On Hausdorff completions of commutative rings in rigid geometry
type: Item
instance of: scholarly article
title: On Hausdorff completions of commutative rings in rigid geometry
zbMATH Open document ID: 1230.13021
author: Kazuhiro Fujiwara
author: Ofer Gabber
author: Fumiharu Kato
published in: Journal of Algebra
publication date: 2011-11-08T00:00:00Z
review text: If \(A\) is a noetherian ring and \(I\) an ideal of \(A\), the lemma of Artin-Rees ensures that the \(I\)-adic completion \(\hat{A}\) of \(A\) satisfies many nice properties. In the article under review, the authors define several classes of non-necessarily-noetherian rings that still behave well. Let us mention only some of them. In the following, \(A\) is a ring endowed with an adic topology defined by a finitely generated ideal \(I\). The ring \(A\) is said to be pseudo-adhesive if it is noetherian outside \(I\) (i.e. Spec\((A) \setminus V(I)\) is noetherian) and if finitely generated \(A\)-modules have bounded \(I\)-torsion. If \(A\) is such a ring, there are canonical isomorphisms \(M\otimes \hat{A} \simeq \hat{M}\) for finitely generated \(A\)-modules \(M\), the completion \(\hat{A}\) is flat over \(A\) and a local criterion of flatness holds in the following form. Let \(B\) be a pseudo-adhesive \(A\)-algebra with respect to \(IB\) and assume that \(1+IB\) sits inside the Jacobson radical of \(B\). Then, a finitely generated \(B\)-module \(M\) is flat over \(A\) if and only if \(M/I^kM\) is flat over \(A/I^k\) for any \(k\). Examples of pseudo-adhesive rings are rigid-noetherian rings: adic (i.e. Hausdorff complete) rings that are noetherian outside \(I\). Those rings enjoy additional nice properties: finitely generated modules over them are complete and their submodules are closed. The authors also introduce refined versions of their conditions. A ring \(A\) is said to be adhesive if it is noetherian outside \(I\) and if finitely generated \(A\)-modules have finitely generated \(I\)-torsion. Then come a universal version (polynomial rings over \(A\) are adhesive) and a topologically universal version (restricted power series rings over \(A\) are universally adhesive). In the last case, if \(A\) is also assumed to be \(I\)-torsion free, then, for any topologically finitely presented \(A\)-algebra \(B\), the structure sheaf of Spf\((B)\) is coherent. As an exam…
reviewed by: Jérôme Poineau
Mathematics Subject Classification ID: 13J10
Mathematics Subject Classification ID: 13E99
Mathematics Subject Classification ID: 13J07
Mathematics Subject Classification ID: 14G22
zbMATH DE Number: 5969529
zbMATH Keywords: adic topology
zbMATH Keywords: rigid analytic geometry
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.02.001
OpenAlex ID: W1994089197
cites work: Q5522002
cites work: Q4405003
cites work: Almost ring theory
cites work: Q3992979
cites work: Critères de platitude et de projectivité. Techniques de ''platification'' d'un module. (Criterial of flatness and projectivity. Technics of ''flatification of a module.)
cites work: Q5602063
DOI: 10.1016/J.JALGEBRA.2011.02.001
description: scientific article; zbMATH DE number 5969529