Fontaine period rings

Definition

Let $K/\mathbb Q_p$ be a $p$ -adic field, let $G_K=\operatorname{Gal}(\overline K/K)$ be its absolute Galois group, and let $K_0$ be the maximal unramified subfield of $K$ . Fontaine's period rings are topological $\mathbb Q_p$ -algebras with continuous $G_K$ -actions and additional structures. The four basic rings are

B_{\mathrm{HT}},\qquad B_{\mathrm{dR}},\qquad B_{\mathrm{cris}},\qquad B_{\mathrm{st}}.

For a finite-dimensional $p$ -adic representation $V$ , taking invariants in $B\otimes_{\mathbb Q_p}V$ produces its $B$ -period module. Having the largest dimension allowed by $V$ is the corresponding admissibility condition.

Structures carried by the rings

$B_{\mathrm{HT}}=\bigoplus_{i\in\mathbb Z}\mathbb C_p(i)$ is graded and detects Hodge–Tate weights.
$B_{\mathrm{dR}}$ is a complete filtered field; its invariants form the filtered $K$ -vector space of de Rham periods.
$B_{\mathrm{cris}}\subset B_{\mathrm{dR}}$ carries Frobenius. Its invariants form a filtered Frobenius module and define crystalline representations.
$B_{\mathrm{st}}$ contains $B_{\mathrm{cris}}$ and carries Frobenius together with a nilpotent monodromy operator $N$ . It defines semistable representations.

The inclusions and extra structures explain the implication

\text{crystalline}\Longrightarrow\text{semistable} \Longrightarrow\text{de Rham}\Longrightarrow\text{Hodge–Tate}.

Ring versus period module

The rings are universal coefficient objects; they do not depend on a particular $V$ . By contrast,

D_B(V)=(B\otimes_{\mathbb Q_p}V)^{G_K}

is a linear-algebraic invariant of $V$ . For $B_{\mathrm{dR}}$ it is a filtered $K$ -vector space, while for $B_{\mathrm{cris}}$ and $B_{\mathrm{st}}$ it is initially a $K_0$ -vector space with Frobenius and, in the semistable case, monodromy. Confusing $B$ with $D_B(V)$ suppresses both the Galois-invariant operation and the base field of the result.

References

Olivier Brinon and Brian Conrad, CMI Summer School Notes on $p$ -adic Hodge Theory, 2009, Chapters 2, 4, and 9. Author notes.
Jean-Marc Fontaine, “Le corps des périodes $p$ -adiques,” Astérisque 223 (1994), 59–111. Numdam.