{"id":"36e5e60f-c8a4-4c6f-a9d6-831f41692ace","arxiv_id":"2511.10312","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A morphism between complexes, with its target already lifted, has an Ext^1 obstruction and an Ext^0 torsor of lifts; this gives a new proof of uniqueness of deformations of semiorthogonal decompositions.","lead":"This paper builds a deformation theory for morphisms between complexes in derived categories, assuming the codomain is already lifted. It uses it to give a new proof that semiorthogonal decompositions deform uniquely in smooth proper families.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Stated small-extension hypothesis (1.1) fails for the m-adic triples (R/m^{n+2}->R/m^{n+1}->k) used in Theorem C's induction (a*b = m^2/m^{n+2} != 0); Corollary B is invoked outside its hypotheses.","rationale":"The central claim of the paper is a morphism-level deformation-obstruction calculus (Theorems A, 2.10, Corollary B) and its application to uniqueness of deformations of semiorthogonal decompositions (Theorem C). The calculus is a direct and explicit extension of Lowen's object-level theory; the proofs are structured, the imports from [7], [9], [6], [3] are standard black boxes, and the authors transparently flag the algebraization-uniqueness issue and the lack of higher structures. The reader's weakest-assumption identification correctly focuses on the small-extension/square-zero hypothesis, since the normalization leading to the degree-1 obstruction class rests on it. But the sharpest formulation, missed by the reader, is that the paper's own application violates the stated hypothesis: the triples (R/m^{n+2} -> R/m^{n+1} -> k) used in the proof of Theorem C do not satisfy a*b = 0 or b^2 = 0 for n >= 1 (respectively n >= 2), even though the effective square-zero condition I^2 = 0 (I = ker(R_tilde -> R)) does hold. So as written, the main theorem's proof is not justified; this is a correctness risk internal to the paper, not a disagreement with consensus. The finding is concrete and easily checked. Because the intended condition is recognizable and the fix is likely local (restate (1.1), or refine the filtration), the appropriate verdict is CONDITIONAL rather than REJECT: the theorem is probably true, but the text as submitted does not prove Theorem C, and Corollary B's hypotheses must be corrected. Agreement with the reader is partial: the same hypothesis is identified as weak, but the decisive failure mode (inapplicability in the main application) is new.","tokens_in":13268,"tokens_out":46805,"duration_ms":434258,"concrete_test":"Analytic check: take R = k[[t]] and inspect step n = 1 of Theorem C's induction, i.e. the triple R_2 = k[[t]]/(t^3) -> R_1 = k[[t]]/(t^2) -> k. Compute a = (t)/(t^3), b = (t)/(t^2); the product in R_2 is a*b = (t^2)/(t^3) != 0, so (1.1) is violated. This settles that Corollary B is applied outside its scope. Then run the decisive fix: re-derive the normalization and (2.9) with the hypothesis I^2 = 0, I = ker(R_tilde -> R), instead of b^2 = 0; if the quadratic term still vanishes (it does, since I^2 = 0 implies ker(Hom_R(R_tilde,-))^2 = 0), restate (1.1) accordingly and the calculus and Theorem C go through. If it does not, Theorem C requires the refined-filtration argument it does not supply.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The small-extension hypothesis (1.1) is the engine of Section 2: it gives ker(Hom_R(R_tilde,-))^2 = 0, which after (2.4) discards the quadratic term beta_s ∘ d_{1,2} ∘ beta_s in (2.9) and forces the normalized form (2.6)-(2.8). The proof of Theorem C (Section 4) says 'starting with n = 0 we can inductively apply Corollary B ... to (R_tilde -> R -> R0) = (R_{n+1} -> R_n -> R0 = k)' with R_n = R/m^{n+1}. For n >= 1, however, this triple does not satisfy (1.1): with a = ker(R_{n+1} -> k) = m/m^{n+2} and b = ker(R_n -> k) = m/m^{n+1}, the preimage of b in R_{n+1} is a itself, so a*b = a^2 = m^2/m^{n+2}, which is nonzero for R = k[[t]], n = 1. The claimed consequence b^2 = 0 fails for n >= 2. The vanishing that actually powers the normalization is I^2 = 0 with I = ker(R_tilde -> R), which the m-adic steps do satisfy (I = m^{n+1}/m^{n+2}, I^2 = 0); so the intended hypothesis is the classical square-zero small extension, and (1.1) is misstated as the stronger a^2 = 0. Hence, as written, the induction in Theorem C invokes Corollary B outside its stated hypotheses; the theorem needs either a corrected statement of (1.1) or an explicit refinement of the m-adic filtration into square-zero substeps.