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. A refinement of the decalage functor $L\eta$ is introduced to accomplish the proof. Further, we 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.