verify that (q), F), (o~(x)) = F~(.~f~ By Proposition 4.3.1 (ii), one has (9. F), (.g'(x;)) = lira F(q~-~(~), T c~ 9-~(q,')). q.'Bz Since 9 induces a homeomorphism of T with S, the limit coincides with F~(~(y)). It follows that the functor 9. is fully faithful. Let now F be a sheaf on (X, S). Then there is a bijection of Gaoc~, --- Gao~,,-sets F, = (9" F)~. We have (9" F)v = (9. q)'(F))~. Therefore, F -% ~. ~'(F). The required statement follows. (ii) By (i), we may assume that T is closed subset of S, and 9 is the canonical morphism (X, T) -+ (X, S). It is clear that for an abelian sheaf F on (X, T) one has (9. F)~ = F~ if x e T and (9. F)~ = 0 if x e S\T. If now F is a sheaf on (X, S) such that F, ---- 0 for all x e S\T, then (q~. q~'(F))~ = (9" F)~ = F~ if x s T, and therefore F -% ?. ?'(F). 9 We remark that from Proposition 4.3.4 it follows that if q~ : (Y, T) ~ (X, S) is a quasi-immersion of germs over k, then the cohomology of (Y, T) with coefficients in an abelian sheaf coincides with the cohomology of (X, ~(T)) with coefficients in the corresponding sheaf. I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 91 A k-germ (X, S) is said to be paracompact if S has a basis of paracompact neigh- borhoods (in this case S is evidently paracompact). For example, k-germs of the form (X, x), where x e X, are paracompact. From this it follows that if X is Hausdorff, then any k-germ (X, S) with compact S is paracompact. Finally, if X is paracompact and S is closed, then (X, S) is paracompact. 4.3.5. Proposition (Continuity Theorem). -- Let (X, S) be a paracompact k-germ, and let F be a sheaf of sets on X. Then the following map H~(ad, F) --% Ha((X, S), F,x,s,), adD8 where all runs through open neighborhoods of S in X, is b~]ective in each of the foUowing cases: (1) q = 0; (2) F is an abelian sheaf and q >1 0; (3) S is locally closed in X, F is a sheaf of groups, q = 1 and one takes ffI~ instead of H ~. Proof. -- Consider the following commutative diagram of morphisms of sites [Xl T,s "% S From Proposition 4.2.2 (resp. 4.2.4) it follows that in the case (1) (resp. (2)) there is an isomorphism of sheaves i*~(~,F) -~s,(F(x,s~) (resp. i~(Rq~,F) -%R~ns.(F~x,s~)). Therefore the case (1) (resp. (2)) follows from the corresponding topological fact [God], II. 3.3.1 (resp. [Gro], 3.10.2). (3) We may assume that S is closed in X. Let ~ ~ffII((X,S),F), and let {(V~, T~)-~ (X, S))i~z be an 6tale covering such that ~ is induced by a cocycle of this coveting. We can refine the coveting and assume that the morphisms f induce finite morphisms V~-+o//,:=f(V,), Ti =f-l(S), the sets q/, are paracompact, and { ~'~ }~ ~ i is a locally finite covering of X. The sets S~ : = S n q/, and their finite intersections have a basis of paracompact neighborhoods. It follows that we can find, for each pair i,j ~ I, an open neighborhood ~j of S~ n S i in ~ n ok'~ and, for each triple i,j, l e I, an open neighborhood q/~ of S~ n Sjn S z in q/~j n q/~ n ~//~ such that a is defined by elements ao ~ F(fTa(q/*J)) , wherefj :V~ � ~ X, that satisfy the cocycle condition in F(f~ l(q/,jt)), where f~ : V, � x Vj � x Vl -+ X. Let now { q/~ ), ~ be an open coveting of X with compact q~ and o~ C q/~. Since S is closed, the sets S~ n ~ are compact. Therefore there exist open neighborhoods q/~' of S~ n q/~ in q/~ H II " qz"C oll~ and q/~' n o//~ n ok'~ C q/o~. Then ~ comes from the group such that q/~ n _~ ffI~(q/, F), where q/= I.J~ e r @~'. 9 li___~m 92 VLADIMIR G. BERKOVICH 4.3.6. Corollary. -- Let q~:Y-+ X be a quasi-immersion of analytic spaces over k. Suppose that the image ~(Y) of Y has a basis of paracompact neighborhoods in X. Then for any abelian sheaf F on X there is a canonical isomorphism Ha(q/, F) --% Ha(Y, ~* F), q >/ 0, where all runs through open neighborhoods of q~(Y) in X. 9 For a Hausdorff k-analytic space X, let P(X~a ) denote the category of abelian presheaves on the category of closed analytic domains in X. Then any abelian presheaf P E P(Xoa ) and any covering r = { V~ }~ei of X by closed analytic domains define a ~ech complex ~(~, P). Its cohomology groups are denoted by ~I"(~, P). One has I~I~(~, P) = R q L~(P), where L,r : P(XGa ) ~ ~r is the left exact functor defined as follows L,~-(P) = Ker(I1 P(V,) -~ II P(V, c~ V~)). 4.3.7. Theorem (Leray spectral sequence). -- Let X be a paracompact k.analyt# space. For F ~ S(X) and q >1 O, let ~(F) denote the abelian preasheaf V ~ Ha(V, F Iv). Then for any locally finite covering r = { V, }~ ~ f of X by closed analytic domains there is a spectral sequence = V). Proof. -- We will show that the required spectral sequence is the Grothendieck spectral sequence ([Gro], 2.4.1) of the composition of functors L~ S(X) -~Q P(X%t ) ~ Mb, where Q is the composition of functors i 7v jP S(X) P(X) 2Z~ P (Xe.d), and the functor j~ is defined as follows jP P(V) = lira p(a//). --..> (Here P(X) and P([ X [) are the categories of abelian presheaves on the dtale and the usual topologies of X, respectively, and ~, is the restriction functor.) We have to verify the following three facts: O) R~ Q(F) = a~q(V); (2) if F is injective, then the presheaf Q.(F) is L,~-acyclic; (3) (L,. o O) (V) = H~ r). (1) From Corollary 4.3.6 it follows that for any closed analytic domain V C X one has 3((q(F) (V) = lim Ha(q/, F). q/DV li___~m ]~TALE COHOMOLOGY FOR. NON-ARCHIMEDEAN ANALYTIC SPACES 93 This means that ~(F) = j P(zc~(~(F))). But the functors =~ and j~ are exact, and the functors i and ~:~ send injectives to injectives. Therefore R ~ Q.(F) = ~(F). (2) Consider the commutative diagram of functors s(x) > P(X) s(Ixl) P(IXl) If F is injective, then ~,(F) is an injective sheaf on [ X [. By Theorem II. 5.2.3 c) from [God], one has fiq(~,j" o i' o rc.(V)) = 0 for q >/ 1. From the above diagram it follows that Q = j~ o r% o i = j~ o i' o r~., i.e., the sheaf Q(F) is L~-acyclic. (3) follows from the above diagram and Theorem II.5.2.2 from [God]. 9 4.3.8. Corollary. -- Let n be an integer prime to char(k). Then for any k-analytic space X the group Ha(X, [z,,) is canonically isomorphic to the group of isomorphism classes of pairs (L, r where L ~ Pic(X~) and r is an isomorphism Oxo -~ L| Proof. -- The homomorpkism from the first group to the second one (let us denote it by 'Hi(X, ~t,)) was constructed in the end of w 4.1, and we know that it is an iso- morphism if X is good. Suppose that X is paracompact, and let r be a locally finite affinoid covering of X. The Leray spectral sequence for r gives an exact sequence (*) 0 -+E~ '~ --*Ha(X, ~,) --->E 0'1 --~E 2'0. If we set 'H~ ~,) := H~ ~,) and define groups 'E~'~, q = 0, 1, in terms of the groups 'Hi(X, ~,), i = 0, 1, in the same way as E~ '~, q = 0, 1, are defined in terms of the groups Hi(X, ~z,), i = 0, 1, then one can show directly that the similar exact sequence (',) takes place even in the more general situation when X is arbitrary and $/" is a quasinet of analytic domains in X. In particular, the homomorphism considered is an isomorphism when X is paracompact. If X is Hausdorff, we use the analogous spectral sequences (,) and (',) for a covering ~r of X by open paracompact subsets. If X is arbitrary, we use the same reasoning for a covering of X by open Hausdorff subsets. 9 4.4. Quasiconstructible sheaves Let (X, S) be a k-germ. A sheaf F on (X, S) is said to be locally constant if there exists an 6tale covering {(Ui, Ti) -+ (X, S)}ie I such that the restriction FIUi,T; ~ of F to every (Us, T~) is a constant sheaf. A locally constant sheaf F is said to be finite if all the above sheaves F(gi, Ti) are defined by finite sets. There is a one-to-one correspondence 94 VLADIMIR G. BERKOVICH between the set of isomorphism classes of finite locally constant sheaves on (X, S) such that their stalks consist of n elements and the set ~Ia(X, Z,), where Z, is the symmetric group of degree n. If F is finite locally constant, then for any sheaf F' on (X, S) and any point x e S one has o~~ F')~ -~om(F,, F~). Furthermore, the full subcategory of S(X, S) consisting of finite locally constant abelian sheaves is abelian and preserved under extensions. (Everything above holds in any topos.) We remark that if : (Y, T) ~ (X, S) is a quasi-immersion of germs over k, then the categories of finite locally constant sheaves on (Y, T) and (X, ~(Y)) are equivalent. 4.4.1. Proposition. -- Let (X, S) be a paracompact k-germ, and suppose that S is locally closed in X. Then any finite locally constant F on (X, S) comes frora a finite locally constant sheaf on an open neighborhood of S in X. Proof. -- For an integer n >t 0, let S, denote the set of all points x e S such that the stalk F, consists of n elements. Then the sets S, are disjoint and open in S, and therefore the k-germs (X, S,) are also paracompact. Replacing S by S,, we may assume that all stalks of F consist of n elements. In this case the required statement follows from Proposition 4.3.5 (the case (3)). 9 4.4.2. Definition. -- A sheaf F on a k-germ (X, S) is said to be quasiconstructible if there is a finite decreasing sequence of closed subsets S ----- S O D S x D... D S, D S,+1 = O such that for all 0~< i~< n the sheaves F~x.s/\sl.0 are finite locally constant. It is clear that the inverse image of a quasiconstructible sheaf under a morphism of germs over k is quasiconstructible. 4.4.8. Proposition. -- The full subcategory of $(X, S) consisting of quasiconstructible sheaves is abelian and preserved under extensions. Furthermore, any quotient sheaf (and therefore any subsheaf) of a quasiconstructible sheaf is quasiconstructible. Proof. ~ The first statement follows from the corresponding properties of finite locally constant sheaves. To verify the second statement, it suffices to show that ff : F ~ G is an epimorphism of abefian sheaves on (X, S) and F is finite locally constant, then G is quasiconstructible. It suffices to assume that F is locally isomorphic in the ~tale topology to (Zip" Z)~x.s~, where p is a prime integer. The set T of the points x e S such that ~ induces an isomorphism F, ~ G, is closed in S. Over the germ (X, S\T), at goes through the quotient sheaf F/p"-x F. By induction there is a decreasing sequence of r t t t closed subsets S\T : S O D S 1 D... D S m D Sin+ 1 0 such that the sheaves G~x,s, ~s'.l~ are finite locally constant. We get a decreasing sequence of closed subsets S : SoD S~D ... D S,,+ID S,,+~ = O, where S~=S;uT for O~<i~<m+l. Since S~\S~+I=S'~kS;+~ for O~<i~<m and S,.+~ = T, the sheaves Gcx, si\s~§ ~ are finite locally constant. 9 ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 95 Here is an example of a quasiconstructible sheaf. Let T be a locally closed subset of S, and denote by j the canonical morphism of k-germs (X, T) -+ (X, S). Then for any sheaf G on (X, T) one can define the following subsheaf of j, G ofj. G (this is a parti- cular case of a construction from w 5.1). If (X', S') ~ (X, S)) ~ l~t(X, S), then j, G(X', S') consists of all elements of G(X',f-I(T)) whose support is closed in S'. It is clear that (j~ G)<x,T~ = G and (Jr G)~x. s\T> = 0. Therefore if the sheaf G is finite locally constant, then the sheaf j, G is quasiconstructible. Moreover, if G is quasiconstructible, then so is j, G. &.4.4. Proposition. -- Any quasiconstruetible abelian sheaf F on (X, S) has a finite filtration whose subsequent quotients are of the form Jx G, where j is the morpkism of k-germs (X, T) --+ (X, S) defined by a locally closed subset T C S and G is a finite locally constant abelian sheaf on (X, T). Proof. -- Let S-----SoDSxD...DS, DS,+I-----O be a decreasing sequence of closed subsets such that the sheaves F(X, sikfli+l ) are finite locally constant. The set T = S\S1 is open in S, and the canonical morphism j : (X, T) -+ (X, S) induces a monomorphism of sheaves j, F<x,T ~ -+ F. The required statement is obtained by applying the induction to the pullback of the quotient sheaf on the germ (X, St). 9 4.&.5. Proposition. -- Any abelian torsion sheaf F on (X, S) is a filtered inductive limit of quasiconstructible sheaves. 4.4.6. Lemma. -- Let 07 : Y ~ X be a finite 6tale morphism of k-analytic spaces. Then for any finite locally constant sheaf F on Y the sheaf q~. F is finite locally constant. Proof. -- We may assume that X is connected. If V is a connected affinoid domain in X, then the rank of the finite morphism of k-affinoid spaces 9-1(V) -+ V does not depend on V. Let rk(?) denote this rank. We prove the statement by induction on rk(~). If rk(q~) = 1, then q~ is an isomorphism, and the statement is trivial. In the general case we consider 9 as an 6tale covering of X and reduce the situation to the morphism Y Z x Y ~ X' : = Y. The latter morphism has a section ~r which is an open immersion. Therefore, Y X x Y = 6(X') H Y'. Replacing X' by a connected component, we reduce the situation to the morphism ~' : Y' -+ X' whose rank is less than rk(~). 9 Proof of Proposition 4.4.5. -- To simplify notation, we consider only the case of a k-analytic space X (instead of the k-germ (X, S)). If { G, ),~x is a finite family of quasiconstructible subsheaves of F, then the sheaf G := Im((~+ G+ -+ F) is also quasi- constructible (Proposition 4.4.3). Therefore it suffices to show that for any point x ~ S and any element f~ F, there exists a quasiconstructible subsheaf G C F with f e G,. Shrinking X, we may assume that there is a finite Etale surjective morphism ~ : U -+ X 96 VLADIMIR G. BERKOVICH such thatfcomes from a torsion element ofF(U). Let nf ---- 0, n/> 1, and let (Z/nZ)v --+ F Iv be the homomorphism that takes 1 to f. It induces a homomorphism ~?,(Z/nZ)v ~ F. By Lemma 4.4.6, the first sheaf is finite locally constant, and therefore, by Propo- sition 4.4.3, its image in F is quasiconstructible. The required statement follows. 9 w 5. Cohomology with compact support 5.1. Cohomology with support A family (I) of closed subsets of a topological space S is said to be a family of supports if it is preserved under finite unions and contains all closed subsets of any set from ~. The family of supports 9 is said to be paracompactifying if any A e ~ is paracompact and has a neighborhood B ~ (I). Furthermore, for a continuous map q~ : T ~ S and families of supports (I) and tF in S and T, respectively, we denote by ~ the family supports in T which consists of all closed subsets A C T such that A e tF and ~(A) e ~. For example, if tF is the family of all closed subsets of T, then q~ = ~-1(~), where q~-l(~) consists of all closed subsets of the sets ~-I(B) for B ~. 5.1.1. Example. -- Let S be a Hausdorff topological space. Then the family C s of all compact subsets of S is a family of supports. If S is locally compact, then C s is paracompactifying. More generally, let ? : T ~ be S a Hausdorff continuous map and assume that each point of S has a compact neighborhood. Then the family C, of all closed subsets A C T such that the induced map A -+ S is compact is a family of supports. If S is paracompact and T is locally compact, then the family C, is paracompactifying. Let (X, S) be a k-germ, and let 9 be a family of supports in S. Then one can define the following left exact functor I'r : S(X, S) -+ rib: re(F) = { s e F(X, S) I Supp(s) e q~ }. The values of its right derived functors are denoted by H~((X, S), F), n >/ 0. For example, if~ is the family of all closed subsets of S, then we get the groups H"((X, S), F). If~ is the family of all closed subsets of a fixed closed subset Z C S, we get the cohomology groups with support i,z Z denoted by H~((X, S), F). If S is Hausdorff and 9 = Cs, then we get the cohomology groups with compact support H~"((X, S), F). We remark that for any family of supports (I) in S and any F e S(X, S) there is a spectral sequence H~(S, R ~ n,(F)) =~ H~ + ~((X, S), F), where ~ is the morphism of sites (X, S)~ t ~ S. Let now ? : (Y, T) ~ (X, S) be a morphism of germs over k. For ((U, R) 5 (X, S)) s t~t(X, S) I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 97 and a morphism (V, P) -~ (U, R) in l~t(X, S) we introduce notations for germs and morphisms by the following diagram with cartesian squares <p (Y, T) ~ (X, S) (Y~, T~) > (U, R) (Y~,T~) ~s~ (V, P) 5.1. B. Definition. -- A q-family of supports is a system 9 of families of supports 9 (f) in T I for all ((U, R) ~ (X, S)) e ]~t(X, S) such that it satisfies the following conditions: (1) for any morphism (V, P) L (U, R) in ]~t(X, S), one has g~-l(q)(f)) C ~(fg); o i (2) if for a closed subset A C T I there exists a covering { (V~, P~) --> (U, R)}~ i in I~t(X,S) such that g~,,()erb(fg~) -1 A for all i eI, then A eq)(f). (ii) The q-family of supports 9 is said to be paracompactifying if, for any ((U, R) ~ (X, S)) ~ ]~t(X, S), each point of R has a neighborhood (V, P) -~ (U, R) in l~t(X, S) such that the family of supports 6p(fg) is paracompactifying. For a family of supports 9 in S and a ~-family of supports ~, we denote by q)~ the family of supports ~F(id) in T. Let now t~ : (Z, P) -+ (Y, T) be a second mor- phism, and let gP (resp. ~) be a q-family (resp. t~-family) of supports. For ((U, R) ->~ (X, S)) e ]~t(X, S),we set (~) (f) ----- q)(f) ~F(f~). Then ~" = {(~F) (f)} is a q+-family of supports. Furthermore, if q) is a q-family of supports and ((U, R) L (X, S)) e gt(X, S), then, for any A e q)(f), each point u e R has an open neighborhood og in R such that the closure of the set f,(A n q)-~(q/)) belongs to ~(id). 5.1.3. Example. -- Let q : (Y, T) ~ (X, S) be a morphism of germs over k such that the induced map T ---> S is Hausdorff. For ((U, R) ~ (X, S)) e ]~t(X, S), we set ~f~(f) ---- C~f. Then ~ ~ { ~(f)} is a q-family of supports. If S and T are locally closed in X and Y, respectively, then the family ~ is paracompactifying. Furthermore, if S is Hausdorff, then ~s ~ ---- C~. If ~ : (Z, P) -~ (Y, T) is a second morphism of germs such that the induced map P --* T is also Hausdorff, then ~, ~.~ = ~. A 9-family of supports 9 defines a left exact functor 9| S(Y, T) ~ 8(X, S) as follows. If F e S(V, T) and ((U, R) _~I (X, S)) e ]~t(X, S), then (~r F) (U, R) = {s ~ F(Y s, Ts)[ Supp(s) cO(f)}. 13 98 VLADIMIR G. BERKOVICH For example, if* is the family of all closed subsets, then we get the functor ?.. If the map T -+ S is Hausdorff and * ---- g',, then we get a left exact functor S(Y, T) ~ S(X, S) which is denoted by ?,. Furthermore, if 9 is a family of supports in S, then there is an isomorphism of functors F,r o ?