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arxiv: 2606.10901 · v1 · pith:CZSXEZECnew · submitted 2026-06-09 · 🧮 math.RT

Spherical modules and the Auslander--Gorenstein condition for Auslander--Yoneda algebras

Finn Bauwens , Norihiro Hanihara , Ren\'e Marczinzik , Marvin Plogmann , Jan Thomm This is my paper

Pith reviewed 2026-06-27 11:21 UTC · model grok-4.3

classification 🧮 math.RT
keywords Auslander-Yoneda algebraAuslander-Gorensteinspherical modulesspherical algebrassplit torsion pairNakayama algebrasreplicated algebras
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The pith

The Auslander-Yoneda algebra is Auslander-Gorenstein if and only if every indecomposable module is spherical.

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

The paper proves that for a finite-dimensional algebra A of finite global dimension, the Auslander-Yoneda algebra is an Auslander-Gorenstein algebra exactly when every indecomposable left and right A-module is spherical. This equivalence motivates the introduction of spherical algebras, those algebras where every indecomposable module satisfies the spherical condition. Spherical algebras are characterized by a natural pair of subcategories forming a split torsion pair. The work further shows that representation-finite spherical algebras are directed, classifies all spherical Nakayama algebras, and proves that replicated algebras of hereditary algebras are spherical. As an application, the spherical condition yields a negative answer to a question of Venjakob on Auslander regular algebras in general, but a positive answer when the condition holds.

Core claim

The Auslander-Yoneda algebra of A is an Auslander-Gorenstein algebra if and only if every indecomposable left and right A-module is spherical in the sense of Auslander and Bridger. Spherical algebras are those for which every indecomposable module is spherical, and they are characterized by a certain natural pair of subcategories being a split torsion pair. Representation-finite spherical algebras are directed, spherical Nakayama algebras admit a full classification, and replicated algebras of hereditary algebras are spherical. The spherical condition gives a negative answer to Venjakob's question on Auslander regular algebras in general but a positive answer when assumed.

What carries the argument

The Auslander-Yoneda algebra, defined as the Yoneda algebra of the direct sum of all indecomposable A-modules, together with the spherical condition on indecomposable modules.

Load-bearing premise

A must be a finite-dimensional algebra of finite global dimension, without which the Yoneda construction and the stated equivalence may not hold.

What would settle it

A finite-dimensional algebra of finite global dimension where the Auslander-Yoneda algebra is Auslander-Gorenstein but some indecomposable module is not spherical, or the converse holds.

read the original abstract

For a finite dimensional algebra $A$ of finite global dimension we study the Auslander--Yoneda algebra defined as the Yoneda algebra of the direct sum of all indecomposable $A$-modules. We show that the Auslander--Yoneda algebra is an Auslander--Gorenstein algebra if and only if every indecomposable left and right $A$-module is spherical in the sense of Auslander and Bridger. This motivates the study of spherical algebras defined by the condition that every indecomposable module is spherical. We characterize spherical algebras by a certain natural pair of subcategories being a split torsion pair. Moreover, we prove that representation-finite algebras which are spherical are directed and give a full classification of spherical Nakayama algebras. Furthermore, we show that replicated algebras of hereditary algebras are spherical. As a final application of the new notion of spherical algebras, we give a negative answer to a question of Venjakob on Auslander regular algebras in general, but show that there is a positive answer when assuming that every indecomposable left $A$-module is spherical.

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. For a finite-dimensional algebra A of finite global dimension, the paper defines the Auslander--Yoneda algebra as the Yoneda algebra of the direct sum of all indecomposable A-modules. It proves that this algebra is Auslander--Gorenstein if and only if every indecomposable left and right A-module is spherical in the Auslander--Bridger sense. It introduces spherical algebras (those where every indecomposable module is spherical), characterizes them by a natural pair of subcategories forming a split torsion pair, proves that representation-finite spherical algebras are directed, gives a full classification of spherical Nakayama algebras, shows that replicated algebras of hereditary algebras are spherical, and uses the notion to give a negative answer to a question of Venjakob on Auslander regular algebras in general while providing a positive answer under the additional assumption that every indecomposable left A-module is spherical.

Significance. If the stated equivalences and classifications hold, the work supplies a precise module-theoretic criterion for the Auslander--Gorenstein property of the Auslander--Yoneda algebra and introduces a new class of algebras (spherical algebras) with concrete characterizations and examples. The classification of spherical Nakayama algebras and the resolution of Venjakob's question (negative in general, positive under the spherical hypothesis) are concrete contributions that connect spherical modules to homological algebra in a falsifiable way.

minor comments (3)
  1. The abstract states the main equivalence but does not indicate the precise section where the 'if' and 'only if' directions are proved; adding a roadmap sentence at the end of the introduction would improve readability.
  2. Notation for the Auslander--Yoneda algebra (presumably denoted something like Λ(A) or similar) should be fixed early and used consistently; the abstract refers to it descriptively without a symbol.
  3. The classification of spherical Nakayama algebras is announced but the statement of the classification (e.g., which quivers or relations appear) is not previewed; a brief sentence in the introduction would help.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and accurate summary of our manuscript, the positive evaluation of its significance, and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper states an explicit if-and-only-if equivalence: the Auslander-Yoneda algebra (defined as the Yoneda algebra of the direct sum of all indecomposables) is Auslander-Gorenstein precisely when every indecomposable module is spherical in the external Auslander-Bridger sense. This equivalence is presented under the standing hypotheses that A is finite-dimensional of finite global dimension; those hypotheses are part of the theorem statement rather than hidden inputs. Subsequent results (characterization via split torsion pairs, classification of spherical Nakayama algebras, replicated hereditary algebras being spherical, and a counterexample to Venjakob's question) are derived from this equivalence and standard representation-theoretic tools. No equation reduces a claimed prediction to a fitted parameter by construction, no load-bearing uniqueness theorem is imported via self-citation, and the spherical notion is not redefined in terms of the new algebra. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claims rest on the definition of spherical modules (from Auslander-Bridger) and the construction of the Auslander-Yoneda algebra; abstract-only review prevents exhaustive listing of all background lemmas.

axioms (1)
  • domain assumption A is a finite dimensional algebra of finite global dimension
    Explicitly stated as the setting in the first sentence of the abstract.

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