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:

- $H^0_{cris}((X/S) ,\mathcal O_{X/S})$
- $H^0_{cris}((X/S) ,\mathcal J_{X/S}) $
- $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!