| -% F,r| If + : (Z, P) -~ (Y, T) is a second morphism of germs and 9 is a +-family of supports, then there is an isomorphism of functors ?| o -% Finally, let (X, S) be a k-germ, and let i be the canonical morphism of k-germs (X, R) -~ (X, S) defined by a closed subset R C S. Then one can define a left exact functor i: : s(x, s) s(x, P.) as follows. Let j denote the canonical morphism (X, T) -+ (X, S), where T = S\R, and let F ~ S(X, S). The stalks of the sheaf F'= Ker(F-+j.j* F) are equal to zero outside R. By Proposition 4.3.4 (ii), F' ~ i.(i* F'). We set i ~ F = i* F'. Since there is an exact sequence 0-~ i.(i: F) -+ F-+j.(j* F), the functor i: is left exact. Its right derived functors are denoted by W~((X, S), F). It is easy to see that the functor i ~ is right adjoint to the functor i.. It follows that the functor i ~ takes injectives to injectives. 5.2. Properties of cohomology with support 5.2.1. Proposition. -- Let ? : (Y, T) ~ (X, S) be a morphism ofgerms over k, and let~ be a q~-farnily of supports. Then for any F ~ S(Y, T) and any n >i 0 the sheaf R" ?o(F)/s associated with the presheaf ((U, R) ~ (X, S)) ~ Hg(1~((Yf, TI), F). Proof. -- Let P" F denote the presheaf considered. We have an exact O-functor { P" },>_.0 : S(Y, T) ~ P(X, S) (see [Gro], w 2.1). If S" F denotes the sheaf associated with the presheaf P" F, then we get an exact 0-functor { S" },>/0 : S(Y, T) ~ S(X, S). Since ?. -% S O and { R" ?. ),>_.0 is a universal 0-functor that extends ?| there is a morphism of O-functors R" ?| ~ S", n >/ 0. It is an isomorphism because the 0-functor { S" },>_-o is exact, and P" F = 0 (and therefore S"F-----0) for any injective sheaf F E S(Y, T) and n >I 1. 9 5.2.2. Theorem (Leray spectral sequence). -- Let q~ : (Y, T) ~ (X, S) be a morphism of germs over k. Let r be a family of supports in S, and let 9 be a ?-family of supports. Suppose that the family 9 is paracompactifying, or that q~ is the family of all closed subsets ofT. Then for any abelian sheaf F on (Y, T) there is a spectral sequence E~" = Hg((X, S), R" ?s,(V)) :~ H.,v ((Y, T), F). Proof. -- Since r| F) = r.~,(F), to apply Theorem 2.4.1 from [Gro], it suffices to verify that the funetor ?,r sends injective sheaves to ro-acyclic sheaves. If 9 is the family of all closed subsets ofT, then this is evident because ?,r = ?. sends injectives to injectives. I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 99 fi.ll.3. Lemma. -- If F is an injective sheaf on (Y, T), then the stalk (~, F), of r F at a point x ~ S is a flabby G2rc,~-module. Proof. -- Since our statement is local, we can shrink X and assume that it is Haus- dorff. For a subset OC T we denote by ZQ the analytic space over k which is a disjoint union of the spaces ~t'(3f'(y)) over all points y ~ Q. The category $(ZQ) is equivalent to the category of families { M(y)}~Q of discrete Gae~u}-modules M(y). Let gQ denote the canonical morphism ZQ ~ (Y, T) of germs over k. Furthermore, for every point y e T we take an embedding of Fv in an injective G~e~u}-module M(y). Let MQ denote the sheaf on $(ZQ) which corresponds to the family { M(y)}u~ Q, and set NQ = IQ.(MQ). It is clear that the sheaves MQ and NQ are injective, and there is a canonical embedding of sheaves F ,-+ N~. Since F and N T are injective, F is a direct summand of N~, and therefore it suffices to verify our statement for the sheaf N~. We claim that the canonical embeddings of sheaves NQ -+ N~ induce an isomorphism lim q~,(NQ) -% ~,(NT). Q E ~F(ld) Indeed, let s s (~,r NT) (U, R), where ((U, R) ~ (X, S)) is a neighborhood of the point x in l~t(X, S), i.e., s e NT(Y I, TI) and Supp(s) e ~(f). For a point u E R we take an open neighborhood q/such that the closure off,(Supp(s) t~ ~7 l(q/),) in T belongs to ~. Denoting this closure by Q, we see that the restriction of s to ~7~(01/, R n o1/) comes from NQ(~71(q/, R n o?/)). The lemma now follows from the fact that the filtered inductive limit of flabby Gar{,}-modules is a flabby Gae~,}-module. 9 Suppose that the family ~ is paracompactifying, and let F be an injective sheaf on (Y, T). From Lemma 5.2.3 and Proposition 4.2.4 it follows that R~ rc.(~r F) = 0 for all q >/ 1, where ~ is the morphism of sites (X, S)6 t -+ S. From the spectral sequence Hg(S, g ~ ~,(~,r r)) = Hg + ~((X, S), q~, F) it follows that Hg((X, S), q~,r F) = Hg(S, ~,(~, F)). Since the family 9 is paracompac- tifying, the restriction of the sheaf q~,(F) to the usual topology of S is Fo-acyclic, by Lemma 3.7.1 from [Gro]. The theorem is proved. 9 Applying Theorem 5.2.2 and Proposition 5.2.1, we get the following 5.~..~. Corollary. -- Let (Z, R) ~* (Y, T) ~ (X, S) be morphisms of germs over k. Let 9 be a ?-family of supports, and let 9 be a +-family of supports. Suppose that the family r is paracompactifying, or that 9 is the family of all closed subsets of R. Then for any abelian sheaf F on (Z, R) there is a spectral sequence R" ~(R ~ d?~,(F)) =~ R'+r (V). 9 Let j : (X, T) -+ (X, S) be the morphism of k-germs defined by a locally closed subset TC S. The functor j~ is exact because the stalk (j, F), coincides with F, if x s T 100 VLADIMIR G. BERKOVICH and is equal to zero if x r T. If T is closed in S, then Jt = J,- If T is open in S, then the functorjt is left adjoint to j*. We remark that for any family of supports 9 in S, one ~j = ~z := ~ It. has 5.2.5. Corollary. -- Let j : (X, T) ~ (X, S) be the morphism of k-germs defined by a locally closed subset T C S, and let 9 be a paracompactifying family of supports in S. Then for any abelian sheaf F on (X, T) and any q >1 0 there is a canonical isomorphism H~((X, S),j: F) --% H~((X, T), V). 9 If 9 is thc family of all closed subsets of T, then ~ = 9 t~ T, wherc ~nT={Ac~T]A~). 5.2.6. Proposition. -- Let (X, S) be a k-germ, F an abelian sheaf on (X, S), * a family of supports in S, T an open subset of S, and R = S\T. (i) There is an exact sequence ... ~ H~-I((X, S), F) ~ H~L89 T), F,x.T,) -+ --+ H~R((X, S), F) ~ H~((X, S), F) ~ H~,,~T((X, T), F,x.T,) -+ ... (ii) If 9 is paracompactifying, then there is an exact sequence ... ~ H~-I((X, S), F) --+ H~~((X, R), Ftx, m ) --+ --+ H~((X, T), F,X,T,) + H~((X, S), F) + Hk((X, R), F,x.a,) +... Proof. -- Consider the following morphisms of k-germs (X, T) ~!~ (X, S) ~- (X, R). (ii) The long exact sequence is obtained from the following exact sequence of sheaves on (X, S) 0 -+j, F<x,T , -+ F ~ i. Fcx, m ~ 0, using the isomorphisms H$((X, S),j, r,x.T,) ---- H~((X, T), V,x,T,), H~((X, S), i. F,x,m ) = Hk((X, R), F,X,R,)- (i) Recall that for any abelian category M the category .Lf(.~) of covariant left exact functors d-+ ~r is abelian (see, for example, [Mit]). Namely, a morphism of functors F -+ G is surjecfive if for any A e ~r and an element a e G(A) there exist a monomorphism A ---> B and an element f~ e F(B) such that the images of the elements 0c and ~ in G(B) coincide. We claim that there is the following short exact sequence of left exact functors on S(X, S) 0 ~ rr ~ r| ~ r| ~0. I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES I01 For this we use the construction from the proof of Lemma 5.2.3. Let F -+ N s be the monomorphism constructed there. It suffices to show that the canonical mapping P~(Ns) -+ rc~T(Ns l,x,T,) is surjective. By the proof of Lemma 5.2.3, the first (resp. second) group coincides with the inductive limit over A e r of the groups Pa(Ns) (resp. FAnT(N s [CX, T~))- But PA(Ns) = Ms(a), Pac~T(Ns I,x,T,) = Ms(A n T), and the mapping Ms(A ) -+ Ms(A c~ T) is evidently surjective. The required exact sequence is induced by the above short exact sequence. 9 5.2.7. Corollary. -- Suppose that (X, S) is a k-germ, R is a closed subset Of S, T = S\R, i : (X, R) -+ (X, S) and j: (X, T) --> (X, S) are the canonical morpkisms, and F e S(X, S). Then (i) there is an exact sequence 0 -+i.(i' F) -+F -+j.j* F -+ i.~((X, S), F) -->0; (ii) for any q >1 1 there is a canonical isomorphism Rqj,(j * F) --% 3f'~+ ~((X, S), r). Proof. -- Let ((U, Q) ~ (X, S)) ~ ~t(X, S). Applying the exact sequence 5.2.6 (i) to the k-germs (U, Q.), (U,f-I(R)), (U,f-I(T)), and to the family of all closed subsets of S, we get a long exact sequence of presheaves on (X, S). It induces a long exact sequence of the associated sheaves. It remains to remark that the sheaf associated with the presheaf (U, Q) ~ Hq((U, Q), F), q >1 1, is equal to zero, and the sheaf associated with the presheaf (U, Q) ~ H~-~m((U, Q), F), q >t 0, coincides with i. ~((X, S), F). 9 5.2.8. Proposition. -- Let (X, S) be a k-germ, and let dp be a paracompactifying family of supports in S. Then for any abelian sheaf F on (X, S) and any q >>. 0 there is a canonical iso- morphism H~T((X , T), F,x,T,) -~ H~((X, S), F), where the limit is taken over the family of all open subsets T C S whose closure belong to dp. Prooof. -- The assertion is evidently true for q = 0. The general case is obtained using Proposition 3.10.1 from [Gro]. 9 5.2.9. Proposition. -- Let (X, S) be a k-germ with Hausdorff X and locally closed S, and let F be an abelian sheaf on (X, S) which # a filtered inductive limit of abelian sheaves, i.e., F = lim F~. Then for any q >1 0 there is a canonical isomorphism lim H~((X, S), F,) ~ H",((X, S), F). lit___~n 102 VLADIMIR G. BERKOVICtt Proof. -- The statement is evidently true for q = 0. Since an inductive system of sheaves has a resolution by inductive systems of injective sheaves, it suffices to veri~ that if all the sheaves F, are injective, then Hg((X, S), F) = 0 for all q >/ 1. But this follows from Proposition 4.2.4 and the corresponding fact for the usual cohomology with compact supports (see [God], II.4.12.1). 9 5.3. The stalks oi" the sheaf R ~ ~?, F 5.3.1. Theorem (Weak Base Change Theorem). -- Let ?:Y-~ X be a Itausdorff morphism of k-analytic spaces, and let F e S(Y) and x e X. We set Y~ = Y, | ~t~ ~ and denote by F, (resp. F;) the inverse image off on Y~ (resp. Y~). Then for any q >~ 0 there is an isomorphism G~et,;modules (R ~ ~, F)z -~ H~(Y~, F~). Proof. -- Since our statement is local, we can decrease X and assume that X and Y are Hausdorff. 5.3.2. Lemma. -- There is an isomorphism (~ F), (gf'(x))~ He(y,,0 Fz). Proof. -- From Proposition 4.2.2 it follows that (q~, F),(gff(x)) = li_~m {s e F(r I the map Supp(s) -+ 0g is compact}. Furthermore, since any compact subset of Y has a basis of paracompact open neigh- borhoods, from Propositions 4.3.4 and 4.3.5 it follows that Hr F,) = lim {s EF(~//')] the set Supp(s) n ?-'(x) is compact}. Therefore our statement follows from the well-known topological fact. 9 5.3.3. Corollary. -- One has (?, F)~ lira o ^ = H,(Y. | K, F.), K/,.,~t~ where K runs through finite extensions of off(x) in ~f'(x) ~ 9 5.3.4. Lernma. -- Let X be a Hausdorff k-analytic space, and let F e S(X). We set X' = X ~ k~ and denote by F' the inverse image of F on X'. Then li_~m H~(X b K, F) -~ H~(X', F'), where K runs through finite extensions of k in k'. ~TALE COHOMOLOGY FOR NON-ARGHIMEDEAN ANALYTIC SPACES 103 Proof. -- Let x' e X'. For a finite extension K of k in k' we denote by x x the image ofx' in X~K. We set x = x k and fix an embedding of fields 9ff(x)*,-+9~(x')* over the embedding 3r ,-. 9~'(x'). Since 9~'(x) k" is everywhere dense in 3(F(x') a, Ca,c,, } -% Ga,c,}, . Therefore there is an exact sequence of Galois groups 0 ~ G~er --~ G#(., -+ G(~f(x) k'/3C~(x)) ~ O. We remark that G(~f(x)kS/~(x)) is a closed subgroup of G k. It follows that ]~,(,~(x')) = lim F~(J~(x~r)). K/k We now claim that for any open neighborhood all' ofx' there exist a finite extension K of k in k* and an open neighborhood og of the point x x such that the preimage of 0//in X' is contained in q/'. Indeed, since the point x has a neighborhood which is a finite union of affinoid domains, the situation is reduced to the case when X = .//t'(d) is k-affinoid. Shrinking d//,, we may assume that ~' = {y s X' ] [f(y) ] < a,, ] gj(y)[ > bj, 1 <~ i <~ n, 1 <~ j <, m }, where f, gj e d | K' for some finite extension K' of k in k% Let K be the maximal subextension of K' separable over k. Ifp = char(k) > 0, then (K')~ts K for some l t> 0, and therefore replacing f, gj, ai, bj by f,t, g~.t, aft, b~.t, respectively, we may assume that f, g~ e d | K. Then the same inequalities define an open neighborhood ~//of x x in X | K whose preimage in X' is og,. 0 t Finally, let s e H,(X, F'). Then for any point x' e Supp(s) there exist a finite subextension k C K C k" and an open neighborhood o// of the point x~: such that the restriction of s to the preimage of q/is induced by an element of H~ F). Since Supp(s) is compact, we can find a finite extension K ofk in k* such that s is induced by an element of H~ Q K, F). It 5.3.5. Corollary. -- In the situation of Lemma 5.3.4, for any q >1 0 there is a canonical isomorphism lim H~(X | K, F) -% H~(X', V'). Proof. -- It suffices to show that ifF is an injective sheaf on X, then H~(X', F') = 0 for all q>l 1. First of all, for any point x' e X', F~, is a flabby G~ei,,~-module. Indeed, if x is the image of x' in X, then F~ is a flabby G~e~,~-module. It follows that F, is a flabby H-module for any closed subgroup H C G~e~,~. Since G~el,,~ is a closed subgroup of G~e(,) and F',. = F,, F~, is a flabby G~e(,,)-module. Consider now the morphism of sites X',t -+ ] X' I- From the previous fact it follows that R q ~, ' F' = 0 for all q 1> 1, and therefore H,(X, q ' F') = H",(] X' 1, re, ' F'). To verify that the latter group is trivial, it suffices to show that for any compact subset X'C X' 104 VLADIMIR G. BERKOVICH and any element s'~ F'(Z') there exists an element t'~ F'(X') which induces s'. By the reasoning from the proof of Lemma 5.3.4, we can find a finite extension K of k in k' and an element s e F(Y.), where Z is the image of Z' in X6 K, which give rise to s'. Since F is injective, there exists an element t e F(X | K) that induces s, and the required fact follows. 9 The case q = 0 of our theorem follows from Corollary 5.3.3 and Lemma 5.3.4. So it suffices to show that if F is an injective sheaf on Y, then H~(Y~, F~) = 0 for all q/> 1 and x e X. By Corollary 5.3.5, it suffices to show that H~(Y,, F) = 0. This is proved, using the reasoning from the proof of Corollary 5.3.5. 9 We say that a morphism of analytic spaces over k,f: X' ~ X, is restricted if for any pair of points x' ~ X' and x ~ X with f(x') = x the canonical embedding of fields ~r176 ~--->~r176 ') extends to an embedding ~r162 ~r a whose image is everywhere dense. For example, quasi-immersions and morphisms of the form X | k~--~ X are restricted. 5.3.6. Corollary. -- Let c? : Y ---> X be a Hausdorff morphism of k-analytic spaces, and let f: X' ---> X be a restricted morphism of analytic spaces over k, which give rise to a cartesian diagram Y *> X T, T, Y' ~'~ X' Then for any abelian sheaf F on Y and any q >>. 0 there is a canonical isomorphism f'(R ~ ~, F) -~ R~ ~;(f'* r). 9 The following is a consequence of Corollary 5.3.5. 5.3.7. Corollary (Hochschild-Serre Spectral Sequence). -- Let X be a Hausdorff k-analytic space, and let F e $(X). We set X' = X | k~ and denote by F' the inverse image of F on X'. Then there is a spectral sequence E,~,~=H~(Gk, ~ ' l-l~,+qr F). 9 Ho(X, F')) =,- _.~ ~._, 5.8.8. Corollary. -- Let ~ : Y -+ X be a Hausdorff morphism of k-analytic spaces, and let F be an abelian torsion sheaf on Y. Then R ~ ~ F = for O all q > 2d, where d is the dimension of~. Proof. -- By Theorem 5.3.1, it suffices to show that if the field k is algebraically closed, and X is a Hausdorff k-analytic space of dimension d, then H~(X, F) = 0 for all q> 2d. By Proposition 5.2.8 and Corollary 5.2.5, one has H~(X, F) = ~ H~((X, ), (j,(r],)),x.~,), q/Cx t~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 105 where q/runs through open subsets with compact closure, j is the canonical embedding ~//~ X. By Proposition 4.3.5, the latter group coincides with the inductive limit of the groups H~(~,j, F I~ ) over all open paracompact neighborhoods 3v" of the compact set ~. The required fact follows from Theorem 4.2.6. 9 Let n be a positive integer, and let ~ : Y ~ X be a Hausdorff morphism of finite dimension. By Corollary 5.3.8, the derived functor R~:: D+(Y, Z/nZ) -+ D+(X, Z/nZ) takes Db(Y, Z/nZ) to Db(X, Z/nZ) and extends to an exact functor RV~ : D(Y, Z/nZ) --> D(X, Z/nZ) which takes D-(Y, Z/nZ) to D-(X, Z/nZ). 5.3.9. Theorem. -- Suppose that F" e D-(X, Z/nZ) and G" e D-(Y, Z/nZ) or that F" e Db(X, Z/nZ) has finite Tor-dimension and G" e D(Y, 7./nZ). Then there is a canonical isomorphism L L F" | R~,(G') -~ R~,(q~*(V') | G'). Proof. -- First of all, for arbitrary abelian sheaves F on X and G on Y there is a canonical homomorphism F @ qh(G) --> T~(q~*(F) @ G). From Theorem 5.3.1 it follows easily that it is an isomorphism. We claim that if F is flat and G is r then the sheaf q~*(F)@ G is also %,-acyclic. Indeed, by Theorem 5.3.1, we may assume that X = ~[(k), where k is algebraically closed. In this case F is a constant sheaf associated with a flat Z/nZ-module M. If M is free of finite rank, then for any q >/ 1 H~(Y, ~*(F) | G) ---- H~(Y, My| G) = H~(Y, G) | M = 0. It follows that the same is true if M is projective of finite rank. Our claim now follows from the facts that any flat module is a filtered inductive limit of projective modules of finite rank, and the functor G ~-* H~(Y, G) commutes with filtered inductive limits (Proposition 5.2.9). In the situation of the theorem we take a flat resolution P" -+ F" of F" and a %-acyclic resolution G'-* I" of G'. In the second case we may assume that P" is bounded. Then ~*(P') -+ ?*(F') is a fiat resolution of q~*(F'). By the previous claim, the complex ~*(P') | I" is %-acyclic, and therefore I, F" | R%(G') = P" | ?,(I') -% ~,(~*(P') | I') = R%(9*(F') | G'). The required statement follows. 9 5.3.10. Corollary. -- If G'e Db(Y, Z/nZ) is of finite Tor-dimension, then Rg~(G') is also of finite Tor-dimension, and for any F" e D(X, Z/nZ) there # a canonical isomorphism I~ L F" e Rg,(G') ~ Rg,(9*(F') | G'). 14 106 VLADIMIR G. BERKOVICH Proof. -- Suppose that H-~(G'| ") = 0 for aU i> m and G' E S(Y, Z/nZ). L L It follows that H-~(q~*(F) | G') ~- 0 and therefore R -~ ~:(q~*(F) | G') = 0 for all i> m and F c S(X,Z/nZ). By Theorem 4.3.9, H-*(F| 0. The second assertion follows from this. 9 If n is prime to char(k), we define for m cZ a sheaf ~.x as follows. If m>~ 0, | - tn V F v then ~.