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 is defined as follows. Let denote the completion of . Let
So an element of is a sequence of elements
such that . There is a natural projection map given by . There is also a multiplicative (but not additive) 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 . The field is just the field of fractions of .