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 quasi-valuation constructed in [Sarussi, S. (2012). Quasi-valuations extending a valuation. J. Algebra. 372:318–364], 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 quasi-valuation defined on it, we prove that the quasi-valuation ring satisfies GU (going up) over O_v. Combining these five properties together, we deduce that any maximal chain of prime ideals of the quasi-valuation ring is lying over $\text{Spec}(O_v)$, in a one-to-one correspondence.

Keywords:

• Quasi-valuations
• going down
• going up
• lying over

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W60
• 13A18
• 13F30