TY - JOUR

T1 - Noncommutative Artin motives

AU - Tabuada, Gonçalo Jorge Trigo Neri

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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 exact sequence and the one constructed by Deligne-Milne. In the mixed world, we introduce the triangulated category NMAM(k) of noncommutative mixed Artin motives and construct a faithful functor from the classical category MAM(k) of mixed Artin motives to it. When k is a finite field this functor is an equivalent. On the other hand, when k is of characteristic zero NMAM(k) is much richer than MAM(k) 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.

AB - 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 exact sequence and the one constructed by Deligne-Milne. In the mixed world, we introduce the triangulated category NMAM(k) of noncommutative mixed Artin motives and construct a faithful functor from the classical category MAM(k) of mixed Artin motives to it. When k is a finite field this functor is an equivalent. On the other hand, when k is of characteristic zero NMAM(k) is much richer than MAM(k) 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.

KW - Artin motives

KW - noncommutative motives

KW - motivic Galois groups

U2 - 10.1007/s00029-013-0131-9

DO - 10.1007/s00029-013-0131-9

M3 - Article

SN - 1022-1824

VL - 20

SP - 315

EP - 358

JO - Selecta Mathematica-New Series

JF - Selecta Mathematica-New Series

IS - 1

ER -