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arxiv: 1907.06448 · v1 · pith:C6SN3D5Enew · submitted 2019-07-15 · 🧮 math.RT

Algebras with finite relative dominant dimension and almost n-precluster tilting modules

Shen Li , Shunhua Zhang This is my paper

Pith reviewed 2026-05-24 21:19 UTC · model grok-4.3

classification 🧮 math.RT
keywords relative dominant dimensionalmost n-precluster tilting modulesalmost n-minimal Auslander-Gorenstein algebrasGorenstein projective modulestilting modulesrepresentation theory of algebras
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The pith

Algebras with finite relative dominant dimension correspond to almost n-precluster tilting modules.

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

The paper examines relative dominant dimension taken with respect to a fixed injective module and identifies the algebras for which this dimension is finite. It defines almost n-precluster tilting modules and proves they stand in one-to-one correspondence with almost n-minimal Auslander-Gorenstein algebras. The same correspondence supplies an explicit description of the Gorenstein projective modules over those algebras. Readers interested in representation theory would see this as a bridge between a homological invariant of the algebra and a special class of modules. The work therefore supplies both a classification tool and a concrete way to compute Gorenstein projectives from the tilting data.

Core claim

We establish a correspondence between almost n-precluster tilting modules and almost n-minimal Auslander-Gorenstein algebras. Moreover, we give a description of the Gorenstein projective modules over almost n-minimal Auslander-Gorenstein algebras in terms of the corresponding almost n-precluster tilting modules. This rests on first characterizing the algebras that possess finite relative dominant dimension with respect to an injective module.

What carries the argument

Relative dominant dimension with respect to an injective module, used to define almost n-minimal Auslander-Gorenstein algebras and to produce the bijection with almost n-precluster tilting modules.

If this is right

  • Gorenstein projective modules over an almost n-minimal Auslander-Gorenstein algebra are recovered directly from the corresponding almost n-precluster tilting module.
  • The finite-relative-dominant-dimension condition classifies a new family of algebras that generalize minimal Auslander-Gorenstein algebras.
  • The bijection preserves the homological properties that define the 'almost n' level of the tilting module.

Where Pith is reading between the lines

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

  • The same relative-dimension invariant might classify higher analogs of cluster-tilting objects once the parameter n is allowed to vary.
  • Descriptions of Gorenstein projectives obtained this way could be compared with existing Auslander-Reiten formulas to test consistency in low-dimensional cases.
  • If the correspondence extends to derived categories, it would give a module-theoretic model for certain objects in higher Auslander-Reiten theory.

Load-bearing premise

Finite relative dominant dimension with respect to an injective module can be used to single out exactly those algebras that admit a corresponding almost n-precluster tilting module.

What would settle it

An explicit algebra whose relative dominant dimension is finite yet which possesses no almost n-precluster tilting module, or an almost n-precluster tilting module whose associated algebra fails to have finite relative dominant dimension.

read the original abstract

In this paper, we investigate the relative dominant dimension with respect to an injective module and characterize the algebras with finite relative dominant dimension. As an application, we introduce the almost n-precluster tilting module and establish a correspondence between almost n-precluster tilting modules and almost n-minimal Auslander-Gorenstein algebras. Moreover, we give a description of the Gorenstein projective modules over almost n-minimal Auslander-Gorenstein algebras in terms of the corresponding almost n-precluster tilting modules.

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 / 4 minor

Summary. The paper investigates the relative dominant dimension of algebras with respect to a fixed injective module. It characterizes algebras of finite relative dominant dimension and, as an application, introduces the notion of almost n-precluster tilting modules. The central results establish a correspondence between almost n-precluster tilting modules and almost n-minimal Auslander-Gorenstein algebras, together with an explicit description of the Gorenstein projective modules over the latter algebras in terms of the corresponding almost n-precluster tilting modules.

Significance. If the stated correspondences hold, the work extends the classical theory of dominant dimension and Auslander-Gorenstein algebras to a relative setting parametrized by an injective module and by an integer n. The new class of almost n-precluster tilting modules supplies a module-theoretic counterpart to the algebraic notion of almost n-minimal Auslander-Gorenstein algebras and yields a concrete description of their Gorenstein-projective modules. Such correspondences are useful for classifying algebras and modules with controlled homological invariants and may serve as a foundation for further results on relative tilting theory.

minor comments (4)
  1. §2: the definition of relative dominant dimension (Definition 2.3) should explicitly record the dependence on the choice of injective module I; the subsequent statements would then read more cleanly when I is fixed.
  2. Theorem 4.7: the statement of the correspondence would benefit from an explicit bijection (or at least a clear statement whether it is one-to-one or many-to-one) between the two classes; the current wording leaves the precise nature of the correspondence implicit.
  3. Notation: the symbol “almost n-” is used both for precluster tilting modules and for minimal Auslander-Gorenstein algebras; a short remark clarifying that the two uses are independent would avoid possible confusion for readers.
  4. References: several standard citations on n-Auslander algebras and relative homological dimensions (e.g., works of Iyama, Marczinzik, or Solberg) are missing from the bibliography; adding them would situate the results more clearly within the existing literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our work, as well as the recommendation for minor revision. No specific major comments appear in the report, so we provide no point-by-point responses below. We remain available to address any minor issues that may be communicated separately.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines the relative dominant dimension with respect to an injective module, characterizes algebras of finite relative dominant dimension, introduces almost n-precluster tilting modules, and establishes a correspondence with almost n-minimal Auslander-Gorenstein algebras plus a description of their Gorenstein-projective modules. These are standard definitional and classificatory steps in representation theory that do not reduce any claimed result to its own inputs by construction, fitted parameters renamed as predictions, or load-bearing self-citation chains. The derivation chain is self-contained against external benchmarks in homological algebra and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper relies on standard assumptions in homological algebra and introduces new entities without independent evidence outside the definitions.

axioms (1)
  • domain assumption The category of modules over an artin algebra is abelian with enough injectives and projectives.
    Standard background assumption in representation theory of algebras.
invented entities (1)
  • almost n-precluster tilting module no independent evidence
    purpose: To establish correspondence with almost n-minimal Auslander-Gorenstein algebras and describe Gorenstein projective modules.
    Newly defined in the paper as an application of the characterization of finite relative dominant dimension.

pith-pipeline@v0.9.0 · 5598 in / 1233 out tokens · 35983 ms · 2026-05-24T21:19:19.057133+00:00 · methodology

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

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