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arxiv: 2605.27263 · v1 · pith:56K6THISnew · submitted 2026-05-26 · 🧮 math.RT · math.CO· math.CT

Relations between categorifications of higher-dimensional type A cluster combinatorics

Mikhail Gorsky , Nicholas J. Williams This is my paper

Pith reviewed 2026-06-29 14:35 UTC · model grok-4.3

classification 🧮 math.RT math.COmath.CT
keywords higher Auslander algebrasd-cluster-tilting objectsd-exangulated categoriescluster combinatoricstype Aderived categoriessilting complexestilting modules
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The pith

The d-almost positive subcategory equals the d-exangulated quotient of the d-exact subcategory and the (d+2)-angulated cluster category by ideals generated by morphisms from injectives to projectives.

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

The paper shows that three categories built from higher Auslander algebras of type A stand in a precise quotient relation: the d-almost positive subcategory of the derived category is obtained from the d-exact subcategory generated by a d-cluster-tilting object together with the (d+2)-angulated cluster category by quotienting out the ideals of morphisms that factor through maps from injective to projective objects. This relation is formulated in the language of d-exangulated categories. A reader might care because the identification supplies a direct algebraic link between two constructions that both aim to categorify the same higher-dimensional cluster combinatorics and accounts for a numerical match between the count of 2-term silting complexes in type A_n and tilting modules in type A_{n+1}. The construction is presented as a concrete instance of a higher-dimensional version of the 0-Auslander correspondence.

Core claim

The d-almost positive subcategory of the derived category is the d-exangulated quotient of the d-exact subcategory of the module category of A^d_{n+1} generated by the d-cluster-tilting object and the (d+2)-angulated cluster category by the ideals generated by morphisms factoring through morphisms from injective to projective objects.

What carries the argument

The d-exangulated quotient by the ideals of morphisms factoring through injective-to-projective maps, which identifies the three categories arising from higher Auslander algebras of type A.

If this is right

  • The identification supplies an algebraic link between two models that both categorify d-dimensional cluster combinatorics in type A.
  • The relation accounts for the observed numerical coincidence between the number of 2-term silting complexes in type A_n and the number of tilting modules in type A_{n+1}.
  • The construction realises a d-exangulated version of a known relation between categories in the classical (d=1) setting.
  • The result positions the higher Auslander algebra construction as a prototypical example of the 0-Auslander correspondence in higher homological algebra.

Where Pith is reading between the lines

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

  • The same quotient construction may apply verbatim to other Dynkin types once the corresponding higher Auslander algebras are defined.
  • Similar d-exangulated quotients could relate additional models of higher cluster combinatorics that have not yet been compared.
  • The numerical coincidence explained here may admit a direct combinatorial proof independent of the categorical quotient.
  • Explicit small-case calculations of the categories could serve as an independent check before attempting general proofs in other settings.

Load-bearing premise

The three categories are constructed exactly as described from the higher Auslander algebras of type A, and the d-exangulated quotient by the given ideals is well-defined and equals the d-almost positive subcategory.

What would settle it

Direct computation of the three categories for d=1 and small n, followed by explicit verification that the quotient category has the same objects and morphism spaces as the d-almost positive subcategory.

Figures

Figures reproduced from arXiv: 2605.27263 by Mikhail Gorsky, Nicholas J. Williams.

Figure 1
Figure 1. Figure 1: Examples of the quivers Q(d,n) 1 2 3 Q(1,3) 13 14 15 24 25 35 Q(2,3) 135 136 137 146 147 157 246 247 257 357 Q(3,3) where 1i := (0, . . . , 0, i 1, 0, . . . , 0). Examples of these quivers are shown in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The category add M(2,3) M135 M136 M137 M146 M147 M157 M246 M247 M257 M357 (6) The unique-up-to-scalar morphisms MA → MB and MB → MC compose to give a non-zero morphism MA → MC if and only if (A − 1) ≀ C. The category add M(d,n) is d-exact in the sense of [Jas16], and therefore d-exangulated by [HLN21, Proposition 4.34]. Indeed, we have that E(MB, MA) = Extd Ad n (MB, MA), with the exact realisation s given… view at source ↗
Figure 3
Figure 3. Figure 3: The category UA2 3 U135 U146 U157 U136 U147 U137 U246 U257 U268 U247 U258 U248 U357 U368 U379 U358 U369 U359 . . . . . . . . . . . . . . . . . . (5) The unique-up-to-scalar morphisms UA → UB and UB → UC compose to give a non-zero mor￾phism UA → UC if and only if a0 − 1 < c0 < a1 − 1 < c1 < · · · < ad − 1 < cd < a0 + n + 2d. By [GKO13, Theorem 1], we have that UAd n is a (d+2)-angulated category. Hence, by … view at source ↗
Figure 4
Figure 4. Figure 4: The category OA2 3 O135 O146 O157 O136 O147 O137 O246 O257 O268 O247 O258 O248 O357 O368 O137 O358 O136 O135 . . . . . . . . . . . . . . . . . . 2.0.5. The d-almost positive category. Given a d-representation-finite d-hereditary algebra Λ with d￾cluster-tilting module M, define the d-almost positive subcategory U {−d,0} Λ to be add(M ⊕ Λ[d]) of UΛ, following [Wil23b]. Remark 2.9. For d = 1, the almost posi… view at source ↗
Figure 5
Figure 5. Figure 5: The category U {−d,0} A2 3 U135 U136 U137 U146 U147 U157 U246 U247 U257 U248 U258 U268 U357 U358 U368 U468 (3) Moreover, if E(UB, UA) ̸= 0, then the exact realisation s is given by scalar multiples of the homotopy equivalence class of the distinguished d-exangle UA → Hd → Hd−1 → · · · → H1 → UB 99K, where Hr = M I⊆{0,1,...,d} mI (A,B)∈˜I d n+2d+1 |I|=r UmI (A,B) , which is not nullhomotopic. (4) The unique… view at source ↗
read the original abstract

