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

TL;DR

This paper investigates the structure of pseudo supports and the non Cohen-Macaulay locus of finitely generated modules over Noetherian local rings, linking these properties to ring catenarity, Serre conditions, and unmixedness.

Contribution

It provides new insights into the relationship between pseudo supports, non Cohen-Macaulay loci, and ring properties like catenarity and unmixedness.

Findings

Pseudo supports are characterized in relation to ring catenarity.

Conditions for the non Cohen-Macaulay locus are established.

Connections between Serre conditions and module properties are clarified.

Abstract

Let $(R,\m)$ be a Noetherian local ring and $M$ a finitely generated $R$-module with $\dim M=d.$ Let $i\geq 0$ be an integer. Following M. Brodmann and R. Y. Sharp \cite{BS1}, the $i$-th pseudo support of $M$ is the set of all prime ideals $\p$ of $R$ such that $ H^{i-\dim (R/\p)}_{\p R_{\p}}(M_{\p})\neq 0.$ In this paper, we study the pseudo supports and the non Cohen-Macaulay locus of $M$ in connections with the catenarity of the ring $R/\Ann_RM$, the Serre conditions on $M$, and the unmixedness of the local rings $R/\p$ for certain prime ideals $\p$ in $\Supp_R (M)$.