{"id":"2d47044d-1a90-4d8d-aa8c-38d80c085fe0","arxiv_id":"2506.12429","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In locally cohomologically stratified tensor triangular categories with noetherian spectrum, the dualizable localizing ideals are exactly the localizing ideals supported on convex subsets of the Balmer spectrum.","lead":"This paper proves a classification theorem in tensor triangular geometry: in a broad class of stable homotopy categories, the localizing ideals that are dualizable correspond exactly to the convex subsets of the Balmer spectrum. It extends a recent theorem of Efimov from derived categories of rings to derived categories of schemes and other stratified tensor-triangulated categories.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.9's hard direction hinges on Efimov's unpublished dualizable Neeman–Thomason theorem (Thm 3.14); if that theorem is misstated or needs compact generation, Lemma 4.21 and the classification collapse.","rationale":"I traced the proof of Theorem 4.9. The easy direction uses Proposition 4.7 and is self-contained. The converse is where the classification lives: the argument localizes at P, reduces to Lemma 4.21, and obtains the needed non-dualizability by producing a compact object t via Theorem 3.14. I checked the surrounding support calculations (Lemma 4.18, Corollary 4.20, and the tensor-product identifications) and they are internally coherent, modulo the usual hypotheses of rigidly-compactly generated categories and local cohomological stratification. The tensor-product step in the final paragraph requires D_{Q'} to be dualizable; this is not stated but follows because D_{Q'} is a retract of the dualizable category D. The one genuinely external load-bearing input is Theorem 3.14, and the paper gives no proof. Its conclusion q(a) ≅ x⊕Σx is stronger than the classical Neeman–Thomason theorem and is not obviously true in the stated generality. Thus I agree with the reader's weakest_assumption. My recommendation is conditional acceptance: the classification should be accepted once Theorem 3.14 is checked against [Efi24] or proved.","tokens_in":13835,"tokens_out":18882,"duration_ms":230335,"concrete_test":"Obtain [Efi24, Proposition 1.18] and compare it word-for-word with Theorem 3.14. Check (i) whether the hypothesis is 'C dualizable' or 'C compactly generated'; (ii) whether the conclusion is q(a) ≅ x⊕Σx or merely 'x is a direct summand of q(a)'. If the stronger conclusion is absent, test Lemma 4.21: can the argument still produce a compact t with the required coproduct-commuting property? A second independent check: re-prove Lemma 4.21 in the affine case C = D(R) using classical Neeman–Thomason; if the affine proof needs the same x⊕Σx conclusion, the dependency is real and must be resolved before Theorem 4.9 is accepted.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim Theorem 4.9 has two directions: convex S ⇒ C_S dualizable (Proposition 4.7, proved in text) and dualizable L ⇒ Supp(L) convex. The second direction is the load-bearing one. Its proof reduces to Lemma 4.21, which shows that in a local cohomologically stratified category a certain ideal C_{S∩{Q}} is not dualizable. The decisive step in Lemma 4.21 is: assuming C_{S1} dualizable, choose a nonzero compact y in C_{S1}/C_{S2} and invoke Theorem 3.14 to produce a compact t ∈ C_{S1} with q(t) ≅ y⊕Σy; then t satisfies Lemma 4.18 and Corollary 4.20, forcing the contradiction. Theorem 3.14 is quoted verbatim from Efimov's unpublished preprint [Efi24] and is not proved or even sketched here. It is also stronger than the classical Neeman–Thomason theorem, which only gives x as a direct summand of q(a), not q(a)≅x⊕Σx. If the theorem is misstated, or if it requires C to be compactly generated rather than merely dualizable, then no compact t is produced, Lemma 4.21 does not go through, and Theorem 4.9 is reduced to the easy direction (convex ⇒ dualizable). The paper's own Remark 4.22 confirms that without Theorem 3.14 the converse is unavailable. This is an external, unpublished dependency, not an internal inconsistency, but it is the single most load-bearing assumption in the proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies rigidly-compactly generated tensor triangulated ∞-categories and their dualizable localizing ideals. The main theorem (Theorem 4.9) asserts that, under local cohomological stratification and a noetherian Balmer spectrum, tensor triangular support induces an inclusion-preserving bijection between dualizable localizing ideals and convex subsets of Spc(Cc), and that such ideals are compactly generated. The proof proceeds by reducing the hard direction to a local lemma (Lemma 4.21), using Efimov's extension of the Neeman–Thomason theorem, BIK support theory, and support computations from the author's earlier work. The paper also applies the result to derived categories of noetherian schemes, generalizing Efimov's affine