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Derived equivalence of posets of torsion classes

T0 review · 2 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read If two finite-dimensional algebras with simple projective modules are related by a 1-APR tilt, then their posets of torsion classes are related by a flip-flop, implying that the incidence algebras of these posets are derived equivalent.

desk verdict This generalizes Ladkani by proving that 1-APR tilts induce flip-flops on torsion-class posets (hence derived equivalences of incidence algebras) via two explicit embeddings, one through silting objects and one through s-torsion pairs. read the letter →

arxiv 2606.21239 v1 pith:XQJNAQPS submitted 2026-06-19 math.RT

Marius Goguet This is my paper · ORCID
classification math.RT
keywords torsionclassesposets1-APRtiltderivedequivalenceincidencealgebrassiltingobjectss-torsionpairsfinitedimensional
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a connection between 1-APR tilting of algebras and the structure of their torsion class posets. It shows that a 1-APR tilt between algebras induces a flip-flop relation on the corresponding posets of torsion classes. This relation in turn implies that the incidence algebras built from these posets are derived equivalent. A reader would care because this provides a way to relate derived categories through combinatorial changes in poset structures arising from tilting.

What carries the argument

The flip-flop relation on posets of torsion classes induced by a 1-APR tilt, which preserves the structure needed for derived equivalence of incidence algebras.

What would settle it

Two algebras related by a 1-APR tilt whose posets of torsion classes are not related by a flip-flop, or whose incidence algebras are not derived equivalent.

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Extended reading notes

Core claim

For finite dimensional algebras admitting a simple projective module, if two such algebras are related by a 1-APR tilt, their posets of torsion classes are related by a flip-flop. This is shown in two proofs: one embedding the posets as subposets of a common poset of silting objects for functorially finite torsion classes, and another embedding into a common poset of s-torsion pairs for arbitrary torsion classes. Consequently, the incidence algebras of the posets are derived equivalent.

Load-bearing premise

The algebras under consideration admit a simple projective module.

Editorial extensions

If this is right

  • The incidence algebras of the two posets are derived equivalent.
  • The result holds for functorially finite torsion classes via embedding into silting posets.
  • The result extends to arbitrary torsion classes via embedding into s-torsion pair posets.
  • This generalizes Ladkani's earlier result on the topic.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the flip-flop preserves more structure, it might induce equivalences on other invariants of the posets.
  • Similar relations could be investigated for other types of tilting or mutations in representation theory.
  • The approach of embedding into larger posets of silting objects or s-torsion pairs may apply to other poset comparisons.
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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

2 major / 2 minor

Summary. The manuscript investigates the poset of torsion classes for finite-dimensional algebras admitting a simple projective module. It generalizes Ladkani's result by proving that a 1-APR tilt between two such algebras induces a flip-flop relation on their torsion-class posets, from which derived equivalence of the corresponding incidence algebras follows. Two proofs are supplied: the first embeds functorially finite torsion-class posets into a common silting poset to realize the flip-flop; the second embeds arbitrary torsion-class posets into a common s-torsion-pair poset.

Significance. If the central claim holds, the work supplies an explicit, functorial link between 1-APR tilting and poset-level flip-flops that yields derived equivalences of incidence algebras, extending Ladkani's theorem to both the functorially finite and general settings. The two distinct embedding constructions (silting and s-torsion pairs) constitute a concrete strength, as they provide verifiable routes to the same relation without relying on fitted parameters or ad-hoc choices.

major comments (2)
  1. [§3.2] §3.2, construction of the common silting poset: the claim that the two torsion-class posets embed as subposets whose order relations realize the flip-flop must be verified by an explicit check that the embedding functors preserve and reflect the covering relations used to define the flip-flop; without this step the passage to incidence-algebra derived equivalence rests on an unverified preservation property.
  2. [§4] §4, s-torsion-pair embedding: the argument that the flip-flop on the embedded posets induces derived equivalence of incidence algebras assumes that the incidence algebra of a subposet is derived-equivalent to a quotient or subalgebra of the ambient incidence algebra; this reduction step is load-bearing for the general (non-functorially-finite) case and requires a precise statement of the functor or equivalence used.
minor comments (2)
  1. The global hypothesis that the algebra admits a simple projective module is stated in the abstract and introduction but its precise use in each embedding construction could be flagged at the relevant lemmas for clarity.
  2. Notation for the flip-flop operation and for s-torsion pairs should be introduced with a short comparison to the conventions in Ladkani's cited work.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and the detailed comments, which help clarify the presentation. We address the two major points below.

read point-by-point responses

  1. Referee: [§3.2] §3.2, construction of the common silting poset: the claim that the two torsion-class posets embed as subposets whose order relations realize the flip-flop must be verified by an explicit check that the embedding functors preserve and reflect the covering relations used to define the flip-flop; without this step the passage to incidence-algebra derived equivalence rests on an unverified preservation property.

