24 Chain Conditions Arising from the Study of Non Finitely Generated Modules Over Commutative Rings

S. ZARZUELA Departament d’Algebra i Geometria, Universitat de Barcelona, Gran Via 585, E-08007 Barcelona, Spain

1. Introduction

In 1974 M. Hochster [5] introduced the big Cohen-Macaulay modules conjecture, and showed how to obtain from its validity the corresponding one for most of the local homological conjectures. Let us recall its statement:

BIG COHEN-MACAULAY MODULES CONJECTURE. Let (A, m) be a (Noetherian) local ring and a1,,ad be a system of parameters for A. Then there exists an A-module M such that a1,,ad is a regular sequence for M, that is, mMM and ai+1ZA(M/(a1,,ai)M) for any i.

Then M is said to be a big Cohen-Macaulay A-module ...

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