{"id":"cf7e67b1-854f-46ef-a6df-b739cf59e61e","arxiv_id":"2607.07211","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"An algebraic d-Auslander extriangulated category satisfying a vanishing condition admits an extriangulated ideal quotient equivalent to a truncated homotopy category of complexes.","lead":"The paper proves that certain algebraic extriangulated categories (d-Auslander categories) are equivalent to categories of truncated chain complexes, and uses this to show some complex categories carry a triangulated structure. This matters because it unifies and extends several recent results in higher homological algebra, answering an open question of Iyama.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"No significant objection identified","rationale":"The reader correctly identified the chain of dependencies (Lemma 4.2 → Corollary 3.6 → Proposition 4.3 → Theorem 4.4) as the most technical part of the argument. However, upon careful examination, the concern about a borderline failure of the global dimension or dominant dimension bounds does not land: the d-Auslander definition ensures these bounds hold by assumption, and the proof uses them exactly where needed. The factorization step in Lemma 4.2 is valid because pd(E) < d+1 follows from the acyclicity and boundedness of the complex, not from any additional unstated hypothesis. The paper is a clean, well-structured piece of homological algebra with detailed proofs. The reader's verdict of ACCEPT with MODERATE confidence is appropriate — the arguments are internally consistent, and while full verification requires specialist expertise, no concrete error or hidden assumption was found.","tokens_in":16553,"tokens_out":731,"duration_ms":2353858,"concrete_test":"Independently verify the inductive step in Lemma 4.2 by explicitly constructing the homotopy for a small test case: take d=1, a specific acyclic complex in K^b(P) over a finite-dimensional algebra (e.g., the A_3 example in Example 4.5), and trace through the factorization of f• through projective-injective objects. Confirm that the morphism j factors through Q and that the resulting f̄• is homotopic to f• with the claimed support properties.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 4.4, which characterizes when an algebraic extriangulated category C admits an extriangulated ideal quotient equivalent to K^{[-d-1,0]}(A). The proof proceeds through the triangulated hull construction (Proposition 2.16), the essential image computation (Theorem 3.7), and the kernel computation (Proposition 4.3). I examined the chain of dependencies the reader flagged: Lemma 4.2's inductive homotopy argument, where the morphism j:E -> P̃_n must factor through a projective-injective object Q via Corollary 3.6. The key step is verifying that pd(E) < d+1, which is what triggers Corollary 3.6. In the proof of Lemma 4.2, E is defined as the cokernel of the inflation g_n ∘ a_n, where a_n: C^{n-1} ↣ P^n is part of an acyclic complex in K^b(P). Since the complex is bounded and acyclic, E is an extension of objects of strictly smaller projective dimension, so pd(E) < d+1 holds. The d-Auslander conditions (Definition 2.7) — global dimension at most d+1 and dominant dimension at least d+1 — are used precisely where needed: global dimension bounds pd(E), and dominant dimension (via Corollary 3.6) provides the projective-injective inflation. The vanishing condition E^k_C(I,P)=0 for 1≤k≤d is used in Lemma 4.1 to control Hom-spaces in the truncated homotopy category. The argument is internally consistent and the logical dependencies are sound. I do not find a load-bearing concern that would undermine the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper establishes a connection between $d$-Auslander extriangulated categories and categories of $(d+2)$-term complexes up to homotopy. The main result (Theorem 4.4) gives a complete homological characterization: an algebraic extriangulated category $C$ admits an extriangulated ideal quotient equivalent to $K^{[-d-1,0]}(A)$ if and only if $C$ is $d$-Auslander and satisfies $E^k_C(I,P)=0$ for $1 le k le d$, $I in I$, $P in P$. The author then applies this to $(d+2)$-cluster-tilting subcategories of triangulated categories (Theorems 5.2, 5.3), and as a corollary answers a question of Iyama by showing that $K^{[-d-1,0]}(N)$ is triangulated when $N$ is a weakly idempotent complete algebraic $(d+4)$-angulated category (Corollary 5.5). The paper concludes with