Equivariant $K$-theory as a global localization

Abstract

We establish a global equivariant refinement of the theorem of Snaith that presents the complex $K$-theory spectrum as a localization of the suspension spectrum of $\mathbb C P^\infty$. We prove that the morphism of ultra-commutative global ring spectra from $Σ^\infty_+ B_{\text{gl}} U(1)$ that classifies the tautological $U(1)$-representation exhibits the global $K$-theory spectrum as a specific localization, in both the $\infty$-categories of ultra-commutative ring spectra and of commutative global ring spectra. The classes to be inverted are certain representation-graded $U(n)$-equivariant homotopy classes of $Σ^\infty_+ B_{\text{gl}} U(1)$, the pre-Bott classes. We also show that for every compact Lie group $G$, the $G$-equivariant $K$-theory spectrum is a telescopic localization of $Σ^\infty_+ B_G U(1)$ by the pre-Bott classes of all unitary $G$-representations.