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  "body": "\n## Statement\n\nFix a [[prime number|01826KMG2H]] $p$.\nLet $K(n)$ be the $n$-th Morava K-theory, with $K(0) = H\\Q$.\nLet $L_n$ be Bousfield localization with respect to the Johnson–Wilson spectrum $E(n)$, as in [[chromatic homotopy theory|300V40FC1B]].\nA finite $p$-local [[spectrum|1GHF4EKBXS]] $X$ has *type* $n$ if $K(n)_* X \\ne 0$ and $K(m)_* X = 0$ for all $m \u003c n$.\nThe following conjectures of Ravenel are theorems.\n\n1. (Nilpotence) A self-map $f \\colon \\Sigma^d X \\to X$ of a finite $p$-local spectrum is nilpotent if $K(n)_*(f)$ is nilpotent for every $n \\ge 0$.\n   This is a form of the [[nilpotence theorem|9N027RT5R6]].\n2. (Periodicity) If $X$ has type $n \\ge 1$, there is a map $v \\colon \\Sigma^d X \\to X$ such that $K(n)_*(v)$ is an isomorphism and $K(m)_*(v)$ is nilpotent for $m \\ne n$.\n3. (Thick subcategories) Let $\\mathcal{C}_n$ be the category of finite $p$-local spectra of type at least $n$.\n   Every thick subcategory of $\\mathcal{C}_0$ is either some $\\mathcal{C}_n$ or the zero subcategory.\n4. (Smash product) For every spectrum $X$, the natural map $X \\wedge L_n S \\to L_n X$ is an equivalence, where $S$ is the $p$-local sphere spectrum.\n5. (Chromatic convergence) For every finite $p$-local spectrum $X$, the map $X \\to \\operatorname{holim}_n L_n X$ is an equivalence.\n\n## Remarks\n\nRavenel posed these conjectures in 1984, together with the telescope conjecture.\nDevinatz, Hopkins and Smith proved the nilpotence theorem, and Hopkins and Smith deduced the periodicity and thick subcategory theorems from it.\nHopkins and Ravenel proved the smash product and chromatic convergence theorems.\nThe [[telescope conjecture|68650624FF]] is the exception: it holds at heights $0$ and $1$ but fails at every height $n \\ge 2$.\n\nAt work: the sphere has type $0$, and the mod $p$ Moore spectrum has type $1$.\nFor $p$ odd, the periodicity theorem is realised by Adams' map $v_1 \\colon \\Sigma^{2p-2} S/p \\to S/p$, whose iterates give infinite families in the stable homotopy groups of spheres.\n\n## Explanations\n\n### L1 human\n\nRavenel predicted a precise structure for the stable homotopy category, organized by the Morava K-theories.\nEvery finite spectrum has a height, and at its height it admits a periodic self-map.\nThe only ways to cut the category into closed pieces are by height.\nAlmost all of these predictions are now theorems.\n\n### L2 human\n\nThe thick subcategory theorem lets one prove a property for all finite spectra of type at least $n$ by checking it on a single example.\nThe periodicity theorem produces the $v_n$-periodic families in the homotopy of spheres that chromatic homotopy theory studies.\nThe smash product theorem says that $L_n$ is a smashing localization, so it is determined by $L_n S$.\nChromatic convergence says that nothing is lost by studying a finite spectrum layer by layer.\n",
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