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

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 $Y$.