Crystalline cohomology

Definition

Let $k$ be a perfect field of characteristic $p > 0$, let $W = W(k)$ be its ring of Witt vectors, and let $W_n = W/p^n W$. Let $X$ be a scheme that is smooth and proper over $k$. A 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 $x \mapsto x^m/m!$. The ideal $pW_n$ of $W_n$ has a canonical one. The crystalline site $\mathrm{Cris}(X/W_n)$ is the site defined as follows.

1. 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 $I$. Here $\delta$ is a divided power structure on $I$ compatible with the one on $pW_n$.
2. Its morphisms are pairs of compatible maps $U \to U'$ and $T \to T'$ that respect the divided powers.
3. 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 $T$.

The structure sheaf $\mathcal{O}_{X/W_n}$ sends $(U, T, \delta)$ to $\Gamma(T, \mathcal{O}_T)$. The crystalline cohomology of $X$ is $$H^i_{\mathrm{cris}}(X/W) = \varprojlim_n H^i\big(\mathrm{Cris}(X/W_n), \mathcal{O}_{X/W_n}\big),$$ an inverse limit of sheaf cohomology groups on these sites.