A note on Hodge-Tate spectral sequences

We prove that the Hodge-Tate spectral sequence of a proper smooth rigid analytic variety can be reconstructed from its infinitesimal $\mathbb{B}_{\text{dR}}<^>+$ -cohomology through the Bialynicki-Birula map. We also give a new proof of the torsion-freeness of the infinitesimal $\mathbb{B}_{\text{dR}}<^>+$ -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.