Hello,


Let $k$ be a field of characteristic $p > 0$, and let $Y$ be a $k$-scheme. Consider the
sites $Y\_{syn}$ and $(Y/W\_n)\_{cris}$ (where $W\_n$ are the Witt vectors of $k$ of length $n$), of $Y$ with its syntomic topology and its crystalline topology. Then the assignment $\mathcal O\_{cris}:Z\mapsto H^0\_{cris}(Y/W\_n)$ is a sheaf on $Y\_{syn}$. It is a fact that
$H^\*\_{syn}(Y,\mathcal O\_{cris})$ is canonically isomorphic to $H^{\*}\_{cris}(Y/W\_n)$, but I don't see how to prove it.


So my question is: How does one prove this fact?


Thanks!



Crystalline cohomology via the syntomic site
A sketch of the proof is as follows:


Consider the site $Y\_{syn-cris}$ where the objects are the same as in $Y\_{cris}$ but the
covering families are surjective syntomic families. Then there are maps of topoi:
$\alpha : Sh(Y\_{syn-cris})\to Sh(Y\_{syn})$ and $\beta : Sh(Y\_{syn-cris})\to Sh((Y/W\_n)\_ {cris})$,
defined by $\beta\_\*(F)(U,T) = F(U, T)$ and $\alpha\_{\*}(F)(U) = H^0\_{syn-cris}(U/W\_n,F)$.


Lemma. $R^i\beta\_\*\mathcal O\_{Y/W\_n} = R^i\alpha\_\*\mathcal O\_{Y/W\_n} = 0$ for $i > 0$.


This implies the result in your question by a standard application of the Leray spectral
sequence.


As for the lemma: To prove that $R^i\beta\_\*\mathcal O\_{Y/W\_n} = 0$ for $i > 0$, it boils
down to checking that if $U$ is an affine in $(Y/W\_n)\_ {cris}$ then $H^i\_{syn}(U,\mathcal O\_U) = 0$
for $i > 0$, and this follows the theory of the Cech complex (it is acyclic because of faithful flatness of the syntomic cover).


To prove that $R^i\alpha\_\*\mathcal O\_{Y/W\_n} = 0$, we have to check that if $U$ is an open of
$Y\_{syn}$, and $s\in H^i\_{cris}(U/W\_n)$ (strictly speaking we have to do the computation with syntomic-cristalline cohomology, but by the previous part, the cohomology groups coincide with the crystalline cohomology groups) then there exists a syntomic cover $U\_i\to U$ such
that $s\mid U\_i = 0\in H^i\_{cris}(U\_i/W\_n)$. Now recall that we can compute this cohomology groups as the hypercohomology groups of the de Rham complex of the divided power envelope of some embedding into a smooth scheme. That means, after shrinking, we can represent $s$ as an $i$-form.
We need to find a syntomic cover such that when we restrict $s$ to this cover, it vanishes. To do this, note that $A[T]\to A[T^{p^{-n}}]$ is a syntomic cover that has the property that the image of
$dT$ is zero.


