# Noncommutative numerical motives, Tannakian structures, and motivic Galois groups

Matilde Marcolli; Gonçalo Tabuada

Journal of the European Mathematical Society (2016)

- Volume: 018, Issue: 3, page 623-655
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topMarcolli, 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 -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.