{"id":"8164ae17-1b3c-42d0-b086-e75e5f621f17","arxiv_id":"2509.01056","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For left artinian rings with a suitable semisimple subring, the singularity category is triangle equivalent to the stable module category of the zeroth component of the Leavitt ring, which is an FC ring.","lead":"This paper proves that the singularity category of an artinian ring can be described as the stable module category over a ring built from relative noncommutative differential 1-forms. The result connects two families of algebraic categories and yields new non-quasi-Frobenius FC rings.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.3's identification of the Leavitt ring with the orbit ring is outsourced to [14, Thm 2.6]; if that colimit presentation has hidden hypotheses or is inapplicable to Ω^nc, the central equivalence D_sg(Λ) ≃ L0-mod breaks.","rationale":"The central claim depends on a chain: D_sg(Λ) ≃ S(Λ-mod, Ω^nc⊗−) (external, [9,31,6]); S ≃ L_Λ(Ω^nc)-grmod (Prop 4.3 + Prop 2.2); strong gradation (Lemma 3.8) yields L0-mod; Frobenius/FC via Lemma 5.1. I checked the internal steps: Lemma 3.8's stabilization argument is sound; Prop 5.2's injectivity argument is sound; the colimit computation in Prop 4.3 is plausible and compatible with the orbit-ring multiplication. The weakest point is exactly the citation of [14, Thm 2.6] for the colimit presentation of the Leavitt ring. This is not a flaw in the present argument's logic but a load-bearing external dependency, and it is self-cited. The reader's weakest_assumption identifies this same step; my stress test agrees. I do not see an internal inconsistency that would warrant REJECT. A conditional accept is appropriate until that theorem is reproduced or its hypotheses are checked for Ω^nc. Hence verdict unchanged.","tokens_in":18216,"tokens_out":31902,"duration_ms":401754,"concrete_test":"Independently re-derive Prop 4.3: define a graded ring structure on C = colim_p (M^*)^⊗p ⊗_R T_R(M) with transition maps Δ_{p,k}=id⊗c⊗id, and prove that the canonical map T_R(M^*⊕M)/(x⊗f−f(x), c−1) → C is a graded ring isomorphism for every ring R and every R-R-bimodule M with _RM finitely generated projective. If the derivation fails (e.g., the relation c=1 or x f=f(x) is not satisfied in C), Prop 4.3 and hence Theorem 5.7 lack proof; if it succeeds, the external dependency is discharged.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 4.3 is the key bridge: it identifies L_R(M) with the orbit ring Γ(S(R);Σ), and Theorem 5.7 builds strong gradedness, FC-ness, and the triangle equivalence on it. The proof, however, does not establish this identification from the defining relations of the Leavitt ring. It cites [14, Theorem 2.6] for the assertion that L_R(M) ≅ colim_p (M^*)^⊗p ⊗_R T_R(M) with transitions id⊗c⊗id. The present text neither states the precise hypotheses of [14, Thm 2.6] nor checks that they hold for arbitrary R-R-bimodules M with only _RM finitely generated projective, nor verifies that the colimit inherits a ring structure making the Cuntz-Krieger relations (x f = f(x), c=1) true on the nose. Since [14] is self-cited, the load-bearing step is not independently confirmed here. If [14, Thm 2.6] were false, or if it required additional hypotheses (e.g., right projectivity of M or a dg setting) not satisfied by Ω^nc, then Prop 4.3 would fail; consequently L_Λ(Ω^nc) would not be known to be strongly graded, L0-mod would not be known Frobenius, and Theorem 5.7 would not be established. No internal contradiction is apparent, but the central claim is conditional on an unverified external theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies, for a left artinian ring Λ with a semisimple subring E such that _EΛ is finitely generated, the stabilization S of Λ-mod by formally inverting the tensor endofunctor Ω^{nc}⊗_Λ−, where Ω^{nc}=Ω^{nc}_{Λ/E} is the bimodule of E-relative noncommutative 1-forms. Building on the authors' earlier Leavitt-ring machinery, the paper proves that S is Frobenius abelian and equivalent to L_0-mod, where L_0 is the degree-zero component of the Leavitt ring L_Λ(Ω^{nc}); it follows that L_0 is an FC ring. The main theorem (Theorem 5.7) then gives a triangle equivalence D_sg(Λ) ≃ L_0-mod. The route is: general stabilization results yielding an orbit-ring description (Sections 2–3), an identification of the Leavitt ring with the orbit ring (Prop. 4.3), and a combination with the Buchweitz–Keller–Vossieck description of the singularity category (Thm. 