On Dubrovin valuation rings in crossed product algebras

Abstract

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