Artin motives in relative Nori and Voevodsky motives

TL;DR

This paper proves the equivalence of Nori and Voevodsky categories of relative Artin motives over schemes of finite type in characteristic zero, with applications to dualisability and motivic Galois groups.

Contribution

It establishes a canonical equivalence between Nori and Voevodsky categories of relative Artin motives and explores their implications for dualisability and motivic Galois groups.

Findings

Nori and Voevodsky categories of relative Artin motives are canonically equivalent.

An Artin motive over a normal base is dualisable iff it is in the thick category generated by finite étale schemes.

An analogue of the étale fundamental group exact sequence is obtained for motivic Galois groups.

Abstract

Over a scheme of finite type over a field of characteristic zero, we prove that Nori an Voevodsky categories of relative Artin motives, that is the full subcategories generated by the motives of \'etale morphisms in relative Nori and Voevodsky motives, are canonically equivalent. As an application, we show that over a normal base of characteristic zero an Artin motive is dualisable if and only if it lies in the thick category spanned by the motives of finite \'etale schemes. We finish with an application to motivic Galois groups and obtain an analogue of the classical exact sequence of \'etale fundamental groups relating a variety over a field and its base change to the algebraic closure.