Symmetric algebras of modules arising from a fixed submatrix of a generic matrix
type: Item
instance of: scholarly article
title: Symmetric algebras of modules arising from a fixed submatrix of a generic matrix
zbMATH Open document ID: 0643.13002
DOI: 10.1016/0022-4049(87)90131-9
author: Winfried Bruns
author: Aron Simis
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let \(K\) be a field and let \(n\leq m\) be positive integers. In the polynomial algebra \(R=K[X]\), where \(X=(X_{ij})\) is a generic \(n\times m\) matrix, let \(I\) be the ideal generated by the \(n\times n\) minors of \(X\) containing the first \(n-1\) columns. In the first part the authors study the divisorial properties of the symmetric algebra \(S(I)\). This is done by first observing that \(S(I)\approx R[T_n,\ldots,T_m]/I_{n+1}(\tilde X)\), where \(\tilde X\) is the matrix obtained by adding the row \((0,\ldots,0,T_n,\ldots,T_m)\) to \(X\) and \(I_{n+1}(\tilde X)\) is the ideal generated by all \((n+1)\times (n+1)\) minors of \(\tilde X\), and then applying the Hochster-Eagon theory of determinantal ideals [\textit{M. Hochster} and \textit{J. A. Eagon}, Am. J. Math. 93, 1020--1058 (1971; Zbl 0244.13012)]. The following results are proved: (1) \(S(I)\) is a Cohen-Macaulay normal domain; (2) for \(n\geq 2\) the divisor class group of \(S(I)\) is \(\mathbb{Z}\oplus \mathbb{Z}\); (3) the canonical module of \(S(I)\) is \(\mathfrak a^{m-n-1}\), where \(\mathfrak a\) is the ideal of \(S(I)\) generated by the \(n\times n\) minors of the first (resp. last) \(n\) columns of \(\tilde X\); (4) the type of \(S(I)\) is \(\binom{m-1}{m-n-1}\). In the second part similar results are proved for the symmetric algebra \(S(M)\), where \(M\) varies over certain submodules of \(\operatorname{coker}(X^*: (R^n)^*\to (R^m)^*)\).
Mathematics Subject Classification ID: 13C13
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 4047844
zbMATH Keywords: generic matrix
zbMATH Keywords: determinantal ideals
zbMATH Keywords: Cohen-Macaulay normal domain
zbMATH Keywords: divisor class group
zbMATH Keywords: symmetric algebra
reviewed by: Balwant Singh
MaRDI profile type: Publication
full work available at URL: 0022-4049(87)90131-9
OpenAlex ID: W1993780625
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description: scientific article; zbMATH DE number 4047844