Some results on the cofiniteness of local cohomology modules

Let R be a commutative Noetherian ring, a an ideal of R, M an R-module and t a non-negative integer. In this paper we show that the class of minimax modules includes the class of AF modules. The main result is that if the R-module Ext _R ^t (R/a,M) is finite (finitely generated), H _a ⁱ (M) is a-cofinite for all i < t and H _a ^t (M) is minimax then H _a ^t (M) is a-cofinite. As a consequence we show that if M and N are finite R-modules and H _a ⁱ (N) is minimax for all i < t then the set of associated prime ideals of the generalized local cohomology module H _a ^t (M,N) is finite.