Quasi-valuations and algebras over valuation domains

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.