nLab
chromatic localization

Contents

Idea

The Bousfield localization of spectra L E(n)L_{E(n)} at nnth Morava E-theory is called chromatic localization. The tower of chromatic localizations as nn ranges is called the chromatic tower, leading to the chromatic filtration. This is the subject of chromatic homotopy theory.

Properties

Relation to formal groups

Chromatic localization on complex oriented cohomology theories is like the restriction to the closed substack

FG n+1 FG×Spec (p) \mathcal{M}_{FG}^{\leq n+1} \hookrightarrow \mathcal{M}_{FG} \times Spec \mathbb{Z}_{(p)}

of the moduli stack of formal groups on those of height n+1\geq n+1.

(e.g. Lurie, lect 22, above theorem 1)

In this way the localization tower at the Morava E-theories exhibits the chromatic filtration in chromatic homotopy theory.

Examples

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum HH \mathbb{Z}HZR-theory
0th Morava K-theoryK(0)K(0)
1complex K-theorycomplex K-theory spectrum KUKUKR-theory
first Morava K-theoryK(1)K(1)
first Morava E-theoryE(1)E(1)
2elliptic cohomologyelliptic spectrum Ell EEll_E
second Morava K-theoryK(2)K(2)
second Morava E-theoryE(2)E(2)
algebraic K-theory of KUK(KU)K(KU)
3 …10K3 cohomologyK3 spectrum
nnnnth Morava K-theoryK(n)K(n)
nnth Morava E-theoryE(n)E(n)BPR-theory
n+1n+1algebraic K-theory applied to chrom. level nnK(E n)K(E_n) (red-shift conjecture)
\inftycomplex cobordism cohomologyMUMR-theory

References