Algebraization of rigid analytic varieties and formal schemes via perfect complexes

2025/01/23 by Matteo Montagnani, Montagnani, Matteo · 1 citation
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2501.13911

openalex publication_date 2025/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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