# Derived Non-archimedean analytic Hilbert space

## TL;DR

This paper proves the representability of the derived Hilbert space for separated k-analytic spaces by combining existing theorems and developing new localization results for formal models in derived non-archimedean geometry.

## Contribution

It introduces a new representability theorem for derived Hilbert spaces in the context of derived non-archimedean analytic geometry.

## Key findings

- Proves the existence of the derived Hilbert space RHilb(X) for separated k-analytic spaces.
- Establishes a localization theorem relating categories of almost perfect complexes.
- Develops results on formal models for almost perfect modules on derived k-analytic spaces.

## Abstract

In this short paper we combine the representability theorem introduced in [17, 18] with the theory of derived formal models introduced in [2] to prove the existence representability of the derived Hilbert space RHilb(X) for a separated k-analytic space X. Such representability results relies on a localization theorem stating that if X is a quasi-compact and quasi-separated formal scheme, then the \infty-category Coh^+(X^rig) of almost perfect complexes over the generic fiber can be realized as a Verdier quotient of the \infty-category Coh^+(X). Along the way, we prove several results concerning the the \infty-categories of formal models for almost perfect modules on derived k-analytic spaces.