{"id":"4fc88178-bf0d-437a-b7dd-c0b234254cb5","arxiv_id":"2601.04053","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every regular Noetherian algebraic stack with quasi-finite diagonal has its derived category generated by a single perfect complex.","lead":"This paper proves that for certain well-behaved geometric objects called regular algebraic stacks, the entire derived category of sheaf complexes is generated by a single perfect complex. If correct, it strengthens earlier results that only showed generation by a collection of complexes, and removes technical restrictions such as finite dimension and separated diagonal.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Thm 1.1 uses Perf=D^b_coh via [DLMP25, Thm 3.7] without a finite-dimension hypothesis; this equality is false for infinite-dimensional regular schemes (Nagata), so the induction step may fail to produce a perfect lift.","rationale":"I read the paper in good faith and found no reason to doubt the broad strategy or the significance of the main theorem. The monomorphic splitting sequence, recollement gluing, and finite-duality machinery are natural and likely reusable. However, the proof of Theorem 1.1 has a single load-bearing dependency on the equality Perf=D^b_coh for the intermediate regular stacks X_c, cited from [DLMP25, Theorem 3.7]. The reader's suspicion is precise and justified: under the paper's stated hypotheses, there is no finite Krull dimension assumption, and the equality is false for infinite-dimensional regular schemes such as Nagata's example. This is an internal correctness risk in the proof as written, not merely a conflict with prevailing expectations. The recommended conditional acceptance is therefore appropriate: the main statement may well be true, but the written induction needs either a finite-dimension hypothesis on the cited theorem, a corrected citation, or an alternative perfect-lifting argument. I agree with the reader's identification of the weakest assumption; the concrete test of inspecting [DLMP25] or testing Nagata's scheme would settle the issue.","tokens_in":11064,"tokens_out":19451,"duration_ms":182837,"concrete_test":"Read the statement of [DLMP25, Theorem 3.7] in arXiv:2504.02813. If it has no finite Krull dimension or concentratedness assumption, test it on Nagata's regular Noetherian ring A of infinite Krull dimension with X=Spec A. Since X is qcqs with quasi-finite diagonal, the equality would force every bounded coherent complex to be perfect; exhibit a maximal ideal m with dim A_m arbitrarily large and show pd_A A/m is infinite (e.g. via Bass's theorem or an explicit free resolution), giving a bounded coherent non-perfect complex. If Theorem 3.7 does assume finite dimension, then the proof of Theorem 1.1 as written has an unstated hypothesis and the induction must be reworked.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the induction step of Theorem 1.1, after setting up the recollement for X_{c-1}⊂X_c, the proof invokes [HLLP25, Prop. B.1] to obtain a Verdier localization sequence on bounded coherent categories and then states: 'However, [DLMP25, Theorem 3.7] tells us that Perf=D^b_coh in each case.' This equality is used to lift the inductive perfect generator G_{c-1}∈Perf(X_{c-1}) to a perfect complex G_c∈Perf(X_c): the localization produces a bounded coherent lift, and only the Perf=D^b_coh identification makes that lift perfect. But Theorem 1.1 does not assume finite Krull dimension, and the equality Perf=D^b_coh is false for regular Noetherian schemes of infinite Krull dimension: Nagata's example has all local rings regular yet infinite global dimension, so there are bounded coherent complexes (e.g. residue fields of maximal ideals of arbitrarily large local dimension) that are not perfect. Since a qcqs scheme has quasi-finite diagonal, such X is in the scope of the theorem. Thus, unless [DLMP25, Thm 3.7] carries an unstated finite-dimensional or concentratedness hypothesis, the cited equality is not available. This is load-bearing: it is exactly the step that converts a bounded-coherent lift into a perfect generator, and the same equality is also used in Corollary 4.7. If the theorem is true in infinite dimension, a different lifting argument is needed; if Theorem 3.7 has a hidden finiteness hypothesis, the paper's hypotheses need amendment.