# Faltings' finiteness dimension of local cohomology modules over local Cohen-Macaulay rings

## TL;DR

This paper studies Faltings' finiteness dimension in local Cohen-Macaulay rings, establishing explicit formulas and conditions for equidimensionality, and providing bounds on the injective dimension of certain local cohomology modules.

## Contribution

It determines the exact value of Faltings' finiteness dimension and conditions for equidimensionality in local Cohen-Macaulay rings, with bounds on local cohomology injective dimension.

## Key findings

- Faltings' finiteness dimension equals max{1, ht I}.
- Certain quotient rings are equidimensional of dimension dim R - 1.
- Provides a lower bound for the injective dimension of local cohomology modules.

## Abstract

Let $(R, \frak m)$ denote a local Cohen-Macaulay ring and $I$ a non-nilpotent ideal of $R$. The purpose of this article is to investigate Faltings' finiteness dimension $f_I(R)$ and equidimensionalness of certain homomorphic image of $R$. As a consequence we deduce that $f_I(R)={\rm max}\{1, {\rm ht}\ I\}$ and if ${\frak m}\mathrm{Ass}_R(R/I)$ is cotained in Ass$_R(R)$, then the ring $R/ I+\cup_{n\geq 1}(0:_RI^n)$ is equidimensional of dimension $\dim R-1$. Moreover, we will obtain a lower bound for injective dimension of the local cohomology module $H^{{\rm ht}\ I}_I(R)$, in the case $(R, \frak m)$ is a complete equidimensional local ring.