Homology, Homotopy and Applications

Volume 13 (2011)

Number 2

Co-representability of the Grothendieck group of endomorphisms functor in the category of noncommutative motives

Pages: 315 – 328

DOI: https://dx.doi.org/10.4310/HHA.2011.v13.n2.a19

Author

Gonçalo Tabuada (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.; Departamento de Matemática, FCT-UNL, Quinta da Torre, Caparica, Portugal)

Abstract

In this article we prove that the additive invariant corepresented by the noncommutative motive $\mathbb{Z}[r]$ is the Grothendieck group of endomorphisms functor $K_0\mathrm{End}$. Making use of Almkvist’s foundational work, we then show that the ring $\mathrm{Nat}(K_0\mathrm{End},K_0\mathrm{End})$ of natural transformations (whose multiplication is given by composition) is naturally isomorphic to the direct sum of $\mathbb{Z}$ with the ring $W_0(\mathbb{Z}[r])$ of fractions of polynomials with coefficients in $\mathbb{Z}[r]$ and constant term 1.

Keywords

$K$-theory of endomorphisms, noncommutative motives, dg categories

2010 Mathematics Subject Classification

18D20, 18F30, 19D99

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Published 25 January 2012