K-theory of endomorphisms via noncommutative motives

Abstract

We extend the K-theory of endomorphisms functor from ordinary rings to (stable) ∞-categories. We show that KEnd(-) descends to the category of noncommutative motives, where it is corepresented by the noncommutative motive associated to the tensor algebra S[t] of the sphere spectrum S. Using this corepresentability result, we classify all the natural transformations of 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 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[subscript 0]-theory of endomorphisms of a connective ring spectrum R equals the K[subscript 0]-theory of endomorphisms of the underlying ordinary ring π[subscript 0]R.