{"id":"b98e874f-4bde-4ba9-8db8-7d293bb9d921","arxiv_id":"2412.19040","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Tilting objects with higher representation infinite endomorphism rings and strict root pairs are constructed for two families of toric singularities, giving cluster equivalences and explicit Calabi-Yau algebras.","lead":"For Veronese subrings and Segre products of polynomial rings, the paper constructs tilting objects in singularity categories whose endomorphism rings are higher representation infinite, yielding equivalences with folded cluster categories and explicit quiver presentations. The proof of the Segre product part rests on a reduction theorem whose stated assumptions appear impossible, calling the second main theorem into question.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption 2.2(2) is untenable: a d-representation infinite algebra over a field has inj.dim_A A = d, so Theorem 2.3 has an empty hypothesis and the proof of Theorem 5.1 collapses. The Segre-product results are unsupported as written; the Veronese proof is independent.","rationale":"The reader identified the same load-bearing concern, and the concern is correct. The paper's own Definition 2.1, together with standard bounds on injective dimension over algebras of finite global dimension, implies that for every d-representation infinite algebra A with d≥1, inj.dim_A A equals d, not less than d. Hence Assumption 2.2(2) is never satisfied in the setting of Theorem 2.3, making that theorem vacuous and unusable for the applications in Section 5. The proof of Theorem 5.1 explicitly relies on Theorem 2.3 to reduce the n-representation infinite algebra B = Kronecker^{⊗n} to an (n−2)-representation infinite quotient; since the hypothesis cannot hold, the reduction is invalid and the Segre-product results collapse. This is not a matter of differing convention or a fixable typo in a bound: the strict inequality is essential to the proof's induction step, so the argument cannot be patched by changing '< d' to '≤ d'. The Veronese part, Theorem 4.1, is proven directly and does not use Theorem 2.3 except in a remark for the special case a=1, so that portion of the paper appears substantially more secure. However, the abstract and the paper's advertised central claims cover both families of singularities, and the Segre-product side is unsupported as written. A revision could potentially rescue the paper by proving the needed idempotent-quotient reduction under a different, satisfiable hypothesis or by giving a direct proof for the Segre algebras, but the current manuscript cannot be accepted in this form. My assessment therefore agrees with the reader's REJECT verdict.","tokens_in":28714,"tokens_out":14973,"duration_ms":140965,"concrete_test":"Take B = (Kronecker)^{⊗3} (or any d-Beilinson algebra) and compute the injective dimension of the regular module, e.g. by showing Ext^3_B(DB,B)≠0. If, as standard theory predicts, inj.dim_B B=3, then Assumption 2.2(2) fails exactly where Theorem 5.1 invokes Theorem 2.3, and the proof of the Segre-product claim cannot proceed as written. For an even smaller check, verify Example 2.9 with d=1: for A the Kronecker algebra, Ext^1_A(DA,A)≠0, so inj.dim_A A=1, contradicting Assumption 2.2(2).","verdict_should_be":"REJECT","load_bearing_attack":"The reader's objection lands. For a d-representation infinite algebra A (d≥1), Definition 2.1 forces gl.dim A = d: the module ν^{-1}_d A = RHom_A(DA,A)[d] is nonzero, since otherwise Hom(-,A) would vanish on the injective cogenerator DA and hence on all modules. Thus Ext^d_A(DA,A)≠0. Since gl.dim A=d, the regular module satisfies inj.dim_A A≤d, and the nonzero Ext^d(DA,A) gives inj.dim_A A≥d; hence equality. Therefore Assumption 2.2(2), 'inj.dim_A A < d', cannot hold for any d-representation infinite algebra. This hypothesis is load-bearing in the proof of Theorem 2.3: it forces ν^{-1}_d(ν^{-i}_{d-1}A) to sit in degree ≤−1, producing the exact sequence that drives the induction; with inj.dim_A A=d that step fails. Example 2.9, which claims the assumption holds for the d-Beilinson algebra, is consequently false. In §5.2, B = Kronecker^{⊗n} is n-representation infinite, so inj.dim_B B = n, and the two applications of Theorems 2.3 and 2.12 in the proof of Theorem 5.1 are invalid. Thus Theorem 5.1(2), Theorem 5.2(3)–(4), and the Segre-product part of the abstract are not established. The Veronese Theorem 4.1 has a self-contained proof and does not depend on this reduction, so that half of the paper may survive a repair.