Varieties without boundary in rigid analytic geometry
Let \(K\) be a field complete with respect to a non trivial non-Archimedean norm and let \(R\) be the valuation ring. Raynaud introduced the functor \(\mathcal X\mapsto \mathcal X_{\text{rig}}\), where \(\mathcal X_{\text{rig}}\) is the generic fibre of \(\mathcal X\), from the category of admissible formal \(R\)-schemes to the category of \(K\)-analytic rigid analytic varieties. He showed that this functor is faithful and turns admissible formal blowing-ups into isomorphisms. The author extends these results, showing that under this functor, there is an equivalence of categories between \(\bullet\) The category of generically paracompact admissible formal \(R\)-schemes localized with respect to formal admissible blowing-ups. \(\bullet\) The category of paracompact, quasi-separated \(K\)-analytic varieties. The author also generalizes the definition of a proper morphism as follows: a morphism \(f\: Y\to X\) of \(K\)-analytic varieties is said to be without boundary, if \(X\) admits an admissible affinoid covering \(\{X_i\}\), and each \(f^{-1}(X_i)\) admits two admissible affinoid coverings \(\{U_{ij}\}\) and \(\{V_{ij}\}\), such that for all \(i\) and \(j\), we have that \(U_{ij}\) is relatively compact in \(V_{ij}\) over \(X_i\). If one requires moreover that \(f\) is separated and the two admissible coverings \(\{U_{ij}\}\) and \(\{V_{ij}\}\) are finite, then we recover Kiehls definition of proper morphism. If the structure morphism \(X\to \text{Sp} K\) is without boundary, then \(X\) is called an analytic variety without boundary. Examples of the latter are proper varieties, algebraic varieties and Stein varieties. Moreover, if \(R\) is assumed to be discretely valued, then any Zariski open subset of a paracompact separated analytic variety without boundary is shown to be without boundary. This is derived as a corollary of the fact that under the above equivalence of categories, locally quasi-compact morphisms for which the special fibre is universally closed, correspond to locally quasi-compact morphisms without boundary. This also enables the author to reconcile his definition of morphism without boundary with an older notion introduced in [\textit{W. Lütkebohmert}, Math. Ann. 286, No. 1-3, 341-347 (1990; Zbl 0716.32022)].
scientific article; zbMATH DE number 1395889
scientific article