Some results on the local cohomology of minimax modules

Cominimaxness of local cohomology modules

“…The following lemma is needed in the proof of the next theorem. We are now ready to state and prove the main results (Theorem 2.7 and the Corollaries 2.8 and 2.10) which are extensions of Bahmanpour-Naghipour's results in [7] and [8] in terms of minimax modules, [1], Corollary 2.3, [2], Theorem 3.4 and Corollaries 3.5 and 3.6, [24], Corollary 2.3, and Hong Quy's result in [32].…”

“…Recently many authors have studied the minimaxness and cominimaxness of local cohomology modules and answered Conjecture 1.1 and Question 1.2 in the class of minimax modules in some cases (see [1], [7], [20], [22] [24], [26]). The purpose of this note is to make a suitable generalization of Conjecture 1.1 and Question 1.2 in terms of minimax modules instead of finitely generated modules.…”

Cominimaxness of certain general local cohomology modules

“…F D 1 (R, I) ethcom ) is an Abelian category.Proof. We prove theorem for the W L (R, I) ethcom case and by using the same proof, the F D1 (R, I) ethcom case follows. Let M, N ∈ W L (R, I) ethcom and let f : M −→ N be an R-homomorphism.…”

“…≤1 R-module and the assertion follows by Corollary 2.11 (i). (ii) Proof is the same as 2.11 (ii).One of the main results of this section is to prove that for an arbitrary ideal I of a Noetherian ring R, the category of W L (R, I) ethcom and F D1 (R, I) ethcom modules are Abelian category. Let I be an ideal of a Noetherian ring R. Let W L (R, I) ethcom (resp.…”

“…The general local cohomology module H j Φ (M ) is defined to be Φ-cominimax if there exists an ideal I ∈ Φ such that Ext i R (R/I, H j Φ (M )) is minimax, for all i, j ≥ 0. Recently many authors studied the minimaxness and cominimaxness of local cohomology modules and answered the Conjecture 1.1 and Question 1.2 in the class of minimax modules in some cases (see [1,3,9,14,24,27,29]). The purpose of this note is to make a suitable generalization of Conjecture 1.1 and Question 1.2 in terms of minimax modules instead of finitely generated modules for general local cohomology modules.…”