Let R be a commutative Noetherian ring and I an ideal of R. Let t ∈ N 0 be an integer, M a finitely generated R-module and X be an R-module such that ExtiR ( R/I, X ) is finitely generated (resp. minimax, weakly Laskerian) for all i ≤ t + 1. We prove that if HiI ( M, X ) is FD≤1 for all i < t, then the R-modules HiI ( M, X) are I-cofinite (resp. I-cominimax, I-weakly cofinite) for all i < t and for any FD ≤0 (or minimax) submodule N of HtI ( M, X), the R-module ExtiR ( R/I, HtI ( M, X ) /N ) is finitely generated (resp. minimax, weakly Laske- rian) for i = 0, 1. In particular the set Ass R ( HtI ( M, X ) /N ) is a finite set.
