𝐾-theory of endomorphisms via noncommutative motives

Blumberg, Andrew; Gepner, David; Tabuada, Gonçalo

doi:10.1090/tran/6507

$K$-theory of endomorphisms via noncommutative motives
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by Andrew J. Blumberg, David Gepner and Gonçalo Tabuada PDF

Trans. Amer. Math. Soc. 368 (2016), 1435-1465 Request permission

Abstract:

We extend the $K$-theory of endomorphisms functor from ordinary rings to (stable) $\infty$-categories. We show that $\mathrm {KEnd}(-)$ descends to the category of noncommutative motives, where it is corepresented by the noncommutative motive associated to the tensor algebra $\mathbb {S}[t]$ of the sphere spectrum $\mathbb {S}$. Using this corepresentability result, we classify all the natural transformations of $\mathrm {KEnd}(-)$ in terms of an integer plus a fraction between polynomials with constant term $1$; this solves a problem raised by Almkvist in the seventies. Finally, making use of the multiplicative coalgebra structure of $\mathbb {S}[t]$, we explain how the (rational) Witt vectors can also be recovered from the symmetric monoidal category of noncommutative motives. Along the way we show that the $K_0$-theory of endomorphisms of a connective ring spectrum $R$ equals the $K_0$-theory of endomorphisms of the underlying ordinary ring $\pi _0R$.

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