Formal models for relative adic spaces

TL;DR

This paper extends Raynaud's formal model theory to a broader class of adic spaces over arbitrary bases, introducing normalized formal blow-ups as a key new concept.

Contribution

It generalizes the theory of formal models to uniform qcqs adic spaces over any Tate affinoid base, removing finite-type restrictions.

Findings

Develops the concept of normalized formal blow-ups

Extends formal model theory to non-finite type adic spaces

Provides a framework for broader applications in nonarchimedean geometry

Abstract

We extend Raynaud's theory of formal models from rigid-analytic spaces over a nonarchimedean field to uniform qcqs adic spaces $X$, with no finite-type assumptions, over an arbitrary Tate affinoid base $S$. The key new ingredient is the notion of a normalized formal blow-up which takes on the role played by admissible formal blow-ups in the classical theory.