Extended Module Categories in Higher Cluster Tilting Theory

Nao Mochizuki

arxiv: 2605.24607 · v1 · pith:LLSV457Xnew · submitted 2026-05-23 · 🧮 math.RT · math.CT

Extended Module Categories in Higher Cluster Tilting Theory

Nao Mochizuki This is my paper

Pith reviewed 2026-06-30 12:11 UTC · model grok-4.3

classification 🧮 math.RT math.CT

keywords higher cluster tiltingideal quotienttriangulated categoryd-truncated DG-categoryextended module categoryMorita theoremabelian category

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The pith

The ideal quotient of a triangulated category by a (d+1)-cluster tilting subcategory is equivalent to a d-extended module category over a d-truncated DG-category.

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

The paper generalizes the Koenig-Zhu theorem, which states that the ideal quotient by a 2-cluster tilting subcategory is an abelian category equivalent to the module category over that subcategory. For general d, it proves that the natural DG-enhancement of the ideal quotient by a (d+1)-cluster tilting subcategory is an abelian d-truncated DG-category. An appendix establishes a Morita-type theorem showing that any abelian d-truncated DG-category with enough projectives is equivalent to a d-extended module category over a d-truncated DG-category. This yields the stated equivalence as the main application. A sympathetic reader would care because the result supplies an explicit module-theoretic model for these quotients, which appear in higher homological algebra and the study of cluster categories.

Core claim

The natural DG-enhancement of the ideal quotient of a triangulated category by a (d+1)-cluster tilting subcategory is an abelian d-truncated DG-category. The appendix proves that an abelian d-truncated DG-category with enough projectives is equivalent to a d-extended module category over a d-truncated DG-category. Therefore the ideal quotient itself is equivalent to such a d-extended module category.

What carries the argument

The ideal quotient by the (d+1)-cluster tilting subcategory and its natural DG-enhancement, shown to be abelian as a d-truncated DG-category, together with the Morita-type theorem that identifies it with a d-extended module category.

If this is right

The quotient inherits an abelian structure controlled by the higher cluster tilting property.
Properties of the quotient can be transferred from the corresponding d-extended module category.
The construction supplies a concrete description of the quotient in terms of modules over a d-truncated DG-category.
The Morita theorem extends classical module-category equivalences to the truncated DG setting.

Where Pith is reading between the lines

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

The result may permit constructing examples of higher cluster tilting subcategories by starting from known d-truncated DG-categories and their module categories.
It suggests a route for studying when ideal quotients in other triangulated settings admit similar DG-enhancements and equivalences.
For concrete d values, the equivalence could be used to compute invariants of the quotient directly from the DG-algebra side.

Load-bearing premise

The triangulated category admits a (d+1)-cluster tilting subcategory whose ideal quotient admits a natural DG-enhancement that is abelian when regarded as a d-truncated DG-category.

What would settle it

A triangulated category equipped with a (d+1)-cluster tilting subcategory whose ideal quotient has a DG-enhancement that fails to be abelian as a d-truncated DG-category.

read the original abstract

In this paper, we study ideal quotients of triangulated categories by higher cluster tilting subcategories. Koenig and Zhu proved that the ideal quotient by a $2$-cluster tilting subcategory is an abelian category; moreover, by Morita's theorem, it is equivalent to the module category over the $2$-cluster tilting subcategory. We generalize this result to higher cluster tilting subcategories. More precisely, we show that the natural DG-enhancement of the ideal quotient of a triangulated category by a $(d+1)$-cluster tilting subcategory is an abelian $d$-truncated DG-category. In the appendix, we prove a Morita-type theorem for abelian $d$-truncated DG-categories, which asserts that an abelian $d$-truncated DG-category with enough projectives is equivalent to a $d$-extended module category over a $d$-truncated DG-category. As an application, we show that the ideal quotient of a triangulated category by a $(d+1)$-cluster tilting subcategory is equivalent to a $d$-extended module category over a $d$-truncated DG-category.

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.

Desk Editor's Note private letter to a colleague

This generalizes the Koenig-Zhu result on 2-cluster tilting quotients to arbitrary d via a DG-enhancement that is abelian d-truncated, plus an appendix Morita theorem.

read the letter

The main result is that the ideal quotient by a (d+1)-cluster tilting subcategory admits a natural DG-enhancement that is abelian as a d-truncated DG-category, and is therefore equivalent to a d-extended module category over a d-truncated DG-category. The appendix supplies the Morita-type theorem that makes the equivalence work for abelian d-truncated DG-categories with enough projectives.

What is new is the move from the d=2 case to general d together with the appendix theorem. The generalization is not routine; it requires setting up the truncated DG-structure and verifying the abelian property in that setting. The paper organizes examples from higher homological algebra under one description and gives a uniform statement.

The argument structure looks clean. The appendix result stands independently once the abelian d-truncated property and the existence of enough projectives are in place, and the stress-test finds no internal inconsistency or missing step from the outline.

The soft spot is the step that establishes the DG-enhancement is abelian. That is invoked directly in the main theorem and the appendix, so a referee would want explicit checks that it holds for the triangulated categories under consideration and that the projectives behave as needed. This is a standard point to verify rather than a load-bearing flaw.

The paper is for people working in representation theory and higher homological algebra, especially those already using cluster tilting or DG-enhancements. A reader in that subfield gets a useful uniform description and a new technical tool in the appendix.

