Motifs et adjoints

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.