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$\lambda$-ring structures on the $K$-theory of algebraic stacks

Roy Joshua and Pablo Pelaez

Vol. 9 (2024), No. 3, 519–581
Abstract

We consider the K-theory of smooth algebraic stacks, establish λ and γ operations, and show that the higher K-theory of such stacks is always a pre-λ-ring, and is a λ-ring if every coherent sheaf is the quotient of a vector bundle. As a consequence, we are able to define Adams operations and absolute cohomology for smooth algebraic stacks satisfying this hypothesis. We also obtain a comparison of the absolute cohomology with the equivariant higher Chow groups in certain special cases.

Keywords
algebraic stacks, $K$-theory, lambda operations
Mathematical Subject Classification
Primary: 14C35, 14D23, 19E08
Milestones
Received: 3 September 2023
Revised: 15 April 2024
Accepted: 14 July 2024
Published: 1 October 2024
Authors
Roy Joshua
Department of Mathematics
Ohio State University
Columbus, OH
United States
Pablo Pelaez
Instituto de Matemáticas
IMATE, UNAM
Ciudad Universitaria
Ciudad de México
Mexico