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  "body": "\n## Definition\n\nLet $k$ be a field and let $X$ be a [[scheme|5A35TYZD02]] that is [[smooth|EXKGG9X6J0]] over $k$.\nFor $p \\ge 0$ let $\\Omega^p_{X/k} = \\bigwedge^p \\Omega^1_{X/k}$, where $\\Omega^1_{X/k}$ is the [[sheaf of differentials|6VC6HQTFSA]], built from [[Kähler differentials|RG6WFFW7K3]].\nThe exterior derivative $d \\colon \\Omega^p_{X/k} \\to \\Omega^{p+1}_{X/k}$ is the unique $k$-linear map with $d(f) = df$ on functions, $d \\circ d = 0$, and $d(\\alpha \\wedge \\beta) = d\\alpha \\wedge \\beta + (-1)^p \\alpha \\wedge d\\beta$ for $\\alpha$ of degree $p$.\nThe complex $\\Omega^\\bullet_{X/k} = (\\Omega^0_{X/k} \\to \\Omega^1_{X/k} \\to \\Omega^2_{X/k} \\to \\cdots)$ is the *algebraic de Rham complex*.\nThe *algebraic de Rham cohomology* of $X$ is the hypercohomology $H^i_{\\mathrm{dR}}(X/k) = \\mathbb{H}^i(X, \\Omega^\\bullet_{X/k})$.\nThis is the $i$-th [[sheaf cohomology|PQ6Q143ARJ]] of the complex $\\Omega^\\bullet_{X/k}$, computed from an injective resolution of the complex.\n\n## Remarks\n\nIf $X = \\operatorname{Spec} A$ is affine, the sheaves $\\Omega^p_{X/k}$ are quasi-coherent and have no higher cohomology.\nThen $H^i_{\\mathrm{dR}}(X/k)$ is the $i$-th cohomology of the complex of $A$-modules $\\Omega^\\bullet_{A/k}$.\nExample: let $k$ have characteristic $0$ and $A = k[t, t^{-1}]$.\nThen $\\Omega^1_{A/k} = A \\, dt$ and $d(t^n) = n t^{n-1} dt$.\nThe kernel of $d$ on $A$ is $k$, and the image of $d$ contains $t^m \\, dt$ for every $m \\ne -1$.\nSo $H^0_{\\mathrm{dR}} = k$ and $H^1_{\\mathrm{dR}}$ is spanned by the class of $dt/t$.\nNon-example of good behaviour: if $k$ has characteristic $p \u003e 0$ and $A = k[t]$, then $d(t^p) = 0$.\nSo $H^0_{\\mathrm{dR}}(\\mathbb{A}^1_k/k) = k[t^p]$ is infinite-dimensional, and the theory does not count connected components as in characteristic $0$.\nGrothendieck's comparison theorem: if $X$ is smooth over $\\C$, then $H^i_{\\mathrm{dR}}(X/\\C) \\cong H^i(X(\\C), \\C)$ for all $i$.\n",
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