# The Cohen-Macaulay property in derived commutative algebra

## TL;DR

This paper extends classical Cohen-Macaulay theory to derived commutative algebra, establishing new inequalities, defining local-Cohen-Macaulay DG-rings, and showing many classical properties generalize to the derived setting.

## Contribution

It introduces the notion of local-Cohen-Macaulay DG-rings, generalizes classical Cohen-Macaulay theory to derived algebraic geometry, and explores examples including local Gorenstein DG-rings.

## Key findings

- Proved amplitude inequalities for finite DG-modules.
- Defined and studied local-Cohen-Macaulay DG-rings.
- Showed classical Cohen-Macaulay properties extend to derived setting.

## Abstract

By extending some basic results of Grothendieck and Foxby about local cohomology to commutative DG-rings, we prove new amplitude inequalities about finite DG-modules of finite injective dimension over commutative local DG-rings, complementing results of J{\o}rgensen 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.