Comparison of Exterior Power Operations on Higher K-Theory of Schemes
Bernhard Köck, Ferdinando Zanchetta
TL;DR
The paper addresses the problem of reconciling exterior power operations on higher $K$-theory across multiple constructions by extending exterior powers to complexes, comparing combinatorial and homotopical definitions, and unifying with Barwick–Glasman–Mathew–Nikolaus via Zan21. It develops a robust framework that carries the operations to $K$-theory spaces via Grayson’s $G$-construction and demonstrates compatibility through a sequence of isomorphisms, culminating in a main theorem that equates distinct constructions. A central achievement is proving the GRR conjecture for composition of exterior powers in higher equivariant $K$-theory, thereby confirming that higher equivariant $K$-theory forms a lambda-ring. The results solidify Grothendieck’s Riemann–Roch program in the higher and equivariant setting and provide a unified foundation across classical and modern approaches to exterior power operations.
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.