{"id":"00e0c169-6f49-4150-90b0-c0f4730bfa0a","arxiv_id":"2507.20359","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every continuous localizing invariant satisfies adelic descent when computed on nuclear modules over the adelic rings of a scheme of finite type over Z.","lead":"Adelic descent, a way to assemble invariants of a scheme from completed local rings at its points, is proved for every continuous localizing invariant using Clausen-Scholze nuclear modules. This extends earlier K-theory and dualizable-category versions and gives a general computational tool for arithmetic schemes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Non-affine reduction in §3.1 rests on an unproved essential surjectivity in pullback square (3.1.0.1) and an unverified preservation of such squares by arbitrary stable localizing invariants.","rationale":"Reading the paper in good faith, the central claim is a plausible extension of the adelic descent for solid quasi-coherent sheaves in [1] to nuclear modules and then to localizing invariants. The affine argument in §3.2–3.3 is internally coherent: the filtration by Nuc≤k(A■) and the induction using the cubical reduction principle fit together, and Proposition 3.2.3 supplies the needed localizations and kernel identifications. The main weakness is precisely the non-affine reduction. The text itself flag the missing essential surjectivity by saying it 'can be proved directly' rather than proving it, and the preservation of the pullback square by arbitrary stable localizing invariants is asserted without a proof or a precise citation. These are not signs of a false theorem, but they are structural proof gaps: the theorem as written claims descent for all finite-type schemes, and the reduction to the affine case is essential. The reader's conditional verdict is therefore appropriate. The gaps are likely fillable from the cited literature, but until explicit proofs or precise references are supplied, Theorem 3.0.1 should be regarded as conditional. No more severe concern, such as an internal inconsistency or a counterexample, is apparent.","tokens_in":14174,"tokens_out":17039,"duration_ms":194477,"concrete_test":"Prove essential surjectivity in (3.1.0.1) for a non-affine test case: X=P^1_Z with U=V=A^1, U∩V=G_m and the flag tuple (0), so L_0O_X is the product of completed local rings at closed points. Construct explicitly the inverse of Nuc(L_0O_X,X■) -> Nuc(L_0O_U,U■) ×_{Nuc(L_0O_{G_m},G_m■)} Nuc(L_0O_V,V■), using the fully faithful right adjoints of Proposition 2.3.7 and the base change formula of [3, Example 13.15]. If such an inverse cannot be constructed, or if the glued object is only a module over L_0O_X and not nuclear, the affine reduction in §3.1 is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.0.1 is proved for finite-type schemes by reducing to the affine case in §3.1. That reduction requires square (3.1.0.1) to be a pullback in CAlg(Pr^L_st) and requires every stable localizing invariant to map that square to a pullback. The text only says the natural functor to the pullback is fully faithful 'by construction' and that essential surjectivity 'can be proved directly using Proposition 2.3.7 and the base change for solid quasi-coherent sheaves ... see Example 13.15 in [3]'. No proof is given. This is load-bearing: if compatible nuclear modules over U and V and their intersection do not glue to a nuclear module over X, the descent statement for non-affine X does not follow from the affine argument. Separately, the paper asserts that because each functor in the square is a strongly continuous localization, any stable localizing invariant preserves the square. This preservation property is not proved and no precise reference is supplied; it may be a consequence of the definition of localizing invariant in [4], but the text does not establish that link. Either gap, if not fillable, would leave Theorem 3.0.1 established only for affine schemes.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a version of adelic descent for continuous localizing invariants. For a scheme X of finite type over Z and an accessible stable localizing invariant F: Pr^dual_st → E, Theorem 3.0.1 asserts that the canonical map F(Nuc(X■)) → lim_{0≤k0<...