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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.

##### MSC:
 14G22 Rigid analytic geometry
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