{"id":"b0481b5c-2535-493f-8c4b-a876bef0bff0","arxiv_id":"2509.01507","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The map Br^der(X) -> Pic(Mot_X) is injective for every qcqs scheme X.","lead":"This paper proves that the derived Brauer group of any qcqs scheme injects into the Picard group of localizing motives, so a universal K-theory-like invariant remembers every Brauer class. It extends earlier contributions by Tabuada and Tabuada-Van den Bergh and introduces a twisted determinant method that works beyond affine or regular schemes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Transfer of [TVdB16, Thm B.15] to qcqs schemes is asserted, not proven; internal slips flag risk","rationale":"The paper's genuine novelty is the determinant/Pic_A construction, which plausibly proves injectivity of H^2_ét(X, Gm) into Pic(Mot_X) via Corollary 2.17 and Theorem 2.14; that portion is internally coherent. The full injectivity theorem, however, hangs on a transfer of [TVdB16, Theorem B.15] to qcqs schemes that is asserted rather than demonstrated. The reader's weakest_assumption identifies exactly this point, and I agree. The concern is not that the theorem is false, but that the proof as written is incomplete at a load-bearing step. The internal slips—the noetherian parenthetical in Corollary 2.18 and the B.5/B.15 discrepancy in Lemma 2.19—reinforce the need for the author to spell out the transfer or point to explicit lines in [TVdB16] being adapted. A referee could reasonably request this before full acceptance. Hence I would change the verdict to CONDITIONAL, not REJECT: the mathematical core is promising and the missing piece is a verification, not a known flaw.","tokens_in":10496,"tokens_out":21823,"duration_ms":234902,"concrete_test":"Download Tabuada–Van den Bergh, arXiv:1508.03675 (or the published version), and read the proof of Theorem B.15 in Appendix B. Enumerate every step and check whether each is valid for an arbitrary qcqs scheme, with the paper's Lemmas 2.19 and 2.20 inserted exactly where the author claims. In particular, find the step where the proof passes from a trivial rational motive to an Azumaya algebra with nowhere vanishing rank; verify that this step uses only the two identified lemmas and no hidden affine-only input (e.g., a single compact generator of Perf(R), noetherian induction, or a finite affine covering argument). If any unaddressed affine/noetherian step exists, Theorem 2.21 is not established; if none is found, the concern is resolved and the original ACCEPT is justified.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 2.21 depends on the claim that the proof of [TVdB16, Theorem B.15] transfers verbatim from affine noetherian schemes to arbitrary qcqs schemes, except for the two gaps filled by Lemmas 2.19 and 2.20. The manuscript does not reproduce that proof, so a reader cannot check whether hidden assumptions of affineness or noetherianness occur in the untransferred steps. In particular, the step that a derived Azumaya algebra with trivial rational motive is Morita equivalent to an algebra with nowhere vanishing rank—the pivotal step in the proof of Theorem 2.21—is not demonstrated. The risk is amplified by internal inconsistencies: (i) in the proof of Corollary 2.18, a leftover parenthetical '(this is where we use that R is noetherian)' contradicts the non-noetherian status of Lemma 2.19; (ii) Lemma 2.19 cites '[TVdB16, Theorem B.5]' while the surrounding text and Corollary 2.18 refer to '[TVdB16, Theorem B.15]'. These slips suggest the generalization from [TVdB16] may not have been fully rechecked.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for every qcqs scheme X the natural map Br^der(X) -> Pic(Mot_X) is injective, extending results of Tabuada and Tabuada–Van den Bergh. The proof introduces, for a derived Azumaya algebra A, an étale sheaf of spectra Pic_A obtained by étale sheafification of the 1-truncated K-theory of Perf(A). The author shows that Pic_A is a twisted form of Pic and that, on the H^2_ét(X,G_m) subgroup of the derived Brauer group, the map to Pic(Mot_X) is injective (Corollary 2.17). Full injectivity is then obtained by combining this subgroup result with a claimed generalization of [TVdB16, Theorem B.15] to arbitrary qcqs schemes, effected in Lemmas 2.19 and 2.20, which reduces a derived Azumaya algebra with trivial localizing motive to a torsion class in H^2_ét(X,G_m).","tokens_in":10743,"tokens_out":15089,"duration_ms":174458,"significance":"If correct, the main theorem settles the injectivity side of a natural question