# nLab chromatic homotopy theory

cohomology

### Theorems

#### Stable Homotopy theory

stable homotopy theory

# Contents

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

By the construction of complex oriented cohomology theories from formal groups, the height filtration of formal groups induces a “chromatic” filtration on complex oriented cohomology theories. Chromatic homotopy theory is the study of stable homotopy theory and specifically of complex oriented cohomology theories by means of and along this chromatic filtration.

More in detail, for each prime $p\in ℕ$ and for each $n\in ℕ$ there is a Bousfield localization of spectra

${L}_{n}≔{L}_{K\left(0\right)\vee \cdots \vee K\left(n\right)}\phantom{\rule{thinmathspace}{0ex}},$L_n \coloneqq L_{K(0)\vee \cdots \vee K(n)} \,,

where $K\left(n\right)$ is the $n$th Morava K-theory (for the given prime $p$). These arrange into the chromatic tower? which for each spectrum $X$ is of the form

$X\to \cdots \to {L}_{n}X\to {L}_{n-1}X\to \cdots \to {L}_{0}X\phantom{\rule{thinmathspace}{0ex}}.$X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X \,.

The chromatic convergence theorem? states mild conditions under which the homotopy limit over this tower is the $p$-localization

$X\to {X}_{\left(p\right)}$X \to X_{(p)}

of $X$.

Since moreover ${L}_{n}X$ is the homotopy fiber product

${L}_{n}X\simeq {L}_{K\left(n\right)}X\underset{{L}_{n-1}{L}_{K\left(n\right)}X}{×}{L}_{n-1}X$L_n X \simeq L_{K(n)}X \underset{L_{n-1}L_{K(n)}X}{\times} L_{n-1}X

it follows that in principle one can study a spectrum $X$ by understanding all its “chromatic pieces” ${L}_{K\left(n\right)}X$.

## Examples

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ring
0ordinary cohomologyEilenberg-MacLane spectrum $Hℤ$
Morava K-theory$K\left(0\right)$
1complex K-theorycomplex K-theory spectrum $\mathrm{KU}$
Morava K-theory$K\left(1\right)$
2tmftmf spectrum
Morava K-theory$K\left(2\right)$

## References

A lighning review of results by Henn with Goerss, Mahowald, Rezk, and Karamanov is in

• Hans-Werner Henn, Recent developments in stable homotopy theory (pdf)

Revised on June 17, 2013 20:54:34 by Urs Schreiber (89.204.138.194)