<km≤d} F(Nuc(L_{k0...km}O_X, X■)) is an isomorphism, where the limit is taken over the cubical diagram of nuclear module categories over adelic rings. The proof reduces to the affine case via Zariski open covers, then uses a filtration by skeletons and Proposition 3.2.3 to split the cubical diagram into fiber sequences. The paper also states and proves a categorical version (Theorem 3.4.1) asserting that the same diagram is a limit in both Pr^L_st and Pr^dual_st, and that it is preserved by stable localizing invariants. A generalization to filtrations by arbitrary specialization-closed subsets is given in Section 3.5.","tokens_in":14320,"tokens_out":5357,"duration_ms":54864,"significance":"If the main theorem is correct, it establishes adelic descent for continuous localizing invariants in the Clausen–Scholze nuclear-module setting, complementing and differing from Kim's earlier adelic descent results ([6], [7]). The affine proof is a systematic and largely explicit argument based on a skeleton filtration, and the paper correctly identifies the key structural input, Proposition 3.2.3, which is proved by a series of lemmas. The categorical adelic descent statement for nuclear sheaves (Theorem 3.4.1) is also a useful result in its own right. However, the paper's transition from the affine case to the general finite-type case in Section 3.1 contains two assertions that are load-bearing and are not proved in the text: the essential surjectivity in the pullback square (3.1.0.1), and the preservation of that pullback square by arbitrary stable localizing invariants. Because Theorem 3.0.1 is stated for all finite-type schemes, these gaps affect the central claim and require a substantial addition or a precise reference.","major_comments":[{"comment":"The essential surjectivity of the functor Nuc(L_{k0...km}O_X, X■) → lim of the restrictions to U, V, and U∩V is asserted but not proved. The text states only that it 'can be proved directly using Proposition 2.3.7 and the base change for solid quasi-coherent sheaves with respect to Zariski open embeddings, see Example 13.15 in [3]'. This step is load-bearing: without a proof that compatible nuclear modules over an open cover glue to a nuclear module over X, the reduction to the affine case fails and Theorem 3.0.1 is only established for affine schemes. Please provide a complete argument, or state the glueing lemma as a separate proposition with a full proof or with a precise citation that covers exactly this statement.","section":"§3.1, square (3.1.0.1)"},{"comment":"The claim that the pullback square (3.1.0.1) is mapped to a pullback by any stable localizing invariant is not demonstrated. The text says that because each functor in the square is a strongly continuous localization, 'it follows' that the square is preserved, but it does not explain which property of stable localizing invariants from [4] is being used. If the intended argument is that the horizontal localizations have equivalent kernels and that localizing invariants convert the induced short exact sequences into fiber sequences, this should be written out explicitly; alternatively, a precise reference to a theorem in [4] should be given. Without this, the induction over affine covers cannot be concluded.","section":"§3.1, paragraph after (3.1.0.1)"},{"comment":"In the proof of Lemma 3.3.4, the step identifying the kernel of Nuc≤k(A■) → Nuc(Γ(S^c_{k−1},O)•) with the kernel of Nuc≤k(A■) → Nuc((LkA)•) relies on the claim that the image of the composite lies in (LkA)•−Mod and that the functor Nuc≤k(A■)⊗Nuc(A■) Nuc((LkA)•) → Nuc((LkA)•) is fully faithful because Nuc((LkA)•) is dualizable over Nuc(A■). This is only sketched; a full justification or a reference for the full faithfulness assertion is needed, since Proposition 3.2.3(3) is used in the induction step of the main theorem.","section":"§3.3, Lemma 3.3.4"}],"minor_comments":[{"comment":"There are several typos: 'thrid' should be 'third', 'Nevetheless' should be 'Nevertheless', and 'out main theorem' should be 'our main theorem'.","section":"§1, Introduction"},{"comment":"The equivalence Nuc(A■) ≃ ModA is invoked without proof or reference; since it is used to identify Nuc≤k(A■) with the category of S_k-torsion modules, a brief justification or a precise citation would improve readability.","section":"§3.2"},{"comment":"The proof of Proposition 2.3.7 uses that j_* preserves nuclear objects and that j_*j^*L_{k0...km}O_X is a retract of L_{k0...km}O_X; this is plausible but the argument would be clearer if the retract statement were stated and proved explicitly, since it depends on the explicit product formula for adelic