Cohen–Macaulay properties for balanced big Cohen–Macaulay modules

Abstract

Let A be a (commutative, Noetherian) local ring (with identity) and let a1,…, an be a system of parameters (s.o.p.) for A. A (not necessarily finitely generated) A-module M is said to be a big Cohen–Macaulay.A-module with respect to a1,…, an if a1,…, an is an M-sequence, that is if M ‡ = (a1,…, an) M and, for each i = 1,…, n,One of the main open problems in commutative algebra at the present time is that of establishing the existence of a big Cohen–Macaulay module with respect to a specified s.o.p. in an arbitrary local ring. The work and writings of Hochster, such as (5), show that, if the existence of such modules could be established, then several conjectures in commutative algebra, some of which are quite long-standing, would be settled. Moreover, Hochster has established the existence of such big Cohen–Macaulay modules whenever the local ring A concerned contains a field as a subring, or has dimension not exceeding 2: see ((5), chapters 4, 5) and (4).