K-theory of endomorphisms via noncommutative motives
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Tabuada_k-theory.pdf
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Author(s)
Blumberg, Andrew J.
Gepner, David
Trigo Neri Tabuada, Goncalo Jorge
Date Issued
July 2015
Journal
Transactions of the American Mathematical Society
Publisher
American Mathematical Society (AMS)
Citation
Blumberg, Andrew J., David Gepner, and Gonçalo Tabuada. “$K$-Theory of Endomorphisms via Noncommutative Motives.” Transactions of the American Mathematical Society 368.2 (2015): 1435–1465.
Version
Final published version
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1090/tran/6507
