#### Vol. 9, No. 2, 2015

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$p$-adic Hodge theory in rigid analytic families

### Rebecca Bellovin

Vol. 9 (2015), No. 2, 371–433
##### Abstract

We study the functors ${D}_{{B}_{\ast }}\left(V\right)$, where ${B}_{\ast }$ is one of Fontaine’s period rings and $V$ is a family of Galois representations with coefficients in an affinoid algebra $A$. We first relate them to $\left(\phi ,\Gamma \right)$-modules, showing that ${D}_{HT}\left(V\right)={\oplus }_{i\in Z}{\left({D}_{Sen}\left(V\right)\cdot {t}^{i}\right)}^{{\Gamma }_{K}}$, ${D}_{dR}\left(V\right)={D}_{dif}{\left(V\right)}^{{\Gamma }_{K}}$, and ${D}_{cris}\left(V\right)={D}_{rig}\left(V\right){\left[1∕t\right]}^{{\Gamma }_{K}}$; this generalizes results of Sen, Fontaine, and Berger. We then deduce that the modules ${D}_{HT}\left(V\right)$ and ${D}_{dR}\left(V\right)$ are coherent sheaves on $Sp\left(A\right)$, and $Sp\left(A\right)$ is stratified by the ranks of submodules ${D}_{HT}^{\left[a,b\right]}\left(V\right)$ and ${D}_{dR}^{\left[a,b\right]}\left(V\right)$ of “periods with Hodge–Tate weights in the interval $\left[a,b\right]@$”. Finally, we construct functorial ${B}_{\ast }$-admissible loci in $Sp\left(A\right)$, generalizing a result of Berger and Colmez to the case where $A$ is not necessarily reduced.

##### Keywords
$p$-adic Hodge theory, rigid analytic geometry
Primary: 11S20
Secondary: 14G22