On Dubrovin Valuation Rings in Crossed Product Algebras

Abstract

Let $F$ be a field and let $V$ be a valuation ring in $F$. If $A$ is a central simple $F$-algebra then $V$ can be extended to a Dubrovin valuation ring in $A$. In this paper we consider the structure of Dubrovin valuation rings with center $V$ in crossed product algebras $(K/F,G,f)$ where $K/F$ is a finite Galois extension with Galois group $G$ unramified over $V$ and $f$ is a normalized two-cocycle. In the case where $V$ is indecomposed in $K$ we introduce a family of orders naturally associated to $f$, examine their basic properties, and determine which of these orders is Dubrovin. In the case where $V$ is decomposed we determine the structure in the case of certain special discrete, finite rank valuations.