On Two Dimensional Semi-Local 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.