Table Of ContentAspects of Mathematics
Roland Huber
Étale Cohomology of
Rigid Analytic Varieties
and Adic Spaces
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*A Publication of the Max·Pianck·lnstitut fur Mathematik, Bonn
Roland Huber
Étale Cohomology of
Rigid Analytic Varieties
and Adic Spaces
I I
VJeweg
Professor Dr. Roland Huber
Fachbereich Mathematik
Bergische Universitat
Gesamthochschule Wuppertal
GauBstr. 20
42097 Wuppertal
Mathematics Subject Classification:
14 F20,32 P 05
All rights reserved
© Springer Fachmedien Wiesbaden 1996
Originally published by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden in 1996
No part of this publication may be reproduced, stored in a retrieval system or
transmitted, mechanical, photocopying or otherwise without prior permission
of the copyright holder.
Cover design: Wolfgang Nieger, Wiesbaden
Printed on acid-free pape:
ISSN 0179-2156
ISBN 978-3-663-09992-5 ISBN 978-3-663-09991-8 (eBook)
DOI 10.1007/978-3-663-09991-8
v
Contents
Introduction vu
0 Etale cohomology of rigid analytic varieties (summary) 1
0.1 The etale site of a rigid analytic variety 1
0.2 Cohomology of constant sheaves . 1
0.3 Base change theorems . . . . . . . 7
0.4 Cohomology with compact support 10
0.5 Finiteness . . . . . . . 19
0.6 Poincare Duality . . . 22
0.7 Comparison theorems 23
1 Adic spaces 36
1.1 Definition of adic spaces . . . . . . . . . . . . 36
1.2 Morphisms of finite type . . . . . . . . . . . . 45
1.3 Separated morphisms and proper morphisms 51
1.4 Finite morphisms . . . . . . . . . . . . . . . . 60
1.5 Quasi-finite morphisms. . . . . . . . . . . . . 66
1.6 Definition of unramified, smooth and etale morphisms 76
1. 7 Etale morphisms . . . . . . . . . . . . . . . . . . . . . 80
1.8 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 90
1.9 The analytic adic space associated with a formal scheme 95
1.10 Pseudo-adic spaces . . . . . . . . . . . . . . . . . . . . . 99
2 The etale site of a rigid analytic variety and an adic space 108
2.1 Comparison between the rigid etale site and the adic etale site 109
2.2 Some examples of etale sheaves . . . . . . . 112
2.3 The etale topos of a subset of an adic space 117
2.4 Projective systems of adic spaces . . . . . . 125
2.5 Strict localizations of adic spaces . . . . . . 130
2.6 The stalks of Rq f., for quasi-compact morphisms 138
2.7 Constructible sheaves on the etale site of an adic space . 145
2.8 Cohomological dimension . . . . . . . . . . . . . . . . . 156
2.9 Constructible sheaves on the etale site of a rigid analytic variety 159
3 Comparison theorems 162
3.1 Henselizations of affinoid rings . . . . . . . . . . . . . . . . . . 163
3.2 Etale cohomology of pro-special subsets of affinoid adic spaces . i 73
3.3 Proof of (3.2.12), part I . . . . . . . . . . . . . . . . . . . 183
3.4 Proof of {3.2.12), part II . . . . . . . . . . . . . . . . . . . 191
3.5 Comparison with the etale cohomology of formal schemes 201
VI Contents
3.6 Proof of (3.5.8)- (3.5.10) . . . . . . . . . . . . . . . . . . . . . 211
3.7 A comparison theorem for proper morphisms . . . . . . . . . . 225
3.8 A comparison theorem for schemes over non-archimedean fields 227
3.9 Cohomological purity . . . . . . . . . . . . . . . . . . . . . . . . 229
4 Base change theorems 238
4.1 Formulation of the base change theorem . . . . . . . . . . . . . . . 238
4.2 A remark on the etale cohomology of schemes over valuation rings 240
4.3 The quasi-compact/generalizing base change theorem. 253
4.4 The proper base change theorem 258
4.5 The smooth base change theorem . 265
5 Cohomology with compact support 269
5.1 Compactifications. . . . . . . . . . . . . . . . . . 271
5.2 The functor !! . . . . . . . . . . . . . . . . . . . . 286
5.3 The functor R+ !! for partially proper morphisms 289
5.4 The functor R+ f! for taut morphisms . . . . . . 294
5.5 The functor Rf! for finite-dimensional taut morphisms 299
5.6 Cohomology with compact support for rigid analytic varieties 309
5.7 Comparison theorems for cohomology with proper support . 316
6 Finiteness 324
6.1 Finiteness of Hn(X, F) . 324
6.2 Finiteness of Rn f!F 327
6.3 Proof of (6.2.2) 330
7 Poincare Duality 360
J' . . . . . . . . .
