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Let R be a commutative Noetherian ring, Φ a system of ideals of R and I ∈ Φ. In this paper among other things we prove that if M is finitely generated and t ∈ N such that the R-module HiΦ(M) is FD≤1 (or weakly Laskerian) for all i < t, then H i Φ(M) is Φ-cofinite for all i < t and for any FD≤0 (or minimax) submodule N of H t Φ(M), the R-modules HomR(R/I,H t Φ(M)/N) and Ext 1 R(R/I,H t Φ(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 ∈ Φ, then the local cohomology module HiΦ(M) is Φ-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.

COFINITENESS OF GENERAL LOCAL COHOMOLOGY MODULES FOR SMALL DIMENSIONS