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 BdR
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 $$(x_1,x_2,\ldots)$$ 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 $$\mathbf{B}_{dR}^+$$.