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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]].\nA *(one-dimensional, commutative) formal group law* over $R$ is a [[formal power series|072W67KFYK]] $F(x, y) \\in R[\\![x, y]\\!]$ such that:\n\n1. $F(x, y) \\equiv x + y$ modulo terms of total degree at least $2$.\n2. $F(x, F(y, z)) = F(F(x, y), z)$ in $R[\\![x, y, z]\\!]$.\n3. $F(x, y) = F(y, x)$.\n\nLet $E$ be a multiplicative [[generalized cohomology theory|H28PSZR7XG]] with coefficient ring $E^* = E^*(\\mathrm{pt})$.\nA *complex orientation* of $E$ is a class $x \\in \\tilde E^2(\\C P^\\infty)$ whose restriction to $\\tilde E^2(\\C P^1) \\cong E^0(\\mathrm{pt})$ is the unit $1$.\nThen $E^*(\\C P^\\infty) = E^*[\\![x]\\!]$ and $E^*(\\C P^\\infty \\times \\C P^\\infty) = E^*[\\![x_1, x_2]\\!]$, with $x_i$ pulled back from the $i$-th factor.\nLet $m \\colon \\C P^\\infty \\times \\C P^\\infty \\to \\C P^\\infty$ classify the tensor product of the two universal line bundles.\nThe *formal group law of $(E, x)$* is the power series $F$ over $E^*$ with $m^*(x) = F(x_1, x_2)$.\nIt satisfies the three conditions above, and it is homogeneous of degree $2$ when $x$, $x_1$ and $x_2$ have degree $2$.\n\n## Remarks\n\nExample: in ordinary cohomology $H^*(-; \\Z)$ with $x$ the first Chern class of the universal line bundle, $c_1(L_1 \\otimes L_2) = c_1(L_1) + c_1(L_2)$, so $F(x, y) = x + y$.\nThe three conditions are clear for this additive law.\n\nExample: in complex K-theory with Bott class $\\beta \\in K^{-2}(\\mathrm{pt})$, take $x = \\beta^{-1}([L] - 1)$ for the universal line bundle $L$.\nSince $[L_1 L_2] - 1 = ([L_1] - 1) + ([L_2] - 1) + ([L_1] - 1)([L_2] - 1)$, the law is $F(x, y) = x + y + \\beta x y$, the multiplicative formal group law.\n\nNon-example: $G(x, y) = x + y + x^2 y$ over $\\Z$ satisfies clause 1 but fails clause 3, since $G(y, x) = x + y + x y^2$.\n\nThere is a universal formal group law over the Lazard ring $L$: formal group laws over $R$ correspond to ring maps $L \\to R$.\n",
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