Altered local uniformization of rigid-analytic spaces

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