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  "body": "\n## Definition\n\nLet $k$ be a perfect field of characteristic $p \u003e 0$, let $W = W(k)$ be its ring of [[Witt vectors|C9Z22J9SQE]], and let $W_n = W/p^n W$.\nLet $X$ be a [[scheme|5A35TYZD02]] that is [[smooth|EXKGG9X6J0]] and [[proper|9537RKBZPP]] over $k$.\nA *divided power structure* on an ideal $I$ of a ring is a family of maps $\\gamma_m \\colon I \\to I$, $m \\ge 1$, with the identities of $x \\mapsto x^m/m!$.\nThe ideal $pW_n$ of $W_n$ has a canonical one.\nThe *crystalline site* $\\mathrm{Cris}(X/W_n)$ is the [[site|NYE0Z94MME]] defined as follows.\n\n1. Its objects are triples $(U, T, \\delta)$ with $U \\subset X$ Zariski open and $U \\hookrightarrow T$ a closed immersion into a $W_n$-scheme $T$ with ideal $I$.\n   Here $\\delta$ is a divided power structure on $I$ compatible with the one on $pW_n$.\n2. Its morphisms are pairs of compatible maps $U \\to U'$ and $T \\to T'$ that respect the divided powers.\n3. Its coverings are the families $\\set{(U_i, T_i, \\delta_i) \\to (U, T, \\delta)}$ in which the maps $T_i \\to T$ are open immersions that cover $T$.\n\nThe *structure sheaf* $\\mathcal{O}_{X/W_n}$ sends $(U, T, \\delta)$ to $\\Gamma(T, \\mathcal{O}_T)$.\nThe *crystalline cohomology* of $X$ is\n$$H^i_{\\mathrm{cris}}(X/W) = \\varprojlim_n H^i\\big(\\mathrm{Cris}(X/W_n), \\mathcal{O}_{X/W_n}\\big),$$\nan [[inverse limit|DZB378CNCK]] of sheaf cohomology groups on these sites.\n\n## Remarks\n\nEach $H^i_{\\mathrm{cris}}(X/W)$ is a finitely generated $W$-module, and it vanishes for $i \u003e 2 \\dim X$.\nIf $\\mathfrak{X}$ is a smooth proper scheme over $W$ with special fiber $X$, then $H^i_{\\mathrm{cris}}(X/W) \\cong H^i_{\\mathrm{dR}}(\\mathfrak{X}/W)$, the [[algebraic de Rham cohomology|13F5JZPDD0]] of the lift.\nExample: for $X = \\mathbb{P}^1_k$, the lift $\\mathbb{P}^1_W$ gives $H^0_{\\mathrm{cris}} \\cong H^2_{\\mathrm{cris}} \\cong W$ and $H^1_{\\mathrm{cris}} = 0$.\nNon-example: replacing $W$ by $k$, that is, taking only $n = 1$, gives the de Rham cohomology $H^i_{\\mathrm{dR}}(X/k)$.\nThis is a vector space over a field of characteristic $p$, so it cannot give coefficients in a field of characteristic $0$.\nAfter tensoring with the fraction field $K$ of $W$, crystalline cohomology is a Weil cohomology theory, with the Frobenius of $X$ acting semilinearly.\n",
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