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a deformation-obstruction calculus for lifting morphisms of complexes when a lift of the codomain is fixed, in the setting of flat nilpotent deformations of abelian categories. The main results are Theorem A (homotopy category of injectives) and its dual Theorem 2.10, with Corollary B for perfect complexes on schemes. For a morphism s: F -> G with a fixed lift of G, an obstruction class in H^1 of an appropriate RHom space is constructed; its vanishing is equivalent to the existence of a lift of s, and under an Ext^{-1}=0 hypothesis the lifts form a torsor under the corresponding H^0 group. As an application, the paper proves Theorem C, giving an alternative proof of the uniqueness of deformations of semiorthogonal decompositions in smooth proper families.","tokens_in":13637,"tokens_out":17818,"duration_ms":164071,"significance":"If correct, this is a significant contribution: it extends Lowen's object-level deformation theory [7] to the morphism level with a fixed codomain lift, in a natural and useful form. The obstruction and torsor descriptions are explicit and the passage from an abstract model-categorical setting to perfect complexes on schemes is carefully executed. The application to semiorthogonal decompositions is already known from [1], but the proof via the new calculus is elegant and demonstrates the framework's reach. The paper is honest about imported technical inputs and leaves some variations to the reader. No circularity or parameter fitting is present.","major_comments":[],"minor_comments":[{"comment":"Proposition 2.9's second isomorphism has the Hom arguments swapped: based on the proof's μ_j (which sends t^j to a map Hom_{R0}(b, J0^i) -> K0^{i+j}), the correct statement should be Hom^•_+(Hom_{R0}(b, I0^•), I0^•) ≃ Hom^•(Hom_{R0}(b, J0^•), K0^•). The printed version (Hom(b,K) -> J) contradicts the obstruction class in Theorem A and should be corrected.","section":"§2, Eq. (2.20)"},{"comment":"The tensor product subscripts are inconsistent: (3.7) should involve derived tensor products over the upper ring R~ (e.g., b⊗^L_{R~} E, R⊗^L_{R~} E), and (3.8) should read b⊗^L_{R~} E ≃ b⊗^L_{R~} (R⊗^L_{R~} E). Please clarify the notation.","section":"§3, Eqs. (3.7)-(3.8)"},{"comment":"The notation '(R↠R↠R0) = (R_{n+1} ↠ R_n ↠ R0 = k)' reuses R for both the upper and middle rings. Use a tilde on the upper ring to match (1.1).","section":"§4, Eq. (4.13)"},{"comment":"The m-adic triples (R_{n+1}->R_n->k) do satisfy hypothesis (1.1): with a = m/m^{n+2} and b = m^{n+1}/m^{n+2}, we have a·b = m^{n+2}/m^{n+2} = 0. Thus the inductive application of Corollary B is within the stated hypotheses.","section":"§4, induction step"},{"comment":"The final paragraph of the proof is very terse: the claimed equivalence between coboundaries in Hom^•_+ and coboundaries in Hom^•(I,I) is not obvious and should be expanded or replaced by a direct argument that the action is free and transitive.","section":"§2, Proposition 2.8"},{"comment":"Some notation switches between R and R~ (e.g., in (1.1), (3.7), (3.8)) should be harmonized for readability.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The stress-test concern raised in the review process does not land: the m-adic filtration steps satisfy (1.1). The paper is mathematically sound; the main issue is the typo in Proposition 2.9, which is local. I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper does what it says. It builds the deformation-obstruction calculus for lifting a morphism F->G when a lift of G is fixed, generalizing Lowen's object-level theory to morphisms, then specializes to perfect complexes. The obstruction class in Ext^1(F0, b⊗H0) with a torsor under Ext^0 is the natural statement, and the proof via model structures and normalization of the lift is coherent. The application—uniqueness of deformations of semiorthogonal decompositions—is explicitly an alternative proof of the authors' own [1, Thm 7.7], so the new value is methodological, not a new geometric theorem. They are upfront about this, which I appreciate.