Uniformly rigid spaces
type: Item
instance of: scholarly article
title: Uniformly rigid spaces
zbMATH Open document ID: 1254.14025
author: Christian Kappen
published in: Algebra \& Number Theory
publication date: 2012-08-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.1056
full work available at URL: p06.xhtml
reviewed by: Jérôme Poineau
Mathematics Subject Classification ID: 14G22
zbMATH DE Number: 6064713
zbMATH Keywords: rigid analytic spaces
zbMATH Keywords: uniformly rigid spaces
zbMATH Keywords: formal schemes
zbMATH Keywords: semiaffinoid
zbMATH Keywords: formally finite type
MaRDI profile type: Publication
OpenAlex ID: W2963829062
arXiv ID: 1009.1056
DOI: 10.2140/ANT.2012.6.341
review text: Let \(R\) be a complete discrete valuation ring and let \(K\) be its fraction field. Let \(\mathfrak{X}\) be the formal spectrum of a quotient \(R\langle T_{1},\dots,T_{n}\rangle/I\). In [Bull. Soc. Math. Fr., Suppl., Mém. 39--40, 319--327 (1974; Zbl 0299.14003)], \textit{M. Raynaud} defined its generic fiber \(\mathfrak{X}^{\mathrm{rig}}\) as the maximal spectrum of \(R\langle T_{1},\dots,T_{n}\rangle/I \otimes_{R} K\). This construction extends to a functor from the category of formal \(R\)-schemes locally of topologically finite type to the category of rigid \(K\)-analytic spaces, which is well understood: adding finiteness conditions and localizing by admissible blowups at the source, we may turn it into an equivalence of categories.NEWLINENEWLINELet us now consider formal \(R\)-schemes locally of formally finite type, i.e. locally formal spectra of quotients of \(R[[S_{1},\dots,S_{m}]]\langle T_{1},\dots,T_{n}\rangle\). Berthelot extended the functor rig to this setting at the cost of losing some of its nice properties: some morphisms between rigid spaces in the image fail to come from morphisms between models. In the text under review, Kappen explains how to recover the desired properties by considering uniformly rigid spaces (a notion that he defines) instead of rigid spaces.NEWLINENEWLINEThe building blocks of uniformly rigid spaces are semiaffinoid spaces, i.e. spectra of quotients of \(R[[S_{1},\dots,S_{m}]]\langle T_{1},\dots,T_{n}\rangle \otimes_{R} K\). Loosely speaking, this means that uniformly rigid spaces (and their functions) are constructed using bounded functions on products of open and closed discs. Those spaces are endowed with a Grothendieck topology and Kappen proves the analogues of the classical results from rigid geometry in this setting: the presheaf of semiaffinoid functions is a sheaf and coherent ideals on semiaffinoid spaces are associated to their ideals of global sections.NEWLINENEWLINEWe refer to the author's dissertation [Uniforml…
description: scientific article; zbMATH DE number 6064713