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  "body": "\n## Statement\n\nLet $R$ be a homotopy associative [[ring spectrum|BAW0T68GWK]], and let $MU$ be the spectrum of [[complex cobordism|5CZHFNY9F3]].\nLet $h \\colon \\pi_* R \\to MU_* R$ be the Hurewicz homomorphism, which sends $\\alpha$ to the image of $\\alpha$ under the unit map $R \\to MU \\wedge R$.\nThen every element of the kernel of $h$ is nilpotent.\n\n## Remarks\n\nThe theorem was proved by Devinatz, Hopkins and Smith in 1988.\nThe proof filters $MU$ by a sequence of Thom spectra $X(n)$ built from $\\Omega SU(n)$ and shows inductively that detection of nilpotence passes from $X(n+1)$ to $X(n)$.\nFor $R$ the sphere spectrum, $MU_* = \\pi_* MU$ is torsion-free while every element of positive degree in $\\pi_* S$ is torsion.\nSo the Hurewicz image of such an element is zero, and the theorem gives back the [[Nishida nilpotence theorem|FQ048Z65F9]].\nEquivalent forms state that a self-map $\\Sigma^d X \\to X$ of a finite spectrum is nilpotent when its $MU$-homology is nilpotent.\n\n## Explanations\n\n### L1 human\n\nComplex cobordism $MU$ is a homology theory that sees a lot of stable homotopy theory.\nThe theorem says that whatever $MU$ cannot see in a ring spectrum must be nilpotent.\nSo $MU$ detects all the elements that can be iterated forever.\n\n### L2 human\n\nThe nilpotence theorem is the foundation of chromatic homotopy theory.\nTogether with the work of Hopkins and Smith it yields the thick subcategory theorem and the periodicity theorem, which classify finite spectra by their Morava $K$-theory type.\nThese results proved most of the Ravenel conjectures.\n",
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