Timeline for Explicit description of O^{cris}_n in Fontaine/Messing
Current License: CC BY-SA 3.0
5 events
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| May 4, 2012 at 15:00 | comment | added | Matthias Kümmerer | 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 $O_n^{cris}(A)$? | |
| May 4, 2012 at 14:47 | comment | added | Matthias Kümmerer | 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. | |
| Apr 25, 2012 at 15:17 | comment | added | Keerthi Madapusi |
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 $p^n=0$.
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| Apr 25, 2012 at 15:17 | comment | added | Filippo Alberto Edoardo | 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 | |
| Apr 25, 2012 at 12:26 | history | asked | Matthias Kümmerer | CC BY-SA 3.0 |