5.4).","tokens_in":18635,"tokens_out":36854,"duration_ms":454301,"significance":"If correct, the main theorem provides a broad and conceptually unified description: the singularity category of any artinian ring (under the stated hypotheses) is a stable module category over an FC ring. This goes beyond the previously known encounters with Leavitt path algebras and gives a new source of non-quasi-Frobenius FC rings. The orbit-ring/stabilization formalism is elegant and likely to be useful beyond the present application. The paper is well organized and the deductions after the key identifications are mostly transparent. The central risk is the reliance on the colimit presentation of the Leavitt ring cited from the authors' earlier work [14, Thm. 2.6]; that step is load-bearing and is not independently verified in the present text.","major_comments":[{"comment":"The proof of Proposition 4.3 is the key bridge: it identifies L_R(M) with the orbit ring Γ(S(R);Σ), and Theorem 5.7 builds strong gradedness, FC-ness, and the triangle equivalence on this identification. However, the proof does not establish this identification directly from the defining relations of the Leavitt ring. It cites [14, Theorem 2.6] for the assertion that L_R(M) is isomorphic to colim_p (M^*)⊗p ⊗_R T_R(M) with transitions id⊗c⊗id, and then shows the multiplication matches. The present text neither states the precise hypotheses of [14, Thm. 2.6] nor verifies that they hold for an arbitrary R-R-bimodule M with only _RM finitely generated projective. Since [14] is self-cited and this theorem is load-bearing for the main claim, I ask the authors to either prove the colimit presentation from the defining relations of the Leavitt ring, or state [14, Thm. 2.6] explicitly and check i","section":"Section 4, Proposition 4.3"},{"comment":"The proof of (2)⇒(3) invokes 'the dual of Lemma 2.3 and its proof', but Lemma 2.3 concerns the equivalence C(P,−) for orbit rings and does not directly yield the contravariant equivalence Hom_R(−,R): R-mod → (R^op-mod)^op. Since Lemma 5.1 is used in Theorem 5.7 to pass from 'L_0-mod is Frobenius abelian' to 'L_0 is an FC ring', this step needs a correct proof or a precise standard reference. The statement is standard, but the proof as written is not sufficient.","section":"Section 5, Lemma 5.1"}],"minor_comments":[{"comment":"The proof says 'The same argument in Lemma 2.3 shows...' but Lemma 2.3 appears later; it should presumably refer to Lemma 2.1.","section":"Section 2, Proposition 2.2"},{"comment":"The proof of Proposition 6.1 is omitted ('We omit the details'). Since this is a nontrivial explicit description of the Leavitt ring, please include at least a sketch of the isomorphism, or state the computation as a known/straightforward consequence with a reference.","section":"Section 6, Proposition 6.1"},{"comment":"The choice of E is not specified. Since the paper's setup allows any semisimple subring E, please clarify that E is taken to be the field K (or state the intended choice). Also, the claimed infinite strictly ascending chain of subobjects of (Λ,0) is asserted without construction; a few sentences indicating the inductive step would make the non-noetherian claim convincing.","section":"Section 5, Example 5.3"},{"comment":"Please fix typographical issues: 'calld' for 'called' (Section 5), the double colon in the proof of Proposition 4.3, the missing exponent 'for some l ≥.' in Lemma 3.8, the garbled title of reference [10] (appears as 'catˇ sˇSgories dˇ sˇSrivˇ sˇSes'), and the header typo 'CA TEGORY' / 'ST ABLE'.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The key external theorem [14, Thm. 2.6] is from the authors' own previous work and is not reproved or stated in this manuscript. Given that Proposition 4.3 is the central bridge to the main theorem, I would recommend that the editor require the authors to provide a self-contained statement and verification of the hypotheses of that theorem, or a direct proof. This is not a rejection of the mathematics, but a request to make the load-bearing step accessible to the reader."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does what the title says: for a left artinian ring Λ containing a semisimple subring E with Λ finitely generated over E, it proves that the singularity category D_sg(Λ) is triangle equivalent to the stable module category over the zeroth component L0 of the Leavitt ring L_Λ(Ω^nc). That is a genuine structural result, and it yields new FC rings that are not quasi-Frobenius. I did not find a load-bearing flaw.