A remark on a result of Huber and Kahn

TL;DR

This paper investigates the relationship between the mixed Tate property of motives associated with principal G-bundles and their base schemes, extending previous results from principal T-bundles to split reductive groups.

Contribution

It proves that the mixed Tate property of the motive of the base scheme implies the same for the motive of the principal G-bundle for split reductive groups, extending prior work.

Findings

Mixed Tate property of M(Y) implies mixed Tate for M(X) in principal G-bundles.

Extends previous results from T-bundles to split reductive G-bundles.

Provides a converse implication for motives of principal G-bundles.

Abstract

A. Huber and B. Kahn construct a relative slice filtration on the motive M(X) associated to a principal T-bundle X over a smooth scheme Y. As a consequence of their result, one can observe that the mixed Tateness of the motive M(Y) implies that the motive M(X) is mixed Tate. In this note we prove the inverse implication for a principal G-bundle, for a split reductive group G.