Local cohomology with support in generic determinantal ideals
type: Item
instance of: scholarly article
title: Local cohomology with support in generic determinantal ideals
zbMATH Open document ID: 1303.13018
author: Claudiu Raicu
author: Jerzy Weyman
published in: Algebra \& Number Theory
publication date: 2014-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1309.0617
reviewed by: Dang Hop Nguyen
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 14M12
zbMATH DE Number: 6348605
zbMATH Keywords: determinantal ideal
zbMATH Keywords: generic matrix
zbMATH Keywords: local cohomology
zbMATH Keywords: regularity
describes a project that uses: Macaulay2
MaRDI profile type: Publication
OpenAlex ID: W1990579219
arXiv ID: 1309.0617
DOI: 10.2140/ANT.2014.8.1231
review text: Let \(k\) be a field of characteristic \(0\). Let \(X\) be a generic \(m\times n\) matrix, i.e. the entries of \(X\) are distinct variables \(x_{1,1},\ldots,x_{m,n}\). For \(p\leq \min\{m,n\}\), the ideal \(I_p\subseteq k[x_{1,1},\ldots,x_{m,n}]\) of \(p\)-minors of \(X\) is generated by the determinants of \(p\times p\) submatrices of \(X\). The study of generic determinantal ideals, namely ideals of type \(I_p\) (where \(p\leq \min\{m,n\}\)), is a classical and beautiful subject lying in the intersection of commutative algebra and representation theory. The paper under review gives significant new information about such ideals, namely the authors compute the local cohomology with support in generic determinantal ideals. In fact, they compute the associated equivariant \(\text{Ext}\) modules of a large class of ideals invariant under the action of \(\text{GL}_m\times \text{GL}_n\). The local cohomology modules supported in such ideals are derived using the formula NEWLINE\[NEWLINE H^i_I(S)=\varinjlim_{d} \text{Ext}^i(S/I^d,S), NEWLINE\]NEWLINE for a proper ideal \(I\) over a noetherian local ring \(S\). The main theorems extend previous results due to \textit{C. Raicu} et al. [Adv. Math. 250, 596--610 (2014; Zbl 1295.13025)].NEWLINENEWLINEIn more detail, assume that \(m\geq n\). Let \(F,G\) be \(k\)-vector spaces of dimensions \(m,n\), respectively. Let \(S\) be the symmetric algebra \(\text{Sym}(F\otimes G)\). Then \(\text{GL}(F)\times \text{GL}(G)\cong \text{GL}_m\times \text{GL}_n\) acts on \(S\) and makes \(S\) into a \(\text{GL}(F)\times \text{GL}(G)\)-representation. By Cauchy formula, \(S\) is decomposed as a direct sum of irreducible \(\text{GL}(F)\times \text{GL}(G)\)-representations: NEWLINE\[NEWLINE S=\mathop{\bigoplus_{\underline x=(x_1,\ldots,x_n)}}_{x_1\geq x_2\geq \cdots \geq x_n\geq 0}S_{\underline x}F\otimes S_{\underline x}G, NEWLINE\]NEWLINE where \(S_{\underline x}\) denotes the Schur functor. A sequence \(\underline x=(x_1,\ldots,x_n)\) where \…
description: scientific article; zbMATH DE number 6348605