The Cohen-Macaulay property in derived commutative algebra

Abstract

By extending some basic results of Grothendieck and Foxby about local cohomology to commutative DG-rings, we prove new amplitude inequalities about finite DGmodules of finite injective dimension over commutative local DG-rings, complementing results of Jørgensen and resolving a recent conjecture of Minamoto. When these inequalities are equalities, we arrive to the notion of a local-Cohen-Macaulay DG-ring. We make a detailed study of this notion, showing that much of the classical theory of Cohen-Macaulay rings and modules can be generalized to the derived setting, and that there are many natural examples of local-Cohen-Macaulay DG-rings. In particular, local Gorenstein DG-rings are local-Cohen-Macaulay. Our work is in a non-positive cohomological situation, allowing the Cohen-Macaulay condition to be introduced to derived algebraic geometry, but we also discuss extensions of it to non-negative DG-rings, which could lead to the concept of Cohen-Macaulayness in topology.

Sequence-regular commutative DG-rings

“…Following [28,Definition 5.8], a prime ideal p ∈ Spec(H 0 (A)) is called an associated prime ideal of M if depth Ap (Mp) = inf(Mp). The set of all associated primes of M is denoted by Ass A (M ).…”

“…By [28,Theorem 4.1], if R is a dualizing DG-module over A, there is always an inequality amp(R) ≥ amp(A).…”

“…The reason for this somewhat cumbersome terminology is that, as demonstrated in [28,Example 8.3], the classical fact that the localization of a Cohen-Macaulay ring is again Cohen-Macaulay does not always generalize to the DG setting. Thus, there exist examples of local-Cohen-Macaulay DG-rings A, and prime ideals p ∈ Spec(H 0 (A)), such that Ap is not local-Cohen-Macaulay.…”

“…The key ingredient to obtain such a characterization is to use Cohen-Macaulay DG methods. The notion of a Cohen-Macaulay DG-ring was first described in [28], and was later extensively developed in [30]. In this paper we will continue the development of this notion, and use the theory of Cohen-Macaulay DG-rings to prove the following characterization of sequence-regualr DG-rings: Theorem B.…”

“…When this ring is local with a maximal ideal m, we say that (A, m) is a noetherian local DG-ring. For noetherian local DG-rings (A, m) with bounded cohomology, the notion of an A-regular sequence in m was introduced and studied, first by Minamoto in [20], and later by the author in [28,30]. It is recalled in Section 1.3 below.…”