On the weakly cofiniteness of local cohomology modules

Let R be a commutative Noetherian ring, a be anideal of R and M be an R-module. In this article, we investigatethe cofiniteness and weakly cofiniteness of local cohomology andgeneralized local cohomology modules. Among other things, weprove that if ExtiR(R/a, M) is weakly Laskerian for all i ≤ dim M,then the R-module ExtiR(N, M) is weakly Laskerian for all i ≥ 0and for any finitely generated R-module N with SuppR(N) ⊆ V (a)and dim N ≤ 1. As a consequence, we show that if M is an Rmodule such that Hia(M) is a-weakly cofinite for all i ≥ 0, thenthe R-module Hia(N, M) is weakly Laskerian for all i ≥ 0 and forany finitely generated R-module N with SuppR(N) ⊆ V (a) anddim N ≤ 1.