# nLab chromatic homotopy theory

Contents

cohomology

### Theorems

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

We use the word ‘chromatic’ here for the following reason. The $n$-th subquotients in the chromatic filtration consists of $v_n$-periodic elements. As illustrated in 2.4.2, these elements fall into periodic families. The chromatic filtration is thus like a spectrum in the astronomical sense in that it resolves the stable homotopy groups of a finite complex into periodic families of various periods. Comparing these to the colors of the rainbow led us to the word ‘chromatic’.

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 “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 abstractly, this filtering is induced by the prime spectrum of a symmetric monoidal stable (∞,1)-category of the (∞,1)-category of spectra for p-local finite spectra, which is labeled by the Morava K-theories.

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 (see at smash product theorem and see this remark at fracture square )

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

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

## 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

### Stable case

Original articles:

A quick idea:

• Mark Mahowald, Doug Ravenel, Section 6 of: Towards a Global Understanding of the Homotopy Groups of Spheres, pages 57-74 in Part II of: The Lefschetz Centennial Conference – Proceedings on Algebraic Topology, Proceedings of the Lefschetz Centennial Conference held December 10-14, 1984, Contemporary Mathematics 58, American Mathematical Society, 1987. (ISBN:978-0-8218-5063-3, pdf, pdf)

Historical introduction:

Comprehensive lecture notes:

Survey:

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

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

• Glossary for stable and chromatic honotopy theory (pdf)

• David Mehrle, Chromatic homotopy theory: Journey to the frontier, Graduate workshop notes, Boulder 16-17 May 2018, (pdf, pdf)

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)

Relation of chromatic homotopy theory to Goodwillie calculus is discussed in

Categorical ultraproducts are used to provide asymptotic approximations in

### Unstable case

There are also chromatic approximations in ordinary (not stabilized) homotopy theory:

Last revised on January 14, 2024 at 05:23:32. See the history of this page for a list of all contributions to it.