Some loci related to Cohen–Macaulayness

Abstract

Let (R, 𝔪) be a Noetherian local ring and M a finitely generated R-module. The pseudo Cohen–Macaulayness (respectively, generalized Cohen–Macaulayness) was introduced by Cuong–Nhan [Pseudo Cohen–Macaulay and pseudo generalized Cohen–Macaulay modules, J. Algebra267 (2003) 156–177] as an extension of the Cohen–Macaulayness (respectively, generalized Cohen–Macaulayness). In this paper, we describe the pseudo Cohen–Macaulay (pseudo CM) locus and pseudo generalized Cohen–Macaulay (pseudo generalized CM) locus of M. We also study the non-Cohen–Macaulay locus and the non-generalized Cohen–Macaulay locus of the canonical module K(M) of M in case where R is a quotient of a Gorenstein local ring.