Graph explorer

Noncommutative Artin motives

In this article we introduce the categories of noncommutative (mixed) Artin motives. In the pure world, we start by proving that the classical category AM(k) of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully-embeds into noncommutative Chow motives. Making use of a refined bridge between pure motives and noncommutative pure motives we then show that the image of this full embedding, which we call the category NAM(k) of noncommutative Artin motives, is invariant under the different equivalence relations and modification of the symmetry isomorphism constraints. As an application, we recover the absolute Galois group of k from the Tannakian formalism applied to NAM(k). Then, we develop the theory of base-change in the world of noncommutative pure motives. As an application, we obtain new tools for the study of motivic decompositions and Schur/Kimura finiteness. Making use of this theory of base-change we construct a short exact sequence relating the absolute Galois group of k with the noncommutative motivic Galois groups of k and of its algebraic closure. Finally, we describe a precise relationship between this short exa