# Contents

## Idea

The Bousfield localization of spectra $L_{E(n)}$ at $n$th Morava E-theory is called chromatic localization. The tower of chromatic localizations as $n$ 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

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

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

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 $H \mathbb{Z}$HZR-theory
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$KR-theory
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomologyelliptic spectrum $Ell_E$
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
3 …10K3 cohomologyK3 spectrum
$n$$n$th Morava K-theory$K(n)$
$n$th Morava E-theory$E(n)$BPR-theory
$n+1$algebraic K-theory applied to chrom. level $n$$K(E_n)$ (red-shift conjecture)
$\infty$complex cobordism cohomologyMUMR-theory

## References

• Mark Hovey, Hal Sadofsky, Invertible spectra in the $E(n)$-local stable homotopy category (pdf)