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  "body": "\n## Summary\n\nLet $X$ be a smooth variety over a field $k$.\nIts algebraic de Rham cohomology is the hypercohomology of the complex of algebraic differential forms on $X$.\nIt replaces smooth differential forms by [[Kähler differentials|7HCFARHG7D]], so it makes sense over any field.\nOver $\\C$, Grothendieck showed that it agrees with the singular cohomology of $X(\\C)$ with complex coefficients.\nIt is the algebraic form of [[de Rham cohomology|GX74C07T0G]] and a model for later cohomology theories such as [[crystalline cohomology|EVXCN06FW3]].\n\n## Explanations\n\n### L1 human\n\nOn a manifold, de Rham cohomology counts closed differential forms that are not derivatives.\nOn a variety, one can do the same with forms built from polynomials, such as $dx$ or $x \\, dy$.\nFor the punctured line $\\C \\setminus \\{0\\}$, with functions $t^n$ for all integers $n$, every form $t^n \\, dt$ with $n \\ne -1$ is the derivative of $t^{n+1}/(n+1)$.\nThe form $dt/t$ is closed but not a derivative, and it spans the first cohomology.\nThis matches the one loop around the origin in the topology of the punctured plane.\n",
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