On semianalytic subsets of rigid analytic varieties
The author first introduces a notion of semianalytic subsets in rigid geometry (i.e. analytic geometry in the sense of Tate, Grauert and Remmert). Second he proves all the required properties: behavior under (quasi-compact) morphisms, when taking closure or interior... The main and interesting application of this notion and its properties is the following: let \(f : X = SpA \to Y = SpB\) be a morphism of affinoid rigid analytic spaces, let \(x \in X\), \(\widehat {\mathcal O}_{X,x}\) and \(\widehat {\mathcal O}_{Y,f(x)}\) be the completion (with respect to the maximal ideals) of the fibers of the rigid analytic structural sheaves and let \(A^h_x\), \(B^h_{f (x)}\) be the henselizations of the localizations of \(A\) and \(B\); then the following assertions are equivalent: (i) \(f\) is open in \(x\), (ii) \(\text{Spec} {\mathcal O}_{X,x} \to \text{Spec} {\mathcal O}_{Y, f(x)}\) is surjective, (iii) \(\text{Spec} \widehat {\mathcal O}_{X,x} \to \text{Spec} \widehat {\mathcal O}_{Y, f(x)}\) is surjective, (iv) \(\text{Spec} A^h_x \to \text{Spec} B^h_{f(x)}\) is surjective.
scientific article; zbMATH DE number 826787