\n\nThe stress-test concern about the induction in Theorem C does not survive contact with the paper. In the triple (R_{n+1}->R_n->k), the paper's b is ker(R_{n+1}->R_n) = m^{n+1}/m^{n+2}, not ker(R_n->k). Then ab = (m/m^{n+2})(m^{n+1}/m^{n+2}) = m^{n+2}/m^{n+2} = 0, so the small-extension hypothesis (1.1) holds at every step. No gap there.\n\nSoft spots: the proof leans heavily on imported machinery—[7] for the object-level calculus, [9] for flat deformations, [6] for algebraization—and some steps are 'by the same reasoning as in [7]' rather than worked out. The H^{-1} vanishing assumption is a real restriction; they explicitly say they don't develop the higher-structure version. The algebraization step borrows [6, Prop 3.6.1] and they note it's not clear whether uniqueness follows from loc. cit., so they prove it separately. These are honest limitations, not hidden ones.\n\nBottom line: the central claim (Theorem A/2.10/Corollary B) is well-supported, the paper is clearly written by people who know the area, and the alternative proof of the SOD deformation theorem is a legitimate payoff. It deserves serious refereeing. I'd probably accept after minor revision; the referee should check the imported black-boxes and the formal geometry step carefully.","headline":"A solid, honest extension of Lowen's deformation theory to morphisms with fixed target lifts; the main application is a repackaged known result, and the apparent induction gap flagged in the stress-test does not hold up.","tokens_in":14211,"tokens_out":3478,"would_cite":true,"duration_ms":31600,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14D15","14F08","18G80"],"pacs":[],"model":"deepseek-v4-flash","headline":"An Ext^1 obstruction class, defined for any morphism with a fixed lift of its target, completely controls whether that morphism lifts to a flat nilpotent deformation.","keywords":["deformation theory","derived categories","obstruction class","flat nilpotent deformations","abelian categories","semiorthogonal decompositions","perfect complexes","torsor"],"falsifier":"Take a specific square-zero extension of a scheme (e.g., the double of the affine line), choose a morphism between two perfect complexes whose Ext^1 obstruction class is nonzero, and verify that no lift exists. Or, more sharply, find a case where the obstruction class vanishes but Ext^{-1} is nonzero and show that the set of lifts is not transitive under the Ext^0-action, contradicting the torsor claim.","tokens_in":83,"feed_emoji":"📐","tokens_out":4793,"duration_ms":99520,"temperature":0.7,"pith_summary":"The paper develops a deformation–obstruction calculus for morphisms between complexes, in the setting of derived categories of flat nilpotent deformations of abelian categories. It shows that, once a lift of the codomain is fixed, the question of whether a given morphism lifts is answered by the vanishing of a single cohomology class in Ext^1. When a mild vanishing hypothesis holds, the isomorphism classes of lifts form a torsor under the corresponding Ext^0 group, meaning lifts are either unique or form a principal homogeneous space. This generalizes earlier object-level deformation theory to the morphism level, and provides a new proof that semiorthogonal decompositions deform uniquely in smooth proper families. The result matters because morphisms between complexes are the basic building blocks of derived geometry, and such a clean criterion makes deformation-theoretic arguments in that setting tractable.","feed_headline":"Ext^1 obstruction controls when a morphism lifts","feed_subtitle":"New theorem makes uniqueness of semiorthogonal decompositions a one-line corollary.","key_machinery":"The key machinery is the normalization of a morphism of complexes via the coderived model structure: up to homotopy, the morphism becomes a degreewise monomorphism J ↪ I, and the square-zero condition on the thickening forces the quadratic term in the deformed differential to vanish. This allows the obstruction to be read off as the (2,1)-component of the conjugated differential, a cocycle in Hom^+ (Hom(b,I),I); its cohomology class is the obstruction. The identification of this complex with RHom_{R0}(F0, b⊗^L H0) transfers the result to the derived category.","core_discovery":"The central discovery is that the liftability of a morphism s:F→G, with a fixed lift of G, is governed by a single obstruction class o(s) living in H^1(RHom(RHom(b,F0),H0)) — in