\n\nThe genuinely new piece is Proposition 4.3, which identifies the Leavitt ring with an orbit ring coming from the stabilization of the module category, and the resulting Theorem 5.7. The proof chain is sound: Lemma 3.8 gives strong gradedness from the descending chain condition, Prop 4.3 transfers it to L, and Prop 5.5 plus the Buchweitz–Keller–Vossieck equivalence yield the stable-category identification. The authors also honestly flag where the restriction to finitely presented modules matters (Remarks 4.6, 4.7).\n\nThe main thing the stress-test flags is that Prop 4.3 cites [14, Thm 2.6] for the colimit presentation of the Leavitt ring. This is a self-citation, but it is to a published Advances in Mathematics paper, and the hypotheses match the stated setting (R-R-bimodule with RM finitely generated projective). The stress-test worry does not land: a published theorem does not need to be reproved here, and the rest of Prop 4.3 checks the ring/composition compatibility directly. If that theorem later turned out to be false, the paper would break, but that is a generic dependency, not a gap.\n\nThe genuine soft spots are minor. Proposition 6.1's proof is basically omitted (\"We omit the details\"), and Example 5.3's infinite ascending chain is asserted rather than demonstrated. The radical-square-zero example depends on a case-by-case description that is a bit too terse. None of this affects the main equivalence.\n\nWho should read this: anyone working on singularity categories, Leavitt path algebras, or FC rings. It connects two active areas and produces a new family of non-QF FC rings. It should go to peer review, and I would expect acceptance after minor revisions. I would bring it to a reading group and would likely cite it.","headline":"Solid, genuinely new structural result; the main theorem holds up, with only minor presentation gaps.","tokens_in":19074,"tokens_out":2257,"would_cite":true,"duration_ms":25667,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16S88","18G80","13D02"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the singularity category of any left artinian ring is triangle equivalent to the stable module category over the degree-zero part of a Leavitt ring, and that this part is an FC ring.","keywords":["singularity category","stable module category","Leavitt ring","FC ring","Frobenius abelian category","stabilization","noncommutative differential forms","artinian ring"],"falsifier":"Take R = K a field and M = K^2, so the Leavitt ring is the Leavitt path algebra of the one-vertex two-loop quiver. Compute the degree-zero component directly on both sides of Proposition 4.3: the colimit of M_{2^p}(K) on the orbit-ring side, and the degree-zero part of the Leavitt path algebra on the other. If the two rings are not isomorphic, the identification of the Leavitt ring with the orbit ring fails, and the main equivalence of Theorem 5.7 cannot hold.","tokens_in":18148,"feed_emoji":"🧩","tokens_out":14267,"duration_ms":145758,"temperature":0.7,"pith_summary":"This paper proves a structural theorem about singularity categories, the derived-category quotients that vanish precisely for rings of finite global dimension and therefore measure homological singularity. For any left artinian ring Λ with a suitable semisimple subring E, the paper forms the bimodule Ω_nc of E-relative noncommutative differential 1-forms, whose tensor functor acts as a syzygy functor on Λ-mod. Stabilizing the module category by formally inverting this functor yields a Frobenius abelian category S, which the paper identifies with the category of finitely presented modules over the zeroth component L_0 of a Leavitt ring. The payoff is a triangle equivalence between the singularity category of Λ and the stable module category over L_0, and the corollary that L_0 is an FC ring—usually not quasi-Frobenius—whose von Neumann regularity is exactly equivalent to Λ having finite global dimension. This recasts a subtle triangulated invariant as an ordinary module category over an explicitly constructed ring.","feed_headline":"Singularity categories become stable module categories of FC rings","feed_subtitle":"For each artinian ring, its singularity category equals the stable module category of an explicitly built FC ring.","key_machinery":"Central object: the stabilization S(Λ-mod, Ω_nc ⊗_Λ −), formed by formally inverting the tensor endofunctor of the relative 1-form bimodule. The load-bearing identity is Proposition 4.3: the Leavitt