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Zbl 0804.32019
Berkovich, Vladimir G.
Étale cohomology for non-Archimedean analytic spaces. (English)
[J] Publ. Math., Inst. Hautes Étud. Sci. 78, 5-161 (1993). ISSN 0073-8301; ISSN 1618-1913

This deep and complete paper is mainly concerned with the construction of an étale cohomology and an étale cohomology with compact support for analytic non archimedean spaces, that is spaces first introduced by the author [math. surveys and monographs No. 33, A.M.S. (1990)] which are here generalized $(\S\S 1$ and 2), to give rise to all reasonable rigid analytic spaces (quasifinite). The $3\sp{rd}$ $\S$ gives definitions and properties of unramified, étale and smooth morphisms (for quasifinite morphisms, def. 3.1.1). These notions are also compared with the algebraic ones: if $\varphi$ is a morphism between schemes of locally finite type over $\text {Spec} ({\cal A})$, where ${\cal A}$ is an affino\" id algebra, then it is unramified (resp. étale, resp. smooth) if and only if the corresponding morphism between the analytified spaces has the same property. The $4\sp{th}$ and $5\sp{th}$ $\S\S$ contain respectively definitions and first properties of the étale cohomology and the étale cohomology with compact support. These definitions are natural and all the expected properties are proved. Drinfeld's interpretation of $H\sp 1\sb{\acute et} (.,\mu\sb n)$, $n$ prime to the characteristic of the base field [Math. USSR Sb. 23 (1974)], is proved here for analytic spaces $(\S 4.3)$. Analytic curves are studied in the $\S 6$. Some results are preparation for the last $\S$ (such as the comparison theorem for étale cohomology with compact support) or are extended later (such as the existence of a trace map...). It is also proved here that any tame finite étale Galois covering of the closed disc is trivial. As a corollary this gives a ``Riemann- existence theorem'' for coverings of degree prime to the residue characteristic of the base field.\par The final results are in the $\S 7$; all main properties expected from a good étale cohomology theory are proved here: comparison theorem for étale cohomology with compact support between a compactifiable scheme and its analytification; existence and properties of a trace map for separated smooth morphisms of pure dimension and for the constant sheaves $\bbfZ/n \bbfZ$, $n$ being prime to the characteristic of the base field... The central result is the ``Poincaré duality theorem'' $(\S 7.3)$ for separated smooth morphisms and for complexes of modules over the constant sheaves $\bbfZ/n \bbfZ$, $n$ prime to the residue characteristic of the base field. A comparison theorem for étale cohomology is also given, a base change theorem for cohomology with compact support...\par Étale cohomology for analytic spaces over an ultrametric field is today the object of many works. One of the main motivations for that is certainly to construct an explicit version of the abstract cohomology of {\it P. Schneider} and {\it V. Stuhler} [Invent. Math. 105, No. 1, 47-122 (1991; Zbl 0751.14016)]. In de Jong-van der Put, an étale cohomology for rigid analytic spaces and overconvergent sheaves is constructed (preprint), which coincides with the one of Schneider-Stuhler. A big preprint of Huber has compared cohomology theories of (Berkovich) analytic spaces and rigid spaces. An explicit proof that étale- cohomology of analytic spaces coincides with the one of Schneider-Stuhler is not explicit in the works of the author, but is known by the experts (it uses results of Berkovich and de Jong-van der Put).
[M.Reversat (Toulouse)]

MSC 2000:: *32P05 Non-Archimedean complex analysis
14F20 Grothendieck cohomology and topology

Keywords: étale cohomology; compact support; analytic spaces; morphisms; sheaves; rigid spaces

Citations: Zbl 0751.14016

Cited in: Zbl 1177.14049 Zbl 1177.14048 Zbl 1161.14001 Zbl 1136.14010 Zbl 1060.32010 Zbl 1029.32009 Zbl 0930.32016 Zbl 0904.32029 Zbl 0988.14004 Zbl 0867.11049 Zbl 0856.14007

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