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Fontaine's period rings

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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 B d R {\displaystyle \mathbf {B} _{dR}} is defined as follows. Let C p {\displaystyle \mathbf {C} _{p}} denote the completion of Q p ¯ {\displaystyle {\overline {\mathbf {Q} _{p}}}} . Let

E ~ + = lim ← x ↦ x p ⁡ O C p / ( p ) {\displaystyle {\tilde {\mathbf {E} }}^{+}=\varprojlim _{x\mapsto x^{p}}{\mathcal {O}}_{\mathbf {C} _{p}}/(p)}

So an element of E ~ + {\displaystyle {\tilde {\mathbf {E} }}^{+}} is a sequence ( x 1 , x 2 , … ) {\displaystyle (x_{1},x_{2},\ldots )} of elements x i ∈ O C p / ( p ) {\displaystyle x_{i}\in {\mathcal {O}}_{\mathbf {C} _{p}}/(p)} such that x i + 1 p ≡ x i ( mod p ) {\displaystyle x_{i+1}^{p}\equiv x_{i}{\pmod {p}}} . There is a natural projection map f : E ~ + → O C p / ( p ) {\displaystyle f:{\tilde {\mathbf {E} }}^{+}\to {\mathcal {O}}_{\mathbf {C} _{p}}/(p)} given by f ( x 1 , x 2 , … ) = x 1 {\displaystyle f(x_{1},x_{2},\dotsc )=x_{1}} . There is also a multiplicative (but not additive) map t : E ~ + → O C p {\displaystyle t:{\tilde {\mathbf {E} }}^{+}\to {\mathcal {O}}_{\mathbf {C} _{p}}} defined by t ( x , x 2 , … ) = lim i → ∞ x ~ i p i {\displaystyle t(x_{,}x_{2},\dotsc )=\lim _{i\to \infty }{\tilde {x}}_{i}^{p^{i}}} , where the x ~ i {\displaystyle {\tilde {x}}_{i}} are arbitrary lifts of the x i {\displaystyle x_{i}} to O C p {\displaystyle {\mathcal {O}}_{\mathbf {C} _{p}}} . The composite of t {\displaystyle t} with the projection O C p → O C p / ( p ) {\displaystyle {\mathcal {O}}_{\mathbf {C} _{p}}\to {\mathcal {O}}_{\mathbf {C} _{p}}/(p)} is just f {\displaystyle f} . The general theory of Witt vectors yields a unique ring homomorphism θ : W ( E ~ + ) → O C p {\displaystyle \theta :W({\tilde {\mathbf {E} }}^{+})\to {\mathcal {O}}_{\mathbf {C} _{p}}} such that θ ( [ x ] ) = t ( x ) {\displaystyle \theta ()=t(x)} for all x ∈ E ~ + {\displaystyle x\in {\tilde {\mathbf {E} }}^{+}} , where [ x ] {\displaystyle } denotes the Teichmüller representative of x {\displaystyle x} . The ring B d R + {\displaystyle \mathbf {B} _{dR}^{+}} is defined to be completion of B ~ + = W ( E ~ + ) [ 1 / p ] {\displaystyle {\tilde {\mathbf {B} }}^{+}=W({\tilde {\mathbf {E} }}^{+})} with respect to the ideal ker ⁡ ( θ : B ~ + → C p ) {\displaystyle \ker \left(\theta :{\tilde {\mathbf {B} }}^{+}\to \mathbf {C} _{p}\right)} . The field B d R {\displaystyle \mathbf {B} _{dR}} is just the field of fractions of B d R + {\displaystyle \mathbf {B} _{dR}^{+}} .

Notes

  1. Fontaine (1982)

References

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