Fontaine's period rings

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

-adic Galois representations.

The ring

{\displaystyle \mathbf {B} _{dR}}

is defined as follows.

denote the completion of

Let An element of

is a sequence

of elements

( mod

There is a natural projection map

There is also a multiplicative (but not additive) map

defined by where the

are arbitrary lifts of the

The composite of

with the projection

The general theory of Witt vectors yields a unique ring homomorphism

θ :

θ ( [ x ] ) = t ( x )

denotes the Teichmüller representative of

The ring

{\displaystyle \mathbf {B} _{dR}^{+}}

is defined to be completion of

with respect to the ideal

ker ⁡ ( θ :

Finally, the field

{\displaystyle \mathbf {B} _{dR}}

is just the field of fractions of

{\displaystyle \mathbf {B} _{dR}^{+}}