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  "body": "\n## Summary\n\nLet $X$ be a variety over a separably closed field, and let $\\ell$ be a prime different from the characteristic.\nThe $\\ell$-adic cohomology of $X$ is the inverse limit of the [[étale cohomology|7YX76G0838]] groups of $X$ with coefficients $\\Z/\\ell^n\\Z$.\nIt gives modules over the [[$\\ell$-adic integers|GBN9F8FAQ3]], and after tensoring, vector spaces over $\\Q_\\ell$, a field of characteristic $0$.\nThese groups behave like the singular cohomology of a complex variety, with Poincaré duality, a Künneth formula and a Lefschetz trace formula.\nThey are the cohomology used in the proof of the [[Weil conjectures|81TXP32VDM]], and the Galois action on them gives [[Galois representations|JQEYEQJY2G]].\n\n## Explanations\n\n### L1 human\n\nÉtale cohomology with finite coefficients $\\Z/n\\Z$ works well, but one wants coefficients in a field of characteristic $0$, like $\\Q$ in topology.\nCoefficients $\\Q$ directly give the wrong answer, so one takes the coefficients $\\Z/\\ell, \\Z/\\ell^2, \\Z/\\ell^3, \\ldots$ and passes to the limit.\nFor an elliptic curve over an algebraically closed field, the first $\\ell$-adic cohomology is a $2$-dimensional space over $\\Q_\\ell$.\nIt is dual to the space built from the [[Tate module|4CSMQFTGD8]], made of the points of order $\\ell^n$ of the curve.\n",
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