rings.","section":"§2.3, Proposition 2.3.7"},{"comment":"The proof of Theorem 3.4.1 says 'By the same reduction as in section 3.1 combined with Proposition 1.87 of [4]', but since Section 3.1 contains the two unproved assertions listed above, this reduction inherits the same gaps.","section":"§3.4"}],"recommendation":"major_revision","confidential_remarks":"The two gaps in Section 3.1 are potentially fixable: the glueing statement is likely a consequence of known properties of nuclear modules over analytic rings, and the preservation of pullback squares is likely a formal consequence of the definition of localizing invariants. However, as written, these assertions are load-bearing for the non-affine case and are not backed by a complete proof or by a reference that covers the exact statement. I would like to see those two points addressed in detail before I can recommend acceptance. The affine part of the paper appears sound and is a meaningful contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves something genuinely new: for any accessible stable localizing invariant F, adelic descent holds for nuclear modules on finite-type Z-schemes. This is not in Kim's papers, which either treat K-theory or work inside QCoh(X)-Mod(Pr^dual_st). The author states the novelty clearly and is honest about what is a formal consequence of Brav–Konovalov [1]. The categorical setup is careful, the affine argument is mostly rigorous, and the paper includes a useful comparison with Efimov and Kim. The proof of the affine case rests on a clean induction using the skeletal filtration; that part looks sound.\n\nThe soft spots are exactly where the reader's stress-test lands. In §3.1, the reduction to affine requires the pullback square (3.1.0.1). Full faithfulness is clear by construction, but essential surjectivity is asserted to follow from Proposition 2.3.7 and base change, with no proof shown. Separately, the preservation of that square by an arbitrary stable localizing invariant is asserted on the grounds that the functors are strongly continuous localizations, but no reference or argument for that preservation is given. These are load-bearing for non-affine X. They may very well be fillable from the cited literature, but the text should say how. There is also a minor instance: the identification Nuc(X■) ≃ QCoh(X) is called well known and used without a citation in Prop 2.3.2.\n\nNone of this looks like circularity or fraud. The central affine theorem is proved in detail, and the gaps are concentrated in the gluing step. The paper's own limitation remarks in §3.5 are honest. If the gaps are fillable, the theorem stands as a useful tool for K-theory and motivic homotopy in the Clausen–Scholze setting.\n\nThis is a paper for specialists. A serious referee should engage with it and push the author to provide the missing proof or precise references for the two asserted steps in §3.1. It deserves peer review, not a desk reject.","headline":"A genuinely new adelic descent statement for all stable localizing invariants in the nuclear setting, with a plausible proof whose main gaps are explicit and likely fillable.","tokens_in":14962,"tokens_out":1402,"would_cite":false,"duration_ms":15093,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Adelic descent holds for every stable localizing invariant when applied to nuclear sheaves on finite-type Z-schemes.","keywords":["adelic descent","localizing invariants","nuclear modules","solid quasi-coherent sheaves","skeletal filtration","Zariski descent","K-theory","stable presentable categories"],"falsifier":"Check the asserted pullback square for a non-affine scheme such as $\\mathbb{P}^1_{\\mathbb{Z}}$ with its standard affine cover: if the functor from nuclear modules over the adelic rings to the pullback over the two affine lines is not essentially surjective, the affine reduction fails. Alternatively, take the invariant to be algebraic K-theory on $\\mathrm{Spec}(\\mathbb{Z})$ and compare $F(\\mathrm{Nuc}(\\mathbb{Z}_\\blacksquare))$ with the limit over $F(\\mathrm{Nuc}(L_0\\mathbb{Z}))$, $F(\\mathrm{Nuc}(L_1\\mathbb{Z}))$, and $F(\\mathrm{Nuc}(L_{0,1}\\mathbb{Z}))$; any disagreement would refute Theorem 3.0.1.","tokens_in":13874,"feed_emoji":"📐","tokens_out":11024,"duration_ms":104111,"temperature":0.7,"pith_summary":"This paper proves that any accessible stable localizing invariant--a class that includes algebraic K-theory and other additive