  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.
