Introduction by Jean-Louis Colliot-Thélène (Université Paris-Sud)
Let K be a complete rank 1 valued field with ring of integers OK, A an adic nœtherian ring and φ: A → OK an adic morphism. We show that if g: X → Y is a proper flat morphism between rigid analytic spaces over K then locally on Y a flat formal model of
g is the pullback of a proper flat morphism between formal
schemes topologically of finite type over A. For this, if S is an affine nœtherian scheme, T0→S affine of finite type and X0→T0
proper flat, we construct a compatible system of versal n-th order deformations of X0→T0 over S. As an application, one can prove that for a proper smooth g and K of characteristic 0, the Hodge to de Rham spectral sequence for g degenerates and the
R q g∗ΩpX/Y are locally free. This is reduced to the case where
K is a finite extension of Qp and Y is a nilpotent thickening of Sp
K, where the result over K was proved by Scholze and follows for
Y by imitating the proof of Deligne over C using a construction of
crystalline cohomology in this case.
