Bounds for codimensions of Fitting ideals
type: Item
instance of: scholarly article
title: Bounds for codimensions of Fitting ideals
zbMATH Open document ID: 0907.13008
published in: Journal of Algebra
publication date: 1999-03-09T00:00:00Z
review text: Let \(R\) be a regular affine \(k\)-algebra where \(k\) is a field of characteristic 0, and \(M\) a finitely generated \(R\)-module such that the symmetric algebra \(\text{Sym}(M)\) of \(M\) is equidimensional. The main result of the paper says that \(\text{codim} Q\leq r(r-\text{rank} M)\) for every minimal prime ideal \(Q\) of the Fitting ideal \(\text{Fitt}_{r-1}(M)\) that is not associated to \(\text{Fitt}_r(M)\). The proof is based on the author's earlier results concerning minimal primes of tensor powers of \(\text{Sym}(M)\). In terms of an \(m\times n\) matrix \(A\) over \(R\) the bound yields \(\text{codim} I_t(A)\leq(m-t+1)(\text{rank} A-t+1)\) for the ideal of \(t\)-minors of \(A\), provided the cokernel of the linear map \(R^n\to R^m\) represented by \(A\) has an equidimensional symmetric algebra. In this case the bound is sharper than the Eagon-Northcott bound and other bounds given by the reviewer [\textit{W. Bruns}, Proc. Am. Math. Soc. 83, 19-26 (1981; Zbl 0478.13006)].
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13D40
zbMATH DE Number: 1066128
zbMATH Keywords: bounds for codimensions
zbMATH Keywords: minors
zbMATH Keywords: symmetric algebra
author: Michal Kwienciński
reviewed by: Winfried Bruns
MaRDI profile type: Publication
full work available at URL: jabr.1996.6999
OpenAlex ID: W2071767530
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DOI: 10.1006/JABR.1996.6999
description: scientific article; zbMATH DE number 1066128