Generalization of a going-down theorem in the category of Chow-Grothendieck motives due to N. Karpenko

TL;DR

This paper extends a theorem in the theory of Chow-Grothendieck motives, showing that certain motive decompositions over an extension field imply similar decompositions over the base field under specific conditions.

Contribution

It generalizes a going-down theorem for motives, providing conditions under which motive summands over an extension field descend to the base field.

Findings

The theorem applies to geometrically split varieties satisfying the nilpotence principle.

Under certain extension conditions, motive summands over the extension field descend to the base field.

The result broadens understanding of motive decompositions in algebraic geometry.

Abstract

Let $\mathbb{M}:=(M(X),p)$ be a direct summand of the motive associated with a geometrically split, geometrically variety over a field $F$ satisfying the nilpotence principle. We show that under some conditions on an extension $E/F$, if $\mathbb{M}$ is a direct summand of another motive $M$ over an extension $E$, then $\mathbb{M}$ is a direct summand of $M$ over $F$.