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A rigid object M encodes its finitely presented objects as 2-term complexes over End(M), matching cluster-tilting with silting.

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2026-08-04 20:57 UTC pith:OVPHLXK7

load-bearing objection 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. the 1 major comments →

arxiv 2509.08246 v1 pith:OVPHLXK7 submitted 2025-09-10 math.RT math.CT

From objects finitely presented by a rigid object in a triangulated category to 2-term complexes

Dong Yang This is my paper
classification math.RT math.CT MSC 16E3518G80
keywords rigid objectstriangulated categories2-term complexessilting objectscluster-tilting objectsextriangulated categoriesself-injective algebrasquivers with potential
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.

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.

Core claim

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

What carries the argument

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

Load-bearing premise

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.

What would settle it

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.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

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

1 major / 4 minor

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.

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 (1)
  1. [Theorem 3.13(c)] 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
minor comments (4)
  1. [Abstract] 'silting complexs' should be 'silting complexes'.
  2. [Theorem 1.1(e)] 'the bijection in (c) restricts' appears to be a cross-reference error; the intended statement is about the bijection in (d).
  3. [Theorem 3.13(d)] 'restricts a bijection' should be 'restricts to a bijection'.
  4. [Section 4.1] After Corollary 4.3, 'Amiot's conjecture hods' should be 'Amiot's conjecture holds'.

Circularity Check

1 steps flagged

No load-bearing circularity; P is constructed and proved explicitly. Minor self-citations are not load-bearing. One non-circular proof gap in Theorem 3.13(c) is flagged.

specific steps
  1. other [Theorem 3.13(c) and its proof, Section 3.7]
    "Theorem 3.13. (c) P induces a bijection between the set of isomorphism classes of relative cluster-tilting objects of pr(M) and the set of isomorphism classes of 2-term silting objects of H^b(projA). ... Proof. It remains to prove that the bijection in (d) commutes with mutations."

    Not a circular step: this is an omitted proof, flagged per review rule. The object-level bijection in (c) is stated without Hom-finiteness, but the proof only addresses mutation compatibility of (d). The route from relative cluster-tilting objects to weakly relative cluster-tilting subcategories (Theorem 3.11) is via Lemma 2.5, which assumes k is a field and pr(M) is Hom-finite. Thus (c) lacks a demonstrated derivation in its stated generality; however, the bijection is not assumed as an input, so this is a correctness/generality gap rather than a reduction of the conclusion to the hypotheses.

full rationale

The central construction is self-contained. P: pr(M) -> H^{[-1,0]}(projA) is built explicitly from chosen presentations (Proposition 3.2), and its fullness, density, kernel, isomorphism/indecomposability detection and extriangle detection are proved (Theorem 3.8, Proposition 3.7), not assumed. The cluster-tilting/silting bijections are derived from these properties together with external results [48, Theorem 5.4], [16, Theorem 3.4], [18, Theorem 4.3], [47, Proposition 3.4]. Self-citations [8], [24], [27], [35], [45] are used as technical tools or for context (e.g., idempotent completeness, recovery of [8] as a special case, compatibility with tau-tilting, cluster category existence, and a remark on a question), and none of them contains the target statements by definition. The appendix by Iyama proves the converse using independent algebra results. The only flagged issue is Theorem 3.13(c): the proof is incomplete in the stated generality, since the bridge via Lemma 2.5 requires Hom-finiteness. This is a proof gap, not a circular reduction: the claimed bijection is not an input to the construction. Accordingly the circularity score is low (2), reflecting minor non-load-bearing self-citations rather than any self-referential derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The central claim rests on standard structural assumptions of the field: algebraic triangulated categories with a rigid object, the extriangulated structure on pr(M), cited characterizations of cluster-tilting objects, and the cluster category construction for quivers with potential. There are no fitted free parameters and no invented entities; the paper is a theorem-and-proof contribution.

