Fontaine's period rings

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

The ring B_dR

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

$\tilde{\mathbf{E}}^+ = \varprojlim_{x\mapsto x^p} \mathcal{O}_{\mathbf{C}_p}/(p)$

So an element of $\tilde{\mathbf{E}}^+$ is a sequence $(x_1,x_2,\cdots)$ of elements $x_i\in \mathcal{O}_{\mathbf{C}_p}/(p)$ such that $x_{i+1}^p \equiv x_i \pmod p$. There is a natural projection map $f:\tilde{\mathbf{E}}^+ \to \mathcal{O}_{\mathbf{C}_p}/(p)$ given by $f(x_1,x_2,\dotsc) = x_1$. There is also a multiplicative (but not additive) map $t:\tilde{\mathbf{E}}^+\to \mathcal{O}_{\mathbf{C}_p}$ defined by $t(x_,x_2,\dotsc) = \lim_{i\to \infty} \tilde x_i^{p^i}$, where the $\tilde x_i$ are arbitrary lifts of the $x_i$ to $\mathcal{O}_{\mathbf{C}_p}$. The composite of $t$ with the projection $\mathcal{O}_{\mathbf{C}_p}\to \mathcal{O}_{\mathbf{C}_p}/(p)$ is just $f$. The general theory of Witt vectors yields a unique ring homomorphism $\theta:W(\tilde{\mathbf{E}}^+) \to \mathcal{O}_{\mathbf{C}_p}$ such that $\theta([x]) = t(x)$ for all $x\in \tilde{\mathbf{E}}^+$, where $[x]$ denotes the Teichmüller representative of $x$. The ring $\mathbf{B}_{dR}^+$ is defined to be completion of $\tilde{\mathbf{B}}^+ = W(\tilde{\mathbf{E}}^+)[1/p]$ with respect to the ideal $\ker\left( \theta : \tilde{\mathbf{B}}^+ \to \mathbf{C}_p \right)$. The field $\mathbf{B}_{dR}$ is just the field of fractions of $\mathbf{B}_{dR}^+$.

References

Secondary sources

• Berger, Laurent (2004), "An introduction to the theory of p-adic representations", Geometric aspects of Dwork theory I, Berlin: Walter de Gruyter GmbH & Co. KG, arXiv:math/0210184, ISBN 978-3-11-017478-6, MR 2023292
• Brinon, Olivier; Conrad, Brian (2009), CMI Summer School notes on p-adic Hodge theory (PDF), retrieved 2010-02-05
• Fontaine, Jean-Marc, ed. (1994), Périodes p-adiques, Astérisque 223, Paris: Société Mathématique de France, MR 1293969