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 ${𝐁}_{dR}$ is defined as follows. Let ${ℂ}_p$ denote the completion of $\overline{{\mathbb{Q}}_p}$. Let

${\widetilde{𝐄}}^{+}=\underset{x\mapsto x^p}{\underleftarrow{\lim }}{𝒪}_{{ℂ}_p}/(p).$

An element of ${\widetilde{𝐄}}^{+}$ is a sequence $(x_1,x_2,\dots )$ of elements $x_i\in {𝒪}_{{ℂ}_p}/(p)$ such that $x_{i+1}^p\equiv x_i(\mathrm{mod}p)$. There is a natural projection map $f:{\widetilde{𝐄}}^{+}\to {𝒪}_{{ℂ}_p}/(p)$ given by $f(x_1,x_2,\dots )=x_1$. There is also a multiplicative (but not additive) map $t:{\widetilde{𝐄}}^{+}\to {𝒪}_{{ℂ}_p}$ defined by

$t(x_{,}{x}_2,\dots )=\underset{i\to \mathrm{\infty }}{\lim }{\tilde{x}}_i^{p^i}$,

where the ${\tilde{x}}_i$ are arbitrary lifts of the $x_i$ to ${𝒪}_{{ℂ}_p}$. The composite of $t$ with the projection ${𝒪}_{{ℂ}_p}\to {𝒪}_{{ℂ}_p}/(p)$ is just $f$.

The general theory of Witt vectors yields a unique ring homomorphism $\theta :W({\widetilde{𝐄}}^{+})\to {𝒪}_{{ℂ}_p}$ such that $\theta ([x])=t(x)$ for all $x\in {\widetilde{𝐄}}^{+}$, where $[x]$ denotes the Teichmüller representative of $x$. The ring ${𝐁}_{dR}^{+}$ is defined to be completion of ${\widetilde{𝐁}}^{+}=W({\widetilde{𝐄}}^{+})[1/p]$ with respect to the ideal $\ker (\theta :{\widetilde{𝐁}}^{+}\to {ℂ}_p)$. Finally, the field ${𝐁}_{dR}$ is just the field of fractions of ${𝐁}_{dR}^{+}$.

References

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