How to calculate zeroth crystalline cohomology

Question

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!

Answer 1

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.