R\'eduction en famille d'espaces affino\"ides

Abstract

Let $k$ be a non-archimedean complete field. We prove a substitute for the reduced fiber theorem (of Bosch, L\"utkebohmert and Raynaud) that holds for every morphism $Y\to X$ flat and with geometrically reduced fibers between $k$-affinoid spaces in the sense of Berkovich, without assuming that $X$ and $Y$ are strict, nor that the relative dimension of $Y$ over $X$ is constant. We do not use the original reduced fiber theorem, nor the language or the techniques of formal geometry. Our statement is formulated in terms of Temkin's graded reduction; our proof rests on a finiteness result of Grauert and Remmert and on Temkin's theory of (graded) reduction of germs of analytic spaces. It will be used for describing the variation of the connected components of the fiber of a quasi-smooth map in a forthcoming work on flattening in the Berkovich setting.