A non-vanishing theorem for local cohomology modules

Assume that (R,m) is a local Noetherian ring and a is an ideal of R. In this paper we introduce a new class of R-modules denoted by weakly finite modules that is a generalization of finitely generated modules and containing the class of Big Cohen-Macaulay modules and a-cofinite modules. We improve the non-vanishing theorem due to Grothendieck for weakly finite modules. Finally we define the notion depthR M and we prove that if M is a weakly finite R-module and Hi m(M) 6= 0 for some i, then depthR (M) ? i ? dimM