Altered Local Uniformization Of Rigid-Analytic Spaces

Bogdan Zavyalov

Published 2021-02-09Version 1

We prove a version of Temkin's local altered uniformization theorem. We show that for any rig-smooth, quasi-compact and quasi-separated admissible formal $\mathcal{O}_K$-model $\mathfrak{X}$, there is a finite extension $K'/K$ such that $\mathfrak{X}_{\mathcal{O}_{K'}}$ locally admits a rig-\'etale morphism $g\colon \mathfrak{X}' \to \mathfrak{X}_{\mathcal{O}_{K'}}$ and a rig-isomorphism $h\colon \mathfrak{X}'' \to \mathfrak{X}'$ with $\mathfrak{X}'$ being a successive semi-stable curve fibration over $\mathcal{O}_{K'}$ and $\mathfrak{X}''$ being a poly-stable formal $\mathcal{O}_{K'}$-scheme. Moreover, $\mathfrak{X}'$ admits an action of a finite group $G$ such that $g\colon \mathfrak{X}' \to \mathfrak{X}_{\mathcal{O}_{K'}}$ is $G$-invariant, and the adic generic fiber $\mathfrak{X}'_{K'}$ becomes a $G$-torsor over its quasi-compact open image $U=g_{K'}(\mathfrak{X}'_{K'})$. Also, we study properties of the quotient map $\mathfrak{X}'/G \to \mathfrak{X}_{\mathcal{O}_{K'}}$ and show that it can be obtained as a composition of open immersions and rig-isomorphisms.