Homology, Homotopy and Applications

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

Goncalo Tabuada

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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.

Article information

Source
Homology Homotopy Appl., Volume 13, Number 2 (2011), 315-328.

Dates
First available in Project Euclid: 30 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.hha/1335806756

Mathematical Reviews number (MathSciNet)
MR2861234

Zentralblatt MATH identifier
1275.18025

Subjects
Primary: 18D20: Enriched categories (over closed or monoidal categories) 18F30: Grothendieck groups [See also 13D15, 16E20, 19Axx] 19D99: None of the above, but in this section

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

Citation

Tabuada, Goncalo. Co-representability of the Grothendieck group of endomorphisms functor in the category of noncommutative motives. Homology Homotopy Appl. 13 (2011), no. 2, 315--328. https://projecteuclid.org/euclid.hha/1335806756


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