Some results on local cohomology modules

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