Betti Numbers and Formal Local Cohomology Modules

TL;DR

This paper explores the connection between Betti numbers and formal local cohomology modules over Noetherian local rings, revealing new links to depth, Artinian properties, and applications to Cohen-Macaulay rings.

Contribution

It establishes a novel relationship between Betti numbers and the structure of formal local cohomology modules, generalizing classical results.

Findings

Betti numbers relate to the Artinian property of formal local cohomology modules

Vanishing of formal local cohomology links to module depth

Applications to Cohen-Macaulay rings are provided

Abstract

This article investigates the relationship between Betti numbers of finitely generated modules over a Noetherian local ring $(R, \mathfrak{m})$ and the structure of formal local cohomology modules. We establish a connection between the vanishing of formal local cohomology and the depth of a module, generalizing classical results. The main theorem links the Betti numbers to the Artinian property of formal local cohomology modules, and we provide applications to Cohen-Macaulay rings. Original proofs are given for all stated results.