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,\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