Maximal covers of chains of prime ideals

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.