Cofiniteness with Respect to Ideals of Small Dimensions

Melkersson condition for extension of Serre subcategories

“…The previous argument implies that Ext i R (R/a, F ) is finitely generated for i = 0, 1 and Supp F ⊆ V (a). Therefore, it follows from [BNS,Lemma 2.2] that F is a-cofinite and consequently Γ a (M ) is a-cofinite.…”

“…R (R/a, F ); and hence it is finitely generated. Now, it follows from [BNS,Theorem 2.5 (i)] that H i aS (F ) is aS-cofinite for all i ≥ 0 and so by virtue of [DM,Proposition 2], H i a (F ) is a-cofinite for all i ≥ 0. Thus Hom R (R/a, H 1 a (F )) is finitely generated and since Supp H 1 a (F ) ⊆ V (m), the module Hom R (R/a, H 1 a (F )) has finite length and the fact that R/m ∈ S forces Hom R (R/a, H 1 a (F )) ∈ S; and hence H 1 a (F ) ∈ S as S satisfies C a condition.…”

“…There is an exact sequence 0 −→ N −→ M −→ F −→ 0 such that N is finitely generated and F has finite support; and hence dim F ≤ 1. The assumption implies that Ext i R (R/a, F ) is finitely generated for i = 0, 1 and hence it follows from [BNS,Theorem 2.5(i)] that Ext i R (R/a, F ) is finitely generated for all i ≥ 0. It now follows from [BNS,Theorem 2.5(ii)] that H i a (M ) is a-cofinite for all i ≥ 0.…”

A criterion for cofiniteness of modules

“…Grothendieck [4], exposé 13, 1.2 conjectured that if M is a finitely generated A-module, then Hom A (A/a, H i a (M )) is finitely generated, where H i a (M ) is the i-th local cohomology of M with respect to the ideal a. The concept of cofiniteness of modules was defined for the first time by Hartshorne [5], giving a negative answer to the Grothendieck's conjecture and later was studied by the numerous authors [1,3,6,7,8,9,10].…”

Cofiniteness of local cohomology modules over Noetherian local rings

“…In particular, if dim R/I = 1 and Supp M ⊆ V(I) then M is I-cofinite if (and only if) Hom R (R/I, M) and Ext 1 R (R/I, M) are finitely generated. Also, in [4], Bahmanpour et al as a generalization of [18,Theorem 2.3]…”