{"id":"f395046a-d885-4cb3-a132-83aff6a6ac79","arxiv_id":"2411.09461","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A locally finite approximable DG-category with a strong generator inside its finitely valued modules is finite-reflexive; this yields reflexivity for proper schemes, Azumaya algebras, and proper connective DG-algebras.","lead":"This paper connects two tools for comparing small derived categories: approximable triangulated categories and reflexive DG-categories. It gives a condition for a proper DG-category to be reflexive and applies it to proper schemes, Azumaya algebras, and proper connective DG-algebras.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.2's filtration argument conflates H^0(A)-bimodule quotients with A^e-modules, ignoring actions of negative-degree cohomology elements; the proof of the DG-algebra application needs repair or a detailed Orlov alternative.","rationale":"The reader's weakest-assumption analysis correctly identified strong generation as the load-bearing input and noted that Lemma 5.2's filtration argument is compressed. My stress-test sharpens this: the specific factor not addressed is the A^e-module structure of H^*(A) coming from cohomology classes in negative degrees. The filtration by powers of rad H^0(A) only controls the H^0(A)-bimodule filtration; after restricting along A^e → H^0(A)^op ⊗ H^0(A)/J, all nonzero-degree elements of A act trivially, whereas the natural H^*(A)-bimodule action is nontrivial in general. The example A = k[ε]/(ε^2), deg ε = -1, d = 0, exhibits exactly this: F_0 restricts to a module with ε acting by zero, while A itself has a nonzero degree -1 action, so A is not in level one of the restricted generator. The lemma's conclusion may still be true and the Orlov-based alternative in Remark 5.3 likely repairs the gap, but the paper as written does not supply the missing argument. This affects Proposition 5.4, one of the central applications, though the scheme and Azumaya applications rest on different external generation inputs and are not affected by this particular gap. I therefore recommend conditional acceptance: the main theorem is sound, but the proof of Lemma 5.2 must be corrected or replaced by a detailed proof of the stated alternative before the DG-algebra application is fully justified.","tokens_in":9527,"tokens_out":53473,"duration_ms":505154,"concrete_test":"Compute the minimal generation level for the two-dimensional DG-algebra A = k[ε]/(ε^2) with deg ε = -1 and zero differential: verify that D(A) = <k>_2 but A ∉ <k>_1. This shows the filtration-by-J argument in Lemma 5.2, if read as a direct substitution, yields the wrong level and that the 'restriction' step is not formally valid. Then check whether a corrected argument, or a fully detailed proof via Orlov's DG-radical as suggested in Remark 5.3, supplies a finite N for general proper connective DG-algebras; if it does, Proposition 5.4 survives with a revised proof, and if not, the application is unsupported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 4.4 is an implication; its proof is sound conditional on the strong-generation hypothesis. The soft spot is the verification of that hypothesis for proper connective DG-algebras in Lemma 5.2. The proof shows A ∈ <H^*(A)>_N in D(A^e), then filters H^*(A) by powers of J = rad H^0(A) and asserts H^*(A) ∈ <F_i>_{N'} in D(H^0(A)^op ⊗ H^0(A)/J). It then says that restricting along A^e → H^0(A)^op ⊗ H^0(A)/J gives A ∈ <F_i>_{N''} in D(A^e). This restriction step is the problem: the F_i, as modules over H^0(A)^op ⊗ H^0(A)/J, carry only degree-zero actions, so all elements of A in nonzero degree act trivially after restriction. But H^*(A) in D(A^e) is the cohomology algebra with the natural A-bimodule action; negative-degree cohomology classes act by nonzero degree-shifting maps and are not captured by the filtration. A concrete witness is A = k[ε]/(ε^2) with deg ε = -1 and d = 0. Here H^0(A) = k, J = 0, F_0 = H^*(A) = k ⊕ k[-1]; the restriction of F_0 to A^e has ε acting by zero, while H^*(A) (which is A) has ε acting by a nonzero degree -1 map. Hence A is not in <F_0>_1 in D(A^e); in fact A ∈ <k>_2. The proof therefore does not establish the claimed generation without an additional argument controlling these higher-degree actions. Remark 5.3 sketches an alternative via Orlov's DG-radical, but it is not spelled out and relies on [Orl20]. Since Proposition 5.4 is a headline application, this gap is load-bearing.