A note on Hodge–Tate spectral sequences

Abstract We prove that the Hodge–Tate spectral sequence of a proper smooth rigid analytic variety can be reconstructed from its infinitesimal B₃ₑ^+ -cohomology through the Bialynicki–Birula map. We also give a new proof of the torsion-freeness of the infinitesimal B₃ₑ^+ -cohomology independent of Conrad–Gabber spreading theorem, and a conceptual explanation that the degeneration of Hodge–Tate spectral sequences is equivalent to that of Hodge–de Rham spectral sequences.