$p$-adic Hodge theory in rigid analytic families

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.