{"id":"cd22a285-cf11-4bb5-83ca-ed0fcdda04bf","arxiv_id":"2512.05026","paper_version":3,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For Noetherian concentrated F-finite algebraic stacks with quasi-finite separated diagonal, Frobenius pushforwards of perfect complexes classically generate the bounded derived category.","lead":"This paper defines F-finiteness for algebraic stacks and proves that sufficiently many Frobenius pushforwards generate bounded derived categories of coherent sheaves. It extends an earlier result for schemes to a broad class of stacks, including Deligne–Mumford stacks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.18's induction hinges on a corrected but unstated version of [HR18, Thm E]; without verifying the exact hypotheses of that dévissage theorem, the proof is not self-contained.","rationale":"The reader's weakest assumption correctly identifies the application of [HR18, Theorem E] as the main risk in Theorem 4.18. I agree that the proof depends on a dévissage theorem whose precise hypotheses are not spelled out. My reading of the paper confirms that the footnote explicitly acknowledges a typo in (I2) of the published theorem and defers to [DLM25, Prop. 5.10] for the corrected version. This is a genuine self-reported limitation: the authors do not state the corrected theorem, nor do they show their class E satisfies all its hypotheses. The three propositions in the paper verify the natural dévissage steps, but without the exact statement of Theorem E it is impossible to rule out hidden conditions (e.g., stability under arbitrary base change, or requirement that the property is étale-local on the target). The reader also mentions Proposition 4.15 relying on a finite flat surjective morphism from an affine scheme; this is partially a non-issue because Proposition 4.15 assumes such a morphism as a hypothesis, and in the dévissage step (2) such a morphism is given as an element of E. However, the flatness of that morphism is essential and the corrected (I2) must guarantee it. Therefore my concern is aligned with the reader's main worry, but I would not focus on the affine-cover existence as the primary gap. The appropriate verdict is CONDITIONAL: the central claim is likely true, but the paper should be accepted only after the authors explicitly state [HR18, Theorem E], prove that their class E and property D satisfy its hypotheses, and address the (I2) flatness issue transparently. This does not change the reader's verdict, but it sharpens the condition.","tokens_in":18166,"tokens_out":18502,"duration_ms":160571,"concrete_test":"Read the exact statement of [HR18, Theorem E] and [DLM25, Prop. 5.10]. Check that: (i) the theorem's hypotheses on S (e.g., quasi-compact, quasi-separated, quasi-finite diagonal) are satisfied by S in Theorem 4.18; (ii) the class E defined in the proof is exactly the class covered by the theorem; (iii) the property 'source satisfies Hypothesis 4.9' satisfies the theorem's required closure conditions (e.g., stability under base change and étale localization); (iv) condition (I2) is finite flat surjective, not merely finite surjective. If any of these fail, the induction in Theorem 4.18 has a gap that must be fixed before the result is accepted.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim (Theorem 4.18) is established by reducing Hypothesis 4.9 to three base cases via [HR18, Theorem E]. The paper neither states [HR18, Theorem E] nor verifies that the class E (morphisms representable by algebraic spaces, separated, finitely presented, quasi-finite, flat) and the property D ('source satisfies Hypothesis 4.9') meet the theorem's closure hypotheses. The only indication of a mismatch is the footnote on p.15: 'there is a minor typo in (I2) of [HR18, Theorem E] which requires finite surjective morphisms from affine scheme which are flat.' This means the authors rely on a corrected version of (I2), imported from [DLM25, Prop. 5.10], without stating the correction. If the corrected theorem has additional hypotheses—e.g., that the property be stable under arbitrary base change, or that the class E be closed under fiber products in the relevant 2-category—then Propositions 4.12, 4.15, and 4.17 may not suffice. In particular, Proposition 4.15 uses flatness of the finite cover to ensure Lf^* lands in D^b_coh; if the published (I2) is only 'finite surjective', the induction breaks without an extra flatness hypothesis. Since the proof is otherwise a black box, this is the weakest link in the central argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a notion of F-finiteness for algebraic stacks: a Noetherian stack over F_p is F-finite if its absolute Frobenius is proper (Definition 3.5). The main theorem (Theorem 4.18) states that if S is a concentrated F-finite algebraic stack with separated quasi-finite diagonal and whose Frobenius is representable by algebraic spaces, then for every closed Z⊆|S| and e≫0 there is G∈Perf_Z(S) such that D^b_coh,Z(S)=⟨RF^e_*G⟩. This generalizes the Ballard–Iyengar–Lank–Mukhopadhyay–Pollitz theorem from schemes to stacks. The proof reduces the desired property to three cases using Hall–Rydh's étale