  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 contrasted with the more classical situation where one deals with a finite
type morphism of schemes. Among various applications, one is a functorial
construction of the "Tate-Safarevic motive" introduced in arXiv:1401.6847
[math.NT]. We also deduce a possible approach to Bloch's conjecture on
surfaces, by reduction to curves.
