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 H-a(i) (M, N) is weakly Laskerian for all i < r. Then we prove that Hom(R)(R/a, H-a(r)(M, N)) is also weakly Laskerian and so Ass(R),(H-a(r) (M, N)) is finite. Moreover, we show that if 8 is a non-negative integer such that Ext(R)(j)(M, H-a(i) (N)) is weakly Laskerian for all i, j >= 0 with i <= s, then Ext(R)(j)(R/a, H-a(i) (M, N)) 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 H-a(n) (N) is weakly Laskerian?