Koszul complexes and hyperdeterminants
type: Item
instance of: scholarly article
title: Koszul complexes and hyperdeterminants
zbMATH Open document ID: 0976.13010
author: Jerzy Weyman
author: Giandomenico Boffi
published in: Journal of Algebra
publication date: 2002-01-06T00:00:00Z
review text: The starting point is the paper ``Hyperdeterminants'' by \textit{I. M. Gelfand}, \textit{M. M. Kapranov} and \textit{A. V. Zelevinsky} [Adv. Math. 96, 226-263 (1992; Zbl 0774.15002)], where the concept of hyperdeterminant \(\text{Det}(A)\) and the Cayley-Koszul complexes \(C(m_1, \dots, m_r,A)\) are introduced, for any \(R\)-linear form \(A\) in the polynomial ring \(R=C[x_1^{(1)} \dots x_{d_1}^{(1)}; \cdots;x_1^{(r)} \dots x_{d_r}^{(r)}]\), \(r\geq 2\), \(d_i\geq 2\) \((i=1,\dots,r)\), with the condition \(d_{i_0}-1\leq \sum_{i\neq i_0}(d_i-1)\), for all \(i_0\in\{1, \dots,r\}\). In the present paper, the authors introduce a Koszul complex \({\mathcal K}_A\) on a free \(R\)-module \(M\) and prove that the Cayley-Koszul complexes are multigrade components of such a \({\mathcal K}_A\). Moreover, they introduce a double complex \({\mathcal D}_A\) of free \(R\)-modules and express \(\text{Det}(A)\) in terms of the determinant of a multigraded component of \({\mathcal D}_A\).
Mathematics Subject Classification ID: 13D25
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 1523165
zbMATH Keywords: Schur complex
zbMATH Keywords: Jacobian ideals
zbMATH Keywords: multigraded components of a complex
zbMATH Keywords: hyperdeterminant
zbMATH Keywords: Koszul complex
reviewed by: Carla Massaza
MaRDI profile type: Publication
full work available at URL: jabr.1999.7646
OpenAlex ID: W1978678670
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cites work: Calculating Discriminants by Higher Direct Images
DOI: 10.1006/JABR.1999.7646
description: scientific article; zbMATH DE number 1523165