applications to higher Auslander algebras, recovering a result of Gorsky-Williams. The proofs proceed through a clear logical chain: an explicit construction of a triangulated hull (Section 2.3), homological lemmas for $d$-Auslander categories (Section 3), and the main equivalence (Section 4).","tokens_in":17145,"tokens_out":1139,"duration_ms":278818,"significance":"The paper makes a solid contribution to the structure theory of extriangulated categories and their relationship to higher homological algebra. The main theorem provides a clean, complete characterization of truncated homotopy categories among algebraic extriangulated categories, generalizing the $0$-Auslander correspondence of [FGPPP23, Che23, Yan25]. The application to cluster-tilting subcategories and the resulting answer to Iyama's question are natural and well-motivated. The recovery of the Gorsky-Williams equivalence in Section 6 demonstrates the applicability of the general framework. The proofs are detailed and the logical dependencies are traceable: the triangulated hull construction (Proposition 2.16), the essential image computation (Theorem 3.7), and the kernel computation (Proposition 4.3) each build on the previous step in a transparent manner.","major_comments":[],"minor_comments":[{"comment":"There is a systematic misuse of 'Theorem' for internal cross-references to lemmas and propositions within proofs (e.g., 'Theorem 2.12' for Lemma 2.12, 'Theorem 3.3' for Lemma 3.3, 'Theorem 3.5' for Proposition 3.5, 'Theorem 3.6' for Corollary 3.6, 'Theorem 4.1' for Lemma 4.1, 'Theorem 4.2' for Lemma 4.2, 'Theorem 4.3' for Proposition 4.3, 'Theorem 5.1' for Proposition 5.1, 'Theorem 5.3' for Corollary 5.3, 'Theorem 5.5' for Corollary 5.5). This should be corrected throughout for precision.","section":null},{"comment":"In the proof of Proposition 5.1, the sentence beginning 'We notice that $X hookrightarrow 0 twoheadrightarrow Sigma X$ is a conflation in $T_M$ for any $X in M * cdots * Sigma^d M$' could benefit from a brief justification of why this implies that injectives must lie in $Sigma^{d+1}M$, as the logical step from conflations of this form to the injective characterization is condensed.","section":null},{"comment":"In the proof of Theorem 3.7, the base case of the induction states that $P^{bullet}_{d+1} := P^{bullet}$ is exact up to degree $d+1$ because this is an empty condition. It would be clearer to briefly note that exactness up to degree $d+1$ is vacuous since the complex is supported in $[-d-1,0]$.","section":null},{"comment":"The abstract states '$d$-Auslander extriangulated categories' without the hyphen used in Definition 2.7 ('$d$-Auslander'). Minor consistency in hyphenation would improve readability.","section":null},{"comment":"In Example 4.5, the category $A = mod(Lambda)/[inj(mod(Lambda))]$ is used but the notation $[inj(mod(Lambda))]$ for the ideal of morphisms factoring through injectives is not explicitly defined in the paper, though it follows the convention of $[I to P]$. A brief clarifying remark would help.","section":null},{"comment":"The reference [ZZZ26] is cited as 'arXiv preprint, to appear.' If possible, the arXiv identifier should be included so readers can access the independent related results mentioned in the acknowledgments.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is mathematically sound and the central claims are well-supported. The minor comments are all presentation issues that do not affect correctness. I note that the independent work of Zhang-Zhou-Zhu [ZZZ26] is acknowledged; the editor may wish to verify the degree of overlap, but based on the manuscript's description, the results appear to be complementary rather than duplicative."},"author_rebuttal":null,"desk_editor":{"model":"glm-5.2","letter":"The main result is Theorem 4.4: an algebraic extriangulated category C admits an extriangulated ideal quotient equivalent to K^{[-d-1,0]}(A) if and only if C is d-Auslander and satisfies the vanishing E^k_C(I,P)=0 for 1≤k≤d. This specializes the 0-Auslander correspondence of [FGPPP23, Che23, Yan25] to arbitrary d, and the paper applies it to answer Iyama's question (Corollary 5.5): K^{[-d-1,0]}(N) is triangulated when N is a weakly idempotent complete algebraic (d+4)-angulated category. The recovery of Theorem 3.1 of [GW26] as a special case is a nice unification check on the framework. The paper does what it sets out to do. The logical chain is clear: Section 2.3 constructs the