{"id":"8261bdde-1df0-4d3a-8bab-a196fd493402","arxiv_id":"2509.08246","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For a rigid object M, the presentation functor P from pr(M) to 2-term complexes over End(M) is full, dense, detects extriangles, and induces a mutation-commuting bijection between basic relative cluster-tilting objects and basic 2-term silting objects.","lead":"A new 'presentation' functor sends objects built from a rigid object in a triangulated category to 2-term complexes of modules over its endomorphism algebra, and matches cluster-tilting objects with silting complexes. It recovers a theorem about self-injective quivers with potential and, in an appendix by Iyama, characterizes exactly when these 2-term complexes form a triangulated category.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.13(c) lacks proof without Hom-finiteness: the only supplied route from relative cluster-tilting objects to Theorem 3.11's weakly relative cluster-tilting subcategories is Lemma 2.5, which assumes Hom-finite.","rationale":"The reader's verdict identified the non-Hom-finite statement of Theorem 3.13(c) as a subtle point deserving a check, but did not treat it as load-bearing; the reader's weakest_assumption instead focused on algebraicity. My reading agrees that the Frobenius-model assumption is explicit and standard, and I do not see a flaw in the construction of P or in the Hom-finite applications. However, Theorem 3.13(c) is a stated central theorem whose proof is missing: the proof of Theorem 3.13 provides no argument for (c), and the only bridge between the paper's two notions of cluster-tilting (relative cluster-tilting vs. weakly relative cluster-tilting) is Lemma 2.5, which assumes Hom-finiteness. This is a concrete missing implication, located in a theorem that is part of the main results. It does not invalidate the Hom-finite mutation-commuting bijection or the 2-Calabi-Yau applications, so the verdict should be conditional rather than reject: the paper should either supply the missing non-Hom-finite argument or restrict the statement to the Hom-finite setting.","tokens_in":28778,"tokens_out":30377,"duration_ms":355955,"concrete_test":"Analytical check: attempt to prove the implication from Definition 2.3(e) to Definition 2.3(d) for add(N) in the setting of Theorem 3.13(c), without invoking Lemma 2.5. In particular, verify whether the contravariant-finiteness condition plus the equality in Definition 2.3(e) forces the generating condition M ⊆ Σ^{-1}N ∗ N when Hom_T(M,M) is not finite-dimensional. If the only available argument passes through Lemma 2.5 or [48, Theorem 3.1] and those require Hom-finiteness, then Theorem 3.13(c) should be restricted to the Hom-finite case and Theorem 1.1(d) adjusted accordingly.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 3.13(c) asserts a bijection between isomorphism classes of relative cluster-tilting objects of pr(M) and 2-term silting objects of H^b(projA) without any Hom-finiteness assumption. Its proof is not supplied: after (a) and (b), the proof says only 'It remains to prove that the bijection in (d) commutes with mutations,' so (c) is taken to follow from Theorem 3.11. But Theorem 3.11 concerns weakly relative cluster-tilting subcategories, whereas Definition 2.3(e) of a relative cluster-tilting object/subcategory requires contravariant finiteness and does not, as stated, imply the generating condition in Definition 2.3(c). The only implication in the paper from relative cluster-tilting to generating/weakly relative cluster-tilting is Lemma 2.5, whose assumptions include 'k is a field and pr(M) is Hom-finite over k' (equivalence (i)⇔(iii)). No argument is given that contravariant finiteness forces generating when Hom-finiteness fails. Thus Theorem 3.13(c), and consequently Theorem 1.1(d), appear to require either an additional proof or a Hom-finite hypothesis. The paper's main applications and Theorem 1.1(e) are Hom-finite, so the core construction and its principal consequences are not undermined; the gap concerns the generality of the stated bijection.