Dimension-Preserving Saturated Embeddings of Finite Posets into the Spectra of Noetherian UFDs

TL;DR

This paper characterizes when finite posets can be embedded into the spectra of local Noetherian UFDs with preserved dimension and saturated chains, expanding understanding of poset embeddings into algebraic spectra.

Contribution

It provides necessary and sufficient conditions for embedding finite posets into spectra of local Noetherian UFDs with preserved dimension and chain structure.

Findings

Many finite posets can be embedded into spectra of local Noetherian UFDs.

Existence of semi-local quasi-excellent rings with saturated embeddings of finite posets.

Embeddings preserve minimal elements and chain lengths in the spectrum.

Abstract

Given a finite poset $X$, we find necessary and sufficient conditions for there to exist a local Noetherian UFD $A$ and a saturated embedding of posets $\phi : X \longrightarrow \mbox{Spec}(A)$ such that $\dim(X)=\dim(A)$. The conditions imposed on $X$ in our characterization are remarkably mild, demonstrating that there is a large class of finite posets that can be embedded into the spectrum of a local Noetherian UFD of the same dimension as $X$ in a way that preserves saturated chains. We also show that given any finite poset $Y$, there exists a semi-local quasi-excellent ring $S$ and a saturated embedding $\psi: Y \longrightarrow \mbox{Spec}(S)$ such that if $z$ is a minimal element of $Y$, then $\psi(z)$ is a minimal prime ideal of $S$ and the coheight of $\psi(z)$ is the same as the length of the longest chain in $Y$ that starts at $z$ and ends at a maximal element of $Y$.