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Specializations of Brauer classes over algebraic function fields

Authors: Burton Fein and Murray Schacher
Journal: Trans. Amer. Math. Soc. 352 (2000), 4355-4369
MSC (2000): Primary 12E15, 12G05, 16K50
Published electronically: May 12, 2000
MathSciNet review: 1661250
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Abstract: Let $F$ be either a number field or a field finitely generated of transcendence degree $\ge 1$ over a Hilbertian field of characteristic 0, let $F(t)$ be the rational function field in one variable over $F$, and let ${\alpha }\in \operatorname {Br}(F(t))$. It is known that there exist infinitely many $a\in F$ such that the specialization $t\to a$ induces a specialization ${\alpha }\to \overline {{\alpha }}\in \operatorname {Br}(F)$, where $\overline {{\alpha }}$ has exponent equal to that of ${\alpha }$. Now let $K$ be a finite extension of $F(t)$ and let ${\beta }=\operatorname {res}_{K/F(t)}({\alpha })$. We give sufficient conditions on ${\alpha }$ and $K$ for there to exist infinitely many $a\in F$ such that the specialization $t\to a$has an extension to $K$ inducing a specialization ${\beta }\to \overline {{\beta }}\in \operatorname {Br}(\overline{K})$, $\overline{K}$ the residue field of $K$, where $\overline {{\beta }}$ has exponent equal to that of ${\beta }$. We also give examples to show that, in general, such $a\in F$ need not exist.

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Additional Information

Burton Fein
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331

Murray Schacher
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90024

Keywords: Brauer group, Brauer-Hilbertian, corestriction, Hilbertian, specializations.
Received by editor(s): March 23, 1998
Received by editor(s) in revised form: October 30, 1998
Published electronically: May 12, 2000
Additional Notes: The authors are grateful for support under NSA Grants MDA904-97-1-0040 and MDA904-97-1-0060, respectively.
Article copyright: © Copyright 2000 American Mathematical Society

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