The geometric K-theory of quotient stacks
Francesco Sala, Laurent Schadeck, Angelo Vistoli
TL;DR
The paper extends Vezzosi–Vistoli's decomposition of rational equivariant K-theory to the K-theory of quotient stacks 𝓧=[X/G] with finite stabilizers over an excellent base, introducing a geometric component of K-theory and proving that the full rational K-theory admits an intrinsic decomposition compatible with stack presentations. It defines a $\mathcal{C}(G)$-decomposition over essential dual cyclic subgroups and a $μ_r$-localization framework, isolating a geometric part $K'_{*}(\mathcal{X})_{\mathbf{g}}$ that captures the quotient geometry and an algebraic part $K'_{*}(\mathcal{X})_{\mathbf{a}}$ that vanishes precisely when 𝓧 is an algebraic space. The authors establish functoriality results: $f^{*}$ preserves the geometric decomposition for representable maps of finite flat dimension, and $f_{*}$ preserves it for proper, relatively tame maps, enabling descent and ascent arguments. A tameness criterion links the vanishing of higher μ_r-parts to the underlying stack being an algebraic space, providing a computationally tractable handle on the rational K-theory of quotient stacks and enabling computation via the geometric component.
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.