~,x = ~z., x. If m< 0, then ~.~,x = (~,x) , where = ~,~ffom(F, Z/nZ). For F" ~ D(X, Z/nZ) we set F'(m) = F" | ~,x- '~ Since ~.,xm is a locally free Z/nZ-module, F'(m) ----- F" | Vt.~,x . One has F'(m) (m') ~ F'(m + m') and ~*(F'(m)) = (q~* F') (m). 5.3.11. Corollary. -- Suppose that n is prime to char(k). Then for F" ~D(X, Z/nZ), G" ~ D(Y, Z/nZ) and m ~ Z there are canonical isomorphisms Rqh(G'(m)) -% (R~,G') (m) and Rqh(q~*F'(m)) -%V'| 9 5.4. The trace mapping for flat quasiflnite morphisms In this subsection, for every separated flat quasifinite morphism q~:Y-+ X of analytic spaces over k and every abelian sheaf F on X, we construct a trace mapping Tr, : q~, ~*(F) -~ F. Suppose that such mappings are already constructed. We say that Tr, are compa- tible with base change if for any separated flat quasifinite morphism of k-analytic spaces q~ : Y -+ X and any morphism f: X' ~ X of analytic spaces over k which give rise to a cartesian diagram Y ~>X T r ? Y' ~' 9 X' the following diagram is commutative ~', ~'*(f* F) 9;f'* ~* F -- f*(~, ~* F) f*F where Tr. is the evident homomorphism induced by Tr.: ~, ~*F-+ F (here we use the canonical isomorphism f* 9, G-% ~?~f'* G, G ~ S(Y), which is easily obtained from the proof of Corollary 4.3.2). ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 107 Furthermore, we say that the Tr~ are compatible with composition if for any separated flat quasifinite morphisms Z ~ Y -~ X the following diagram is commutative (~q;), (~q~)* F ~:(+, q;') qr ITr ~,~ ~, ~*(F) where Tr, is the evident homomorphism induced by Tr, : tp, qJ*(~* F) ~ ~* F. 5.4.1. Theorem. -- To every separated fiat quasifinite morphisra ~ : Y ~ X aM every abelian sheaf on X one can axsign a trace mapping Tr~ : Vt r -+ r. These mappings have the following properties and are uniquely determined by them: a) the Tr~ are functional on F; b) the Tr, are compatible with base change; c) the Tr~ are compatible with composition; d) if ~ is finite of constant rank d, then the composition homomorphism Tr~ F-+ e, ~*(F) = ~, ~'(F) ~ F is the multiplication by d. Proof. -- 1) Suppose that X = J/(K), where K is a non-Archimedean field over k. Then Y = ,~(L), where L is a finite product [[~ e i L~ of finite local Artinien K-algebras. Let K~ be the maximal separable extension of K in L~. We set Y~ = ~r and Y~ = ~r The categories S(Y~) and S(Y~) are canonically equivalent, Let ~ denote the morphism Y~ ~ X. From b)-d) it follows that the following equality should hold Tr, = Y~ [~: Kj Tr,,. If now L is a finite separable extension of K, then F can be regarded as a GK-module, and ~. ~*(F) is the induced module IndGG~(F) which is the set of all continuous maps f: G,~ ~ F such that f(hx) = hf(x) for h e G~. with the action (g f) (x) =f(xg) for g e Gx. In this case from b) it follows that Tr, should coincide with the homomorphism of G,r-modules Ind~(r) -~ F :f~ Z xf(x-X). z E O~Ob We remark that if ? is arbitrary and some mapping T : ~, 9*(F) -+ F induces on stalks the above homomorphisms, then T should coincide with Tr,. 108 VLADIMIR G. BERKOVICH 2) Suppose that ~ is finite. For (U ~ X) e ]~t(X), let { V,}ie x be the connected components of Yt = Y � x U. Then the induced morphisms ~ : V, -+ U are finite. We set P(U) = O~i F(U). The correspondence U ~ P(U) defines an abelian presheaf on X. Furthermore, the canonical homomorphisms P(U) ---- ~]~),ei F(U) -* O, ei(qh" 9: F) (U) = (~. ~" F) (U) define a homomorphism of presheaves P-+ 9. 9* F. We claim that it induces an iso- morphism of sheaves aP-% t?, 9" F. Indeed, it suffices to verify that it induces an iso- morphism on stalks, but this is evident. Let now d~ be the rank of q~ (recall that V~ is connected). For (s,),e I e O,e~ F(U) = P(U) we set Tv((s,),e ~) ---- ,~, d, s, ~ F(U). It is easy to see that the mappings T v define a homomorphism of presheaves P -+ F, and therefore a mapping % ~'(F) -§ F. Considering its stalks, we see that it should coincide with Tr,. 3) If ~ is an open embedding, then the functor ~, is left adjoint to ~?*. It is clear that Tr, should coincide with the adjunction mapping ~, ~'(F) -+ F. More generally, suppose that ~ can be represented as a composition yd~y' 2~ X' d~ X, where j and j' are open embeddings and ~' is finite. Then we define Tr, as the unique mapping which satisfies c). Considering the stalks, we see that Tr, does not depend on j, j' and ?'. 4) Let ~ be arbitrary. Take an open covering { q/] }~ e i of Y such that ~ induces finite morphisms ~--~ ~?(q/]~). We denote by ~?~ the morphism r ~ X and by '~, the canonical embedding ~ ~ Y. There is an exact sequence of sheaves on Y @~. ~x v.s,((e* F)l,%) ~ @*ex ~',((~?* F)I~,) ~ q0" V -. 0, where r = ~ c~ ~ and v u is the canonical embedding ~r Since the functor 9~ is exact, there is an exact sequence @,. ,e~ ~,,. ~;,(F) --* @)~e. ~,, ?;(F) -+ ~. ~*(F) --. O, where ~o are the induced morphisms ~r ~ X. From 3) it follows that the mapping Oier Tr~;: GIEI q~, q~(F) ~F I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 109 is zero on the image of @~,~ei?i~': ?~i(F) 9 Therefore it induces a mapping Tr~ : ~., ?*(F) -+ F. Considering its stalks, we see that it does not depend on the choice of the covering. Moreover, the mapping constructed has all the properties a)-d). 9 All the mappings which are induced by the trace mapping Try, will also be denoted by Tr,. For example, the induced homomorphisms H~(Y, q; F) = H~(X, % q?(F)) --+ H~(X, F) are examples of such mappings. 5.4.2. Remarks. -- (i) Let q~:Y ~ X be a flat finite morphism. Then ?.(0u is a locally free sheaf of Ox-algebras. Therefore one can define in a standard way the norm homomorphism N : ?.(r -+ Cx" This homomorphism extends naturally to a homomorphism of sheaves on the 4tale site of X N : q~.(Gm, y) -+ Gm, x. From Theorem 5.4.1 it follows easily that, for any n/> 1 prime to char(k), there is a commutative diagram of abelian sheaves on X 0 > q0,(~tn, y) > q0,(Gm, y) > q~,(Om, y) ) 0 0------+ ~z., x > Gin, x --- > Gin, x , 0 (ii) If q~ : Y ~ X is an 6tale morphism, then ?: is left adjoint to the functor 9". In this case the trace mapping Tr~ coincides with the adjunction mapping ?: q;(F) --* F. w 6. Calculation of cohomology for curves 6.1. The Comparison Theorem for projective curves Recall that a scheme 5F over k is called compactifiable if there exists an open immersion j : Y" ~ 5~ of W in a proper scheme ff over k. For such a scheme ~r and an abelian sheaf ~" on W one defines the dtale cohomology groups with compact support as follows: H~(Y', o~-) = H~(s o~-) (this definition does not depend on the choice of j). Recall also that, by Nagata's Theorem, any separated scheme of finite type over k is compactifiable. The following statement is the starting point for the induction in the proof of the Comparison Theorem for 110 VLADIMIR G. BERKOVICH Cohomology with Compact Support 7.1.1 (The proof of the latter theorem may be read immediately after the proof of 6.1.1.) 6.1.1. Theorem. -- Let ~" be a separated algebraic curve of finite type over k, and let o~" be an abelian torsion sheaf on Ys Then for any q >1 0 there is a canonical isomorphism H:(~', ~r) _~ g~(~, ~"). Proof. -- Denote the homomorphism considered by 0 ~. First we want to reduce the situation to the case when k is algebraically closed, ~ is projective, and ~" is finite constant. 1) We may assume that ~" is projective. -- Indeed, ifj : Y" ,-. ~ is an open embedding in a projective curve, then HI(~, ~') = H'(~j, ~'), H~(:~ ", ~-'*) = H'(:~",j~ #"~) and (j, #v),~ __%j~ ~'.~. 2) We may assume that 0~" is constructible. -- This is because any abelian torsion sheaf on ~ is a filtered inductive limit of constructible sheaves and the cohomology of s and ~ commutes with filtered inductive limits (Proposition 5.2.9). 3) We may assume that ~" is finite constant. -- Indeed, assume that 0 q are isomorphisms for such sheaves. Then 0 ~ are isomorphisms for any sheaf of the form #"---- %((Z/nZ)~) for some finite morphism ~? : ~' -+ ~ because in this case H~(~, ~') = Hq(~t, Z/nZ) and H~(~, ~-'~)-----H~(~/", Z/nZ) (Corollary 4.3.2). Furthermore, an arbitrary constructible sheaf ~" can be embedded in a finite direct sum of sheaves of the above form. It follows that there is an exact sequence 0 ~ ~- ~ ~-0 ~ o,-a ~ ... such that 0 a are isomorphisms for each #-~, i >1 0. Therefore 0 ~ are isomorphisms for #'. 4) We may assume that k is algebraically closed. -- Indeed, let ~' ---- s | k" and 2Y" = ~ | k~, and let ~" and #'" denote the pullbacks of #- on f' and s respectively. Then Hq(s #r,)__% H~(~,,, ~',,) because the fields k' and k~ are separably closed. Since ~""= ~" | k*, the required fact follows from the homomorphism of Hochschield- Serre spectral sequences H,(Gk, H~(R "', ~")) 9 H,+,(~, ~') H'(Gk, H'(~'", ~" .... )) :- H'+'(~ ", ~-") We remark that since the cohomological dimension of :~" and :~" is at most two, then the 0 ~ are isomorphisms for q ~> 2. If ~- = (Z/p" Z)~, where p = char(k), then the Artin-Schreier exact sequences for ~ and :~'~, the coincidence of the ~tale cohomology with the usual cohomology for coherent sheaves (Theorem 4.2.7) and GAGA (I-Bet], 3.4.10) imply that the {}~ are isomorphisms. Suppose that ~" = (Z/nZ)~r, where n is prime to char(k). This sheaf is isomorphic to ~t,,a~. Then 0 ~ is an isomorphism because ~o(s r) = ~0(:Lr~), by [Ber], 3.4.8 and 3.5.1. ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 111 Furthermore, the Kummer exact sequences for 5~ and ~'~, the Hilbert Theorem 90 (Proposition 4.1.10) and GAGA imply that 01 is an isomorphism and that for the veri- fication of the fact that 02 is an isomorphism it suffices to show that the n-torsion of the group H*(~ n, Gin) is trivial. This follows from the following lemma. 6.1.2. Lemma. -- Suppose that k is algebraically closed. Let X be a paracompact good one-dimensional k-analytic space. Then the torsion of the group H2(X, GN) /s p-torsion, where p = char(k). If char(k) = 0, then Hz(X, Gin) = 0. Proof. -- The spectral sequence of the morphism of sites ~z : X~t ---~[ X [, Lemma 4.2.8 and Proposition 1.2.18 imply that H2(X, Gin) = H~ X [, R2~. Gin, x). Thus, to prove the lemma it suffices to show that the sheaf R 2 ~z. Gm, x is p-torsion. By Proposition 4.2.4 and Lemma 4.2.8, for a point x ~ X one has (R 2 ~, Gin, x)~ = H2(G~,.,, (d~,,)*). The latter is the Brauer group of the local Henselian ring Ox, ~ and is isomorphic to the Brauer group of 1r (see [Gro2]). It is equal to zero, by Theorem 2.5.1. 9 The following fact is a corollary of Lemma 6.1.2. It will be used in w 6.2 and w 6.3 and will be proved in w 6.4 for arbitrary one-dimensional affinoid spaces and for integers n prime to char(~). 6.1.3. Corollary. -- Suppose that k is algebraically closed. Let X be an affinoid domain in the analytification ~ of a projective curve ~. Then for any integer n prime to char(k) one has Ha(X, ~,) = 0. Proof. -- The group H~(X, V,) does not change if we replace X by its inverse image in the normalization of the reduction of W. Therefore we may assume that s is smooth connected and X is connected. By Lemma 6.1.2 and the Kummer exact sequence, one has Hs(X, ~,) = Pic(X)/n Pic(X). We claim that the group Pic(X) is divisible. Since X is connected, there exists a point x ~/~'(k) = ~(k) which does not belong to X. Then ~' =/~\{ x } is an irreducible affine curve, and it is known that the group Pic(/~") is divisible. Therefore it suffices to show that the canonical map Pic(~') ~ Pie(X) is surjective. One has Pic(~') = Div(~')lDivo(~'), where Div(:~') (resp. Div0(:~')) is the group of divisors (resp. principal divisors) on ~'. Similarly, if q/= Spec(.~), where X =.lr then Pic(X)= Pic(q]), by Kiehi's Theorem, and one has Pie(q/) = Div(q/)/Div0(q/). Our claim follows from the evident fact that the canonical map Div(~ r') ~ Div(q/) is surjective. 9 112 VLADIMIR G. BERKOVICH 6.2. The trace mapping for curves For brevity a separated k-analytic space of pure dimension one will be called a k-analytic curve. 6.2.1. Theorem. -- Suppose that k is algebraically closed, and let n be an integer prime to char(k). Then one can assign to every smooth k-analytic curve X a trace mapping Tr x : H~(X, ~z,) -+ Z/nZ. These mappings have the following properties and are uniquely determined by them: a) for any flat quasifinite morphism ~ : Y ~ X the following diagram is commutative Tr~p> H~(Y, V,) H~(X, [z,) \ f,x Try "~ Z/nZ b) Trp, is the canonical mapping H2(P 1, pin) -~ Z/nZ which is induced by the degree homo- morphism deg : Pic(P 1) -% Z. Furthermore, the Tr x are compatible with algebraically closed extensions of the ground field and are surjective. If n is prime to char(~) and X is connected, then Tr x is an isomorphism. Proof. -- Let A be the closed annulus A(a;r,r)----{x ~A1 [ [(T- a)(x)[ = r}, and let f= Y~=_ ~ a~(T -- a) ~ ~ 0(A). 6.2.2. Lemma. --fc0(A)* if and only if there exists m with [ a,, [r'~> [a, [r' for all i oe m. Proof. -- If r r ], then 0(A) is a field, and our statement is evident. Suppose that r~[k*[. In this case we may assume that r= 1, []f[[ = 1 and a=0. Since then the element j~IT, 1] is invertible. It follows thatff-----~T ~ for 1, ]jf ltl some a ~, m ~ Z, and we are done. 9 The complement of A in p1 is a disjoint union of the open discs D(a, r) and PI\E(a, r) which are called the complementary open discs of A. Let D be one of these discs. For f~ O(A)* we set degD(f) = m, ifD ---- D(a, r), and degD(f) = -- m, ifD = PikE(a , r), where m is from Lemma 6.2.2. We get a homomorphism deg.: O(A)* -+Z. The notation can be motivated as follows. There is a rational function g with [If-- g ]1 < I[f][- For such a g one has g ~ 0(A)*, and therefore the divisor (g) is concen- ]~TALE COHOMOLOGY FOR NON-AR.CHIMEDEAN ANALYTIC SPACES 113 trated outside A. If (g), denotes the part of the divisor which is concentrated on D, then degD(f) = deg(g)D. We remark that if D' is the second complementary disc, then degD(f) + degv,(f) = 0. 6.2.3. Lemma. -- Let f ~ 0(A). Tken f(A) is an annulus if and only if there exists b ~ k* such that f-- b ~ 0(a)* and degD( f- b) 4= 0. Proof. -- Assume that f(A) = A(b; R, R). Then the function f-- b is invertible and, by Lemma 6.2.2, one has f-- b = a,,(T -- a)" (1 + ?), where ~ ~ 0(A) with II ~ II < x and R = [ a,~ [r = = Ilf-- b I[. If m = 0, then f-- b = a0(l + ?) and that is impossible because in this casef(A) C D(b, R). Conversely, iff = b + a,,(T -- a)"* (1 + q~), where II ~ I1 < 1 and m 4 = 0, then f(A) = A(b, ]a,, [rm). 9 Letf ~ 0(A) and assume that A' =f(A) is an annulus. For a complementary open disc D of A we denote byf(D) the complementary open disc of A' which is of the same type as D if m > 0 (resp. of the opposite type if m < 0), where m is from Lemma 6.2.2. (We say that two open discs in p1 are of the same type if they contain or do not contain the infinity simultaneously.) For example, iff comes from O(E), where E = D u A, then f(D) is the usual image of D under f. We remark that the induced morphism f: A -+ A' is flat and finite. Let N be the norm homomorphism 0(A)* ~ 0(A')*. 6.8.4. Lemma. ~ For any g ~ O(A)* one has deglm~(N(g)) = deg(g). Proof. -- Since both sides of the equality do not change under extensions of the ground field, we may assume that r = 1 and [[fl] = 1. Of course, we may also assume that a = b = 0 and [[ g [] = 1. The morphismfinduces a flat finite morphism of reduc- tions d7: A = Spec (~[T, I])--~ A'= Spec (~[T',I]), and the following diagram \ L ~J! \ L ~J/ is commutative O(A)O" ~> O(A,)O" ~- r If degv(g ) = 0, then ~ is constant on A. This implies that N(~) = N(g) is constant on A', and therefore deglm~(N(g)) = 0. Since both sides of the equality are additive with respect to g, we may assume that g = T. One has T' = a~ T"(1 + ~), where In,,,[ = 1, ][~?[[< 1 and m40. If re>O, then f(D) =D and'N(T) =h~ -1T'. If ~- T'-I In both m < 0, then f(D) is the second complementary disc and N(T) ---- a,, 1 9 cases we have the required equality. 9 15 114 VLADIMIR G. BERKOVICH Let 7D denote the composition of the following surjective homomorphisms Trpl $(A)" --~ Ha(A, ~t.) --> H~(D, ~t.) --> H2(P a, ~t.) "% Z/nZ. The first homomorphism is obtained from the Kummer exact sequence (it is surjective because Pic(A)= 0). The second one is obtained from the cohomological exact sequence associated with the embeddings D r E = D u A <-- A (it is surjective because H2(E, bt.) = 0, by Corollary 6.1.3). The third one is surjective because H2(Pa\D, yr.) = 0. We remark that if n is prime to char(~), then 0(A)*/0(A)*"--% Z[nZ. 6. B. 5. Lemma. -- For any g ~ 0(A)* one has 7D(g) = -- degD(g) (mod n). Proof. -- It suffices to consider the case D = D(a, r). First we claim that it suffices to verify the equality only for the function g = T -- a. Indeed, take a rational function h with sufficiently small [] g -- h [] so that g - h(mod 0(A)*") and degD(g ) = degD(h ). Then 7D(g) : YD(h), and therefore we may assume that g is a rational function. Since 7D(g) and degD(g ) are additive with respect to g, it suffices to assume that g = T -- = for some ~ ~ k. If, r D, then g comes from 0(E)*. Therefore the image of T -- ~ in Ha(A, ft,) comes from Ha(E, ~t,) in the exact sequence HI(E, it,) -+ Hi(A, ~z,) --> H~(D, ~z.) asso- ciated with the embeddings D r E ~- A. It follows that yD(T -- 0~) = 0. Thus, we may assume thatg = T -- a. We remark that this function belongs to 0~.(A)" = H~ i" G~. ~.). Consider the commutative diagram 0 0 0 0 > J, ~.,D ~ ~.,~ 9 i. ~.,A ~ 0 0 9 j, Om, D 9 G..,E , i.i'Gm, z , 0 0 0 0 whose columns are Kummer exact sequences. It induces the anticommutative diagram $1(A)" , HX,(D, G.) H'(A, ~.) , I-I~(D, t~.) I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 115 Therefore it suffices to verify that the image of the element 8(T- a) in Pic(P x) = HI(p 1, G~) has degree one. The group H~(D, G,,) has the following two descriptions. It is the group of equi- valences classes of pairs (L, 9), where L is an invertible sheaf on E (resp. p1) and ~ is an isomorphism O~. ---> L in a neighborhood of A (resp. an isomorphism 01,~ -+ L in a neighborhood of PI\D). If h e 0E(A)* , then 8(h)= (0E, d)~ 0~.). On the other hand one has a commutative diagram H~ ~iv.) > H~.(D, G~) H~ 1, Nivp,) > H~(P ~, G~) where ~ivD and ~ivp1 are the sheaves of Cartier divisors. If d = Y~n,(a~) e H~ ~iVD), then v(d) = (Oz(d), O~. ---> O~.