Formal models for relative adic spaces

Dimitri Dine

REVIEW 3 major objections 3 minor 54 references

Formal models for relative adic spaces

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Every uniform qcqs adic space over a Tate affinoid base has an integrally closed formal model, and morphisms lift uniquely after normalized formal blow-ups.

desk verdict Real mathematics with a plausible, high-stakes main theorem, but the localized category in Theorem 6.4 is never actually constructed — referee time yes, acceptance conditional on a repair. read the letter →

arxiv 2507.11073 v1 pith:35ERAJCG submitted 2025-07-15 math.AG math.NT

Dimitri Dine This is my paper · ORCID

classification math.AGmath.NT MSC 14G22

keywords adicspacesformalmodelsnormalizedblow-upsuniformgenericfiberfunctorschemesTateaffinoidbaserigidanalyticgeometry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a formal-model theory for uniform quasi-compact quasi-separated (qcqs) adic spaces over any Tate affinoid base, without imposing finite-type or Noetherian hypotheses on the spaces or the base. The central claim is that the generic-fiber functor gives an equivalence between the category of integrally closed formal schemes localized by a new class of arrows, the normalized formal blow-ups, and the category of uniform qcqs adic spaces over the base. Concretely, this says every such adic space can be obtained as the generic fiber of a formal scheme, and every morphism of such spaces comes from a unique morphism of formal models after a suitable normalized blow-up. This matters because many adic spaces of current interest, such as perfectoid spaces and their relatives, fall outside the classical Noetherian and finite-type settings where formal models were previously known to work.

What carries the argument

The key new object is the normalized formal blow-up: the composition of a $\varpi$-torsion-free admissible formal blow-up with the normalization of the resulting formal scheme inside its generic fiber. The normalization is formed by taking the formal spectrum of the direct image $sp_{\mathfrak{S},\mathfrak{S},*}O^+_{\mathfrak{S}}$ of the integral structure sheaf of the generic fiber, an adically quasi-coherent algebra, and this operation refines a formal model without changing its generic fiber while forcing integral closedness. Integral closedness is exactly what allows morphisms of adic spaces to lift to morphisms of formal models. The surrounding machinery includes the specialization map from the generic fiber to the formal model and a global inverse-limit description of the generic fiber as the limit of all admissible formal blow-ups of a formal model.

What would settle it

Exhibit a uniform qcqs adic space over $S=\operatorname{Spa}(R[\varpi^{-1}], \overline{R})$ with no $\varpi$-torsion-free integrally closed formal model, or exhibit a morphism between two such spaces that lifts to no morphism of formal models after any normalized formal blow-up. Either would falsify Theorem 6.4; alternatively, two composable normalized formal blow-ups whose composite is not a normalized formal blow-up would break the localization underlying the theorem.

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Extended reading notes

Core claim

The paper's central discovery is Theorem 6.4: for $R$ a complete adic ring whose ideal of definition is generated by a non-zero-divisor $\varpi$, with $R[\varpi^{-1}]$ sheafy and $S=\operatorname{Spa}(R[\varpi^{-1}], \overline{R})$, the functor $\mathfrak{X}\mapsto\mathfrak{X}^{ad}_{\eta}$ is an equivalence of categories between (1) locally stably uniform, $\varpi$-torsion-free, qcqs adic formal $R$-schemes that are integrally closed in their generic fibers, localized by normalized formal blow-ups, and (2) uniform qcqs adic spaces over $S$. The proof follows the classical three-step pattern: morphisms of formal models are determined by their generic fibers, morphisms between generic fibers lift uniquely after a normalized formal blow-up of the source model, and every uniform qcqs adic space admits an integrally closed formal model. A finiteness-restricted variant (Theorem 6.13) recovers the classical description using admissible formal blow-ups for adic spaces of finite type over $S$.

Load-bearing premise

The load-bearing premise is that normalized formal blow-ups form a class of arrows one can localize by: composing two of them should again give one, or at least the class should admit a calculus of fractions, and the paper does not construct that localized category before using it.