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a k-linear functor P: pr(M) -> H^{[-1,0]}(proj A) for a rigid object M in an algebraic triangulated category T, where pr(M) is the subcategory of objects finitely presented by M and A = End_T(M). The functor sends an object to its presentation complex. The main results are: (a) P is full and dense, with kernel the ideal of morphisms factoring through a morphism ΣM_1 -> M_2, so P induces an equivalence modulo that ideal; (b) P detects isomorphisms, indecomposability and extriangles, and induces a bijection on isomorphism classes; (c) in the Hom-finite case, P induces a mutation-commuting bijection between basic relative cluster-tilting objects of pr(M) and basic 2-term silting objects of H^b(proj A); (d) in the 2-Calabi-Yau cluster-tilting case, P is an equivalence iff A is self-injective, and then H^{[-1,0]}(proj A) admits a triangle structure. The paper applies these results to self-injective quivers with potential, recovering a theorem of Mizuno, and Iyama's appendix proves a converse: if H^{[-1,0]}(proj A) has a triangle structure, then A is self-injective.","tokens_in":29103,"tokens_out":8858,"duration_ms":105591,"significance":"If the general statements hold, the paper gives a unified and explicit framework that recovers and extends earlier results of Buan--Yang, Iyama--Yang, and the tau-tilting correspondence. The construction is hands-on, and the proof of the main functor properties is unusually detailed: the liftable-presentation formalism in Proposition 3.2, the fullness and kernel description in Theorem 3.8, and the extriangle detection in Proposition 3.7 all appear with complete diagram chases. The appendix by Iyama is a strong addition, providing a converse characterization. The main caveat is a gap in the proof of the non-Hom-finite relative cluster-tilting bijection; because the paper's applications and its Hom-finite mutation statement are unaffected, the core contribution remains valuable, but the stated generality of Theorems 1.1(d) and 3.13(c) needs repair.","major_comments":[{"comment":"The bijection between relative cluster-tilting objects of pr(M) and 2-term silting objects of H^b(projA) is not proved in the stated generality. The proof of Theorem 3.13 says only 'It remains to prove that the bijection in (d) commutes with mutations', so (c) is being deduced from Theorem 3.11. But Theorem 3.11 concerns weakly relative cluster-tilting subcategories, which by Definition 2.3(d) are required to be generating. Definition 2.3(e) of a relative cluster-tilting object/subcategory does not include the generating condition. The only bridge in the paper from relative cluster-tilting to generating/weakly relative cluster-tilting is Lemma 2.5, whose hypotheses include Hom-finiteness. No argument is given that contravariant finiteness plus the equality condition forces generating when Hom-finiteness fails. Thus Theorem 3.13(c), and consequently Theorem 1.1(d), require either an addit","section":"Theorem 3.13(c)"}],"minor_comments":[{"comment":"'silting complexs' should be 'silting complexes'.","section":"Abstract"},{"comment":"'the bijection in (c) restricts' appears to be a cross-reference error; the intended statement is about the bijection in (d).","section":"Theorem 1.1(e)"},{"comment":"'restricts a bijection' should be 'restricts to a bijection'.","section":"Theorem 3.13(d)"},{"comment":"After Corollary 4.3, 'Amiot's conjecture hods' should be 'Amiot's conjecture holds'.","section":"Section 4.1"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The honest summary: this is a good paper, and the reader's ACCEPT is right, provided the author fixes one overstatement. The construction of the functor P from pr(M) to 2-term complexes is new and done carefully; the kernel description (morphisms factoring through Sigma M -> M'), the extriangle detection, and the mutation-compatible bijection in the Hom-finite setting are all genuine contributions. The application to Mizuno's theorem and the Iyama appendix are nice. The introduction is transparent about simultaneous work by [15] and [12].\n\nThe soft spot is exactly the one the stress-test flags. Theorem 3.13(c) states a bijection between isomorphism classes of relative cluster-tilting objects and 2-term silting objects, with no Hom-finiteness assumption. The proof after (a),(b) simply says \"it remains to prove that the bijection in (d) commutes with mutations,\" leaving (c) to be inferred from Theorem 3.11. But Theorem 3.11 is about weakly relative cluster-tilting subcategories, and the only bridge from relative cluster-tilting to weakly relative cluster-tilting in the paper is Lemma 2.5, which assumes k is a field and pr(M) is Hom-finite. No argument is given that contravariant finiteness forces generating without Hom-finiteness. So (c), and consequently Theorem 1.1(d), is not justified as stated. The main applications are Hom-finite, so this does not sink the paper, but it is a real statement that needs either a proof or a Hom-finite hypothesis.