Noncommutative numerical motives, Tannakian structures, and motivic Galois groups
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Tabuada_Noncommutative motives.pdf
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Author(s)
Marcolli, Matilde
Trigo Neri Tabuada, Goncalo Jorge
Date Issued
2016
Journal
Journal of the European Mathematical Society
Publisher
European Mathematical Society Publishing House
Citation
Marcolli, Matilde, and Gonçalo Tabuada. “Noncommutative Numerical Motives, Tannakian Structures, and Motivic Galois Groups.” J. Eur. Math. Soc. 18, no. 3 (2016): 623–655.
Version
Original manuscript
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 [over-bar HP∗]on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues CNC and DNC of Grothendieck's standard conjectures C and D. Assuming C[subscript NC], we prove that NNum(k)F can be made into a Tannakian category NNum[superscript †](k)F by modifying its symmetry isomorphism constraints. By further assuming D[subscript NC], we neutralize the Tannakian category Num†(k)F using [over-bar 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.4171/jems/598
