24 Chain Conditions Arising from the Study of Non Finitely Generated Modules Over Commutative Rings
In 1974 M. Hochster  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 be a system of parameters for A. Then there exists an A-module M such that is a regular sequence for M, that is, and for any i.
Then M is said to be a big Cohen-Macaulay A-module ...