Lambda-ring structures on the K-theory of algebraic stacks

TL;DR

This paper develops lambda-ring structures on the K-theory of smooth algebraic stacks, establishing new operations and comparing cohomology theories, thereby advancing the understanding of algebraic stacks' K-theoretic properties.

Contribution

It introduces lambda and gamma operations on the K-theory of smooth algebraic stacks and characterizes when the higher K-theory forms a lambda-ring, enabling new cohomology definitions.

Findings

Higher K-theory of smooth stacks is a pre-lambda-ring.

Under certain conditions, the K-theory forms a lambda-ring.

Comparison between absolute cohomology and equivariant higher Chow groups.

Abstract

In this paper we consider the K-theory of smooth algebraic stacks, establish lambda and gamma operations, and show that the higher K-theory of such stacks is always a pre-lambda-ring, and is a lambda-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.