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

Abstract

Let $(R,\mathfrak{m})$ 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 $f_I(R)=\max \{1,\mathrm{ht}I\}$ and if $\mathfrak{m}{\mathrm{Ass}}_R(R/I)$ is cotained in Ass${}_R(R)$, then the ring $R/I+{\cup }_{n\ge 1}(0{:}_R{I}^n)$ is equidimensional of dimension $\dim R-1$. Moreover, we will obtain a lower bound for injective dimension of the local cohomology module $H_I^{\mathrm{ht}I}(R)$, in the case $(R,\mathfrak{m})$ is a complete equidimensional local ring.

arXiv comment11 pages, To appear in Canadian Mathematical Bulletin