Comparison of exterior power operations on higher K-theory of schemes
Comparison of exterior power operations on higher K-theory of schemes
Exterior power operations provide an additional structure on K-groups of schemes which lies at the heart of Grothendieck's Riemann-Roch theory. Over the past decades, various authors have constructed such operations on higher K-theory. In this paper, we prove that these constructions actually yield the same operations, ultimately matching up the explicit combinatorial description by Harris, the first author and Taelman on the one hand and the recent, conceptually clear-cut construction by Barwick, Glasman, Mathew and Nikolaus on the other hand. This also leads to the proof of a conjecture by the first author about composition of these operations in the equivariant context, completing the proof that higher equivariant K-groups satisfy all axioms of a lambda-ring.
exterior power operations; higher K-groups; homotopy fibre; Dold-Puppe construction; categories with weak equivalences; lambda ring; equivariant K-theory; Köck's conjecture
Koeck, Bernhard
84d11519-7828-43a6-852b-0c1b80edeef9
Zanchetta, Ferdinando
9bbe0774-d23e-4bf5-95f9-a09fc23d9db7
21 February 2025
Koeck, Bernhard
84d11519-7828-43a6-852b-0c1b80edeef9
Zanchetta, Ferdinando
9bbe0774-d23e-4bf5-95f9-a09fc23d9db7
Koeck, Bernhard and Zanchetta, Ferdinando
(2025)
Comparison of exterior power operations on higher K-theory of schemes.
Mathematische Zeitschrift, 309.
(doi:10.1007/s00209-025-03681-2).
Abstract
Exterior power operations provide an additional structure on K-groups of schemes which lies at the heart of Grothendieck's Riemann-Roch theory. Over the past decades, various authors have constructed such operations on higher K-theory. In this paper, we prove that these constructions actually yield the same operations, ultimately matching up the explicit combinatorial description by Harris, the first author and Taelman on the one hand and the recent, conceptually clear-cut construction by Barwick, Glasman, Mathew and Nikolaus on the other hand. This also leads to the proof of a conjecture by the first author about composition of these operations in the equivariant context, completing the proof that higher equivariant K-groups satisfy all axioms of a lambda-ring.
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Accepted/In Press date: 24 December 2024
Published date: 21 February 2025
Keywords:
exterior power operations; higher K-groups; homotopy fibre; Dold-Puppe construction; categories with weak equivalences; lambda ring; equivariant K-theory; Köck's conjecture
Identifiers
Local EPrints ID: 452484
URI: http://eprints.soton.ac.uk/id/eprint/452484
ISSN: 0025-5874
PURE UUID: f494be37-310c-466b-90b9-e367b1345887
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Date deposited: 11 Dec 2021 11:17
Last modified: 25 Feb 2025 02:38
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Author:
Ferdinando Zanchetta
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