{"id":"6c5acb5c-932d-46bf-be93-4ffbae4c0005","arxiv_id":"2602.20742","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Separated tame Deligne–Mumford stacks of finite presentation over a field admit dualizing complexes; the advertised t-structure classification is absent from the body.","lead":"This paper proves that a large class of geometric objects called tame Deligne–Mumford stacks always have a 'dualizing complex,' a tool used to study duality and singularities. The result would give algebraic geometers a standard duality package for moduli and birational-geometry problems, but the advertised application to t-structures is missing from the text.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main theorem rests on [Ryd26, Theorem B] for Nagata compactification of tame DM stacks; if that forthcoming theorem has hidden hypotheses or is unavailable, Lemma 4.3 and Prop 4.7 collapse.","rationale":"The central claim that must be secure is Theorem 4.8/Corollary 1.2. After reading the proof, the least secure point is not any internal derivation: the category S_e is built expressly to make Neeman's f^! formalism available, and once Nagata compactification is granted, the reduction in Prop 4.7 is plausible. The single step that cannot be checked inside the paper is the existence of the factorization Y→Y'→X for every separated finite-type morphism between tame DM stacks, cited to [Ryd26, Theorem B]. Lemma 4.3 relies on it, and Prop 4.7's first sentence 'As f admits a Nagata compactification' is the load-bearing input. If Ryd26's theorem is absent or narrower than stated, the proof does not merely require a tweak; the main reduction to the quasi-proper case disappears. The same is true for [LM22, Lemma 2.7], though that concerns algebraic spaces and is more likely established. I also agree with the reader that the abstract's t-structure classification is missing from the body; this is an overclaim that must be fixed, but it does not support the existence theorem, so I do not treat it as the primary load-bearing concern. A targeted check of the Ryd26 manuscript will settle the matter.","tokens_in":10379,"tokens_out":19221,"duration_ms":177994,"concrete_test":"Download the cited PDF [Ryd26] from Rydh's webpage and verify Theorem B in full: (1) Does it state that every separated finite-type morphism between tame DM stacks (over a field, equicharacteristic) admits a Nagata compactification with j dominant flat monomorphism and p universally quasi-proper, with no extra quasi-projectivity or diagonal assumptions beyond separated diagonal? (2) Does its 'strictly tame' condition indeed coincide with tameness in equicharacteristic as used in Lemma 4.3? (3) Check that the morphisms in the compactification belong to the category S_e defined in Lemma 4.3. If any of these fail, produce a separated tame DM k-stack whose structural morphism lacks such a compactification; Prop 4.7's reduction would not apply, so Theorem 4.8 would not follow.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The construction of S_e in Lemma 4.3 and the reduction in Prop 4.7 both depend on [Ryd26, Theorem B], a forthcoming compactification theorem for tame DM stacks. In Notation 4.1 a Nagata compactification of f:Y→X is a factorization Y --j--> Y' --p--> X with j a dominant flat monomorphism and p universally quasi-proper, all morphisms in S_e. Lemma 4.3 asserts that for separated finite-type morphisms between tame Noetherian DM k-stacks with separated diagonal this exists, citing only [Ryd26, Theorem B]; it also asserts 'tameness coincides with strictly tameness' in the equicharacteristic setting. Prop 4.7 then uses exactly this factorization: it reduces to the quasi-proper case, applies Lemma 4.5 to j^!g^!K, and identifies j^!g^!K ≅ f^!K via Remark 4.4(1). If [Ryd26, Theorem B] is unavailable, not yet published, or has additional hypotheses (e.g. characteristic zero, quasi-projective diagonal, or 'strictly tame' rather than tame), Lemma 4.3's S_e is not known to satisfy Notation 4.1, and the compactification step in Prop 4.7 — hence Theorem 4.8 and Corollary 1.2 — has no proof. This is a load-bearing reliance on an unstated external result, not an internal inconsistency. Separately, the abstract's promised classification of tensor t-structures on D^b_coh is absent from the body; that is a serious overclaim but not a step in the existence proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a notion of dualizing complex on algebraic stacks via the lisse-étale site (Definition 3.4) and proves, under a 2-categorical framework S_e admitting Nagata compactifications (Notation 4.1), that for a Deligne–Mumford morphism f: Y → X in S_e of finite presentation, f^! preserves dualizing complexes (Theorem 4.8). The main special case is Theorem 1.1: for a separated finite-type morphism between tame Deligne–Mumford k-stacks with X Noetherian and separated diagonal, f^! of