the geometric case, Ext^1_{X0}(F0, b⊗^L H0). This class is constructed by embedding the morphism as a degreewise monomorphism, lifting the embedding, and measuring the failure of the lifted map to be a subcomplex via a cocycle in a subcomplex of homomorphisms. Its vanishing is necessary and sufficient for the existence of a lift. Once the obstruction vanishes, and if the corresponding Ext^{-1} vanishes, the set of isomorphism classes of lifts is a torsor under Ext^0. This provides a uniform deformation–obstruction pi","pith_inferences":["The same obstruction–torsor pattern likely extends to higher-order thickenings if one tracks the quadratic and higher terms; the square-zero case treated here is the first rung of a general 'derived' deformation theory with higher obstructions.","The theorem suggests that the space of lifts of a morphism is, in the unobstructed case, a gerbe (or at least a principal bundle) under the derived Hom group, and that the Ext^{-1} condition is the shadow of a full derived structure that the paper does not develop.","One could test the calculus on concrete examples of morphisms between perfect complexes over, say, a double point, to see whether the Ext^1 class reproduces the classical obstruction to lifting a morphism of vector bundles; this would provide a transparent check of the formalism.","The algebraization step in the application suggests a general principle: formal liftability of a morphism, combined with a Grothendieck-existence-type comparison for Hom-spaces, gives actual liftability over complete local bases."],"forward_implications":["If the obstruction class vanishes, a lift exists, and when Ext^{-1} vanishes, all lifts are determined up to isomorphism by an Ext^0-torsor.","The morphism-level theory subsumes the object-level theory: taking the target to be zero recovers the known obstruction theory for lifting objects.","The geometric version (Corollary B) applies to perfect complexes on flat families of schemes, giving a clean criterion for lifting morphisms between perfect complexes along the pullback functor.","As a corollary, semiorthogonal decompositions of categories of perfect complexes deform uniquely in smooth proper families, recovering and reproving a key ingredient in the construction of moduli spaces of semiorthogonal decompositions.","The framework is flexible enough to obtain further applications whenever one studies morphisms in derived categories over Artinian or complete local bases."],"fun_headline_variants":["Ext^1 class blocks or allows morphism lifts","One obstruction class rules morphism lifting","Obstruction in Ext^1 decides if morphism lifts","New proof: Ext^1 obstruction governs morphism lifts","Morphism liftability reduced to a single Ext^1 class"],"cache_read_input_tokens":15360,"weakest_assumption_plain":"The whole obstruction calculus rests on the assumption that the ring extension is 'small', meaning the square of the defining ideal is zero; this is what kills the quadratic term and makes the obstruction a plain cohomology class. If the thickening is not square-zero, the proposed class may not exist or may not control liftability.","fun_headline_variants_meta":{"raw":{"variants":["Ext^1 class blocks or allows morphism lifts","One obstruction class rules morphism lifting","Obstruction in Ext^1 decides if morphism lifts","New proof: Ext^1 obstruction governs morphism lifts","Morphism liftability reduced to a single Ext^1 class"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000148,"raw_usage":{"total_tokens":951,"prompt_tokens":592,"completion_tokens":359,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":336,"completion_tokens_details":{"reasoning_tokens":281}},"tokens_in":336,"tokens_out":359,"duration_ms":3806,"temperature":1.0,"reasoning_tokens":281,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T22:27:40.832962+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a specific square-zero extension of a scheme (e.g., the double of the affine line), choose a morphism between two perfect complexes whose Ext^1 obstruction class is nonzero, and verify that no lift exists. Or, more sharply, find a case where the obstruction class vanishes but Ext^{-1} is nonzero and show that the set of lifts is not transitive under the Ext^0-action, contradicting the torsor claim.","supporting_citations":[],"review_version":1}