Algebraization of rigid analytic varieties and formal schemes via perfect complexes

TL;DR

By extending Toën–Vaquié's algebraization framework to non-Archimedean and formal contexts, the paper proves that a smooth, proper rigid analytic variety $X$ is algebraizable if and only if its ∞-category of perfect complexes $\mathrm{Perf}(X)$ is smooth and proper. The approach builds a Morita-style moduli theory for $k$-linear stable ∞-categories, introduces analytic moduli of pseudo-perfect objects, and relates them to the analytification of algebraic moduli via descent and Tate acyclicity. A key step embeds $X$ into the coarse moduli space of simple objects and uses non-Archimedean Moishezon-space results to deduce algebraizability; the formal setting yields analogous statements and clarifies when $\mathrm{Coh}^{b}$ fails to be smooth. The findings reveal that smoothness of categories behaves essentially as an algebraic notion in non-Archimedean and formal geometry and provide a precise bridge between algebraizability and categorical smoothness with concrete moduli-theoretic constructions.

Abstract

In this paper, we extend a theorem of Toën and Vaquié 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.

Theorems & Definitions (119)

• Definition 1
• Theorem 2
• Theorem 3
• Corollary 4
• Theorem 1.1
• Proposition 1.2
• proof
• Definition 1.3
• Remark 1.4
• Lemma 1.5