Fontaine's period rings

In mathematics, Fontaine's period rings are a collection of commutative rings first defined by Jean-Marc Fontaine^[1] that are used to classify ${\displaystyle p}$-adic Galois representations.

The ring BdR

The ring ${\displaystyle \mathbf {B} _{dR}}$ is defined as follows. Let ${\displaystyle \mathbb {C} _{p}}$ denote the completion of ${\displaystyle {\overline {\mathbb {Q} _{p}}}}$. Let

${\displaystyle {\tilde {\mathbf {E} }}^{+}=\varprojlim _{x\mapsto x^{p}}{\mathcal {O}}_{\mathbb {C} _{p}}/(p).}$

An element of ${\displaystyle {\tilde {\mathbf {E} }}^{+}}$ is a sequence ${\displaystyle (x_{1},x_{2},\ldots )}$ of elements ${\displaystyle x_{i}\in {\mathcal {O}}_{\mathbb {C} _{p}}/(p)}$ such that ${\displaystyle x_{i+1}^{p}\equiv x_{i}\!\!\!{\pmod {p}}}$. There is a natural projection map ${\displaystyle f:{\tilde {\mathbf {E} }}^{+}\to {\mathcal {O}}_{\mathbb {C} _{p}}/(p)}$ given by ${\displaystyle f(x_{1},x_{2},\dotsc )=x_{1}}$. There is also a multiplicative (but not additive) map ${\displaystyle t:{\tilde {\mathbf {E} }}^{+}\to {\mathcal {O}}_{\mathbb {C} _{p}}}$ defined by

${\displaystyle t(x_{,}x_{2},\dotsc )=\lim _{i\to \infty }{\tilde {x}}_{i}^{p^{i}}}$,

where the ${\displaystyle {\tilde {x}}_{i}}$ are arbitrary lifts of the ${\displaystyle x_{i}}$ to ${\displaystyle {\mathcal {O}}_{\mathbb {C} _{p}}}$. The composite of ${\displaystyle t}$ with the projection ${\displaystyle {\mathcal {O}}_{\mathbb {C} _{p}}\to {\mathcal {O}}_{\mathbb {C} _{p}}/(p)}$ is just ${\displaystyle f}$.

The general theory of Witt vectors yields a unique ring homomorphism ${\displaystyle \theta :W({\tilde {\mathbf {E} }}^{+})\to {\mathcal {O}}_{\mathbb {C} _{p}}}$ such that ${\displaystyle \theta ([x])=t(x)}$ for all ${\displaystyle x\in {\tilde {\mathbf {E} }}^{+}}$, where ${\displaystyle [x]}$ denotes the Teichmüller representative of ${\displaystyle x}$. The ring ${\displaystyle \mathbf {B} _{dR}^{+}}$ is defined to be completion of ${\displaystyle {\tilde {\mathbf {B} }}^{+}=W({\tilde {\mathbf {E} }}^{+})[1/p]}$ with respect to the ideal ${\displaystyle \ker(\theta$ :{\tilde {\mathbf {B} }}^{+}\to \mathbb {C} _{p})} . Finally, the field ${\displaystyle \mathbf {B} _{dR}}$ is just the field of fractions of ${\displaystyle \mathbf {B} _{dR}^{+}}$.