On the generalization of Faltings' Annihilator Theorem

TL;DR

This paper extends Faltings' Annihilator Theorem to a broader class of rings using Gorenstein dimensions, establishing equality of certain invariants related to local cohomology and module support.

Contribution

It generalizes Faltings' Annihilator Theorem to rings that are homomorphic images of Gorenstein rings, using Gorenstein dimensions.

Findings

Equality of invariants for local cohomology supports

Extension of Faltings' Theorem to new ring classes

Use of Gorenstein dimensions in the proof

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 $\inf\{\lambda_{\fa R_{\p}}^{\fb R_{\p}}(M_{\p})|\,\p\in {\rm Spec} \, R \text{and} \dim R/ \p\geq n\}$ are equal, for every finitely generated $R$-module $M$ and for every ideals $\frak a, \frak b$ of $R$ with $\frak b\subseteq \frak a$. This generalizes the Faltings' Annihilator Theorem [G. Faltings, {\it \"Uber die Annulatoren lokaler Kohomologiegruppen}, Arch. Math. {\bf30} (1978) 473-476].