Let R be a commutative Noetherian ring, Φ a system of ideals of R and I ∈ Φ. Let t ∈ N0 be an integer and M an R-module such that ExtiR (R/I, M ) is minimax for all i ≤ t+1. We prove that if the R-module Hi Φ (M ) is FD≤1 (or weakly Laskerian) for all i < t, then Hi Φ (M ) is Φ-cominimax for all i < t and for any FD≤0 (or minimax) submodule N of Ht Φ(M ), the R-modules HomR (R/I, Ht Φ(M )/N ) and Ext1 R (R/I, Ht Φ (M )/N ) are minimax. 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-cominimax for all i and j whenever M is minimax and Hi Φ(M ) is FD≤1 (or weakly Laskerian) for all i.