dévissage theorem [HR18, Theorem E]: open immersions, finite flat surjective affine covers, and étale neighborhoods. The paper also gives an explicit bound on the number of Frobenius iterates for Deligne–Mumford stacks (Proposition 4.21) and discusses a relative Kunz criterion and a stacky-curve application (Proposition 4.24).","tokens_in":18506,"tokens_out":16335,"duration_ms":140154,"significance":"If the main theorem is correct, it is a substantial advance: it identifies explicit generators for bounded derived categories of coherent sheaves on a large class of algebraic stacks, including Deligne–Mumford stacks, and it provides a structural refinement of Kunz's theorem in the stack setting. The paper is clearly structured and makes appropriate use of prior results such as compact approximation [HLLP25] and the Hall–Rydh dévissage framework. The relative formulation around Hypothesis 4.9 is natural and likely to be useful for future work. However, the central induction is presented as a black box: [HR18, Theorem E] is not stated, and footnote 2 acknowledges a correction to item (I2) without giving the corrected statement. Several lemmas in the induction have proofs that are too terse to verify the exact hypotheses. With these details supplied, the paper would be a solid contribution.","major_comments":[{"comment":"The proof of the main theorem is entirely an application of [HR18, Theorem E], but that theorem is not stated and its hypotheses are not verified beyond three bullet points. Footnote 2 acknowledges a 'minor typo' in (I2) and refers to [DLM25, Prop. 5.10], but does not state the corrected condition. Since the validity of the induction depends on matching the corrected statement (notably the flatness of the finite surjective affine cover), please state the theorem and its correction, and verify explicitly that Propositions 4.12, 4.15 and 4.17 satisfy all closure conditions.","section":"§4.2, Theorem 4.18 proof"},{"comment":"The proof of Lemma 3.7 shows only that F:U→U is locally of finite type, not that it is proper. In the factorization F=f∘F_{U/X}, properness of f does not imply properness of the composition unless F_{U/X} is proper (or finite). The proof should supply the missing argument that F_{U/X} is proper/finite for a smooth morphism from a scheme, or cite a result. This lemma is used in Proposition 3.9(1)=>(3) and Corollary 3.12, and hence underpins the observations in Theorem 4.18.","section":"§3.2, Lemma 3.7"},{"comment":"The proof establishes that E is a direct summand of Rf_*Lf^*E, not that E is isomorphic to an object in the essential image. If 'essentially dense' is intended in the usual triangulated-category sense that every object is a direct summand of an object in the image, this should be stated, and Proposition 4.15 should explain why this suffices for thick generation. Otherwise, a stronger statement is required for the finite flat dévissage step.","section":"§4.2, Lemma 4.14"},{"comment":"In step (2), the sentence 'we see that (s')_*G is a classical generator' is not what is proved: the proof immediately argues for L(s')^*G and uses L(s')^*G throughout. If the pushforward statement is intended, it needs a separate justification; if not, the text should be corrected to say L(s')^*G. This step is needed for the claimed explicit bound in the Deligne–Mumford case.","section":"§4.2, Proposition 4.21"}],"minor_comments":[{"comment":"Typo: 'convience' should be 'convenience'.","section":"Footnote 2, §4.2"},{"comment":"Typo: 'Lema 4.8(3)' should be 'Lemma 4.8(3)'. In addition, the proof of Proposition 4.17 contains the garbled expression 'D^b_coh,Z(X)⊆⟨RF^e_*Perf_{Z∩W}(X)⊕RF^e_*Perf_Z(X)⟩' which should be rewritten for clarity.","section":"§4.2, Proposition 4.21"},{"comment":"The titles '[DLMV25]' and '[LMV25]' contain 'Meausuring' instead of 'Measuring'.","section":"References"},{"comment":"The proof appears to use essential density in the wrong direction: if Rπ_* is essentially dense, it does not in general follow that pushing forward a strong generator yields a strong generator. Please clarify or revise this argument if the proposition is to be retained.","section":"§4.2, Proposition 4.24"}],"recommendation":"major_revision","confidential_remarks":"This is a promising paper and the main theorem seems plausible, but the referee's main reservation is the unstated use of [HR18, Theorem E] with the known correction to (I2). The readers' stress-test note and my own reading agree that this is the weakest link. If the authors supply the exact statement and verify all hypotheses, I would support acceptance. I did not see circularity or a hidden dependence on the paper's own claims; the reliance on [HR18], [HLLP25], and related prior work is legitimate. The paper is within the scope of Math.AG."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know that this paper does what it claims: it generalizes Ballard–Iyengar–Lank–Mukhopadhyay–Pollitz from F-finite schemes to concentrated F-finite algebraic stacks with quasi-finite separated diagonal. The new definition of F-finiteness via a proper Frobenius is natural, and Example 3.2 (BG_m) explains convincingly why the naive finite-morphism notion is too restrictive. The main theorem is Theorem 4.18, and the proof strategy via étale dévissage is appropriate for stacks, where local rings are not available.