ON DUBROVIN VALUATION RINGS IN CROSSED PRODUCT ALGEBRAS

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.