classification.","tokens_in":14112,"tokens_out":16527,"duration_ms":200612,"significance":"If correct, the paper gives a clean structural classification in tensor triangular geometry that unifies Neeman-type and Efimov-type results at a high level of generality. The proof is coherent and does not assume the conclusion: it uses stratification, local-to-global principles, and support identification as inputs, and the author explicitly records in Remark 4.22 that without Efimov's dualizable Neeman–Thomason theorem only the easier direction is available. The main caveat is that the hard direction depends on Theorem 3.14, which is quoted from an unpublished preprint and not proved in the manuscript.","major_comments":[{"comment":"The hard direction of Theorem 4.9 depends on Theorem 3.14, Efimov's extension of the Neeman–Thomason localization theorem to dualizable categories. The theorem is quoted from the unpublished preprint [Efi24] and no proof is supplied. It is stronger than the classical Neeman–Thomason theorem because it asserts q(a) ≅ x ⊕ Σx rather than only that x is a direct summand of q(a), and it is precisely what produces the compact object t in Lemma 4.21. The authors should prove this theorem in the paper, or give a precise, publicly verifiable reference with all hypotheses checked; as Remark 4.22 acknowledges, without it the converse implication in Theorem 4.9 is unavailable.","section":"§3.14, Lemma 4.21"},{"comment":"Several support-theoretic inputs are taken from the author's unpublished preprint [Zou23]: Theorem 9.3 (identification of tt-support with BIK support), Example 6.1 (localization at a prime), and Corollary 5.30 (support under base change). These statements are used at load-bearing points in Lemma 4.18, Proposition 3.23, and the proof of Theorem 4.9. The paper should state these results with precise hypotheses or give proofs for the cases used, so that the reader can verify the translation between BIK support and tensor triangular support.","section":"§3.18, §3.23, Theorem 4.9 proof"}],"minor_comments":[{"comment":"The word 'spetrum' in Example 3.12 should be 'spectrum'.","section":"§3.12"},{"comment":"In the proof of Lemma 4.12, 'To prove thatα is an isomorphism' is missing a space; it should read 'To prove that α is an isomorphism'.","section":"§4.12"},{"comment":"In Example 4.25, 'stratifed' should be 'stratified'.","section":"§4.25"},{"comment":"In Remark 4.22, 'the the extension' contains a duplicated article and should be corrected.","section":"§4.22"},{"comment":"The notation T^⊥ in equation (3.7) is used without being defined; please add a sentence defining it as the right orthogonal of T_{S2} in T.","section":"Equation (3.7)"},{"comment":"The Brown representability step in Lemma 4.12 would be easier to check if the author briefly noted that the functor Hom_R(H^*_1(-), I) sends coproducts to products because the unit is compact.","section":"§4.12"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is mathematically coherent and the main theorem is likely correct, but the central proof relies on an unpublished theorem of Efimov and on several unpublished support-identification results from the author's own preprint. If the editors are willing to accept references to arXiv preprints for load-bearing statements, the paper could be accepted after a statement/proof check; otherwise the authors should supply proofs or published references."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves the expected classification: in a rigidly-compactly generated tt-∞-category that is locally cohomologically stratified with noetherian Balmer spectrum, dualizable localizing ideals correspond exactly to convex subsets of the spectrum. This genuinely extends Efimov's theorem from affine derived categories to derived categories of noetherian schemes and other examples, and the statement is clean. The proof is largely coherent and transparent about its inputs. It reduces the hard direction to a local statement, uses BIK support to show certain objects have specialization-closed support, and then derives the contradiction. I especially liked Corollary 4.20; the argument via injective cogeneration is neat.\n\nThe main soft spot is real: the hard direction of Theorem 4.9 passes through Lemma 4.21, which uses Efimov's dualizable Neeman–Thomason theorem (Theorem 3.14) to produce a compact object t with q(t) ≅ y ⊕ Σy. That theorem is quoted from a preprint and not proved in this paper. If it is misstated, or if it needs compact generation rather than dualizability, Lemma 4.21 collapses and the classification reduces to the easy convex ⇒ dualizable direction. The paper is honest about this in Remark 4.22, but a referee should verify Theorem 3.14 carefully before signing off. There is also reliance on the author's own [Zou23] for support identifications; those are used as tools, and I see no circularity there.