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 GaitsgoryStudyII. More specifically, given a morphism f X Y of (derived) k-analytic spaces we construct the non-archimedean deformation to the normal cone associated to f. The latter can be thought as an A1k-parametrized deformation whose fiber at 1 ∈ A1k coincides with the formal completion of f and the fiber at 0 ∈ A1k 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.