# 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 (via the Landweber exact functor theorem), the height filtration of formal groups induces a “chromaticfiltration 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 $p \in \mathbb{N}$ and for each $n \in \mathbb{N}$ there is a Bousfield localization of spectra

$L_n \coloneqq L_{K(0)\vee \cdots \vee K(n)} \,,$

where $K(n)$ 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

$X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X \,.$

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

$X \to X_{(p)}$

of $X$.

Since moreover $L_n X$ is the homotopy fiber product

$L_n X \simeq L_{K(n)}X \underset{L_{n-1}L_{K(n)}X}{\times} L_{n-1}X$

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

## Examples

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ring
0ordinary cohomologyEilenberg-MacLane spectrum $H \mathbb{Z}$
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomology$Ell_E$
tmftmf spectrum
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
$n$$n$th Morava K-theory$K(n)$
$n$th Morava E-theory$E(n)$
$n+1$algebraic K-theory applied to chrom. level $n$$K(E_n)$ (red-shift conjecture)
$\infty$

## References

A quick idea is given in section 6 of

A good historical introduction is in

Comprehensive lecture notes are in

Brief surveys include

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

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

A collection of various lecture notes and other material is kept at

Random useful discussion is in

Discussion of generalization of elliptic cohomology to higher chromatic homotopy theory is discussed in

• Doug Ravenel, Toward higher chromatic analogs of elliptic cohomology pdf

• Doug Ravenel, Toward higher chromatic analogs of elliptic cohomology II, Homology, Homotopy and Applications, vol. 10(1), 2008, pp.1-36 (pdf, pdf slides)

Revised on January 30, 2014 12:47:36 by Urs Schreiber (82.113.121.9)