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...

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Bibliographic Details
Main Authors: Blumberg, Andrew J. (Author), Gepner, David (Author), Trigo Neri Tabuada, Goncalo Jorge (Contributor)
Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
Format: Article
Language:English
Published: American Mathematical Society (AMS), 2016-11-04T18:45:57Z.
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Online Access:Get fulltext
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100 1 0 |a Blumberg, Andrew J.  |e author 
100 1 0 |a Massachusetts Institute of Technology. Department of Mathematics  |e contributor 
100 1 0 |a Trigo Neri Tabuada, Goncalo Jorge  |e contributor 
700 1 0 |a Gepner, David  |e author 
700 1 0 |a Trigo Neri Tabuada, Goncalo Jorge  |e author 
245 0 0 |a K-theory of endomorphisms via noncommutative motives 
260 |b American Mathematical Society (AMS),   |c 2016-11-04T18:45:57Z. 
856 |z Get fulltext  |u http://hdl.handle.net/1721.1/105205 
520 |a 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. 
546 |a en_US 
655 7 |a Article 
773 |t Transactions of the American Mathematical Society