Altered Local Uniformization Of Rigid-Analytic Spaces

Abstract

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 $K^{\prime }/K$ such that ${\mathfrak{X}}_{{\mathcal{O}}_{K^{\prime }}}$ locally admits a rig-étale morphism $g:{\mathfrak{X}}^{\prime }\to {\mathfrak{X}}_{{\mathcal{O}}_{K^{\prime }}}$ and a rig-isomorphism $h:\mathfrak{X}"\to {\mathfrak{X}}^{\prime }$ with ${\mathfrak{X}}^{\prime }$ being a successive semi-stable curve fibration over ${\mathcal{O}}_{K^{\prime }}$ and $\mathfrak{X}"$ being a poly-stable formal ${\mathcal{O}}_{K^{\prime }}$-scheme. Moreover, ${\mathfrak{X}}^{\prime }$ admits an action of a finite group $G$ such that $g:{\mathfrak{X}}^{\prime }\to {\mathfrak{X}}_{{\mathcal{O}}_{K^{\prime }}}$ is $G$-invariant, and the adic generic fiber ${\mathfrak{X}}_{K^{\prime }}^{\prime }$ becomes a $G$-torsor over its quasi-compact open image $U=g_{K^{\prime }}({\mathfrak{X}}_{K^{\prime }}^{\prime })$. Also, we study properties of the quotient map ${\mathfrak{X}}^{\prime }/G\to {\mathfrak{X}}_{{\mathcal{O}}_{K^{\prime }}}$ and show that it can be obtained as a composition of open immersions and rig-isomorphisms.

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