We prove certain weak versions of some celebrated results due to Alexander Vishik comparing rationality of algebraic cycles over the function field of a quadric and over the base field. The original proofs use Vishik’s symmetric operations in the algebraic cobordism theory and work only in characteristic 0. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2. Our weak versions are still sufficient for existing applications. In particular, Vishik’s construction of fields of u-invariant 2r + 1, for r ≥ 3, is extended to arbitrary characteristic ≠ 2.
@article{bwmeta1.element.doi-10_2478_s11533-013-0228-6,
author = {Nikita Karpenko},
title = {Variations on a theme of rationality of cycles},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {1056-1067},
zbl = {1300.14008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0228-6}
}
Nikita Karpenko. Variations on a theme of rationality of cycles. Open Mathematics, Tome 11 (2013) pp. 1056-1067. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0228-6/
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