On the cofiniteness of local cohomology modules

In this note we show that if I is an ideal of a Noetherian ring R and M is a finitely generated R-module, then for any minimax submodule N of H(I)(t) (M) the R-module Hom(R)(R/I, H(I)(t) (M)/N) is finitely generated, whenever the modules H(I)(0)(M), H(I)(1) (M),..., H(I)(t-1) (M) are minimax. As a consequence, it follows that the associated primes of H(I)(t) (M)/N are finite. This generalizes the main result of Brodmann and Lashgari (2000).