of Tabuada in full generality, showing that the localizing motive U^X_loc(Perf(A)) determines a derived Azumaya algebra up to Morita equivalence. The determinant/Pic_A construction is elegant, new, and yields a genuinely softer proof of restricted injectivity for H^2_ét. The paper is clearly written and properly credits prior work. Its main weakness is that the final step of the proof of Theorem 2.21 rests on an asserted verbatim transfer of a substantial external theorem from the affine noetherian setting to arbitrary qcqs schemes, without the reader being able to verify the absent steps; internal inconsistencies in the references and a leftover noetherian remark reinforce this concern.","major_comments":[{"comment":"The proof of Theorem 2.21 rests on the assertion that the proof of [TVdB16, Theorem B.15] 'works with no modification' on arbitrary qcqs schemes until one concludes that A is Morita equivalent to an Azumaya algebra with nowhere vanishing rank. This is a load-bearing transfer of an external result, but it is not demonstrated in the manuscript. The relevant parts of [TVdB16, Appendix B] are not reproduced, and the reader cannot check whether affineness or noetherianness enters in the untransferred steps. In particular, the step producing the algebra B with nowhere vanishing rank is exactly what allows the use of [TVdB16, Theorem B.11] and the reduction to H^2_ét(X,G_m). Please either give a self-contained proof of this reduction, or state a precise qcqs version of [TVdB16, Theorem B.15] with a proof (or with a citation to an existing statement that covers qcqs schemes) and identify each pl","section":"Proof of Theorem 2.21"},{"comment":"The non-noetherian generalization in Lemma 2.19 is stated as the conclusion of '[TVdB16, Theorem B.5]', but its proof and Corollary 2.18 refer to '[TVdB16, Theorem B.15]'. Moreover, the proof of Corollary 2.18 contains the leftover parenthetical '(this is where we use that R is noetherian)' immediately after invoking the non-noetherian Lemma 2.19. These internal inconsistencies suggest that the adaptation of [TVdB16] was not fully rechecked. Since Lemma 2.19 is load-bearing for affine injectivity (Corollary 2.18), the correct reference and a careful proof of the transferred statement are needed; the current proof only sketches the replacement of [TVdB16, Lemma B.16] and does not address the rest of the translated argument.","section":"Lemma 2.19 and Corollary 2.18"}],"minor_comments":[{"comment":"The sentence 'the vertical maps are isomorphisms upon sheafification' is terse; it would help to spell out the identification Z^× ≅ Z/2 and the sign convention giving det(O) = ΣO with K-theory class -1.","section":"Lemma 2.7"},{"comment":"'essentially a priori' appears to be a typo; probably 'essentially, a priori,' or simply 'a priori'.","section":"Remark 2.12"},{"comment":"The displayed quotation of [TVdB15, Theorem 2.1] and [TVdB16, Theorem B.15] could be confusing because the second bullet is later attributed only to [TVdB16, Theorem B.15]. It would help to state precisely which part is used where.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"This is a well-written short paper with a nice determinant construction, and the restricted injectivity result (Corollary 2.17) seems solid. The main theorem, however, depends on a transfer of [TVdB16, Theorem B.15] that is asserted rather than proven. The inconsistencies between Theorem B.5 and Theorem B.15, together with the leftover noetherian remark, currently undermine the credibility of that transfer. These issues are fixable within the manuscript's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me cut to it: the main theorem is new and, as far as I can see, correct. The injectivity of Br^der(X) -> Pic(Mot_X) for every qcqs scheme removes the affine/noetherian/regular hypotheses from earlier work, and the proof route is original: the twisted determinant Pic_A is a good idea, and Cor 2.17 is a clean way to get the H^2_et part. The local analysis in Section 2 is the strongest part of the paper. The author also deserves credit for being explicit about what is imported and for supplying Lemmas 2.19 and 2.20 to handle the non-noetherian and non-affine steps.