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

Thanks for any answers!

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

Thanks for any answers!

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

Thanks for any answers!

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user141691
user141691

Fontaine-Fargues curve and period rings and untilt

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

Thanks for any answers!