On the free character of the first koszul homology module
ISSN: 0213-8743
Year of publication: 1991
Volume: 6
Issue: 2-3
Pages: 126-128
Type: Article
More publications in: Extracta mathematicae
Abstract
Let (A,M,K) denote a local noetherian ring A with maximal ideal M and residue field K. Let I be an ideal of A and E the Koszul complex generated over A by a system of generators of I. The condition: H1(E) is a free A/I-module, appears in several important results of Commutative Algebra. For instance: - (Gulliksen [3, Proposition 1.4.9]): The ideal I is generated by a regular sequence if and only if I has finite projective dimension and H1(E) is a free A/I-module. - (André [2]): Assume that A is a complete intersection. Then, A/I is complete intersection if and only if H1(E)2 = H2(E) and H1(E) is a free module. The purpose of this note is to generalize both results.