Stefan Schwede: Equivariant K-theory as a coherent localization

Abstract.

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.