{"id":"4d9e59ea-01f9-4132-ad40-22e38f523216","arxiv_id":"2607.15586","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The localizing tensor ideals of D(U) for cyclic p-groups, elementary abelian p-groups, and FI-modules are classified by homological support, yielding the telescope conjecture.","lead":"This paper classifies all localizing tensor ideals in derived categories of global representations for cyclic and elementary abelian p-groups and for FI-modules, and infers the telescope conjecture in those categories. It tests whether support-theoretic classification can work when compact objects are not rigid, a long-standing obstacle in tensor triangular geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the alleged failure of Hom_{D(E_p)}(χ_i, Σ^s e_n)=0 is not supported; naturality to larger elementary abelian groups annihilates the maps.","rationale":"The stress-test pass did not find a load-bearing flaw in the central argument. The reader's strongest concern is a concrete computational objection to the orthogonality used in Corollary 5.5, but that objection ignores the naturality condition: a nonzero value in e_n(E_i) does not automatically give a natural transformation χ_i→e_n because maps to larger elementary abelian groups must compose to zero. For the free objects e_n, those transition maps are injective, so no nonzero natural transformation exists. With injectivity of e_n, the derived vanishing claimed in Corollary 5.5 holds. The same argument transfers to FI-modules via injectivity of M_n. Thus the advertised telescope conjecture for D(E_p), VI-modules, and FI-modules is not refuted by the cited issue. The paper's remaining unproved lemmas and 'same argument' references are gaps in exposition rather than identified mathematical falsehoods; they may warrant caution but do not constitute a concrete load-bearing objection. Hence no significant objection is identified.","tokens_in":21848,"tokens_out":42981,"duration_ms":479070,"concrete_test":"Compute Hom_{A(E_p)}(χ_i,e_n) directly for i=n. A component x∈e_n(E_n) must satisfy e_n(f)(x)=0 for the transition map induced by a fixed surjection f:E_{n+1}→E_n. Since e_n(f) is injective on the basis of surjections E_n→E_n, this forces x=0. More generally, verify that e_n is injective in A(E_p) (e.g., via [5, Lemma 7.10]) so the derived Homs reduce to this abelian Hom; if this verification passes, the identity used in Corollary 5.5 stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's central objection is that Corollary 5.5 uses the identity Hom_{D(E_p)}(χ_{i,k}, Σ^s e_n)=0 and that this is false for i≥n because e_n((Z/p)^i)≠0. This misses the naturality condition defining Hom in the functor category A(E_p). A natural transformation χ_i→e_n is determined by an element x∈e_n(E_i) that is annihilated by the transition maps e_n(E_i)→e_n(E_j) for every surjection E_j→E_i with j>i. For e_n, these transition maps are injective: they send a basis element β:E_i→E_n to β∘α, where α:E_j→E_i is a fixed surjection, and this is injective on the basis of surjections. Hence the only possible x is 0. Thus Hom_{A(E_p)}(χ_i,e_n)=0 for all i,n. Since e_n is injective and projective (as used explicitly in the proof of Theorem 5.3, citing [5, Lemma 7.10]), the higher Ext groups vanish as well, so Hom_{D(E_p)}(χ_i,Σ^s e_n)=0 for all s. The same reasoning applies to the FI-module case, where M_n is injective by Lemma 6.5. Therefore the alleged counterexample does not land, and the proof of the telescope conjecture for D(E_p) and D_FI is not undermined by this point. The remaining omitted proofs (e.g., Lemma 6.12) are standard and not obviously load-bearing.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the derived categories D(U) of global representations over a characteristic-zero field k for several infinite families U, as well as the derived category of FI-modules. It claims a complete classification of localizing tensor ideals by homological support (Theorems 3.11, 4.15, 5.3, 6.13) and deduces the telescope conjecture for D(C_pr), D(C_p), D(E_p), the derived category of VI-modules, and the derived category of FI-modules (Theorem 1.4 and Corollaries 5.6, 6.14). The strategy is to separate each localizing ideal into a torsion part generated by characteristic objects and a non-torsion part controlled by a cofinite support and a single compact object such as e_\\Delta or M_\\Delta.","tokens_in":22209,"tokens_out":51602,"duration_ms":508157,"significance":"If the results were correct, this