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes a bridge between approximable triangulated categories and reflexive DG-categories. Theorem 4.3 shows that for an approximable locally finite DG-category A, the subcategory D(A)^b_c is exactly D^{fvd}(A). Theorem 4.4 then proves that if D(A) is strongly generated by an object of D^{fvd}(A), then A is finite-reflexive. The authors apply this to three families: proper schemes over a commutative Noetherian ring, proper connective DG-algebras over a field, and Azumaya algebras over proper schemes. An appendix by Raedschelders and Stevenson proves that every proper connective DG-algebra over any field admits a finite-dimensional DG-model.","tokens_in":9950,"tokens_out":2899,"duration_ms":27450,"significance":"If the results hold, the paper gives a uniform sufficient condition for finite-reflexivity that covers and extends the main examples of Kuznetsov and Shinder, and it clarifies the relationship between approximability and reflexivity. The conditional Theorem 4.4 is clean and its proof is sound, and the identification D(A)^b_c = D^{fvd}(A) is a useful structural result. The appendix is a valuable standalone contribution. However, one of the three headline applications, Proposition 5.4 on proper connective DG-algebras, currently rests on a flawed proof step in Lemma 5.2, so the full set of applications is not yet established as written.","major_comments":[{"comment":"The proof of Lemma 5.2 contains an invalid restriction step. After showing H^*(A) ∈ ⟨F_i⟩_{N'} in D(H^0(A)^op ⊗ H^0(A)/J), the proof asserts that restricting along A^e → H^0(A)^op ⊗ H^0(A)/J gives A ∈ ⟨F_i⟩_{N''} in D(A^e). This is not justified: the objects F_i are modules over H^0(A)^op ⊗ H^0(A)/J, so when viewed as A^e-modules via this restriction, all elements of A in nonzero cohomological degree act trivially. But A ∈ D(A^e) has the natural A-bimodule action, which is generally not captured by the restriction. A concrete witness is A = k[ε]/(ε^2) with deg ε = -1 and d = 0. Here H^0(A) = k, J = 0, and F_0 = H^*(A) = k ⊕ k[-1]. The restriction of F_0 to A^e makes ε act by zero, whereas A (which is H^*(A)) has ε acting by a nonzero degree -1 map; indeed A ∈ ⟨k⟩_2 but not in ⟨F_0⟩_1 in D(A^e). The proof therefore does not establish the strong generation hypothesis needed for Proposition 5.4. Remark 5.3 sketches an alternative via Orlov's DG-radical, but it is only a sketch and relies on [Orl20], so Proposition 5.4 is currently unsupported.","section":"§5, Lemma 5.2"},{"comment":"Since Proposition 5.4 is a headline application and depends entirely on Lemma 5.2, the gap in Lemma 5.2 is load-bearing. The paper's central theoretical result, Theorem 4.4, is conditional on a strong generation hypothesis, and for proper connective DG-algebras that hypothesis is verified only through Lemma 5.2. The appendix (Theorem A.3) provides finite-dimensional models but does not, by itself, supply the missing generation statement; the connection to Orlov's DG-radical in Remark 5.3 would need to be written out in detail. Until this is repaired, the claim that all proper connective DG-algebras over a field are reflexive is not proven.","section":"§5, Proposition 5.4"}],"minor_comments":[{"comment":"There are several typos: 'the the theory' in the introduction, 'reminisint' for 'reminiscent', and 'A zumaya' in the abstract.","section":"§1, abstract and introduction"},{"comment":"The phrase 'don't depend' should be 'does not depend'; also the remark could state explicitly that