If the socle fits
type: Item
instance of: scholarly article
title: If the socle fits
zbMATH Open document ID: 0762.13004
DOI: 10.1016/0021-8693(92)90252-H
author: Bernd Ulrich
author: Andrew Kustin
published in: Journal of Algebra
publication date: 1992-06-28T00:00:00Z
review text: In studying ideals that are obtained in some fashion from minors of matrices one is frequently in the position of having two ideals \(K\subset J\) for which the generators of \(K\) are known, perfection of \(J\) is known, and there is a strong suspicion of equality. The authors prove that for graded artinian \(k\)-algebras a simple comparison of socle dimensions suffices: if \(\dim_ k\operatorname{socle}(R/K)_ i\leq \dim _ k \operatorname{socle}(R/J)_ i\) for all \(i\), then \(K=J\). The principal new application is to identify a set of generators for the generic residual intersection of a generic grade three Gorenstein ideal; but the power of the method is also illustrated by a new proof (using residual intersection theory) of the perfection of the determinantal ideal generated by the maximal minors of a generic matrix.
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 13E10
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 37121
zbMATH Keywords: graded artinian algebras
zbMATH Keywords: socle dimensions
zbMATH Keywords: set of generators
zbMATH Keywords: generic grade three Gorenstein ideal
zbMATH Keywords: determinantal ideal
zbMATH Keywords: residual intersection
MaRDI profile type: Publication
cites work: Residual intersections in Cohen-Macaulay rings
cites work: The Koszul algebra of a codimension 2 embedding
cites work: The resolution of the generic residual intersection of a complete intersection
cites work: Determinantal rings
cites work: Ideals defined by matrices and a certain complex associated with them
cites work: Q3222956
cites work: On the canonical module of the Rees algebra and the associated graded ring of an ideal
cites work: Cohen-Macaulay Rings, Invariant Theory, and the Generic Perfection of Determinantal Loci
cites work: Linkage and the Koszul Homology of Ideals
cites work: Strongly Cohen-Macaulay Schemes and Residual Intersections
cites work: Residual intersections.
cites work: Generating a residual intersection
cites work: A family of complexes associated to an almost alternating map, with applications to residual intersections
cites work: SEMI-REGULAR RINGS AND SEMI-REGULAR IDEALS
cites work: Liaison des variétés algébriques. I
Recommended article: Q4496059
description: scientific article; zbMATH DE number 37121