\n\nThe citation pattern looks normal; self-citation is for work the author actually did. The math appears solid conditional on the external theorem, and the paper is well organized apart from minor typos. This is not a desk-reject. It deserves a serious referee, and the referee's main task is to check that one external input. I would cite this if I worked in the area, and I would bring it to a reading group with the caveat that Efimov's theorem needs scrutiny before relying on it.","headline":"Clean generalization of Efimov's dualizable ideal classification, but its hard direction leans on an unproved external theorem that a referee must check.","tokens_in":14646,"tokens_out":2613,"would_cite":true,"duration_ms":31860,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18G80","18N60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Convex subsets classify dualizable localizing ideals in tt-geometry","keywords":["tensor triangular geometry","Balmer spectrum","localizing ideals","dualizable categories","convex subsets","cohomological stratification","Lurie tensor product","derived categories of schemes"],"falsifier":"Construct a rigidly-compactly generated tt-$\\infty$-category that is locally cohomologically stratified with noetherian Balmer spectrum, and exhibit a dualizable localizing ideal whose support is not convex; Theorem 4.9 asserts none exists. Equivalently, find a convex subset $S$ of such a spectrum for which $\\mathcal{C}_S$ is not dualizable. A concrete place to look is the local Lemma 4.21: a subset containing the unique closed point and a point $Q$ with $\\overline{\\{Q\\}} \\nsubseteq S$ but $S \\cap (\\overline{\\{Q\\}}\\setminus\\{Q\\})$ specialization closed should force $\\mathcal{C}_{S\\cap\\overline{\\{Q\\}}}$ to be non-dualizable, so producing a category where it is dualizable would refute the claim.","tokens_in":13593,"feed_emoji":"🔺","tokens_out":7516,"duration_ms":76352,"temperature":0.7,"pith_summary":"The paper establishes a classification theorem for a large class of tensor triangular categories: rigidly-compactly generated tensor triangular $\\infty$-categories that are locally cohomologically stratified and have noetherian Balmer spectrum. In such a category, the dualizable localizing ideals---those with a well-behaved dual under the Lurie tensor product---are in inclusion-preserving bijection with the convex subsets of the Balmer spectrum. The bijection sends an ideal to its tensor triangular support, and the inverse sends a convex subset to the localizing ideal of objects supported there. If the paper is right, this fills the middle rung of a three-level hierarchy of ideal classifications, between all localizing ideals (arbitrary subsets) and compactly generated localizing ideals (specialization closed subsets), and extends a known affine classification to settings such as derived categories of noetherian schemes.","feed_headline":"Convex subsets classify dualizable localizing ideals","feed_subtitle":"A broad family of tensor triangular categories, including noetherian schemes, now has a convex-set classification.","key_machinery":"The argument turns on four interacting pieces. Local cohomological stratification lets the paper compute supports through local cohomology of the endomorphism ring. A quoted dualizable version of the Neeman-Thomason localization theorem, stated as Theorem 3.14, supplies the compact object needed to detect a contradiction in the local case. The characterization of convex subsets as differences $S_1 \\setminus S_2$ of specialization closed subsets connects convexity to finite localizations. The tensor product of presentable stable categories, whose dualizability is stable under base change and localization by Lemmas 2.12 and 2.13, transports the nonconvexity obstruction from a local category back to the original one. Tensor triangular support itself, the set of primes where an object remains nonzero after tensoring with the idempotent $g_P$, is the map that carries the bijection.","core_discovery":"The central claim is Theorem 4.9. It asserts that for a rigidly-compactly generated tt-$\\infty$-category $\\mathcal{C}$ that is locally cohomologically stratified and has noetherian $\\mathrm{Spc}(\\mathcal{C}^c)$, tensor triangular support induces an inclusion-preserving bijection from dualizable localizing ideals of $\\mathcal{C}$ to convex subsets of $\\mathrm{Spc}(\\mathcal{C}^c)$, with inverse $S \\mapsto \\mathcal{C}_S$. The theorem also asserts that each dualizable localizing ideal is itself compactly generated as a stable $\\infty$-category. The proof first reduces to the case where the category is stratified, then treats a local category with a unique closed point: there, Lemma 4.21 shows that an ideal supported on a nonconvex 'punctured closure' slice cannot be dualizable, and global nonconvexity is pushed forward through finite localizations and relative tensor products to produce a contradiction.","pith_inferences":["Editorial inference: the theorem suggests that dualizability is the categorical finiteness condition