Around rationality of cycles

TL;DR

This paper investigates the rationality of algebraic cycles over function fields of quadrics, extending Vishik's characteristic zero results to arbitrary characteristics not equal to 2 using Steenrod operations.

Contribution

It generalizes Vishik's results by employing Steenrod operations in Chow theory, applicable in any characteristic other than 2, broadening the understanding of cycle rationality.

Findings

Proves rationality comparison results over various characteristics

Utilizes Steenrod operations in Chow theory for the proofs

Extends previous characteristic zero results to characteristic ≠ 2

Abstract

In this article we prove certain results comparing rationality of algebraic cycles over the function field of a quadric and over the base field. Those results have already been proved by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic different from 2.