Derived ${\mathcal{O}}_k$-adic geometry and derived Raynaud localization theorem

Abstract

The goal of the present text is to state and prove a generalization of Raynaud localization theorem in the setting of derived geometry. More explicitly, we show that the $\infty$-category of quasi-paracompact and quasi-separated derived $k$-analytic spaces can be realized as a localization of the $\infty$-category of admissible derived formal schemes. We construct a derived rigidification functor generalizing Raynaud rigidification functor. In order to construct the latter we will need to formalize derived formal ${\mathcal{O}}_k$-adic formal geometry via a structured spaces approach. We prove that ${\mathcal{O}}_k$-adic Postnikov towers of derived ${\mathcal{O}}_k$-adic Deligne-Mumford stacks decompose and we relate these to Postnikov towers of derived $k$-analytic spaces. This is possible by a precise comparison between the ${\mathcal{O}}_k$-adic cotangent complex and the $k$-analytic cotangent complex.

References in corpus (6)

• Higher Topos Theory
• Higher and derived stacks: a global overview
• Proper local complete intersection morphisms preserve perfect complexes
• Derived non-archimedean analytic spaces
• Moduli of $p$-adic representations of a profinite group
• Formal gluing along non-linear flags

Cited by in corpus (2)

• Moduli of $p$-adic representations of a profinite group
• Moduli of $\ell$-adic pro-étale local systems for smooth non-proper schemes