A generalization of formal schemes and rigid analytic varieties
type: Item
instance of: scholarly article
title: A generalization of formal schemes and rigid analytic varieties
zbMATH Open document ID: 0814.14024
DOI: 10.1007/BF02571959
author: Roland Huber
published in: Mathematische Zeitschrift
publication date: 1995-06-20T00:00:00Z
full work available at URL: https://eudml.org/doc/174705
review text: There is a natural class of topological rings which contains both the affinoid rings (from rigid analytic geometry) and the noetherian adic rings. Namely the class of topological rings which have an open adic subring with a finitely generated ideal of definition. We call such a ring \(f\)-adic. For every \(f\)-adic ring \(A\) the topological space \(\text{Cont }A\) of all continuous valuations of \(A\) is a spectral space. If \(A\) has an open noetherian adic subring or if for every \(n \in N\) the ring \(A\langle X_ 1,\dots, X_ n\rangle\) of restricted power series in \(n\) variables over \(A\) is noetherian then there is a natural sheaf \({\mathcal O}_ A\) of topological rings on \(\text{Cont }A\). All stalks of \({\mathcal O}_ A\) are local rings. We call a topologically and locally ringed space which is locally isomorphic to some \(\text{Spa }A := (\text{Cont }A,{\mathcal O}_ A)\) adic. There are natural fully faithful functors from the category of rigid analytic varieties and the category of locally noetherian formal schemes to the category of adic spaces. In the first case one assigns to an affinoid rigid analytic variety \(\text{Sp }A\) the adic space \(\text{Spa }A\) and in the latter case one assigns to an affine noetherian formal scheme \(\text{Spf }A\) the adic space \(\text{Spa }A\).
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 13J20
zbMATH DE Number: 737152
zbMATH Keywords: affinoid rings
zbMATH Keywords: adic rings
zbMATH Keywords: \(f\)-adic ring
zbMATH Keywords: continuous valuations
zbMATH Keywords: formal schemes
zbMATH Keywords: rigid analytic variety
reviewed by: Frank Herrlich
MaRDI profile type: Publication
cites work: Q3325851
cites work: Sur quelques points d'algèbre homologique
cites work: Continuous valuations
cites work: Q5651328
cites work: Rigid analytic spaces
OpenAlex ID: W2073964955
description: scientific article; zbMATH DE number 737152