Editorial extensions

If this is right

Every uniform qcqs adic space over $S=\operatorname{Spa}(R[\varpi^{-1}], \overline{R})$ has a $\varpi$-torsion-free, locally stably uniform formal model that is integrally closed in its generic fiber.
Every morphism $f:Y\to X$ of such spaces is, after replacing $Y$'s formal model by a normalized formal blow-up, induced by a unique morphism of formal models; if $f$ is an isomorphism, the lifted model morphism is an isomorphism.
Uniform qcqs adic spaces over a Tate affinoid base can be studied through formal schemes localized by normalized formal blow-ups, giving a non-Noetherian analogue of the classical rigid-geometry formal-model setup.
Under topologically finite-type hypotheses, the classical admissible-blow-up description is recovered, and formal modifications between finite-type models are dominated by admissible formal blow-ups when the generic fiber is a strong adic space.
The generic fiber of a formal scheme is recovered as the inverse limit of all its admissible formal blow-ups, a global version of the Zariski-Riemann-space description.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If Theorem 6.4 holds, formal-model techniques such as descent for coherent sheaves should transfer to uniform qcqs adic spaces without Noetherian hypotheses, with the main remaining obstruction being the exact behavior of the localized category of normalized formal blow-ups.
The normalized formal blow-up class suggests a birational geometry for non-Noetherian formal schemes, and one could test whether the localized category is itself described by a Zariski-Riemann-type site built from integrally closed formal models.
A concrete extension to try is whether the equivalence restricts to a subcategory with better finiteness or compactness properties for sousperfectoid or strongly rigid-Noetherian adic spaces, making the formal models more computable.
The finite-type comparison result could potentially be promoted to a full identification of normalized formal blow-ups with admissible formal blow-ups whenever the base is strongly noetherian.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing you should know: this is not a crank paper. The author has a serious attack on a real problem, and the main result would be foundational if correct. The equivalence between uniform qcqs adic spaces and integrally closed formal models localized by normalized formal blow-ups is exactly the missing Raynaud theory for perfectoid spaces. The normalization construction via Fujiwara–Kato's adically quasi-coherent sheaves is a good idea, and the paper is full of careful, useful technical work: the global version of Bhatt's Zariski–Riemann space theorem, Proposition 6.7 comparing normalized and admissible blow-ups, and the examples including the perfectoid disk and Fargues–Fontaine curve. I give it credit for being explicit about what [3] does and doesn't provide.

Now the soft spots, in proportion. The stress-test is on target, though I would phrase it as a gap in the statement rather than a fatal contradiction. Theorem 6.4 says 'localized by normalized formal blow-ups,' but the paper never defines that localization. To make it a category you need the class W of normalized formal blow-ups to admit a right calculus of fractions, or at least to show compositions of W-morphisms stay in W. Definition 5.15 and Lemma 5.16 do not do this. Lemma 6.1 gives faithfulness only for the unlocalized category; Lemma 6.2 gives a V → Z span for each generic fiber morphism. Without proving W is a multiplicative system (or giving a concrete morphism set via spans up to refinement), the claimed equivalence is not fully established. I think this is repairable — one could bypass localization by defining morphisms as spans with an equivalence relation, following Bosch–Lütkebohmert — but as written it is a real gap, and a referee should ask for a fix, not just stylistic changes.

Second, Theorem 6.13 has a qcqs-versus-finite-type mismatch: the statement says 'quasi-separated' but the proof of faithfulness invokes Lemma 6.1, which requires qcqs. This is probably harmless because topologically finite type over an affine base often forces qcqs, but it should be checked and stated cleanly.

Third, the gluing step in Theorem 6.3 is a bit sketchy: it asserts that two integrally closed models can be glued after normalized blow-ups, but the verification that the glued object is an adic formal scheme is buried in the induction. Again, plausible but under-explained.

Who is this paper for? Anyone working on perfectoid Shimura varieties, relative Fargues–Fontaine curves, or nonarchimedean geometry beyond finite type. It deserves a serious referee — the potential payoff is high and the flaws are not dead ends. I would send it to refereeing, with a request that the author construct the localized category and check the finite-type hypotheses.

Recommendation: engage with the work; in principle accept, but only after a revision that nails down the localization. If the author cannot fix it, the main theorem should be downgraded to a construction of the normalization functor plus full faith and fullness for the unlocalized category.

Referee Report

3 major / 3 minor

Summary. The paper proposes an analog of Raynaud's formal-model theory for uniform qcqs adic spaces over a Tate affinoid base, without Noetherian or finite-type assumptions. It develops a generic fiber functor and specialization map for locally rig-sheafy formal schemes, proves a global inverse-limit description of the generic fiber via admissible formal blow-ups (Theorem 4.13), introduces integral closure of formal models in their generic fiber, defines normalized formal blow-ups, and states a categorical equivalence between integrally closed formal models localized by normalized formal blow-ups and uniform qcqs adic spaces (Theorem 6.4), plus a finite-type variant (Theorem 6.13). The method follows the classical Bosch-Lütkebohmert strategy, replacing admissible formal blow-ups by normalized formal blow-ups.

