On the generalization of Faltings’ Annihilator Theorem

Abstract

Let R be a commutative Noetherian ring and let n be a non-negative integer. In this article, by using the theory of Gorenstein dimensions, it is shown that whenever R is a homomorphic image of a Noetherian Gorenstein ring, then the in-M )) ≥ n for all t ∈ N 0 } and inf{λ bRp aRp (M p )| p ∈ Spec R and dim R/p ≥ n} are equal, for every finitely generated R-module M and for every ideals a, b of R with b ⊆ a. This generalizes the Faltings' Annihilator Theorem [G. Faltings,Über die Annulatoren lokaler Kohomologiegruppen, Arch. Math. 30 (1978) 473-476].

Faltings' annihilator theorem and almost Cohen-Macaulay rings

“…In 2008, Kawasaki proved a general theorem [13,Theorem 1.1], which implies that Faltings' annihilator theorem holds over a homomorphic image of a Cohen-Macaulay ring. Recently, Doustimehr and Naghipour [11] In this paper, we investigate the relationship between f b a (M ) n and λ b a (M ) n over an almost Cohen-Macaulay ring by using the results about large restricted flat dimension given in [1]. The main result is the following theorem; recall that the Cohen-Macaulay defect of R is defined by cmd R := sup{ht p − depth R p | p ∈ Spec R}, and that R is called almost Cohen-Macaulay if cmd R 1.…”

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

“…In [8], the author and Naghipour defined the nth b-finiteness dimension f b a (M) n of M relative to a by In section 3, we establish a generalization of Faltings' Annihilator Theorem for the finiteness dimensions. More precisely, as a second main result, we prove the following.…”

“…q ) γ. As dim R/q n, it therefore follows from[8, Theorem 2.10] and the definition ofλ b a (M) n that f b a (M) n = λ b a (M) n λ bRq aRq (M q ) γ. Now, we prove the converse inequality.…”

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

“…The result in Theorem 1.3 is proved in Theorem 3.1. Our method is based on the notion of the nth b-minimum a-adjusted depth of M (see [12])…”