Schur finiteness and nilpotency

TL;DR

This paper generalizes the concept of Kimura finiteness to special Schur-finite objects within tensor categories, establishing equivalences with nilpotency of certain endomorphism ideals in the context of Chow motives.

Contribution

It extends Kimura's finiteness notion to special Schur-finite objects and explores their implications in Chow motives under the homological sign conjecture.

Findings

Finiteness of numerically trivial endomorphisms implies nilpotency.

Equivalence of Kimura-finiteness, Schur-finiteness, and nilpotency conditions in Chow motives.

Generalization of Kimura's result to a broader class of objects.

Abstract

Let A be a Q-linear pseudo-abelian rigid tensor category. A notion of finiteness due to Kimura and (independently) O'Sullivan guarantees that the ideal of numerically trivial endomorphism of an object is nilpotent. We generalize this result to special Schur-finite objects. In particular, in the category of Chow motives, if X is a smooth projective variety which satisfies the homological sign conjecture, then Kimura-finiteness, a special Schur-finiteness, and the nilpotency of CH^{ni}(X^i\times X^i)_{num} for all i (where n=dim X) are all equivalent.