K-theory of endomorphisms via noncommutative motives
Author(s)Blumberg, Andrew J.; Gepner, David; Trigo Neri Tabuada, Goncalo Jorge
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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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Transactions of the American Mathematical Society
American Mathematical Society (AMS)
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.
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