The case of the submaximal minors: A finishing touch
type: Item
instance of: scholarly article
title: The case of the submaximal minors: A finishing touch
zbMATH Open document ID: 0643.13005
DOI: 10.1016/0001-8708(88)90008-4
author: Giandomenico Boffi
published in: Advances in Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let \(F_ 0\) and \(G_ 0\) be the free modules over the commutative ring \(R_ 0\) with finite bases \(\{f_ 1,...,f_ m\}\), \(\{g_ 1,...,g_ n\}\) respectively, where \(m\geq n\). Let R denote the symmetric algebra \(S(F_ 0\otimes G_ 0)\) over \(R_ 0\). Then R is a polynomial ring \(R_ 0[X_{ij}]\) where the indeterminate \(X_{ij}\) corresponds to \(f_ i\otimes g_ i\) for each i, j. This paper is concerned with the free resolutions of the ideals \(I_ p\) generated by the minors of order p of the generic matrix \((X_{ij})\). A construction of a minimal resolution of the ideal of the submaximal minors was given by \textit{K. Akin}, \textit{D. A. Buchsbaum} and \textit{J. Weyman} [Adv. Math. 39, 1-30 (1981; Zbl 0474.14035)]. However this construction gave no description of the boundary map of the complex resolving \(I_{n-1}\). Knowledge of this boundary map would be helpful in tackling the resolutions of \(I_{n-2}\), \(I_{n-3}\), etc. The author provides an explicit description of the boundary map and then illustrates this in the specific case when \(m=n+1\).
Mathematics Subject Classification ID: 13D25
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 4047847
zbMATH Keywords: determinantal ideal
zbMATH Keywords: free modules
zbMATH Keywords: polynomial ring
zbMATH Keywords: free resolutions
zbMATH Keywords: boundary map
reviewed by: John Clark
MaRDI profile type: Publication
full work available at URL: 0001-8708(88)90008-4
OpenAlex ID: W2001330497
cites work: Resolutions of determinantal ideals: The submaximal minors
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cites work: A new construction of the Eagon-Northcott complex
cites work: What makes a complex exact?
cites work: Cohen-Macaulay Rings, Invariant Theory, and the Generic Perfection of Determinantal Loci
cites work: Q4149875
description: scientific article; zbMATH DE number 4047847