When I read the paper "[THE FARGUES–FONTAINE CURVE AND DIAMONDS][1]" 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.

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][2].

Thanks for any answers!


  [1]: http://www.bourbaki.ens.fr/TEXTES/Exp1150-Morrow.pdf
  [2]: https://www.math.u-psud.fr/~fontaine/galoisrep.pdf