triangulated hull explicitly (a special case of [Che26, Prop-Def 3.14]), Section 3 computes the essential image under the d-Auslander hypotheses, and Section 4 identifies the kernel of the quotient functor as [I→P]. The proofs of Lemmas 4.1–4.2 and Proposition 4.3 are given in full, and the inductive homotopy argument in Lemma 4.2 checks out: the key step where j:E→P̃_n factors through a projective-injective Q via Corollary 3.6 is justified because E, being a cokernel in a bounded acyclic complex, has pd(E)<d+1 by the global dimension bound. The stress-test concern about this chain does not land — the dependencies are sound. The applications in Section 5 (cluster-tilting subcategories yielding d-Auslander categories, the vosnex property giving the right vanishing) are natural and well-motivated. Section 6 on higher Auslander algebras of type A is mostly a consistency check rather than new content, but it is short and does no harm. The independent and concurrent work of Zhang–Zhou–Zhu [ZZZ26] is acknowledged honestly. One minor concern: the paper relies on [Kva26, Theorem 7.5] for the passage from (d+4)-angulated categories to cluster-tilting subcategories in Corollary 5.5. If that result has issues, the corollary weakens, but the rest of the paper stands independently. This is a paper for specialists in extriangulated and higher homological algebra. The core results are correct and the writing is clear. It deserves a serious referee who can check the homological algebra in detail, particularly the triangulated hull construction and the kernel computation in Proposition 4.3.","headline":"Clean characterization of truncated homotopy categories among d-Auslander extriangulated categories, answering a question of Iyama.","tokens_in":17533,"tokens_out":665,"would_cite":true,"duration_ms":203953,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"When do extriangulated categories become truncated homotopy categories?","keywords":[],"falsifier":"A counterexample would be an algebraic extriangulated category C that is d-Auslander and satisfies E^k_C(I,P)=0 for 1<=k<=d, but where the functor rho from D^{[-d-1,0]} to K^{[-d-1,0]}(P/[Q]) has a kernel strictly larger than [I->P]. This would arise if Lemma 4.2's homotopy reduction fails: specifically, if there exists a morphism between acyclic bounded complexes in K^b(P) that factors through [Q] componentwise but cannot be homotoped to a single component in [Q], which would happen if the factoring through projective-injectives in Lemma 4.1 or Corollary 3.6 fails for some object at the edge","tokens_in":16749,"feed_emoji":"🔗","tokens_out":1311,"duration_ms":159461,"temperature":0.7,"pith_summary":"The paper proves a precise equivalence: an algebraic extriangulated category C admits an extriangulated ideal quotient equivalent to the category of (d+2)-term complexes up to homotopy K^{[-d-1,0]}(A) if and only if C is d-Auslander (meaning it has enough projectives, global dimension at most d+1, and dominant dimension at least d+1) and satisfies the vanishing condition E^k_C(I,P)=0 for 1<=k<=d on extensions between injectives I and projectives P. When these conditions hold, the additive category A is determined as P/[Q] (projectives modulo projective-injectives) and the quotient C/[I->P] (morphisms factoring through injective-domain, projective-codomain maps) is equivalent to K^{[-d-1,0]}(A). This generalizes the d=0 case established in prior work. The paper then shows that (d+2)-cluster-tilting subcategories of triangulated categories provide a natural supply of d-Auslander extriangulated categories, yielding concrete equivalences between quotient categories and truncated homotopy categories. As a corollary, it proves that K^{[-d-1,0]}(N) carries a triangulated structure when N is a weakly idempotent complete algebraic (d+4)-angulated category, answering a question of Iyama.","feed_headline":"Truncated homotopy categories characterized as d-Auslander quotients","feed_subtitle":"An if-and-only-if condition tells us exactly when an extriangulated category quotients to (d+2)-term complexes, settling a question of Iyama","key_machinery":"The proof embeds C into a triangulated hull D=K^-(P)/K^b(Q) using the canonical functor from an exact model E of C. For a d-Auslander category, the essential image of this embedding is shown to be D^{[-d-1,0]} (the bounded part of the triangulated quotient). A functor rho from D^{[-d-1,0]} to K^{[-d-1,0]}(P/[Q]) is then constructed. Lemma 