The first syzygies of determinantal ideals
type: Item
instance of: scholarly article
title: The first syzygies of determinantal ideals
zbMATH Open document ID: 0777.13010
DOI: 10.1016/0022-4049(93)90172-P
author: Yonghao Ma
published in: Journal of Pure and Applied Algebra
publication date: 1993-05-16T00:00:00Z
review text: Let \(A\) be a commutative ring, and \(S=A[X]\) the polynomial ring over \(A\) in the indeterminates \(X_{ij}\) forming the \(m\times n\) matrix \(X\). Denote the ideal of \(S\) generated by the \(t\)-minors of \(X\) by \(I_ t\). A free resolution of \(S/I_ t\) as an \(S\)-module was constructed by Eagon and Northcott in the case \(t=\min(m,n)\) and by Akin, Buchsbaum, and Weyman for \(t=\min(m,n)-1\). Furthermore Lascoux describes resolutions for arbitrary \(t\) in the case in which \(A\) contains the rational numbers. Without any assumption on the characteristic of \(A\), \textit{D. W. Sharpe} [Proc. Lond. Math. Soc., III. Ser. 15, 645-679 (1965; Zbl 0119.036)] showed that for \(t=2\) the first module of syzygies of \(I_ t\) has a system of generators whose coefficients are linear in the \(X_{ij}\). For arbitrary \(t\) this was proved by \textit{K. Kurano} [J. Algebra 124, No. 2, 414-436 (1989; Zbl 0703.13011)] using the characteristic free representation theory of the general linear group. In particular the second Betti number of \(S/I_ t\) is independent of the characteristic of \(A\); this is wrong for the third Betti number [\textit{M. Hashimoto}, Nagoya Math. J. 118, 203-216 (1990; Zbl 0707.13005)]. In the paper under review the author derives Kurano's theorem in an `elementary' manner, using a Gröbner basis approach. The heart of the paper is a direct, algorithmic proof of the fact that the \(t\)-minors of \(X\) form a Gröbner basis of \(I_ t\) with respect to a natural term order. [Other proofs of this fact use counting arguments; see for examples \textit{B. Sturmfels}, Math. Z. 205, No. 1, 137-144 (1990; Zbl 0685.13005)].
reviewed by: Winfried Bruns
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 166183
zbMATH Keywords: determinantal ideal
zbMATH Keywords: syzygies
zbMATH Keywords: system of generators
zbMATH Keywords: Betti number
zbMATH Keywords: Gröbner basis
MaRDI profile type: Publication
cites work: Resolutions of determinantal ideals: The submaximal minors
cites work: Schur functors and Schur complexes
cites work: Q3208084
cites work: A new construction of the Eagon-Northcott complex
cites work: Cohen-Macaulay Rings, Invariant Theory, and the Generic Perfection of Determinantal Loci
cites work: Ideals defined by matrices and a certain complex associated with them
cites work: Syzygies des variétés déterminantales
cites work: Complexes associated with trace and evaluation. Another approach to Lascoux's resolution
cites work: ON CERTAIN POLYNOMIAL IDEALS DEFINED BY MATRICES
cites work: The Syzygies and Semi-Regularity of Certain Ideals Defined by Matrices
full work available at URL: 0022-4049(93)90172-p
OpenAlex ID: W1980987560
description: scientific article; zbMATH DE number 166183