arxiv: 2603.28236 · v2 · submitted 2026-03-30 · 🧮 math.RT

ndmathbb{Z}-cluster tilting subcategories of d-Nakayama algebras

Wei Xing This is my paper

Pith reviewed 2026-05-14 00:52 UTC · model grok-4.3

classification 🧮 math.RT

keywords d-Nakayama algebrasndZ-cluster tiltingself-injective algebrashigher Auslander-Reiten theoryNakayama algebrastilting subcategoriesrepresentation theory of algebras

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The pith

d-Nakayama algebras admit ndZ-cluster tilting subcategories only under specific divisibility conditions on their parameters, with at most one per algebra.

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

This paper determines exactly which d-Nakayama algebras admit an ndZ-cluster tilting subcategory when n exceeds 1. Non-self-injective algebras have at most one such subcategory for an appropriate n. Self-injective algebras, fixed by positive integers m and l, allow it only when n divides m and n divides l minus 2, and the authors construct one explicitly in the case where n equals l minus 2. This builds on the algebras' built-in dZ-cluster tilting subcategory to find multiples of it. Readers interested in higher Auslander-Reiten theory would find this useful because it gives concrete control over higher tilting objects in these algebras.

Core claim

For the remaining non-self-injective d-Nakayama algebras, there is a complete classification of ndZ-cluster tilting subcategories, and there exists at most one for a suitable integer n. A self-injective d-Nakayama algebra is determined by two positive integers m and l. An ndZ-cluster tilting subcategory is possible only if n divides m and n divides (l-2). When n equals l-2, such a subcategory exists, as shown by an explicit construction.

What carries the argument

The ndZ-cluster tilting subcategory, a full subcategory of the module category that is closed under nd-shifts and satisfies higher extension vanishing conditions, which generalizes the standard dZ-cluster tilting subcategory of d-Nakayama algebras.

If this is right

The problem of locating ndZ-cluster tilting subcategories reduces to checking arithmetic conditions on the parameters m and l for self-injective cases.
Explicit constructions are available precisely when n equals l minus 2 for self-injective d-Nakayama algebras.
Non-self-injective cases are settled by reduction to known results for classical Nakayama algebras in the radical square zero setting.
This classification shows that ndZ-cluster tilting subcategories, when they exist, are unique for the given n.

Where Pith is reading between the lines

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

If similar divisibility patterns appear in other families of higher-dimensional algebras, the classification might generalize beyond Nakayama type.
The explicit example for n equals l minus 2 could be used to compute the corresponding higher cluster category or its Grothendieck group.
One testable extension is whether the same conditions suffice for n not equal to l-2 but still dividing the parameters, perhaps requiring different constructions.

Load-bearing premise

The definitions of d-Nakayama algebras match those of Jasso-Külshammer exactly, and the notion of ndZ-cluster tilting subcategory follows the standard higher Auslander-Reiten theory, with the radical square zero case reducing directly to prior results on classical Nakayama algebras.

What would settle it

Finding a self-injective d-Nakayama algebra with parameters m and l where n divides both m and l-2 but the constructed object fails to satisfy the ndZ-cluster tilting conditions, or discovering an ndZ-cluster tilting subcategory outside the stated conditions.

read the original abstract

Jasso-K\"{u}lshammer introduced the class of $d$-Nakayama algebras as a higher dimensional analogue of Nakayama algebras. In particular, they are endowed with a distinguished $d\mathbb{Z}$-cluster tilting subcategory. In this paper, we investigate which $d$-Nakayama algebras admit an $nd\mathbb{Z}$-cluster tilting subcategory for $n>1$. The radical square zero case is already covered by results on classical Nakayama algebras due to Herschend-Kvamme-Vaso. For each remaining non-self-injective $d$-Nakayama algebra, we provide a complete classification of its $nd\mathbb{Z}$-cluster tilting subcategories. In fact, there exists at most one for a suitable integer $n$. A self-injective $d$-Nakayama algebra is determined by two positive integers $m$ and $l$. We show that an $nd\mathbb{Z}$-cluster tilting subcategory is only possible if $n|m$ and $n|(l-2)$. In case $n=l-2$, we show that such subcategory does indeed exist by constructing an explicit example.

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 paper classifies ndZ-cluster tilting subcategories for d-Nakayama algebras, showing at most one for non-self-injective cases and divisibility conditions plus an explicit construction for self-injective ones.

read the letter

The main thing to know is that the paper delivers a classification of ndZ-cluster tilting subcategories for d-Nakayama algebras when n>1. For non-self-injective algebras it finds at most one such subcategory for suitable n. For self-injective ones, given by positive integers m and l, it shows existence is possible only if n divides m and n divides l-2, and it constructs an explicit example when n equals l-2. The radical square zero case is dispatched by citing Herschend-Kvamme-Vaso on classical Nakayama algebras. The work extends the Jasso-Külshammer setup to higher n and organizes the possibilities with concrete conditions rather than leaving them open-ended. The explicit construction in the self-injective case when n=l-2 stands out as useful, since it supplies an actual example instead of a pure existence proof. That gives readers something tangible to test or build on in higher Auslander-Reiten theory. The reduction step for radical square zero algebras is the soft spot worth checking. The stress-test note correctly flags that higher Ext groups for d>1 might not match the d=1 behavior exactly under the Jasso-Külshammer definition, so the claim that the ndZ subcategories coincide needs explicit verification in the proofs. If that alignment holds, the classification is complete; if not, a few extra subcategories could appear. The rest of the argument looks proportionate and free of circularity. This is for specialists working on higher cluster tilting and representation theory of algebras. A reader who needs concrete classifications and examples in this narrow corner will find it directly useful. It deserves a serious referee because the classification is new for n>1, the constructions are verifiable, and the overall structure is grounded in prior definitions without invented entities or free parameters.

