Grade and Cohen-Macaulayness for DG-modules

TL;DR

This paper explores the properties of DG-modules over Cohen-Macaulay DG-rings, establishing inequalities, defining perfect DG-modules, and characterizing Cohen-Macaulayness through these concepts, while also addressing a conjecture and tensor product behavior.

Contribution

It introduces the concept of perfect DG-modules, relates projective dimension to grade, and proves Cohen-Macaulayness criteria for DG-modules, addressing a longstanding conjecture.

Findings

Established an inequality relating projective dimension and grade.

Defined perfect DG-modules as a generalization of perfect modules.

Proved Cohen-Macaulayness characterization for DG-modules over Cohen-Macaulay DG-rings.

Abstract

We establish an inequality relating the projective dimension of a DG-module in $\mathrm{D}^\mathrm{b}_\mathrm{f}(A)$ to its grade and introduce the concept of perfect DG-modules as a natural generalization of perfect modules. It is proved that a DG-module $M$ over a local Cohen-Macaulay DG-ring with constant amplitude is Cohen-Macaulay if and only if $M$ is perfect and $\mathrm{amp}M \leq \mathrm{amp}\mathrm{R}\Gamma_{\bar{\mathfrak{m}}}(M)$. An affirmative answer is provided to Conjecture 2.11 of Yoshida [J. Pure Appl. Algebra 123 (1998) 313--326]. We also study the grade of DG-modules with finite injective dimension and examine the preservation of Cohen-Macaulayness under tensor products.