Fontaine's period rings explained

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

The ring B_dR

The ring

B _{dR} is defined as follows. Let C _{p} denote the completion of \overline{ Q _{p}} . Let\tilde{ E }^+ = \varprojlim_ \mathcal_/(p)

So an element of

\tilde{ E }^+ is a sequence (x _{1,x} _{2,\ldots)} of elements x _{i\in} l{O} _{C _{p}} /(p) such that x ^{p} _{i +1} \equiv x _{i} \pmod p . There is a natural projection map f:\tilde{ E }^+ \to \mathcal_/(p) given by f(x _{1,x} _{2,...c)} = x _{1} . There is also a multiplicative (but not additive) map t:\tilde{ E }^+\to \mathcal_ defined by t(x _{,x} _{2,...c)} = \lim _{i\to} \tilde x ^{p ^{i}} _{i} , where the \tilde x _{i} are arbitrary lifts of the x _{i} to l{O} _{C _{p}} . The composite of t with the projection l{O} _{C _{p}} \to l{O} _{C _{p}} /(p) is just f . The general theory of Witt vectors yields a unique ring homomorphism \theta:W(\tilde{ E }^+) \to \mathcal_ such that \theta([x]) = t(x) for all x\in \tilde{ E }^+, where [x] denotes the Teichmüller representative of x . The ring B ^{+} _{dR} is defined to be completion of \tilde{ B }^+ = W(\tilde^+)[1/p] with respect to the ideal \ker\left( \theta : \tilde{ B }^+ \to \mathbf_p \right). The field B _{dR} is just the field of fractions of B ^{+} _{dR} .

Notes and References

1. Fontaine (1982)