Noncommutative numerical motives, Tannakian structures, and motivic Galois groups

Marcolli, Matilde, and Tabuada, Gonçalo. "Noncommutative numerical motives, Tannakian structures, and motivic Galois groups." Journal of the European Mathematical Society 018.3 (2016): 623-655. <http://eudml.org/doc/277232>.

@article{Marcolli2016,
abstract = {In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the change-of-coefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schur-finiteness, we prove that the category NNum$(k)_F$ of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum$(k)_F$ is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric monoidal functor $\overline\{HP\}_*$ on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues $C_\{NC\}$ and $D_\{NC\}$ of Grothendieck’s standard conjectures $C$ and $D$. Assuming $C?\{NC\}$, we prove that NNum$(k)_F$ can be made into a Tannakian category NNum†$(k)_F$ by modifying its symmetry isomorphism constraints. By further assuming $D_\{NC\}$, we neutralize the Tannakian category Num†$(k)_F$ using $\overline\{HP\}_*$. Via the (super-)Tannakian formalism, we then obtain well-defined noncommutative motivic Galois (super-)groups. Finally, making use of Deligne-Milne’s theory of Tate triples, we construct explicit morphisms relating these noncommutative motivic Galois (super-)groups with the classical ones as suggested by Kontsevich.},
author = {Marcolli, Matilde, Tabuada, Gonçalo},
journal = {Journal of the European Mathematical Society},
keywords = {noncommutative algebraic geometry; noncommutative motives; periodic cyclic homology; Tannakian formalism; motivic Galois groups; noncommutative algebraic geometry; noncommutative motives; periodic cyclic homology; Tannakian formalism; motivic Galois groups},
language = {eng},
number = {3},
pages = {623-655},
publisher = {European Mathematical Society Publishing House},
title = {Noncommutative numerical motives, Tannakian structures, and motivic Galois groups},
url = {http://eudml.org/doc/277232},
volume = {018},
year = {2016},
}

TY - JOUR
AU - Marcolli, Matilde
AU - Tabuada, Gonçalo
TI - Noncommutative numerical motives, Tannakian structures, and motivic Galois groups
JO - Journal of the European Mathematical Society
PY - 2016
PB - European Mathematical Society Publishing House
VL - 018
IS - 3
SP - 623
EP - 655
AB - In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the change-of-coefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schur-finiteness, we prove that the category NNum$(k)_F$ of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum$(k)_F$ is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric monoidal functor $\overline{HP}_*$ on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues $C_{NC}$ and $D_{NC}$ of Grothendieck’s standard conjectures $C$ and $D$. Assuming $C?{NC}$, we prove that NNum$(k)_F$ can be made into a Tannakian category NNum†$(k)_F$ by modifying its symmetry isomorphism constraints. By further assuming $D_{NC}$, we neutralize the Tannakian category Num†$(k)_F$ using $\overline{HP}_*$. Via the (super-)Tannakian formalism, we then obtain well-defined noncommutative motivic Galois (super-)groups. Finally, making use of Deligne-Milne’s theory of Tate triples, we construct explicit morphisms relating these noncommutative motivic Galois (super-)groups with the classical ones as suggested by Kontsevich.
LA - eng
KW - noncommutative algebraic geometry; noncommutative motives; periodic cyclic homology; Tannakian formalism; motivic Galois groups; noncommutative algebraic geometry; noncommutative motives; periodic cyclic homology; Tannakian formalism; motivic Galois groups
UR - http://eudml.org/doc/277232
ER -