\n\nThat said, there are real soft spots. The proof of Theorem 2.21 rests on the assertion that the proof of [TVdB16, Thm B.15] works over qcqs schemes with no modification except for the two steps addressed. That is load-bearing, and the manuscript does not reproduce the part of the argument showing that a derived Azumaya algebra with trivial rational motive is Morita equivalent to an algebra with nowhere vanishing rank. I cannot check that step from this paper alone. It may be true, but a referee needs to verify it. The internal slips do not inspire full confidence: in the proof of Cor 2.18 there is a leftover parenthetical 'this is where we use that R is noetherian' that contradicts the non-noetherian statement of Lemma 2.19, and Lemma 2.19 cites Theorem B.5 while the text and Cor 2.18 refer to Theorem B.15. These are minor and fixable, but they suggest the transfer was not fully rechecked line by line.\n\nOn the substance, I found no circularity and no invented entities. The reliance on TVdB16 is external dependence, not circularity. The determinant construction is independent of the conclusion. The result is a natural generalization, and the paper reads as an honest, well-organized contribution. The reader's ACCEPT verdict is about right, with moderate confidence. I would not desk-reject this; I would send it to a referee who knows the TVdB16 proof and the higher-categorical machinery, and ask them to write out the transfer in detail or confirm it. If the transfer checks out, this is a solid paper worth publishing. The audience is people working in noncommutative motives and Brauer groups; they will want to read it.","headline":"A real new result, with a proof that leans on an unverified transfer of a heavy theorem; worth a careful referee.","tokens_in":11243,"tokens_out":2801,"would_cite":true,"duration_ms":28731,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F22","18N60"],"pacs":[],"model":"deepseek-v4-flash","headline":"A derived Azumaya algebra over a qcqs scheme is Morita equivalent to the structure sheaf exactly when its category of perfect modules has the unit localizing motive.","keywords":["derived Brauer group","Azumaya algebras","localizing motives","qcqs schemes","twisted determinant","étale cohomology","Morita equivalence","noncommutative motives"],"falsifier":"Compute U^X_loc(Perf(A)) for the standard quaternion algebra over Spec R, a nontrivial class in H^2_ét(Spec R,G_m); the theorem entails that this motive is not the unit motive. An explicit K-theory calculation, using the semiorthogonal decomposition of the associated twisted projective space, showing U^X_loc(Perf(A)) ≃ U^X_loc(Perf(X)) would refute the main theorem.","tokens_in":10361,"feed_emoji":"🧩","tokens_out":19145,"duration_ms":215267,"temperature":0.7,"pith_summary":"The paper proves that the derived Brauer group of any quasi-compact quasi-separated scheme embeds into the Picard group of X-linear localizing motives: a derived Azumaya algebra is Morita equivalent to the structure sheaf as soon as its category of perfect modules carries the unit localizing motive. This matters because localizing motives are the common target of all localizing invariants, so no localizing invariant can confuse two Morita-distinct derived Azumaya algebras. The proof combines a new twisted determinant sheaf, which recovers the H^2_ét(X,G_m) part of a Brauer class from the motive, with a transfer of the known affine noetherian trivial-motive theorem to arbitrary qcqs schemes for the remaining H^1_ét(X,Z) part.","feed_headline":"Derived Brauer groups embed into motives","feed_subtitle":"On any quasi-compact quasi-separated scheme, the localizing motive fixes the Morita class of every Azumaya algebra.","key_machinery":"The central object is the twisted determinant sheaf Pic_A: the étale sheafification of U ↦ τ_[0,1] K(Perf(A) ⊗_{Perf(X)} Perf(U)), i.e. the 1-truncated K-theory of the A-twisted perfect complexes, equipped with its natural module structure over the Picard sheaf Pic. Étale-locally Pic_A ≃ Pic, so it is a twisted form; its π0 is locally Z and its higher part is a twisted form of BG_m. The paper shows this sheaf is a functor of the localizing motive U^X_loc(A), and that its class in H^1_ét(X,Z/2), together with the G_m-fiber under the equivalence B^ét 2Pic ≃ B^ét 2Z × B^2_ét G_m, determines the original Brauer class. This is the mechanism that splits the comparison map on the H^2_ét(X,G_m) side","core_discovery":"The paper's main theorem is injectivity of Br^der(X) → Pic(Mot_X) for every qcqs scheme X: if a derived Azumaya algebra A satisfies U^X_loc(Perf(A)) ≃ U^X_loc(Perf(X)), then A is Morita equivalent to X. The proof splits a Brauer class into an H^2_ét(X,G_m) part and an H^1_ét(X,Z) part. For the H^2 part it builds, from the motive of A, an étale sheaf Pic_A that is locally a twisted form of the Picard sheaf; the cohomological invariants of that twisted form recover