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 invariants $\inf\{i\in\nat_0|\, {\dim\Supp}(\fb^tH_{\fa}^i(M))\geq n\text{for all} t\in\nat_0\}$ and $\inf\{λ_{\fa R_{\p}}^{\fb R_{\p}}(M_{\p})|\,\p\in {\rm Spec} \, R \text{and} \dim R/ \p\geq n\}$ are equal, for every finitely generated $R$-module $M$ and for every ideals $\mathfrak{a},\mathfrak{b}$ of $R$ with $\mathfrak{b}\subseteq \mathfrak{a}$. This generalizes the Faltings' Annihilator Theorem [G. Faltings, {\it Über die Annulatoren lokaler Kohomologiegruppen}, Arch. Math. {\bf30} (1978) 473-476].

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