invariants of dualizable stable categories--of a scheme $X$ of finite type over $\\mathbb{Z}$ is determined by the same invariant applied to a finite cubical diagram of nuclear module categories built from the adelic rings of $X$. The adelic ring $L_{k_0...k_m}\\mathcal{O}_X$ is the iterated localization-and-completion of the structure sheaf at a chain of points of increasing dimension. Replacing the large category of solid quasi-coherent sheaves by the smaller, dualizable categories of nuclear modules is what makes the statement true for localizing invariants. If correct, the result gives a canonical adelic decomposition of invariants such as K-theory over the arithmetic point set of $X$.","feed_headline":"Adelic descent works for all stable localizing invariants","feed_subtitle":"On finite-type Z-schemes, invariants are captured by nuclear modules over adelic rings.","key_machinery":"The machinery is the nuclear-object construction $\\mathrm{Nuc}(-)$, applied to module categories over the idempotent adelic algebras $L_{k_0\\cdots k_m}\\mathcal{O}_X$ in $\\mathrm{QCoh}(X_\\blacksquare)$. An object is nuclear when every map from a compact object is trace-class; nuclear module categories are dualizable and inherit the symmetric monoidal structure. The supporting apparatus--rigidification, the description of $\\mathrm{Nuc}(A^{\\sim}_{T,E})$ as generated by a single completed direct-sum object, and the recollement of $\\mathrm{Nuc}(A_\\blacksquare)$ by $S_k$-torsion subcategories--lets the author split the adelic cube into fiber sequences and apply the cubical reduction principle.","core_discovery":"The central claim is Theorem 3.0.1: for $X$ of finite type over $\\mathbb{Z}$ and any accessible stable localizing invariant $F$ from dualizable stable presentable categories to an accessible stable category, the canonical map $F(\\mathrm{Nuc}(X_\\blacksquare)) \\to \\lim_{0\\le k_0<\\cdots<k_m\\le d} F(\\mathrm{Nuc}(L_{k_0\\cdots k_m}\\mathcal{O}_X,X_\\blacksquare))$ is an isomorphism. The limit runs over all strictly increasing chains of dimensions appearing in the skeletal filtration of $X$. A companion statement, Theorem 3.4.1, says the diagram itself is a limit in both $\\mathrm{Pr}^{\\mathrm{L}}_{\\mathrm{st}}$ and $\\mathrm{Pr}^{\\mathrm{dual}}_{\\mathrm{st}}$ before $F$ is applied, so the invariant version follows from the fact that stable localizing invariants preserve limits of strongly continuous localizations.","pith_inferences":["If the main theorem is right, the adelic cube is a universal finite coordinate system for stable localizing invariants on finite-type $\\mathbb{Z}$-schemes, suggesting that such invariants can be computed by a spectral sequence assembled from flags of points.","Theorem 3.4.1 may imply that the descent property holds for any invariant preserving pullbacks along strongly continuous localizations, even if it is not a full localizing invariant; testing this would delimit the role of the accessibility and dualizability hypotheses.","A natural experimental check is to specialize to algebraic K-theory and compare the nuclear-module diagram with earlier K-theoretic adelic descent results; agreement would show the nuclear formulation is equivalent for K-theory, and a mismatch would localize where the difference lies.","The author's remark on working over a countable field suggests the proof structure could transfer verbatim to finite-type schemes over such fields once the base solid abelian groups are replaced by ultrasolid vector spaces, giving a non-arithmetic variant of the theorem."],"forward_implications":["For any accessible stable localizing invariant $F$, the value on a scheme $X$ can be computed from the finite cubical diagram of $F$ applied to nuclear modules over the adelic rings $L_{k_0...k_m}\\mathcal{O}_X$.","The same adelic cube is a limit in both $\\mathrm{Pr}^{\\mathrm{L}}_{\\mathrm{st}}$ and $\\mathrm{Pr}^{\\mathrm{dual}}_{\\mathrm{st}}$, so the descent statement is independent of the choice of invariant.","When $X=\\mathrm{Spec}(A)$ is affine, the proof gives an explicit splitting of the cube into fiber sequences indexed by the skeletal filtration, which is what makes the general descent tractable.","Theorem 3.5.1 extends the descent statement to any filtration of $X$ by specialization closed subsets, though the entries are not generally computable from that filtration alone.","As a consequence, K-theory and other continuous localizing