K-theory of endomorphisms via noncommutative motives
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 cor...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
American Mathematical Society (AMS)
2016
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| Online Access: | http://hdl.handle.net/1721.1/105205 https://orcid.org/0000-0001-5558-9236 |
| Summary: | 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. |
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