When I read the paper "[THE FARGUES–FONTAINE CURVE AND DIAMONDS](http://www.bourbaki.ens.fr/TEXTES/Exp1150-Morrow.pdf)" of Matthew Morrow, I have a question on page 11.


Question: The arthur said that the de Rham and crystalline period rings implicitly depended on having chosen an untilt of $F$, where $F=C\_p^b$ is the tilt of the p-adic complex field $C\_p$. I can't understand this. If the author just defines the $\infty$ point on Fontaine-Fargues curve to be the class $[C\_p]$, why the author mention the period rings at the bottom of page11 when he defines the $\infty$ point?


Definition: The definitions of the de Rham and crystalline period rings I know are $B\_{dR}:=Frac(lim W(R)[\frac{1}{p}]/{(ker\theta)}^n])$, $B\_{cris}:=Frac(A\_{cris}[\frac{1}{p}])$, $A\_{cris}:=limA^0\_{cris}/p^nA^0\_{cris}$, and $A^0\_{cris}$ is just the sub $W(R)$-module of $W(R)[\frac{1}{p}]$ generated by the $\frac{\xi^n}{n!}$ where $n$ takes all positive integers. These definitions come from Fontaine's readable book [Theory of p-adic Galois Representations](https://www.math.u-psud.fr/~fontaine/galoisrep.pdf).


Thanks for any answers!



Fontaine-Fargues curve and period rings and untilt
As Daniel Litt says, the choice of $C$ is actually an untilt. The "classical" approach to period rings, which you might have in mind, was to start with a certain complete, algebraically closed field $C\_p$, then to construct $R$ out of the quotient $O\_{C\_p}/p$ and out of its Witt vectors to construct the period rings.


The content of Morrow's Proposition 5.1 is that $R$ may arise from many other fields $C$ other than from your initially chosen $C\_p$ and all these form, modulo $\varphi^\mathbb{Z}$, all the equivalence classes of untilts. But to produce $\theta$ you need a target, hence you need to pick one of these choices. By Theorem 2.3 *ibid.* this choice is a point on The Curve, call it $\infty$: the construction of the line bundle depends on this point. Morrow's remark that a choice has been made means the following: suppose $B\_e$ could be constructed independently of the choice of an untilt. Then the construction in (6) would give the same result irrespectively of the chosen point, providing you with a canonical line bundle on it, or equivalently with a prefered point on $\mathbb{P}^1$, which is absurd. On the other hand, the two choices match with each other: a point on $\mathbb{P}^1$ (rather, on $X^{FF}$) call it $\infty$ and an untilt, for instance $C\_p$.


