Weak proregularity, weak stability, and the noncommutative MGM equivalence
type: Item
instance of: scholarly article
title: Weak proregularity, weak stability, and the noncommutative MGM equivalence
zbMATH Open document ID: 1401.18036
author: Rishi Vyas
author: Amnon Yekutieli
published in: Journal of Algebra
publication date: 2018-09-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1608.03543
review text: Let \(R\) be a commutative ring, and let \(\mathfrak{a}\) be a finitely generated ideal in \(R\). There are two functors on the category of \(R\)-modules that the ideal \(\mathfrak{a}\) determines: the \(\mathfrak{a}\)-torsion functor \(\Gamma_{\mathfrak a}\), and the \(\mathfrak{a}\)-adic completion functor \(\Lambda_{\mathfrak a}\). The most known general condition under which the functors \(\Gamma_{\mathfrak a}\) and \(\Lambda_{\mathfrak a}\) are adjoint to each other is when the ideal \(\mathfrak{a}\) is \textit{weakly proregular}. In particular, the \(MGM\) \textit{Equivalence} holds. The goal of this paper is to find a noncommutative analogue of weak proregularity, and to prove a suitable version of the \(MGM\) Equivalence. To aim that, the authors define a new condition called \textit{weak stability} on a torsion class \(T\) in a module category. Then, the first main theorem says that in the commutative case, the ideal \(\mathfrak{a}\) is weakly proregular if and only if the corresponding torsion class \(T\) is weakly stable. Weak stability of torsion classes in module categories over noncommutative rings is studied and there are three main theorems in this context: Theorem 7.13, Theorem 8.4 and Theorem 9.7 (it is a correction of Theorem 1.23 by \textit{A. Yekutieli} and \textit{J. J. Zhang} [J. Algebra 259, No. 2, 451--493 (2003; Zbl 1035.16004)]).
reviewed by: Marek Golasiński
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 16E35
zbMATH DE Number: 6942900
zbMATH Keywords: torsion classes
zbMATH Keywords: derived torsion
zbMATH Keywords: derived completion
Wikidata QID: Q112881682
MaRDI profile type: Publication
OpenAlex ID: W2517678992
arXiv ID: 1608.03543
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DOI: 10.1016/J.JALGEBRA.2018.07.023
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description: scientific article