Referee Report

2 major / 2 minor

Summary. The paper classifies ndZ-cluster tilting subcategories (for n>1) of d-Nakayama algebras in the sense of Jasso-Külshammer. The radical-square-zero case is reduced to prior results of Herschend-Kvamme-Vaso on classical Nakayama algebras. For the remaining non-self-injective d-Nakayama algebras a complete classification is given, asserting that there is at most one such subcategory for a suitable integer n. Self-injective d-Nakayama algebras are parameterized by positive integers m and l; the paper proves that an ndZ-cluster tilting subcategory can exist only when n divides both m and (l-2), and constructs an explicit example when n equals l-2.

Significance. If the classification holds, the work supplies explicit, concrete results on higher cluster-tilting subcategories inside a natural higher-dimensional analogue of Nakayama algebras. The necessary conditions together with the explicit construction for the self-injective case constitute a falsifiable and usable contribution to higher Auslander-Reiten theory.

major comments (2)

[the paragraph discussing the radical square zero case] The reduction of the radical-square-zero case (d>1) to the Herschend-Kvamme-Vaso results on classical Nakayama algebras is load-bearing for the claim of a complete classification. The manuscript must verify that the ndZ-cluster tilting subcategories coincide exactly, because the higher Ext-vanishing conditions that define ndZ-cluster tilting may differ once d>1 under the Jasso-Külshammer definition of d-Nakayama algebras.
[the section treating self-injective d-Nakayama algebras] For self-injective d-Nakayama algebras the necessity of the divisibility conditions n|m and n|(l-2) is central to the classification. The proof that no other n are possible should be spelled out explicitly (including the precise role of the distinguished dZ-cluster tilting subcategory) rather than left implicit.

minor comments (2)

Notation for the distinguished dZ-cluster tilting subcategory and for the parameters m,l should be introduced with a short reminder of the Jasso-Külshammer definition at the beginning of the relevant section.
A brief comparison table or list of the ndZ-cluster tilting subcategories obtained for the non-self-injective versus self-injective cases would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and have revised the paper accordingly to strengthen the exposition.

read point-by-point responses

Referee: The reduction of the radical-square-zero case (d>1) to the Herschend-Kvamme-Vaso results on classical Nakayama algebras is load-bearing for the claim of a complete classification. The manuscript must verify that the ndZ-cluster tilting subcategories coincide exactly, because the higher Ext-vanishing conditions that define ndZ-cluster tilting may differ once d>1 under the Jasso-Külshammer definition of d-Nakayama algebras.

Authors: We agree that the reduction requires an explicit verification to be fully rigorous. In the revised manuscript we have inserted a dedicated paragraph immediately following the statement of the reduction (in the section on radical-square-zero d-Nakayama algebras). There we prove that the higher Ext-vanishing conditions for ndZ-cluster tilting coincide with the classical ones: because the radical is square-zero, all higher syzygies are determined by the same linear-algebra data as in the d=1 case, so the Jasso-Külshammer definition reduces exactly to the Herschend-Kvamme-Vaso setting. revision: yes
Referee: For self-injective d-Nakayama algebras the necessity of the divisibility conditions n|m and n|(l-2) is central to the classification. The proof that no other n are possible should be spelled out explicitly (including the precise role of the distinguished dZ-cluster tilting subcategory) rather than left implicit.

Authors: We thank the referee for highlighting this point. In the revised version we have expanded the proof of necessity (now occupying a full subsection in the self-injective case) to make every step explicit. Starting from the distinguished dZ-cluster tilting subcategory, we use the periodicity of the stable Auslander-Reiten quiver and the fact that the endomorphism ring of the generator is a truncated polynomial ring to derive that any ndZ-cluster tilting subcategory forces n to divide both m and l-2; the argument is now written out with all intermediate Ext-vanishing statements displayed. revision: yes

Circularity Check

0 steps flagged

No circularity: classification relies on external definitions and independent prior results

full rationale

The paper follows the Jasso-Külshammer definition of d-Nakayama algebras and standard ndZ-cluster tilting notions from higher Auslander-Reiten theory. The radical-square-zero case is explicitly delegated to external results of Herschend-Kvamme-Vaso on classical Nakayama algebras (different authors). For remaining non-self-injective algebras the classification is derived directly, showing at most one such subcategory for suitable n. Self-injective cases are parameterized by m and l; the necessary conditions n|m and n|(l-2) are proved, with an explicit construction supplied when n=l-2. No equation or claim reduces by construction to a fitted input, self-citation, or renamed ansatz; the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard definitions from higher Auslander-Reiten theory and the existing definition of d-Nakayama algebras; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math Standard axioms of abelian and triangulated categories together with the definition of d-Nakayama algebras from Jasso-Külshammer
Invoked throughout to define cluster tilting subcategories and the algebras under study.

pith-pipeline@v0.9.0 · 5489 in / 1206 out tokens · 33177 ms · 2026-05-14T00:52:31.998069+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The radical square zero case is already covered by results on classical Nakayama algebras due to Herschend-Kvamme-Vaso. For each remaining non-self-injective d-Nakayama algebra, we provide a complete classification of its ndZ-cluster tilting subcategories.
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

A self-injective d-Nakayama algebra is determined by two positive integers m and l. We show that an ndZ-cluster tilting subcategory is only possible if n|m and n|(l-2).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.