Fontaine's Period Rings

In mathematics, Fontaine's period rings are a collection of commutative rings first defined by Jean-Marc FontaineFontaine (1982) that are used to classify $p$-adic Galois representations.

The ring B_dR

The ring $\mathbf_$ is defined as follows. Let $\C_p$ denote the completion of $\overline$. Let

:$\tilde^+ = \varprojlim_ \mathcal_/(p).$

An element of $\tilde^+$ is a sequence $(x_1,x_2,\ldots)$ of elements
$x_i\in \mathcal_/(p)$ such that $x_^p \equiv x_i \!\!\!\pmod p$. There is a natural projection map $f:\tilde^+ \to \mathcal_/(p)$ given by $f(x_1,x_2,\dotsc) = x_1$. There is also a multiplicative (but not additive) map $t:\tilde^+\to \mathcal_$ defined by
:$t(x_,x_2,\dotsc) = \lim_ \tilde x_i^$,
where the $\tilde x_i$ are arbitrary lifts of the $x_i$ to $\mathcal_$. The composite of $t$ with the projection $\mathcal_\to \mathcal_/(p)$ is just $f$.

The general theory of Witt vectors yields a unique ring homomorphism $\theta:W(\tilde^+) \to \mathcal_$ such that \theta( = t(x) for all $x\in \tilde^+$, where /math> denotes the Teichmüller representative of $x$. The ring $\mathbf_^+$ is defined to be completion of \tilde^+ = W(\tilde^+) /p/math> with respect to the ideal $\ker( \theta : \tilde^+ \to \C_p)$. Finally, the field $\mathbf_$ is just the field of fractions of $\mathbf_^+$.

Notes

References

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*{{Citation
, editor-last=Fontaine
, editor-first=Jean-Marc
, editor-link=Jean-Marc Fontaine
, title=Périodes p-adiques
, publisher=Société Mathématique de France
, location=Paris
, year=1994
, mr=1293969
, series=Astérisque
, volume=223

Algebraic number theory
Galois theory
Representation theory of groups
Hodge theory