\(q\)-torsion freeness of symmetric powers
type: Item
instance of: scholarly article
title: \(q\)-torsion freeness of symmetric powers
zbMATH Open document ID: 0899.13015
DOI: 10.1007/BF02844276
author: Gaetana Restuccia
author: Cristodor-Paul Ionescu
publication date: 1998-05-13T00:00:00Z
review text: Let \(R\) be a commutative noetherian unitary ring and \(E\) a finitely generated \(R\)-module. Let \(S_R(E)\) be the symmetric algebra of \(E\) over \(R\) and \(S_R(R^n)= R[X_1,\dots, X_n]\) the polynomial ring over \(R\). If \(E\) is of rank \(e\) and of finite projective dimension then there is an epimorphism between the cycles of the Koszul complex on the kernel of the canonical morphism \(S_R(R^n)\to S_R(E)\) and the cycles of the complex \(Z(E)\). Hence, if the Koszul complex \(K(f_1,\dots, f_m;S)\) of the ideal of relations of \(S(E)\) is exact then so is \(Z(E)\) (theorem 2.11.); here \(Z_r(E)= \text{Ker }\partial_r\) with \(\partial_R: \bigwedge^r R_1^n\to \bigwedge^{r-1} R^n\bigotimes_R E\) gives the complex \(Z(E)\): \[ 0\to Z_n(E)\otimes S[-n]\to Z_{n-1}(E)\otimes S[-n+1]\to \dots\to Z_0(E)\otimes S\to 0. \] For an integer \(q\) the module \(E\) is said to be \(q\)-torsion free if every \(R\)-regular sequence of length \(q\) is also \(E\)-regular. In the last part several results concerning the \(q\)-torsion freeness of the symmetric powers of \(E\) are proved.
Mathematics Subject Classification ID: 13D30
Mathematics Subject Classification ID: 13C12
zbMATH DE Number: 1140075
zbMATH Keywords: symmetric algebra
zbMATH Keywords: Koszul complex
zbMATH Keywords: \(q\)-torsion free module
reviewed by: Ladislav Bican
MaRDI profile type: Publication
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published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 1140075