Families of Berkovich spaces
  This text is devoted to the systematic study of relative properties in the context of Berkovich analytic spaces. We first develop a theory of flatness in this setting. After having shown through a counter-example that naive flatness cannot be the right notion because it is not stable under base change, we define flatness by {\em requiring} invariance under base change, and we study a first important class of 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 {\em boundaryless}, naively flat morphism is flat. After that we prove that the image of a compact analytic space by a 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; we thus recover the result by Raynaud telling that the image of a compact analytic space under a flat morphism is a compact analytic domain of the target. In the last part of this work we study some validity loci. We first prove that the flatness locus of a given morphism of analytic spaces is a Zariski-open subset of the source. We then look at the {\em fiberwise} validity loci of the usual commutative algebra properties. We prove that the results we could expect actually hold: the fiberwise validity locus of such a property is Zariski-constructible, and Zariski-open under suitable extra assumptions (flatness, and also sometimes equidimentionality). 
Antoine Ducros
math.AG