Faltings' Local-global Principle and Annihilator Theorem for the finiteness dimensions

TL;DR

This paper extends Faltings' Local-global Principle for local cohomology modules, establishing new equivalences and invariants related to the finiteness and structure of these modules over Noetherian rings, especially Gorenstein images.

Contribution

It generalizes Faltings' principle and introduces new invariants that relate local and global properties of local cohomology modules over certain rings.

Findings

Equivalence of local and global finiteness conditions for local cohomology modules.

Equality of specific invariants related to support and depth of modules.

Determination of the least degree where local cohomology ceases to be minimax or weakly Laskerian.

Abstract

Let $R$ be a commutative Noetherian ring, $M$ a finitely generated $R$-module and $n$ be a non-negative integer. In this article, it is shown that there is a finitely generated submodule $N_i$ of $H_{\frak a}^i(M)$ such that $\dim{\rm Supp } H_{\frak a}^i(M)/N_i<n$ for all $i<t$ if and only if there is a finitely generated submodule $N_{i,{\frak p}}$ of $H_{{\frak a} R_{\frak p}}^i(M_{\frak p})$ such that $\dim{\rm Supp } H_{{\frak a} R_{\frak p}}^i(M_{\frak p})/N_{i,{\frak p}}<n$ for all $i<t$. This generalizes Faltings' Local-global Principle for the finiteness of local cohomology modules (Faltings' in Math. Ann. 255:45-56, 1981). Also, it is shown that whenever $R$ is a homomorphic image of a Gorenstein local ring, then the invariants $\inf\{i\in\mathbb N_0\mid\dim{\rm Supp}({\frak b}^tH_{\frak a}^i(M))\geq n\text{ for all } t\in\mathbb N_0\}$ and $\inf\{{\rm depth } M_{\frak p}+{\rm ht}({\frak a}+{\frak p})/{\frak p}\mid{\frak p}\in{\rm Spec } R\setminus V({\frak b}) \text{ and } \dim R/({\frak a}+{\frak p})\geqslant n\}$ are equal, for every finitely generated $R$-module $M$ and for all ideals $\frak a, \frak b$ of $R$ with ${\frak b}\subseteq {\frak a}$. As a consequence, we determine the least integer $i$ where the local cohomology module $H_{\frak a}^i(M)$ is not minimax (resp. weakly laskerian).