\n\nThe proofs are otherwise detailed and consistent with the surrounding literature. The algebraicity assumption is standard and clearly flagged. The self-citations are to the author's own earlier work that this genuinely generalizes; I see no circularity. I did not see any unreported overlap or post-hoc fitting. I would not be surprised if the non-Hom-finite claim is true, but the paper does not currently demonstrate it.\n\nRecommendation: send it to a good referee. Ask the referee to check Theorem 3.13(c) and request the author either supply the missing argument or restrict the statement. After that, it deserves to be cited.","headline":"Solid paper with a real contribution, but the non-Hom-finite bijection in Theorem 3.13(c) is missing a proof step and should be fixed before publication.","tokens_in":29638,"tokens_out":4359,"would_cite":true,"duration_ms":51158,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16E35","18G80"],"pacs":[],"model":"deepseek-v4-flash","headline":"A rigid object M encodes its finitely presented objects as 2-term complexes over End(M), matching cluster-tilting with silting.","keywords":["rigid objects","triangulated categories","2-term complexes","silting objects","cluster-tilting objects","extriangulated categories","self-injective algebras","quivers with potential"],"falsifier":"Find a finite-dimensional algebra A over a field such that H^{[-1,0]}(proj A) admits a triangulated structure while A is not self-injective; the appendix proves this cannot happen, so a single such example would refute the converse half. A second decisive test: in an algebraic triangulated T with rigid M and Hom(M,Σ^{-1}M)=0, exhibit a morphism in the kernel of P that is not homotopic to zero, contradicting the claimed equivalence criterion.","tokens_in":28619,"feed_emoji":"🔄","tokens_out":10292,"duration_ms":105523,"temperature":0.7,"pith_summary":"This paper sets up a direct dictionary between objects built from a rigid object M in an algebraic triangulated category and two-term complexes of projective modules over the endomorphism algebra A of M. The dictionary is a functor P that sends an object X, presented by a triangle M^{-1}→M^0→X→ΣM^{-1}, to the complex M^{-1}→M^0; the hard part is defining P on morphisms by lifting them to presentations. P is proved full and dense, with kernel precisely the morphisms factoring through some ΣM_1→M_2, so it becomes an equivalence exactly when Hom(M,Σ^{-1}M)=0; it also detects indecomposability and extriangles. In the Hom-finite case, P matches relative cluster-tilting objects of pr(M) with two-term silting complexes of H^b(proj A), and this bijection commutes with mutations. In the 2-Calabi-Yau cluster-tilting case, the equivalence is governed by self-injectivity of A, and the two-term projective homotopy category carries a triangle structure precisely in that case.","feed_headline":"Rigid objects become 2-term complexes, matching tilting to silting","feed_subtitle":"A mutation-commuting bijection links cluster-tilting objects with 2-term silting complexes.","key_machinery":"The load-bearing object is the presentation functor P and its lifting lemma. For X in pr(M), the paper fixes a triangle M^{-1}→M^0→X→ΣM^{-1} and sets P(X)=(M^{-1}→M^0); for a morphism f:X→Y, it defines P(f) as the homotopy class of a liftable pair of maps between the chosen presentations. 'Liftable' is defined through a Frobenius model of the algebraic triangulated category: every map in T is represented by a map in the model, and the Lifting Lemma 3.1 lifts such maps to levelwise maps between the M-terms. The kernel ideal I, consisting of morphisms factoring through some ΣM_1→M_2, is what must be quotiented out, and the relation I²=0 is what makes P detect isomorphisms and indecomposability","core_discovery":"The central claim is Theorem 1.1: for a rigid object M in an algebraic triangulated category T and A=End_T(M), the presentation functor P:pr(M)→H^{[-1,0]}(proj A) is full and dense, and two morphisms have the same image exactly when their difference factors through a morphism of the form ΣM_1→M_2 with M_1,M_2 in add(M). Hence P induces an equivalence pr(M)/I ≅ H^{[-1,0]}(proj A), and P itself is an equivalence iff Hom_T(M,Σ^{-1}M)=0. P detects isomorphisms, indecomposability, and extriangles, and in the Hom-finite Krull-Schmidt case it induces a bijection between basic relative cluster-tilting objects and basic two-term silting objects that commutes with mutation. When T is 2-Calabi-Yau and","pith_inferences":["The functor P gives a homological reason for the known correspondence between support τ-tilting pairs and two-term silting objects: the A-module Hom(M,–) is roughly the 0th cohomology of P(–), so the complex remembers extension data the module forgets.","Since the paper notes that a morphic enhancement of T could play the role of the Frobenius model, the main