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that for every quasi-compact quasi-separated regular algebraic stack X with quasi-finite diagonal, the derived category D_qc(X) of complexes with quasi-coherent cohomology is generated by a single perfect complex (Theorem 1.1). In the concentrated case, it further claims that D_qc(X) is singly compactly generated (Corollary 1.2). The proof uses Hall–Rydh's Nisnevich presentations with monomorphic splitting sequences, establishes a 1-Thomason property on the strata, and glues perfect generators along recollements via Proposition 3.7. The main local ingredients are a finite-duality lemma and the assertion that Perf = D^b_coh for regular stacks, used to lift bounded coherent generators to perfect ones.","tokens_in":11345,"tokens_out":46126,"duration_ms":469370,"significance":"If the main theorem is correct, it is a substantial strengthening of known results: it removes separated diagonal, concentratedness, and finite Krull dimension assumptions for generation by a single perfect complex, and in the concentrated case upgrades compact generation to single-object compact generation. The recollement-glueing proposition is clean and potentially reusable. The paper is concise and builds on standard, mostly cited machinery; however, the central proof depends on an unqualified equality Perf = D^b_coh that needs careful verification.","major_comments":[{"comment":"The induction step invokes [DLMP25, Theorem 3.7] to assert Perf(X_c)=D^b_coh(X_c), and Corollary 4.7 uses the same equality for X itself. The manuscript states no hypotheses for this cited theorem. This is load-bearing: the localization step produces a bounded coherent lift of the generator, and only the Perf=D^b_coh identification makes that lift perfect. As written, the equality is used with no finite Krull dimension or concentratedness assumption. This is not harmless: for a regular Noetherian scheme of infinite Krull dimension (Nagata's example), the inclusion Perf ⊂ D^b_coh is strict, since there exist bounded coherent sheaves of infinite projective dimension; such a scheme is within the scope of Theorem 1.1. The authors must either quote the precise hypotheses of [DLMP25, Theorem 3.7] and verify them, supply a proof, or amend the main theorem's hypotheses. Without this, the inducti","section":"Proof of Theorem 1.1 and Corollary 4.7"},{"comment":"The conclusion that some B∈B has support equal to Z does not follow as written for arbitrary β. The proof argues that if the support of every Rf_*(B⊗P) were properly contained in Z, then the support of every object of D_qc,Z(X) would be properly contained in Z. This is false when B has more than one element: a finite or infinite coproduct of generators can have support equal to the union of their supports, which may be all of Z even if each individual support is proper. The argument is valid when β=1, because the generating collection is a single object, and this is the only case used in the proof of Theorem 1.1; however, the proposition as stated is not proved. Please restrict the statement to β=1 or supply a correct argument for the general case.","section":"Proposition 4.3, final paragraph"}],"minor_comments":[{"comment":"In the statement, 'B⊆D^b_coh(X)' should presumably be 'B⊆D^b_coh(Y)'. As written, the notation is inconsistent with the proof and with the intended use in the proof of Theorem 1.1.","section":"Lemma 4.1"},{"comment":"The proof uses t^{-1}(Z')=Z for a closed subset Z, where Z' is the closure of t(Z), and justifies this by injectivity of t on underlying topological spaces. Injectivity alone is not sufficient for arbitrary quasi-affine morphisms; the equality holds for monomorphisms. Since the application in Theorem 1.1 uses a monomorphism, the hypothesis should be strengthened to 't is a quasi-affine monomorphism'.","section":"Lemma 4.4"},{"comment":"The application of [HLLP25, Proposition B.1] should state the hypotheses needed for the Verdier localization sequence on bounded coherent categories. The paper currently invokes it without indicating what conditions on X_c are required, making the argument difficult to verify independently.","section":"Proof of Theorem 1.1"},{"comment":"There are several typographical errors, e.g., 'Specfically' in the proof of Theorem 1.1, 'T ying' in the proof of Proposition 3.1, and 'containes' in Section 4. These do not affect the mathematics but should be corrected.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the recollement strategy is attractive, but the proof's reliance on [DLMP25, Theorem 3.7] without any quoted hypotheses is a serious verification risk. The paper also depends on three recent preprints with overlapping authorship ([HLLP25], [DLMP25], [DLM25]); the editor may wish to ensure these are available and that the specific statements used are correct as cited. The Proposition 4.3 gap is less severe because only the β=1 case is needed, but it should still be fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Pat proves a genuinely new theorem: for a qcqs regular algebraic stack with quasi-finite diagonal, D_qc is generated by a single perfect complex. This improves the set-wise generation results of Hall–Rydh and removes the finite-dimension assumption in Hall's concentrated case. The proof strategy is clean: use the Hall–Rydh étale devissage to build a monomorphic splitting sequence, then glue generators along recollements. The recollement lemmas (3.5–3.7) and the finite duality argument in Lemma 4.1 are well-executed and likely reusable.