p-adic Hodge theory in rigid analytic families

We study the functors D-B* (V), where B-* 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 (phi. Gamma)-modules, showing that D-HT(V) = circle plus(i is an element of Z) (D-Sen (V) . t(i))(Gamma K) , D-dR(V) = D-dif(V)(Gamma K) , and D-cris (V) = D-rig(V) [1/t](Gamma K) this generalizes results of Sen, Fontaine, and Berger. We then deduce that the 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-*-admissible loci in Sp(A), generalizing a result of Berger and Colmez to the case where A is not necessarily reduced.