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 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 (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 . Finally, the field is just the field of fractions of .