Varieties without boundary in rigid analytic geometry
type: Item
instance of: scholarly article
title: Varieties without boundary in rigid analytic geometry
zbMATH Open document ID: 0947.32011
author: R. Lamjoun
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2000-03-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108507
review text: 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, c…
reviewed by: Hans Schoutens
Mathematics Subject Classification ID: 32P05
zbMATH DE Number: 1395889
zbMATH Keywords: proper morphism
zbMATH Keywords: variety without boundary
zbMATH Keywords: rigid analytic geometry
MaRDI profile type: Publication
cites work: Q3998951
cites work: Q3325851
cites work: Formal and rigid geometry. I: Rigid spaces
cites work: Q5602063
cites work: Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Havard 1963/64. Appendix: Cohomology with supports and the construction of the \(f^!\) functor by P. Deligne
cites work: Formal-algebraic and rigid-analytic geometry
cites work: Serre duality for rigid analytic spaces
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description: scientific article; zbMATH DE number 1395889