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

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 $R$.