By Ofer Gabber
Let $K$ be a complete rank 1 valued field with ring of integers $\mathcal O_K$, $A$ an adic nœtherian ring and $\phi: A\rightarrow \mathcal O_K$ an adic morphism. We show that if $g: X\rightarrow Y$ 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, $T_0\rightarrow S$ affine of finite type and $X_0\rightarrow T_0$ proper flat, we construct a compatible system of versal n-th order deformations of $X_0\rightarrow T_0$ 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 $R^q g_∗\Omega^p_{X/Y}$ are locally free. This is reduced to the case where $K$ is a finite extension of $Q_p$ and $Y$ is a nilpotent thickening of $\mathrm{Sp} K$, where the result over $K$ was proved by Scholze and follows forY by imitating the proof of Deligne over C using a construction ofcrystalline cohomology in this case.
By Ofer Gabber
By Geordie Williamson
By Mark Kisin
By Yves André
By Dennis Gaitsgory
By Ofer Gabber
By Yakov Varshavsky
By Fabrice Orgogozo
By Takeshi Saito
By Ofer Gabber
By Francesca Arici
By Shirly Geffen
By Juan Camilo Arosemena Serrato
By Amine Marrakchi
By Pieter Spaas
