{"id":"6180778b-4668-4631-bcbb-50ba0750cb3c","arxiv_id":"2507.11073","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Uniform qcqs adic spaces over any Tate affinoid base are shown to be equivalent to integrally closed formal models up to normalized formal blow-ups.","lead":"This paper extends Raynaud's older dictionary between analytic and formal spaces to spaces with no finiteness restrictions, including the perfectoid spaces used in modern number theory. The new method replaces ordinary admissible blow-ups with normalized formal blow-ups, and gives a precise equivalence between formal models and uniform adic spaces.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6.4's localized category is never constructed: normalized formal blow-ups are not shown to form a calculus of fractions, so the stated equivalence lacks a defined domain.","rationale":"The reader's weakest assumption identifies exactly the gap that is most load-bearing: the localized category in Theorem 6.4 is never constructed, and the class of normalized formal blow-ups is not shown to be a multiplicative system. Without this, the domain of the claimed equivalence is not a well-defined category, and the faithfulness/fullness arguments in Section 6 do not apply to morphisms of that category. The rest of the proof is largely a translation of the Bosch-Lutkebohmert argument and is plausible: Lemma 6.2 correctly reduces lifting of morphisms to the affine integrally-closed case, and Theorem 6.3 is a reasonable gluing argument. The missing piece is categorical bookkeeping, not a mathematical contradiction; the theorem may well be true, but the present text does not establish it. This supports keeping the CONDITIONAL verdict rather than upgrading to ACCEPT, and the concrete test above would settle whether the gap is only expositional or substantive.","tokens_in":45861,"tokens_out":12257,"duration_ms":145496,"concrete_test":"Define the localized category C[W^{-1}] with morphisms as equivalence classes of spans (Z' -> Z, Z' -> X), where the left leg is a normalized formal blow-up and equivalence is via a common normalized refinement. Then verify: (a) W is closed under composition, i.e., for any normalized formal blow-ups Z'' -> Z' and Z' -> Z, construct a normalized formal blow-up W -> Z with factorizations W -> Z'' -> Z' -> Z and W -> Z' -> Z; (b) the right Ore condition: for phi: Z' -> Z in W and any f: X -> Z, find psi: Y -> X in W and g: Y -> Z' with phi o g = f o psi. If these fail, Theorem 6.4's category (1) is undefined and the proof of full faithfulness is incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 6.4 claims an equivalence between the category of integrally closed formal models localized by normalized formal blow-ups and the category of uniform qcqs adic spaces. However, the paper never defines this localized category. Definition 5.15 introduces normalized formal blow-ups, but the proof of Theorem 6.4 (Section 6) simply asserts that Lemma 6.1 gives faithfulness and Lemma 6.2 gives fullness. This is only valid if morphisms in the localized category are spans modulo a common refinement and if the class W of normalized formal blow-ups admits a right calculus of fractions. In particular, one needs: (i) W is closed under composition; (ii) every diagram Z' -> Z <- X can be completed to a square with left leg in W; and (iii) two morphisms equalized by a W-arrow are equalized after precomposition with a W-arrow. Lemma 5.16 only proves that a normalized formal blow-up of an open subscheme can be extended to the whole scheme; it does not establish composition closure or the Ore condition. Without these, the domain category in (1) is not well-defined, and faithfulness of the functor on the localized category is not established: two different spans with the same generic fiber morphism might not be identified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":46113,"tokens_out":6281,"duration_ms":80015,"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":[{"comment":"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.","section":"Definition 5.15 / Theorem 6.4"},{"comment":"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.","section":"Theorem 6.13"},{"comment":"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.","section":"Lemma 6.2"}],"minor_comments":[{"comment":"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.","section":"Section 2, Remark 2.11 and Lemma 2.12"},{"comment":"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.","section":"Definition 4.6"},{"comment":"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.","section":"Definition 6.6 and Proposition 6.7"}],"recommendation":"major_revision","confidential_remarks":"The paper is serious and well-informed, and the technical apparatus around generic fibers, specialization maps, and integral closure is largely convincing. The main obstacle is the undefined localization appearing in Theorem 6.4; this is a repairable gap but it is load-bearing. The finite-type theorem also needs its hypotheses aligned with the qcqs results used in its proof. I would encourage the editor to send the manuscript back for a revision addressing these two points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nNow 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.\n\nSecond, 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.\n\nThird, 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.\n\nWho 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.