ring L_R(M) is isomorphic to the orbit ring Γ(S(R);Σ) built from the Σ-progenerator S(R)=(R,0) in the stabilization. Strong gradedness of this orbit ring — equivalent to add(R)=add(ΣR), forced by artinian stabilization of add(Ω_nc^{⊗n}⊗Λ) — collapses the graded Leavitt module category to L_0-mod. The Frobenius property of the stabilization then yields the triangulated structure and the FC property of L_0.","core_discovery":"The main result (Theorem 5.7): for a left artinian ring Λ with a semisimple subring E over which Λ is finitely generated, with Ω_nc the bimodule of E-relative noncommutative 1-forms, the Leavitt ring L_Λ(Ω_nc) is strongly graded; its zeroth component L_0 is an FC ring; and D_sg(Λ) is triangle equivalent to the stable module category L_0-mod. The key identity is Proposition 4.3: the Leavitt ring of any bimodule M over a ring R is isomorphic to the orbit ring of the stabilization of R-mod by M ⊗_R −, taken at the generator R. The stabilization using Ω_nc is Frobenius abelian, which transfers a Frobenius module-category structure to L_0-mod and forces L_0 to be FC. Corollary 5.9: Λ has finite g","pith_inferences":["The identification in Proposition 4.3 is not restricted to the 1-form setting, so the same orbit-ring machinery could be applied to other loop functors whose descending chains stabilize; for non-artinian rings the chain need not stabilize, and the method would then yield only a graded equivalence, not one against an ungraded module category.","The construction depends on a choice of the semisimple subring E; different choices may give different FC rings L_0, and whether those rings are Morita equivalent is not discussed in the paper and would clarify how canonical the output is.","Section 6's description of L_0 as a trivial extension of a von Neumann regular ring suggests that a better understanding of the bimodule V_0 in that presentation would make the FC-ring structure computable in wider classes of examples, such as monomial or gentle algebras.","Because FC rings are coherent analogues of quasi-Frobenius rings, the theorem indicates that every singularity category of an artinian ring sits inside the world of coherent rings; testing whether invariants of L_0 such as its flat or coherent dimensions match known invariants of Λ could give new derived invariants."],"forward_implications":["The singularity category D_sg(Λ) of every artinian ring is triangle equivalent to the stable module category of the explicitly constructed FC ring L_0.","L_0 is an FC ring that is usually not noetherian, hence not quasi-Frobenius; the construction yields a new family of FC rings.","Λ has finite global dimension if and only if L_0 is von Neumann regular; the failure of regularity of L_0 is exactly the homological singularity of Λ.","The stabilization S is a Frobenius abelian category, giving an explicit Frobenius enhancement of D_sg(Λ).","For radical-square-zero algebras, L_0 is isomorphic to a trivial extension of a von Neumann regular ring, making the FC ring in that case concrete."],"supporting_citations":[{"why":"Provides the definition of the Leavitt ring and the colimit presentation (its Theorem 2.6) that Proposition 4.3 uses to identify it with the orbit ring.","marker":"[14]"},{"why":"Supplies the singularity category and the theorem that the stabilization of the module category by the syzygy endofunctor is triangle equivalent to it.","marker":"[9]"},{"why":"Extends that stabilization–singularity equivalence to general (not necessarily Gorenstein) artinian rings, used in Theorem 5.4.","marker":"[31]"},{"why":"Gives the detailed proof of the stabilization equivalence and the left-triangulated structure that induces the triangulation of the stabilization.","marker":"[6]"},{"why":"Defines FC rings and provides the criterion that a ring is FC exactly when its finitely presented module category is Frobenius abelian, used to conclude L_0 is FC.","marker":"[19]"},{"why":"Introduces the bimodule of relative noncommutative differential 1-forms and proves it is projective on both sides, making the syzygy identification valid.","marker":"[17]"},{"why":"Establishes that the stable category of any Frobenius exact category is canonically triangulated, which makes D_sg(Λ) triangulated via the Frobenius enhancement.","marker":"[22]"},{"why":"Supplies the theory of strongly graded rings and the equivalence between Γ-grmod and Γ_0-mod that collapses the graded Leavitt module category to L_0-mod.","marker":"[18]"},{"why":"Introduces the orbit ring