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When do extriangulated categories become truncated homotopy categories?

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2026-07-09 17:07 UTC pith:4VKYDHYP

load-bearing objection Clean characterization of truncated homotopy categories among d-Auslander extriangulated categories, answering a question of Iyama.

arxiv 2607.07211 v1 pith:4VKYDHYP submitted 2026-07-08 math.RT

Extriangulated ideal quotients and d-Auslander categories

Lior Silberberg This is my paper
classification math.RT
keywords categoriesextriangulatedmathcalauslanderadmitsalgebraiccategoryideal
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

Core claim

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

What carries the argument

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]).

Load-bearing premise

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

What would settle it

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

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If this is right

  • 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 6 minor

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).

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.

minor comments (6)
  1. 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.
  2. 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.
  3. 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 $[-d-1,0]$.
  4. The abstract states '$d$-Auslander extriangulated categories' without the hyphen used in Definition 2.7 ('$d$-Auslander'). Minor consistency in hyphenation would improve readability.
  5. 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.
  6. 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.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper is a pure mathematics result with a deductive proof structure. The central theorem (Theorem 4.4) is derived from first principles within the established framework of extriangulated categories. The proof chain proceeds through the triangulated hull construction (Proposition 2.16), the essential image computation (Theorem 3.7), and the kernel computation (Proposition 4.3). Each step introduces new mathematical content: the d-Auslander conditions (Definition 2.7) provide global dimension and dominant dimension bounds, the vanishing condition E^k_C(I,P)=0 is used in Lemma 4.1 to control Hom-spaces, and the inductive homotopy argument in Lemma 4.2 uses Corollary 3.6 (which relies on pd(E)<d+1) to factor morphisms through projective-injective objects. The cited results ([Che26] for the triangulated hull, [Kva26] for Theorem 7.5 used in Corollary 5.5, [FGPPP23] for ideal quotients) are used as building blocks but are standard mathematical citations, not self-citations by the same author. The main derivation does not reduce to a tautology or depend on fitted parameters. The logical dependencies are sound and the result has independent mathematical content beyond its inputs. No circularity patterns (self-definitional, fitted-input-as-prediction, self-citation load-bearing, uniqueness imported, ansatz smuggling, or renaming) are present.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper introduces no new physical entities or free parameters. It works entirely within the established framework of extriangulated categories, triangulated categories, and homological algebra. The axioms are standard results from the cited literature or standard domain assumptions for algebraic extriangulated categories.

axioms (4)
  • domain assumption Existence of enough projectives and injectives in the extriangulated category C
    Definition 2.7 requires enough projectives as part of the d-Auslander condition. The construction of the triangulated hull in Section 2.3 and the proofs in Sections 3-4 depend on the existence of projective resolutions.
  • domain assumption The category C is algebraic, i.e., equivalent to E/[Q_0] for some exact category E with projective-injective subcategory Q_0
    Stated at the beginning of Section 2.3 and used throughout. This is necessary for the embedding into K^-(P)/K^b(Q_0).
  • standard math Theorem 7.5 of [Kva26]: A weakly idempotent complete algebraic (d+4)-angulated category is additively equivalent to a (d+2)Z-cluster-tilting subcategory of an algebraic triangulated category
    Used in the proof of Corollary 5.5 to reduce the triangulated structure question to Theorem 5.3.
  • standard math Proposition 3.19 of [HLN21]: Relative extriangulated structures from additive subcategories
    Used in Proposition 2.6 and Section 5 to define the extriangulated structure T_M on a triangulated category T with cluster-tilting subcategory M.

reviewed 2026-07-09 · how reviews work

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Cite this review

Pith. "Pith review of Extriangulated ideal quotients and $d$-Auslander categories." pith.science (2026). https://pith.science/paper/4VKYDHYP

@misc{pith2026260707211,
  author       = {Pith},
  title        = {Pith review of: Extriangulated ideal quotients and $d$-Auslander categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4VKYDHYP}},
  note         = {Machine review of arXiv:2607.07211}
}
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read the original abstract

Building on recent studies of 0-Auslander categories, we establish a connection between $d$-Auslander extriangulated categories and categories of $(d+2)$-term complexes up to homotopy. We give a precise homological condition under which an algebraic extriangulated category admits an extriangulated ideal quotient equivalent to $\mathcal{K}^{[-d-1,0]}(\mathcal{A})$. We then demonstrate that $d$-cluster-tilting subcategories in triangulated categories serve as a key source of $d$-Auslander extriangulated categories. Using these structural results, we answer a question posed by Iyama in the Appendix of arXiv:2509.08246 by proving that $\mathcal{K}^{[-d-1,0]}(\mathcal{N})$ admits a triangulated structure when $\mathcal{N}$ is a weakly idempotent complete algebraic $(d+4)$-angulated category.

Figures

Figures reproduced from arXiv: 2607.07211 by Lior Silberberg.

Figure 1
Figure 1. Figure 1: On the left, the Auslander–Reiten quiver of the module category mod(Γ), the [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The case d = 0. On the top, the Auslander–Reiten quiver of the derived category Db (Λ), the subcategory U represented by black nodes. On the bottom, the Auslander–Reiten quiver of the category of 2-term complexes up to homotopy in U. We can now give a solution to a question raised by Iyama in [Yan25]. Corollary 5.5. Let N be a weakly idempotent complete algebraic (d + 4)-angulated category. Then K[−d−1,0](… view at source ↗

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This paper was first reviewed by glm-5.2 on July 9, 2026.