p-adic Hodge theory in rigid analytic families

Abstract

We study the functors _(V), where 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 (V)=i∈((V)· ti)K, (V)=(V)K, and (V)=(V)[1/t]K, generalizing results of Sen, Fontaine, and Berger. The modules (V) and (V) are coherent sheaves on (A), and (A) is stratified by the ranks of submodules [a,b](V) and [a,b](V) of "periods with Hodge-Tate weights in the interval [a,b]". Finally, we construct functorial -admissible loci in (A), generalizing a result of Berger-Colmez to the case where A is not necessarily reduced.