On the free character of the first koszul homology module

• Autores: Antonio G. Rodicio
• Localización: Extracta mathematicae, ISSN-e 0213-8743, Vol. 6, Nº 2-3, 1991, págs. 126-128
• Idioma: inglés
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• Resumen

  • 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.