Continuit\'e des racines d'apr\`es Rabinoff et Berkovich

TL;DR

This paper generalizes Rabinoff's theorem on the continuity of roots in non-archimedean analytic geometry, establishing conditions under which a finite morphism is flat and finite with constant fiber cardinality.

Contribution

It extends Rabinoff and Berkovich's results to broader settings, providing new criteria for flatness and finiteness of morphisms in non-archimedean analytic spaces.

Findings

Generalization of Rabinoff's theorem to wider contexts

Conditions ensuring flatness and finiteness of morphisms

Constant cardinality of finite fibers under new hypotheses

Abstract

The content of this paper is a generalization of a the theorem 9.2 of the paper arXiv:1007.2665 written by Joseph Rabinoff : if $\mathcal{P}$ is a finite family of polyhedra in $N_{\mathbb{R}}$ such that there exists a fan in $N_{\mathbb{R}}$ that contains all the recession cones of the polyhedra of $\mathcal{P}$, if $k$ is a complete non-archimedean field, if $S$ is a connected and regular $k$-analytic space and $Y$ is a closed $k$-analytic subset of $U_{\mathcal{P}} \times_k S$ which is relative complete intersection and contained in the relative interior of $U_{\mathcal{P}} \times_k S$ over $S$, then the quasifiniteness of $\pi : Y \to S$ implies its flatness and its finiteness ; moreover, all the finite fibres of $\pi$ have the same cardinality.