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

TL;DR

This paper presents a new version of the reduced fiber theorem for non-archimedean analytic spaces, removing previous restrictions and using graded reduction techniques, with applications to the study of fiber connected components.

Contribution

It provides a generalized reduced fiber theorem for Berkovich spaces without strictness or constant dimension assumptions, using Temkin's graded reduction and avoiding formal geometry.

Findings

Established a substitute for the reduced fiber theorem applicable to all flat morphisms with geometrically reduced fibers.

Utilized Temkin's graded reduction and Grauert-Remmert finiteness to prove the result.

Set the stage for future work on fiber component variation in Berkovich spaces.

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.