\n\nWhat is actually new: the F-finiteness notion for stacks, the relative version Proposition 4.8 (plus the Kunz-style Remark 4.19), and the main generation theorem. Corollary 4.20 upgrades to strong generation for separated Deligne–Mumford stacks, and Proposition 4.24 on stacky curves is a nice application. The paper is honest about what it recovers and cites prior work properly, including the authors' own BIL+23.\n\nThe soft spot is exactly what the stress-test note flags: the reliance on [HR18, Theorem E] in Theorem 4.18 without stating the theorem or verifying all its hypotheses in the text. But the footnote on p.15 explicitly mentions the typo in (I2) and refers to the corrected flat version in [DLM25, Prop 5.10]. Since the class E in the proof is defined to include flat morphisms, the correction aligns with their setup. So this is a presentational gap, not a load-bearing flaw. Still, a referee should ask them to spell out the exact statement of [HR18, Theorem E] and why each hypothesis is satisfied. Proposition 4.15 also assumes the existence of a finite flat surjective cover from an affine scheme; that comes from the dévissage theorem, so it is justified if one accepts that theorem.\n\nI cannot fully verify every step—several proofs cite lengthy external results (e.g., Lemma 4.14 relies on a splitting result from DLMRP25, and approximation by compacts is imported from HLLP25). But that is normal for this area, and the cited results are independently published. I see no red flag that would make the argument circular or depend on a false statement.\n\nWho should read this: people working on derived categories of algebraic stacks or prime-characteristic geometry. It deserves a serious referee. I would likely cite it if I worked in the area. The referee should focus on the dévissage verification and the flatness correction, but the paper is in good shape.","headline":"A genuine extension of Frobenius generation from schemes to stacks; the proof leans on Hall–Rydh dévissage and the flatness correction is flagged in a footnote, so the stress-test worry is real but not fatal.","tokens_in":18989,"tokens_out":3373,"would_cite":true,"duration_ms":31383,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14A30","14A20","13A35","14F08"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that on a broad class of positive-characteristic algebraic stacks, enough Frobenius pushforwards of a single perfect complex generate the entire bounded derived category of coherent sheaves with prescribed support.","keywords":["Frobenius pushforward","derived categories","algebraic stacks","F-finiteness","classical generators","strong generators","positive characteristic","Deligne–Mumford stacks"],"falsifier":"For a concrete test, let S be a genus-one tame stacky curve over an algebraically closed field of characteristic p, e.g. a quotient of an elliptic curve by a finite group. Compute, for e≫0, the minimal n with ⟨RF^e_*Perf(S)⟩_{n+1}=D^b_coh(S). If no finite n exists, the main theorem fails on a stack satisfying its hypotheses; if n is finite, compare it with the bound ceil(log_p N) from Proposition 4.21 to test sharpness.","tokens_in":18035,"feed_emoji":"🔁","tokens_out":7755,"duration_ms":68459,"temperature":0.7,"pith_summary":"This paper establishes that, on a large class of algebraic stacks in positive characteristic, Frobenius pushforwards of a single perfect complex generate the whole bounded derived category of coherent sheaves supported on any closed substack, provided one pushes forward enough times. To state this, the authors introduce a notion of F-finiteness for stacks based on properness of the Frobenius morphism, since finiteness fails for basic examples such as classifying stacks. The main theorem covers concentrated F-finite stacks with separated quasi-finite diagonal whose Frobenius is representable by algebraic spaces; Deligne–Mumford stacks of finite type over an F-finite scheme are the principal examples. A sympathetic reader should care because this gives an explicit recipe for generators in settings—like good moduli spaces—where no such recipe existed, and it generalizes the known scheme-level statement.","feed_headline":"Frobenius pushforwards generate stack derived categories","feed_subtitle":"A single perfect complex, pushed along Frobenius, generates the full derived category—explicit generators for stacks where none were known.","key_machinery":"The load-bearing objects are the absolute Frobenius morphism F:S→S and its iterated pushforward RF^e_* on the derived category. The paper's new notion of F-finiteness—Noetherian plus proper Frobenius—replaces the scheme-level notion of finite Frobenius because on stacks Frobenius need not be representable by schemes; properness still yields an exact enough functor. The proof is carried by an étale dévissage theorem that lets the authors reduce generation from an arbitrary stack to three étale-local situations, and by the relative categories D^b_coh,Z, which record