On Two Dimensional Semi-Local Noetherian Spectra

TL;DR

This paper introduces a technique to decompose certain two-dimensional partially ordered sets, providing partial answers to a conjecture about the structure of semi-local Noetherian spectra, and demonstrating many such posets are realizable as spectra of Noetherian rings.

Contribution

The paper develops a novel splitting technique for posets and applies it to show many two-dimensional semi-local Noetherian spectra are realizable as ring spectra.

Findings

Many two-dimensional posets with finitely many height two nodes are spectra of Noetherian rings.

The technique simplifies the structure of complex posets to analyze their realizability.

Partial progress on a conjecture by Wiegand and Wiegand regarding Noetherian spectra.

Abstract

In this article, we develop a technique to "split" certain types of partially ordered sets into simpler ones and use that technique to give a partial answer to a conjecture by R. Wiegand and S. Wiegand on the structure of semi-local, two-dimensional Noetherian spectra in their exposition on Noetherian prime ideals. Specifically, we show that very many two-dimensional posets with finitely many height two nodes that satisfy the necessary conditions of being the spectrum of a Noetherian ring, along with a very mild cardinality assumption, are in fact the spectrum of a Noetherian ring.