𝐾-theory of endomorphisms via noncommutative motives

Abstract

We extend the K K -theory of endomorphisms functor from ordinary rings to (stable) ∞ \infty -categories. We show that K E n d ( − ) \mathrm {KEnd}(-) descends to the category of noncommutative motives, where it is corepresented by the noncommutative motive associated to the tensor algebra S [ t ] \mathbb {S}[t] of the sphere spectrum S \mathbb {S} . Using this corepresentability result, we classify all the natural transformations of K E n d ( − ) \mathrm {KEnd}(-) in terms of an integer plus a fraction between polynomials with constant term 1 1 ; this solves a problem raised by Almkvist in the seventies. Finally, making use of the multiplicative coalgebra structure of S [ t ] \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 K_0 -theory of endomorphisms of a connective ring spectrum R R equals the K 0 K_0 -theory of endomorphisms of the underlying ordinary ring π 0 R \pi _0R .