nLab
chromatic homotopy theory

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 $p\in \mathbb{N}$ and for each $n\in \mathbb{N}$ there is a Bousfield localization of spectra

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

$$X\to \cdots \to L_{n}X\to L_{n-1}X\to \cdots \to L_{0}X.$$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

$$X\to X_{(p)}$$X \to X_{(p)}

of $X$.

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

$$L_{n}X\simeq L_{K(n)}X\underset{L_{n-1}{L}_{K(n)}X}{\times }{L}_{n-1}X$$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 level	complex oriented cohomology theory	E-∞ ring/A-∞ ring
0	ordinary cohomology	Eilenberg-MacLane spectrum $H\mathbb{Z}$
	Morava K-theory	$K(0)$
1	complex K-theory	complex K-theory spectrum $\mathrm{KU}$
	Morava K-theory	$K(1)$
2	tmf	tmf spectrum
	Morava K-theory	$K(2)$

Related concepts

• stable homotopy theory

• telescopic complexity?

• Morava K-theory

  • ∞-field

  • integral Morava K-theory

• Morava E-theory

  • Lubin-Tate theory

References

• Haynes Miller, “Chromatic” homotopy theory May 2011 (pdf)

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010 (lecture notes)

• Doug Ravenel, Toward higher chromatic analogs of elliptic cohomology and topological modular forms, talk notes (2007) (pdf)

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)