would be a significant advance: the telescope conjecture is usually proved in rigidly-compactly generated settings, while the categories considered here are generally not rigid. A complete classification of localizing ideals is also stronger than the telescope conjecture itself. The sections on C_pr, C_p, and E_p are developed in detail and appear to contain substantial work. However, the FI-section contains a false generation lemma (Lemma 6.6(4) and its consequence Lemma 6.9), and because the proof of the FI classification and telescope conjecture relies essentially on that lemma, the advertised Theorem 1.4 for D_FI is not established as written.","major_comments":[{"comment":"Lemma 6.6(4) states D(FI_{\\ge n}) = Loc<M_n>. This is false. Let \\chi_2 be the FI-module with \\chi_2(2)=k (trivial S_2-action) and \\chi_2(m)=0 for m\\ne 2. In D(FI_{\\ge 2}), the object \\chi_2 is nonzero and has homological support {2}. The class of objects with cofinite-or-empty homological support is a localizing subcategory; it contains M_2 because hsupp(M_2)=\\{m\\ge 2\\} is cofinite in {m\\ge 2}. Hence every object of Loc<M_2> has cofinite-or-empty support. But {2} is not cofinite, so \\chi_2\\notin Loc<M_2>. Thus D(FI_{\\ge 2})\\ne Loc<M_2>. The same example disproves Lemma 6.9: in D_FI, min hsupp(\\chi_2)=2, yet \\chi_2\\notin Loc<M_2>. Lemma 6.9 is used essentially in Proposition 6.11 and again in the proof of Theorem 6.13, so the classification of localizing ideals and the telescope conjecture for D_FI are unsupported.","section":"§6, Lemma 6.6(4) and Lemma 6.9"},{"comment":"The proof of Theorem 6.13 relies on the false assertion that (j_m)_!(j_m)^* M_{\\Delta} \\in Loc<M_m>; this is justified only by Lemma 6.6(4). Since that lemma is false, the containment is not established. Likewise, Proposition 6.11, which is used to show M_{\\Delta}\\in Loc<m_L> in Corollary 6.14, depends on Lemma 6.9. Consequently the FI-module part of the main theorem, including the telescope conjecture for D_FI, does not follow from the arguments given. A replacement argument for the FI case would be needed.","section":"§6, Theorem 6.13 and Corollary 6.14"},{"comment":"Lemma 6.12 is stated without proof: 'We omit the proof of the following lemma, which is the FI-analogue of Lemma 5.2.' This lemma is load-bearing in Theorem 6.13, where it is used to assert that a non-torsion localizing ideal has cofinite support. If it is standard, a precise reference should be supplied; otherwise a proof is required. The current text leaves an essential step unsupported.","section":"§6, Lemma 6.12"}],"minor_comments":[{"comment":"There is a typo in the sentence 'Hence L^\\perp=D(C_p)': it should be D(E_p).","section":"§5, Corollary 5.5"},{"comment":"The displayed diagram in the proof is garbled in the text; the triangle and the map g_n are hard to read. Please reformat.","section":"§3, Proposition 3.2"},{"comment":"The statement 'All categories considered in this paper are assumed to be small' is in tension with constructions such as Fun(U^op, Mod_k), which is not small. This is probably a harmless convention, but it should be stated precisely.","section":"§2, Conventions"},{"comment":"Parts (1)-(3) and (5) of Lemma 6.6 are standard, but part (4) is false as discussed. The 'proof omitted' note is not acceptable for a false statement.","section":"§6, Lemma 6.6"}],"recommendation":"reject","confidential_remarks":"The reader's central objection to Corollary 5.5 does not appear to land: a natural transformation \\chi_i\\to e_n is forced to be zero by naturality, since the transition maps of e_n are injective, and higher Ext vanishes because e_n is injective. My rejection is based instead on a different, concrete false statement in Section 6. The E_p and C_p portions may be salvageable, but the advertised FI-module results are unsupported by a false generation lemma; this is not a local fix within the current proof strategy."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this paper is a genuine advance, and I don't agree with the reader's rejection. The one objection that looked load-bearing — Hom(χ_i, Σ^s e_n)=0 failing when i≥n — does not land. A map χ_i→e_n in the functor category is an element x∈e_n(E_i) satisfying α_*x=0 for every surjection α:E_j→E_i with j>i. The induced map α_*:e_n(E_i)→e_n(E_j) sends a basis surjection f to fα, and that is injective on the basis, so x must be zero. Since e_n is injective and projective, higher Exts vanish too. The same argument goes through for FI with M_n. So Corollaries 5.5 and 6.14 are not undermined by the reported counterexample.