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arxiv: 2604.19582 · v1 · submitted 2026-04-21 · 🧮 math.RT

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Fractionally Calabi-Yau algebras and cluster tilting

Aaron Chan, Osamu Iyama, Rene Marczinzik

Pith reviewed 2026-05-10 00:54 UTC · model grok-4.3

classification 🧮 math.RT
keywords twisted fractionally Calabi-Yau algebrashigher Auslander algebrasd-cluster tiltingreplication of algebrashigher representation-finite algebrasstable endomorphism algebrasderived equivalences
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The pith

An algebra of finite global dimension is twisted fractionally Calabi-Yau if and only if there exists i such that the replicated algebra A^{(i)} is a higher Auslander algebra if and only if there exist infinitely many i such that A^{(i)} is

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

The paper shows that the class of twisted fractionally Calabi-Yau algebras of finite global dimension coincides with the stable endomorphism algebras of d-cluster tilting modules over d-representation-finite algebras. This follows from the main result that an algebra A of finite global dimension is twisted fractionally Calabi-Yau precisely when some replication A^{(i)} is a higher Auslander algebra, and equivalently when infinitely many such replications are higher Auslander algebras. A sympathetic reader cares because this creates a direct link between higher Auslander-Reiten theory and the study of fractionally Calabi-Yau algebras, allowing constructions and properties to transfer between the two. The paper also derives several applications including characterisations and derived equivalences.

Core claim

An algebra A of finite global dimension is twisted fractionally Calabi-Yau if and only if there exists i such that the replicated algebra A^{(i)} is a higher Auslander algebra if and only if there exist infinitely many i such that A^{(i)} is a higher Auslander algebra. This gives a new connection between the study of higher Auslander-Reiten theory and twisted fractionally Calabi-Yau algebras, and provides a new construction of large classes of higher Auslander algebras and higher representation-finite algebras. We give several applications such as an explicit characterisation of twisted n/2-Calabi-Yau algebras, and a triangle equivalence between the bounded derived category of a twisted

What carries the argument

The replication process producing the algebras A^{(i)}, which translates the twisted fractionally Calabi-Yau property into the higher Auslander algebra property.

If this is right

  • The class of twisted fractionally Calabi-Yau algebras of finite global dimension coincides with the stable endomorphism algebras of d-cluster tilting modules over d-representation-finite algebras.
  • This provides an explicit characterisation of twisted n/2-Calabi-Yau algebras.
  • A triangle equivalence holds between the bounded derived category of a twisted fractionally Calabi-Yau algebra of finite global dimension and the Z-graded stable module category of an associated higher preprojective algebra.
  • New large classes of higher Auslander algebras and higher representation-finite algebras can be constructed using this equivalence.

Where Pith is reading between the lines

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

  • Properties of d-cluster tilting modules can now be used to study twisted fractionally Calabi-Yau algebras and vice versa.
  • The replication construction may be applied to known examples in one area to generate examples in the other.
  • The infinite replication condition suggests that the property is stable under further replication once it holds for one i.
  • This unification may help in classifying algebras with finite global dimension that satisfy either property.

Load-bearing premise

The definitions of twisted fractionally Calabi-Yau algebras and the replication process A^{(i)} are such that the equivalence to higher Auslander algebras holds without additional hidden constraints on the base field or the algebra structure.

What would settle it

A concrete counterexample would be an algebra of finite global dimension that is twisted fractionally Calabi-Yau but has no replication A^{(i)} that is a higher Auslander algebra.

read the original abstract

We show that the class of twisted fractionally Calabi-Yau algebras of finite global dimension coincides with the stable endomorphism algebras of $d$-cluster tilting modules over $d$-representation-finite algebras. This is an application of our main result stating that an algebra $A$ of finite global dimension is twisted fractionally Calabi-Yau if and only if there exists $i$ such that the replicated algebra $A^{(i)}$ is a higher Auslander algebra if and only if there exist infinitely many $i$ such that $A^{(i)}$ is a higher Auslander algebra. This gives a new connection between the study of higher Auslander-Reiten theory and twisted fractionally Calabi-Yau algebras, and provides a new construction of large classes of higher Auslander algebras and higher representation-finite algebras. We give several applications such as an explicit characterisation of twisted $\frac{n}{2}$-Calabi-Yau algebras, and a triangle equivalence between the bounded derived category of a twisted fractionally Calabi-Yau algebra of finite global dimension and the $\mathbb{Z}$-graded stable module category of an associated higher preprojective 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 / 3 minor

Summary. The paper claims that an algebra A of finite global dimension is twisted fractionally Calabi-Yau if and only if there exists i such that the replicated algebra A^{(i)} is a higher Auslander algebra if and only if there exist infinitely many such i. As an application, the class of twisted fractionally Calabi-Yau algebras of finite global dimension coincides with the stable endomorphism algebras of d-cluster tilting modules over d-representation-finite algebras. Further results include an explicit characterisation of twisted n/2-Calabi-Yau algebras and a triangle equivalence between the bounded derived category of such an A and the Z-graded stable module category of an associated higher preprojective algebra.

Significance. If the main equivalences hold, the work forges a direct link between twisted fractionally Calabi-Yau algebras and higher Auslander-Reiten theory through the replication construction, yielding new constructions of higher Auslander algebras and higher representation-finite algebras. The applications to d-cluster tilting and derived equivalences supply concrete tools for studying these objects, with the iff statements offering falsifiable predictions via explicit examples.

minor comments (3)
  1. [Abstract] Abstract: the replication construction A^{(i)} is invoked without a one-sentence reminder of its definition; a brief parenthetical would improve accessibility for readers outside the immediate subfield.
  2. [Main theorem section] The statement that the equivalences hold 'without additional hidden constraints on the base field' (as assumed in the main theorem) should be explicitly confirmed in the statement of Theorem X.Y, including whether k is required to be algebraically closed.
  3. [Applications] Figure or diagram illustrating the replication process for a small example (e.g., a hereditary algebra) would clarify the passage from A to A^{(i)} and strengthen the applications section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. We are pleased that the main equivalences are recognized as forging a direct link between twisted fractionally Calabi-Yau algebras and higher Auslander-Reiten theory, with the applications to d-cluster tilting and derived equivalences noted as useful tools.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proves a main theorem establishing logical equivalences between three independently defined notions for algebras A of finite global dimension: being twisted fractionally Calabi-Yau, having some replication A^{(i)} that is a higher Auslander algebra, and having infinitely many such replications. These equivalences are derived as theorems rather than holding by definitional fiat or by fitting parameters to data. The replication construction, the definition of twisted fCY algebras, and the definition of higher Auslander algebras are introduced separately; the paper then shows they coincide for the stated class. Applications such as explicit characterizations of twisted n/2-Calabi-Yau algebras and triangle equivalences with graded stable categories supply independent content. Although the authors work in higher Auslander-Reiten theory and may cite related prior results, no load-bearing step reduces the central claim to a self-citation chain, a renaming of a known pattern, or an ansatz smuggled in from earlier work. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard definitions and properties from higher Auslander-Reiten theory and homological algebra; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard properties of bounded derived categories, stable module categories, and replication of algebras hold as in prior literature on Auslander-Reiten theory.
    Invoked implicitly in the statements about triangle equivalences and the behavior of replicated algebras A^{(i)}.

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Reference graph

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