construction used to build the graded ring Γ(P;Σ) in Proposition 2.2 and to identify it with the Leavitt ring.","marker":"[33]"}],"fun_headline_variants":["FC rings encode singularity categories of artinian rings","Singularity category of artinian ring equals stable modules of FC ring","Leavitt ring links singularity categories to stable modules","Artinian singularity categories become stable module categories"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The chain of equivalences rests on the cited theorem that the Leavitt ring is the colimit of the tensor powers of the dual bimodule; if that colimit presentation were false, the identification in Proposition 4.3 would break and the main equivalence would not follow.","fun_headline_variants_meta":{"raw":{"variants":["FC rings encode singularity categories of artinian rings","Singularity category of artinian ring equals stable modules of FC ring","Leavitt ring links singularity categories to stable modules","Artinian singularity categories become stable module categories"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0002,"raw_usage":{"total_tokens":1190,"prompt_tokens":700,"completion_tokens":490,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":444,"completion_tokens_details":{"reasoning_tokens":426}},"tokens_in":444,"tokens_out":490,"duration_ms":5898,"temperature":1.0,"reasoning_tokens":426,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T12:59:11.350391+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take R = K a field and M = K^2, so the Leavitt ring is the Leavitt path algebra of the one-vertex two-loop quiver. Compute the degree-zero component directly on both sides of Proposition 4.3: the colimit of M_{2^p}(K) on the orbit-ring side, and the degree-zero part of the Leavitt path algebra on the other. If the two rings are not isomorphic, the identification of the Leavitt ring with the orbit ring fails, and the main equivalence of Theorem 5.7 cannot hold.","supporting_citations":[{"cited_title":"Chen, and Z","cited_arxiv_id":null,"evidence_quote":"Provides the definition of the Leavitt ring and the colimit presentation (its Theorem 2.6) that Proposition 4.3 uses to identify it with the orbit ring."},{"cited_title":"Buchweitz , Maximal Cohen-Macaulay Modules and Tate-cohomology over Goren- stein Rings, with appendices by L.L","cited_arxiv_id":null,"evidence_quote":"Supplies the singularity category and the theorem that the stabilization of the module category by the syzygy endofunctor is triangle equivalent to it."},{"cited_title":"Keller, and D","cited_arxiv_id":null,"evidence_quote":"Extends that stabilization–singularity equivalence to general (not necessarily Gorenstein) artinian rings, used in Theorem 5.4."},{"cited_title":"Beligiannis, The homological theory of contravariantly finite subcatego ries: Auslander- Buchweitz contexts, Gorenstein categories and (co-)stabi lization, Comm","cited_arxiv_id":null,"evidence_quote":"Gives the detailed proof of the stabilization equivalence and the left-triangulated structure that induces the triangulation of the stabilization."},{"cited_title":"Damiano , Coflat rings and modules , Pacific J","cited_arxiv_id":null,"evidence_quote":"Defines FC rings and provides the criterion that a ring is FC exactly when its finitely presented module category is Frobenius abelian, used to conclude L_0 is FC."},{"cited_title":"Cuntz, and D","cited_arxiv_id":null,"evidence_quote":"Introduces the bimodule of relative noncommutative differential 1-forms and proves it is projective on both sides, making the syzygy identification valid."},{"cited_title":"Happel , Triangulated Categories in the Representation Theory of F inite Dimensional Algebras, London Math","cited_arxiv_id":null,"evidence_quote":"Establishes that the stable category of any Frobenius exact category is canonically triangulated, which makes D_sg(Λ) triangulated via the Frobenius enhancement."},{"cited_title":"Dade , Group-graded rings and modules , Math","cited_arxiv_id":null,"evidence_quote":"Supplies the theory of strongly graded rings and the equivalence between Γ-grmod and Γ_0-mod that collapses the graded Leavitt module category to L_0-mod."},{"cited_title":"Lenzing , Wild canonical algebras and rings of automorphic forms , in: Finite- Dimensional Algebras and Related Topics, NATO ASI Ser","cited_arxiv_id":null,"evidence_quote":"Introduces the orbit ring construction used to build the graded ring Γ(P;Σ) in Proposition 2.2 and to identify it with the Leavitt ring."}],"review_version":1}