Rationality of cycles over function fields of $SL_1(A)$-torsors

TL;DR

This paper investigates how the rationality of algebraic cycles over the function field of an $SL_1(A)$-torsor relates to their rationality over the base field, providing insights into algebraic cycle behavior in this context.

Contribution

It establishes a comparison result for the rationality of algebraic cycles over the function field of an $SL_1(A)$-torsor and the base field, advancing understanding in algebraic geometry.

Findings

Proves a comparison theorem for cycle rationality

Links cycle rationality over function fields and base fields

Enhances understanding of algebraic cycles over torsors

Abstract

In this note we prove a result comparing rationality of algebraic cycles over the function field of a $SL_1(A)$-torsor for a central simple algebra $A$ and over the base field.