Every Finite Poset is Isomorphic to a Saturated Subset of the Spectrum of a Noetherian UFD

TL;DR

This paper demonstrates that any finite partially ordered set can be represented as a saturated subset within the spectrum of certain well-behaved algebraic domains, specifically Noetherian UFDs and quasi-excellent domains.

Contribution

It establishes a universal representation of finite posets in algebraic geometry spectra, extending known correspondences to broader classes of domains.

Findings

Every finite poset is isomorphic to a saturated subset of the spectrum of a Noetherian UFD.

Every finite poset is isomorphic to a saturated subset of the spectrum of a quasi-excellent domain.

Abstract

We show that every finite poset is isomorphic to a saturated subset of the spectrum of a Noetherian unique factorization domain. In addition, we show that every finite poset is isomorphic to a saturated subset of the spectrum of a quasi-excellent domain.