theorem likely extends to non-algebraic triangulated categories with such an enhancement; the proof strategy is not intrinsically about algebraicity.","The appendix's twisted 4-periodicity suggests that any triangulated structure on H^{[-1,0]}(proj A) forces the algebra to be very special; a natural test is whether that structure is unique and whether it always arises from a 4-angulated structure on proj A.","The mutation-commuting bijection offers an algorithmic route: compute cluster-tilting mutations in the two-term silting category, where complexes and their endomorphism algebras are more explicit, then transfer the result back to the cluster category."],"forward_implications":["Relative cluster-tilting objects in pr(M) and two-term silting objects in H^b(proj A) form the same combinatorial object: the bijection preserves isomorphism classes and mutation.","Because P detects extriangles, the quotient pr(M)/I is extriangulated equivalent to H^{[-1,0]}(proj A), not just additively equivalent.","In the 2-Calabi-Yau cluster-tilting case with A self-injective, the cluster category is additively encoded by two-term complexes, so cluster-tilting objects with the same endomorphism algebra give triangle-equivalent categories under mild hypotheses.","For self-injective quivers with potential, endomorphism algebras of iterated silting mutations in the two-term category are exactly the Jacobian algebras of the mutated quivers with potential, recovering the known theorem on such mutations.","A triangle structure on H^{[-1,0]}(proj A) forces A to be self-injective (and, under separability, twisted 4-periodic), so such structures are rare and come with strong periodicity constraints."],"supporting_citations":[{"why":"gives the criterion Hom_T(M,Σ^{-1}M)=0 iff A is self-injective, used to turn the equivalence criterion into a self-injectivity statement.","marker":"[26]"},{"why":"supplies the extriangulated structure on pr(M) via its Lemma 4.57, framing the extriangle detection results.","marker":"[42]"},{"why":"defines relative cluster-tilting objects and mutations in triangulated categories, the objects whose classification is transferred by P.","marker":"[47]"},{"why":"contains the earlier analogous statements for silting objects that Theorem 1.1 generalizes and completes.","marker":"[8]"},{"why":"provides the notion of silting mutation used to state that the bijection commutes with mutations.","marker":"[2]"},{"why":"supplies the cluster category of a quiver with potential and the mutation behaviour needed for the application section.","marker":"[35]"},{"why":"states the theorem on endomorphism algebras of silting mutations that the paper recovers as an application.","marker":"[40]"}],"fun_headline_variants":["Rigid objects to 2-term complexes: a tilting-silting bijection","Presentation functor links rigid objects and 2-term silting","From rigid objects to silting: full and dense, with bijection","Rigid objects yield 2-term complexes and a mutation-preserving map","Cluster-tilting to silting: the rigid-object presentation bridge"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The construction assumes T is an algebraic, idempotent-complete triangulated category with a Frobenius model in which 'liftable presentations' can be defined; without such a model, the functor P is not constructed here.","fun_headline_variants_meta":{"raw":{"variants":["Rigid objects to 2-term complexes: a tilting-silting bijection","Presentation functor links rigid objects and 2-term silting","From rigid objects to silting: full and dense, with bijection","Rigid objects yield 2-term complexes and a mutation-preserving map","Cluster-tilting to silting: the rigid-object presentation bridge"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001227,"raw_usage":{"total_tokens":4988,"prompt_tokens":958,"completion_tokens":4030,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":702,"completion_tokens_details":{"reasoning_tokens":3932}},"tokens_in":702,"tokens_out":4030,"duration_ms":35602,"temperature":1.0,"reasoning_tokens":3932,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T20:57:25.696780+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a finite-dimensional algebra A over a field such that H^{[-1,0]}(proj A) admits a triangulated structure while A is not self-injective; the appendix proves this cannot happen, so a single such example would refute the converse half. A second decisive test: in an algebraic triangulated T with rigid M and Hom(M,Σ^{-1}M)=0, exhibit a morphism in the kernel of P that is not homotopic to zero, contradicting the claimed equivalence criterion.","supporting_citations":[],"review_version":1}