4.1 controls morphisms from objects of bounded projective dimension to projectives, and Lemma 4.2 provides a homotopy reduction for morphisms between acyclic bounded complexes. Together these identify the kernel of rho as exactly [I->P], yielding the equivalence C/[I->P] = K^{[-d-1,0]}(P/[Q]).","core_discovery":"The central result is a complete homological characterization: the truncated homotopy categories K^{[-d-1,0]}(A) are, up to extriangulated ideal quotient, exactly the algebraic d-Auslander extriangulated categories satisfying the extension-vanishing condition E^k(I,P)=0 for 1<=k<=d. The mechanism is an explicit embedding of an algebraic extriangulated category into a triangulated quotient K^-(P)/K^b(Q) via a triangulated hull construction, followed by a careful analysis of which morphisms survive the further quotient to K^{[-d-1,0]}(P/[Q]). The kernel of this quotient functor is precisely the ideal [I->P] of morphisms factoring through injective-domain, projective-codomain maps, and this is究","pith_inferences":[],"forward_implications":["Any (d+2)-cluster-tilting subcategory M with the vosnex property in an algebraic triangulated category T yields an equivalence T_M/[Sigma^{d+1}M -> M] = K^{[-d-1,0]}(M), connecting cluster-tilting theory to truncated homotopy categories.","When M is (d+2)Z-cluster-tilting (i.e., Sigma^{d+2}M=M), the ideal [Sigma^{d+1}M -> M] vanishes, giving a direct equivalence T_M = K^{[-d-1,0]}(M) without any quotient.","The category K^{[-d-1,0]}(N) inherits a triangulated structure whenever N is a weakly idempotent complete algebraic (d+4)-angulated category, since such N arises as a (d+2)Z-cluster-tilting subcategory.","For d-Auslander algebras Gamma satisfying Ext^k_Gamma(DGamma, Gamma)=0 for 1<=k<=d, the quotient mod(Gamma)/[DGamma->Gamma] is equivalent to K^{[-d-1,0]}(add(M)/[inj(M)]), recovering and generalizing results on higher Auslander algebras of type A."],"fun_headline_variants":["Extension vanishing pins down d-Auslander quotient structure","Truncated homotopy categories arise exactly as d-Auslander quotients","Homological criterion classifies d-Auslander extriangulated quotients","Iyama's question answered: truncated complexes inherit triangulated structure","d-cluster-tilting subcategories generate d-Auslander extriangulated structure"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The proof depends on the triangulated hull embedding and the claim that a certain functor rho has kernel exactly [I->P]. This relies on Lemma 4.2, which uses an inductive homotopy argument factoring morphisms through projective-injective objects. The factoring step invokes Corollary 3.6, which requires the global dimension and dominant dimension bounds from the d-Auslander definition to hold. If these bounds fail in a borderline case, the kernel identification breaks and the主","fun_headline_variants_meta":{"raw":{"variants":["Extension vanishing pins down d-Auslander quotient structure","Truncated homotopy categories arise exactly as d-Auslander quotients","Homological criterion classifies d-Auslander extriangulated quotients","Iyama's question answered: truncated complexes inherit triangulated structure","d-cluster-tilting subcategories generate d-Auslander extriangulated structure","When extriangulated quotients match truncated homotopy categories","Extriangulated ideal quotients characterized by extension vanishing","Complete if-and-only-if: truncated complexes as d-Auslander quotients"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":2047,"prompt_tokens":553,"completion_tokens":1494,"prompt_tokens_details":null},"tokens_in":553,"tokens_out":1494,"duration_ms":50383,"temperature":1.0,"reasoning_tokens":1409,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T17:07:21.124023+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"A counterexample would be an algebraic extriangulated category C that is d-Auslander and satisfies E^k_C(I,P)=0 for 1<=k<=d, but where the functor rho from D^{[-d-1,0]} to K^{[-d-1,0]}(P/[Q]) has a kernel strictly larger than [I->P]. This would arise if Lemma 4.2's homotopy reduction fails: specifically, if there exists a morphism between acyclic bounded complexes in K^b(P) that factors through [Q] componentwise but cannot be homotoped to a single component in [Q], which would happen if the factoring through projective-injectives in Lemma 4.1 or Corollary 3.6 fails for some object at the edge","supporting_citations":[],"review_version":1}