7.1 The functor R+ 361
7.2 H'1(X, Jln) for a smooth curve X 366
7.3 The trace morphism . . . . . 376
7.4 Deligne's fundamental lemma 382
7.5 Poincare Duality . . . . . . . 389
8 Partially proper sites of rigid analytic varieties and adic spaces 399
8.1 The partially proper topology . . . . . . . . 399
8.2 The partially proper etale site . . . . . . . . 407
8.3 Comparison with Berkovich' analytic spaces 422
A Appendix 435
Bibliography 443
Index of notations 446
Index of terminology 448
Vll
Introduction
Tate introduced analytic spaces over non-archimedean fields which are called rigid
analytic varieties ([T]). Raynaud realized that the category of quasi-compact quasi
separated rigid analytic varieties over a non-archimedean field k is equivalent to
the localization of the category of formal schemes of finite type over the valuation
ring k0 of k with respect to the class of admissible formal blowing-ups ([RI]). This
approach to rigid analytic geometry can be extended by localizing a more general
class of formal schemes with respect to admissible formal blowing-ups. The result
ing category is called the category of relative rigid spaces ([BL]). But one can also
extend Tate's definition of analytic spaces. We call this more general analytic spaces
analytic adic spaces ([Hu]). The category of analytic adic spaces is a full subcategory
of the category of locally and topologically ringed spaces with a distinguished valu
ation on every residue field of the structure sheaf. There is a naturally fully faithful
functor d from the category of relative rigid spaces to the category of analytic adic
spaces.
The aim of this text is to develop basic properties of the etale cohomology of
torsion sheaves on analytic adic spaces: base change theorems, Poincare duality,
finiteness, comparison theorems. For a rigid analytic variety X, or more general
relative rigid space X, the etale topos Xd of X is naturally equivalent to the etale
topos d(X)it of the analytic adic space d(X) associated with X. Hence the results
on etale cohomology mentioned above also hold for relative rigid spaces.
Etale cohomology of rigid analytic varieties was first defined by Drinfeld ([D]),
Fresnel and van der Put ([FP]) and Schneider and Stuhler ([SS]). Berkovich intro
duced in [Ber] non-archimedean analytic spaces. These spaces are different from
rigid analytic varieties but his results on etale cohomology of his spaces ([Berl])
can be applied to rigid analytic varieties (see below). Fujiwara proved comparison
theorems for the etale cohomology of relative rigid spaces ([F]). In [JP] de Jong and
van der Put give an introduction to the etale cohomology of rigid analytic varieties,
in particular they prove a base change theorem.
A big part of this book is devoted to etale cohomology with compact support.
Similarly as for schemes, in order to define the direct image functor with proper
support R/! for morphisms f: X -+ Y of analytic adic spaces (or rigid analytic
varieties) one has to use compactifictions. This can be seen already in the simplest
example: Let k be an algebraically closed complete non-archimedean field, let 1$ be
the rigid analytic variety {x E k llxl ::S 1} and let m be a natural number which is
prime to the characteristic char( k0 / k00) of the residue class field of the valuation
= =
ring k0• Then H0(Im,'ll/m'll) 'll/m'll and Hi(Im,'ll/m'll) 0 fori> 0. Since 1$ is
smooth, H c(B, 7l /m'll) should satisfy a Poincare duality. So we expect
Vlll Introduction
(1) Hi(Tll> 71/ 71)=~ { 0 for i==J 2
c llll, Lt.. miL. .:Zfm.:Z for i 2.
The right derived functor of the global section functor with compact support
r c(lffi,-) does not give (1). But if j: lffi ~ IJD denotes the open embedding of lffi into
the projective line considered as a rigid analytic variety over k then
Hi(IJD ,J•! (71/ miL71. )),..=.., { .0:Z fm.:Z ffoorr ii ==J 22.
Lt..
Every morphism of analytic adic spaces /:X ---+ Y which is locally of finite type,
separated and taut (i.e., for every quasi-compact quasi-separated open subset V ofY
and every quasi-compact open subset U of f-1(V), the closure U of U in f-1(V) is
quasi-compact) has a compactification. Furthermore, there is a proper base change
theorem for the etale cohomology of analytic adic spaces. Therefore one can define,
for separated taut morphisms locally of finite type between analytic adic spaces, the
direct image functor with proper support R+ /! by the general construction given
by Deligne in [SGA 4, XVII.3].
The main point in the proof of the Poincare duality is to show that, for every
separated smooth connected curve X over a geometric adic point S and for every
natural number m which is prime to char+(s),
(2) H'l(X, fl.m) = .:Zjm.:Z.
