On the connectedness of the spectrum of forcing algebras

TL;DR

This paper investigates the connectedness of forcing algebra spectra over noetherian rings, providing geometric criteria and characterizations, especially for specific base rings, and explores the property’s local nature.

Contribution

It introduces a geometric criterion for connectedness of forcing algebra spectra over integral rings and relates it to the integral closure of ideals.

Findings

Connectedness criterion for forcing algebra spectra over integral rings

Simplifications for local, one-dimensional, or factorial base rings

Characterization of integral closure via universal connectedness

Abstract

We study the connectedness property of the spectrum of forcing algebras over a noetherian ring. In particular we present for an integral base ring a geometric criterion for connectedness in terms of horizontal and vertical components of the forcing algebra. This criterion allows further simplifications when the base ring is local, or one-dimensional, or factorial. Besides, we discuss whether the connectedness is a local property. Finally, we present a characterization of the integral closure of an ideal by means of the universal connectedness of the forcing algebra