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 -adic Galois representations.

The ring BdR

The ring is defined as follows. Let denote the completion of. Let
An element of is a sequence of elements
such that. There is a natural projection map given by. There is also a multiplicative map defined by
where the are arbitrary lifts of the to. The composite of with the projection is just.
The general theory of Witt vectors yields a unique ring homomorphism such that for all, where denotes the Teichmüller representative of. The ring is defined to be completion of with respect to the ideal. Finally, the field is just the field of fractions of.