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 \({{\cal O}_K}\)-model \(\mathfrak{X}\), there is a finite extension K′/K such that \({\mathfrak{X}_{{{\cal O}_{{K^\prime }}}}}\) locally admits a rig-étale morphism \(g:{\mathfrak{X}^\prime } \to {\mathfrak{X}_{{{\cal O}_{{K^\prime }}}}}\) and a rig-isomorphism \(h:{\mathfrak{X}^{\prime \prime }} \to {\mathfrak{X}^\prime}\) with \({\mathfrak{X}^\prime }\) being a successive semi-stable curve fibration over \({{\cal O}_{{K^\prime }}}\) and \({\mathfrak{X}^{\prime \prime }}\) being a polystable formal \({{\cal O}_{{K^\prime }}}\)-scheme. Moreover, \({\mathfrak{X}^\prime }\) admits an action of a finite group G such that \(g:{\mathfrak{X}^\prime } \to {\mathfrak{X}_{{{\cal O}_{{K^\prime }}}}}\) 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 \({\mathfrak{X}^\prime }/G \to {\mathfrak{X}_{{{\cal O}_{{K^\prime }}}}}\) and show that it can be obtained as a composition of open immersions and rig-isomorphisms.