We consider three categories arising from the higher Auslander algebras of type $A$ in relation to $d$-dimensional cluster combinatorics: $d$-exact subcategory of the module category of $A^d_{n+1}$ generated by the $d$-cluster-tilting object, the $(d+2)$-angulated cluster category, and the $d$-almost positive subcategory of the derived category (the higher analogue of the category of two-term complexes of projectives). We show that the third one, introduced by the second-named author, is the $d$-exangulated quotient of the other two, introduced by Oppermann and Thomas, by the ideals generated by morphisms factoring through morphisms from injective to projective objects, thus providing an algebraic connection between the two models of Oppermann-Thomas. This is a $d$-exangulated version in type $A$ of a result of Br\"ustle and Yang and its interpretation by the first-named author together with Fang, Palu, Plamondon and Pressland. It also explains a well-known coincidence between the number of 2-term silting complexes in type $A_{n}$ and of tilting modules in type $A_{n+1}$ from the $0$-Auslander perspective. We expect this to serve as a prototypical example of the $0$-Auslander correspondence in higher homological 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

0 major / 2 minor

Summary. The paper considers three categories from higher Auslander algebras of type A in d-dimensional cluster combinatorics: the d-exact subcategory of mod(A^d_{n+1}) generated by the d-cluster-tilting object (Oppermann-Thomas), the (d+2)-angulated cluster category (Oppermann-Thomas), and the d-almost positive subcategory of the derived category (introduced by the second author). It claims to prove that the third is the d-exangulated quotient of the first two by the ideals generated by morphisms factoring through injective-to-projective maps. This supplies a d-exangulated type-A analogue of the Brüstle-Yang result (and its 0-Auslander interpretation) and accounts for the numerical coincidence between 2-term silting complexes in A_n and tilting modules in A_{n+1}.

Significance. If the identification holds, the result supplies an explicit algebraic link between two Oppermann-Thomas models via a quotient construction that is well-defined in the d-exangulated setting. It extends the Brüstle-Yang correspondence to higher homological algebra without introducing free parameters or circular definitions, and it furnishes a concrete 0-Auslander perspective on the noted numerical coincidence. The manuscript positions the construction as a prototypical example for future higher-dimensional correspondences.

minor comments (2)
  1. The abstract and introduction refer to the 'd-exangulated quotient' without an explicit citation to the precise definition of d-exangulated categories or the quotient construction used; adding a reference to the relevant prior work on d-exangulated structures would improve readability.
  2. Notation for the three categories (e.g., the precise symbol for the d-almost positive subcategory) is introduced only in the abstract; a short notational table or consistent symbols in §1 would aid cross-reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of its contents, and the recommendation of minor revision. We will incorporate any minor suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity; minor self-citation not load-bearing

full rationale

The paper establishes an algebraic connection showing that the d-almost positive subcategory equals the d-exangulated quotient of the d-exact subcategory and the (d+2)-angulated cluster category by the ideal of morphisms factoring through injective-to-projective maps. This is framed as a higher-dimensional type A analogue of the Brüstle-Yang result (with interpretation in prior joint work by the first author), but the central constructions originate from independent sources (Oppermann-Thomas for two of the categories, and standard higher Auslander algebra definitions for the third). No step reduces a prediction or uniqueness claim to a self-citation chain, fitted parameter, or definitional tautology; the quotient is asserted to be well-defined on the given inputs without circular reduction. The single self-citation is contextual and does not bear the load of the new relation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract only; no explicit free parameters, invented entities, or ad-hoc axioms are mentioned. The result relies on standard definitions and properties of higher Auslander algebras, d-exangulated categories, and type A constructions from prior works.

axioms (1)
  • domain assumption The d-exact subcategory, (d+2)-angulated cluster category, and d-almost positive subcategory are constructed from higher Auslander algebras of type A as introduced in the cited works of Oppermann-Thomas and the second author.
    The paper invokes these prior constructions without re-deriving them.

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