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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

$K$-theory of endomorphisms via noncommutative motives


Authors: Andrew J. Blumberg, David Gepner and Gonçalo Tabuada
Journal: Trans. Amer. Math. Soc. 368 (2016), 1435-1465
MSC (2010): Primary 19D10, 19D25, 19D55, 18D20, 55N15
DOI: https://doi.org/10.1090/tran/6507
Published electronically: July 10, 2015
MathSciNet review: 3430369
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Abstract: We extend the $K$-theory of endomorphisms functor from ordinary rings to (stable) $\infty$-categories. We show that $\mathrm {KEnd}(-)$ descends to the category of noncommutative motives, where it is corepresented by the noncommutative motive associated to the tensor algebra $\mathbb {S}[t]$ of the sphere spectrum $\mathbb {S}$. Using this corepresentability result, we classify all the natural transformations of $\mathrm {KEnd}(-)$ 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 $\mathbb {S}[t]$, 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 $K_0$-theory of endomorphisms of a connective ring spectrum $R$ equals the $K_0$-theory of endomorphisms of the underlying ordinary ring $\pi _0R$.


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Additional Information

Andrew J. Blumberg
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78703
MR Author ID: 648837
Email: blumberg@math.utexas.edu

David Gepner
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
MR Author ID: 880977
Email: dgepner@purdue.edu

Gonçalo Tabuada
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 – and – Departamento de Matemática, FCT, UNL, Portugal, Centro de Matemática e Aplicações (CMA), FCT, UNL, Portugal
MR Author ID: 751291
Email: tabuada@math.mit.edu

Keywords: $K$-theory of endomorphisms, noncommutative motives, stable $\infty$-categories, Witt vectors
Received by editor(s): February 20, 2013
Received by editor(s) in revised form: March 3, 2014, and June 19, 2014
Published electronically: July 10, 2015
Additional Notes: The first author was partially supported by the NSF grant DMS-0906105
The third author was partially supported by the National Science Foundation CAREER Award #1350472 and by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project grant UID/MAT/00297/2013 (Centro de Matemática e Aplicações)
Article copyright: © Copyright 2015 American Mathematical Society