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    pp-adic Hodge theory in rigid analytic families

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    We study the functors \D_{\B_\ast}(V), where \B_\ast is one of Fontaine's period rings and VV is a family of Galois representations with coefficients in an affinoid algebra AA. 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][a,b]". Finally, we construct functorial \B_\ast-admissible loci in \Sp(A), generalizing a result of Berger-Colmez to the case where AA is not necessarily reduced.Comment: Final version. 44 page
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