Commutative Ring Theory

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 $a_1,\dots ,a_d$ be a system of parameters for A. Then there exists an A-module M such that $a_1,\dots ,a_d$ is a regular sequence for M, that is, $\text{m}M\ne M$ and $a_{i+1}\notin Z_A\left(M/\left(a_1,\dots ,a_i\right)M\right)$ for any i.

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