Variations on a theme of rationality of cycles

Abstract

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 -invariant 2 + 1, for ≥ 3, is extended to arbitrary characteristic = 2. MSC: 14C25, 11E04Keywords:The main results of this note are Theorem 1.1 (the basic result) with its enhancement Theorem 2.1, Proposition 3.1 with its enhancement Proposition 4.1 implying Theorems 3.2 and 3.3 (which go a little bit beyond the basic result), and (a quite special) Proposition 5.3 (going in a special situation even more beyond the basic result). The main application is Theorem 5.1.In characteristic 0, all of this has been proved several years ago by Alexander Vishik in [7,9] (exact references are given right before each statement) with a help of the algebraic cobordism theory and especially symmetric operations of [8].In fact, the original versions of the most results are stronger: they do not require to mod out 2-torsion elements of Chow groups (as our weak versions do). In particular, they do not require the assumption that the group CH(Y ) (notation introduced in the beginning of Section 1) is 2-torsion-free (as do our very weak versions).Note that it has been explained in [7, Remark on p. 370] that the weak versions can be obtained in characteristic 0 with a help of the Landweber-Novikov operations (in the algebraic cobordism theory) replacing the symmetric operations.Although the very weak versions are already sufficient for existing applications, we prove the weak versions as well, see Theorem 2.1 and Proposition 4.1. The proofs here are only a bit more complicated (than in the very weak case) *

Motivic Steenrod operations in characteristic $p$

“…In [30], Vishik constructed characteristic 0 fields of u-invariant 2 r +1 for all r ≥ 3. Karpenko used Steenrod squares on mod 2 Chow groups to show that for any r ≥ 3 and any field F of characteristic = 2, F is contained in a field of u-invariant 2 r + 1 [19]. Karpenko's constructions in [19] now extend to fields of characteristic 2 through the use of the Steenrod squares Sq 2n k defined in this paper for k of characteristic 2.…”

Motivic Steenrod operations in characteristicp

“…In [32], Vishik constructed characteristic 0 fields of -invariant 2 + 1 for all ≥ 3. Karpenko used Steenrod squares on mod 2 Chow groups to show that for any ≥ 3 and any field of characteristic ≠ 2, is contained in a field of -invariant 2 + 1 [19]. Karpenko's constructions now extend to fields of characteristic 2 through the use of the Steenrod squares Sq 2 defined in this article for of characteristic 2.…”