Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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!

share|improve this question

1 Answer 1

up vote 7 down vote accepted

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.

share|improve this answer
1  
Thanks! Do you know any other simple example besides the B_cris one? What if, say, X is the affine space $A^n$ or a quotient of $A^n$ by some "nice" ideals? –  natura Dec 26 '12 at 15:20

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.