objects supported on a closed substack. For Deligne–Mumford stacks, the number of iterates is controlled by γ(F_*O_U), the suprem","core_discovery":"The central claim is Theorem 4.18: if S is a concentrated F-finite algebraic stack with separated quasi-finite diagonal and the absolute Frobenius F:S→S is representable by algebraic spaces, then for every closed subset Z⊆|S| and all e sufficiently large there exists a perfect complex G supported on Z such that the thick closure of the Frobenius pushforward RF^e_*G is the entire bounded derived category D^b_coh,Z(S). Equivalently, the Frobenius pushforwards of perfect complexes classically generate the bounded derived category of coherent sheaves with support in Z. The proof proceeds by an étale dévissage that reduces the statement to three cases—open immersions, finite flat surjective cover","pith_inferences":["The étale dévissage strategy suggests that Frobenius generation is an étale-local property: if the dévissage theorem were unavailable in some broader class, one might still expect generation to descend from étale covers by other means, so the theorem's hypotheses are likely not optimal.","Because BG_m is F-finite but its Frobenius is not representable by algebraic spaces, the theorem sharply separates stacks with representable Frobenius from those without; testing generation on such classifying stacks would show whether the representability hypothesis is genuinely needed.","One can test the bound in Proposition 4.21 on explicit tamely stacky curves: computing γ(F_*O_U) should give the exact minimal e, and comparing it with the actual generation threshold would measure how sharp the log_p bound is.","The genus-zero conclusion for strong generation by F_*O_X on stacky curves suggests that the structure sheaf is a much weaker generator than a well-chosen perfect complex; understanding this gap for higher-genus stacky curves would clarify the role of the generator G in the main theorem."],"forward_implications":["For every concentrated F-finite Deligne–Mumford stack with separated diagonal, bounded derived categories of coherent sheaves supported on a closed substack admit an explicit classical generator of the form RF^e_*G with G perfect.","The same conclusion holds for the broader class of Artin stacks with Frobenius representable by algebraic spaces; the theorem's hypotheses are satisfied, for instance, by Deligne–Mumford stacks of finite presentation over an F-finite scheme.","On a Deligne–Mumford stack, the number of Frobenius iterates required is at most ceil(log_p(N)), where N is the minimal number of local sections generating F_*O_U over an étale affine cover; N is finite and independent of the chosen closed substack.","A regularity dichotomy holds: on such stacks, a closed substack lies in the regular locus exactly when Frobenius pushforwards of a generator remain perfect complexes.","For separated Deligne–Mumford stacks, a classical generator of the perfect complexes becomes, after sufficiently many Frobenius pushforwards, a strong generator of D^b_coh.","For separated Deligne–Mumford stacks, a classical generator of the perfect complexes becomes, after sufficiently many Frobenius pushforwards, a strong generator of D^b_coh."],"fun_headline_variants":["Frobenius pushforward of one complex generates stack derived categories","Large Frobenius powers give a single generator for stack categories","Frobenius pushforwards generate derived categories on stacks","Frobenius gives explicit generators for stack derived categories"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof assumes that the étale dévissage theorem applies to the class of morphisms that are representable by algebraic spaces, separated, finitely presented, quasi-finite and flat, and in particular that every stack in sight admits a finite flat surjective cover by an affine scheme; if that dévissage step fails, the induction collapses.","fun_headline_variants_meta":{"raw":{"variants":["Frobenius pushforward of one complex generates stack derived categories","Large Frobenius powers give a single generator for stack categories","Frobenius pushforwards generate derived categories on stacks","Frobenius gives explicit generators for stack derived categories"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000984,"raw_usage":{"total_tokens":3941,"prompt_tokens":604,"completion_tokens":3337,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":348,"completion_tokens_details":{"reasoning_tokens":3275}},"tokens_in":348,"tokens_out":3337,"duration_ms":26039,"temperature":1.0,"reasoning_tokens":3275,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T06:36:09.449487+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a concrete test, let S be a genus-one tame stacky curve over an algebraically closed field of characteristic p, e.g. a quotient of an elliptic curve by a finite group. Compute, for e≫0, the minimal n with ⟨RF^e_*Perf(S)⟩_{n+1}=D^b_coh(S). If no finite n exists, the main theorem fails on a stack satisfying its hypotheses; if n is finite, compare it with the bound ceil(log_p N) from Proposition 4.21 to test sharpness.","supporting_citations":[],"review_version":1}