Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On Dubrovin valuation rings in crossed product algebras


Authors: Darrell Haile and Patrick Morandi
Journal: Trans. Amer. Math. Soc. 338 (1993), 723-751
MSC: Primary 16H05; Secondary 12G05, 13F30, 16G30, 16W60
MathSciNet review: 1104201
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1104201-X
Article copyright: © Copyright 1993 American Mathematical Society