Tor and torsion on a complete intersection
type: Item
instance of: scholarly article
title: Tor and torsion on a complete intersection
zbMATH Open document ID: 0898.13008
author: David A. Jorgensen
published in: Journal of Algebra
publication date: 1998-11-01T00:00:00Z
review text: Let \((R, {\mathfrak m})\) be a complete intersection ring, i.e. \(\widehat R=Q/(x)\) with a regular local ring \((Q, {\mathfrak n},k)\) and a \(Q\)-regular sequence \(x=(x_1, \dots, x_r)\), and let \(M,N\) be two finitely generated \(R\)-modules. The author studies the vanishing of \(\text{Tor}^R_i (M,N)\) for \(i\gg 0\) and its relation to the existence of torsion elements in \(M \otimes_RN\). His first result says that \(\text{Tor}^R_i (M,N)=0\) for \(i\gg 0\) implies the triviality of the intersection \({\mathcal V} (Q,x,M) \cap {\mathcal V} (Q,x,N)= \{0\}\). Here \({\mathcal V} (Q,x,M)\) is the cohomological variety \[ {\mathcal V} (\text{Ann}_{\mathcal R} (\text{Ext}^*_R(M,k)))\subseteq \mathbb{A}^r_k \] of the module \(M\) defined by \textit{L. L. Avramov} [cf. Invent. Math. 96, No. 1, 71-101 (1989; Zbl 0677.13004)]. As an application, the author shows that if \(M\) is a generically free maximal Cohen-Macaulay \(R\)-module of complexity \(c=\dim {\mathcal V} (Q,x,M)\), and if \(S_i\) is an \(i\)-th syzygy module of \(M\) for \(i=1, \dots, c\in \{1,r\}\), then the tensor product \((M \oplus S_1 \oplus \cdots \oplus S_c) \otimes_R M^*\) has torsion elements. Finally, the author also shows a sufficient condition for the vanishing of \(\text{Tor}^R_i (M,N)\) for \(i\gg 0\). More precisely, he proves that this is the case, if \(M\) lifts to \(Q/(x_1, \dots, x_i)\) and \(N\) to \(Q/(x_j, \dots, x_r)\) for \(1\leq i<j\leq r\geq 2\). He applies this condition to provide a source of maximal Cohen-Macaulay tensor products \(M\otimes_RN\) in case \(r\geq 2\), where neither \(M\) nor \(N\) need to be free.
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 13D30
Mathematics Subject Classification ID: 18G15
Mathematics Subject Classification ID: 14M10
zbMATH DE Number: 1088613
zbMATH Keywords: Tor
zbMATH Keywords: complete intersection ring
zbMATH Keywords: torsion elements
zbMATH Keywords: syzygy module
zbMATH Keywords: maximal Cohen-Macaulay tensor products
reviewed by: Martin Kreuzer
MaRDI profile type: Publication
full work available at URL: jabr.1997.7064
OpenAlex ID: W2020175994
cites work: Modules of finite virtual projective dimension
cites work: Modules over unramified regular local rings
cites work: Q5661996
cites work: Q4274969
cites work: Q4317713
cites work: Q5602063
cites work: A Change of Ring Theorem with Applications to Poincaré Series and Intersection Multiplicity.
cites work: Tensor products of modules and the rigidity of Tor
cites work: Tensor products of modules, rigidity and local cohomology.
cites work: Complexity and Tor on a complete intersection
cites work: Q3882615
DOI: 10.1006/JABR.1997.7064
description: scientific article; zbMATH DE number 1088613