approximability is independent of the t-structure, not just of the generator.","section":"§2, Remark 2.3"},{"comment":"The notation for ⟨S⟩^{[a,b]}_n and ⟨S⟩^{[a,b]}_1 versus ⟨S⟩_{[a,b]} etc. is dense; a short example or a reference to the original convention would help readability.","section":"§2, Definition 2.1"},{"comment":"In the statement of Corollary 5.1, the notation Dperf(A) appears, but A is not defined; it should be Dperf(X). Similarly, in Corollary 5.10 the final sentence writes 'Dperf(A)' where the context is Dperf(A,X).","section":"§5, Corollary 5.1"},{"comment":"The filtration notation H^*(A)J^i is not defined explicitly; it would help to state that J is the radical of H^0(A) and that H^*(A)J^i denotes the sub-bimodule generated by J^i on appropriate factors.","section":"§5, Lemma 5.2"}],"recommendation":"major_revision","confidential_remarks":"The paper's main theorem is sound and the applications to schemes and Azumaya algebras appear to go through, but the application to proper connective DG-algebras is not proven due to the flaw in Lemma 5.2. The acknowledgments mention that a previous version of Lemma 5.2 contained a mistake; the current version still has a problem in the restriction step. I would encourage the editor to ask for a full repair of Lemma 5.2, or a detailed version of the Orlov-based argument in Remark 5.3, before publication. The self-citation [Goo24] in Remark 3.4 is not load-bearing and need not be a concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the main thing: Theorem 4.4 is a genuine and clean statement. If A is locally finite and approximable and D(A) is strongly generated by an object in D^fvd(A), then A is finite-reflexive. The proof via Neeman's representability is coherent, and Theorem 4.3, identifying D(A)^b_c with D^fvd(A), is a useful companion. The applications to proper schemes and Azumaya algebras over proper schemes look solid, assuming the cited generation results from Neeman and De Deyn-Lank-Rahul; that is a fair use of the literature.\n\nThe soft spot is Lemma 5.2, which supplies the strong generation hypothesis for proper connective DG-algebras. The stress-test lands. The filtration argument filters H^*(A) by powers of rad H^0(A), then restricts along A^e → H^0(A)^op ⊗ H^0(A)/J. The problem is that the subquotients F_i only carry H^0-level actions; after restriction, negative-degree classes of A act by zero on them. But H^*(A) as an A^e-module has nonzero actions by those classes. The example A = k[ε]/(ε^2), deg ε = -1, is a clear witness: the proof would put A in <k>_1 in D(A^e), while in fact A is only in <k>_2 there. So the argument as written does not establish the claimed generation.\n\nRemark 5.3 gestures at a repair via Orlov's DG-radical and the finite-dimensional models in the appendix, but it is only a sketch. Since Proposition 5.4 is a headline application, that repair needs to be made fully explicit.\n\nI don't see circularity or fitting elsewhere. The reliance on deep representability theorems is acceptable, and the appendix is a genuinely useful addition. This paper is for people in noncommutative derived geometry and anyone working with Kuznetsov-Shinder reflexivity. It deserves referee time; the main theorem is likely correct and the scope of examples is broader than the existing literature. I would send it out, with a request to repair Lemma 5.2 before acceptance.","headline":"Solid criterion for reflexivity; the connective DG-algebra proof has a genuine gap that needs a rewrite.","tokens_in":10483,"tokens_out":9499,"would_cite":true,"duration_ms":90376,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F08","18G80","16E45"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a locally finite approximable DG-category whose derived category is strongly generated by a finite-valued object is finite-reflexive: the double-dual evaluation functor is an equivalence.","keywords":["approximable triangulated categories","reflexive DG-categories","proper DG-algebras","finite-valued