A Non-Commutative Approach to Rigid Analytic Geometry

This thesis consists mainly of three chapters, based respectively on [Mon25; Mon26; MP25]. In Chapter 1, 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. We will then use this result to prove an analog statement for formal schemes. In particular, in this chapter we highlight that the notion of smooth and proper categories is, classically, intrinsically algebraic. In Chapter 2, we introduce a notion of smooth and proper categories adapted to the analytic setting. We achieve this by first studying the notion of smooth categories within the framework of a six-functor formalism. Subsequently, using the theory of condensed mathematics and analytic stacks, we demonstrate that a rigid analytic variety is smooth if and only if its associated category of nuclear sheaves is smooth. Furthermore, we relate the atomic generation of the category of nuclear sheaves to the algebraization of the rigid analytic variety. We then employ these results to obtain an example of a non-atomically generated but internally smooth category. In Chapter 3, which is joint work with Emanuele Pavia, we develop a general frame- work in which to study the notion of 1-affineness, originally introduced by Gaitsgory in [Gai15]. We then apply this framework to establish 1-affineness for sheaves of nu- clear categories over rigid analytic varieties, for the Betti stack (as studied in [Sch24]) of a finite-dimensional metrizable compact Hausdorff space, and, as a corollary, for the analytic de Rham stack of a compact complex manifold.