\n\nThe soft spot is exactly where the reviewer put it. In the induction step, the proof needs the equality Perf = D^b_coh for the intermediate stacks X_c, and it cites [DLMP25, Theorem 3.7] for this. But Theorem 1.1 has no finite Krull dimension hypothesis, and for a regular Noetherian scheme of infinite Krull dimension (Nagata's example), Perf is strictly contained in D^b_coh — residue fields at points of arbitrarily large local dimension are bounded coherent but not perfect. Since such a scheme is a regular qcqs stack with quasi-finite diagonal, it is in the theorem's scope. Unless [DLMP25] carries a hidden finiteness hypothesis, the cited equality is false in the stated generality. That makes the induction step that lifts the inductive generator depend on an unjustified premise.\n\nThis is a real gap, but it is localized. The rest of the proof — the Thomason condition lifting, the recollement glueing, the finite duality — is not affected. The fix is either to add a finite-dimension (or quasi-excellence) hypothesis to the statement, or to find a way to lift the generator directly to Perf without the full equality. Since the author is also a coauthor of the cited preprint, I suspect a missing hypothesis there rather than a fundamental issue with the main theorem.\n\nThe paper is for people who care about compact generation of derived categories of stacks. The recollement glueing proposition is worth knowing on its own. I would send this to a serious referee; it deserves a careful read, and the gap is addressable. My own verdict would be conditional acceptance pending clarification of the Perf = D^b_coh step.","headline":"New single-perfect-generator theorem for regular stacks; the proof has a real but localized gap where it cites Perf = D^b_coh without a finite-dimension hypothesis.","tokens_in":11954,"tokens_out":5475,"would_cite":true,"duration_ms":46869,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14A30","14D23","14F08","18G80"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single perfect complex generates the derived category of every regular algebraic stack with quasi-finite diagonal.","keywords":["algebraic stacks","perfect complexes","derived categories","compact generation","regular stacks","quasi-finite diagonal","Thomason condition","recollement"],"falsifier":"A concrete test: examine a regular Noetherian scheme of infinite Krull dimension (such a scheme exists) and check whether the cited regularity theorem equating perfect and bounded coherent complexes holds there. If it fails, run the paper's induction on a stack whose monomorphic splitting sequence has an infinite-dimensional regular scheme as an intermediate piece; the induction would produce a bounded coherent complex that is not perfect, contradicting Theorem 1.1's conclusion that the generator is perfect.","tokens_in":10799,"feed_emoji":"🧩","tokens_out":10226,"duration_ms":90126,"temperature":0.7,"pith_summary":"On any regular algebraic stack with quasi-finite diagonal — a geometric space that may have finite automorphism symmetries at points — the paper proves that the entire derived category of quasi-coherent sheaves is generated by one perfect complex. Perfect complexes are the bounded, locally free-like objects that play the role of 'finite resolutions'; having a single one generate everything means every complex can be built from it by shifts, sums, and cones. The proof works for stacks that are not concentrated, do not have separated diagonal, and may have infinite Krull dimension, all of which were previously restrictive assumptions. When the stack is concentrated, the single generator can be chosen compact, making the derived category singly compactly generated. This matters because a one-object generator turns questions about the whole category into questions about one concrete object.","feed_headline":"A single perfect complex generates any regular stack's derived category","feed_subtitle":"One object builds every sheaf complex, even for infinite-dimensional stacks with non-separated diagonal.","key_machinery":"The central mechanism is the recollement associated to a quasi-compact open immersion j: U → X with closed complement Z. A recollement is a decomposition of a triangulated category into three categories — the open part D_qc(U), the whole D_qc(X), and the closed-support part D_qc,Z(X) — with three adjoint pairs of functors, so that every object in the middle is built from an object of the closed piece and an object of the open piece. Proposition 3.7 shows that a generator of the open part plus a generator of the closed part gives a single generator of the whole. The theorem uses this to glue along a monomorphic splitting sequence of a Nisnevich covering, a finite chain of open substacks on wh","core_discovery":"The paper proves Theorem 1.1: for any quasi-compact quasi-separated regular algebraic stack X with