Significance. If the central equivalence is fully established, this would be a substantial extension of Raynaud theory to non-Noetherian, non-finite-type situations, with potential applications to perfectoid Shimura varieties, relative Fargues-Fontaine curves, and other analytic adic spaces. The paper contains detailed constructions, a global Bhatt-type theorem for the Zariski-Riemann space, and useful examples; it introduces no fitted parameters or circular dependencies. However, the main theorem's domain category is not actually constructed, and the finite-type theorem has a qcqs-versus-locally-finite-type mismatch in its hypotheses and proof, so the central claims are not yet fully supported.

major comments (3)

[Definition 5.15 / Theorem 6.4] The statement of Theorem 6.4 uses a localization of the category of integrally closed formal schemes by normalized formal blow-ups, but this localized category is never constructed. Definition 5.15 fixes a class W, Lemma 5.16 only proves that a W-morphism on an open subscheme extends to the whole scheme, and Section 6 does not show that W is closed under composition or satisfies the Ore/calculus-of-fractions conditions. Consequently Lemma 6.1 and Lemma 6.2 establish faithfulness and fullness of the generic-fiber functor only on the unlocalized category: in a localized category a morphism is a span modulo a refinement relation, and equality of generic fibers of representing morphisms does not by itself identify spans. The domain in (1) is therefore not a well-defined category as stated, and Theorem 6.4 is not established until a localization construction, or an equivalent universal-property formulation, is supplied.
[Theorem 6.13] The hypotheses of Theorem 6.13 are 'quasi-separated adic formal R-schemes topologically of finite type' and 'quasi-separated adic spaces of finite type', but the proof relies throughout on results stated for qcqs objects: Lemma 6.1 uses Corollary 4.16, whose hypotheses include quasi-compact and quasi-separated; Theorem 6.3, invoked for essential surjectivity, assumes qcqs; and the reduction to affine objects in the fullness part needs a finite covering. As written the theorem does not cover all quasi-separated finite-type objects, and the proof does not indicate how to pass from finite local affinoid covers to non-quasi-compact spaces. Either qcqs should be added to the hypotheses or separate arguments must be supplied for the non-qcqs case.
[Lemma 6.2] In the gluing step of Lemma 6.2 the proof starts with a cover (U_i) of X by rig-sheafy affine open subschemes and an affinoid open cover (V_i) of Z_ad_eta with f(V_i) subset U_i, then invokes Corollary 4.18 to obtain an admissible formal blow-up of Z. Corollary 4.18 requires a finite open cover by quasi-compact subsets; the proof does not state that the cover has been taken finite. Although qcqs of Z_ad_eta makes such a finite refinement possible, the gap between the stated cover and the cited corollary should be made explicit.

minor comments (3)

[Section 2, Remark 2.11 and Lemma 2.12] The notation A⟨ f_1^n,...,f_r^n / ϖ ⟩ is used both for the completed affine blow-up algebra and for a rational localization, and the exponential notation in Lemma 2.12 is inconsistent in places (e.g., f^{msn} vs. f^{sn}); these points should be clarified or corrected.
[Definition 4.6] The definition of sp_{X,X'} depends on a choice of an affine rig-sheafy neighbourhood and an integer n; the claim that this is independent of choices is asserted rather than proved, and a short verification would improve the exposition.
[Definition 6.6 and Proposition 6.7] The construction of the n-dimensional ball over a strong adic space via gluing along the B^n_{V_ij} is terse: the compatibility data for the gluing are not spelled out, and the role of the 'strong' hypothesis in Proposition 6.7 could be stated more explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is derived from the generic fiber construction, normalization inside the generic fiber, and external foundations, with no fitted inputs or self-derived target result.

