On pseudo supports and non-Cohen–Macaulay locus of finitely generated modules

Let (R,m) be a Noetherian local ring and M a finitely generated R-module with . Let i⩾0 be an integer. Following M. Brodmann and R.Y. Sharp (2002) [BS1], the i-th pseudo support of M is the set of all prime ideals p of R such that . In this paper, we study the pseudo supports and the non-Cohen–Macaulay locus of M in connections with the catenarity of the ring R/AnnRM, the Serre conditions on M, and the unmixedness of the local rings R/p for certain prime ideals p in SuppR(M).