Let R be a commutative Noetherian ring, a system of ideals of R and I 2 . In this paper among other things we prove that if M is finitely generated and t 2 N such that the R-module Hi (M) is FD≤1 (or weakly Laskerian) for all i < t, then Hi (M) is -cofinite for all i < t and for any FD≤0 (or minimax) submodule N of Ht (M), the R-modules HomR(R/I, Ht (M)/N) and Ext1 R(R/I, Ht (M)/N) are finitely generated. Also it is shown that if cd I = 1 or dimM/IM 1 (e.g., dimR/I 1) for all I 2 , then the local cohomology module Hi (M) is -cofinite for all i 0. These generalize the main results of Aghapournahr and Bahmanpour [2], Bahmanpour and Naghipour [6, 7]. Also we study cominimaxness and weakly cofiniteness of local cohomology modules with respect to a system of ideals. 1.