a dualizing complex is dualizing. Corollary 1.2 concludes existence for separated tame DM k-stacks of finite presentation, and Corollary 1.3 gives a proper version using f^×. The proof reduces to algebraic spaces via étale/smooth descent and uses base-change isomorphisms from [Nee23, Theorem 1.8] and the algebraic-space result [LM22, Lemma 2.7]. The abstract also promises a classification of all tensor t-structures on D^b_coh, but no such classification appears in the body.","tokens_in":10738,"tokens_out":6159,"duration_ms":58448,"significance":"If the existence results are correct, they constitute a significant advance: dualizing complexes are obtained for separated tame Deligne–Mumford stacks without properness constraints, a natural and useful extension of the scheme/algebraic-space theory. The strategy of reducing to algebraic spaces through Neeman's f^! formalism is elegant and potentially influential. The paper is clearly written and carefully distinguishes the f^! and f^× functors. However, the proof rests on two substantial external inputs—[Ryd26, Theorem B] (a forthcoming compactification theorem) and [LM22, Lemma 2.7] (an algebraic-space duality preservation result)—neither of which is stated or proved. The main proof also contains a terse truncation argument in Lemma 4.6 that appears insufficient as written. These issues are fixable but currently prevent full verification.","major_comments":[{"comment":"The construction of the 2-category S_e in Lemma 4.3 and the compactification reduction in Proposition 4.7 depend entirely on [Ryd26, Theorem B], a forthcoming and not yet published result. In particular, Lemma 4.3 asserts that S_e of tame Noetherian DM k-stacks satisfies Notation 4.1, and the proof says only that Nagata compactifications exist by [Ryd26, Theorem B] and that tameness coincides with 'strictly tameness' in the equicharacteristic setting. None of these hypotheses or statements is spelled out. If [Ryd26, Theorem B] has additional hidden hypotheses (e.g., characteristic zero, quasi-projective diagonal, or only for strictly tame stacks), then Lemma 4.3 is false and the factorization f = g∘j used in Proposition 4.7 collapses. Since this step is the bridge from the quasi-proper case to the general case, Theorem 4.8 and Corollary 1.2 are not verifiable from the material supplied.","section":"Lemma 4.3 and Proposition 4.7"},{"comment":"The algebraic-space input [LM22, Lemma 2.7] is used in a load-bearing way: in Lemma 4.5 it is the step that upgrades (f')^! of a dualizing complex on U to a dualizing complex on Y×_X U, and in Proposition 4.7 it is used again for the étale fiber products. The lemma is not stated, and its hypotheses are not explained. Without a statement or proof, the reduction to the algebraic-space case is not self-contained. This is especially important because the entire proof strategy is to reduce to algebraic spaces; the reader must be able to check that the cited lemma applies to the morphisms in question.","section":"Lemma 4.5 and Proposition 4.7"},{"comment":"The proof of Lemma 4.6 does not convincingly establish that f^× preserves D^+_qc. The argument assumes 'L ∈ D^+_qc(Y) satisfying H^j(L)=0 if j≥c', i.e., that L is bounded above, which is not the condition for membership in D^+_qc; the final conclusion H^i(f^×L)=0 for i≤c−B does not follow from the displayed Hom-vanishing in the way stated. Since Lemma 4.6 is used in Proposition 4.7 to conclude f^×K ∈ D^+_qc(X) before identifying f^×K with f^!K, this gap affects a step in the main proof. The statement is a standard fact (cf. [Sta26, Tag 0E56]), but the proof needs to be rewritten correctly or replaced by a precise citation.","section":"Lemma 4.6"},{"comment":"The abstract states as an application that the paper 'classifies all tensor t-structures on their bounded derived category of coherent sheaves'. No such classification, or even a statement of a theorem about t-structures, appears anywhere in the body of the manuscript. This is a serious overclaim. The authors should either add the promised classification theorem (with proof) or remove the sentence from the abstract. Since the existence of dualizing complexes does not by itself classify all tensor t-structures, this is not a harmless omission.","section":"Abstract"}],"minor_comments":[{"comment":"Typo: 'of presentation' should be 'of finite presentation'. Also, the phrase 'Deligne-Mumford morphism (f:X→S)∈S_e' uses X for the source while the rest of the paper uses Y; this is confusing and should be standardized.","section":"Proposition 4.7"},{"comment":"The references to [Ryd26, Theorem B] and [Ryd26, Theorem F] are both used, but the numbering and content of these theorems are not described. Since one is a key dependency, please give a precise statement or at least a summary of the hypotheses and conclusions.","section":"Example 4.2 and Lemma 4.3"},{"comment":"This remark compiles