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

TL;DR

This paper investigates the relationship between the Cohen-Macaulayness of formal fibers of a local ring and the dimensions of certain local cohomology modules, providing characterizations and applications to ring structure and module loci.

Contribution

It establishes a criterion linking the dimension of a specific ideal to the Cohen-Macaulayness of formal fibers and explores implications for local ring structures and non Cohen-Macaulay loci.

Findings

Dimension of R/a(M) is less than d iff R/p is unmixed and its generic formal fiber is Cohen-Macaulay for all p with dim(R/p)=d.

R/p is unmixed and has Cohen-Macaulay generic formal fiber for all p with dim(R/p) > dim(R/a(M)).

Applications include insights into the structure of local rings and the dimension and closedness of non Cohen-Macaulay loci.

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 $\mathfrak{a}_i(M):={\rm Ann}_RH^i_{\mathfrak{m}}(M)$ for $i\geq 0$. In this paper, we study the Cohen-Macaulayness of formal fibers of $R$ in the relation with the dimension ${\rm dim} (R/\mathfrak{a}(M)).$ We prove that ${\rm dim} (R/\mathfrak{a}(M))<d$ if and only if $R/\mathfrak{p}$ is unmixed and the generic formal fiber of $R/\mathfrak{p}$ is Cohen-Macaulay for all $\mathfrak{p}\in{\rm Supp}_R(M)$ with ${\rm dim} (R/\mathfrak{p})=d.$ In general, $R/\mathfrak{p}$ is unmixed and the generic formal fiber of $R/\mathfrak{p}$ is Cohen-Macaulay for all $\mathfrak{p}\in{\rm Supp}_R(M)$ with ${\rm dim} (R/\mathfrak{p})>{\rm dim} (R/\mathfrak{a}(M)).$ As applications, we explore the structure of local rings and the dimension, the closedness of non Cohen-Macaulay locus of finitely generated modules.