(d)) (the latter is an isomorphism in a neighborhood of A which does not meet the support of d). Thus we have ~(T -- a) = (0~., 0z T-,> OE ) and v(a) = (O~(a), 9 E ~ O~.(a)). These pairs are equivalent because there is a commutative diagram T--a lid /. The lemma is proved. 9 Let now B be the open annulus B(a; r, R) ={x eAllr< [(T -- a) (x)[ < R). We set A = A(a; r, R), A 1 = A(a; R, R) and A~ ----- A(a; r, r). Let -(~ denote the compo- sition of the following surjective homomorphisms O(A1)* 9 r --> H~(A~, ~z,) | H~(A,, [z,) ----> HZ(B, ~z,) Trp1 --~ H~(p1, ~.) ~> Z/nZ. The first homomorphism is obtained from the Kummer exact sequence. The second one is obtained from the cohomological exact sequence associated with the embeddings B ~ A <- A 1 H A~ (it is surjective because H2(A, ~,) = 0, by Corollary 6.1.3). The third one is surjective because H~(PI\B, V,)= 0. We also set D 1 ----D(a, R) and D 2 = PI\E(a, r). One has B = D1 c~ Dz. 6.9..6. Lemma. -- For (gl, g2) e 0(Ax)*| 0(A~)* one has v~(gl, g,) = -- degDl(gl) -- degD,(g,) (mod n). 116 VLADIMIR G. BERKOVICH Proof. -- We set E I = E(a, R), E z = PI\D(a, r), A I = Ex\E(a, r), and A 2 = DI\D(a , r). For i = I, 2 one has a commutative diagram of embeddings B < 9 ~. < AIIIA,. T T B c >A~< A~ Dir N< A~. It induces a commutative diagram of homomorphisms d)(Aa)* (~ 0(A2)* > Hm(Ax, Vt,,) (~ Hi(A2, ~,) > HI(B, ~.) , H~(P m, ~.) 0(A~)* > Hm(&, ~.) , HI(B, ~.) > H~( PI, t~.) d~(A~)" > Hi(A,, ~.) > HI(D,, V-.) > H'( P:, V..) The required statement now follows from Lemma 6.2.5. 9 For an open subset X C A 1 we define Tr x = Trrl o Tr~ : HI(X , ~t.) ---> Z/nZ, where j is the embedding X ~ p1. 6.2.7. Lemma. -- Let X be an open disc or an open annulus in A 1. Then for any nonconstant function f ~ 0(X) the following diagram is commutative HI(X, yr.) T'S> H2(p1, ~,) Z/nZ Proof. -- Consider first the case X = D = D(a, r). Then f= 2~~ o ai(T -- a)!. Since the group H~(D, ~,) is an inductive limit of the groups H~(D(a, r'), ~.), it suffices to verify the statement for D(a, r') instead of D, where r' is sufficiently close to r. Therefore we may assume that the function f satisfies the following conditions: a) [ a, l ricO for i~oo; b) there exists m/> 1 with ] % [ rm > a~ I [ r' for all it> 1 with i 4= m. The property a) means thatfextends to the closed disc E = E(a, r), and b) means that, for A = A(a; r, r), A' =f(A) is an annulus and D' =f(D) is a complementary ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 117 open disc of A'. Since the homomorphism O(A)* -+ HI(D , yr.) is surjective, it suffices to verify the commutativity of the diagram r ~, O(A')* Z/nZ For g e d~(A)* one has 7D,(N(g)) - -- deg,,(N(g)) = -- degD(g ) - 7D(g) (mod n). Consider now the case X = B ---- B(a; r, R). Then f----- 2~,~_ | a,(T -- a)'. As above we may assume that f extends to the closed annulus A = A(a; r, R) and, if A 1 = A(a; R, R) and A s ---- A(a; r, r), then A i =f(A1) and A~ =f(A~) are annuli. It follows that one can find r < t 1 < ... < t~ < R, l >1 0, such that for each 0 <~ j.< l there is m with [a.,[t"> lasl # for all te]tj, t~+l[ and i4=m (we set t 0=r and t,+l = R). Let xj =p(E(a, ti)) and Z ={Xl, ...,x~). Then the open set q/=X\Z is a disjoint union of the open annuli B~ = B(a; t~, t~+l), 0~<j~< l, and of an infinite number of open discs. By Theorem 2.5.1, one has H2((B, X), ~,) = @~=1 H2(G,,,p, Yr,) = 0. Therefore the embeddings ad ~ B <- (B, Z) induce a surjection H~(~ ~.) ---> HI(B , ~t,). Since the lemma is true for open discs, it suffices to verify it for the annuli B~. Thus, we may assume in addition that for some m eZ one has f = a~(T -- a)" (1 + ~), where q~ ~ O(A) and [I q~ Ilx < 1. In particular, B' -----f(B) and A' =f(A) are also annuli. Furthermore, if D x = D(a, R) and D~---- pl\E(a,r), then B' = D i c~D'2, where D~-----f(D,). Since the homomorphism d~(A~)*@0(Az)*-->H~(B, bt,) is surjective, it suffices to verify the commutativity of the diagram r O 0(k2)" lq 9 d~(A,1). ~) d~(A~)" Z/nZ For (gl, g~) e 0(A1)* @ ~(Az)* one has 7~,(N(gx), N(ga)) = yDi(N(gl)) + yD~(N(g,)) = 7xh(gl) + 7D,(g~) = 7~(gx, g,)" The lemma is proved. 9 Suppose that X is an elementary k-analytic curve (see w 3.6), and let f be a non- constant analytic function on it. It gives rise to a flat quasifinite morphism f: X --> PL We set Tr x = TrF~ o Tr I : H~(X, ~t,) -+ Z/nZ. 118 VLADIMIR G. BERKOVICH 6.2.8. Corollary. -- The mapping Tr x does not depend on the choice off. Proof. -- Assume that X is not isomorphic to an open disc or an open annulus. Then one can find a point x ~ X such that the open set q/= X\( x } is a disjoint union of a finite number of open annuli and of an infinite number of open discs (see Remark 3.6.3 (ii)). Since the homomorphism H~(ql, ~t,)-+H~(X, ~t,) is surjective, the required statement follows from Lemma 6.2.7. 9 Let now X be an arbitrary smooth k-analytic curve. By Proposition 3.6.1, we can find an open covering { q/, }~ ~ r of X by elementary open subsets and, for each pair, i,j e I, an open covering { q/~},Ei0, of ~ r~ ~ also by elementary open subsets. If ,~ and "~o~ denote the open embeddings q/, ~ X and q/~j~ ~-> X, then one has an exact sequence which induces a commutative diagram with exact rows 2 2 H~(X, ~.) , 0 O,, ~,, Ho(q/ol, ~.) , O, H,(q/,, ~.) > > O, ZlnZ , ZlnZ , o The right vertical arrow is the definition of Tr x. By Corollary 6.2.8, it does not depend on the choice of the coverings. Thus, the trace mappings are defined. The verification of the necessary properties of Tr x is now trivial. 9 6.2.9. Corollary. -- If X = t~ "~, where ~ is a separated smooth algebraic curve of finite type over k, then tke diagram H~(~V, ~,) "~, H~(X, ~,) Z/nZ is commutative. 9 6.2.10. Remark. -- The last statement of the theorem is not true without the assumption that n is prime to char(k). For example, assume that char(k)= 0 and p = char(~)> 0, and let X = D be the open unit disc in A 1. Then the embeddings D '-+ pl+_ E = PI\D induce an exact sequence 0 -+ El(E, [~,) -+ H2,(D, ~,) -+ H2(P 1, liT) -+ 0. The group HX(E, ~,) is huge, its cardinality is at least the cardinality of ~ (see Remark 6.4.2). I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 119 6.3. Tame ~r coverings of curves In this subsection the ground field k is assumed to be algebraically closed, and the characteristic of the residue field k of k is denoted by p. Let q~ : Y ~ X be an dtale morphism. The geometric ramification index of ~ at a point y ~ Y is the number v~(y) ~-- [Jt~ :~(x)], where x ---- ~(y). The morphism ~ is said to be tame at y if v,(y) is not divisible by p. It is said to be tame if it is tame at all points of Y. We remark that ifX (and therefore Y) is good, then, by Proposition 2.4.1, one has v,(y) = [K(y) :*:(x)]. 6.3.1. Remarks. -- (i) It follows from the definition that ify belongs to an analytic subdomain X' s X, then ,,(y) ~-- ~,,(y), where ~' is the induced morphism ~-x(X') --+ X'. We note also that if ~ is tame at y, then ~(y) = [~(y):~(x)] []~(y)* [ : [~(x)* [] (Proposition 2.4.7 and Lemma 2.4.8). (ii) If ~ is an dtale Galois covering with Galois group G, then v,(y) is equal to the order of the stabilizer ofy in G. In particular, if the order of G is not divisible by p, then ~ is tame. (iii) Suppose that X is connected and q~ is finite. Then the sheaf ~,(0y) is a locally free 0x-module. Its rank is said to be the degree deg(~) of ~. In this case for any x e X one has deg(q~)---- ~ ~(y). (iv) The use of the word " geometric " can be explained as follows. Suppose that the set of k-points X(k) is everywhere dense in X (this is so, for example, if the valua- tion of k is nontrivial and X is strictly k-analytic). Then ~(y) is equal to the maximal integer n such that for any open neighborhood 3r p ofy there exists a point x e X having at least n inverse images in ~. 6.3.2. Theorem. -- Any tame finite gtale Galois covering of the one-dimensional disc is trivial. Proof. -- If the valuation of k is trivial, the statement is easily verified. So we assume that the valuation of k is nontrivial. Let ?:Y ~ X = E(O, r) be a tame finite Etale Galois covering. We assume that Y is connected and the number n = deg(~?) is bigger than one. Since X is simply connected (and even contractible), it suffices to show that any point x ~ X has exactly n inverse images. Suppose that this is not so, and let Y~ denote the set of points x ~ X having at most n -- 1 inverse images. It is clear that ~ is closed. We now use the classi- 120 VLADIMIR G. BERKOVICH fication of points of E(0, r) from [Ber], 1.4.4 (see w 3.6). Ifa point x is of type (1) or (4), then the field o~(x'-~----k is algebraically closed, and the group [~t~(x)*[- - [k*[ is divisible. From Remark 6.3. I (i) it follows that x r Z. Hence, Y~ may consists only of points of types (2) or (3). Consider the following partial ordering on X: x~y if If(x) l<<. If(y)[ for all f ~ k[T]. The restriction of this ordering to Z satisfies the conditions of Zorn's Lemma, and therefore there exists a minimal point x e Z. Let x =p(E(a, r')) for some a ek and r' > 0. Since E(a, r') ~ { x' e X [ x' ~< x }, we may replace X by E(a, r') and assume that x is the maximal point of X and Y~ = { x }. We claim that the preimage of x in Y consists of one point. Indeed, the set X\{ x } is a disjoint union of open discs. Let D be such an open disc. Since D is simply connected, ~-I(D) is a disjoint union ]-I~= i D~, where all D i are isomorphic to D. For the closure D, of D, in Y one has D,=D~u{yi} (see Remark 6.3.4 (i)). It is clear that ~-1(x) = {Yl, .-.,Y, }. Since Y is arcwise connected, it follows thatyl ..... y, =y. One has n = [~(y) : .~f(x)] ---- [~(y) :g(x)] [[ ~(y)* [ : ~(x)* I I]. Suppose first that x is of type (3), i.e., r elk* 1. In this case X = D u{x}, where D = D(a, r), and the group [~(x)* [ is generated by [ k* [ and r. Since ~(x) = k, we have [I ~(Y)* [ : [~(x)* 1] = n. We now remark that the group [a~(y)" ] is generated by the values of the spectral norm on 8, where Y = ~r becausey is the maximal point of Y. But for any f e :~ one has p(f) = sup [f(Y')[, ~' G ~-I(D) and we know that @-1(D) is a disjoint union of n copies of D ~-- D(a, r). Since a non- zero analytic function on Y has at most a finite number of zeroes, it follows that the number in the right hand side of the equality belongs to the group generated by [ k ~ I and r. This contradicts to the equality [I ~(Y)* I : I~(x)" I] = n. Suppose now that x is of type (2), i.e., r ~[k* I. We then may assume that r ---- 1. In this case X = Spec(~), where ~r ~ k { T }, is the affine line over ~, and ~(x) = ~(T). By Remark 6.3.1 (i), [~(y) :~(x)] = n. The field ~f'(y) is the field of rational func- tions of the affine curve Y -~ Spec(~) (see [Ber], 2.4.4). We claim that the induced morphism ~ : Y -> X is dtale. For this it suffices to show that the any k~-point ~ ~ X(~) has exactly n inverse images in ~{(~). Let ~ (resp. ~') denote the reduction map X --~ (resp. Y ---~ ~). Then ~-~(~) is the unit open disc D, and ~-I(D) is a disjoint union of n copies of D. But ~-1 (D) = 7~ '-1 (~-~ (~')). By a result of Bosch ([Bos]), all the sets n'-I (y), where y e Y(k), are connected. It follows that ~ has exactly n inverse images. Thus we get a nontrivial finite dtale Galois covering of the affine line over k whose degree is prime to p. This is impossible. 9 I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 121 6.3.3. Corollary. -- Any finite gtale Galois covering of the one-dimensional disc, whose degree is prime to p, is trivial. 9 6.3.4. Remarks. -- (i) The following fact was used in the proof of 6.3.2 and will of 6.3.9. Let ~g be an open subset of a k-affinoid space X, and be used in the proof assume that there is an isomorphism q~ : ~//~ D = D(0, r) C AL Then ~ = Y/u { x }, where x r X(k), and for any open neighborhood ~ of the point x the set ?(~//n ~r contains the annulus B(r', r) = D\E(a, r') for some 0 < r' < r. This fact easily follows from the description of k-analytic curves in [Ber], w 4, but here is its simple explanation. Let x ~ ~\0//. A basis of open neighborhoods of the point x is formed by the sets of the form r {y ~ X I If(Y)] < a~, ] g~(Y) l > b~, 1 ~< i ~< n, 1 ~< j ~< m }. But itfis a nonzero analytic function on X, it has at most a finite number of zeroes on ~. It follows that the set p({y ~ ~ ] If(Y) l < a}) is a disjoint union of open discs in D, and the set ~({Y ~ ~/[If(Y)[ > a }) is the complement of a disjoint union of closed discs in D. The first set is relatively compact in D, and the second one contains the annulus B(r', r) for some 0 < r' < r. It follows that the set q~(Y/c~ r contains the annulus B(r', r) for some 0 < r'< r. The point x does not belong to X(k) because a basis of open neigh- borhoods of a k-point is formed by sets of the form {y e X ] tf(y) ] < a }. The required fact follows. (ii) A morphism of k-analytic spaces q~ : Y -+ X is said to be a covering if every point x e X has an open neighborhood q/such that ~0-1(~//) is a disjoint union of non- empty spaces ~ such that the induced morphisms ~---> o-// are finite. It is easy to deduce from Theorem 6.3.2 and its proof that any tame ~tale Galois covering of the one-dimensional disc is trivial. For 0< r<~ R< ~ we denote by A(r,R) the annulus{xeA ~[r~< IT(x) I~<R}" Let p, denote the finite morphism A(r ~/", R a/") ---> A(r, R) : z ~ z". If n is prime to char(k), then p, is a finite 6tale Galois covering. A finite ~tale covering of A(r, R) is said to be standard if it is isomorphic to p, for some n. If n is prime to p, then q~, is tame. 6.3.5. Theorem. -- Any tame finite r Galois covering p:Y-+ X : A(r, R) with connected Y is standard. Proof. -- We set n-----deg(q~). Suppose first that r = R and set x----p(E(0, r)). If r r k* [ [, then X = { x }, and our statement follows from Proposition 2.4.4. If r ~ k* [ [, and the we may assume that r= 1. In this case X =~r where ~r =k{T,T], reduction :~ is the complement to zero in the affine line over ~. The set X\{ ~ } is a disjoint union of open unit discs. By Theorem 6.3.2, for such a disc D the space ~0-I(D) is a disjoint union of n spaces isomorphic to D. Since Y is arcwise connected, we have ~-l(x) ={y}. Let Y ~-Jg(~). We claim that ~/T e~. Indeed, as in the proof of Theorem 6.3.2 one shows that the induced morphism ~:~/-->.X is 6tale and 16 122 VLADIMIR G. BERKOVICH deg(~) = n. Therefore ~ = ff["~-T], and there exists an elementfe ~ with Ilfll = I and [If" - T II < 1. The element fis invertible in ~ because T is invertible. We have Since II (T --f")[f" II < 1 and p ,~n, ~ e ~. From this it follows that 9 is isomorphic to 9.. In the general case we denote by m the product of all integers between 1 and n which are prime to p, and set X' = A(r v", Rl/"). It suffices to show that there exists a finite morphism + :X'-+Y such that 9d? = %~. For this we consider the induced morphism 9' : Y' -~ Y � x X' --~ X'. Since X' is simply connected (and even contrac- tible), it suffices to show that vr 1 for any point y' e Y'. We set e ={p(E(O,t))lr<<. t<~ R}C X. Ify r (9~ 9') -~ (t), then the required fact follows from Theorem 6.3.2 because X\t is a disjoint union of open discs. Let x'= 9'(Y') and suppose that the point x = %,(x') belongs to t, i.e., x = p(E(0, t)), r ~< t ~< R. By Remark 6.3.1 (i), %,(y') does not change if we replace X by the annulus A(t, t). But for such annuli the required fact is already established. 9 6.3.6. Corollary. -- Let D be an open disc with center at zero, and set D" = D\E(0, r), where 0 ~< r < r(D). Then any tame finite ~tale Galois covering 9" : Y* ~ D" extends to a finite flat covering 9 : Y ~ D, which is gtale outside zero. 9 6.3.7. Theorem (Riemann Existence Theorem). --Let Yf be an algebraic curve of locally finite type over k. Then the functor ~/ ~-~ ~/~n defines an equivalence between the category of finite ~tale Galois coverings of ~r, whose degree is prime to p, and the category of similar coverings of ~r~.. 6.3.8. Remark. ~ If the valuation on k is trivial, then the Riemann Existence Theorem is true for arbitrary schemes of locally finite type over k and for arbitrary finite ~tale coverings. This follows from [Ber], 3.5.1 (iii). Proof. -- 1) Thefunctor is fully faithful (this is true for schemes of arbitrary dimension and for arbitrary finite 6tale coverings). Let ~r, ~ ~ and ~t,, _+ ~ be finite 6tale cove- rings of ~. We may assume that Y/' is connected. Then the set Hom~r(q/', Y/") corres- ponds bijectively to the set of connected components ~t of Y/' xm ~/" such that the canonical morphism ~/~ ~ ~t, is an isomorphism. The similar fact is true for the set Hom~r~(~/'~, ~/"~"). Therefore the bijectivity of the map Hom~r(~/', q/") ~ Hom~r=(~ t'=, ~"~) follows from [Ber], 3.4.6 (9) and 3.4.8 (iii). 2) Thefunctor is essentially surjective. -- We may assume that ~ is separated, reduced and irreducible. If Y is projective, the assertion follows from GAGA (see [Ber], 3.4.14). ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 123 In the general case, let W ~ 5F' be an open immersion of/~" in a projective curve ~' such that all the points xl, 9 9 x, from the complement to/Y are smooth. Then we can find suff� small open neighborhoods D~ of x~, which are isomorphic to open discs and disjoint. Let Y-+ 5F =n be a tame finite dtale covering of W an. Applying Corol- lary 6.3.6, we can construct a finite flat covering ~' : Y' ~ ~'~n. By GAGA, ~' comes from a finite flat covering q~ : if/' -+~'. Hence ~ comes from the covering q~-l(~) ~/~- 9 6.3.9. Theorem. -- Let ~ be a projective curve over k, and let X be an affinoid subdomain of Yf~ such that ~'~\X is a disjoint union U~= 1 D~, where each D, is isomorphic to an open disc. Pick points x~ ~ D~(k), 1 <~ i <~ n, and set 5F' = Yf\~ xz, ..., x, }. Then the functor ~r ~_~ ~t,~ � er, a~ X defines an equivalence between the category of finite gtale Galois coverings of Yt", whose degree is prime to p, and the category of similar coverings of X. 6.8.10. Remark. -- It is very likely that any pure one-dimensional reduced k-affinoid space X can be identified with an affinoid subdomain of the analytification 5F =" of a projective curve ~ such that the condition of Theorem 6.3.9 holds. This is true at least in the following cases: 1) the valuation on k is nontrivial, and X is a normal strictly k-affinoid space (M. Van der Put [Put]); 2) the valuation on k is trivial, and X is irreducible and contains a point x for which the field .