Quasi-valuations and algebras over valuation domains

TL;DR

This paper investigates properties of algebras over valuation domains, establishing conditions for various prime ideal extension properties and showing a correspondence between chains of prime ideals in certain algebraic structures.

Contribution

It provides new necessary and sufficient conditions for prime ideal extension properties in algebras over valuation domains, including the construction of a filter-based ext{qv} ring with specific properties.

Findings

R satisfies SGB over O_v.

Conditions for R to satisfy LO over O_v.

R satisfies GD if torsion-free over O_v.

Abstract

Suppose $F$ is a field with valuation $v$ and valuation domain $O_{v}$, and $R$ is an $O_{v}-$algebra. We prove that $R$ satisfies SGB (strong going between) over $O_{v}$. We give a necessary and sufficient condition for $R$ to satisfy LO (lying over) over $O_{v}$. Using the filter \qv constructed in [Sa1], we show that if $R$ is torsion-free over $O_{v}$ then $R$ satisfies GD (going down) over $O_{v}$. In particular, if $R$ is torsion-free and $(R^{\times} \cap O_{v}) \subseteq O_{v}^{\times}$, then for any chain in $\text{Spec}(O_v)$ there exists a chain in $\text{Spec}(R)$ covering it. Assuming $R$ is torsion-free over $O_{v}$ and $[R \otimes_{O_{v}}F:F]< \infty$, we prove that $R$ satisfies INC (incomparabilty) over $O_{v}$. Assuming in addition that $(R^{\times} \cap O_{v}) \subseteq O_{v}^{\times}$, we deduce that $R$ and $O_{v}$ have the same Krull dimension and a bound on the size of the prime spectrum of $R$ is given. Under certain assumptions on $R$ and a \qv defined on it, we prove that the \qv ring satisfies GU (going up) over $O_{v}$. Combining these five properties together, we deduce that any maximal chain of prime ideals of the \qv ring is lying over $\text{Spec}(O_{v})$, in a one-to-one correspondence.