    Authors: We agree that an explicit verification of preservation and reflection of covering relations is desirable for full rigor. The embeddings are constructed so that covering relations in the torsion-class posets correspond exactly to irreducible silting mutations in the ambient poset; this correspondence is functorial and therefore preserves the flip-flop structure by construction. Nevertheless, to make the argument self-contained, we will insert a short lemma in the revised §3.2 that directly checks the covering relations under both embeddings. revision: yes

  2. Referee: [§4] §4, s-torsion-pair embedding: the argument that the flip-flop on the embedded posets induces derived equivalence of incidence algebras assumes that the incidence algebra of a subposet is derived-equivalent to a quotient or subalgebra of the ambient incidence algebra; this reduction step is load-bearing for the general (non-functorially-finite) case and requires a precise statement of the functor or equivalence used.

    Authors: The reduction uses the standard fact that the incidence algebra of a subposet is the quotient of the ambient incidence algebra by the two-sided ideal generated by the basis elements outside the subposet; the quotient map induces a derived equivalence because the ideal is generated by idempotents corresponding to the omitted elements. We will add an explicit statement of this functor (the natural projection) together with a reference to the relevant property of incidence algebras in the revised §4. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The derivation relies on two explicit, independent constructions: (1) functorial embedding of torsion-class posets into a common silting poset to realize the flip-flop for the functorially finite case, and (2) embedding into a common s-torsion-pair poset for arbitrary torsion classes. Both are stated to yield the same flip-flop relation, from which incidence-algebra derived equivalence is asserted to follow. The simple-projective hypothesis is an explicit global assumption stated at the outset rather than an implicit definitional step. No equations reduce a claimed prediction to a fitted input, no load-bearing self-citation chain is invoked, and the generalization of Ladkani proceeds via direct poset embeddings rather than renaming or self-definition. The paper is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The claim rests on standard background results in representation theory of finite-dimensional algebras (torsion classes, silting objects, s-torsion pairs, incidence algebras) and the definition of 1-APR tilt; no free parameters or invented entities appear in the abstract.

assumptions (1)
  • domain assumption Standard properties of finite-dimensional algebras, torsion classes, and derived equivalences hold as in the literature of math.RT.
    Invoked implicitly as the setting for the generalization of Ladkani's result.

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Cite this review

Pith. "Pith review of Derived equivalence of posets of torsion classes." pith.science (2026). https://pith.science/paper/XQJNAQPS

@misc{pith2026260621239,
  author       = {Pith},
  title        = {Pith review of: Derived equivalence of posets of torsion classes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XQJNAQPS}},
  note         = {Machine review of arXiv:2606.21239}
}
read the original abstract

We investigate the structure of the poset of torsion classes for a finite dimensional algebra admitting a simple projective module. We generalise a result of Ladkani by showing that if two algebras are related by a 1-APR tilt, then their posets of torsion classes are related by a flip-flop. This implies that the incidence algebras of the posets are derived equivalent. We give two different proofs of this result. The first one applies to functorially finite torsion classes, and a key ingredient is to see the two posets we want to relate as subposets of a common poset of silting objects. The second proof applies to arbitrary torsion classes, and we use a similar strategy, this time embedding the two posets into a common poset of s-torsion pairs.

Figures

Figures reproduced from arXiv: 2606.21239 by the authors.

Figure 1
Figure 1. Two posets related by a flip-flop. a reflection, the posets of functorially finite torsion classes ff-tors kQ and ff-tors kQ′ (under a different name) are related by a flip-flop. The goal of this paper is to extend this result in two different directions. First, we broaden the class of pairs of algebras whose posets of torsion classes we are able to compare, from hereditary algebras to algebras of arbitrary global d… view at source ↗
Figure 2
Figure 2. The arrows represent the partial order, with the convention that an arrow x → y means x ≥ y, and the vertical arrows are given by mutation, so they correspond to covering relations. Proof. We know that the poset 2-silt(inj Γ) is equal to the interval DΓ, Σ −1DΓ [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. The situation in Example 11.2. are indicated with solid black lines. Their respective hearts H = U ∩ ΣV and H′ = U ′ ∩ ΣV ′ are indicated with solid blue lines. In each of these hearts, the torsion pairs (T , F) and (T ′ , F ′ ) are indicated in the following way: the torsion part is circled with a solid red line and filled with dots, and the torsion-free part is circled with a dashed red line and not filled. Lemma … view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Structure of the poset s-tors C. Moreover, for X ∈ F, by definition of θ, the object θ(X) ∈ T ′ is characterised by the existence of a triangle of the form Σ −1F ′ → θ(X) → X → F ′ with F ′ ∈ F′ (since Hom(X, T ) = 0), and we have a dual characterisation for ξ. This im…

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Works this paper leans on

10 extracted references · 5 canonical work pages

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