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Étale cohomology for non-Archimedean analytic spaces. (English) Zbl 0804.32019
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 \(3^{rd}\) \(\S\) 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 \(\text{Spec} ({\mathcal A})\), where \({\mathcal A}\) is an affinoïd 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 \(4^{th}\) and \(5^{th}\) \(\S\S\) 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 \(H^ 1_{\acute et} (.,\mu_ n)\), \(n\) 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.
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 \(\mathbb{Z}/n \mathbb{Z}\), \(n\) 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 \(\mathbb{Z}/n \mathbb{Z}\), \(n\) 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...
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 P. Schneider and 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).

MSC:
32P05 Non-Archimedean analysis
14F20 Étale and other Grothendieck topologies and (co)homologies
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