Let R be a commutative Noetherian ring, Φ a system of ideals of R and I ∈ Φ. Let M be an R-module (not necessary I-torsion) such that dim M ≤ 1, then the R- module ExtiR (R/I, M ) is weakly Laskerian, for all i ≥ 0, if and only if the R-module ExtiR (R/I, M ) is weakly Laskerian for i = 0, 1. Let t ∈ N0 be an integer and M an R-module such that ExtiR (R/I, M ) is weakly Laskerian for all i ≤ t + 1. We prove that if the R-module HΦ i (M ) is FD≤1 for all i < t, then Hi Φ (M ) is Φ-weakly cofinite for all i < t, and for any FD≤0 (or minimax) submodule N of HΦ t (M ), the R-modules HomR (R/I, Ht Φ (M )/N ) and Ext1 R (R/I, Ht Φ (M )/N ) are weakly Laskerian. Let N be a finitely generated R-module. We also prove that ExtjR (N, Hi Φ (M )) and TorR j (N, HΦ i (M )) are Φ-weakly cofinite for all i and j whenever M is weakly Laskerian and HΦ i (M ) is FD≤1 for all i. Similar results are true for ordinary local cohomology modules and local cohomology modules defined by a pair of ideals.
