Mathematics – Algebraic Geometry
Scientific paper
2011-07-21
Mathematics
Algebraic Geometry
Scientific paper
This text is devoted to the systematic study of flatness in the context of Berkovich analytic spaces. After having shown through a counter-example that naive flatness in that context is not stable under base change, we introduce the notion of {\em universal} flatness and we study a first important class of universally flat morphisms, that of quasi-smooth ones. We then show the existence of local {\em d\'evissages} (in the spirit of Raynaud and Gruson) for coherent sheaves, which we use, together with a study of the local rings of 'generic fibers' of morphisms, to prove that a flat, boundaryless morphism is universally flat. After that we prove that the image of a compact analytic space by a universally flat morphism can be covered by a compact, relatively Cohen-Macaulay and zero-dimensional multisection, and the image of the latter is shown to be a compact analytic domain of the target. It follows that the image of a compact analytic space by a universally flat morphism is a compact analytic domain of the target. This was first proved in the rigid-analytic context by Raynaud, but our proof is completely different: it is based upon Temkin's theory of the reduction of analytic germs and quantifiers elimination in the theory of non-trivially valued algebraically closed fields, and it uses neither formal models nor flattening techniques. We end the paper by showing, using Kiehl's method, Zariski-openness of the universal flatness and of the quasi-smoothness loci of a given morphism.
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