A characterization of Cohen-Macaulay rings in terms of levels of perfect complexes

TL;DR

This paper characterizes Cohen-Macaulay rings by examining the finiteness of levels of perfect complexes relative to Gorenstein $C$-projective modules, linking homological properties to ring structure.

Contribution

It provides a new characterization of Cohen-Macaulay rings using levels of complexes with respect to Gorenstein $C$-projective modules, extending previous Gorensteinness results.

Findings

Characterizes Cohen-Macaulayness via levels of perfect complexes.

Recovers a recent theorem on Gorensteinness of rings.

Links homological levels to ring properties.

Abstract

Let $R$ be a commutative noetherian ring, and let $C$ be a semidualizing $R$-module. In this paper, we study levels of bounded complexes of finitely generated $R$-modules with respect to the full subcategory $\mathsf{G}_{C}(R)$ consisting of Gorenstein $C$-projective $R$-modules. Our main result provides a characterization of the Cohen-Macaulayness of $R$ in terms of the finiteness of levels of perfect complexes with respect to $\mathsf{G}_{C}(R)$. This recovers a recent theorem of Christensen, Kekkou, Lyle and Soto Levins on the Gorensteinness of $R$.