Comparison of Exterior Power Operations on Higher K-Theory of Schemes

TL;DR

This paper proves that various constructions of exterior power operations on higher K-theory of schemes are equivalent, confirming their consistency and completing the lambda-ring structure in equivariant K-theory.

Contribution

It demonstrates the equivalence of different constructions of exterior power operations and confirms the lambda-ring axioms for higher equivariant K-groups.

Findings

All constructions of exterior power operations coincide.

Confirmed the lambda-ring axioms for higher equivariant K-groups.

Proved a conjecture on composition of operations in equivariant K-theory.

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.