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# 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: http://dx.doi.org/10.4310/HHA.2011.v13.n2.a19

#### Author

#### 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

Published 25 January 2012