Nikita A. Karpenko: Variations on a theme of rationality of cycles

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Submission: 2011, Sep 21

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 different from 2. Our weak versions are still sufficient for all existing applications. In particular, Vishik's construction of fields of u-invariant 2r+1 (for r>2) is extended to arbitrary characteristic not 2.

2010 Mathematics Subject Classification: 14C25; 11E04

Keywords and Phrases: Chow groups, quadrics, Steenrod operations, u-invariant.

Full text: dvi.gz 28 k, dvi 75 k, ps.gz 838 k, pdf.gz 153 k, pdf 189 k.

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