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Berkovich spaces are excellent. (Les espaces de berkovich sont excellents.) (French) Zbl 1177.14049
The paper under review in devoted to the study of some basic properties of Berkovich’s analytic spaces over non-archimedean fields [see V. G. Berkovich, Spectral theory and analytic geometry over non-archimedean fields. Mathematical Surveys and Monographs, 33. Providence, RI: American Mathematical Society (AMS). (1990; Zbl 0715.14013)] and [Publ. Math., Inst. Hautes Étud. Sci. 78, 5–161 (1993; Zbl 0804.32019)]).
The first results concern the local rings of Berkovich spaces. The author shows that they are excellent when the space is affinoid (théorème 2.13). In the rigid analytic setting, this result is due to B. Conrad [Ann. Inst. Fourier 49, No. 2, 473–541 (1999; Zbl 0928.32011)]. Next, the classical algebraic properties of the local rings (\(R_{m}\), \(S_{m}\), being regular, Cohen-Macaulay, Gorenstein, complete intersection) are investigated. The author studies their behaviours with regard to analytic extensions of the base field (théorème 3.1 and §3.4 in the non-good case). For this purpose, in the case of property \(R_{m}\) and regularity, he has to introduce the notion of analytically separable field extension (§1). Then he turns to the behaviour under analytification (théorème 3.4 and §3.4 in the non-good case). Let \({\mathcal X}\) be a scheme of finite type over an affinoid algebra and \(X\) be its analytification. For any of the aforementioned properties, the locus where it holds is a Zariski open subset of \({\mathcal X}\) and it holds at a point \(x\) of \(X\) if and only if it holds at its image in \({\mathcal X}\).
The rest of the paper is devoted to global properties of Berkovich spaces, especially normality, connectedness and irreductibility. The Zariski topology of an affinoid space being Noetherian, one may define its irreducible components in the classical way. This is no longer possible for more general analytic spaces. Let’s mention that B. Conrad has defined the irreducible components of rigid spaces through normalization [loc. cit.]. Here the author proceeds in another way and gives a direct definition of the irreducible components of an analytic space \(X\) as the Zariski closures of the irreducible components of the affinoid domains of \(X\). He shows that they coincide with the maximal irreducible Zariski subsets (théorème 4.20). He also proves that analytic spaces admit normalizations (théorème 5.13) and that the connected components of the normalization of a space correspond bijectively to its irreducible components (théorème 5.17). A property is said to be geometric if it holds after any analytic extension of the base field. The author gives several characterizations of geometric connectedness (théorème 7.11) and geometric irreducibility (théorème 7.12).
Last, the author considers product of spaces and proves that if a geometric property holds for a couple of spaces, it still holds for their product (théorème 8.1 for local properties and 8.3 for global ones).
Let’s also mention that there is a very nice and detailed introduction, which is all the more useful as the paper is quite long. At the end of it, the author explains how his results compare to what was known before, mainly the works of R. Kiehl see [J. Reine Angew. Math. 234, 89–98 (1969; Zbl 0169.36501)], B. Conrad [loc. cit.] and V. Berkovich [loc. cit.].
The paper is thorough and self-contained (the author reproves most of what he needs). It will undoubtedly become a reference work for people interested in Berkovich theory.

MSC:
14G22 Rigid analytic geometry
14A99 Foundations of algebraic geometry
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References:
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