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 cristalline 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 to prove it?

Thanks!