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arxiv: 2509.12983 · v2 · submitted 2025-09-16 · 🧮 math.RT · math.AC· math.CT

Detecting derived equivalences with the CHZ criterion

Sergio Pavon This is my paper

Pith reviewed 2026-05-18 16:40 UTC · model grok-4.3

classification 🧮 math.RT math.ACmath.CT MSC 18E3016G10
keywords CHZ criterionHRS-tilttorsion pairderived equivalencestable torsion pairArtin algebrasilting mutationtilting complex
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The pith

Stable torsion pairs always induce derived equivalences via HRS-tilt in any abelian category.

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

The paper applies the CHZ criterion from 2018 to determine when the HRS-tilt of a torsion pair produces a derived equivalence. It proves that every stable torsion pair in an arbitrary abelian category satisfies the criterion and therefore always induces such an equivalence. The work further applies the criterion to abelian categories of global dimension at most two, gives a purely combinatorial criterion for (co)hereditary torsion pairs over Artin algebras, and examines whether irreducible silting mutation acts transitively on two-term tilting complexes over finite dimensional algebras. A sympathetic reader would care because these results supply practical tests for detecting derived equivalences in homological algebra without computing the entire derived category.

Core claim

The author uses the CHZ criterion to establish that the HRS-tilt at any stable torsion pair in an arbitrary abelian category induces a derived equivalence. The paper also provides applications to categories of global dimension at most two, a combinatorial criterion for derived equivalence in the case of (co)hereditary torsion pairs over Artin algebras, and an analysis of transitivity questions for irreducible silting mutation on two-term tilting complexes.

What carries the argument

The CHZ criterion, which checks whether the HRS-tilt of a given torsion pair induces a derived equivalence.

If this is right

  • Stable torsion pairs in arbitrary abelian categories always induce derived equivalences under HRS-tilt.
  • The criterion applies directly in abelian categories of global dimension at most two.
  • For (co)hereditary torsion pairs over Artin algebras, derived equivalence reduces to a purely combinatorial condition.
  • The criterion can be used to investigate whether irreducible silting mutation acts transitively on two-term tilting complexes over finite dimensional algebras.

Where Pith is reading between the lines

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

  • Derived equivalences may occur more frequently than expected among stable torsion pairs across general abelian settings.
  • The combinatorial criterion for Artin algebras could enable computational classification of tilting phenomena in representation theory.
  • Similar detection methods might extend to questions of equivalence in triangulated or higher homological settings.

Load-bearing premise

The CHZ criterion applies verbatim to the HRS-tilt constructed from the torsion pair in the stated settings of abelian categories and Artin algebras.

What would settle it

An explicit counterexample of a stable torsion pair in some abelian category where the corresponding HRS-tilt fails to induce a derived equivalence.

Figures

Figures reproduced from arXiv: 2509.12983 by Sergio Pavon.

Figure 1
Figure 1. Figure 1: The Hasse quiver of tors(Λ), for the algebra Λ = k(• → • → •). Each torsion pair is represented with the shape of the Auslander–Reiten quiver of Λ, marking the torsion modules with •, the torsion-free modules with ◦ and the others with ∗. If we choose M to be the direct sum of the indecomposable injectives which are not projective, the corresponding partition of tors(Λ) into the sets tors(M/ε) is shaded in… view at source ↗
Figure 2
Figure 2. Figure 2: The Hasse quiver of the lattice tors(Λ), for the Kronecker algebra Λ = k(• ⇒ •) (see e.g. [30, Ex. 1.3] for an explicit description of the lattice points). The finitely many intervals of the partition given by the sets tors((Λ ⊕ DΛ)/ε) are highlighted, with the ones lying in torsd (Λ) shaded in gray. To obtain this, one can inspect the actual torsion pairs, which shows that, except for the one in the cente… view at source ↗
read the original abstract

In 2018, Chen, Han and Zhou introduced a criterion to determine whether the HRS-tilt at a given torsion pair induces derived equivalence. We showcase four applications of this criterion: to stable torsion pairs in arbitrary abelian categories (which we prove to always induce derived equivalence), to abelian categories of global dimension at most two, to (co)hereditary torsion pairs over artin algebras (for which we give a purely combinatorial criterion for derived equivalence), and to study whether irreducible silting mutation acts transitively on two-term tilting complexes over a finite dimensional algebra.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript applies the 2018 Chen-Han-Zhou (CHZ) criterion to detect when the HRS-tilt of a torsion pair induces a derived equivalence. It proves that every stable torsion pair in an arbitrary abelian category yields such an equivalence, supplies a combinatorial criterion for (co)hereditary torsion pairs over Artin algebras, treats abelian categories of global dimension at most two, and examines transitivity of irreducible silting mutation on two-term tilting complexes over finite-dimensional algebras.

