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  "body": "\n## Summary\n\nÉtale cohomology is a cohomology theory for schemes, built from sheaves on the étale site instead of the Zariski topology.\nGrothendieck and Artin introduced it to obtain, for varieties over any field, groups that behave like the singular cohomology of a complex variety.\nIt contains [[Galois cohomology|9MPCZ3EAJM]] as the case of a point.\nIt is the setting of [[ℓ-adic cohomology|E1DVHN06CR]] and of Deligne's proof of the [[Weil conjectures|81TXP32VDM]].\n\n## Explanations\n\n### L1 human\n\nThe Zariski topology of a variety has too few open sets: on an irreducible variety, any two nonempty open sets meet.\nSo sheaf cohomology with constant coefficients in that topology sees almost nothing.\nÉtale cohomology allows \"open sets\" that are étale maps $U \\to X$, which look like local isomorphisms, such as finite unramified covers.\nThese play the role of small discs in topology.\nFor a smooth projective complex curve of genus $g$, the first étale cohomology with coefficients $\\Z/n\\Z$ is $(\\Z/n\\Z)^{2g}$, as in topology.\n",
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