Fontaine's period rings

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_dR

The ring ${\mathbf {B}}_{{dR}}$ is defined as follows. Let $\mathbf {C} _{p}$ denote the completion of $\overline {{\mathbf {Q}}_{p}}$ . Let

{\tilde {{\mathbf {E}}}}^{+}=\varprojlim _{{x\mapsto x^{p}}}{\mathcal {O}}_{{{\mathbf {C}}_{p}}}/(p)

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

References

Secondary sources

Berger, Laurent (2004), "An introduction to the theory of p-adic representations", Geometric aspects of Dwork theory, I, Berlin: Walter de Gruyter GmbH & Co. KG, arXiv:math/0210184, ISBN 978-3-11-017478-6, MR 2023292
Brinon, Olivier; Conrad, Brian (2009), CMI Summer School notes on p-adic Hodge theory (PDF), retrieved 2010-02-05
Fontaine, Jean-Marc, ed. (1994), Périodes p-adiques, Astérisque, 223, Paris: Société Mathématique de France, MR 1293969

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Fontaine's period rings

The ring BdR

References

Secondary sources

The ring B_dR