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies graded and ungraded singularity categories of two classes of commutative Gorenstein toric singularities: Veronese subrings of polynomial rings and Segre products of copies of k[x,y]. For Veronese subrings the author constructs an explicit tilting object T whose endomorphism ring is claimed to be higher representation infinite and equipped with a strict root pair, yielding equivalences with folded cluster categories and explicit quiver presentations of the associated Calabi-Yau completions. For Segre products the author constructs a tilting object ⊕_{0<v<1} M_v, claims its endomorphism ring is (n−2)-representation infinite, and, for n=3, gives a hereditary endomorphism ring with a strict root pair and an alternating quiver presentation. The core technical tool is a reduction theorem (Theorem 2.3) asserting that a certain idempotent quotient of a d-representation infinite algebra is (d−1)-representation infinite. The Veronese arguments appear detailed and internally coherent, but the Segre-product proof depends on an assumption that is impossible for d-representation infinite algebras, and that part of the paper is not established as written.","tokens_in":29046,"tokens_out":14478,"duration_ms":152855,"significance":"If the Veronese results are correct, they provide explicit, noncommutative-resolution-style descriptions of graded singularity categories, with tilting objects whose endomorphism rings are higher representation infinite and whose root pairs are strict; the quiver-and-relations descriptions of the twisted Calabi-Yau algebras are concrete and valuable. The proposed reduction theorem for idempotent quotients would also be of independent interest if a correct hypothesis could be supplied. The Segre-product claims, however, rest on Theorem 2.3, whose Assumption 2.2(2) is inconsistent with the definition of higher representation infiniteness. Since the abstract and Section 5 present the Segre results as a main contribution, the paper in its current form cannot be accepted, although the Veronese half may survive a substantial repair.","major_comments":[{"comment":"Assumption 2.2(2), \"inj.dim_A A < d\", cannot hold for any d-representation infinite algebra A with d≥1. By Definition 2.1, ν_d^{-1}A = RHom_A(DA,A)[d] lies in mod A; it is nonzero because Hom_A(DA,A)≠0 (A is an injective cogenerator), so Ext^d_A(DA,A)≠0. Hence inj.dim_A A ≥ d, while gl.dim A ≤ d gives inj.dim_A A ≤ d, forcing equality. Thus the hypothesis of Theorem 2.3 is empty, and the proof's key step — that ν_d^{-1}(ν^{-i}_{d-1}A) has cohomology only in degree ≤−1 — uses exactly the false inequality. Consequently Example 2.9's assertion that the d-Beilinson algebra satisfies Assumption 2.2(2) is false. The theorem must be replaced or repaired with a viable hypothesis; the surrounding argument suggests the author may have intended an inequality involving the quotient A/(e), but that corrected statement and proof are not what appears.","section":"Section 2, Assumption 2.2(2) and Theorem 2.3"},{"comment":"The proof that End^Z_{sg R}(T) is (n−2)-representation infinite applies Theorem 2.3 to B = (Kronecker)^{⊗n} and then Theorem 2.12 to B/(e0). But B is n-representation infinite, so by the argument in the previous comment inj.dim_B B = n, not < n; Theorem 2.3 therefore cannot be applied to (B,e0). Similarly, B/(e0), if it is (n−1)-representation infinite, satisfies proj.dim_{B/(e0)}D(B/(e0)) = n−1, not < n−1, so the dual Theorem 2.12 cannot be applied either. The claim that these applications are valid is the load-bearing step for Theorem 5.1(2), Theorem 5.2(3)–(4), and the corresponding statements in the abstract. The tilting-object part of Theorem 5.1 and the computations in Theorem 5.2(1)–(2) do not depend on Section 2, but the higher-representation-infinite and strict-root-pair conclusions are unsupported as written.","section":"Section 5.2, proof of Theorem 5.1 and Theorem 5.2"}],"minor_comments":[{"comment":"There are numerous typographical slips, e.g. \"monimial\" in Theorem 1.1, \"in genral\" and \"exsistence\" in