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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Value functions and associated graded rings for semisimple algebras


Authors: J.-P. Tignol and A. R. Wadsworth
Journal: Trans. Amer. Math. Soc. 362 (2010), 687-726
MSC (2000): Primary 16W60; Secondary 16K20, 16W70
Published electronically: September 18, 2009
MathSciNet review: 2551503
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Abstract: We introduce a type of value function $ y$ called a gauge on a finite-dimensional semisimple algebra $ A$ over a field $ F$ with valuation $ v$. The filtration on $ A$ induced by $ y$ yields an associated graded ring $ \textsl{gr}_y(A)$ which is a graded algebra over the graded field $ \textsl{gr}_v(F)$. Key requirements for $ y$ to be a gauge are that $ \textsl{gr}_y(A)$ be graded semisimple and that $ \dim_{\textsl{gr}_v(F)}(\textsl{gr}_y(A)) = \dim_F(A)$. It is shown that gauges behave well with respect to scalar extensions and tensor products. When $ v$ is Henselian and $ A$ is central simple over $ F$, it is shown that $ \textsl{gr}_y(A)$ is simple and graded Brauer equivalent to $ \textsl{gr}_w(D)$, where $ D$ is the division algebra Brauer equivalent to $ A$ and $ w$ is the valuation on $ D$ extending $ v$ on $ F$. The utility of having a good notion of value function for central simple algebras, not just division algebras, and with good functorial properties, is demonstrated by giving new and greatly simplified proofs of some difficult earlier results on valued division algebras.


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Additional Information

J.-P. Tignol
Affiliation: Institut de Mathématique Pure et Appliquée, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium
Email: jean-pierre.tignol@uclouvain.be

A. R. Wadsworth
Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112
Email: arwadsworth@ucsd.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04681-9
PII: S 0002-9947(09)04681-9
Received by editor(s): May 2, 2007
Received by editor(s) in revised form: November 6, 2007
Published electronically: September 18, 2009
Additional Notes: The first author was partially supported by the National Fund for Scientific Research (Belgium) and by the European Community under contract HPRN-CT-2002-00287, KTAGS. The second author would like to thank the first author and UCL for their hospitality while the work for this paper was carried out.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.