DG-modules","strong generation","proper schemes","Azumaya algebras","derived categories"],"falsifier":"Test Theorem 4.4 by taking a locally finite approximable DG-category A over a Noetherian ring k with $D(A)=\\langle G\\rangle_n$ for some $G\\in D^{fvd}(A)$ and checking whether the double-dual map $D^{\\mathrm{perf}}(A)\\to D^{fvd}(D^{fvd}(A)^{\\mathrm{op}})^{\\mathrm{op}}$ is essentially surjective; a single failure would refute the theorem. Natural places to look are proper connective DG-algebras over non-perfect fields and proper schemes over non-regular rings, where all hypotheses can be verified explicitly.","tokens_in":9326,"feed_emoji":"🔄","tokens_out":12317,"duration_ms":98255,"temperature":0.7,"pith_summary":"Reflexive DG-categories formalise the duality between perfect complexes and the bounded derived category of coherent sheaves: the double-dual functor is an equivalence. This paper finds a sufficient condition for that equivalence, phrased entirely in the language of approximable triangulated categories. The condition is that the DG-category A be locally finite and approximable, and that its derived category D(A) be generated in finitely many steps by one finite-valued DG-module G in $D^{{fvd}}$(A). When those hypotheses hold, Theorem 4.4 concludes that A is finite-reflexive; the paper then verifies the generation hypothesis for proper schemes, proper connective DG-algebras, and Azumaya algebras over proper schemes.","feed_headline":"One finite object forces a DG-category to be reflexive","feed_subtitle":"Proper schemes, connective DG-algebras, and Azumaya algebras all satisfy the new reflexivity criterion.","key_machinery":"The machinery is the theory of approximable triangulated categories: a triangulated category with coproducts, a compact generator G, and a t-structure such that every object can be approximated in finitely many stages by objects built from G. The key move is the completion–duality match: Lemma 4.1 characterizes $T^b_c$ as the objects whose Hom-sets into all compacts are finitely generated k-modules, and Theorem 4.3 converts this into the equality $D(A)^b_c = D^{fvd}(A)$ for locally finite approximable DG-categories. The reflexive conclusion then follows from the representability theorems in [Nee21c] and [Nee18], which produce $M\\in D^{\\mathrm{perf}}(A)$ representing any finite homological functor on $D(A)^b_c$.","core_discovery":"The central discovery is the identification of the completion of a locally finite approximable DG-category with its finite-valued modules: Theorem 4.3 shows $D(A)^b_c = D^{fvd}(A)$. Combining this with the representability theorems in [Nee21c] and [Nee18], Theorem 4.4 proves that a locally finite approximable DG-category A is finite-reflexive whenever there exists $G\\in D^{fvd}(A)$ with $D(A)=\\langle G\\rangle_n$. Finite-reflexivity means the functor $D^{\\mathrm{perf}}(A)\\to D^{fvd}(D^{fvd}(A)^{\\mathrm{op}})^{\\mathrm{op}}$ given by $M\\mapsto \\mathrm{RHom}_A(M,-)$ is an equivalence. The paper obtains the strong generation input in three settings: proper schemes over a Noetherian ring via [Nee21b], proper connective DG-algebras over any field via a radical filtration of $H^0(A)$ or via finite-dimensional models from the appendix, and Azumaya algebras over proper schemes via [DLR24b].","pith_inferences":["The paper's sufficient condition may in fact be near-necessary: any locally finite approximable DG-category whose double dual is an equivalence is likely to admit a finite-valued strong generator, so the reflexive and strongly-generated phenomena may coincide.","Over a non-regular base ring, the distinction between finite-reflexivity and Morita-reflexivity matters; the paper's results are for finite-reflexivity, leaving open whether the perfect complexes of the same schemes or algebras are Morita-reflexive over such bases.","The appendix's finite-dimensional model theorem over arbitrary fields makes Orlov's DG-radical construction available uniformly, so Lemma 