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  "body": "\n## Summary\n\nBrown–Peterson cohomology is a [[generalized cohomology theory|H28PSZR7XG]] attached to a prime $p$, represented by a [[ring spectrum|BAW0T68GWK]] $BP$.\nAfter [[complex cobordism|5CZHFNY9F3]] $MU$ is localized at $p$, it splits as a wedge of suspensions of copies of $BP$.\nSo $BP$ keeps all the $p$-local information of $MU$ in a much smaller object, with coefficient ring $\\pi_* BP = \\Z_{(p)}[v_1, v_2, \\ldots]$ and $\\abs{v_i} = 2(p^i - 1)$.\nIt is the basic tool of the Adams–Novikov spectral sequence and of chromatic homotopy theory.\n\n## Explanations\n\n### L1 human\n\nComplex cobordism is a very powerful cohomology theory, but it is large and hard to compute with.\nWhen one only cares about one prime $p$, most of it repeats the same pattern many times.\nBrown–Peterson cohomology is one copy of this pattern.\nIts coefficients are polynomials in variables $v_1, v_2, \\ldots$, and for $p = 2$ these sit in degrees $2, 6, 14, 30, \\ldots$, that is $2(2^i - 1)$.\nThe variable $v_n$ is the source of the \"height $n$\" layer studied in chromatic homotopy theory.\n",
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