~ff(x) is bigger than k and has trivial valuation (see the proof of Theorem 6.4.1). Proof. -- For a fixed i, take a point x ~ Y'(k) which does not belong to b~ (such a point evidently exists). Iff is a nonconstant rational function on ~ regular outside x, then the set { z ~ Y'~" [ If(z)[ < a} is an affinoid domain in ~"~ and, for a sufficiently large a, contains D~. Therefore we can apply Remark 6.3.4 (i) to D~. It follows that D, = D~ u { zi }, where z~ ~ X\X(k), and for a sufficiently small open neighborhood of the point z, its intersection with D~ is the annulus D~\E,, where E~ is a closed disc in D~ with center at x~. (We remark that some of the points z, may coincide.) Let now : Y ~ X be a finite fitale Galois covering, whose degree is prime to p. By Corollary 3.4.2 applied to the points z,, ~ can be extended to a finite dtale Galois covering ~' : Y' ~ X', where X' = X u U~=I(D,\E,) and E, is a closed disc in D~ with center at x~. From Corollary 6.3.6 it follows that q~' extends to a finite dtale Galois covering of 5F "~. The required statement now follows from Theorem 6.3.7. 9 The following statement is a particular case of the Comparison Theorem 7.5.1 (it will not be used in the sequel). 6.3.11. Corollary. (Comparison Theorem for curves). -- Let ~ be an algebraic curve of locally finite type over k, and let .~ be an abelian constructible sheaf on Y~ with torsion orders prime to p. Then for any q >>. 0 there is a canonical isomorphism H~(~, ~') -~ H"(~ -, ~'). 124 VLADIMIR G. BERKOVICH Proof. -- Using the spectral sequences of an open affine covering { Y'~ }~ e i of and of the corresponding open covering { 5~," }~ e ~ of ~", we reduce the statement to the case when Y" is affine of finite type over k. Furthermore, we may assume that is reduced. Finally, we may assume that ~'--~ (Z/nZ)~, p Xn (see the proof of Theorem 6.1.1). The homomorphism considered is an isomorphism for q = 0, because n0(W) = n0(W~"), and for q = 1, by the Riemann Existence Theorem 6.3.7. Since Hq(~, Z/nZ) = 0 for q/> 2, it remains to show that H~(~ ~, Z/nZ) = 0. Let 5~-+ be an open embedding off in a projective curve ~ such that ~\Y" = { Xl, ..., x,~ } are smooth k-points. Then the k-analytic space ~" is a union of an increasing sequence of affinoid domains X~, i 1> 1, whose complements in Y'~" are disjoint unions of m open discs with centers at the point xa, ..., x,,. By Theorem 6.3.9, H~(Y ", Z/nZ) ~ HI(X~, Z/nZ) and, by Corollary 6.1.3, H~(X~, Z/nZ) = 0 for all i t> 1. Therefore the required fact follows from the following lemma which is an analog of Proposition 3. I0.2 from [Gro] (the field k is not assumed to be algebraically closed). 6.3.12. Lemma. -- Let X be a paracompact k-analytic space, and suppose that X is a union of an increasing sequence of closed or open analytic domains X~, i >1 1. Let F be an abelian sheaf on X, and let q >i 1. Assume that for each i >1 1 the image of the group H ~- I rX~ i+1, F) in Hq-a(X~__~, F) under the restriction homomorphism coincides with the image of the group H~-~(X F). Then there is a canon#al isomorphism k i+2, H~(X, F) ~ lira H~(X~, F). Proof. -- First of all we remark that ifJ is an injective abelian sheaf on X and Y is a closed or open analytic domain in X, then the pullback of J on V is acyclic and the homomorphism J(X) ~J(Y) is surjective. Take an injective resolution of F, 0 -+ F __>j0 __>j~ .__> --., and consider, for i/> 1, the commutative diagram 0 > J~ , J~(X) > J2(X) > ... 0 9 j0(x~) > ji(x 3 , y(x~) > ... The first row gives the cohomology groups of X, and from the above remark it follows that the second row gives the cohomology groups of X~ and the vertical arrows are surjections. That the homomorphism considered is surjective is easy. Suppose that 0~ ~Jq(X) is such that d0c ----- 0 and the image of 0~ in each Hq(X~, F) is zero. We have to construct an element ~ ~Jq-l(X) with d~ = x. Since y-l(X)= li~_my-l(X,), it suffices to construct a system of elements ~i ey-l(X~), i/> 1, such that ~r = d~, and ~i+alxi = ~i" Suppose that, for some i/> 1, we already constructed elements ~j eJ"-l(Xj), 1 ~<j~ i, and ~+~ eJ~-l(X~+~) with ~+l[xj = ~ for j~< i and I It q--1 " tt ~[xi+l =d~i+l" Take an element ~,+2 eJ (X~+2) with e[xi+, =d~,+2" Then the ~+2 Xi+l ~+~ gives rise to an element of Hq-l(Xi+l F). By hypothesis, element 6" -- ' 126 VLADIMIR G. BERKOVICH Consider now the general case. Let K be a bigger algebraically closed non-Archi- medean field K over k, and let ~ denote the canonical morpkism X' =- X | K --* X, To prove the theorem, it suffices to show that for any point x ~ X the following property holds (.) = o and (R = O. Wc remark that if, for any ~tale morphism g:Y ~ X with g-l(x) =-(y}, the pointy has a basis of affinoid neighborhoods such that the theorem is true for them, then the property (.) holds for the point x. For example, this is the case when the valuation on k is nontrivial and X is strictly k-affinoid. Furthermore, we remark that if Y is an affinoid domain in X and x ~Y, then (Rq~.(Z/nZ)x,)~-~ (R~z~y,(Z/nZ)y,),. Therefore the validity of the property (.) does not change if we replace X by a smaller or a bigger k-affinoid space. There are the following two possibilities: (11 I~(x)* I = Ik*l; (2) o~~ k_ and the group [o~ff(x)* [ is generated by [k*[ and a number rr I. 1) If x ~ X(k), then (.) evidently holds. Suppose that x r X(k). Take an admis- sible epimorphism q~ : k{ r~-X T1, ..., r~lT,, ) -+ d, where X = egg(d), and set f = ~?(Ti). We assume that f 4= 0. Suppose first that the valuation on k is nontrivial. In this case we can find numbers r; E[k*[ with If(x)]<<.r;<~r,, l<~i<~m. Then the Weierstrass domain X(r'l-lf~, ...,r',,-af,,) is strictly k-affinoid and contains the point x. By the above remark, the property (*) holds for the point x. Suppose now that the valuation on k is trivial. Then the valuation on o~ff(x) is also trivial, and therefore If(x)] >/ 1. In particular, r, >i 1 and the algebra ~/is finitely generated over k. We replace X by the Weierstrass domain X(f~, ...,f,). If ~ is the projectivization of the affine curve Spec(~r then X is an affinoid domain in ~ and the complement of X is a finite disjoint union of open discs. Moreover, the point x has a basis of affinoid neighborhoods of the same type. Finally, if g : Y ~ X is an ~tale mor- phism with g-~(x) = {y }, then the similar facts are true for the pointy. By the above remark, the property (*) holds for the point x. 2) Shrinking X, we can findf ~ ~r with If(x) ] = r. Consider the induced morphism f: X ~A 1. The image of the point x is the point y =p(E(0, r)) because r elk* [. Since the fibres of X are discrete, we may shrink X and assume thatf-~(y) = { x }. But Y = {y } is an affinoid domain in A t (it is the annulus A(r, r)). Therefore { x } =f-l(y) is an affinoid domain in X. Thus, replacing X by the reduction of{ x }, we get a morphism f: X = { x } --* Y = {y }. The field M'(x) is a zero-dimensional oct~ algebra. I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 127 It follows that the extension ~t~(x)/oW(y) is finite, and therefore f is a finite morphism. One has ~(y) = { 2~,~_ o~ a, T']] a, I r* ~ 0 as i --> oo } and [ ~(y)* [ = I 9f~(x) * [. Suppose first that the valuation on k is trivial. Then [ ~(y)* [ is a discrete subgroup of R~_, and from this it follows that 9if(y) -% ~f~(x). This means that f is an isomorphism X-% A(r, r). Moreover, any finite dtale covering of X is of the same type. Therefore the property (*) holds for the point x. Suppose now that the valuation on k is nontrivial. Then the extension ~,~(x)/~,W(y) = ,=-o~ a~T ~ ~f(y) and q~V~ EjtO(x)\~f,(y). is separable. Indeed, suppose that , 2] ~ We may assume that a~ = 0 for i divisible by p. Then I ~~ = ]a, I r" for some n prime to p. We get [ q~l/~ [ = (] a, ] r) a/v r [~Yt~ [ = la~'(y)* I" Thus, f: X ~ Y is a finite dtale morphism. By Corollary 3.4.2,fextends to a finite dtale morphismf: X -~ Y = A(r', r") r H k* for some r' < r < r" with r', r ] [. Since Y is strictly k-affinoid, then so is X, and we are done. 9 6.4.2. Remark. -- Both statements of Theorem 6.4.1 are not true without the assumption that n is prime to char(~). For example, assume that char(k) = 0 and p = char(~) > 0, and let X be the closed unit disc in %1. Since Pic(X) = 0, the Kummer exact sequence implies that HI( X, ~v) = k { T }*/k{ T }*P. One has k { T }* = {f ~ k { T } [If-- [ f(0) I I < [If I[ }" Therefore the correspondence f~f'(O)/f(O) gives a surjective homomorphism from k { T}* to the maximal ideal k ~176 of the ring of integers k ~ It induces a surjective homomorphism Hi(X, koo/pkoo. If now K is an algebraically closed non-Archimedean field over k for which K~176 ~176 is bigger than k~176 ~176 then the group HI(X ~ K, Vtv) does not coincide with Hx(X, W~)- By the way, in Drinfeld's calculation of the group Hi(X, ~) for a standard affinoid domain in A x ([Dr], 10.1) one also should assume that n is prime to char(kl, otherwise the result stated is not true. (The same is repeated in [FrPu], V. 3.7.) w 7. Main Theorems 7.1. The Comparison Theorem for Cohomology with Compact Support Recall that a morphism of schemes q~:~/--> ~ is called compactifiable if there is a commutative diagram c~ C~> 6~j ~r 128 VLADIMIR G. BERKOVICH where ~ is a proper morphism and j is an open immersion. For a sheaf ~ on ~t one defines R q q~, ff = R ~ ~.(j: N). By Nagata's Theorem, any separated morphism of finite type is compactifiable. 7.1.1. Theorem. -- Let ~ be a compactifiable scheme over k, and let ~" be an abelian torsion sheaf on ~. Then for any q >1 0 there is a canonical isomorphism H~(Y', if) --% H~(Y "~", ,~~"). Proof. -- Let q0 : ~/-+ ~ be a compactifiable morphism between schemes of locally finite type over Spec(d), where d is a k-affinoid algebra (the situation is slightly more general for a further use). For a point x e 5F we denote by ~x the fibre of q0 at x and set ~/~ = ~x | k(x)% and for an abelian torsion sheaf fr on ~/, we denote by Nx and fr the pullbacks of N on ~,t and ~r respectively. By the Base Change Theorem for Coho- mology with Compact Support for schemes, one has ( R~ q~: N)i = H~(~i, Ni)- Furthermore, for a point x e :y~n over x, we fix an embedding of fields k(x)" ,-+~(x) ~. It gives rise to an isomorphism ;v * Since the cohomology with compact support of schemes are preserved under separably closed extensions of the ground field, one has H.(~i, (r ~ q Finally, the Weak Base Change Theorem 5.3.1 tells that R q?~ (~),= ~,_~, ~,. We use the above remarks to establish the following two facts. 7.1.2. Lemma. -- In the above situation assume that the dimension of ~ is at most one. Then for any q >1 0 one has (R ~ qo~ fg)~ --% R ~ q~.~ r#~,. Furthermore, if in addition ~ and ~' are compactifiable over k and the theorem is true for ~, then it is also true for ~'. #,TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 129 Proof. -- The first statement follows from Theorem 6.1.1 and the above remarks, and the homomorphism of Leray spectral sequences H~(~, R r ~: ~) :- H~+'(~, if) H~(~'~, R r ~ ~) ___;. 14,+4(.oS,- shows that the second statement is also true. 9 7.1.8. Lemma. -- In the above situation assume that the morphism ~ is finite and surjective. Then if :~ and q/ are compactifiable over k and the theorem is true for q/, then it is also true for ~. Proof. -- Let #v be an abelian torsion sheaf on ~. Since ? is finite and surjective, the canonical homomorphism ~"-+?.(?* ~') is injective. Furthermore, by Corol- lary 4.3.2, for an abelian sheaf f# on q/ one has H~(~, ?~ ff~n)--% H~(~'n, ~n). In particular, the hypothesis implies that the theorem is true for all sheaves of the form ?. ~. Thus, one can construct a resolution of ~', 0--> ~-~ o*'~ #-1 ~ ..., such that the theorem is true for all ~'~. The homomorphism of spectral sequences H~(H~(~ ", ,~")) > --~14~ + el,r,=, .~') H,(H~(,Z ~n, ,.~~')) :. H~+ q(,~ ~, ,~'~) shows that the theorem is also true for ~-. 9 We are now ready to prove the theorem. 1) The theorem is true for the direct product (p1),, where PI is a the projective line over k. -- This follows from Eemma 7.1.2. 2) The theorem is true for the projective space P" over k. -- Indeed, there exists a finite surjective morphism (P~)" -+ P", and we can use Lemma 7.1.3. 3) The theorem is true for any projective scheme over k. -- This follows from Lemma 7.1.2. 4) The theorem is true for any affine scheme of finite type over k. -- This is so because such a scheme is isomorphic to an open subscheme of a projective scheme. 5) The theorem is true for any proper scheme over k. -- For a proper scheme ~ we can find an open everywhere dense affine subscheme q/, and so the dimension of the closed subscheme q/= ~\q/is strictly less than the dimension of 5F. Therefore the statement is obtained by induction using the exact cohomological sequences associated with the embeddings q/,-+ ~ <--- q/ and q#n ,_+ s162 +_ ~n and the five-lemma. 6) The theorem is true for any compactifiable scheme. -- This is already clear. 9 7.1.4. Corollary. -- Let r : ~-+ YC be a compactifiable morphism between schemes of locally finite type over Spec(~'), where sd is a k-affinoid algebra, and let fr be an abelian torsion sheaf on q/. Then for any q >1 0 there is a canonical isomorphism (R ~ q~ ~)~n __% R ~ ~.~ ~,. 17 130 VLADIMIR G. BBRKOVICIt Proof. -- The statement is obtained from Theorem 7.1.1, using the remarks from the beginning of its proof. 9 Let ~ : Y ~ X be a morphism of k-analytic spaces, and let G be a sheaf on Y. We say that the pair (% G) is quasialgebraic if, for each point x e X, one has Y, = ~ean and G, = ff~", where ~s is a compacfifiable scheme over the field ocg'(x) and ff is a sheaf on ~L~. For example, for the canonical projection pr : Y = X � ~ra= __~ X, where ~r is a compactifiable scheme over k, the pair (pr, t~,, r) is quasialgebraic. 7.1.5. Corollary. -- Let ~ : Y -+ X be a Hausdorff morphism of k-analytic spaces, and let f: X' ~ X be a morphism of analytic spaces over k, which give rise to a cartesian diagram Y ~X T, T, Y' ~'~ X' Furthermore, let G be an abelian torsion sheaf on Y and assume that the pair (% G) is quasialgebraic. Then for any q >1 0 there is a canonical isomorphism if(R" qh G) -% R q ~;(f'~* G). Proof. -- Let x ~ X and x' E X' be a pair of points with x =f(x'). By hypothesis, one can find a compactifiable scheme ~Lr over ~(x) and a sheaf ~ on ~ with Y, -- ~ and G~ = (Can. One has Y', = (~| an. The statement follows from the Weak Base Change Theorem 5.3.1, Theorem 7.1.1 and the fact that the cohomology with compact support of schemes are preserved under separably closed extensions of the ground field. 9 7.2. The trace mapping In this subsection we fix an integer n which is prime to the characteristic of the field k. Our goal is to extend the construction of the trace mapping from w 5.4 and w 6.2 to any separated smooth morphism q~ : Y ~ X of pure dimension d, i.e., to construct a canonical homomorphism of sheaves : R y) (Z/nZ)x. As in the Theorems 5.4.1 and 6.2.1, we will characterize the trace mapping by certain properties. Let ~ : Y ~ X be a Hausdorff morphism of k-analytic spaces, and let f: X' ~ X be a morphism of analytic spaces over k. They give rise to a cartesian diagram Y ~X T, ? Y' ~'~ X' I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 131 2d d Suppose we are given two mappings 0r %(ft.,y)--->(Z/nZ)x and r R ~ r y) -+ (z/~Z)x,. We say that = and =' are compatible with base change if the following diagram is commutative 9 ~a d R ~ , d f (R ?,(B.,y)) > 9:(B.,y,) f*((Z/nZ)x ) "~ _~ (Z/nZ)x, Here the upper arrow is the base change morphism, and the lower isomorphism is the canonical one. Furthermore, let q~ : Y ~ X and + : Z -+ Y be Hausdorff morphisms whose dimen- sions are at most d and e, respectively. Suppose we are given three homomorphisms : R ~ %(~fl.,y) -+ (Z/~Z)x, ~ : R 2' +,(tz'.,z) --> (Z/nZ)~ and v: R~+"(~+), ,~.,z Jr ~+,, _+ (Z/~Z)~. Using the Leray spectral sequence 5.2.2 and Corollary 5.3.8, we get an isomorphism R=~+ *l(~b), ~.,zt" a§ _% R~n ~, (R" +,(t*., )).a+' z Corollary 5.3.11 gives an isomorphism R 2, +(~.,z.a+,,) _~ R~ +,(~'.,z) | ~a,y. We get a mapping R2~a+,l(~+)~ , ~+,, _%R ~ (R ~, ~ d t~.,z ) ~, +,(~..z) | ~..Y) -+ R ~ 9,(~'. Y) -* (Z/.Z)x The composition mapping is denoted by ~ [] [3. We remark that if 0t and ~ are isomor- phisms (resp. epimorphisms), then ~ [] ~ is also an isomorphism (resp. epimorphism). We say that the mappings % ~ and y are compatible with composition if y = a [] }. Note that the operation J is transitive. 7.2.1. Theorem. -- One can assign to every separated smooth morphism ~ : Y ---> X of pure dimension d a trace mapping Tr~ : R ~ % (~a y) ~ (Z/nZ)x. These mappings have the following properties and are uniquely determined by them: a) Tr~ are compatible with base change; b) Tr~ are compatible with composition; 132 VLADIMIR G. BERKOVICH c) if d = 0 (i.e., 9 is gtale), then Tr, is the trace mapping q~.,(Z/nZ)r r -+ (Z/nZ)x from w 5.4; d) if X ----o;[(k), k is algebraically closed and d = 1, then Tr~ is the trace mapping Tr r : H~(Y, ~t,) ~ Z/nZ from w 6.2. Furthermore, if the fibres of ~ are nonempty, then Tr~ is an epimorphism. If in addition the geometric fibres of ~ are nonempty and connected and n is prime to char(k), then Tr~ is an isomorphism. Proof. -- First of all, let q~ be the morphism r~:~ -+.~r It is the ana- lytification of the morphism of schemes +:An-+ Spec(k). One has a trace mapping Tr+: R ~ ~b, (~t,a,, #) -+ (Z/nZ)s~k~ (which is an isomorphism). Its analytificafion (Corol- lary 7.1.4) gives rise to a trace mapping Tr#:R~.,(~a,,#)-+ (Z/nZ)~,, (which is also an isomorphism). Furthermore, if q~ is the morphism ~ : A~ = X � ~ -+ X, then we define Tr=~ as the base change of Tr# (Corollary 7.1.5). 7. B. 2. Lemma. -- Let ~ : Y -+ X be a separated smooth morphism which can be repre- sented as a composition of an dtale morphism f: Y -+ Aax with the projection rc~ : Aax -+ X. Then the mapping Tr, = Tr~x~ o Tr s : R ~ qg., ([zan. s --~ (Z/nZ)x does not depend on the representation. Proof. -- We may increase the field k and assume that its valuation is nontrivial. If d = 1, the statement is obtained from the case of curves (Theorem 6.2.1), using the Weak Base Change Theorem 5.3.1. Suppose that d/> 2. The Weak Base Change Theorem 5.3.1 reduces the situation to the case when we are given a separated connected smooth k-analytic space X for algebraically closed k and elements fi,-..,fa ~ 0(X) such that the morphism f: X -+ A a that they define is dtale. We have to verify that the mapping Try'. ..... :a, = Tr# o Tri: H~(X, ~ta,) ~ H2afA n" ~ , ~a) __~ Z/nZ does not depend on the choice of the elementsfi, ..