Some results on local cohomology modules

Abstract.

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