S is given by a pair (I<, A) where I< is an algebraically closed field and A is a
valuation ring of K. If rank(A) = 1 then X can be considered as a rigid analytic
variety but if rank(A) > 1 then X can no longer be considered as a rigid analytic
variety. Even if one wants to prove the Poincare duality only for rigid analytic va
rieties one has to prove (2) for adic curves X over arbitrary geometric adic points
Sand it is not enough to prove (2) only for rigid analytic curves X.
One ingredient of the proof of (2) is the following comparison theorem for coho
mology with compact support which we formulate here only for valuation rings of
rank 1 but which holds analogously also for valuation rings of higher rank: Let A
be a complete valuation ring of rank 1 such that the quotient field k = qf(A) is
algebraically closed. Let X be a compactifiable scheme over A and let xrig be the
rigid analytic variety over k associated with the formal completion X of X along
the closed subscheme X,:= X 0A A/sA where s # 0 is an element of the maximal
ideal m of A. Let F be a Z::/m.:Z-module on (X 0A k).;t with m prime to char(A/m),
let RIJT(F) E D((X. ).;t, Z::/m.:Z) b'e the complex of vanishing cycles ofF, and let F'
be the inverse image ofF on (Xrig)e't· Then, for every n E Z::,
=
H~(Xrig, F') H~(X,, RIJT(F)).
=
(Remark. We also have Hn(xrig, F') Hn(x., RIJT(F))).
IX
Beside the Poincare duality our main result on cohomology with proper support
is a finiteness theorem for Rn f, which says that iff: X --+ Y is a separated smooth
quasi-compact morphism of analytic adic spaces and F is a constructible Z/m7l
module on Xet with m prime to char+(Y) then, for every n E No, Rn j,F is a
constructible Z / m7l-module on Yet. As a corollary we obtain that, for proper smooth
morphism f and locally constant F, all Rn f,. F are locally constant.
There are three base change theorems for the etale cohomology of analytic adic
spaces: a smooth base change theorem, a proper base change theorem and a quasi
compact/generalizing base change theorem. In [JP] there is proved the quasi-compact
base change theorem for overconvergent sheaves on rigid analytic varieties. There are
interesting sheaves which are not overconvergent (for example, the sheaf j,(Z/mZ)
n;
on lP' et which we used above in order to define (JIB, Z / m"ff.)). Therefore in this book
we do not restrict ourselves to overconvergent sheaves. In particular, we prove the
base change theorems for arbitrary sheaves of Z/m7l-modules.
In [Berl] Berkovich defines, for a complete non-archimedean field k, k-analytic
spaces and constructs a fully faithful functor s from the category of hausdorff strictly
k-analytic spaces to the category of rigid analytic varieties over k. We will study
this functor s. The image of s is the class of all rigid analytic varieties X such that
X is quasi-separated and, for every quasi-compact admissible open subset U of X,
there is a quasi-compact admissible open subset U' of X such that { U', X - U} is
an admissible open covering of X (note that since X is quasi-separated and U is
quasi-compact, X-U is an admissible open subset of X). For a hausdorff strictly k
analytic space X and associated rigid analytic variety s(X), the category of sheaves
on Xet is equivalent to the category of overconvergent sheaves on s( X)e't, and,
for an abelian sheaf F on Xet and corresponding sheaf F' on s(X)e·t, the natural
mappings nn(X, F) --+ Hn(s(X), F') are bijective. For a compact morphism of
hausdorff strictly k-analytic spaces f: X --+ Y, the functors R+ f,. and R+ s(f),. are
compatible but the functors R+ f, and R+ s(f), in general are not compatible (since
R+ f,. = R+ j, but in general R+ s(f),. f; R+ s(f),). For an arbitrary morphism of
hausdorff strictly k-analytic spaces f: X --+ Y, the functors R+ f, and R+ s(f), are
compatible if and only iff is closed (in the sense of [Ber1, 1.5.3.iii]).
In Chapter 1 we introduce adic spaces and develop some basic notions for these
spaces. Furthermore in this Chapter we compare adic spaces with rigid analytic
varieties and relative rigid spaces. In Chapter 2 we define etale cohomology of adic
spaces and rigid analytic varieties, and in Chapter 5 we define etale cohomology with
compact support for analytic adic spaces and rigid analytic varieties. All results of
this book are formulated and proved for adic spaces. In Chapter 0 we explain these
results especially for rigid analytic varieties.
It is a pleasure to thank W. Liitkebohmert who directed my attention to the
reduced fibre theorem ([BLR]) and compactifications of smooth families of affinoid
rigid analytic curves ([L2]).
This book was written during a stay at the Sonderforschungsbereich of the Univer
sity of Bielefeld and a stay at the University of Wuppertal. I would like to thank
X Introduction
both institutions for the hospitality and support; especially I am grateful to Th.
Zink and M. Rapoport who made possible these visits. I thank Mrs. R. Bonn and
Mrs. P. Herdieckerhoff for typesetting of this text.