-,fa. First of all, this mapping is independent of the ordering of the elements fi, 9 9 9 ,fa since the group GLa(k ) acts trivially on H~(A a, bta,) = H~(A a, ~ta,). Let gi, 9 9 -, gn be another system of elements in 0(X) for which the corresponding morphism g : X -+ A n is dtale. Take an arbitrary point x ~ X(k) (such a point exists because the field k is algebraically closed, and its valuation is nontrivial). Replacing f byf --f(x) and g~ by gi -- g,(x), we may assume that the elementsfi, ... ,fa, gl, ..., ga are contained in the maximal ideal m~ of the local ring 0x, ~. But then (fi, .... fa) and (gi, ...,gn) are regular systems of parameters for Ox,~, i.e., they form two bases I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 133 of the k-vector space mJm~. Applying Steinitz Exchange Theorem for these bases, we can find a finite chain of systems of elements (A, ..-,A) = (A '1', ..-,A'I'), (A'2', 9 (A'", ...,A'") = (gx, ..-,gd) such that a) each r is one of the elements fx ,.fd, gl, gd; dj , ...... , b) each system gives rise to a basis of mJm~; c) each system (fill+l), ...... ,ff+ll) arises from (~l*), ,adr by the replacement of just one element. By Proposition 3.3 .... 10, each of the systems (f~"), ,aar gives rise to an ~tale morphzsmj 9 "~ : X' -~ A d, where X' is a nonempty Zariski open subset of X. We remark now that ifY = X\X', then dim(Y) ~< d -- 1, and therefore the canonical homomorphism H~(X', ~z,) ---> H~(X, ~,) is bijective. Thus it suffices to show that if fz,...,.fd-1, g andfa, .-.,fa-1, h are two systems of elements in 0(X) which give rise to 6tale mor- phisms ~ : X ~A d and + : X -+A d, respectively, then Try1 ..... ~d-~,g~ = Tr~ ..... tu_~,h~. Let n be the projection &d ~ A n- 1 on the first d -- 1 coordinates. Since 7~ o q~ = 7~ o +, it follows, by the case d = 1, that Tr~ o Tr,-----Tr~ o Tr4. We have Tr~ ..... Sd_~,g) __-- (TrAd_ ~ [] Try) o Tr~ : TrAd_~ [] (Tr~ o Try) = TrA~_~ [] (Tr~ o Try) : (TrAd_~ [] Try) o Tr~ The lemma is proved. 9 The construction of the trace mapping for arbitrary q~ is obtained from Lemma 7.2.2 in the same way as the corresponding construction in Theorem 6.2.1 is obtained from Corollary 6.2.8. It remains to show that if the geometric fibres of 9 are nonempty and connected and n is prime to char(~), then Tr~ is an isomorphism. By the Weak Base Change Theorem 5.3.1, it suffices to show that if k is algebraically closed and X is a separated connected smooth k-analytic space of dimension d, then Tr x : H2afX~ ~ , ~d) ~ Z/nZ. We remark that it suffices to find for such a space X an 6tale covering (U~ ---> X)~ez with separated U~ such that all Trv~ are isomorphisms. This is verified by induction. To use the induction, it suffices to show that for any point x ~ X there exists a separated Etale morphism f: U ~ X and a smooth morphism q0 : U ~ V of pure dimension one to a separated smooth k-analytic space V such that x ~f(U) and the geometric fibres of "r are nonempty and connected. For this we shrink X and take an 6tale morphism X -+ A d. Let + be the composition of the latter morphism with the projection A d ~ A ~- ~. The morphism + : X -+ Ad-a is smooth of pure dimension one. Applying Theorem 3.7.2, we get the required morphisms f and q~. 9 134 VLADIMIR G. BERKOVICH 7.2.8. Corollary. -- Let ~ : q/ -~ Y" be a separated smooth morphism of pure dimension d between schemes of locally finite type over Spec(d), where d is a k-affinoid algebra. Then the diagram (R ~ ~,(~',.~))" "~, R u ~?"(/,,~ao) is commutative. 9 Let q~ : Y -+ X be a separated smooth morphism of pure dimension d. By Corol- lary 5.3.11, for any F" e D(X,Z[nZ) there is a canonical isomorphism R%(~* F'(d) [2d]) -% F" | R~,(~td.,r) [2d]. Theorem 7.2.1 gives a morphism (in D(X, Z/nZ)) R% (Vt~,, y) [2d] -+ (Z/nZ)x. Therefore we get a morphism Tr, : R%(~* F'(d) [2d]) ~ F" which will also be called a trace mapping. For F e S(X, Z/nZ) the latter morphism is induced by a homomorphism of sheaves R u %(9* F(d)) -+ F. It is an isomorphism if the geometric fibres of q~ are nonempty and connected and n is prime to char(~) because R ~a 9,(9" V(d)) = F| R ~a %(pta,,, y). 7.3. Poincar~ Duality First of all we want to fix notation. Let X be a k-analytic space, and let A be a ring (in practice, A = Z/nZ). The functors of homomorphisms Hom and of germs of ,r (between Ax-modules ) have derived functors homomorphisms Hom : D(X, A) ~ � D+(X, A) --> D(A) and ~r : D(X, A) ~ � D+(X, A) -+ D(X, A), where D(A) is the derived category of A-modules. Recall that if G" is bounded below and G" -+J" is an injective resolution of G', then Hom(F', G') = Hom'(F', J') and af~om(F ", G') = Ygom'(F',J'). One sets G') = H (H0m(F ", G')) and g"xt~(F ", O') = H~(~Ct%m(F ", G')). I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 135 For sheaves F, G c $(X, A) these are the usual functors Ext and ~xt, and g'xtq(F, G) is the sheaf associated with the presheaf (U ---> X) ~ Extq(F [o, G [.). We remark that if a A-module G is injective, then the sheaf ~0m(F, G) is flabby. Let now ~ : Y -+ X be a separated smooth morphism of pure dimension d, and let n be an integer. By functoriality of R~z, for any complexes G" and G" of sheaves of (Z]nZ)y-modules which are bounded above and below, respectively, there is a cano- nical morphism of complexes %(3f'om'(G', G")) ~Wom'(Rg, G', R~?, G"). Let G'" -+J" be an injective resolution of G". Then the complexes ~ffom'(G',J') on Y and q~ J" on X consist of flabby sheaves. Therefore there is the following canonical morphisms in D+(X, Z/nZ) R~,(.~om(G', G")) = ~,(~0m'(G',J')) -*.~om'(R% G', R% J') = ~om(R% G', Rcp. G"). Assume now that n is prime to char(k). Then applying this morphism to complexes G" of the form 9* F'(d) [2d] and using the trace mapping Rg~(q~* F'(d) [2d]) ~ F', we obtain for any G" ~ D- (Y, Z/nZ) and F" E D + (X, Z/nZ) a duality morphism Rq~.(gf%m(G', q~* F'(d) [2d])) --~om(Re?, G', F'). '/.a.1. Theorem (Poincard Duality Theorem). -- Suppose that n is prime to char(k). Then the duality morphism is an isomorphism. We remark that the theorem is equivalent to the fact that, for all q e Z, the duality morphism induces isomorphisms Extq(O ", q~* F'(d) [2d]) ~ Extq(R~., G', r'). By Remark 6.2.10, the theorem is not true without the assumption that n is prime to char(T). Proof. -- Fixing one of the complexes G" or F', one gets an exact functor with respect to the second complex which is way out right (see [Hal], w I. 7). Therefore it suf- fices to verify the theorem only for complexes of the form G" = G, where G e S(Y, Z/nZ), and F" = F(-- d) [-- 2d], where F e $(X, Z/nZ). Thus it suffices to show that for all q >t 0 the canonical mappings q~q(G, F) := ExU(G, q0* F) --~ q~(G, F) := Uom(Rq~, G, F(-- d) [q -- 2d]) are isomorphisms. For example, in the case d = 0 the theorem follows from the fact that the functor % is left adjoint to the functor q0*. First of all we reduce the situation to the case when the space X (and therefore Y) is good. Indeed, assume that our statement is true in this case. Since the statement is 136 VLADIMIR G. BERKOVICH local with respect to X, we can shrink X and assume that X = U~= 1 X~, where X~ are closed analytic domains isomorphic to open subsets ofk-affinoid spaces (in particular, the X~ are good). Since any F has an embedding in the direct sum O~n=l 'r Fi, where F~ = v~ F and v~ is the canonical embedding X~ ~ X, it suffices to verify that '~(G, ~,. F,) ~ a'~(G, ~,. F,), q >/0. If v~ and q~ are defined by the cartesian diagram Y ~)X %,Xi. there is a canonical isomorphism of sheaves ~* v~. F~-~ v~. q~* F~ (the stalks of these sheaves are isomorphic). Therefore, Oq(G, v~. F~) -=-- Oq(v~" G, Fi). Furthermore, the morphism v~:X~--~X satisfies the condition of Corollary 5.3.6, and therefore v*(Rqb G) --% Rq~, (v~* G). It follows that W~(G, v~. F~) ----- ~"(v~* G, F~), and since the space X i is good, the assumption gives an isomorphism ~(G, vi. Fi) --% W~(G, vi. Fi). Thus, we may assume that the spaces X and Y are good. Suppose that d/> 1. We fix the sheaf F and set (I)~(G) = Oq(G, F) and W~(G) = tFq(G, F). Then {Oq}q~>0 and { W q }q~0 are exact contravariant O-functors from S(Y, Z/nZ) to ~qCb, the functors ~~ = Hom(G, ~* F) and W~ = Hom(R 2a q~ G, F(-- d)) are left exact, and 9 ~ are right satellites of the functor 9 ~ Thus to prove the theorem it suffices (and is necessary) to show that a) 0 ~ W ~ and b) the functors XFq, q >1 1, are right satellites of the functor W ~ The statement b) is equivalent to the fact that the functors XF~, q >/ 1, are effaceable, i.e., for any G e S(Y, Z/nZ) and ~ ~ ~q(G) there exists an epimorphism of sheaves G' -~ G such that 0~ goes to zero under the induced homomorphism ~F~(G) --~ ~(G'). Step 1. -- O0 ~ ~F o. Since the functors O0 and W ~ are left exact, it suffices to find a family of sheaves ~ C S(Y, Z/nZ) such that for any G ~ S(Y, Z/nZ) there exists an epimorphism O~ ~ x M~ ~ G with M~ ~./1 and, for any M e.//, one has O~ ~- W~ We take for ~g the class of sheaves of the form g~ (Z/nZ) v, where g : V ~ Y is separated a 6tale morphism for which there exists a commutative diagram Y ~X V ~U such that f is a separated 6tale morphism and ~b is a smooth morphism whose geometric fibres are nonempty and connected. From Corollary 3.7.3 (see also the proof of Pro- ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 137 position 4.4.5) it follows that any G is the epimorphic image of a direct sum of some sheaves from .#g. Let M = g,(Z/nZ)v. Then in the above notation we have O~ = Hom(g,(Z/nZ)v , q~* F) = Hom((Z/nZ)v, ~* F[v ) = (p* F(V) and q~~ = Hom(R ~a ?,(g,(Z[nZ)v), F(-- d)) = Hom(f (R sa d?,(Z/nZ)v), F(-- d)) = Hom(R ~ +, (Z/nZ)v , F(-- d)Iu) = Hom((Z/nZ)u (-- d), F(-- d)[u) = F(U) because the geometric fibres of q2 are nonempty and connected and n is prime to char(~) (Theorem 7.2.1). Thus, it suffices to show that if ~p:Y-->X is a separated smooth morphism of pure dimension d with nonempty and connected geometric fibres, then for F e 8(X, Z/nZ) the mapping Z: q~* F(Y) -- Hom((Z/nZ)y, q~* F) F(d)) = F(X) _+ Hom(R~a a R2a ~p* ~ (~,, y), the composition of the is an isomorphism. Since R ~ ~i r F(d) ~ F | 2a R a canonical mapping F(X) -+ ~* F(Y) with X is the identity on F(X). Therefore the required fact follows from the following statement. 7.8.2. Proposition. -- Let ~p : Y ~ X be a morphism of k-analytic spaces, and suppose that any gtale base change of 9 is an open map with nonempty and connected fibres. Then for any sheaf of sets F on X one has F -~ 9. 9* F. 7.8.8. Lemma. -- Any k-analytic space X contains a point x such that, for some embedding of fields k s ~-~ ~(x)', the image of the induced homomorphism of Galois groups G ~(,) ~ G k has finite index in G k. Proof. -- We may assume that X is k-affinoid. If X is strictly k-affinoid, then we take an arbitrary point x with [~(x) : k] < oo (i.e., x ~ X0). Therefore it suffices to show that if K = K,, where r r %/[ k* [, and the statement is true for the K-affinoid space X' = X | K, then it is also true for X. For this we take a point x' ~ X' such that, for some embedding of fields K* r the image of the induced homomorphism Gw(,,~ ~ G x has finite index. Furthermore, we fix an embedding of fields k"--> K ~. (We remark that the induced homomorphism Gr ~ G k is surjective.) Let x be the image of the point x' in X. The above embeddings of fields induce an embedding k ' r and we get a commutative diagram of homomorphisms G~r174 > G k T T Gw~,, ~ :, G K where the right homomorphism is surjective and the image of the lower homomorphism has finite index. The required statement follows. 9 18 138 VLADIMIR G. BERKOVICH Proof of Proposition 7.3.2. -- For a sheaf of sets G on Y and two elements f, g e G(Y) we set Supp(f--g) ----{y e Y ]f~ 4 g~}. It is a closed subset of Y. We claim that iff, g e ~0" F(Y), then Supp(f--g) = ?-l(E) for some set Z C X. (From this it follows that the set Y is closed because q0 is an open map.) Indeed, it suffices to show that if X = J/(k) andf:~ g, then Supp(f-- g) = Y. Assume that fu + gu at some point y e Y. By the definition of q0* F, there is an 6tale neighborhood h : V -+ Y of the pointy such that V is a K-analytic space, where K is a finite separable extension of k, and the elements f[ v and g{v come from some elements of F(K). Clearly, the latter elements are not equal. It follows that f] v and g Iv do not coincide at all points of V, and therefore fr + gr for all y'e h(V). Since h is an open map, it follows that the set Supp(f--g) is open, and therefore it coincides with Y because Y is connected. Let now f e ?* F(Y). It suffices to show that every point x e X has an 6tale neighborhood X'-+ X such that the restriction of f on Y � X' comes from an element g e F(X'). By Lemma 7.3.3, we can replace X by an 6tale neighborhood of the point x and assume that there exists a pointy e ?-x (x) for which the homomorphism Gav~u I --~ Gael~l, induced by an embedding og((x)" ,_.~yf(y)8, is surjective. It follows that F~(~(x)) --% (?* F)u (3r Therefore we can shrink X and find an element g e F(X) whose image in (?* F)u (~(y)) coincides with f~. We have Supp(f-- g) = ~-I(Z) for some closed set E C X. Since x r E, we can find an open neighborhood q/of x such that f]~-l~ = g. 9 Step 2. -- The functors W ~, q >1 1, are effaceable for d = 1. To show this we need the following fact which is an analog of the Fundamental Lemma 1.6.9 from [SGA4], Exp. XVIII. 7.3.4. Fundamental Lemma. -- Let q~ : Y --~ X be a separated smooth morphism of pure dimension one, and suppose that X is good. Then for any point y e Y there exist separated dtale morphisms f: X' ~ X and g : Y" ~ Y' = Y � x X' Y ~> X Y' r X' y~ such that y ~f'(g(Y")) and (i) the homomorphism RXq~i'(~,,y,,)-+R~ ?'~(~t,,y ,) /s zero; (ii) the geometric fibres of ?" are nonempty, noncompact and connected. Proof. -- We replace X by an affinoid neighborhood of the point x ----- ~(y). Let x = ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 139 1) One can shrink Y and assume that there is a commutative diagram y c J > ~an where + : ~ -+ ~ = Spec(zff) is a smooth affine curve of finite type over ~, and j is an open immersion. This is Lemma 3.6.2. We remark that from 1) it follows that q~i(~,,Y) = 0 and the same is true for any ~tale open subset of Y. Let Y/= Spec(B). 2) One can shrink Y and @' and assume that there exist ft, .... fm~ B and ct, .... r > 0 such that Y ={Y' E~r=[ If(Y')] < ~,, 1~ i~ m}. For this we remark that a basis of open sets in @r~ is formed by sets of the form {Y' ~ Y/= I If(f)[ < a,, [ g,(y')[ > b~, 1 ~< i~< n, 1 ~< j< m}, where f, gj r B and a~, bj > 0. Replacing B by the localization Bg~... g=, the second inequality can be rewritten as ]gT~(y')[ < b-f ~, and we are done. 3) In the situation of 2) the canonical homomorphism R x ~ R ~ +~(~t.,~.) is injective. (We remark that, by Corollary 7.1.4, the latter sheaf coincides with (R 1 +~ F,,g)~n.) By the Weak Base Change Theorem 5.3.1, it suffices to verify the fol- lowing fact. Suppose that 5F is an affine curve of finite type over k, fl, ...,f,~ ~ O(YT), and ~ = { x ~n ~ ] if(x)[ < r 1 ~< i ~< m } for some r > 0. Then the canonical homo- morphism Hlo(~, ~,) ~ HI0(;T =, ~t,) is injective. We may assume that the set S = Y'~a\q/ is nonempty. By Proposition 5.2.6 (ii), there is an exact sequence H~ S), t~.) -+ H~,(q/, F.) ~H~o( ~, ~.)- We claim that H~ "~, S), ~q) = 0. Indeed, it suffices to verify that if S is contained in a disjoint union of open sets r and ~ and the set S c~ ~ is nonempty, then it is noncompact. Suppose that S n 3r is nonempty and compact. Then we can find its affinoid neighborhood V in r (see [Ber], 2.6.3). By the Maximum Modulus Principle [Ber], 2.5.20, every of the functions f takes its maximum at the topological boundary 0(V/r of the set V in r Since 0(V/3r r3 S = O, then 0(V/r C qA From this it follows that If(x) ] < r for all x c V and 1 ~< i ~< m. This contradicts to the supposition that S n~4=O. 140 VLADIMIR G. BERKOVICH We are now ready to prove the lemma. By the Fundamental Lemma 1.6.9 from [SGA4], Exp. XVIII, one can find a separated &ale morphism of finite type g : @" ~ ~t T'/ o~, such that y e g(~t'~ (y is the image of the point y in ~t) and the homomorphism RI +;(t~.,~,) --~ R1 +~(t~.,~) is zero. Let Y' be the inverse image of Y in ~/'~, and let ~' denote the restriction of +'~ to Y'. Consider the commutative diagram R 1 9,(~.,~) > (R 1 +, ~t.,~,)" T T By 3), the upper arrow is injective. Since the right arrow is zero, it follows that the left arrow is also zero. Finally, to satisfy the condition (ii), it suffices to apply Theorem 3.7.2 to the morphism q~':Y'-+ Y. The Fundamental Lemma is proved. 9 The following is Lemma 2.14.2 from [SGA4], Exp. XVIII. 7.3.5. Lemma. -- Suppose that ~t is an abelian category, m >~ 0, { F; }0<~+~2.~ are objects of Db(~r such that H'(F;)=0 for qr f:F;-+F~+ 1 (0~<i~<2m--l) are morphisms, and f is the# composition F; -+ F~.,. Assume that H'(f) = 0 for q < m. Then there exists in Db(d) a morphism r : H"(F;) [-- m] --~ F~., such that the diagram Fo , t > F~., \% H~(Fo) [-- m] is commutative. 