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Value functions and Dubrovin valuation rings on simple algebras

Authors: Mauricio A. Ferreira and Adrian R. Wadsworth
Journal: Trans. Amer. Math. Soc. 368 (2016), 1681-1734
MSC (2010): Primary 16W60; Secondary 16K20, 16W70, 16L30
Published electronically: June 17, 2015
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Abstract: In this paper we prove relationships between two generalizations of commutative valuation theory for noncommutative central simple algebras: (1) Dubrovin valuation rings; and (2) the value functions called gauges introduced by Tignol and Wadsworth. We show that if $ v$ is a valuation on a field $ F$ with associated valuation ring $ V$ and $ v$ is defectless in a central simple $ F$-algebra $ A$, and $ C$ is a subring of $ A$, then the following are equivalent: (a) $ C$  is the gauge ring of some minimal $ v$-gauge on $ A$, i.e., a gauge with the minimal number of simple components of $ C/J(C)$; (b) $ C$ is integral over $ V$ with $ C = B_1 \cap \ldots \cap B_\xi $ where each $ B_i$ is a Dubrovin valuation ring of $ A$ with center $ V$, and the $ B_i$ satisfy Gräter's Intersection Property. Along the way we prove the existence of minimal gauges whenever possible and we show how gauges on simple algebras are built from gauges on central simple algebras.

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Mauricio A. Ferreira
Affiliation: Departamento de Ciências Exatas, Universidade Estadual de Feira de Santana, Avenida Transnordestina, S/N, Novo Horizonte, Feira de Santana, Bahia 44036-900 Brazil

Adrian R. Wadsworth
Affiliation: Department of Mathematics 0112, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112

Received by editor(s): July 20, 2013
Received by editor(s) in revised form: December 31, 2013
Published electronically: June 17, 2015
Additional Notes: Some of the results in this paper are based on the first author’s doctoral dissertation written under the supervision of the second author and Antonio José Engler. The first author was partially supported by FAPESP, Brazil (Grant 06/00157-3) during his graduate studies. This author would like to thank the second author and UCSD for their hospitality during his visit in 2009.
Some of the research for this paper was carried out during the second author’s visit to the University of Campinas, Brazil, during July-August, 2010, which was made possible by a grant from FAPESP, Brazil. The second author would like to thank Professor Antonio José Engler and the first author for their hospitality during that visit.
Article copyright: © Copyright 2015 American Mathematical Society