full rationale

I walked the derivation chain of Theorem 6.4. The proof has three parts: faithfulness (Lemma 6.1), fullness (Lemma 6.2), and essential surjectivity (Theorem 6.3). Lemma 6.1 uses the specialization map and Corollary 4.16, not the desired equivalence. Lemma 6.2 proves existence of formal models of morphisms by reducing to the affine case via the identity O_Z(Z) = O^+_{Z_η}(Z_η) for integrally closed formal models, then glues using normalized formal blow-ups; it does not assume the theorem. Theorem 6.3 constructs formal models inductively from the affinoid model Spf(A^+) for a uniform affinoid Spa(A,A^+), and glues using Lemma 6.2 and Lemma 5.16; the affinoid case is by construction of the generic fiber, not by the theorem. The category in Theorem 6.4(1) is defined independently as integrally closed qcqs formal R-schemes localized by normalized formal blow-ups, where normalized formal blow-ups are defined via admissible blow-ups followed by normalization (Definition 5.15). No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the author's own prior work, and no load-bearing self-citation occurs. Citations to Fujiwara-Kato, Buzzard-Verberkmoes-Mihara, Zavyalov, and Pilloni-Stroh are external foundational inputs. The skeptic's concern that the localized category is not explicitly constructed (e.g., no calculus of fractions is verified for normalized formal blow-ups) is a possible rigor gap in the proof of Theorem 6.4, but it is not circularity: even a missing Ore condition would be an unproven technical hypothesis, not a reduction of the conclusion to its own input. Therefore the paper exhibits no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities. The main theorem rests on standard foundations and on Fujiwara-Kato's theory of adically quasi-coherent sheaves and admissible formal blow-ups. The central new notion, normalized formal blow-up, is a composition of existing operations: an admissible blow-up followed by normalization inside the generic fiber.

assumptions (4)

domain assumption Fujiwara-Kato foundations for adically quasi-coherent sheaves, formal spectra, and admissible formal blow-ups without Noetherian assumptions
Used throughout Sections 4 and 5; Theorem 4.13 and the existence of normalizations depend on these external results.
domain assumption Huber's Lemma 2.4.3(iv) on rational localizations of integrally closed rings
Used in Proposition 5.3 to prove that sp_* O^+_S is adically quasi-coherent, which is the base of the normalization construction.
domain assumption Buzzard-Verberkmoes and Mihara: stably uniform Tate rings are sheafy
Ensures locally stably uniform adic formal R-schemes have well-defined generic fibers; used in Section 5 and Lemma 5.14.
domain assumption R is ϖ-adically complete, ϖ is a non-zero-divisor, and R[ϖ^{-1}] is sheafy
This is the standing setup for the paper and defines the base affinoid adic space S.

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Cite this review

Pith. "Pith review of Formal models for relative adic spaces." pith.science (2026). https://pith.science/paper/35ERAJCG

@misc{pith2026250711073,
  author       = {Pith},
  title        = {Pith review of: Formal models for relative adic spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/35ERAJCG}},
  note         = {Machine review of arXiv:2507.11073}
}

abstract

We extend Raynaud's theory of formal models from rigid-analytic spaces over a nonarchimedean field to uniform qcqs adic spaces $X$, with no finite-type assumptions, over an arbitrary Tate affinoid base $S$. The key new ingredient is the notion of a normalized formal blow-up which takes on the role played by admissible formal blow-ups in the classical theory.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [3]

Ayoub, M

J. Ayoub, M. Gallauer, and A. Vezzani. The six-functor formalism for rigid analytic motives. Forum of Mathematics, Sigma , 10(e61):1–182, 2022

work page 2022

[2] [1]

Andreatta and C

F. Andreatta and C. Gasbarri. The canonical subgroup for families of abelian vari- eties. Compositio Math., 143:566–602, 2007

work page 2007

[3] [2]

Ardakov and S

K. Ardakov and S. Wadsley. bD-modules on rigid analytic spaces I. Journal f¨ ur die reine und angewandte Mathematik , 747:221–275, 2019

work page 2019

[4] [4]

Berkovich

V. Berkovich. Spectral Theory and Analytic Geometry over Non-Archimedean Fields, volume 33 of Surveys and Monographs . Amer. Math. Soc., Providence, 1990. 51

work page 1990

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Berthelot

P. Berthelot. Cohomologie rigide et cohomologie rigide ` a supports propres. Preprint 96-03, Institut de Recherche Math. de Rennes, 1996

work page 1996

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B. Bhatt. Lecture notes for a class on perfectoid spaces. Available at http://www- personal.umich.edu/ bhattb/teaching/mat679w17/lectures.pdf, 2017

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B. Bhatt. On the direct summand conjecture and its derived variant. Inventiones mathematicae, 212(2):297–317, 2018

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Bhatt and D

B. Bhatt and D. Hansen. The six functors for Zariski-constructible sheaves in rigid geometry. Compositio Math., 158(2):437–482, 2022

work page 2022