Cohen-Macaulayness of formal fibers and dimension of local cohomology modules

Abstract

Let $(R,\mathfrak{m})$ be a Noetherian local ring, $M$ a finitely generated $R$-module of dimension $d$. Set $\mathfrak{a}(M):={\mathfrak{a}}_0(M)\cdots {\mathfrak{a}}_{d-1}(M)$, where ${\mathfrak{a}}_i(M):={\mathrm{Ann}}_R{H}_{\mathfrak{m}}^i(M)$ for $i\ge 0$. In this paper, we study the Cohen-Macaulayness of formal fibers of $R$ in the relation with the dimension $\dim (R/\mathfrak{a}(M)).$ We prove that $\dim (R/\mathfrak{a}(M))<d$ if and only if $R/\mathfrak{p}$ is unmixed and the generic formal fiber of $R/\mathfrak{p}$ is Cohen-Macaulay for all $\mathfrak{p}\in {\mathrm{Supp}}_R(M)$ with $\dim (R/\mathfrak{p})=d.$ In general, $R/\mathfrak{p}$ is unmixed and the generic formal fiber of $R/\mathfrak{p}$ is Cohen-Macaulay for all $\mathfrak{p}\in {\mathrm{Supp}}_R(M)$ with $\dim (R/\mathfrak{p})>\dim (R/\mathfrak{a}(M)).$ As applications, we explore the structure of local rings and the dimension, the closedness of non Cohen-Macaulay locus of finitely generated modules.

Published online in Communications in Algebra