| We establish some properties of (ϕ,Γ)-modules associated to absolutely crystalline representations. As a corollary, we can answer (in the “unramified case”) two questions of Fontaine. First, we show that a Zp-representation, which is a limit of crystalline Zp-representations with bounded Hodge-Tate weights is itself crystalline. Second, we show that every admissible filtered ϕ-module can be constructed from a (ϕ, Γ F)-module of finite q-height (that is, the functor i* : is essentially surjective). The main ingredient is the computation of an explicit bound for the annihilator of the cokernel of the inclusion . |
