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  "body": "\n## Summary\n\nChromatic homotopy theory studies the [[stable homotopy category|26GTNMAF7B]], one prime $p$ at a time, by splitting it into layers indexed by a height $n = 0, 1, 2, \\ldots$.\nThe height comes from the height of formal groups, through [[Quillen's theorem|4E518M24MA]] and [[Brown–Peterson cohomology|S8S72AA9C8]].\nEach layer is seen by a Morava K-theory $K(n)$ and is governed by $v_n$-periodic phenomena.\nThe [[nilpotence theorem|9N027RT5R6]] and the other [[Ravenel conjectures|B8XP2K7RS1]] proved by Devinatz, Hopkins, Ravenel and Smith are its foundations.\n\n## Explanations\n\n### L1 human\n\nThe stable homotopy groups of spheres look chaotic, but they contain families that repeat with fixed periods.\nChromatic homotopy theory sorts these families by a number $n$, like splitting white light into colours with a prism.\nHeight $0$ is rational information, which is simple.\nHeight $1$ is seen by K-theory and contains the image of the J-homomorphism, whose groups repeat with period $8$ at the prime $2$.\nHigher heights contain more subtle periodic families, studied with Morava K-theories and Morava E-theories.\n",
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