Noncommutative mixed (Artin) motives and their motivic Hopf dg algebras

TL;DR

This paper develops a theory of noncommutative mixed motives over any ring, constructs associated motivic Hopf dg algebras, and relates them to classical Galois groups and Voevodsky's motives.

Contribution

It introduces a new framework for noncommutative mixed motives, constructs their motivic Hopf dg algebras, and connects these to classical motives and Galois groups.

Findings

Explicit description of motivic Hopf dg algebra via Hochschild homology

Computation of Hopf dg algebra for NC mixed Artin motives using classical motives

Establishment of a short exact sequence linking Galois group functions and motivic Hopf dg algebras

Abstract

We start by developing a theory of noncommutative (=NC) mixed motives with coefficients in any commutative ring. In particular, we construct a symmetric monoidal triangulated category of NC mixed motives, over a base field k, and a full subcategory of NC mixed Artin motives. Making use of Hochschild homology, we then apply Ayoub's weak Tannakian formalism to these motivic categories. In the case of NC mixed motives, we obtain a motivic Hopf dg algebra, which we describe explicitly in terms of Hochschild homology and complexes of exact cubes. In the case of NC mixed Artin motives, we compute the associated Hopf dg algebra using solely the classical category of mixed Artin-Tate motives. Finally, we establish a short exact sequence relating the Hopf algebra of continuous functions on the absolute Galois group with the motivic Hopf dg algebras of the base field k and of its algebraic closure. Along the way, we describe the behavior of Ayoub's weak Tannakian formalism with respect to orbit categories and relate the category of NC mixed motives with Voevodsky's category of mixed motives.