9 7.3.6. Corollary. -- Let r : Y ~ X be a separated smooth morphism of pure dimension one, and suppose that X is good. Then for any point y E Y there exist separated ltale morphisms f: X' ~ Xand g : Y" ~ Y' ~ Y � X' Y +>X T,' T, Y' +'> X' T; ~ y,, I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 141 such that y ef'(g(Y")) and there is a factorization R~',' (~t,, y,,) > R~', (~t., y,) (z/.Z) x, where t is induced by the trace mapping. Proof (see [SGA4], Exp. XVIII, 2.14.4). -- We set % = ~p : Y0 = Y -+ X0 = X and construct by induction four diagrams Y, ,X+ T T Y++I N+, + ",'++y' Y, + 1 which satisfy the conclusion of the Fundamental Lemma. Furthermore, we set X' = X, and Y~' = Y, � xi X'. If ~b, denotes the canonical morphism Y~' --~ X', then the mappings R+ +++ll(~,,Y~;l) ->" R+ ~b,.,(~,.y+,) are zeroes for q = 0, 1 and the mapping Tr: R 2 ++,(~,,Yi') -> (Z/nZ)x' is an isomorphism. By Lemma 7.3.5, the required statement is true for Y" = Y'4'- 9 We now prove the statement of Step 2. Let ~ be the family of sheaves on Y of the form ht(Z/nZ)v , where h:V--+ Y is a separated +tale morphism. Since any G e S(Y, Z/nZ) is the epimorphic image of a direct sum of some sheaves from ~r it suffices to show that for any M = ht(Z]nZ)v and 0r e u/q(M) there exists an epimorphism O+ e t M+ + M with M+ e ~ such that the image of m in all ~q(M+) is zero. Let + denote the morphism h o ~p : V --+ X. One has 9 q(M) = Horn(R?, M, F(-- 1) [q -- 2]) = Hom(R+. b%,v, F[q -- 2]). We apply Corollary 7.3.6 to the morphism + and an arbitrary point y e V. It follows that one can find separated dtale morphisms f: X' ~ X and g:V" ~ V' = V Xx X' V --+~X V' +'> X' T,~ 142 VLADIMIR G. BERKOVICH such that y ~f'(g(V")) and there is a factorization R+i'(~.,v,,) , R+i(~.,v,) (z/nZ)x, [- 2] We set M'= (hf'g)~ (Z/nZ)v,,. The image of ~ in tF~(M') = Hom(RqJi'(~t,,v,,), Fix, [q - 2]) is a morphism which goes through a morphism (Z/nZ)x, [-- 2] -+ F Ix' [q -- 2]. The latter morphism is represented by an element ~5 ~Ext~((Z/nZ)x,,FIx, ) = H~(X',F). Let x' = q~(y'), where y' ~ V' and f'(y') =y. Since q >/ 1, we can find a separated 6tale neighborhood X" -+ X' of the point x' such that the element ~ goes to zero under the canonical homomorphism H~(X ', F) ~ H"(X", r). Thus, if we replace the above objects by their base change under the morphism X" -+ X', the image of ~ in hV~(M ') will be zero. We remark that from Steps 1 and 2 it follows that the theorem is true for d = 1. Step 3. -- The functors tF~, q >1 1, are effaceable for d/> 2. First of all we remark that to verify the statement for a morphism 9:Y ~ X it suffices to find an dtale covering { Y, -~ Y }, e ~ such that it is true for all of the morphisms q~, = V o g~ : Y, ~ X. Indeed, suppose we have G e S(Y, Z/nZ), q/> 1 and ~ e tFq(G). Let G~ be the pullback of G on Y, and let ~ be the image of oc in ~F~(G~) = Hom(Rq~, G,, F(-- d) [q -- 2d]). By hypothesis, for any i e I there exists an epimorphism G~ -+ G, such that the image q t of ~, in ~F~(Gi) is zero. Then we get an epimorphism of sheaves @ieigi~(G~) ~G q t q t such that the image of e in any ~ (g,~(G,)) = u/i(Gi) is zero. We prove our statement by induction. Assume that it is true for d- 1. By the above remark it suffices to verify the statement for a morphism Z : Z -+ X which is a composition of separated smooth morphisms d/: Z -~ Y of pure dimension d- 1 and q~ : Y ~ X of pure dimension one. Since the theorem is true for d = 1, we have 9 ~(G) = Hom(Rz, G, F(-- d) [q -- 2d]) = Hom(R~,(Rd?, G(d -- 1) [2d -- 2]), F(-- I) [q -- 2]) = Ext"(R+, G(d -- I) [2a - 2], ~* F) = Hom(R+, G, ~* F(-- d + 1) [q -- 2d + 2]) __ q - ~(,,~.r,(G) where the latter denotes the analogous functor for the morphism + and the sheaf 9" F. By q ! induction, there exists an epimorphism G' ~ G such that the image of ~ in ~F(r ) is zero. Since the latter group coincides with ~Fq(G'), we get the required statement. The theorem is proved. 9 ]~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 143 7.4. Applications of Polncar6 Duality Let n be an integer prime to char(~). 7.4.1. Theorem. -- Let ~ : Y ~ X be a separated smooth morphism of pure dimension d. Then for any F e S(X, Z/nZ) there is a canonical isomorphism R~0,(c?* F) -~%m(R%(Z/nZ)y, F(-- d) [-- 2d]). Proof. -- Since Z/nZ is a free Z/nZ-module, then gxt~((Z/nZ)y, ~* F) = 0 for all q i> 1, and therefore ~om((Z/nZ)y, ~* F) --% c?* F. The required statement is obtained by applying Poincar~ Duality to the complexes G" = (Z/nZ)y and F" = F(-- d) [-- 2d]. 9 We say that a morphism of k-analytic spaces ~:Y-+ X is acyclic if for any F e S(X, Z/nZ) one has F-~ ?,(~?* F) and R ~ ~,(?* F) = 0, q >1 1. 7.4.2. Corollary. -- Let ~ : Y -+ X be a separated smooth morphism of pure dimension d, and suppose that the geometric fibres of ~ are nonempty and connected and have trivial cohomology groups H~ with coefficients in Z/nZ for q < 2d. Then the morphism ~ is acyclic. For example, the morphisms X � A a --> X and X � D -+ X, where D is an open disc in A a, are acyclic. 9 The following is a straightforward consequence of Poincard Duality. 7.4.3. Theorem. -- Suppose that k is algebraically closed, and let X be a separated smooth k-analytic space of pure dimension d. Then for any F ~ S(X, Z/nZ) and q >1 0 there is a canonical isomorphism Ext'( F, bta,, Y) --%-~cH2a- F) v. In particular, if F is finite locally constant, then one has H~(X, VV(d)) --% H~-q(X, F) v. 9 7.4.4. Corollary. -- Suppose that k is algebraically closed, and let X be a proper smooth k-analytic space. Tken for any finite locally constant sheaf F s S(X, Z/nZ) the groups He(X, F) are finite. 9 Let S be a k-analytic space. A smooth S-pair (Y, X) is a commutative diagram of morphisms of k-analytic spaces Y ~ X ~f3C~-~, 144 VLADIMIR G. BERKOVICH where f and g are smooth, and i is a closed immersion. The codimension of (Y, X) at a pointy e Y is the codimension at y of the fibre Y~ in Xs, where s -~ g(y). One can asso- ciate with a smooth S-pair (Y, X) the following commutative diagram Y i~X+ 2-j~ U l,s where U = X\Y, and j is the canonical open immersion. 7.4.5. Theorem (Cohomological Purity Theorem). -- Let (Y, X) be a smooth S-pair of codimension c, and let F be an abelian sheaf on X which is locally isomorphic (in the/tale topology) to a sheaf of the form f* F', where F'~ S(S, Z/nZ) (for example, F is a locally constant (Z/nZ)x-module). Then (i) ~f}(X, F) = 0 for q ~: 2c; (ii) there is a canonical isomorphism ~~ F)~ i* F(--c). Proof. -- We remark that if we are given a cartesian diagram Y > X T T Y' 9 X' with 6tale ~, then for any q/> 0 there is a canonical isomorphism v)IY.- (x', Fix,). Therefore we may assume that F ----f* F' for some F' ~ S(S, Z/nZ). Letfbe of pure dimension d. Then g is of pure dimension e := d -- c. By Poincar6 Duality, for any G e S(Y, Z/nZ) there is a canonical isomorphism Hom(G, g* F'(e) [2e]) -~ Hom(Rg, G, F'). Since Rg, G = Rf(i. G), it follows by Poincar6 Duality applied to the morphism f that one has Hom(Rg, G, F') -~ Hom(i. G,f* F'(d) [2d]). Furthermore, since the functor i. is left adjoint to the functor i', the latter group is isomorphic to Hom(G, Ri'(f* F') (d) [2d]). Thus, we get an isomorphism Hom(G, g* F'(e) [2e]) --~ Horn(G, Ri'(f* F') (d) [2d]), and this isomorphism is functorial on G. It follows that it is induced by an isomorphism of complexes in D(Y, Z/nZ) g* F'(e) [2e] ~ Ri'(f* F') (d) [2d]. Hence, Ri I (F) -~ i* F(-- c) [-- 2c], and the theorem follows. 9 In the following three corollaries, the situation is the same as in Theorem 7.4.5. I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 145 7.4.6. Corollary. -- H~(X, F) = 0for 0 ~< q ~< 2c -- 1 and H~(X, F) --% Hq-2'(Y, i* F(-- c)) for q >~2v. 9 7.4.7. Corollary. -- F -~3",(Flu), RZ'-lj,(FIo) ~ i,(i* F(-- c)), and Rqj,(FIv ) = 0 4: 0,2c-- 1. 9 for q Applying the spectral sequence RPf.(Raj.(FIv)) :~ R~+qh.(FIo), we get 7.4.8. Corollary (Gysin sequence). -- Suppose that the relative dimension off is equal to& (i) R~f, F ~ R a h,(FIv ) for 0 ~< q ~< 2c -- 2, and there is an exact sequence 0 -+ R2'-lf, F --> R 2~-1 h,(F[v ) -+g,(i* F(-- c)) -+ R2*f, F -+ R2* h,(FIv) ~ Rag,(/* F(-- c)) -+... (ii) Suppose that k is algebraically closed and S = Jl(k). Then H~(X, F) --% Ha(U, F) for 0 ~ q <<. 2c -- 2, and there is an exact sequence 0 -+ H2'-I(X, F) --> H~'-'(U, F) -+ H~ i* F(-- c)) ---> -+ n2~(X, F) -+ H~(U, F) ~ H'(Y, i* F(-- c)) -+ ... -~ H 2'a- ')(Y, i* F(- c)) -+ Hm(X, F) --> H~(U, F) --> 0. The following statement will be used in the proof of the Comparison Theorem 7.5.1. 7.4.9. Theorem. -- Let r : Y ---> X be a separated smooth morphism of pure dimension d, and suppose that F is a finite locally constant (Z/nZ)y-module such that all the sheaves R q r q >1 O, are finite locally constant. Then for any q >>. 0 there is a canonical isomorphism R, r. F -U (R v (-- d). 7.4.10. Lemma. -- Let X be a k-analytic space. Suppose that the cohomology sheaves of a complex F" e D-(X, Z/nZ) are finite locally constant, and let G be a locally free sheaf of (Z/nZ)x-modules of finite rank. Then for any q e Z there is a canonical isomorphism 8xf(F', G) --% ~om(H-~(F'), G). Proof. -- If F is finite locally constant, then gxt~(F, G) = 0 for all q >/ 1 because Z[nZ is an injective Z/nZ-module. Our statement now follows from the spectral sequence (see [Gro], 2.4.2) gxt~(H-q(F'), G) = gxt~+~(F ", G). ,' Proof of Theorem 7.4.9. -- Since G v is also finite locally constant, Lemma 7.4.10 gives an isomorphism ~om(F v, ~ti, y) = ~om(F v, ~ti, y) = O(d). 19 146 VLADIMIR G. BERKOVICH By Poincar~ Duality and Lemma 7.4.10, we have (R q ~. F) (d) = R q ~.(o~f0m(F v, Ba.y)) = d~xtq-~(Rq),(FV), (Z/nZ)x) = (R2a-q ~,(FV)) v The required statement follows. 9 7.5. The Comparison Theorem 7.5.1. Theorem. -- Suppose that 5" is a scheme of finite type over Spec(~), where ~ is a k-affinoid algebra, f: Y" ---> 50 and ~ : ~' --> f are morphisms of finite type, and o~" is a cons- tructible abelian sheaf on ~t with torsion orders prime to char(k). Then there exists an everywhere dense open subset all C 50 such that the following properties hold. (i) The sheaves R q ~. ~ ]~_,,~/~ are constructible and equal to zero except a finite number of them. (ii) The formation of the sheaves R q cO. ~" is compatible with any base change 50' --> 5 ~ such that the image of 50' is contained in ~ll. (iii) In (ii) assume that 5O' is a scheme of locally finite type over Spec(~), where d~ is an affinoid d-algebra, and that the morphism 5O' --> 5O is a composition 5O' --> 5 ~ | d~ -~ 5O. Let 9' be the morphism ~/' = ~t � s~ 5O, --> y[', = ys x s," 5O', and let ~" be the inverse image of o~ on ~/'. Then for any q >1 0 there is a canonical isomorphism The condition on the morphism 5O' ~ ,9~ implies that for any scheme ~ of locally finite type over 5O one has 5O'),,,, ~ ..r 5O',-,. (..r Xs~ X s~,,,, The existence of an everywhere dense open subset ~ C 5" which possesses the properties (i) and (ii) is guaranteed by Deligne's " generic " theorem 1.9 from [SGA489 Th. finitude. In fact the proof of Theorem 7.5.1 follows closely the proof of Deligne's theorem and uses it. Moreover, the proof is a purely formal reasoning which works over the field of complex numbers C as well (of course, in this case one should assume that ~' = ~ ---- 13). The other main ingredients of the proof are the Comparison Theorem for Cohomology with Compact Support, Poincard Duality for schemes and analytic spaces, and the constructibility of the sheaves R ~ q~., ~- ([SGA4], Exp. XVII, 5.3.6). Proof. -- By Deligne's theorem, we can shrink 5 ~ and assume that the properties (i) and (ii) hold for ~ = 5". And so it remains to show that there exists an everywhere dense open subset 0//C 5O for which the property (iii) holds. Since the property (iii) is local with respect to ,9~ and f, we may assume that they are separated. Furthermore, if (~ti --~gi ~r is a finite ~tale covering of ~t, then it I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 147 suffices to verify (iii) for all of the morphisms ?g~ : q/i -+ R'. Therefore we may assume that q/ is also separated. Finally, since the scheme q/ is Noetherian, we can find an integer n prime to char(~) with no~- = 0. Therefore we can work with the categories of sheaves of Z/nZ-modules. 7.5.2. Lemma. -- The property (fix) holds when Ri" = 50, ~ is smooth, and o~" is locally constant. Proof. -- We may assume that ~ is smooth of pure dimension d. Since the sheaf ~-v is also locally constant, all the sheaves R ~ V: (fly), q i> 0, are constructible. Furthermore, since they are equal to zero for q > 2d, we can shrink 5# and assume that all the sheaves R ~ ~(5~-v), q >/ 0, are actually locally constant. Theorem 7.4.9 and its analog for schemes give isomorphisms R -X (R (-- a), Rq q~.(o~z.~.) _,% (R2a-~ q~,~(~-~v))v (_ d). The required statement now follows from the Comparison Theorem for Cohomology with Compact Support (Corollaries 7.1.4 and 7.1.5). 9 We remark that it suffices to prove the theorem in the case when ~ is an open immersion with everywhere dense image. Indeed, we may assume that q/ is affine. Then there is a factorization V-~ ~j, where j:q/~ ~ is an open immersion, and : ~e ~ y- is a proper morphism. Hence R% = R+, Rj, and R% ~" = R+**" Rj, ~, and the required statement follows from the Comparison Theorem for Cohomology with Compact Support. We remark also that it suffices to assume that 5: is the spectrum of an integral domain. Let ~ be the generic point of 5:. We prove the following statement by induction. (*)a The property (iii) holds when dim(~) <~ d and q~ is an open immersion with every- where dense image. 1) (*)0 is true. It suffices to show that there exists an everywhere dense open subset o//C 5# with f-l(://) C ~. For this we may assume that Y" is affine, and we can find an open immersion with everywhere dense image ~ ~ ~ in a proper 5P-scheme ~. It follows that q/is everywhere dense in ~Y_ and therefore ~ = Y/,. The latter means that the point ~ is not contained in the closed set { s ~ 5: [ dim(~8) t> 1 }. Thus, we can shrink 5: and assume that 5~ is finite over 5#. Finally, since 5~ = ~/~, the image of the closed set ~\~ is closed and nowhere dense in 5 #. Therefore, shrinking 5 #, we get Suppose now that d>~ 1 and assume that (*)a-1 is true. 2) One can shrink 5f and find an open subset .~f C Y" such that its complement ~1 is finite over 50 and the property (iii) holds for the open immersion q/ n ~ -+ ~f. 148 VLADIMIR G. BERKOVICH Since the statement is local with respect to ~, we may assume that ~ is a closed subscheme of the affine space A~. Let zq denote the i-th projection ~-+ A~. The induction hypotheses applied to the diagrams \/, gives open subsets q/~ C A~ such that the property (iii) holds for the open immersions ~/o rc~-l(q/~) ___~ ~-1(0//~) and the sets q/~. It follows that if we set .~ = I.J~= a ~-x(q/~), then it also holds for the open immersion ~r o .o~ ~ ~ and the set Se. Finally, the reaso- ning from 1) shows that we can shrink oq ~ and assume that the morphism A~\q/~ ~ Sr are finite. It follows that the morphism ~1 = ['l~=a ~-l(A~\q/i) ~ S~ is also finite. 3) (*)a is true if ff/ is smooth over ,9~ and ~" is locally constant. Since the statement is local with respect to ~, we may assume that ~ is a closed subscheme of the affine space A~. After that we can replace ~ by its closure in the pro- jective space P~. In particular, we may assume that the morphismf: ~r ~ 5~ is proper. We shrink 6 a and take the open immersion j: ~r ~ Y" and the closed immersion i:~ ~ ~ which are guaranteed by 2). Consider the commutative diagram (" ~> ~ < ~1 By construction, f is proper, fl is finite, and the property (iii) holds for the open immersion ~r c3 .o~ ~ ~e and the set q/= Sr By hypotheses, g is smooth, and there- fore, by Lemma 7.5.2, we can shrink S e and assume that the property (iii) holds for the morphism g : ~ ~ 5". To verify (iii), we may assume that the base change is already done. (The only thing we should take care of is that our further manipulations do not change the scheme oq'.) Consider the exact sequence (1) 0 -+j,j* R% ~- ~ Rg. 5 ~ i. i* R?. ~- ~ 0 and the induced morphisms of exact sequences (1)~-+ (1 ~) 0 ~ jy(j* Rq~, .,~')" ~- (Rq~,.~')~" , i,~(i * R%.~')~ ~ 0 0 , j~(j,.* R?~f ~'~) > Repel ~'~ , i.~(i "** Rep. ~ ~"~) , 0 Since the first vertical arrow is an isomorphism, then to show that 0(~') is an isomorphism, it suffices to verify that (i" R~,. 5) ~ ~ '~" i R~. "~ ~r~. I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 149 For this we apply Rf, and Rf, "n to the sequences (1) and (pn), respectively. We obtain a morphism of exact triangles 9 (Rf, j.,j*R~,~') "n ~ (Rg,.~') "n ) f~n(i*R~,~')'n > , Rf~nj~(j ,n* Rq~: n 5z'~) ) Rg,n ~-,n , f~(i "n* R~ ~-'~) , The first and the second vertical arrows are isomorphisms. Therefore the third arrow is also an isomorphism. Since fz is finite, it follows that (i* R~. ~')'n ~ i '~" R~." ~"~. 4) (*)a is true in the general case. Decreasing 5P, we can find a finite radicial surjective morphism 5r --+ 5r such that the scheme (~ X s~ ~9~ has an everywhere dense open subset which is smooth over St'. Since such a manipulation does not change the 6tale cohomology, we may assume that ~/has an everywhere dense subset ~e which is smooth over 5~. Shrinking ~e, we may assume that ~-is locally constant over ~e. Let j denote the open immersion ~e ,_+ ~. By 3), we can shrink 5P and assume that the theorem is true for the pairs of morphisms (f: At -+ ,9 ~ ~ = ~j : ~e ~ At) and (g : ~ -+ 5P, j : ~e ~ if/) and the sheaf j* ~-. It follows that if we define if" by the exact triangle ~.~- ~ Rj, j* 5 -~ if" ~ (2) then there is an exact triangle (2 "n) Furthermore, the sheaves Hq(ff ") are constructible and equal to zero except a finite number of them, and the formation of Hr ") is compatible with any base change. Finally, the sheaves Hr ") have support in the closed subset zr q/\Y', and one has dim(C/f~) ~< d -- 1. Since the canonical morphism zr ~ At is a composition of the open immersion ~qP,-+ ~, where ~ is the Zariski closure of Yf" in At, and the closed immersion Of" -+ At, it follows, by inductional hypotheses, that we can shrink 5p and assume that the property (iii) holds for the morphism q~ : ~t _+ At and the sheaves Hq(ff'). As in the proof of 3), to verify (iii) we may assume that the base change is already done. Applying Rq~. and Rq~. = to the triangles (2) and (2~n), respectively, we obtain a morphism of triangles 10`'' 1 9 Rq~,,n .~',n > Rq~',nj'n* .2~"n > R~ ff'~" > Since the second vertical arrow is an isomorphism, to show that 0(~') is an isomorphism, it suffices to verify that (R% ff.)~l ~. ff~.. 150 VLADIMIR G. BERKOVICH But this follows from the homomorphism of spectral sequences (R v %(H~(~'))) ~" ==> (R'+q~, if')" because the left arrows are isomorphisms. The theorem is proved. 9 7.5.8. Corollary. -- Let ~ : q/ -+ ~ be a morpkism of finite type between schemes of locally finite type over k, and let ~ be a constructible abelian skeaf on q/ with torsion orders prime to char(~). Then for any q >1 0 tkere is a canonical isomorphism (R r ~. ~r)~n -% R ~ ~p ~,~. Proof. -- Since the statement is local with respect to ~, we may assume that (and therefore ~') is of finite type over k, and therefore we can apply Theorem 7.5.1 for d = k and ~ = Spec(k). 