statements from [Nee23, Theorem 1.8] but does not indicate which parts require D^+_qc and which hold on all D_qc. Adding that information would clarify the use of Lemma 4.6.","section":"Remark 4.4"},{"comment":"The definition of dualizing complex on a stack is given 'smooth locally', but the paper does not discuss how this behaves under arbitrary base change or how it depends on the choice of smooth cover. Proposition 3.8 addresses the cover independence, but a brief discussion would improve readability.","section":"Section 3, Definition 3.4"}],"recommendation":"major_revision","confidential_remarks":"The central existence theorem is plausible and the overall strategy is sound, but the manuscript's verification depends critically on two external results that are neither stated nor proved. In particular, the unpublished [Ryd26, Theorem B] is the foundation of Lemma 4.3 and Proposition 4.7; without a precise statement and confirmation of its hypotheses, the main theorem cannot be accepted. The Lemma 4.6 proof also needs correction. Once these are addressed—and the abstract's unsupported t-structure claim is removed or substantiated—the paper would be suitable for publication. I would not recommend rejection, but the current version does not allow a referee to check the proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main new thing here is Theorem 4.8/Cor 1.2: for a separated tame DM stack of finite presentation over a field, dualizing complexes exist, with no properness assumption. As far as I can tell that's genuinely new. The proof is a formal reduction to algebraic spaces using Neeman's f^! formalism, and the smooth-local definition of dualizing complex on the lisse-étale site is handled carefully. If the dependencies hold, the result is solid and will be useful for people doing duality or MMP-type programs on stacks over Q.\n\nThat said, there are three soft spots.\n\nFirst, the abstract announces a classification of all tensor t-structures on D^b_coh. The body never returns to this. No definition, no statement, no proof. That is a serious overclaim and has to be removed or substantiated. Right now it misrepresents the content.\n\nSecond, the proof rests on [Ryd26, Theorem B], a forthcoming compactification theorem for tame DM stacks. The hypotheses are not stated in the paper beyond a citation, and the key reduction in Prop 4.7 collapses if that theorem doesn't hold in the claimed generality. This is a standard dependency on unpublished work, but it's load-bearing and should be made explicit or the theorem stated as conditional.\n\nThird, Lemma 4.6 is too terse. The spectral sequence argument does not justify the claimed bound for bounded-below complexes; it looks repairable along the lines of Stacks 0E56, but as written it's not convincing. This is a minor gap in a supporting lemma, not in the main theorem.\n\nThe self-citation [HLLP25] is just for a standard fact about morphisms between tame DM stacks being concentrated; that's fine.\n\nNet: the main existence theorem is a real contribution and deserves a serious referee. The authors need to fix the abstract, clarify the dependency on Rydh, and expand Lemma 4.6. I'd send it out, but with a clear request for those revisions before acceptance.","headline":"Novel existence theorem for dualizing complexes on tame DM stacks, but the abstract overclaims a t-structure classification absent from the body, and the proof leans on unpublished compactification results.","tokens_in":11246,"tokens_out":3541,"would_cite":true,"duration_ms":33356,"reading_group":"maybe","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":"For a separated tame Deligne–Mumford stack of finite presentation over a field, a dualizing complex always exists.","keywords":["algebraic stacks","dualizing complexes","Deligne–Mumford stacks","tame stacks","Grothendieck duality","Nagata compactification","upper shriek functor","t-structures"],"falsifier":"Find a separated tame Deligne–Mumford stack of finite presentation over a field that provably has no dualizing complex; Corollary 1.2 would fail. Short of that, exhibiting a morphism in S_e that lacks the required Nagata compactification would undercut the construction of f^!.","tokens_in":10210,"feed_emoji":"🧮","tokens_out":5920,"duration_ms":52309,"temperature":0.7,"pith_summary":"The paper establishes that dualizing complexes exist for all separated tame Deligne–Mumford stacks of finite presentation over a field, with no properness assumption. The engine is a transfer theorem: whenever f: Y → X is a finite-presentation, separated Deligne–Mumford morphism between stacks in a suitable 2-category and K is dualizing on X, the upper-shriek pullback f^!K is dualizing on Y. The proof obtains f^! through Nagata compactification, reducing the question to algebraic spaces and then descending along smooth covers. If correct, this gives Grothendieck duality a firm