An extension of a theorem of Hartshorne

TL;DR

This paper extends Hartshorne's classical theorem on the connectedness of the punctured spectrum of a local ring by examining the homology of a related simplicial complex.

Contribution

It introduces a novel approach linking homology groups of simplicial complexes to the connectedness properties of local rings.

Findings

Connectedness characterized by homology groups

New criteria for punctured spectrum connectedness

Extended theorem applicable to broader classes of rings

Abstract

We extend a classical theorem of Hartshorne concerning the connectedness of the punctured spectrum of a local ring by analyzing the homology groups of a simplicial complex associated with the minimal primes of a local ring.