On Weakly Laskerian and Weakly Cofinite Modules

Abstract

Let R be a commutative Noetherian ring with non-zero identity, 𝔞 an ideal of R and M a finitely generated R-module. We assume that N is a weakly Laskerian R-module and r is a non-negative integer such that the generalized local cohomology module [Formula: see text] is weakly Laskerian for all i < r. Then we prove that [Formula: see text] is also weakly Laskerian and so [Formula: see text] is finite. Moreover, we show that if s is a non-negative integer such that [Formula: see text] is weakly Laskerian for all i, j ≥ 0 with i ≤ s, then [Formula: see text] is weakly Laskerian for all i ≤ s and j ≥ 0. Also, over a Gorenstein local ring R of finite Krull dimension, we study the question when the socle of [Formula: see text] is weakly Laskerian?