I am just learning crystalline cohomology, so I understand the basic set-ups. But I can't really do any calculations.


For example, let's choose the base $S=W(k)/p^n$, and let $X$ be an **affine** scheme over $S$, so it is represented by a $W(k)/p^n$-algebra $A$. Let's just consider the structural sheaf $\mathcal O\_{X/S}$ and the sheaf of PD ideal $\mathcal J\_{X/S}$ . Now, what is the zero-th cohomology:


1. $H^0\_{cris}((X/S) ,\mathcal O\_{X/S})$
2. $H^0\_{cris}((X/S) ,\mathcal J\_{X/S}) $
3. $H^0\_{cris}((X/S) ,\mathcal J\_{X/S}^{[i]}) $


I believe that there must be some books explaining this basic example, but I still can't find the reference..


Thanks!



How to calculate zeroth crystalline cohomology
In this generality, this is unfortunately not so easy because it requires to compute universal PD-envelopes. The case where $X/S$ is embedded in a smooth scheme $Y/S$ is explained in the book by Pierre Berthelot, *Cohomologie cristalline des schémas de caractéristique $p>0$*, Lecture notes in mathematics 407, 1974 (see chapter V). 


You may also look at a paper by Jean-Marc Fontaine (« Cohomologie de de Rham, cohomologie cristalline et représentations $p$-adiques », in *Algebraic geometry Kyoto-Tokyo*,
Lecture notes in mathematics 1016, 1983, p. 86-108). There, he proves that the 0th crystalline cohomology of $O\_{\bar K}$ ($K$ local field) is the ring $B\_{\rm{cris}}$ he had defined earlier.