\n\nWhat is genuinely new: the localizing ideal classifications for D(C_pr), D(C_p), D(E_p), and D_FI, and the resulting telescope conjecture for non-rigid categories. Previous work mostly settled thick ideals in the compact part or prime spectra; the step to all localizing ideals is substantial. The torsion-generic quotient is a real extension, and the homological-support bijections are likely to be useful.\n\nThe soft spots are real but smaller than the reader thinks. Lemma 6.12 and parts of Lemma 6.6 are left to the reader, and Corollary 6.14 leans on 'the same argument' from Corollary 5.5. That is sloppy for a paper with this much machinery; a referee should ask for the arguments in full. The dependence on [26] is heavy and worth making explicit, but it is not circular: [26] supplies tensor-abelian geometry, not the localizing-ideal classifications. The paper also has places where 'by the same proof' conceals bookkeeping, but I did not find a mathematical gap after the naturality check.\n\nWho it is for: people in tensor triangular geometry and representation stability. It is a serious paper with significant results, and it deserves a serious referee rather than a desk rejection. I would send it out, with a request that the omitted lemmas be proved and the reliance on [26] be stated cleanly.","headline":"The classifications are new and the telescope proofs hold up — the reader's main objection dissolves once naturality in A(E_p) is taken seriously; this deserves a serious referee.","tokens_in":22688,"tokens_out":3263,"would_cite":true,"duration_ms":37381,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18G80","20C99"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper classifies the localizing ideals of derived categories of global representations over several infinite families of finite groups and uses the classification to prove the telescope conjecture for these categories and for derived F","keywords":["telescope conjecture","global representations","localizing ideals","FI-modules","VI-modules","tensor triangular geometry","homological support","derived categories"],"falsifier":"Compute Hom_{D(E_p)}(χ_{i,k}, e_n) for i≥n in the derived category of global representations of elementary abelian p-groups. The evaluation e_n((Z/p)^i) is nonzero, giving a nonzero map χ_{i,k} → e_n in degree zero, contradicting the identity used in the proof of Corollary 5.5; the same computation with FI-modules (replacing e_n by M_n and χ_{i,k} by the FI-characteristic object) tests Corollary 6.14.","tokens_in":21711,"feed_emoji":"🔭","tokens_out":6664,"duration_ms":57560,"temperature":0.7,"pith_summary":"The paper aims to settle the telescope conjecture for derived categories of global representations over three infinite families of finite groups — cyclic groups of prime order plus the trivial group, cyclic p-groups, and elementary abelian p-groups — and for the derived category of FI-modules. A sympathetic reading: it claims that in each of these categories every smashing tensor ideal is generated by compact objects, and that the full lattice of localizing tensor ideals is classified by homological support into subsets (or open subsets) of an explicit spectrum. This matters because these categories are not rigidly compactly generated, so the standard stratification machinery that proves the telescope conjecture in rigid settings does not apply; the paper offers a different mechanism based on characteristic objects, torsion theory, and a generic quotient. If correct, the result also covers derived VI-modules via Pontryagin duality and provides the first large tensor-triangular classification in a non-rigid global representation setting.","feed_headline":"Telescope conjecture proven for global reps and FI-modules","feed_subtitle":"Classification of localizing ideals shows smashing ideals are compactly generated, extending support theory beyond rigid settings.","key_machinery":"The load-bearing machinery is the homological support hsupp(X) = {G | X(G) ≠ 0}, together with characteristic objects χ_{G,V} concentrated at a single group. For essentially finite families a localizing lemma shows any object is generated by the χ_{G,k} for G in its support. For the infinite families, the argument separates torsion from non-torsion localizing ideals: torsion ideals are generated by χ_{i,k}, while non-torsion ideals contain a compact generator e_Δ or M_Δ and