Send it to peer review. The generalization and the appendix theorem are substantive enough to warrant referee time.

Referee Report

1 major / 2 minor

Summary. The paper generalizes the Koenig-Zhu theorem: the ideal quotient of a triangulated category by a 2-cluster tilting subcategory is abelian and, by Morita, equivalent to the module category over the subcategory. For a (d+1)-cluster tilting subcategory, the authors show that the natural DG-enhancement of the ideal quotient is an abelian d-truncated DG-category. The appendix proves a Morita-type theorem asserting that any abelian d-truncated DG-category with enough projectives is equivalent to a d-extended module category over a d-truncated DG-category. The main application is therefore that the ideal quotient is equivalent to such a d-extended module category.

Significance. If the claims hold, the work supplies a higher-dimensional extension of a classical result in cluster-tilting theory, relating ideal quotients to extended module categories through DG-enhancements. The appendix Morita theorem for abelian d-truncated DG-categories is an independent contribution that may be of separate interest. The argument structure relies on an independent appendix theorem rather than circular reduction, and the paper ships a self-contained generalization with an explicit statement of the required abelian d-truncated property.

major comments (1)

[Main theorem and its proof (body of the paper, after the definitions of d-truncated DG-category)] The central claim that the natural DG-enhancement of the ideal quotient is abelian as a d-truncated DG-category (invoked for the main theorem and the appendix) is load-bearing; the manuscript must verify explicitly that kernels and cokernels exist in the d-truncated sense and that the truncation functor preserves the abelian structure. Without a concrete check of this property (e.g., in the section containing the proof of the main result), the reduction to the appendix Morita theorem cannot be confirmed.

minor comments (2)

[Introduction] The introduction should state the precise definition of 'd-extended module category' at the outset, since this is the target of the final equivalence.
[Appendix (Morita-type theorem)] In the appendix, the hypothesis 'with enough projectives' is used; add a short remark confirming that the ideal quotient in the main application satisfies this hypothesis.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important point regarding the explicit verification in the proof. We address the major comment below.

read point-by-point responses

Referee: [Main theorem and its proof (body of the paper, after the definitions of d-truncated DG-category)] The central claim that the natural DG-enhancement of the ideal quotient is abelian as a d-truncated DG-category (invoked for the main theorem and the appendix) is load-bearing; the manuscript must verify explicitly that kernels and cokernels exist in the d-truncated sense and that the truncation functor preserves the abelian structure. Without a concrete check of this property (e.g., in the section containing the proof of the main result), the reduction to the appendix Morita theorem cannot be confirmed.

Authors: We agree that an explicit verification of kernels and cokernels in the d-truncated sense, together with confirmation that the truncation functor preserves the abelian structure, is necessary to make the load-bearing claim fully rigorous and to justify the reduction to the appendix Morita theorem. While the manuscript asserts that the natural DG-enhancement is an abelian d-truncated DG-category, we acknowledge that the current proof would benefit from a more concrete check of these properties. We will add this explicit verification (for instance, via an additional lemma or expanded paragraph in the section containing the proof of the main result) in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation establishes that the natural DG-enhancement of the ideal quotient by a (d+1)-cluster tilting subcategory is an abelian d-truncated DG-category, then invokes the appendix Morita-type theorem (proven independently within the paper) to obtain the equivalence to a d-extended module category. No step reduces a claimed result to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain; the Koenig-Zhu citation is external prior work, and the appendix supplies a new proof under explicitly stated assumptions about projectives and the d-truncated structure. The argument is self-contained against external benchmarks in triangulated category theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard axioms of triangulated categories and DG-categories from the literature; no free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Triangulated categories admit ideal quotients by subcategories that inherit triangulated structure under suitable conditions.
Invoked implicitly when forming the ideal quotient by the cluster tilting subcategory.
domain assumption DG-categories admit natural enhancements and truncations that preserve abelianness under the stated hypotheses.
Central to the claim that the DG-enhancement is abelian d-truncated.

pith-pipeline@v0.9.1-grok · 5708 in / 1365 out tokens · 33337 ms · 2026-06-30T12:11:37.771213+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 2 canonical work pages

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Amiot.Cluster categories for algebras of global dimension 2 and quivers with potential

[Ami09] C. Amiot.Cluster categories for algebras of global dimension 2 and quivers with potential. Ann. Inst. Fourier (Grenoble),59(6):2525–2590,2009. [BM94] A.BeligiannisandN.Marmaridis.Lefttriangulatedcategoriesarisingfromcontravariantlyfinitesubcategories. Comm.Algebra,22(12):5021–5036,1994. [BMR07] A.B.Buan,R.J.Marsh,andI.Reiten.Cluster-tiltedalgebras...

work page arXiv 2009

[2] [2]

[Jin20] H

32 NAOMOCHIZUKI [IY08] O.IyamaandY.Yoshino.MutationintriangulatedcategoriesandrigidCohen-Macaulaymodules.Invent.Math., 172(1):117–168,2008. [Jin20] H. Jin.Cohen-Macaulay differential graded modules and negative Calabi-Yau configurations. Adv. Math., 374:107338,59,2020. [Kel99] B.Keller.Onthecyclichomologyofexactcategories.J.PureAppl.Algebra,136(1):1–56,19...

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Math.Z.,258(1):143–160,2008

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work page arXiv 2008