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  "body": "\n## Summary\n\nCrystalline cohomology is a cohomology theory for varieties over a perfect field $k$ of characteristic $p \u003e 0$.\nIts values are modules over the ring $W(k)$ of [[Witt vectors|6N3X2QX4W9]].\nGrothendieck proposed it and Berthelot developed it, to obtain a $p$-adic cohomology in characteristic $p$, where [[ℓ-adic cohomology|E1DVHN06CR]] with $\\ell = p$ fails.\nWhen the variety lifts to $W(k)$, it is the [[algebraic de Rham cohomology|13F5JZPDD0]] of the lift.\nSo it is a de Rham theory that does not depend on the choice of a lift.\nIt carries a Frobenius action and is a main object of $p$-adic Hodge theory.\n\n## Explanations\n\n### L1 human\n\nDe Rham cohomology over a field of characteristic $p$ gives vector spaces over that field, and it loses information, because differentiating $t^p$ gives $0$.\nOne would like to lift the variety to the $p$-adic integers, or more generally to $W(k)$, and compute de Rham cohomology there.\nLifts do not always exist and are not unique.\nCrystalline cohomology computes what the answer would be from the variety alone, using all its thickenings at once.\nFor the projective line over $\\F_p$ it gives $\\Z_p$ in degrees $0$ and $2$, as the de Rham cohomology of the projective line over $\\Z_p$ does.\n",
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