Vol. 2, No. 3, 2017

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Exterior power operations on higher $K$-groups via binary complexes

Tom Harris, Bernhard Köck and Lenny Taelman

Vol. 2 (2017), No. 3, 409–450
Abstract

We use Grayson’s binary multicomplex presentation of algebraic K-theory to give a new construction of exterior power operations on the higher K-groups of a (quasicompact) scheme. We show that these operations satisfy the axioms of a λ-ring, including the product and composition laws. To prove the latter we show that the Grothendieck group of the exact category of integral polynomial functors is the universal λ-ring on one generator.

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Keywords
exterior power operations, binary complexes, higher algebraic $K$-theory, lambda ring, Dold–Kan correspondence, Dold–Puppe construction, simplicial tensor product, plethysm problem, polynomial functor, Schur algebra
Mathematical Subject Classification 2010
Primary: 19D99
Secondary: 13D15, 14F99, 19E08, 20G05
Milestones
Received: 20 July 2016
Revised: 5 September 2016
Accepted: 16 October 2016
Published: 1 June 2017
Authors
Tom Harris
University Printing House
Shaftesbury Avenue
Cambridge
CB2 8BS
United Kingdom
Bernhard Köck
Mathematical Sciences
University of Southampton
Highfield
Southampton
SO17 1BJ
United Kingdom
Lenny Taelman
Korteweg-de Vries Instituut
Universiteit van Amsterdam
P.O. Box 94248
1090 GE Amsterdam
Netherlands