Faltings’ Local-global Principle for the Finiteness of Local Cohomology Modules over Noetherian Rings

Abstract

Let R denote a commutative Noetherian (not necessarily local) ring, an ideal of R and M a finitely generated R-module. The purpose of this article is to show that f n M = inf 0 ≤ i ∈ dim H i M /N ≥ n for any finitely generated submodule N ⊆ H i M , where n is a non-negative integer and the invariant f n M = inf f R M ∈ SuppM/ M and dim R/ ≥ n is the nth finiteness dimension of M relative to . As a consequence, it follows that the set Ass R ⊕ f n M i=0 H i M ∩ ∈ SpecR dim R/ ≥ n is finite. This generalizes the main result of Quy [10], Brodmann and Lashgari [5], and Asadollahi and Naghipour [1].

Faltings’ local–global principle for the in dimension <n of local cohomology modules

“…Therefore, in view of [21,Corollary 2.14], M/(0 : M a) is in dimension < n. Conversely, if M/(0 : M a) is in dimension < n and g : M t −→ aM be the R-epimorphism for which…”

“…is obviously true by [21,Proposition 2.12]. In order to show (ii) =⇒ (i), we proceed by induction on s. If s = 1, then for some integere t ≥ 1, Before we shall state the next result, we have to recall the notion of the b-minimaxness…”

“…then g induces an R-epimorphism g * : (M/(0 : M a)) t −→ aM for which [21,Proposition 2.12]. In order to show (ii) =⇒ (i), we proceed by induction on s. If s = 1, then for some integere t ≥ 1,…”

“…) is also in dimension < n, and so by [21,Lemma 2.6] the set (Ass R Hom R (R/a, H 2 a ( M ))) ≥n is finite and consequently the set (Ass R H 2 a ( M))) ≥n is finite, and so for every non-negative integer t, the set (Ass R b t H 2 a ( M))) ≥n is also finite. Thus for all t ∈ N 0 , the set (Supp b t H 2 a ( M )) ≥n is a closed subset of Spec(R) (in the Zariski topology), and so the descending chain As a consequence of Theorem 2.23, the following corollary shows that the local-global principle for the annihilation of local cohomology modules holds at level 2 over R and at all levels whenever dim R ≤ 3.…”

Faltings’ local–global principle and annihilator theorem for the finiteness dimensions

“…Let i < f n a (M). By [14,Theorem 2.10], H i a (M) is in dimension < n. Hence, there exists a finitely generated submodule N i of H i a (M) such that dim Supp H i a (M)/N i < n. Since N i is finitely generated, there is t ∈ N such that a t N i = 0, and so N i ⊆ (0 : H i a (M ) a t ). It therefore follows from the exact sequence H i a (M)/N i −→ H i a (M)/(0 : H i a (M ) a t ) −→ 0 that dim Supp H i a (M)/(0 : H i a (M ) a t ) < n. Hence, for all prime ideal p with dim R/p n, (a t H i a (M)) p = (a t (0 : H i a (M ) a t )) p = 0.…”