footing for moduli and birational geometry on tame stacks in all characteristics.","feed_headline":"Dualizing complexes exist for tame Deligne–Mumford stacks","feed_subtitle":"No properness required: Grothendieck duality now covers separated tame stacks over any field.","key_machinery":"The load-bearing object is the upper-shriek functor f^! on the 2-category S_e, constructed from the right adjoint f^× of Rf_* by Neeman's formalism for concentrated morphisms, together with the identification f^× ≅ f^! for universally quasi-proper morphisms. The argument uses Nagata compactification to factor f as a dominant flat monomorphism followed by a universally quasi-proper finite-type morphism, then uses base-change isomorphisms and the reduction to algebraic spaces. A dualizing complex is defined smooth-locally on the lisse-étale site, so the proof checks the property on étale schemes covering the stack and descends.","core_discovery":"The central claim, Theorem 4.8, is that the operation K ↦ f^!K preserves dualizing complexes for Deligne–Mumford morphisms of finite presentation within the 2-category S_e of Noetherian algebraic stacks with quasi-affine diagonals that have Nagata compactifications. Theorem 1.1 is the special case where S_e consists of tame Noetherian Deligne–Mumford k-stacks with separated diagonal. From it, Corollary 1.2 concludes that every separated tame Deligne–Mumford k-stack of finite presentation over a field admits a dualizing complex; in characteristic zero this covers every separated Deligne–Mumford stack of finite presentation. A relative variant (Corollary 1.3) states that a tame proper Deligne–","pith_inferences":["The abstract advertises a classification of all tensor t-structures on D^b_coh, but the body contains no proof of that classification; this extraction only treats the dualizing-complex existence results, and the classification should be regarded as unproved here.","If the quoted forthcoming Nagata compactification theorem for tame Deligne–Mumford stacks is not available in the stated generality, the main reduction collapses; a natural test is to find an independent proof of that compactification or a counterexample.","The same strategy could apply to other classes of stacks once Nagata compactifications and étale descent for dualizing complexes are known; tameness is used for linearly reductive stabilizers, so non-tame positive-characteristic stacks are a natural boundary."],"forward_implications":["Every separated tame Deligne–Mumford stack of finite presentation over a field carries a dualizing complex, even when not proper.","In characteristic zero, every separated Deligne–Mumford stack of finite presentation over a field carries a dualizing complex.","For tame proper Deligne–Mumford morphisms, the right adjoint f^× of derived pushforward preserves dualizing complexes (Corollary 1.3).","Because these complexes are pseudocoherent with bounded cohomology, they can serve as input for duality, residue, and singularity-theoretic tools on stacks.","The existence result removes properness restrictions that limited earlier approaches."],"fun_headline_variants":["Dualizing complexes exist for tame DM stacks without properness","t-structures on tame DM stacks classified via dualizing complexes","Tame Deligne–Mumford stacks admit dualizing complexes—new proof","No properness needed: dualizing complexes for tame stacks","Dualizing complexes and tensor t-structures for tame DM stacks"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof depends on a Nagata compactification theorem for tame Deligne–Mumford stacks quoted from a forthcoming paper; if that theorem is not available in the stated generality, the factorization that defines f^! and drives the reduction is missing.","fun_headline_variants_meta":{"raw":{"variants":["Dualizing complexes exist for tame DM stacks without properness","t-structures on tame DM stacks classified via dualizing complexes","Tame Deligne–Mumford stacks admit dualizing complexes—new proof","No properness needed: dualizing complexes for tame stacks","Dualizing complexes and tensor t-structures for tame DM stacks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000185,"raw_usage":{"total_tokens":1072,"prompt_tokens":570,"completion_tokens":502,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":314,"completion_tokens_details":{"reasoning_tokens":412}},"tokens_in":314,"tokens_out":502,"duration_ms":4797,"temperature":1.0,"reasoning_tokens":412,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T21:14:50.593145+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a separated tame Deligne–Mumford stack of finite presentation over a field that provably has no dualizing complex; Corollary 1.2 would fail. Short of that, exhibiting a morphism in S_e that lacks the required Nagata compactification would undercut the construction of f^!.","supporting_citations":[],"review_version":1}