Around rationality of integral cycles

TL;DR

This paper establishes a new integral comparison of algebraic cycle rationality over function fields of quadrics and base fields, extending known modulo 2 results to integral coefficients in any characteristic not 2.

Contribution

It provides the first integral version of rationality comparison for algebraic cycles over quadrics, generalizing Vishik's characteristic zero results using Steenrod operations.

Findings

Proves an integral rationality comparison theorem for algebraic cycles.

Extends results from modulo 2 coefficients to integral coefficients.

Works in any characteristic different from 2.

Abstract

In this article we prove a result comparing rationality of integral algebraic cycles over the function field of a quadric and over the base field. This is an integral version of the result known for coefficients modulo 2. 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.