Comparison of the Categories of Motives defined by Voevodsky and Nori

Abstract

In this thesis we compare V. Voevodsky's geometric motives to the derived category of M. Nori's abelian category of mixed motives by constructing a triangulated tensor functor between them. It will be compatible with the Betti realizations on both sides. We allow an arbitrary noetherian ring of coefficients, but require it to be a field or a Dedekind domain for the tensor structure on derived Nori motives to exist. There are three key ingredients: we present a theory of Nisnevich covers on finite acyclic diagrams of finite correspondences, explain, following D. Rydh, how to interpret finite correspondences as multivalued morphisms and elaborate on M. Nori's cohomological cell structures. For the first two, we will be working over an arbitrary regular scheme, but the last one will require that we restrict ourselves to a subfield of the complex numbers. On the way we also show that smooth commutative group schemes over a normal base automatically admit transfers, generalizing a result by M. Spiess and T. Szamuely.