Symmetric monoidal structure on non-commutative motives

By Denis-Charles Cisinski, Gonçalo Tabuada. Journal of K-Theory, 2010

81 citations recorded by the source corpus.

In this article we further the study of non-commutative motives, initiated in [12, 43]. Our main result is the construction of a symmetric monoidal structure on the localizing motivator Motlocdg of dg categories. As an application, we obtain : (1) a computation of the spectra of morphisms in Motlocdg in terms of non-connective algebraic K-theory; (2) a fully-faithful embedding of Kontsevich's category KMMk of non-commutative mixed motives into the base category Motlocdg(e) of the localizing motivator; (3) a simple construction of the Chern character maps from non-connective algebraic K-theory to negative and periodic cyclic homology; (4) a precise connection between Toën's secondary K-theory and the Grothendieck ring of KMMk; (5) a description of the Euler characteristic in KMMk in terms of Hochschild homology.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f66d348fc9d2938f80b91b96c0541c59825d65fd

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Citations in the corpus

Chow motives versus non-commutative motives (2011, background): “1) admits an appropriate symmetric monoidal enhancement; see [16].”
Brauer groups and étale cohomology in derived algebraic geometry (2012, background): “The theory of smooth and proper algebras is also fundamental in the theory of noncommutative motives, and has been studied in that setting by Cisinski and Tabuada [18] and Blumberg, Gepner and Tabuada [12].”
A universal characterization of higher algebraic K-theory (2010, background): “In this vein, over the past few years the third author and Cisinski have carried out a program [20; 21; 68] of providing universal characterizations of algebraic K–theory in the setting of dg–categories, using the formalism of derivators.”
Differential Graded Categories are k-linear Stable Infinity Categories (2013, methodology): “del category structures). This is one reason why our comparison between dg categories and klinear stable ∞-categories is not completely trivial. To resolve this issue, we use ideas inspired by [To1], [CT], and [BGT2] to derive the tensor product of dg categories via “ﬂat” dg categories. 2 Following ”
Topological K-theory of complex noncommutative spaces (2012, background): “nd exact sequences to distinguished triangles. The model category M loc(k) is then a SpS(k)-model category with SpS(k)-enriched Hom denoted by Hom Mloc(k) with a derived version RHom M loc(k), and by [3] it is also a monoidal model category in the sense of [17]. The monoidal structure is denoted by ”
Higher traces, noncommutative motives, and the categorified Chern character (2015, methodology): “We are grateful to Clark Barwick, Kai Behrend, David Ben-Zvi, Andrew Blumberg, David Carchedi, Rune Haugseng, Claudia Scheimbauer, Chris Schommer-Pries, Jay Shah, Gonçalo Tabuada, Bertrand Toën, and Jesse Wolfson for useful conversations and for their interest in this project.”