Value functions and Dubrovin valuation rings on simple algebras

Abstract:

In this paper we prove relationships between two generalizations of commutative valuation theory for noncommutative central simple algebras: (1) Dubrovin valuation rings; and (2) the value functions called gauges introduced by Tignol and Wadsworth. We show that if $v$ is a valuation on a field $F$ with associated valuation ring $V$ and $v$ is defectless in a central simple $F$-algebra $A$, and $C$ is a subring of $A$, then the following are equivalent: (a) $C$ is the gauge ring of some minimal $v$-gauge on $A$, i.e., a gauge with the minimal number of simple components of $C/J(C)$; (b) $C$ is integral over $V$ with $C = B_1 \cap \ldots \cap B_\xi$ where each $B_i$ is a Dubrovin valuation ring of $A$ with center $V$, and the $B_i$ satisfy Gräter’s Intersection Property. Along the way we prove the existence of minimal gauges whenever possible and we show how gauges on simple algebras are built from gauges on central simple algebras.