Numerical functors on Voevodsky's motives

TL;DR

This paper explores two different definitions of a functor from Chow motives to numerical motives, analyzing their properties and differences in the context of Voevodsky's motives.

Contribution

It introduces and compares two natural definitions of the quotient functor, highlighting their distinct properties and implications in motive theory.

Findings

The two functors are fundamentally different in behavior.

One functor is full and conservative, while the other is not.

The study clarifies the conditions under which these functors preserve structures.

Abstract

We study mixed versions of the classical quotient functor from Chow motives to numerical motives. We compare two natural definitions, which turn out to be very different. We investigate fullness, conservativity and exactness of these two functors.