Faltings' local-global principle for the finiteness of local cohomology modules

TL;DR

This paper generalizes Faltings' local-global principle by establishing a relationship between local cohomology modules' dimension properties and finiteness dimensions in complete local rings.

Contribution

It extends previous results by showing the equivalence of the least integer where local cohomology fails to be in a certain dimension and the finiteness dimension of modules.

Findings

Equivalence of local cohomology dimension and finiteness dimension

Generalization of Quy and Brodmann-Lashgari's main result

Applicable to complete local rings and finitely generated modules

Abstract

Let (R,m) be a complete local ring, a an ideal of R and M a finitely generated R-module. The aim of this paper is to show that for any non-negative integer n, the least integer i such that the i-th local cohomology with respect to a is not in dimension <n, is equal to the n-th finiteness dimension of M relative to a. This generalizes the main result of Quy and Brodmann-Lashgari.