Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2018; 55(1): 311-317

Online first article September 8, 2017      Printed January 1, 2018

https://doi.org/10.4134/BKMS.b161017

Copyright © The Korean Mathematical Society.

On the cohomological dimension of finitely generated modules

Kamal Bahmanpour, Masoud Seidali Samani

Institute for Research in Fundamental Sciences (IPM), University of Mohaghegh Ardabili

Abstract

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring and $I$ be an ideal of $R$. In this paper it is shown that if ${\rm cd}(I,R)=t>0$ and the $R$-module $Hom_R(R/I, H^t_I(R))$ is finitely generated, then
\begin{align}t={\rm sup}\,\{\dim \widehat{\widehat{R}_{\mathfrak P}}/\mathfrak Q:~&\mathfrak P\in V(I\widehat{R}),\,\,\mathfrak Q\in \rm{mAss}_{\widehat{\widehat{R}_{\mathfrak P}}}\widehat{\widehat{R}_{\mathfrak P}}\,\, \rm{ and}\\ &\mathfrak P\widehat{\widehat{R}_{\mathfrak P}}=\rm{Rad}(I\widehat{\widehat{R}_{\mathfrak P}}+\mathfrak Q)\}.\end{align}
Moreover, some other results concerning the cohomological dimension of ideals with respect to the rings extension $R\subset R[X]$ will be included.

Keywords: attached prime, cofinite module, cohomological dimension, local cohomology, Noetherian ring

MSC numbers: 13D45, 14B15, 13E05