Continuité des racines d'après Rabinoff et Berkovich

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 $π: Y \to S$ implies its flatness and its finiteness ; moreover, all the finite fibres of $π$ have the same cardinality.