Noncommutative Artin motives

Abstract

In this article, we introduce the category of noncommutative Artin motives as well as the category of noncommutative mixed Artin motives. In the pure world, we start by proving that the classical category $${{\mathrm{AM}}}(k)_\mathbb Q $$ 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 $${{\mathrm{NAM}}}(k)_\mathbb Q $$ 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 $$\mathrm{Gal}(\overline{k}/k)$$ from the Tannakian formalism applied to $${{\mathrm{NAM}}}(k)_\mathbb Q $$ . Then, we develop the base-change formalism 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 then construct a short exact sequence relating $$\mathrm{Gal}(\overline{k}/k)$$ with the noncommutative motivic Galois groups of k and $$\overline{k}$$ . Finally, we describe a precise relationship between this short exact sequence and the one constructed by Deligne–Milne. In the mixed world, we introduce the triangulated category $${{\mathrm{NMAM}}}(k)_\mathbb Q $$ of noncommutative mixed Artin motives and construct a faithful functor from the classical category $${{\mathrm{MAM}}}(k)_\mathbb Q $$ of mixed Artin motives to it. When k is a finite field, this functor is an equivalence. On the other hand, when k is of characteristic zero $${{\mathrm{NMAM}}}(k)_\mathbb Q $$ is much richer than $${{\mathrm{MAM}}}(k)_\mathbb Q $$ since its higher Ext-groups encode all the (rationalized) higher algebraic $$K$$ -theory of finite etale k-schemes. In the appendix, we establish a general result about short exact sequences of Galois groups which is of independent interest. As an application, we obtain a new proof of Deligne–Milne’s short exact sequence.