the introduction, \"oridnary\" in Section 3, \"In particuar\" in Section 5.3, and \"potention\" in reference [1]. These should be corrected.","section":"Throughout"},{"comment":"The abstract and Theorem 4.1(2) refer to a strict root pair of ν^{-1}_{d-a-1}, while the theorem statement says \"for ν_{d-a-1}\"; the notation should be made consistent, since the root pair is for the inverse shifted Serre functor.","section":"Abstract vs. Theorem 4.1(2)"},{"comment":"The sentence \"It is easy to see that the injective dimension of the simple B-module corresponding to the summand M_v is n−∑ v_i\" should be expanded into a proof and, more importantly, connected explicitly to whatever corrected version of Assumption 2.2(2) is used after Theorem 2.3 is repaired.","section":"Section 5.2"}],"recommendation":"major_revision","confidential_remarks":"The Veronese part of the paper is substantial and appears to be developed carefully; if the author can supply a corrected reduction theorem (or prove the Segre endomorphism-ring claims by a different method), the paper may become publishable. As it stands, however, a central assumption of Section 2 is impossible, and the entire Segre-product half of the paper rests on that assumption. I would not recommend rejection if the author can repair this, but the current manuscript is not correct in its stated form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper has a strong Veronese half and a Segre half that currently does not hold together. The new tilting objects with strict root pairs are a genuine advance over the earlier tilting objects in [28] and the cluster tilting objects in [23]; the explicit quiver and relations for Γ in Theorem 4.12 is a concrete payoff. The reduction theorem (2.3) would be independently useful if its hypotheses were right.\n\nThe problem is Assumption 2.2(2), inj.dim_A A < d. For a d-representation infinite algebra over a field, gl.dim A = d and ν^{-1}_d A = RHom_A(DA,A)[d] is nonzero, so Ext^d_A(DA,A) ≠ 0. That gives inj.dim_A A ≥ d, hence equality. The strict inequality is impossible. Example 2.9, which claims the assumption holds for the d-Beilinson algebra, is false. The proof of 2.3 uses the strict inequality exactly to force the complex ν^{-1}_d(ν^{-i}_{d-1}A) to sit in degree -1; without it the induction step fails. Since the proof of Theorem 5.1 invokes 2.3 twice for B = Kronecker^{⊗n}, which is n-representation infinite and has inj.dim_B B = n, the Segre product theorem is not proven as written. That includes Theorem 5.2 and the abstract's claim about Segre products.\n\nThe Veronese part is independent of 2.3: Theorem 4.1 is proved directly, and Remark 4.7 only mentions 2.3 as an alternative for a=1. I did not verify every Ext computation, but the lemmas in Section 4 are detailed and the argument is coherent. So that half looks like it should survive a repair.\n\nFor the reader: this paper is for singularity-category people and higher representation theorists. If the reduction theorem can be corrected, the Segre results are likely recoverable and the paper would be solid. As is, the main Segre theorem is unsupported. I would still send this to a serious referee: the error is concrete, localized, and fixable, and the Veronese portion is substantial enough to deserve referee time. I would not desk reject.","headline":"Good Veronese half, broken Segre half: the reduction theorem's key hypothesis is impossible, so the Segre results fail as written, but the paper deserves a careful referee.","tokens_in":29600,"tokens_out":3163,"would_cite":false,"duration_ms":119084,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13C14","16E35","16S38","13D02","16G10","14A22","16G60"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs tilting objects in the graded singularity categories of two families of toric singularities, with endomorphism rings that are higher representation infinite and carry strict root pairs of the shifted Serre functor.","keywords":["Veronese subring","Segre product","n-representation infinite algebra","singularity category","cluster category","strict root pair","twisted Calabi-Yau algebra","toric singularity"],"falsifier":"Compute the self-injective dimension of the standard $d$-representation infinite algebra of the paper's Example 2.9 (presented by a linear quiver with $d+1$ arrows and commutativity relations): if it equals $d$ rather than being strictly less than $d$, then Assumption 2.2(2) cannot hold for a $d$-representation infinite algebra, and the proof of Theorem 5.1 via the reduction theorem would need correction.","tokens_in":28440,"feed_emoji":"","tokens_out":19359,"duration_ms":151595,"temperature":0.7,"pith_summary":"This paper proves that for two families of commutative Gorenstein toric singularities—Veronese subrings of polynomial rings and Segre products of polynomial rings—the graded singularity category admits an explicit tilting object whose endomorphism ring is higher representation infinite. The tilting objects are constructed so that they carry strict root pairs of the shifted Serre functor, which yields equivalences between the singularity categories and (folded) cluster categories that preserve cluster tilting objects. These equivalences also yield explicit twisted Calabi-Yau algebras as Calabi-Yau completions, with quiver and relation presentations given in the paper.","feed_headline":"Tilting objects constructed for Veronese and Segre singularities","feed_subtitle":"They give equivalences to (folded) cluster categories and explicit Calabi-Yau algebras.","key_machinery":"The main machinery is a reduction theorem: under Assumption 2.2, an idempotent quotient $A/(e)$ of a $d$-representation infinite algebra $A$ is $(d-1)$-representation infinite. This reduction is used together with a semi-orthogonal decomposition theorem for derived categories of coherent sheaves and singularity categories, and with the theory of strict root pairs of the shifted Serre functor, to compute the endomorphism rings of the constructed tilting objects and to organize them into orbit decompositions.","core_discovery":"For the Veronese subring $R = k[x_1,\\ldots,x_d]^{(n)}$ with $d = an$, the object $T = S \\oplus \\Omega S(1) \\oplus \\cdots \\oplus \\Omega^{a-1}S(a-1)$ in the graded singularity category $\\mathrm{sg}^{\\mathbb{Z}} R$ is tilting, its endomorphism ring $A$ is $(d-a-1)$-representation infinite, and $A$ has a strict $a$-th root pair $(U,P)$ of $\\nu_{d-a-1}^{-1}$. For the Segre product $R = k[x_1,y_1]\\#\\cdots\\#k[x_n,y_n]$, the object $T = \\bigoplus_{0<v<1} M_v$ is tilting in $\\mathrm{sg}^{\\mathbb{Z}} R$, its endomorphism ring is $(n-2)$-representation infinite, and for $n=3$ it is hereditary with a strict square root pair of $\\nu_1$. These equivalences respect (folded) cluster tilting subcategories and yield twisted Calabi-Yau algebras whose quivers and relations are described explicitly.","pith_inferences":["The orbit construction $T = \\bigoplus_{i=0}^{a-1} F^i T_0$ suggests a general recipe for producing tilting objects compatible with cluster structure in other graded Gorenstein singularities with an algebraic root of the shifted Serre functor.","The explicit quiver presentations of the endomorphism rings and Calabi-Yau completions give concrete models that could be used to compute cohomology or invariants of these singularity categories, and to study their noncommutative geometry directly.","The reduction theorem for idempotent quotients may hold under weaker hypotheses than Assumption 2.2(2), which would make the construction available for a broader class of higher representation infinite algebras.","The equivalences with folded cluster categories suggest a combinatorial interpretation of the cluster tilting subcategories via the quivers, possibly yielding new cluster structures on these singularity categories."],"forward_implications":["The singularity categories of these toric singularities are equivalent as triangulated categories to derived categories of explicit finite-dimensional algebras that are higher representation infinite.","The equivalences are compatible with canonical cluster tilting subcategories, so cluster tilting objects correspond under the equivalences.","For these rings, there are explicit quiver and relation presentations of the endomorphism rings and the associated twisted Calabi-Yau algebras.","The study of folded cluster categories is enriched by examples arising from actual singularities, going beyond formal constructions.","The results provide new examples of Gorenstein rings of hereditary representation type."],"supporting_citations":[{"why":"It provides the method of obtaining