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.
@article{urn:eudml:doc:39932,
title = {On the free character of the first Koszul homology module.},
journal = {Extracta Mathematicae},
volume = {6},
year = {1991},
pages = {126-128},
mrnumber = {MR1185357},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:39932}
}
García Rodicio, Antonio. On the free character of the first Koszul homology module.. Extracta Mathematicae, Tome 6 (1991) pp. 126-128. http://gdmltest.u-ga.fr/item/urn:eudml:doc:39932/