Stellar Black Hole Binaries in the Gravitational Universe - Part 2
By Monica Colpi
Quantum Mechanics and Quantum Field Theory from Algebraic and Geometric Viewpoints (4/4)
By Albert Schwarz
By Ofer Gabber
Appears in collection : Arithmetic and Algebraic Geometry: A conference in honor of Ofer Gabber on the occasion of his 60th birthday
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.