Equivariant K-theory as a global localization

A celebrated theorem of Snaith shows that complex K-theory can be obtained from the unreduced suspension spectrum $\mathbb C P^\infty$ by inverting the Bott class. A contemporary reformulation of this result is that complex K-theory classifies orientations of the multiplicative group in derived algebraic geometry.

In want to explain a highly structured, globally-equivariant refinement of Snaith's localization result: a specific morphism of ultra-commutative global ring spectra from the unreduced suspension spectrum of the global classifying space of $U(1)$ to the global K-theory spectrum is a localization away from the "pre-Bott classes", certain representation-graded $U(n)$-equivariant homotopy classes for $n>0$.

The localization property holds in different, but interrelated senses:

- as universal morphisms in the $\infty$-categories of ultra-commutative ring spectra and of commutative global ring spectra;
- for every compact Lie group $G$, as universal examples in the $\infty$-categories of ultra-commutative G-ring spectra and of commutative $G$-ring spectra;
- for every compact Lie group $G$, at the level of $G$-equivariant cohomology theories.

The construction of the pre-Bott classes exploits the global functoriality and power operations.