Koszul cycles and Eliahou-Kervaire type resolutions
type: Item
instance of: scholarly article
title: Koszul cycles and Eliahou-Kervaire type resolutions
zbMATH Open document ID: 0868.13014
published in: Journal of Algebra
publication date: 1996-07-09T00:00:00Z
review text: The aim of this paper is to compute a minimal free resolution of a finitely generated \(R\) module \(M\), in terms of a basis of \(H(M)\), where \((R, {\mathfrak M})\) is a regular local ring containing its residue class field, \(H(M)\) is the homology of the Koszul complex \(K (M)\) of \(M\) with respect to a minimal system \((x_1, \dots, x_n)\) of generators of \({\mathfrak M}\). Such a problem was already considered in previous papers by the same authors, but here it is treated in a more general situation. A difficult point in the procedure is the computation of a basis for \(H(M)\). So the authors consider the special case of stable ideals, in which it is easy to compute the Koszul cycles, finding again the Eliahou-Kervaire resolution. A relation between stability of \(I\) and proper sequences suggests to consider modules \(M\) for which \(x_n, \dots, x_1\) is proper: For such modules the author are able to describe a basis of \(H(M)\) and, as a consequence, to apply their results. Finally, they give an explicit resolution of the \(k\)-algebras for which \(x_n,\dots,x_1\) is ``strongly proper''.
Mathematics Subject Classification ID: 13D25
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 13D03
zbMATH DE Number: 883764
zbMATH Keywords: basis of homology of the Koszul complex
zbMATH Keywords: minimal free resolution
zbMATH Keywords: stable ideals
author: Annetta G. Aramova
author: Jürgen Herzog
reviewed by: Carla Massaza
MaRDI profile type: Publication
full work available at URL: jabr.1996.0124
OpenAlex ID: W2022451954
DOI: 10.1006/JABR.1996.0124
description: scientific article; zbMATH DE number 883764