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
B
d
R
is defined as follows. Let
C
p
denote the completion of
Q
p
¯
. Let
E
~
+
=
lim
←
x
↦
x
p
O
C
p
/
(
p
)
So an element of
E
~
+
is a sequence
(
x
1
,
x
2
,
…
)
of elements
x
i
∈
O
C
p
/
(
p
)
such that
x
i
+
1
p
≡
x
i
(
mod
p
)
. There is a natural projection map
f
:
E
~
+
→
O
C
p
/
(
p
)
given by
f
(
x
1
,
x
2
,
…
)
=
x
1
. There is also a multiplicative (but not additive) map
t
:
E
~
+
→
O
C
p
defined by
t
(
x
,
x
2
,
…
)
=
lim
i
→
∞
x
~
i
p
i
, where the
x
~
i
are arbitrary lifts of the
x
i
to
O
C
p
. The composite of
t
with the projection
O
C
p
→
O
C
p
/
(
p
)
is just
f
. The general theory of Witt vectors yields a unique ring homomorphism
θ
:
W
(
E
~
+
)
→
O
C
p
such that
θ
(
[
x
]
)
=
t
(
x
)
for all
x
∈
E
~
+
, where
[
x
]
denotes the Teichmüller representative of
x
. The ring
B
d
R
+
is defined to be completion of
B
~
+
=
W
(
E
~
+
)
[
1
/
p
]
with respect to the ideal
ker
(
θ
:
B
~
+
→
C
p
)
. The field
B
d
R
is just the field of fractions of
B
d
R
+
.
