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 ℂ P∞. We prove that the morphism of ultra-commutative global ring spectra from Σ∞+ Bgl U(1) that classifies the tautological U(1)-representation exhibits the global K-theory spectrum as a specific localization, in both the ∞-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∞+ Bgl 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∞+ BG U(1) by the pre-Bott classes of all unitary G-representations.
Related
- An elementary proof of rational injectivity in equivariant KK-theory
- Graded Representation Theory of Equivariant Neural Networks
- Equivariant deformation theory and deformations of homogeneous varieties
- Twisted ambidexterity in equivariant homotopy theory
- Complex Projective Spaces in Rational SO(2)-Equivariant Stable Homotopy Theory
BibTeX
@misc{schwede2026equivariant,
title = {{Equivariant $K$-theory as a global localization}},
author = {Stefan Schwede},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21675},
eprint = {2610.21675},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21675},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Equivariant K-theory as a global localization
AU - Stefan Schwede
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21675
UR - https://arxiv.org/abs/2610.21675
AB - 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 ℂ P^∞. We prove that the morphism of ultra-commutative global ring spectra from Σ^∞_+ B_gl U(1) that classifies the tautological U(1)-representation exhibits the global K-theory spectrum as a specific localization, in both the ∞-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^∞_+ B_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^∞_+ B_G U(1) by the pre-Bott classes of all unitary G-representations.
ER -