Some results on local cohomology modules

Let $R$ be a commutative Noetherian ring, $\text{fa}$ an ideal of $R$, and let $M$ be a finitely generated $R$-module. For a non-negative integer $t$, we prove that $H_{\text{fa}}^t(M)$ is $\text{fa}$-cofinite whenever $H_{\text{fa}}^t(M)$ is Artinian and $H_{\text{fa}}^i(M)$ is $\text{fa}$-cofinite for all $i<t$. This result, in particular, characterizes the $\text{fa}$-cofiniteness property of local cohomology modules of certain regular local rings. Also, we show that for a local ring $(R,\text{fm})$, $f-\text{depth}(\text{fa},M)$ is the least integer $i$ such that $H_{\fa}^i(M)\ncong H_{\fm}^i(M)$. This result in conjunction with the first one, yields some interesting consequences. Finally, we extend the non- vanishing Grothendieck’s Theorem to $\text{fa}$-cofinite modules.