that turns arbitrary localizing-ideal classifications into convex-set classifications, so convexity could play a similar role in any stratified tensor triangular category with a closed tensor product.","A testable extension would be to drop local cohomological stratification and ask whether plain stratification plus a noetherian spectrum already forces the classification; the proof leans on support computations from local cohomology, so that hypothesis is the first one to probe.","The paper's Remark 4.23 indicates the dualizable-convex bijection can survive in some non-noetherian settings; identifying precisely which non-noetherian rings satisfy it would sharpen the boundary of the theorem."],"forward_implications":["For every noetherian scheme $X$, the dualizable localizing ideals of $\\mathcal{D}_{\\mathrm{qc}}(X)$ correspond to the convex subsets of $X$.","Every dualizable localizing ideal in the classified categories is compactly generated, refining the classical description of compactly generated ideals in terms of specialization closed subsets.","The affine classification for commutative noetherian rings is recovered as the special case $\\mathcal{C} = \\mathcal{D}(R)$.","Cohomologically stratified categories, including many examples from modular representation theory and equivariant homotopy theory, fall under the theorem via Corollary 4.24.","The inclusion-preserving bijection means convexity is exactly the finiteness property that dualizability imposes on localizing ideals in this setting."],"supporting_citations":[{"why":"Supplies the dualizable version of the Neeman-Thomason localization theorem stated as Theorem 3.14, which the proof uses to find compact objects in localizations of dualizable categories.","marker":"[Efi24]"},{"why":"Provides the stratification framework, the definition and theory of cohomological stratification, and the local-to-global principle used in Proposition 3.23.","marker":"[BHS23]"},{"why":"Provides the local cohomology and support results used in Lemma 4.18 and Corollary 4.20 to show certain supports are specialization closed.","marker":"[BIK08]"},{"why":"Supplies the comparison between tensor triangular support and Benson-Iyengar-Krause support used throughout the proof, including behaviour under localization.","marker":"[Zou23]"},{"why":"Provides the definitions and properties of locally rigid and dualizable categories, including the base-change lemma that preserves fully faithful morphisms.","marker":"[Ram24]"},{"why":"Gives the classical Neeman-Thomason localization theorem for compactly generated categories, which the dualizable version extends.","marker":"[Nee96]"},{"why":"Introduces the Balmer spectrum and its universal support theory, which underlies the entire ideal-subset correspondence.","marker":"[Bal05]"}],"fun_headline_variants":["Convex subsets classify dualizable ideals in tt-geometry","Dualizable localizing ideals equal convex subsets of Balmer spectrum","Convexity gives dualizable ideal classification for tt-categories","For noetherian schemes, convex sets classify dualizable ideals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is a quoted theorem, not proved here, saying that localizations of dualizable stable categories behave like the classical Neeman-Thomason localization theorem; if that statement fails at the stated level of generality, the argument that a nonconvex support forces a non-dualizable ideal breaks.","fun_headline_variants_meta":{"raw":{"variants":["Convex subsets classify dualizable ideals in tt-geometry","Dualizable localizing ideals equal convex subsets of Balmer spectrum","Convexity gives dualizable ideal classification for tt-categories","For noetherian schemes, convex sets classify dualizable ideals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000788,"raw_usage":{"total_tokens":3423,"prompt_tokens":844,"completion_tokens":2579,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":460,"completion_tokens_details":{"reasoning_tokens":2508}},"tokens_in":460,"tokens_out":2579,"duration_ms":96611,"temperature":1.0,"reasoning_tokens":2508,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:51:02.793892+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a rigidly-compactly generated tt-$\\infty$-category that is locally cohomologically stratified with noetherian Balmer spectrum, and exhibit a dualizable localizing ideal whose support is not convex; Theorem 4.9 asserts none exists. Equivalently, find a convex subset $S$ of such a spectrum for which $\\mathcal{C}_S$ is not dualizable. A concrete place to look is the local Lemma 4.21: a subset containing the unique closed point and a point $Q$ with $\\overline{\\{Q\\}} \\nsubseteq S$ but $S \\cap (\\overline{\\{Q\\}}\\setminus\\{Q\\})$ specialization closed should force $\\mathcal{C}_{S\\cap\\overline{\\{Q\\}}}$ to be non-dualizable, so producing a category where it is dualizable would refute the claim.","supporting_citations":[],"review_version":1}