Specializations of Brauer classes over algebraic function fields
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- by Burton Fein and Murray Schacher PDF
- Trans. Amer. Math. Soc. 352 (2000), 4355-4369 Request permission
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.References
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Additional Information
- Burton Fein
- Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
- Email: fein@math.orst.edu
- Murray Schacher
- Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90024
- Email: mms@math.ucla.edu
- 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.
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 4355-4369
- MSC (2000): Primary 12E15, 12G05, 16K50
- DOI: https://doi.org/10.1090/S0002-9947-00-02474-0
- MathSciNet review: 1661250