Spreading out the Hodge filtration in non-archimedean geometry

Abstract

The goal of the current text is to study non-archimedean analytic derived de Rham cohomology by means of formal completions. Our approach is inspired by the deformation to the normal cone provided in \cite{Gaitsgory_Study_II}. More specifically, given a morphism $f:X\to Y$ of (derived) $k$-analytic spaces we construct the \emph{non-archimedean deformation to the normal cone} associated to $f$. The latter can be thought as an ${\mathbf{A}}_k^1$-parametrized deformation whose fiber at $1\in {\mathbf{A}}_k^1$ coincides with the formal completion of $f$ and the fiber at $0\in {\mathbf{A}}_k^1$ with the (derived) normal cone associated to $f$. We further show that such deformation can be endowed with a natural filtration which spreads out the usual Hodge filtration on the (completed shifted) analytic tangent bundle to the formal completion. Such filtration agrees with the $I$-adic filtration in the case where $f$ is a locally complete intersection morphism between (derived) $k$-affinoid spaces. Along the way we develop the theory of (ind-inf)-$k$-analytic spaces or in other words $k$-analytic formal moduli problems.

References in corpus (3)

• Completions and derived de Rham cohomology
• Analytification of mapping stacks
• Moduli of $\ell$-adic pro-étale local systems for smooth non-proper schemes