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Let $(R,𝔪)$ be a commutative Noetherian regular local ring of dimension $d$ and $I$ be a proper ideal of $R$ such that ${\mathrm{mAss}}_R(R/I)={\mathrm{Assh}}_R\left(I\right)$. It is shown that the $R$-module $H_I^{\mathrm{ht}\left(I\right)}\left(R\right)$ is $I$-cofinite if and only if $\mathrm{cd}(I,R)=\mathrm{ht}\left(I\right)$. Also we present a sufficient condition under which this condition the $R$-module $H_I^i\left(R\right)$ is finitely generated if and only if it vanishes.

Let R be a commutative Noetherian ring, I a proper ideal of R, and M be a finitely generated R-module. We provide bounds for the cohomological dimension of the R-module M with respect to the ideal I in several cases.

Let $(R,𝔪)$ be a complete Noetherian local ring, $I$ an ideal of $R$ and $M$ a nonzero Artinian $R$-module. In this paper it is shown that if $𝔭$ is a prime ideal of $R$ such that $dimR/𝔭=1$ and $\left(0{:}_M𝔭\right)$ is not finitely generated and for each $i\ge 2$ the $R$-module ${\mathrm{Ext}}_R^i(M,R/𝔭)$ is of finite length, then the $R$-module ${\mathrm{Ext}}_R^1(M,R/𝔭)$ is not of finite length. Using this result, it is shown that for all finitely generated $R$-modules $N$ with $Supp\left(N\right)\subseteq V\left(I\right)$ and for all integers $i\ge 0$, the $R$-modules ${\mathrm{Ext}}_R^i(N,M)$ are of finite length, if and only if, for all finitely generated $R$-modules $N$ with $Supp\left(N\right)\subseteq V\left(I\right)$ and...