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