zbMATH — the first resource for mathematics

Semi-stable models for rigid-analytic spaces. (English) Zbl 1099.14010
Summary: Let \(R\) be a complete discrete valuation ring with field of fractions \(K\) and let \(X_K\) be a smooth, quasi-compact rigid-analytic space over \(\text{Sp}\,K\).
We show that there exists a finite separable field extension \(K'\) of \(K\), a rigid-analytic space \(X_{K'}'\), over \(\text{Sp} \,K'\) having a strictly semi-stable formal model over the ring of integers of \(K'\), and an étale, surjective morphism \(f:X_{K'}'\to X_K\) of rigid-analytic spaces over \(\text{Sp}\,K\). This is different from the alteration result of A. J. de Jong [Publ. Math., Inst. Hautes Étud. Sci. 83, 51–93 (1996; Zbl 0916.14005)] who does not obtain that \(f\) is étale. To achieve this property we have to work locally on \(X_K\), i.e. our \(f\) is not proper and hence not an alteration.

14G22 Rigid analytic geometry
Full Text: DOI