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Retract Rationality and Algebraic Tori

Part of: Representation theory of groups Birational geometry Linear algebraic groups and related topics Forms and linear algebraic groups

Published online by Cambridge University Press:  18 July 2019

Federico Scavia
Federico Scavia*
Affiliation:
University of British Columbia, Vancouver, BC V6T1Z4 Email: [email protected]
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Abstract

For any prime number $p$ and field $k$, we characterize the $p$-retract rationality of an algebraic $k$-torus in terms of its character lattice. We show that a $k$-torus is retract rational if and only if it is $p$-retract rational for every prime $p$, and that the Noether problem for retract rationality for a group of multiplicative type $G$ has an affirmative answer for $G$ if and only if the Noether problem for $p$-retract rationality for $G$ has a positive answer for all $p$. For every finite set of primes $S$ we give examples of tori that are $p$-retract rational if and only if $p\notin S$.

Keywords

algebraic toriretract rationalityNoether probleminvertible lattice

MSC classification

Primary: 11E72: Galois cohomology of linear algebraic groups
Secondary: 14E08: Rationality questions 20C10: Integral representations of finite groups 20G15: Linear algebraic groups over arbitrary fields
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 1 , March 2020 , pp. 173 - 186
Copyright
© Canadian Mathematical Society 2019

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