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Power structures on the Grothendieck–Witt ring and the motivic Euler characteristic

Jesse Pajwani and Ambrus Pál
Vol. 10 (2025), No. 2, 123–152
DOI: 10.2140/akt.2025.10.123
Abstract

For k a field, we construct a power structure on the Grothendieck–Witt ring of k which has the potential to be compatible with symmetric powers of varieties and the motivic Euler characteristic. We then show our power structure is compatible with the variety power structure when we restrict to varieties of dimension 0, using techniques of Garibaldi, Merkurjev and Serre about cohomological invariants.

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Keywords
symmetric powers, motivic homotopy, power structures, Euler characteristic
Mathematical Subject Classification
Primary: 11E04, 11E70, 11E81, 19G12
Milestones
Received: 13 February 2024
Revised: 20 January 2025
Accepted: 20 February 2025
Published: 19 March 2025
Authors
Jesse Pajwani
Department of Mathematical Sciences
University of Bath
Bath
United Kingdom
Ambrus Pál
Department of Mathematics
Imperial College
London
United Kingdom
© 2025 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers).