Artin motives in relative Nori and Voevodsky motives

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.