are generated by it plus the χ_{i,k} in the support. A key technical point is the identification, for noetherian families C_p, E_p, and FI, of the parameter space as the tensor abelian spectrum Spc(A^c) rather than the Ba","core_discovery":"Homological support — the set of groups G with X(G) ≠ 0 — is shown to completely classify localizing tensor ideals. For cyclic prime-order groups plus the trivial group, every localizing ideal is generated by characteristic objects χ_{G,k} for G in a subset of P^*, recovering arbitrary subsets of the Balmer spectrum. For cyclic p-groups, only open subsets occur. For elementary abelian p-groups and FI-modules, the classifying space is the tensor abelian spectrum of the noetherian heart, and every non-torsion localizing ideal is generated by one compact generator together with the χ_{i,k} in its support. The telescope conjecture, that every smashing ideal is compactly generated, follows in eac","pith_inferences":["If correct, the support-theoretic form of the classification suggests that for non-rigid categories the relevant geometry is the tensor abelian spectrum of the noetherian heart; this may be the right invariant for other global families and representation-stability categories.","The same torsion/non-torsion dichotomy might extend to any widely closed family U with a locally noetherian heart and a semisimple generic quotient, which would make the telescope conjecture a consequence of two ingredients: compact generation of non-torsion ideals and an orthogonality statement for characteristic objects.","A direct computation of Hom_{D(E_p)}(χ_{i,k}, e_n) for i≥n would test the non-smashing torsion step; since e_n((Z/p)^i) is nonzero, this is a concrete and inexpensive check."],"forward_implications":["If Theorem 1.4 is correct, the telescope conjecture holds for the derived categories of global representations over C_p^r, C_p, E_p, and FI-modules, and via Pontryagin duality for VI-modules.","The bijections in Theorems 3.11, 4.15, 5.3, and 6.13 mean the entire lattice of localizing tensor ideals is explicitly parametrized by subsets or open subsets of a known spectrum, not just the smashing ones.","Because the classification is support-theoretic, it provides a substitute for Balmer–Favi stratification in these non-rigid categories, where the compact objects are not rigid.","The Balmer spectrum of D(E_p)^c is not N^*, so the big category's geometry is governed by the tensor abelian spectrum; this changes how one predicts smashing ideals from compact data.","Every smashing ideal in these categories is generated by compact objects, matching the predicted form of the telescope conjecture."],"fun_headline_variants":["Telescope conjecture solved for global reps","Global reps and FI-modules: telescope conjecture proven","Localizing ideals classified, telescope conjecture follows","Smashing ideals compactly generated: telescope conjecture proven"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof that torsion localizing ideals in D(E_p) and D(FI) are not smashing depends on the assertion that Hom_{D(E_p)}(χ_{i,k}, Σ^s e_n)=0 for all i,n,s; but this appears false for i≥n because e_n((Z/p)^i) ≠ 0, so the argument for Corollary 5.5 (and its FI analogue in Corollary 6.14, which invokes 'the same argument') has no support as written.","fun_headline_variants_meta":{"raw":{"variants":["Telescope conjecture solved for global reps","Global reps and FI-modules: telescope conjecture proven","Localizing ideals classified, telescope conjecture follows","Smashing ideals compactly generated: telescope conjecture proven"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000145,"raw_usage":{"total_tokens":961,"prompt_tokens":635,"completion_tokens":326,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":379,"completion_tokens_details":{"reasoning_tokens":267}},"tokens_in":379,"tokens_out":326,"duration_ms":3970,"temperature":1.0,"reasoning_tokens":267,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T22:53:36.653105+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute Hom_{D(E_p)}(χ_{i,k}, e_n) for i≥n in the derived category of global representations of elementary abelian p-groups. The evaluation e_n((Z/p)^i) is nonzero, giving a nonzero map χ_{i,k} → e_n in degree zero, contradicting the identity used in the proof of Corollary 5.5; the same computation with FI-modules (replacing e_n by M_n and χ_{i,k} by the FI-characteristic object) tests Corollary 6.14.","supporting_citations":[],"review_version":1}