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# Faltings' Finiteness Dimension of Local Cohomology Modules Over Local Cohen-Macaulay Rings

Published:2017-02-16
Printed: Jun 2017
• Kamal Bahmanpour,
Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran
• Reza Naghipour,
Department of Mathematics, University of Tabriz, Tabriz, Iran
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## 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 $f_I(R)=\operatorname{max}\{1, \operatorname{ht} I\}$ and if $\operatorname{mAss}_R(R/I)$ is contained in $\operatorname{Ass}_R(R)$, 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 $H^{\operatorname{ht} I}_I(R)$, in the case $(R, \frak m)$ is a complete equidimensional local ring.
 Keywords: Cohen Macaulay ring, equidimensional ring, finiteness dimension, local cohomology