On tensor products of rings and extension conjectures
type: Item
instance of: scholarly article
title: On tensor products of rings and extension conjectures
zbMATH Open document ID: 1189.13019
DOI: 10.1216/JCA-2009-1-4-635
published in: Journal of Commutative Algebra
publication date: 2010-02-19T00:00:00Z
review text: In recent years several authors have considered a commutative local version of two important conjectures in module and representation theory: i) (Tachikawa) Let \(A\) be a Cohen-Macualy local ring. If \(A\) has a canonical module \(\omega\) and \(\mathrm{Ext}_A^i(\omega , A)=0\) for all \(i>0\), then \(A\) is Gorenstein, i.e. \(\omega = A\); and ii) (Auslander-Reiten) Let \(A\) commutative noetherian local ring, and let \(M\) be a finitely generated \(A\)-module. If \(\mathrm{Ext}e_A^i(M,M\oplus A)=0\) for all \(i>i\), then \(M\) is free. In this paper, the author shows that if two local rings \(R_1\) and \(R_2\) essentially of finite type over the same field \(k\), with maximal ideals \(M_1\) and \(M_2\) respectively, are Cohen-Macualay and the Tachikawa conjecture holds for both of them, then it also holds for \((R_1\otimes_kR_2)_M\), where \(M:=M_1\otimes R_2+R_1\otimes_kM_2\). Moreover, if the Auslander-Reiten conjecture holds for classes \({\mathcal C}_i\) of maximal Cohen-Macaulay modules over Cohen-Macualay rings \(R_i\), \(i=1,2\), then the Auslander-Reiten conjecture holds for \(R\)-modules with a syzygy belonging to the class of all \(R\)-modules wich are finite direct sums of modules of the form \((X_1\otimes X_2)_M\) with \(X_i\in {\mathcal C}_i\), \(i=1,2\). The paper finishes analyzing the last condition.
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 5673550
zbMATH Keywords: Tachikawa conjecture
zbMATH Keywords: local rings
zbMATH Keywords: Cohen-Macaulay rings
zbMATH Keywords: Gorenstein rings
zbMATH Keywords: Auslander-Reiten conjecture
zbMATH Keywords: tensor products of rings
Wikidata QID: Q123139158
author: David A. Jorgensen
reviewed by: Blas Torrecillas
MaRDI profile type: Publication
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full work available at URL: jca-2009-1-4-635
OpenAlex ID: W1989804157
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description: scientific article; zbMATH DE number 5673550