Algebraization of rigid analytic varieties and formal schemes via perfect complexes

TL;DR

This paper extends a theorem relating algebraizability of rigid analytic varieties to the smoothness of their perfect complexes category, also applying to formal schemes, highlighting differences in derived categories.

Contribution

It generalizes To"en and Vaquié's theorem to non-Archimedean and formal contexts, establishing criteria for algebraizability via perfect complexes.

Findings

A smooth and proper rigid analytic variety is algebraizable iff its perfect complexes category is smooth and proper.

An analogous statement holds for formal schemes.

The bounded derived category of coherent sheaves on a formal scheme is generally not smooth.

Abstract

In this paper, we extend a theorem of To\"en and Vaqui\'e to the non-Archimedean and formal settings. More precisely, we prove that a smooth and proper rigid analytic variety is algebraizable if and only if its category of perfect complexes is smooth and proper. As a corollary, we deduce an analogous statement for formal schemes and demonstrate that, in general, the bounded derived category of coherent sheaves on a formal scheme is not smooth.