TY - JOUR

T1 - Noncommutative numerical motives, Tannakian structures, and motivic Galois groups

AU - Tabuada, Gonçalo Jorge Trigo Neri

N1 - Sem PDF.
NSF (DMS-0901221; DMS-1007207; DMS-1201512; PHY-1205440)
National Science Foundation CAREER Award (1350472)
Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) (UID/MAT/00297/2013)

PY - 2016

Y1 - 2016

N2 - 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 (HP*) over bar on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues C-NC and DNC of Grothendieck's standard conjectures C and D. Assuming CNC, we prove that NNum (k)(F) can be made into a Tannakian category NNum(dagger)(k)(F) by modifying its symmetry isomorphism constraints. By further assuming D-NC, we neutralize the Tannakian category Num(dagger) (k)(F) using (HP*) over bar. 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 to the classical ones as suggested by Kontsevich.

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 (HP*) over bar on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues C-NC and DNC of Grothendieck's standard conjectures C and D. Assuming CNC, we prove that NNum (k)(F) can be made into a Tannakian category NNum(dagger)(k)(F) by modifying its symmetry isomorphism constraints. By further assuming D-NC, we neutralize the Tannakian category Num(dagger) (k)(F) using (HP*) over bar. 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 to the classical ones as suggested by Kontsevich.

KW - Noncommutative algebraic geometry

KW - noncommutative motives

KW - periodic cyclic homology

KW - Tannakian formalism

KW - motivic Galois groups

U2 - 10.4171/JEMS/598

DO - 10.4171/JEMS/598

M3 - Article

VL - 18

SP - 623

EP - 655

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 3

ER -