The geometric K-theory of quotient stacks

TL;DR

This paper refines the geometric K-theory of quotient stacks, establishing its properties and its relation to the K-theory of the stack itself, extending previous conjectures to a broader setting.

Contribution

It extends the geometric K-theory construction to quotient stacks over arbitrary excellent bases and proves its intrinsic decomposition and computational properties.

Findings

Refined geometric K-theory for quotient stacks over arbitrary bases.

Proved the intrinsic decomposition of K-theory of quotient stacks.

Established properties facilitating computations of K-theory.

Abstract

Given a quotient of a regular noetherian separated algebraic space $X$ over a field by an affine algebraic group $G$ having finite stabilizers (with some mild technical conditions), G. Vezzosi and A. Vistoli defined the geometric part of the rational equivariant K-theory $K(X,G)$ and conjectured that it is isomorphic to the rational K-theory of the quotient $X/G$. In this paper we refine the construction of geometric K-theory to the rational K-theory of a quotient stack $[X/G]$ over an arbitrary excellent base; we show that it is part of an intrinsic decomposition of the K-theory of the stack and prove many properties that make it amenable to computations.