the original H^2 class. For the H^1 part it transfers a known affine theorem: a trivial rational motive forces A to be Morita equivalent to an Azumaya algebra of nowhere vanishing rank, hence torsion, and H^1_ét(X,Z) is torsion-free","pith_inferences":["Because the determinant step is étale-local and functorial in the motive, the same construction should yield explicit cohomological invariants of Pic_A for concrete Azumaya algebras, for instance via semiorthogonal decompositions of twisted projective spaces; the paper does not carry out such computations.","The proof leaves open whether suspensions of the unit motive generate Pic(Mot_X). If they do, the injection would upgrade to a description of the derived Brauer group as the torsion part of the motive Picard group, giving a fresh handle on the surjectivity question.","A natural stress test is to replace qcqs schemes by derived schemes or stacks, where the étale-local description of the Brauer group is less automatic; whether the injection survives there is not addressed in this paper."],"forward_implications":["For every qcqs scheme, the localizing motive is a complete invariant of the Morita class of a derived Azumaya algebra.","The subgroup H^2_ét(X,G_m) of the derived Brauer group embeds into Pic(Mot_X), so nonzero cohomological Brauer classes always change the motive.","The affine version holds for arbitrary commutative rings, with no noetherian hypothesis.","A classical non-2-torsion Azumaya algebra is never motive-equivalent to its opposite algebra.","A derived Azumaya algebra whose rational motive is trivial is Morita equivalent to a classical Azumaya algebra of nowhere vanishing rank, and is torsion in the Brauer group."],"supporting_citations":[{"why":"Provides the derived Brauer group as π0 of an étale sheaf of spectra and the étale-local triviality of Azumaya algebras, the foundation for the paper's local analysis.","marker":"[Toë12]"},{"why":"Constructs the determinant functor from K-theory to Pic, which Proposition 2.2 sheafifies to build the twisted determinant Pic_A.","marker":"[KM76]"},{"why":"Gives the affine noetherian result that a derived Azumaya algebra with trivial rational motive is Morita equivalent to a classical one; Theorem 2.21 transfers this to qcqs schemes.","marker":"[TVdB16, Theorem B.15]"},{"why":"Shows an Azumaya algebra with nowhere vanishing rank is torsion in the Brauer group, used to kill the H^1_ét(X,Z) component.","marker":"[TVdB16, Theorem B.11]"},{"why":"Classifies thick ⊗-ideals of perfect complexes, used in Lemma 2.20 to prove a perfect sheaf of nowhere vanishing rank generates QCoh(X).","marker":"[Tho97]"},{"why":"Provides the framework for X-linear localizing motives and the description of K-theory as mapping spaces in Mot_X.","marker":"[HSS17]"}],"fun_headline_variants":["Motive pins down Azumaya Morita class","Derived Brauer injects into motive Picard","Localizing motive distinguishes Azumaya algebras","Brauer class encoded by localizing motive"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The argument assumes that the known affine noetherian theorem — a derived Azumaya algebra with trivial rational motive is Morita equivalent to a classical one — transfers to arbitrary qcqs schemes once Lemmas 2.19 and 2.20 are added; the paper does not re-prove the untransferred middle steps of that theorem, and if they fail outside the affine noetherian setting the H^1_ét(X,Z) part of the main theorem has no support.","fun_headline_variants_meta":{"raw":{"variants":["Motive pins down Azumaya Morita class","Derived Brauer injects into motive Picard","Localizing motive distinguishes Azumaya algebras","Brauer class encoded by localizing motive"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000238,"raw_usage":{"total_tokens":1278,"prompt_tokens":606,"completion_tokens":672,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":350,"completion_tokens_details":{"reasoning_tokens":613}},"tokens_in":350,"tokens_out":672,"duration_ms":8824,"temperature":1.0,"reasoning_tokens":613,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T12:29:03.635847+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute U^X_loc(Perf(A)) for the standard quaternion algebra over Spec R, a nontrivial class in H^2_ét(Spec R,G_m); the theorem entails that this motive is not the unit motive. An explicit K-theory calculation, using the semiorthogonal decomposition of the associated twisted projective space, showing U^X_loc(Perf(A)) ≃ U^X_loc(Perf(X)) would refute the main theorem.","supporting_citations":[],"review_version":1}