axioms (6)
  • domain assumption T is an idempotent-complete algebraic triangulated k-category, i.e. admits a Frobenius category model F with T = F stable.
    Stated at the start of Section 3 and used throughout the liftable-presentation machinery (Lemma 3.1, Prop 3.2, Theorem 3.8, Prop 3.7).
  • domain assumption M is a rigid subcategory (or rigid object) closed under finite direct sums and direct summands, with Hom_T(M, Sigma M) = 0.
    Rigidity supplies the Ext^1 vanishings in Lemma 3.1 and the kernel description I^2 = 0 in Section 3.4.
  • standard math pr(M) carries the [Sigma M]-extriangle structure of Nakaoka-Palu, via [42, Lemma 4.57], making it an extriangulated category.
    Used implicitly in Section 2.2 for extriangles and for the mutation Lemma 2.6.
  • standard math In the Hom-finite case, [48, Theorem 3.1] and [18, Theorem 4.3] give equivalence of relative cluster-tilting, maximal relative rigid, generating and |M| = |N|.
    Lemma 2.5, used for the bijections in Theorems 3.11, 3.13 and 3.16.
  • standard math Hom_T(M, Sigma^{-1}M) = 0 iff A is self-injective, in the 2-Calabi-Yau cluster-tilting setting ([26, Proposition 3.6]).
    Used in Theorem 3.16(a) and in the hypotheses of Lemma 3.10.
  • standard math Existence of cluster categories C_{(Q,W)} of quivers with potential, with a basic cluster-tilting object whose endomorphism algebra is the complete Jacobian algebra ([5], [35]).
    Used in Section 4 to reach Mizuno's theorem.

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Pith. "Pith review of From objects finitely presented by a rigid object in a triangulated category to 2-term complexes." pith.science (2026). https://pith.science/paper/OVPHLXK7

@misc{pith2026250908246,
  author       = {Pith},
  title        = {Pith review of: From objects finitely presented by a rigid object in a triangulated category to 2-term complexes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OVPHLXK7}},
  note         = {Machine review of arXiv:2509.08246}
}
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read the original abstract

For a rigid object $M$ in an algebraic triangulated category $\mathcal{T}$, a functor pr$(M)\to\mathcal{H}^{[-1,0]}({\rm proj}\, A)$ is constructed, which essentially takes an object to its `presentation', where pr$(M)$ is the full subcategory of $\mathcal{T}$ of objects finitely presented by $M$, $A$ is the endomorphism algebra of $M$ and $\mathcal{H}^{[-1,0]}({\rm proj}\, A)$ is the homotopy category of complexes of finitely projective $A$-modules concentrated in degrees $-1$ and $0$. This functor is shown to be full and dense and its kernel is described. It detects isomorphisms, indecomposability and extriangles. In the Hom-finite case, it induces a bijection from the set of isomorphism classes of basic relative cluster-tilting objects of pr$(M)$ to that of basic silting complexs of $\mathcal{H}^{[-1,0]}({\rm proj}\, A)$, which commutes with mutations. These results are applied to cluster categories of self-injective quivers with potential to recover a theorem of Mizuno on the endomorphism algebras of certain 2-term silting complexes. As an interesting consequence of the main results, if $\mathcal{T}$ is a 2-Calabi--Yau triangulated category and $M$ is a cluster-tilting object such that $A$ is self-injective, then $\mathbb{P}$ is an equivalence, in particular, $\mathcal{H}^{[-1,0]}({\rm proj}\, A)$ admits a triangle structure. In the appendix by Iyama it is shown that for a finite-dimensional algebra $A$, if $\mathcal{H}^{[-1,0]}({\rm proj}\, A)$ admits a triangle structure, then $A$ is necessarily self-injective.

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Forward citations

Cited by 2 Pith papers

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    An extriangulated functor from (n+1)-term subcategories to (n+1)-term complexes is constructed, with equivalent conditions for fullness yielding an extriangle equivalence and a mutation-compatible silting bijection.

  2. Extriangulated ideal quotients and $d$-Auslander categories

    math.RT 2026-07 accept novelty 6.0

    An algebraic d-Auslander extriangulated category satisfying a vanishing condition admits an extriangulated ideal quotient equivalent to a truncated homotopy category of complexes.

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.