Les espaces de Berkovich sont excellents
By A. Ducros. 2007
60 citations recorded by the source corpus.
In this paper, we first study the local rings of a Berkovich analytic space from the point of view of commutative algebra. We show that those rings are excellent ; we introduce the notion of a an analytically separable extension of non-archimedean complete fields (it includes the case of the finite separable extensions, and also the case of any complete extension of a perfect complete non-archimedean field) and show that the usual commutative algebra properties (Rm, Sm, Gorenstein, Cohen-Macaulay, Complete Intersection) are stable under analytically separable ground field extensions; we also establish a GAGA principle with respect to those properties for any finitely generated scheme over an affinoid algebra. A second part of the paper deals with more global geometric notions : we define, show the existence and establish basic properties of the irreducible components of analytic space ; we define, show the existence and establish basic properties of its normalization ; and we study the behaviour of connectedness and irreducibility with respect to base change.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/b7568fbd26b8c22b9cafe4c7fab1507162e4f16a
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In Algebra and Number Theory
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case (2009, background): “ to rig-smoothness, and in general this means that the morphism becomes rig-smooth after large enough ground field extension). 5.2.3. Excellence. Recall that Berkovich analytic spaces are excellent by [Duc1]. As for complex analytic spaces X, let us say that a Stein compact V ⊂ X is excellent if so i”