A Hodge–Tate decomposition with rigid analytic coefficients

Let X be a smooth proper rigid analytic space over a complete algebraically closed field extension K of Qₚ . We establish a Hodge–Tate decomposition for X with G -coefficients, where G is any commutative locally p -divisible rigid group. This generalizes the Hodge–Tate decomposition of Faltings and Scholze, which is the case G=Gₐ . For this, we introduce geometric analogs of the Hodge–Tate spectral sequence with general locally p -divisible coefficients. We prove that these spectral sequences degenerate at E₂ . Our results apply more generally to a class of smooth families of commutative adic groups over X and in the relative setting of smooth proper morphisms X→ S of seminormal rigid spaces. We deduce applications to analytic Brauer groups and the geometric p -adic Simpson correspondence.