Motifs et adjoints

Abstract

We show in many cases the existence of adjoints to extension of scalars on categories of motivic nature, in the framework of field extensions. This is to be contrastedwith themore classical situationwhere one dealswith a finite typemorphism of schemes. Among various applications, one is a construction of a “Tate–Šafarevič motive” attached to an abelian variety over a function field. We also deduce a possible approach to Bloch’s conjecture on surfaces, by reduction to curves.