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.