quasi-finite diagonal, there is a perfect complex P on X such that every nonzero object E in D_qc(X) has a nonzero morphism from some shift P[n]. Under the extra concentratedness hypothesis, P can be chosen compact, so D_qc(X) is singly compactly generated. The proof is inductive: it uses a monomorphic splitting sequence of a Nisnevich covering to filter X into locally closed pieces, shows each piece carries a single perfect generator via finite duality and the regularity hypothesis, and then glues these generators using recollement diagrams. The novelty is that no separated-diagonal or finite-","pith_inferences":["If the cited regularity theorem (perfect complexes coincide with bounded coherent complexes) can be established for regular stacks without any finite-dimension hypothesis, the same induction would likely upgrade Theorem 1.1 to single compact generation for all quasi-compact quasi-separated regular stacks with quasi-finite diagonal, not just concentrated ones.","The recollement gluing lemma is not specific to this setting; it could be applied to other triangulated categories with open-closed decompositions, such as equivariant derived categories or categories of matrix factorizations, whenever each piece is known to admit a single generator.","A quantitative strengthening is plausible: the single generator is constructed from finitely many piecewise generators, so its complexity (e.g., the number of terms in a perfect resolution) should be bounded by the length of the monomorphic splitting sequence; making this explicit would give a concrete handle on generation time.","The non-compactness phenomenon is likely tied to infinite Krull dimension: on infinite-dimensional regular schemes, bounded coherent complexes need not be perfect, so the induction may produce a perfect but non-compact generator; isolating a stack where this happens would mark the boundary of the concentrated case."],"forward_implications":["For concentrated regular stacks with quasi-finite diagonal, D_qc(X) is singly compactly generated, and the stack satisfies the 1-Thomason condition (Corollary 1.2 and its proof).","The result removes the finite-Krull-dimension hypothesis and the separated-diagonal hypothesis from earlier generation theorems for regular stacks.","It applies to smooth, finitely presented, quasi-Deligne–Mumford stacks over a DVR, including mixed-characteristic cases, whenever stabilizers are affine and 'nice'.","Even when the generator is not compact, the theorem supplies a single perfect complex that generates the whole derived category — a genuinely new class of examples among regular Deligne–Mumford stacks in arbitrary characteristic.","The recollement-gluing proposition gives a reusable recipe for combining generators of open and closed pieces into a generator of a stack."],"fun_headline_variants":["One perfect complex does the job for every regular stack","Perfect generation: one complex suffices for regular stacks","Regular stacks: a single perfect complex generates all","Derived category of any regular stack has a perfect generator","A lone perfect complex builds every sheaf complex on regular stacks"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The induction step in the proof of Theorem 1.1 assumes that a cited regularity theorem — perfect complexes coincide with bounded coherent complexes — applies to each intermediate stack in the monomorphic splitting sequence; if that theorem secretly requires finite Krull dimension, the induction could produce only a bounded coherent complex, not a perfect one.","fun_headline_variants_meta":{"raw":{"variants":["One perfect complex does the job for every regular stack","Perfect generation: one complex suffices for regular stacks","Regular stacks: a single perfect complex generates all","Derived category of any regular stack has a perfect generator","A lone perfect complex builds every sheaf complex on regular stacks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000318,"raw_usage":{"total_tokens":1547,"prompt_tokens":571,"completion_tokens":976,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":315,"completion_tokens_details":{"reasoning_tokens":898}},"tokens_in":315,"tokens_out":976,"duration_ms":9090,"temperature":1.0,"reasoning_tokens":898,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T12:09:48.295600+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test: examine a regular Noetherian scheme of infinite Krull dimension (such a scheme exists) and check whether the cited regularity theorem equating perfect and bounded coherent complexes holds there. If it fails, run the paper's induction on a stack whose monomorphic splitting sequence has an infinite-dimensional regular scheme as an intermediate piece; the induction would produce a bounded coherent complex that is not perfect, contradicting Theorem 1.1's conclusion that the generator is perfect.","supporting_citations":[],"review_version":1}