Cominimaxness of certain general local cohomology modules

Abstract

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

Lower bounds of certain general local cohomology modules

“…It is easy to see that the above definition of system of ideals and general local cohomology modules is equivalent to the [12, Definition 2.1.10 and Notation 2.2.2]. General local cohomology modules was studied by several authors in [10,11,15,4,1,2,12].…”