equivalences between singularity categories and cluster categories from a single tilting object.","marker":"[20]"},{"why":"It supplies the definition and basic theory of n-representation infinite algebras.","marker":"[21]"},{"why":"It gives the semi-orthogonal decomposition theorem for derived categories of coherent sheaves and singularity categories used in Section 5.","marker":"[37]"},{"why":"It provides the cluster tilting object in the singularity category of the Veronese subring.","marker":"[23]"},{"why":"It introduces root pairs and Calabi-Yau completions for roots of bimodule complexes.","marker":"[19]"},{"why":"It introduces Calabi-Yau completions used to construct the twisted Calabi-Yau algebras.","marker":"[33]"},{"why":"It provides cluster tilting subcategories for noncommutative projective schemes.","marker":"[16]"},{"why":"It gives cluster tilting objects for the Segre product rings.","marker":"[22]"},{"why":"It supplies the context of Cohen-Macaulay rings of hereditary representation type.","marker":"[18]"},{"why":"It constructs earlier tilting objects in the graded singularity category of the Veronese subring, which are refined here.","marker":"[28]"}],"fun_headline_variants":["Tilting objects bridge Veronese and Segre to cluster categories","Calabi-Yau algebras from toric singularity tilting objects","Tilting objects yield cluster equivalences and Calabi-Yau algebras","Explicit Calabi-Yau algebras via tilting in toric singularity categories"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The Segre product proof depends on the assumption that a certain finite-dimensional algebra can have self-injective dimension strictly smaller than the global dimension bound $d$, a condition required by the paper's reduction theorem for idempotent quotients.","fun_headline_variants_meta":{"raw":{"variants":["Tilting objects bridge Veronese and Segre to cluster categories","Calabi-Yau algebras from toric singularity tilting objects","Tilting objects yield cluster equivalences and Calabi-Yau algebras","Explicit Calabi-Yau algebras via tilting in toric singularity categories"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000526,"raw_usage":{"total_tokens":2554,"prompt_tokens":972,"completion_tokens":1582,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":1505}},"tokens_in":588,"tokens_out":1582,"duration_ms":12581,"temperature":1.0,"reasoning_tokens":1505,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:58:48.525130+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the self-injective dimension of the standard $d$-representation infinite algebra of the paper's Example 2.9 (presented by a linear quiver with $d+1$ arrows and commutativity relations): if it equals $d$ rather than being strictly less than $d$, then Assumption 2.2(2) cannot hold for a $d$-representation infinite algebra, and the proof of Theorem 5.1 via the reduction theorem would need correction.","supporting_citations":[{"cited_title":"Herschend, O","cited_arxiv_id":null,"evidence_quote":"It supplies the definition and basic theory of n-representation infinite algebras."},{"cited_title":"Orlov, Derived categories of coherent sheaves and tr iangulated categories of singularities, Algebra, arithme tic, and geom- etry: in honor of Yu","cited_arxiv_id":null,"evidence_quote":"It gives the semi-orthogonal decomposition theorem for derived categories of coherent sheaves and singularity categories used in Section 5."},{"cited_title":"Hanihara, Cluster categories of formal DG algebras a nd singularity categories, Forum of Mathematics, Sigma (20 22), Vol","cited_arxiv_id":null,"evidence_quote":"It provides cluster tilting subcategories for noncommutative projective schemes."},{"cited_title":"Higashitani and Y","cited_arxiv_id":null,"evidence_quote":"It gives cluster tilting objects for the Segre product rings."},{"cited_title":"Hanihara, Non-commutative resolutions for Segre pr oducts and Cohen-Macaulay rings of hereditary representat ion type, Trans","cited_arxiv_id":null,"evidence_quote":"It supplies the context of Cohen-Macaulay rings of hereditary representation type."},{"cited_title":"Iyama and R","cited_arxiv_id":null,"evidence_quote":"It constructs earlier tilting objects in the graded singularity category of the Veronese subring, which are refined here."}],"review_version":1}