Goncalo Tabuada

Chow motives versus noncommutative motives

In this article we formalize and enhance Kontsevichs beautiful insight that Chow motives can be embedded into non-commutative ones after factoring out by the action of the Tate object. We illustrate the potential of this result by developing three of its manyfold applications: (1) the notions of Schur and Kimura finiteness admit an adequate extension to the realm of non-commutative motives; (2) Gillet-Soules motivic measure admits an extension to the Grothendieck ring of non-commutative motives; (3) certain motivic zeta functions admit an intrinsic construction inside the category of non-commutative motives.

Noncommutative motives of Azumaya algebras

G. Tabuada was partially supported by the NEC Award-2742738 and by the Portuguese Foundation for Science and Technology through the project PEst-OE/MAT/UI0297/2014 (CMA). This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 0932078 000, undertaken while the authors were in residence at the Mathematical Science Research Institute (MSRI) in Berkeley, California, during the spring semester of 2013.

Noncommutative numerical motives, Tannakian structures, and motivic Galois groups

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* on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues C_(NC) and D_(NC) of Grothendiecks 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 HP*. Via the (super-)Tannakian formalism, we then obtain well-defined noncommutative motivic Galois (super-)groups. Finally, making use of Deligne-Milnes theory of Tate triples, we construct explicit morphisms relating these noncommutative motivic Galois (super-)groups with the classical ones as suggested by Kontsevich.

Noncommutative motives, numerical equivalence, and semi-simplicity

Making use of Hochschild homology, we introduce the correct category

Kontsevich’s noncommutative numerical motives

{\rm NNum}(k)_F

Generalized spectral categories, topological Hochschild homology and trace maps

of noncommutative {\it numerical} motives (over a base ring

A universal characterization of the Chern character maps

k

Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) non-commutative motives

and with coefficients in a field

Lefschetz and Hirzebruch-Riemann-Roch formulas via noncommutative motives

F

Universal suspension via noncommutative motives

). We prove that