Spreading out the Hodge filtration in non-archimedean geometry

TL;DR

This paper introduces a non-archimedean deformation to the normal cone for morphisms of derived $k$-analytic spaces, providing a new way to analyze the Hodge filtration in non-archimedean geometry.

Contribution

It constructs a deformation to the normal cone in non-archimedean analytic geometry and develops the theory of $k$-analytic formal moduli problems.

Findings

Deformation to the normal cone interpolates between formal completion and derived normal cone.

The introduced filtration extends the Hodge filtration on the tangent bundle.

The filtration matches the $I$-adic filtration for locally complete intersection morphisms.

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 $f \colon X \to Y$ 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 $\mathbf A^1_k$-parametrized deformation whose fiber at $1 \in \mathbf A^1_k$ coincides with the formal completion of $f$ and the fiber at $0 \in \mathbf A^1_k$ 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.