Motifs et adjoints

TL;DR

This paper investigates the existence of adjoint functors in motivic categories over field extensions, providing new constructions and applications, including a functorial approach to the Tate-Shafarevich motive and insights into Bloch's conjecture.

Contribution

It establishes conditions for adjoints to scalar extension functors in motivic categories and applies these results to construct the Tate-Shafarevich motive and approach Bloch's conjecture.

Findings

Existence of adjoints in motivic categories over fields.

Functorial construction of the Tate-Shafarevich motive.

Reduction of Bloch's conjecture to curves.

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 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.