9 7.5.4. Corollary. -- Let ~ be a scheme of locally finite type over k, and let ,~ be a cons- tructible abelian sheaf on ~ with torsion orders prime to char(~). Then for any q >i 0 there is a canonical isomorphism Ha(y ", ~-) -% H~(~ -, ~'~-). Proof. -- First of all we remark that it suffices to consider the case when the scheme is affine and of finite type over k. Indeed, if this is so, then in the general case we can take a covering q/-- { U, }~ e ~ of ~ by open affine subschemes of finite type over k and use the homomorphism of spectral sequences ~I'(q//Y', W~(o~')) :. H,+q(y-, ~z-) ; l Thus, we may assume that f is affine and of finite type over k. In this case we apply Corollary 7.5.3, the homomorphism of the Leray spectral sequences associated with the morphisms Y" ~ Spec(k) and ~'~" -+,,/t'(k), and the fact that the required state- ment is true for f = Spec(k). 9 The invariance of cohomology under extensions of the ground field 7.6. 7.6.1. Theorem. -- Let K/k be an extension of algebraically closed non-Archimedean Let X be a k-analytic space, and let F be an abelian torsion sheaf on X with torsion orders fields. prime to char(~). We set X' = X ~ K and denote by F' the inverse image of F on X'. Then for >1 0 there is a canonical isomorphism any q Hq(X, F) -% H~(X ', F'). P, TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 151 Proof. -- All the sheaves considered below are supposed to be abelian torsion with torsion orders prime to char(~). We remark that if the theorem is true for k-affinoid spaces, it is also true for arbitrary k-analytic spaces. Indeed, if X is separated paracompact, we can find a locally finite affinoid covering ~ = { V,~ }~ e i of X and apply the homomorphism of the spectral sequences 4.3.7 ~Iv(~, ~q(F)) > H ~ +q(X, F) l + ~I~(zr :', ~r ---*- H v+"(X', F') where ~r = { V, 6 K }r e x- If X is arbitrary paracompact, we use the same reasoning and the fact that the intersection of two affinoid domains is a compact analytic domain. If X is Hausdorff, we use the similar reasoning for a covering of X by open paracompact subsets and the fact that the intersection of two open paracompact subsets in a locally compact space is paracompact. If X is arbitrary, we use the same reasoning for a covering of X by open Hausdorff subsets. Furthermore, we remark that it suffices to prove the theorem for one-dimensional k-affinoid spaces. Indeed, let Y = .Ar be a k-affinoid space of dimension d >1 2, and suppose that the theorem is true for k-affinoid spaces of smaller dimension. Take an element f e ~ which is not constant at any irreducible component of Y, and consider the induced morphismf: Y -+ A 1. Let X = Jt'(M) be a closed disc in A 1 which contains the image of Y. We get a morphism ofk-affinoid spaces ~ : Y -+ X such that dim(X) = 1 and dim(Y,) < d for all points x E X. Consider the following commutative diagram Y ~>X Y' ~'~ X' where X'=X| and Y'=Y~K=Y X xX'. To prove the theorem for Y, it suffices to show that for any sheaf F on Y and any q >/ 0 there is a canonical isomorphism (R" ~. F)' -~ R ~ ~: F'. But this is obtained, by induction, from the Weak Base Change Theorem 5.3. I. Indeed, if x E X and x' ~ X' is a pair of points with x = ~(x'), then a fixed embedding of fields ~(x) a c->~~ ~ induces an isomorphism of analytic spaces Y~ @~e~: ~(x ) --* Y~,. Thus, we may assume that X in the theorem is a one-dimensional k-affinoid space. To prove the theorem for X, we consider the following more general situation. Let S be a locally closed subset of X, and F be a sheaf on the k-germ (X, S). We set 152 VLADIMIR G. BERKOVICH (X', S') = (X, S)(~ K and denote by F' the inverse image of F on (X', S'). Further- more we denote by 0~((X, S), F) and 0g((X, S), F), respectively, the canonical homo- morphisms H~((X, S), F) -+ H~((X ', S'), F') and H~((X, S), F) -+ H~((X', S'), F'). Our starting point is Theorem 6.4.1 (ii) which implies that 0~(X, F) is an isomorphism if the sheaf F is finite constant. And here is a continuation. 1) 0~(X, F) is an isomorphism, when F is finite locally constant. -- Indeed, one can find a finite ~tale covering Y ~ X such that the sheaf FIg is constant, and there- fore one can use the spectral sequences associated with the 6tale coverings (Y ~ X) and (Y ~ K ~ X'). 2) 0q((X, S), F) is an isomorphism, when S is closed and F is finite locally constant. -- Indeed, by Proposition 4.4.1, the sheaf F extends to a finite locally constant sheaf on an open neighborhood of S. Our statement now follows from Proposition 4.3.5 because affinoid neighborhoods of S form a basis of its neighborhoods. 3) 0~((X, S), F) /s an isomorphism when F extends to a finite locally constant sheaf on (X, S) (S is the closure ors in X). Indeed, we can apply 2), the exact cohomological sequences associated with the embeddings (x, s) 4. (x, g)" (x, g\s) and (X', S') ,~ (X', (S)') '~- (X', (S)'\S'), where (X', (S)') ---- (X, S) ~ K, and the five-lemma. 4) 0g((X, S), F) is an isomorphism for an arbitrary finite locally constant sheaf F. Indeed, by Proposition 5.2.8, one has Hq~((X, S), F) = lim H~((X, T), F) and Hg((X', S'), F') = lim Hg((X', T'), F'), where T runs through open subsets of S with compact closure. Therefore we can apply 3). 5) 0~(X, F) is an isomorphism, when F is of the form j, G, where j is the morphism of germs (X, S) -+ X determined be a locally closed subset S C X, and G is a finite locally constant sheaf on (X, S). From Corollary 5.2.5 it follows that Hq(X,j: G) -- Hg((X, S), G) and Uq(X',jl G') = H~((X', S'), G'). Since j~ G' ---- (j, G)', we can apply 4). 6) 0~(X, F) is an isomorphism, when F is quasiconstructible. This is obtained from 5), by induction, using Proposition 4.4.4 and the five-lemma. 7) 0~(X, F) is an isomorphism for arbitrary F. This follows from 6) and Proposi- tions 4.4.5 and 5.2.9. 9 I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 153 7.6.2. Corollary. -- In the situation of Theorem 7.6.1 assume that X is Hausdorff. Then for any q >>. 0 there is a canonical isomorphism H~(X, F) --% H~(X', F'). Proof. -- From Proposition 5.2.8 and Corollary 5.2.5 it follows that H~(X, F) = lim H"(X,j~<,(F ]e)) and H~(X', F') = lin>~ H'(X,j~,.,(F' le')), where the limit is taken over all open subset o//C X with compact closure, ~' = ~//6 K, and je (resp. j~,) denotes the canonical open immersion q/r X (resp. q/' ~ X'). The required statement follows from Theorem 7.6.1. 9 7.7. The Base Change Theorem for Cohomology with Compact Support 7.7.1. Theorem. -- Let ~ : Y ~ X be a Hausdorff morpkism of h-analytic spaces and a morphism of analytic spaces over k which give rise to a cartesian let f: X' ~X be diagram Y > X T" T, Y' *'> X' Then for any abelian torsion sheaf F on Y with torsion orders prime to char(~) and any q >I 0 there is a canonical isomorphism f*(R ~ ~., F) -% R q q~;(f'* F). Proof. -- The Weak Base Change Theorem 5.3.1 reduces the situation to the case when X = ~'(k) and X'= ~r where K/k is an extension of algebraically closed fields. In this case our statement follows from Corollary 7.6.2. 9 7.7.2. Corollary. -- In the situation of Theorem 7.7.1 suppose that ~? is of finite dimension, and let n be an integer prime to char(k). Then for any F" ~ D-(X', Z/nZ) and G" ~ D-(Y, Z/nZ) there is a canonical isomorphism V" | G') r" of" G'). Proof. -- From Theorem 7.7.1 it follows that there is a canonical isomorphism /*(R~, G') ~ R~j(f'* G'). Therefore our statement is obtained by applying Theorem 5.3.9 to f':Y' ~ X'. 9 20 154 VLADIMIR G. BERKOVICH 7.7.3. Corollary (Kiinneth Formula). -- Given a cartesian diagram of Hausdorff morphisms of k-analytic spaces XxsY X ~h Y suppose that f and g are of finite dimensions, and let n be an integer prime to char(~). Then for any F" ~ D-(X, Z/nZ) and G" ~ D-(Y, Z/nZ) there is a canonical isomorphism L L Rf ,. F" @ Rg, G" -% Rh,. (g'* F" Of'* G'). Proof. -- Applying Theorem 5.3.9 to the morphismf and Corollary 7.7.2, one has L L L R pt t, Rf,. F" | Rg, G" -% Rf,.(F" | G')) -% Rf,. g,.[g F" | G') --~ Rh, (g'* F" | G',). 9 7.8. The Smooth Base Change Theorem We say that a morphism of k-analytic spaces ~ : Y ~ X is almost smooth if, for any point x ~ X, there exist an open Hausdorff neighborhood ~ of x and affinoid domains V1, ..., V, C ~ such that V 1 u ... u V, is a neighborhood of x and all the induced morphisms ~-I(V~) -+ V, are smooth. Or course, smooth morphisms are almost smooth. (And we believe that the converse implication is also true.) 7.8.1. Theorem. -- Let f: X' --> X be an almost smooth morphism of k-analytic spaces, and let ~ : Y -+ X be a morphism of analytic spaces over k, which give rise to a cartesian diagram Y ~ X Y' ~'> X' Then for any (Z/nZ)y-module F, where n is an integer prime to char(~), and for any q >I 0 there is a canonical isomorphism f*(P,q ~, F) --% R q ~'.(f'* r). Proof. -- First of all, the reasoning from the beginning of the proof of Theorem 7.3.1 reduces the situation to the case whenfis a smooth morphism of good k-analytic spaces. I~TALE COHOMOLOGY FOR NON-ARCI-IIMEDEAN ANALYTIC SPACES 155 Furthermore, since the statement is local with respect to X' and is evidently true whenf is 6tale, it suffices to assume that f is of pure dimension one. Let x' be a point of X', and set x = q~(x'). One has (f*(R = q~. F)),, = (R ~ ~, F), = lira Hq(Y X x U, F), where the limit is taken over all 6tale morphisms U --+ X with a fixed point u e U over x and with a fixed embedding of fields Xt~ ~-+ ~(x)" over ~(x). Similarly, one has (R q ~;(f'* F))~, = lim H'(Y' X x, W,f'* F), where the limit is taken over all dtale morphisms W -~ X' with a fixed point w e W over x' and with a fixed embedding of fields W(w) ,-~3f'(x')* over W(x'). Therefore, it suffices to show that for any ~tale morphism (W, w) ~ (X', x') there exist 6tale mor- phisms g : U -+ X and h : U" ~ U' ---- W � x U such that the point w is contained in the image of U" and for any q >t 0 one has Im(Hq(V', f'* F) -+ H'(V", f'* F)) C Im(Hq(V, F) ~ Hq(V '', f'* F)), = U". We can shrink W and where V Y � V'=Y � and V"=Y Xx assume that W-+ X is a separated smooth morphism of pure dimension one. By Corollary 7.3.6, there exist separated 6tale morphisms g:U-+ X and h:U"-+U' =W xxU W > X T T" U' *>U T.Y U" such that the point w is contained in the image of U" and there is a factorization RZ,(~., ~,, ) , R+,(~.,o,) (Z/nZ)v [-- 2] where t is induced by the trace mapping. Consider the fibre product of the previous diagram with Y over X Y � W > Y T T, V' ~' ~V V" 156 VLADIMIR G. BERKOVICH We claim that for any q >i 0 there is a factorizafion Sq(V',f '* F) > S'(V",f'* F) \/ H'(V, F) where the homomorphism Hq(V, F) -+ Hq(V",f '* F) is the canonical one. Indeed, by the Base Change Theorem for Cohomology with Compact Sup- port 7.7.1, there is a factorization RZ;(~,, v,, ) , R+;(V,, v,) (z/nz)v [- 2] where t' is induced by the trace mapping. By Theorem 7.4.1, we have R+:(+'*(FIv)) -~5~~ F]v [- 2]), R)~:()(*(FIv)) ---~om(Rx;(~.,v,,), F[v [- 2]). 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[Ha2] [Kie] KIEHL, R., Ausgezeichnete Ringe in der nichtarchimedischen analytischen Geometric, J. Reine Angew. Math., 234 (1969), 89-98. MATSUMUm% H., Commutative ring theory, Cambridge University Press, 1980. [Mat] [Mit] MXTCHELL, B., Theo~7 of Categories, New York-London, Academic Press, 1965. VAN DER PUT, M., The class group of a one-dimenslonal affinoid space, Ann. Inst. Fourier, 80 (1980), [Put] 155-164. RAX'NAUD, M., Anneaux locaux hensfiliens, Lecture Notes in Math., 169, Berlin-Heidelberg-New York, [Ray] Springer, 1970. GROTn~NDIECK, A., S~minaire de g~omfitrie algfibrique, I. Revetements ~tales et groupe fondamental, [SGA1] Lecture Notes in Math., 9.24, Berlin-Heidelberg-New York, Springer, 1971. [SGA4] ART~N, M., GROTHESDI~CK, A., Vsm)mR, ]'.-L., Th~orie des topos et cohomologie ~tale des schemas, Lecture Notes in Math., 9.69, 270, -q05, Berlin-Heidelberg-New York, Springer, 1972-1973. [SGA489 D~LXON~, P. et al., Cohomologie ~tale, Lectures Notes in Math., 569, Berlin-Heidelberg-New York, Springer, SCm~D~R, P., STUHLER, U., The cohomology of p-adic symmetric spaces, Invent. math., 105 (1991), [ScSt] 47-122. [Ser] S~,J.-P., Cohomologie galoisienne, Lectures Notes in Math., 5, Berlin-Heidelberg-New York, Springer, Z~a~ISKI, O., SAMUEL, P., Commutative algebra, Berlin-Heidelberg-New York, Springer, 1975. [ZaSa] GROTHENDIECK, LOTKEBOHMEnT, 158 VLADIMIR G. BERKOVICH INDEX OF NOTATIONS x[y : 1.1 (13). O, OK (system of affinoid spaces) : 1.2 (15). d, ~v/u : 1.2.3 (16). (X, d, x) : 1.2 (17). ~Pk-.~n, Ok-.~Cn : 1.2 (17). u a~: 1.2 (17). 9V/V, : 1.2.7 (18). 9, 901 U, : 1.2 (18). + o 9, +9 : 1.2 (18). ..<v : 1.2 (20). ~, ~: 1.2 (21). ]X[ : 1.2 (22). k-sin, st-k-dn : 1.2 (22). dim(X) : 1.2 (23). X O (X analytic space) : 1.3 (25). An, E(0;q ..... r.) : 1.3 (25). ~xo, tVxo(M), Supp(F) : 1.3 (25). Mod(Xo), Coh(Xo), Pic(XG) : 1.3 (25). x~, x ~ : 1.3 (25). Ox, Ox(M) : 1.3 (25, 26). Mod(X), Coh(X), Pie(X) : 1.3 (26). F O (F coherent Ox-module ) : 1.3 (26). 9 O (9 morphism of analytic spaces) : 1.3 (27). d~Yo/X o, .fl/'Y/X : 1.3 (28). X @ K : 1.4 (30). O-,~nk, ,~n~ : 1.4 (30). .~(x) (x point of an analytic space) : 1.4 (30). Yx, dlm(9) (9 morphism of analytic spaces) : 1.4 (30). Ay/x, ~YG/XG, ~Y/X : 1.4 (30). Int(Y/X), O(Y/X), Int(Y), 0(Y) : 1.5 (34). Xo, ~xo (X Hausdorff strictly analytic space) : 1.6 (35). Xo ~, F o (F coherent Ox-module) : 1.6 (37). Mod(Xo) , Coh(Xo) , Pic(Xo) : 1.6 (37). ~x,=, re.x, K(x) : 2.1 (38). Y', x, ~'x, k(x) : 2.1 (38). G(L/K), Gar, K s, K a, K : 2.4 (45). K ~ : 2.4 (46). I(L/K), W(L]K) : 2.4 (46). K nr, K mr, O~ r, O~ r, MK, W K : 2.4 (48). cd~(G) : 2.5 (49). s(K/k), t(Klk), d(K/k) : 2.5 (50). ~a, 02/o : 2.6 (51, 52). ~/~', Der~r M) : 3.3 (60). l~t(X) : 3.3 (64). (X,S), (X,x) : 3.4 (64). k-(~erms, ~erms k : 3.4 (64, 65). Et(X, S), F&(X,x), F~t(K) : 3.4 (65). A~: : 3.5 (67/. I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 159 A(X) (the skeleton of X) : 3.6 (71). E(a, r), D(a, r), B(a; r, R), p(E(a, r)) : 3.6 (71). Pu!Q: 3.7 (79). X~t , Xr $(X), S(X, A), D(X), D(X, A), Hq(X, F) : 4.1 (82). o~n : 4.1 (82). Xfqf, k-d~CnlX : 4.1 (82). (F coherent 0xcmodule) : 4.1 (83). Ax, Ga, x, Gin, x, ~tn, x : 4.1 (84). Hq(X, A), Hq(X, Ga), Hq(X, Cm), Hq(X, ~.), ill(X, F), PHS(G/X) : 4.1 (85). exit : 4.1 (85). (x, s)6t, (x, s)~, s(x, s), P(x, s) : 4.2 (86). i(x, s), F(X, s), F(X, S) (F sheaf on X) : 4.2 (86). Ka, I~ : 4.2 (87). Fx, Fz(K ) (K extension of o~/"(x) in .~(x) s) : 4.2 (87). fx, Supp(f) : 4.2 (87). cdt(X ) (X analytic space) : 4.2 (88). sh Cx,~ : 4.2 (89). P(Xoa), ~r162 P), Hq(~, P), Lr Jg'q(F) : 4.3 (92). j: G : 4.4 (95). ~, CW, 9-x(r : 5.1 (96). Cs, Cr : 5.1.1 (96). Fr H~((X, S), F), H~((X, S), F), Hen((X, S), F) : 5. I (96). ~= {Cr )} : 5.1.3 (97). 9~, q~: : 5.1 (97, 98). i ~, .Ji~R((X, S), F) : 5.1 (98). ~T, ~t'~T : 5.2 (100). Y~, F~ : 5.3 (102). F'| : 5.3 (105). ~n,x, m F'(m), F v : 5.3 (106). Tr~ : 5.4 (106). Trx : 6.2 (112). degD,f(D) : 6.2 (112, 113). Y9 : 6.2 (114). 7B : 6.2 (115). '~o(Y) : 6.3 (119). deg(9) : 6.3.1 (iii) (119). k ~176 : 6.4.2 (127). [] [$ : 7.2 (131). -x a, r~ X : 7.2 (132). Try1 ..... Ia): 7.2 (132). Horn, oVg'om, Horn', ,~Fom*, Horn, .;r Extq, d'xtq : 7.3 (134). Cq(G, F), q~'q(G, F) : 7.3 (135). (I)q(G), ~Fq(G) : 7.3 (136). oq((x, s), F), 0.q((X, S), F) : 7.6 (152). 160 VLADIMIR G. BERKOVICH INDEX OF TERMINOLOGY Affmoid atlas : 1.2.3 (16). ~-Affinoid domain : 1.2 (21). ~-Analytie domain : 1.3 (23). Analytic space : 1.2 (17), k-analytic space : 1.2 (22), analytic space over k : 1.4 (30), strictly k-analytic space : 1.2 (22). Boundary of an analytic space : 1.5.4 (34). Closed immersion : 1.3 (28). Coherent sheaf : 1.3 (25, 26). Compact map : 1.1 (14). Conormal sheaf : 1.3 (28). Dense class of affinoid spaces : 1.2 (16). Dense collection of subsets : 1.1 (13). Dimension of an analytic space : 1.2 (23). Dimension of a morphism : 1.4 (30). Duality morphism : 7.3 (135). Elementary curve : 3.6 (71). Elementary fibration : 3.7.1 (76). l~tale topology on an analytic space : 4.1 (82). l~tale topology on a germ : 4.2 (86). ~p-Famity of supports : 5.1.2 (97). Fibre of a morphism at a point : 1.4 (30). Flabby sheaf : 4.2 (88). Geometric ramification index of an ~tale morphism : 6.3 (119). Germ of k-analytic space : 3.4 (64). Germ of analytic space over k : 3.4 (65). Gluing of analytic spaces : 1.3 (24). Good analytic space : 1.2 (22). Ground field extension functor : 1.4 (30). Hausdorff map : 1.1 (14). Interior of an analytic space : 1.5.4 (34). Locally (G-locally) closed immersion : 1.3 (28). Locally Hausdorff topological space : 1.1 (13). Morphism of analytic spaces : 1.2 (20), almost smooth : 7.8 (154), closed : 1.5.3 (34), ~tale : 3.3.4 (63), finite : 1.3 (28), fnite at a point : 3.1.1 (54), flat quasifinite : 3.2.5 (59), good : 1.5.3 (34), locally separated : 1.4 (30), proper : 1.5.3 (34), quasifinite : 3.1.1 (54), restricted : 5.3 (104), separated : 1.4 (30,) smooth : 3.5.1 (67), tame ~tale : 6.3 (119), unramified : 3.3.4 (63). I~TALE COHOMOLOGY FOR NON-ARCHIMEDEAN ANALYTIC SPACES 161 Net : 1.1 (14). Open immersion : 1.3 (24). Paracompact germ : 4.3 (91). Paracompactifying q~-family of supports : 5.1.2 (97). Picard group : 1.3 (25, 26). Quasicomplete field : 2.3.1 (43). Quasiconstruetible sheaf : 4.4.2 (94). Quasi-immersion : 4.3.3 (90). Quasi-isomorphism : 1.2.9 (19). Quasinet : 1.1 (13). Relative boundary of a morphism : 1.5.4 (34). Relative interior of a morphism "- 1.5.4 (34). Sheaf of differentials : 1.4 (30). Smooth pair : 7.4 (143). "~-Special covering : 1.2 (20). R-Special domain : 1.2 (21). -~-Special set : 1.2 (20). Stalk of a sheaf : 4.2 (87). Strong morphism : 1.2.7 (18). Structural sheaf of an analytic space : 1.3 (25). Support of a coherent sheaf : 1.3 (25). Support of a section of a sheaf : 4.2 (87). G-Topology on an analytic space : 1.3 (25). Trace mapping : 5.4 (106), 6.2 (112), 7.2 (130). Zariski closed (open) subset : 1.3 (28). Department of Theoretical Mathematics The Weizmann Institute of Science P.O.B. 26, 76100 Rehovot, Israel Manuscrit refu le 28 janvier 1991. Rgvisg le 13 mai 1992.
Publications mathématiques de l'IHÉS – Springer Journals
Published: Sep 26, 2007
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