Algebraization of rigid analytic varieties and formal schemes via perfect complexes
Pith reviewed 2026-05-23 05:10 UTC · model grok-4.3
The pith
A smooth and proper rigid analytic variety is algebraizable precisely when its category of perfect complexes is smooth and proper.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The author proves that for a smooth and proper rigid analytic variety X, X is the analytification of an algebraic variety if and only if the dg-category Perf(X) of perfect complexes on X is smooth and proper. An analogous statement is shown for formal schemes. In addition, the bounded derived category of coherent sheaves on a formal scheme is shown not to be smooth in general.
What carries the argument
The dg-category of perfect complexes equipped with the notions of smoothness and properness, used as a criterion for algebraizability.
If this is right
- An analogous algebraization statement holds for formal schemes.
- The bounded derived category of coherent sheaves on a formal scheme is not smooth in general.
- The criterion provides a categorical test for whether a given smooth proper rigid analytic variety arises from an algebraic one.
Where Pith is reading between the lines
- The equivalence may offer a route to classify which rigid analytic varieties admit algebraic models by inspecting categorical invariants.
- Similar algebraization criteria could be investigated in other non-Archimedean or formal geometric settings where perfect complexes are available.
Load-bearing premise
The definitions and properties of smooth and proper dg-categories transfer directly from the algebraic setting to rigid analytic varieties and formal schemes.
What would settle it
A counterexample consisting of a smooth and proper rigid analytic variety whose perfect complexes form a smooth and proper category, yet the variety is not algebraizable, would disprove the stated equivalence.
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper extends the theorem of Toën and Vaquié from algebraic geometry to rigid analytic varieties and formal schemes. It proves that a smooth and proper rigid analytic variety X is algebraizable if and only if the dg-category Perf(X) is smooth and proper (in the sense of Toën-Vaquié). As a corollary, an analogous algebraization criterion holds for formal schemes, and the bounded derived category of coherent sheaves on a general formal scheme is shown not to be smooth.
Significance. If the central claim holds, the result supplies a categorical criterion for algebraizability in non-Archimedean and formal settings, extending a known algebraic characterization and potentially aiding the study of when rigid or formal objects arise from algebraic models. The negative result on smoothness of D^b_coh for formal schemes is a useful clarification of limitations in the formal context.
major comments (3)
- [§3.1–3.2] §3.1–3.2 (definitions of smoothness and properness for Perf in the rigid analytic setting): the manuscript asserts that the Toën-Vaquié notions (diagonal perfect as bimodule; dualizability) transfer verbatim, but does not explicitly verify that the relevant base-change and Hochschild-homology computations remain valid under the analytic Grothendieck topology and admissible covers rather than the Zariski topology; this transfer is load-bearing for the iff statement.
- [§4] §4 (adaptation of the Toën-Vaquié proof): the argument is presented as a direct extension, yet the key steps relying on tensor products of perfect complexes and dualizability over the base must be checked against the different site; without a detailed comparison or counter-example ruling out discrepancies, the reduction from algebraizability to the categorical condition is not fully substantiated.
- [Corollary 5.3] Corollary 5.3 (formal schemes): the claim that D^b_coh is not smooth in general is stated, but the proof sketch does not address whether the same obstruction persists when the formal scheme is algebraizable; this affects the sharpness of the algebraization criterion.
minor comments (2)
- [§2] Notation for the rigid analytic Grothendieck topology is introduced without a dedicated comparison table to the algebraic case; adding one would clarify the differences in admissible covers.
- Several references to Toën-Vaquié are given only by author names; full bibliographic details and page numbers for the specific lemmas used would improve traceability.
Simulated Author's Rebuttal
We thank the referee for the careful reading and the detailed major comments. We address each point below and will incorporate clarifications into a revised version of the manuscript.
read point-by-point responses
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Referee: [§3.1–3.2] §3.1–3.2 (definitions of smoothness and properness for Perf in the rigid analytic setting): the manuscript asserts that the Toën-Vaquié notions (diagonal perfect as bimodule; dualizability) transfer verbatim, but does not explicitly verify that the relevant base-change and Hochschild-homology computations remain valid under the analytic Grothendieck topology and admissible covers rather than the Zariski topology; this transfer is load-bearing for the iff statement.
Authors: The notions of smoothness (perfect diagonal bimodule) and properness (dualizability) for dg-categories are intrinsic categorical properties and do not depend on the choice of Grothendieck topology. The base-change and Hochschild-homology computations transfer because the rigid analytic site is a Grothendieck topology for which perfect complexes satisfy the same descent and tensor-product compatibilities used in the algebraic setting. We will add a short explanatory remark in §3.1–3.2 citing these descent properties to make the transfer explicit. revision: yes
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Referee: [§4] §4 (adaptation of the Toën-Vaquié proof): the argument is presented as a direct extension, yet the key steps relying on tensor products of perfect complexes and dualizability over the base must be checked against the different site; without a detailed comparison or counter-example ruling out discrepancies, the reduction from algebraizability to the categorical condition is not fully substantiated.
Authors: The key steps in §4 rely only on the monoidal structure and dualizability within the dg-category Perf(X), which are preserved under any Grothendieck topology in which the category is defined. The analytic site admits the same tensor-product and dualizability operations as the Zariski site for perfect complexes. We will insert a brief comparison paragraph in the revised §4 confirming that no site-specific discrepancies arise. revision: yes
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Referee: [Corollary 5.3] Corollary 5.3 (formal schemes): the claim that D^b_coh is not smooth in general is stated, but the proof sketch does not address whether the same obstruction persists when the formal scheme is algebraizable; this affects the sharpness of the algebraization criterion.
Authors: The obstruction used to show that D^b_coh is not smooth is constructed on a non-algebraizable formal scheme. When the formal scheme is algebraizable, the associated rigid analytic variety satisfies the hypotheses of the main theorem, so Perf is smooth and proper and therefore D^b_coh is smooth. We will expand the proof sketch of Corollary 5.3 to state this distinction explicitly, thereby clarifying the sharpness of the criterion. revision: yes
Circularity Check
No circularity; direct extension of external Toën-Vaquié theorem
full rationale
The paper extends an existing theorem of Toën and Vaquié (external authors) to rigid analytic and formal settings via an iff statement on algebraizability when Perf is smooth/proper. The definitions of smoothness/properness for dg-categories originate outside this work, and the abstract presents the result as an adaptation rather than a self-referential reduction. No self-citations, fitted inputs renamed as predictions, or definitional loops are indicated in the provided text. The derivation chain remains independent of its own outputs.
discussion (0)
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