Uniformly rigid spaces

TL;DR

This paper introduces uniformly rigid spaces, a new category of non-archimedean analytic spaces that extend rigid spaces and relate closely to their classical counterparts, enabling transfer of key geometric results.

Contribution

It defines uniformly rigid spaces, establishes their relation to classical rigid spaces, and proves an analog of Kiehl's patching theorem within this new framework.

Findings

Uniformly rigid spaces extend rigid spaces and are described via bounded functions.

A new notion of uniformly rigid generic fiber is introduced, closely linked to models.

An analog of Kiehl's patching theorem is proved for uniformly rigid spaces.

Abstract

We define a new category of non-archimedean analytic spaces over a complete discretely valued field, which we call uniformly rigid. It extends the category of rigid spaces, and it can be described in terms of bounded functions on products of open and closed polydiscs. We relate uniformly rigid spaces to their associated classical rigid spaces, and we transfer various constructions and results from rigid geometry to the uniformly rigid setting. In particular, we prove an analog of Kiehl's patching theorem for coherent ideals, and we define the uniformly rigid generic fiber of a formal scheme of formally finite type. This uniformly rigid generic fiber is more intimately linked to its model than the classical rigid generic fiber obtained via Berthelot's construction.