{"id":"f193e055-2b38-4b54-b0d1-51c067c47678","arxiv_id":"2605.26057","paper_version":3,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A Noetherian scheme is locally a complete intersection iff every object of D^b_coh(X) is t-⊗-proxy small, with classifications of ⊗-suspended subcategories for hypersurface and complete-intersection cases.","lead":"The paper defines t-proxy smallness for t-structures with tensor actions on derived categories of Noetherian schemes. It gives a new t-structural characterization of locally complete intersection schemes and classifies certain preaisles via singularity categories and Thomason filtrations.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The Reader correctly isolates Lemma 6.8 as the single load-bearing step that upgrades the known affine characterization to arbitrary Noetherian schemes and removes the separatedness hypothesis of earlier work. The proof of that lemma is written out in full and relies only on standard properties of sheaf Hom and the fact that compact generators of K(Inj(X)) are images under Q_𝜌 of objects of D^b_coh(X). No circularity or unstated global assumption appears. Consequently the equivalence (1)\rightleftharpoons(3) of Proposition 1.2 stands, the classifications that follow from it remain valid, and the Reader’s ACCEPT verdict with high confidence requires no adjustment.","tokens_in":34474,"tokens_out":471,"duration_ms":5092,"concrete_test":"Independently verify the key isomorphism Hom_K(Qcoh(X))(S,E) ≅ Hom_K(Inj(X))(Q_𝜌(S),E) used in the first paragraph of the proof of Lemma 6.8 (via the cone of S\to Q_𝜌(S) being bounded-below acyclic) for a non-separated Noetherian scheme (e.g., the affine line with doubled origin); if the isomorphism fails, the locality claim would need re-examination.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central equivalence (Proposition 1.2) rests on the stalk-locality of membership in compactly generated ⊙-aisles of K(Inj(X)) (Lemma 6.8). The proof of that lemma is self-contained: it reduces Hom-vanishing to vanishing of cohomology of the sheaf-Hom complex, uses that the latter is stalkwise, and recovers global vanishing of Hom from compact generators by the same stalkwise argument. The reduction from the global t-⊗-proxy-small claim to the local complete-intersection case (Proposition 6.9) then follows by applying the locality statement to Q_𝜆(P). No hidden global hypothesis (e.g., separatedness) is used, and the earlier affine results of Pollitz and Letz are invoked only after the locality step. The argument therefore appears tight.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper introduces t-proxy smallness (and its tensor refinement t-⊗-proxy smallness) for objects and preaisles in triangulated categories associated to a Noetherian scheme X, working with tensor actions of Perf≤0(X) on Dqc(X), K(Inj(X)) and Sqc(X). The main structural result (Theorem 5.4 / Proposition 5.5) embeds the collection of t-⊗-proxy-small ⊗-suspended subcategories of Dbcoh(X) into pairs consisting of a ★-suspended subcategory of the singularity category and a Thomason filtration on X. As a geometric consequence, the authors prove that X is locally a complete intersection if and only if every object of Dbcoh(X) is ⊗-proxy small if and only if every object is t-⊗-proxy small (Proposition 1.2). When X has only hypersurface singularities, or arises as a zero locus of a section of a vector bundle on a regular scheme (Setup 7.6), the embedding becomes a bijection, yielding topological classifications of ⊗-suspended subcategories of Dbcoh(X) (Theorems 1.3 and 1.5). Intermediate technical results include the compact generation of ⊗-aisles generated by pseudocoherent complexes (Theorem 3.5) and a stalk-locality statement for membership in compactly generated ⊙-aisles of K(Inj(X)) (Lemma 6.8).","tokens_in":34660,"tokens_out":882,"duration_ms":9085,"significance":"The work supplies a genuine t-structural refinement of the proxy-small characterizations of local complete intersections due to Pollitz and Briggs–Iyengar–Letz–Pollitz, and upgrades the latter from the separated to the arbitrary Noetherian setting. The stalk-locality lemma for aisles on K(Inj(X)) is a useful technical contribution that may find further applications. The classification theorems give a non-affine generalization of Takahashi’s recent results on suspended subcategories and are independent of his methods. The distinction between proxy smallness and t-proxy smallness is illustrated by concrete examples (Example 4.11), showing that the new notion is strictly finer. Overall the paper advances the interaction of tensor-triangular geometry, t-structures and singularity categories in a coherent and well-motivated way.","major_comments":[],"minor_comments":[{"comment":"In the introduction (p. 3) the phrase “Producing examples of t-proxy small objects of Dbcoh(X) which are not t-proxy small” is clearly a slip; the intended contrast is with ordinary proxy-small objects.","section":"Introduction"},{"comment":"Notation for the three tensor actions (⊗, ⊙, ★) is introduced only in Example 2.14; a brief forward reference earlier in §2.4 would help the reader.","section":"§2.4"},{"comment":"The proof of Lemma 6.8 invokes [Har13, III.6.8] for stalkwise vanishing of sheaf-Hom cohomology; a parenthetical reminder that the complexes are quasi-coherent would make the citation self-contained.","section":"Lemma 6.8"},{"comment":"Several arXiv preprints are cited with temporary identifiers (e.g., HHLG26); once final versions appear they should be updated.","section":"References"}],"recommendation":"accept","confidential_remarks":"The manuscript is technically solid and the central claims are correctly proved. The novelty relative to the existing proxy-small literature is genuine, and the paper fits well within the scope of a general algebraic-geometry or homological-algebra journal. I see no reason to request further major changes."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The main takeaway is a clean t-structural refinement of the known proxy-small characterizations of local complete intersections, plus explicit bijections classifying ⊗-suspended subcategories of D^b_coh for hypersurface and complete-intersection schemes. Proposition 1.2 equates three conditions: X locally CI, every object of D^b_coh ⊗-proxy small, and every object t-⊗-proxy small. That upgrades BILP22 by dropping separatedness and by inserting the t-structure language.