Maximal covers of chains of prime ideals

TL;DR

This paper investigates the relationships between chains of prime ideals in rings connected by a homomorphism, establishing conditions under which maximal chains in the target ring correspond to those in the source ring.

Contribution

It provides new necessary and sufficient conditions for properties like going down, going up, and strong going between in terms of maximal chains, and characterizes when maximal chains have the same cardinality.

Findings

Characterizes when maximal chains in R cover chains in S.

Provides conditions for properties GD, GU, and SGB in ring homomorphisms.

Shows that satisfying all properties leads to perfect maximal covers.

Abstract

Suppose $f:S \rightarrow R$ is a ring homomorphism such that $f[S] $ is contained in the center of $R$. We study the connections between chains in $\text{Spec} (S)$ and chains in $\text{Spec} (R)$. We focus on the properties LO (lying over), INC (incomparability), GD (going down), GU (going up) and SGB (strong going between). %we define the notion $\mathcal D$-chain which is a chain $\mathcal C \subseteq \text{Spec} (S)$ such that for all $Q \in \mathcal C$, $f^{-1}[Q] \in \mathcal D$. We provide a sufficient condition for every maximal chain in $\text{Spec} (R)$ to cover a maximal chain in $\text{Spec} (S)$. We prove some necessary and sufficient conditions for $f$ to satisfy each of the properties GD, GU and SGB, in terms of maximal $\mathcal D$-chains, where $\mathcal D \subseteq \text{Spec} (S)$ is a nonempty chain. We show that if $f$ satisfies all of the properties above, then every maximal $\mathcal D$-chain is a perfect maximal cover of $\mathcal D$. Our main result is Corollary \ref{equivalent conditions}, in which we give equivalent conditions for the following property: for every chain $\mathcal D \subseteq {\text Spec} (S)$ and for every maximal $\mathcal D$-chain $\mathcal C \subseteq {\text Spec} (R)$, $\mathcal C$ and $\mathcal D$ are of the same cardinality.