Significance. If the central claims hold, the work usefully broadens the reach of the CHZ criterion, especially via the general result on stable torsion pairs. The combinatorial test for Artin algebras provides a concrete, checkable condition that avoids direct computation of derived categories, while the silting-mutation application connects to ongoing work on tilting complexes. These results could serve as a practical toolkit for homological algebra and representation theory.

major comments (2)
  1. [§3, Theorem 3.1] §3, Theorem 3.1: The claim that stable torsion pairs in arbitrary abelian categories always induce derived equivalence rests on the CHZ criterion (presumably Thm. 1.1 or equivalent) applying verbatim to the HRS-tilt. The manuscript does not explicitly verify that the tilted category satisfies CHZ hypotheses such as the existence of a compact generator or boundedness of Hom-spaces, which are not part of the axioms of a general abelian category; stability of the torsion pair alone may not transfer these properties.
  2. [§5] §5, combinatorial criterion: The reduction to a purely combinatorial test for (co)hereditary torsion pairs over Artin algebras is presented as following directly from CHZ, but the manuscript should include an explicit check that no additional hidden assumptions (e.g., finite-dimensionality of Ext groups) are used in the translation from the CHZ conditions to the combinatorial statement.
minor comments (2)
  1. [§2–§3] Notation for torsion pairs and their tilts is not fully standardized between §2 and §3; a single table of symbols would improve readability.
  2. [Introduction] The abstract states four applications but the introduction does not clearly delineate which results are new versus direct consequences of CHZ; a short roadmap paragraph would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive evaluation of its significance. We address the two major comments point by point below, indicating where revisions will be made to strengthen the exposition.

read point-by-point responses

  1. Referee: [§3, Theorem 3.1] The claim that stable torsion pairs in arbitrary abelian categories always induce derived equivalence rests on the CHZ criterion applying verbatim to the HRS-tilt. The manuscript does not explicitly verify that the tilted category satisfies CHZ hypotheses such as the existence of a compact generator or boundedness of Hom-spaces, which are not part of the axioms of a general abelian category; stability of the torsion pair alone may not transfer these properties.

    Authors: We appreciate this observation. The CHZ criterion is invoked on the derived category D(A) of the original abelian category A, where the standard t-structure and the HRS-tilt are considered. Stability of the torsion pair ensures that the tilted heart is again an abelian category whose derived category coincides with D(A), thereby inheriting any compact generator or boundedness properties that D(A) possesses from the original setting. Nevertheless, to make the application fully transparent, we will add a short paragraph immediately after the statement of Theorem 3.1 that explicitly recalls the relevant hypotheses of the CHZ criterion and verifies that they hold for the HRS-tilt of any stable torsion pair in an arbitrary abelian category. revision: yes

  2. Referee: [§5] The reduction to a purely combinatorial test for (co)hereditary torsion pairs over Artin algebras is presented as following directly from CHZ, but the manuscript should include an explicit check that no additional hidden assumptions (e.g., finite-dimensionality of Ext groups) are used in the translation from the CHZ conditions to the combinatorial statement.

    Authors: We agree that an explicit remark would improve clarity. Over an Artin algebra, all finitely generated modules have finite length, so Hom and Ext groups between them are finite-dimensional vector spaces over the residue field; this is a standard feature of the category and is already implicit in the setup of the paper. The combinatorial criterion is obtained simply by rewriting the vanishing/non-vanishing conditions of the CHZ criterion in terms of the support of the torsion pair on the Auslander-Reiten quiver or the quiver of the algebra. We will insert a brief explanatory paragraph at the beginning of §5 that records this translation step and confirms that no extra hypotheses beyond those of the CHZ criterion and the Artin-algebra setting are required. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external CHZ criterion to new cases

full rationale

The paper's central claims rest on verifying the hypotheses of the 2018 CHZ criterion (an external result by different authors) when applied to HRS-tilts arising from stable torsion pairs, abelian categories of global dimension ≤2, (co)hereditary torsion pairs over Artin algebras, and silting mutations. No equations, parameters, or steps within the paper reduce the claimed derived equivalences to quantities defined by fitting or self-reference; the work instead checks applicability of the cited theorem in fresh settings. This is self-contained against the external benchmark and exhibits no self-definitional, fitted-prediction, or self-citation-load-bearing reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on the standard definitions of abelian categories, torsion pairs, HRS-tilts, and the CHZ criterion itself; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • standard math Standard definitions and closure properties of abelian categories, torsion pairs, and HRS-tilts as developed in the homological algebra literature.
    The paper invokes these background notions without re-deriving them.

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Reference graph

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