\n\nRecommendation: 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.","headline":"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.","tokens_in":46586,"tokens_out":3483,"would_cite":true,"duration_ms":46818,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14G22"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["adic spaces","formal models","normalized formal blow-ups","uniform adic spaces","generic fiber functor","formal schemes","Tate affinoid base","rigid analytic geometry"],"falsifier":"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.","tokens_in":45692,"feed_emoji":"📐","tokens_out":8689,"duration_ms":94220,"temperature":0.7,"pith_summary":"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.","feed_headline":"Every uniform adic space has an integrally closed formal model","feed_subtitle":"A new normalized blow-up notion removes Noetherian and finite-type restrictions from formal-model theory.","key_machinery":"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.","core_discovery":"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 $S$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the classical formal-model and rigid-geometry dictionary that the paper extends to uniform adic spaces.","marker":"[46]"},{"why":"Supplies the non-Noetherian foundations: adically quasi-coherent sheaves, admissible formal blow-ups, and relative formal spectra used throughout.","marker":"[24]"},{"why":"Provides the three-step proof pattern of faithfulness, fullness, and essential surjectivity that the proof of Theorem 6.4 follows.","marker":"[13]"},{"why":"Constructs an embedding of uniform analytic adic spaces into Fujiwara-Kato rigid spaces, the prior existence result that Theorem 6.3 also yields and that the paper compares with its categorical statement.","marker":"[3]"},{"why":"Shows that stably uniform Tate rings are sheafy, so locally stably uniform formal schemes are locally rig-sheafy.","marker":"[15]"},{"why":"Also proves sheafiness of stably uniform Tate rings, providing the same bridge from stability to rig-sheafiness.","marker":"[41]"},{"why":"Inspires the normalization-inside-the-generic-fiber construction used to build integrally closed formal models.","marker":"[45]"},{"why":"Supplies the inverse-limit description of the generic fiber over admissible formal blow-ups, which the paper globalizes.","marker":"[6]"},{"why":"Provides the classical generic-fiber construction for formal schemes over a discrete valuation ring that the paper adapts to the adic setting.","marker":"[5]"},{"why":"Supplies the foundational theory of adic spaces and rational localizations used throughout the argument.","marker":"[33]"}],"fun_headline_variants":["Formal models for all uniform adic spaces via normalized blow-ups","No finite type needed: formal models for relative adic spaces","Normalized blow-ups extend Raynaud's theory to adic spaces","Uniform qcqs adic spaces get integrally closed formal models","New blow-up notion removes restrictions in formal model theory"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Formal models for all uniform adic spaces via normalized blow-ups","No finite type needed: formal models for relative adic spaces","Normalized blow-ups extend Raynaud's theory to adic spaces","Uniform qcqs adic spaces get integrally closed formal models","New blow-up notion removes restrictions in formal model theory"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000474,"raw_usage":{"total_tokens":2295,"prompt_tokens":825,"completion_tokens":1470,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":441,"completion_tokens_details":{"reasoning_tokens":1383}},"tokens_in":441,"tokens_out":1470,"duration_ms":12899,"temperature":1.0,"reasoning_tokens":1383,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:18:05.578213+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the classical formal-model and rigid-geometry dictionary that the paper extends to uniform adic spaces."},{"cited_title":"Fujiwara and F","cited_arxiv_id":null,"evidence_quote":"Supplies the non-Noetherian foundations: adically quasi-coherent sheaves, admissible formal blow-ups, and relative formal spectra used throughout."},{"cited_title":"Bosch and W","cited_arxiv_id":null,"evidence_quote":"Provides the three-step proof pattern of faithfulness, fullness, and essential surjectivity that the proof of Theorem 6.4 follows."},{"cited_title":"Ayoub, M","cited_arxiv_id":null,"evidence_quote":"Constructs an embedding of uniform analytic adic spaces into Fujiwara-Kato rigid spaces, the prior existence result that Theorem 6.3 also yields and that the paper compares with its categorical statement."},{"cited_title":"Buzzard and A","cited_arxiv_id":null,"evidence_quote":"Shows that stably uniform Tate rings are sheafy, so locally stably uniform formal schemes are locally rig-sheafy."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Also proves sheafiness of stably uniform Tate rings, providing the same bridge from stability to rig-sheafiness."},{"cited_title":"Pilloni and B","cited_arxiv_id":null,"evidence_quote":"Inspires the normalization-inside-the-generic-fiber construction used to build integrally closed formal models."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the inverse-limit description of the generic fiber over admissible formal blow-ups, which the paper globalizes."},{"cited_title":"Berthelot","cited_arxiv_id":null,"evidence_quote":"Provides the classical generic-fiber construction for formal schemes over a discrete valuation ring that the paper adapts to the adic setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the foundational theory of adic spaces and rational localizations used throughout the argument."}],"review_version":1}