Holmstrom Noncommutative motives

Noncommutative motives

arXiv:1108.3787 A guided tour through the garden of noncommutative motives from arXiv Front: math.KT by Goncalo Tabuada These are the extended notes of a survey talk on noncommutative motives given at the 3era Escuela de Inverno Luis Santalo - CIMPA Research School: Topics in Noncommutative Geometry, Buenos Aires, July 26 to August 6, 2010.

Cisinski and Tabuada: Symmetric monoidal structure on noncommutative motives

Check all things of Cisinski and/or Tabuada, and maybe Kontsevich, for the complete picture

Noncommutative motives

arXiv:1206.3645 Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes from arXiv Front: math.AG by Marco Robalo Let $\V$ be a symmetric monoidal model category and let $X$ be an object in $\V$. From this we can construct a new symmetric monoidal model category $Sp^{\Sigma}(\V,X)$ of symmetric spectra objects in $\V$ with respect to $X$, together with a left Quillen monoidal map $\V\to Sp^{\Sigma}(\V,X)$ sending $X$ to an invertible object. In this paper we use the recent developments in the subject of Higher Algebra to understand the nature of this construction. Every symmetric monoidal model category has an underlying symmetric monoidal $(\infty,1)$-category and the first notion should be understood as a mere “presentation” of the second. Our main result is the characterization of the underlying symmetric monoidal $\infty$-category of $Sp^{\Sigma}(\V,X)$, by means of a universal property inside the world of symmetric monoidal $(\infty,1)$-categories. In the process we also describe the link between the construction of ordinary spectra and the one of symmetric spectra. As a corollary, we obtain a precise universal characterization for the motivic stable homotopy theory of schemes with its symmetric monoidal structure. This characterization trivializes the problem of finding motivic monoidal realizations and opens the way to compare the motivic theory of schemes with other motivic theories.

arXiv:1103.0200 Chow motives versus non-commutative motives from arXiv Front: math.KT by Goncalo Tabuada In this article we formalize and enhance Kontsevich’s 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-Soule’s 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.

arXiv:1108.3785 Kontsevich’s noncommutative numerical motives from arXiv Front: math.KT by Matilde Marcolli, Goncalo Tabuada In this note we prove that Kontsevich’s category NCnum of noncommutative numerical motives is equivalent to the one constructed by the authors. As a consequence, we conclude that NCnum is abelian semi-simple as conjectured by Kontsevich.

nLab page on Noncommutative motives