\n\nWhat is new is the pair of definitions (t-proxy small and its tensor version) and the way they sit inside Krause’s recollement. Theorem 3.5 (pseudocoherent complexes generate compactly generated ⊗-aisles) is a useful technical step that extends the affine result of Alonso–Jeremías–Saorín. The classification theorems (1.3, 1.5, 7.4, 7.7) give a non-affine generalization of Takahashi with an independent proof that routes through Stevenson’s support theory; the image of Θ is described cleanly by a support condition on the singularity category plus a Thomason filtration.\n\nThe soft spot the reader flagged—Lemma 6.8, stalk-locality of membership in compactly generated ⊙-aisles on K(Inj(X))—is actually proved carefully. It reduces Hom-vanishing to vanishing of cohomology of the sheaf-Hom complex, uses that the latter is stalkwise, and recovers global vanishing from compact generators by the same stalkwise argument. No hidden separatedness hypothesis appears, and the reduction from global t-⊗-proxy smallness to the local Pollitz-type argument then goes through. The rest of the paper is standard, carefully written pure math; citations look appropriate and the circularity burden is low.\n\nThis is for people already working with singularity categories, tensor-triangular geometry, or generation questions in D_qc and D_sg. It is not a broad-impact paper, but it is a reliable advance inside that circle. I would send it to a serious referee without hesitation.","headline":"Solid upgrade of proxy-smallness characterizations of local complete intersections, plus usable classifications of ⊗-preaisles; the stalk-locality lemma holds up.","tokens_in":35268,"tokens_out":577,"would_cite":true,"duration_ms":7346,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14A30","14F08","13D09","18G80","14B05"],"pacs":[],"model":"grok-4.5","headline":"A scheme is locally a complete intersection exactly when every bounded coherent complex is t-⊗-proxy small.","keywords":["proxy smallness","t-structures","tensor actions","local complete intersection","preaisles","Thomason filtrations","singularity category","pseudocoherent generation"],"falsifier":"Exhibit a Noetherian local ring that is not a complete intersection yet every object of its bounded coherent derived category is t-proxy small, or find a compactly generated aisle on K(Inj(X)) that fails to be stalk-local.","tokens_in":35372,"feed_emoji":"📐","tokens_out":664,"duration_ms":6455,"temperature":0.7,"pith_summary":"The paper refines the classical notion of proxy smallness by restricting the allowed operations to those of a t-structure (non-negative shifts, extensions, and direct summands) and by incorporating tensor actions of perfect complexes of non-positive amplitude. It proves that, for any Noetherian scheme, three conditions are equivalent: the scheme is locally a complete intersection; every object of the bounded derived category of coherent sheaves is ⊗-proxy small; and every such object is t-⊗-proxy small. Along the way the authors classify certain ⊗-preaisles by pairs consisting of a suspended subcategory of the singularity category and a Thomason filtration of the underlying space. The result upgrades earlier characterizations from the affine or separated setting to arbitrary Noetherian schemes and supplies an independent proof of recent classification theorems for hypersurface and complete-intersection singularities.","feed_headline":"Local CI schemes detected by t-proxy smallness","feed_subtitle":"Every bounded coherent complex is t-⊗-proxy small exactly when the scheme is locally a complete intersection","key_machinery":"t-⊗-proxy smallness: an object P is t-⊗-proxy small when the smallest cocomplete ⊗-preaisle it generates is compactly generated and the compact objects inside that preaisle already lie in the ordinary ⊗-preaisle generated by P. The notion is characterized by a gluing condition for the standard t-structure along Krause’s recollement.","core_discovery":"A Noetherian scheme X is locally a complete intersection if and only if every object of D^b_coh(X) is t-⊗-proxy small (equivalently, ⊗-proxy small). The same framework yields an injective map from the lattice of ⊗-suspended subcategories of D^b_coh(X) into the product of the lattice of ★-suspended subcategories of the singularity category and the set of Thomason filtrations on X; the map becomes bijective when X has only hypersurface singularities or arises as a zero locus of a section of a vector bundle on a regular scheme.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Local CI schemes via every D^b_coh object being t-⊗-proxy small","t-proxy smallness detects schemes that are locally complete intersections","Proxy smallness of coherent complexes characterises local CI schemes","t-structures classify local CI schemes by ⊗-proxy smallness","Preaisles of D^b_coh classified topologically via t-proxy smallness"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"Membership of an object in a compactly generated aisle of the homotopy category of injectives is completely determined by its stalks.","fun_headline_variants_meta":{"raw":{"variants":["Local CI schemes via every D^b_coh object being t-⊗-proxy small","t-proxy smallness detects schemes that are locally complete intersections","Proxy smallness of coherent complexes characterises local CI schemes","t-structures classify local CI schemes by ⊗-proxy smallness","Preaisles of D^b_coh classified topologically via t-proxy smallness"]},"model":"grok-4.5","effort":"low","cost_usd":0.006728,"raw_usage":{"total_tokens":1618,"prompt_tokens":649,"num_sources_used":0,"completion_tokens":102,"cost_in_usd_ticks":67280000,"prompt_tokens_details":{"text_tokens":649,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":867,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":649,"tokens_out":102,"duration_ms":8837,"temperature":1.0,"reasoning_tokens":867,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T18:41:40.675339+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a Noetherian local ring that is not a complete intersection yet every object of its bounded coherent derived category is t-proxy small, or find a compactly generated aisle on K(Inj(X)) that fails to be stalk-local.","supporting_citations":[],"review_version":2}