On the cohomological dimension of finitely generated modules
Bull. Korean Math. Soc. 2018 Vol. 55, No. 1, 311-317
Published online January 1, 2018
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
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