5.2 could be proved by a single radical-filtration argument rather than case-by-case."],"forward_implications":["For a proper scheme X over k, the paper yields $D^b_{\\mathrm{coh}}(X)\\simeq D^{fvd}(D^{\\mathrm{perf}}(X))$ and finite-reflexivity of $D^{\\mathrm{perf}}(X)$.","For a proper connective DG-algebra A over any field, $D^{\\mathrm{perf}}(A)$ is reflexive, generalising the perfect-field examples.","For an Azumaya algebra $(A,X)$ over a scheme X proper over k, $D^{fvd}(D^{\\mathrm{perf}}(A,X))\\simeq D^b_{\\mathrm{coh}}(A,X)$ and $D^{\\mathrm{perf}}(A,X)$ is finite-reflexive.","In any reflexive case, the semi-orthogonal decompositions and derived autoequivalence groups of $D^{\\mathrm{perf}}(A)$ and $D^{fvd}(A)$ coincide, by the cited results in [KS22].","The appendix lets every proper connective DG-algebra over an arbitrary field be replaced by a finite-dimensional model, so the strong generation lemma has a uniform proof over all fields."],"supporting_citations":[{"why":"Introduces reflexive DG-categories and the double-dual functor, and provides the full-faithfulness half of Theorem 4.4.","marker":"[KS22]"},{"why":"Provides the approximability machinery and the representability theorem (Theorem 2.11(1)) used to identify $T^b_c$ and finite functors.","marker":"[Nee21c]"},{"why":"Supplies the representability theorem (Theorem 2.11(2)) for finite functors on $T^b_c$, used to find $M\\in D^{\\mathrm{perf}}(A)$ in Theorem 4.4.","marker":"[Nee18]"},{"why":"Provides the strong generator of $D^b_{\\mathrm{coh}}(X)$ that satisfies the hypothesis of Theorem 4.4 for proper schemes.","marker":"[Nee21b]"},{"why":"Provides the strong generation result for Azumaya algebras over proper schemes used in Corollary 5.10.","marker":"[DLR24b]"},{"why":"Establishes approximability of $D_{\\mathrm{Qcoh}}(A,X)$ and the equality with $D^b_{\\mathrm{coh}}(A,X)$ in the Azumaya case.","marker":"[DLR24a]"},{"why":"Shows $D^{\\mathrm{perf}}(X)$ is locally finite for proper schemes, feeding Theorem 4.3 in Corollary 5.1.","marker":"[Orl16]"},{"why":"Supplies the lemma used inside Lemma 4.1 to conclude that objects with finite Hom-spaces lie in $T^b$.","marker":"[BNP23]"}],"fun_headline_variants":["One finite object forces DG-category reflexivity","Strong generator implies reflexive DG-categories","A single finite object criterion for reflexivity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument's load-bearing premise is that each DG-category A considered has a single finite-valued object G such that $D(A)=\\langle G\\rangle_n$, the full subcategory built from G by finitely many cones and direct summands; the paper verifies this separately for each application, and if that generation input fails the reflexivity conclusion does not follow.","fun_headline_variants_meta":{"raw":{"variants":["One finite object forces DG-category reflexivity","Strong generator implies reflexive DG-categories","A single finite object criterion for reflexivity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000581,"raw_usage":{"total_tokens":2690,"prompt_tokens":851,"completion_tokens":1839,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":467,"completion_tokens_details":{"reasoning_tokens":1805}},"tokens_in":467,"tokens_out":1839,"duration_ms":59458,"temperature":1.0,"reasoning_tokens":1805,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T20:37:00.915696+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test Theorem 4.4 by taking a locally finite approximable DG-category A over a Noetherian ring k with $D(A)=\\langle G\\rangle_n$ for some $G\\in D^{fvd}(A)$ and checking whether the double-dual map $D^{\\mathrm{perf}}(A)\\to D^{fvd}(D^{fvd}(A)^{\\mathrm{op}})^{\\mathrm{op}}$ is essentially surjective; a single failure would refute the theorem. Natural places to look are proper connective DG-algebras over non-perfect fields and proper schemes over non-regular rings, where all hypotheses can be verified explicitly.","supporting_citations":[],"review_version":1}