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  "body": "\n## Definition\n\nLet $p$ be a [[prime number|01826KMG2H]], and let $\\Z_{(p)}$ be the ring of rational numbers whose denominators are prime to $p$.\nLet $MU_{(p)}$ be the $p$-localization of the [[complex cobordism|5CZHFNY9F3]] spectrum.\nQuillen's idempotent is a map of [[ring spectra|BAW0T68GWK]] $e \\colon MU_{(p)} \\to MU_{(p)}$ with $e \\circ e = e$.\nOn coefficients it sends the [[formal group law|TNHP7XXYZT]] of $MU_{(p)}$ to a $p$-typical one.\nA formal group law is $p$-typical if its logarithm over $\\Q$ has the form $\\sum_{i \\ge 0} \\ell_i x^{p^i}$ with $\\ell_0 = 1$.\nThe *Brown–Peterson spectrum* $BP$ (at $p$) is the image of $e$, that is, the homotopy colimit of $MU_{(p)} \\xrightarrow{e} MU_{(p)} \\xrightarrow{e} \\cdots$.\n*Brown–Peterson cohomology* is the [[generalized cohomology theory|H28PSZR7XG]] $BP^*(-)$ represented by $BP$.\n\n## Remarks\n\nThe coefficient ring is $\\pi_* BP \\cong \\Z_{(p)}[v_1, v_2, \\ldots]$ with $\\abs{v_i} = 2(p^i - 1)$.\nWith Hazewinkel's generators, the logarithm coefficients satisfy $p \\ell_n = \\sum_{i=0}^{n-1} \\ell_i v_{n-i}^{p^i}$.\nFor example, $p \\ell_1 = v_1$, so $\\ell_1 = v_1 / p$.\nThe formal group law of $BP$ is the universal $p$-typical formal group law over a $\\Z_{(p)}$-algebra.\n\nExample: $MU_{(p)}$ is equivalent to a wedge $\\bigvee_j \\Sigma^{2 d_j} BP$, so $BP$ is a retract of $MU_{(p)}$, as an image of an idempotent should be.\n\nNon-example: $MU_{(p)}$ itself is not $BP$, because $\\pi_2 MU_{(p)} \\cong \\Z_{(p)}$ is nonzero for every $p$, while $\\pi_2 BP = 0$ for $p$ odd since the first generator $v_1$ has degree $2(p - 1) \\ge 4$.\n\nBrown and Peterson first built $BP$ in 1966 by a different construction; Quillen's idempotent gives the description above.\n",
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