invariants of a finite-type $\\mathbb{Z}$-scheme are determined by completed local data at chains of points, rather than by the full category of quasi-coherent sheaves."],"supporting_citations":[{"why":"defines the adelic rings $L_{k_0...k_m}\\mathcal{O}_X$ and proves adelic descent for solid quasi-coherent sheaves, the input diagram this paper passes to nuclear objects","marker":"[1]"},{"why":"introduces nuclear objects and nuclear modules, the categories for which the descent statement is formulated","marker":"[2]"},{"why":"supplies the duality, base-change, and nuclearity facts used to pass from solid modules to nuclear modules","marker":"[3]"},{"why":"sets up localizing invariants on dualizable presentable categories and the preservation of dualizable limits","marker":"[4]"},{"why":"provides rigidification comparisons identifying nuclear modules over the adelic rings with induced rigid categories","marker":"[5]"},{"why":"contributes the cubical reduction principle that splits the adelic cube into fiber sequences","marker":"[6]"},{"why":"gives Zariski descent for solid quasi-coherent sheaves, used to reduce to affine open covers","marker":"[13]"}],"fun_headline_variants":["Adelic descent proven for continuous localizing invariants","Continuous localizing invariants obey adelic descent on Z-schemes","Adelic descent for all continuous localizing invariants","New: adelic descent for every continuous localizing invariant","Continuous invariants descend adeically on finite-type Z-schemes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, for a Zariski open cover, the square of nuclear module categories attached to the adelic rings is a pullback and that every stable localizing invariant preserves this square because its functors are strongly continuous localizations; the paper proves full faithfulness and asserts the remaining half and the preservation claim.","fun_headline_variants_meta":{"raw":{"variants":["Adelic descent proven for continuous localizing invariants","Continuous localizing invariants obey adelic descent on Z-schemes","Adelic descent for all continuous localizing invariants","New: adelic descent for every continuous localizing invariant","Continuous invariants descend adeically on finite-type Z-schemes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000604,"raw_usage":{"total_tokens":2709,"prompt_tokens":728,"completion_tokens":1981,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":344,"completion_tokens_details":{"reasoning_tokens":1896}},"tokens_in":344,"tokens_out":1981,"duration_ms":15379,"temperature":1.0,"reasoning_tokens":1896,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:44:44.565734+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the asserted pullback square for a non-affine scheme such as $\\mathbb{P}^1_{\\mathbb{Z}}$ with its standard affine cover: if the functor from nuclear modules over the adelic rings to the pullback over the two affine lines is not essentially surjective, the affine reduction fails. Alternatively, take the invariant to be algebraic K-theory on $\\mathrm{Spec}(\\mathbb{Z})$ and compare $F(\\mathrm{Nuc}(\\mathbb{Z}_\\blacksquare))$ with the limit over $F(\\mathrm{Nuc}(L_0\\mathbb{Z}))$, $F(\\mathrm{Nuc}(L_1\\mathbb{Z}))$, and $F(\\mathrm{Nuc}(L_{0,1}\\mathbb{Z}))$; any disagreement would refute Theorem 3.0.1.","supporting_citations":[{"cited_title":"Beilinson-Parshin adeles via solid algebraic geometry","cited_arxiv_id":"2403.08472","evidence_quote":"defines the adelic rings $L_{k_0...k_m}\\mathcal{O}_X$ and proves adelic descent for solid quasi-coherent sheaves, the input diagram this paper passes to nuclear objects"},{"cited_title":"Condensed mathematics and complex geometry,","cited_arxiv_id":null,"evidence_quote":"introduces nuclear objects and nuclear modules, the categories for which the descent statement is formulated"},{"cited_title":"Lectures on analytic geometry,","cited_arxiv_id":null,"evidence_quote":"supplies the duality, base-change, and nuclearity facts used to pass from solid modules to nuclear modules"},{"cited_title":"Adelic descent for K-theory","cited_arxiv_id":"2111.07202","evidence_quote":"contributes the cubical reduction principle that splits the adelic cube into fiber sequences"},{"cited_title":"Lectures on condensed mathematics,","cited_arxiv_id":null,"evidence_quote":"gives Zariski descent for solid quasi-coherent sheaves, used to reduce to affine open covers"}],"review_version":2}