---
tag: 300V40FC1B
kind: concept
name: Chromatic homotopy theory
msc: [55P42, 55Q51, 55P60]
sources:
  - url: https://en.wikipedia.org/wiki/Chromatic_homotopy_theory
same_as:
  wikidata: Q16958142
  wikipedia: Chromatic homotopy theory
---

## Summary

Chromatic homotopy theory studies the [[stable homotopy category|26GTNMAF7B]], one prime $p$ at a time, by splitting it into layers indexed by a height $n = 0, 1, 2, \ldots$.
The height comes from the height of formal groups, through [[Quillen's theorem|4E518M24MA]] and [[Brown–Peterson cohomology|S8S72AA9C8]].
Each layer is seen by a Morava K-theory $K(n)$ and is governed by $v_n$-periodic phenomena.
The [[nilpotence theorem|9N027RT5R6]] and the other [[Ravenel conjectures|B8XP2K7RS1]] proved by Devinatz, Hopkins, Ravenel and Smith are its foundations.

## Explanations

### L1 human

The stable homotopy groups of spheres look chaotic, but they contain families that repeat with fixed periods.
Chromatic homotopy theory sorts these families by a number $n$, like splitting white light into colours with a prism.
Height $0$ is rational information, which is simple.
Height $1$ is seen by K-theory and contains the image of the J-homomorphism, whose groups repeat with period $8$ at the prime $2$.
Higher heights contain more subtle periodic families, studied with Morava K-theories and Morava E-theories.