  We study the functors $\D_{\B_\ast}(V)$, 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 show that
$\D_{\HT}(V)=\oplus_{i\in\Z}(\D_{\Sen}(V)\cdot t^i)^{\Gamma_K}$,
$\D_{\dR}(V)=\D_{\dif}(V)^{\Gamma_K}$, and
$\D_{\cris}(V)=\D_{\rig}(V)[1/t]^{\Gamma_K}$, generalizing results of Sen,
Fontaine, and Berger. The modules $\D_{\HT}(V)$ and $\D_{\dR}(V)$ are coherent
sheaves on $\Sp(A)$, and $\Sp(A)$ is stratified by the ranks of submodules
$\D_{\HT}^{[a,b]}(V)$ and $\D_{\dR}^{[a,b]}(V)$ of "periods with Hodge-Tate
weights in the interval $[a,b]$". Finally, we construct functorial
$\B_\ast$-admissible loci in $\Sp(A)$, generalizing a result of Berger-Colmez
to the case where $A$ is not necessarily reduced.
