# Explicit description of O^{cris}_n in Fontaine/Messing

Let $k$ be a perfect field of characteristic $p$, $W(k)$ the Witt ring and $K$ its quotient field. In their article "$p$-adic periods and $p$-adic etale cohomology" Fontaine and Messing give in II.1.4 an explicit description of the syntomic sheaf $\mathcal{O}^{cris}_n$ which is defined as $\mathcal{O}^{cris}_n(Z) = H^0((Z/W_{n}(k))_{cris},\mathcal{O}_{Z/W_{n}(k)})$ for $Z$ a $k$-Scheme.

In order to do so, for a $k$-Algebra $A$ they construct a morphism $W_n(A) \longrightarrow O^{cris}_n(A)$ via $(a_0, \dots, a_{n-1}) \mapsto \hat a_0^{p^n} + \dots + p^{n-1}\hat a_{n-1}^{p}$, where the $\hat a_i$ should be liftings of the $a_i$ in $\mathcal{O}^{cris}_n(A)$.

As a special case they gave in I.1.3 the morphism $W_n(\mathcal{O}_{\bar K}/p) \to \mathcal{O}_{\bar K}/p^n$. Now, in this special case there is obviously a surjective morphism $\mathcal{O}_{\bar K}/p^n \to \mathcal{O}_{\bar K}/p$ which allows the lifting.

But how does this generalize to $k$-algebras: Why is $\mathcal{O}^{cris}_n(\mathcal{O}_{\bar K}/p) = \mathcal{O}_{\bar K}/p^n$ and why is there a surjective morphism $\mathcal{O}^{cris}_n(A)\to A$ which makes the definition well-defined?

Some thoughts I had my self: $\mathcal{O}^{cris}_n(Z)$ is a crystalline global section and thus a compatible system of sections for all nilpotent thickenings $U \to T$ in the crystalline site. By definition of the crystalline structure sheaf, these sections are just $\mathcal{O}_T(T)$ for a thickening $U \to T$. I guess this morphism is just the projection of the global section to $\mathcal{O}_A(A) = A$. The same morphism is obtained, if one takes for an arbitrary thickening $\text{Spec} A \to T$ the morphism on global sections $\mathcal{O}_T(T) \to A$, which is dividing out the nilpotent ideal. But I do not see, why this morphism is surjective, i.e. why we can choose a compatible system of liftings in all nilpotent thickenings.

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Hi Matthias! Welcome to MO. Are you sure they want to do this for all $k$-algebras? I think thay they prove it only for $A_{cris}(\mathcal{O}_{\bar{K}})$ and they use some universal property of this $A_{cris}$ as discussed in Fontaine's paper in Périodes $p$-adiques. Or may be I do not understand your question: but in Sections I.1.3 and I.1.4 they do not work with general $A$. Filippo – Filippo Alberto Edoardo Apr 25 '12 at 15:17
The presentation seems a little confused here. I would suggest reading Fontaine's first article in Asterisque 223, where he explains very clearly an explicit construction of $\mathcal{O}^{cris}_n(\mathcal{O}_{\overline{K}}/p^n)$: it is the universal PD thickening of $\mathcal{O}_{\overline{K}}/p^n$ in which $p^n=0$. – Keerthi Madapusi Pera Apr 25 '12 at 15:17
Hi! Thank you both for your answers. I wasn't aware of the universal PD thickenings presented in Périodes $p$-adiques. That helps me a lot. Filippo: In II.1.4 (not in I.1.4) Fontaine does this contruction for all $k$-algebras, but his constructions gives an isomorphism only if the frobenius is surjective on A. But his should be possible with the constructions in Périodes $p$-adiques. – Matthias Kümmerer May 4 '12 at 14:47
Keerthi: I am not yet sure why Asterisque 223 shows that $\mathcal{O}^{cris}_n(\mathcal{O}_{\bar K}/p)=\mathcal{O}_{\bar K}/p^n$. Probably I am missing some important fact, but as I understand Asterisque 223 this object should be some $W_n(R_{\mathcal{O}_{\bar K}/p})$. Also, the universal PD-thickening only exists if the Frobenius on $A$ is surjective. But Fontains remark II.1.4 says that he constructs an morphism $W_n(A) \to \mathcal{O}^{cris}_n(A)$ for every $A$, it just may not give rise to an isomorphism. But how should this morphism be constructed as long as we don't know $O_n^{cris}(A)$? – Matthias Kümmerer May 4 '12 at 15:00