Interpolating Hodge-Tate and de Rham Periods

We study the interpolation of Hodge-Tate and de Rham periods over rigid analytic families of Galois representations. Given a Galois representation on a coherent locally free sheaf over a reduced rigid space and a bounded range of weights, we obtain a stratification of this space by locally closed subvarieties where the Hodge-Tate and bounded de Rham periods (within this range) as well as 1-cocycles form locally free sheaves. We also prove strong vanishing results for higher cohomology. Together, these results give a simultaneous generalization of results of Sen, Kisin, and Berger-Colmez. The main result has been applied by Varma in her proof of geometricity of Harris-Lan-Taylor-Thorne Galois representations as well as in several works of Ding.