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arxiv: 2301.01105 · v7 · pith:P6SESCGZnew · submitted 2023-01-03 · 🧮 math.AC

Notes on modules of finite injective dimension

Mohsen Asgharzadeh This is my paper

Pith reviewed 2026-05-24 10:09 UTC · model grok-4.3

classification 🧮 math.AC
keywords finite injective dimensionBass conjectureCohen-Macaulay ringGorenstein ringmodule reflexivityNoetherian ringsyzygyquasi-normal ring
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The pith

Existence of finitely generated modules of finite injective dimension forces the ring to be reduced, normal, an integral domain, complete intersection or Gorenstein beyond Cohen-Macaulay.

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

The paper studies the ring-theoretic consequences of the existence of nonzero finitely generated modules of finite injective dimension over commutative Noetherian rings. Motivated by the Bass conjecture, it shows that such modules impose conditions stronger than Cohen-Macaulay, including reducedness, normality, and being an integral domain, with further restrictions such as complete intersection or Gorenstein in some cases. It further proves that reflexivity or torsionlessness of these modules forces the ring to be quasi-normal, examines the injective dimension of their tensor products and endomorphism rings, and analyzes surjections onto them from high syzygies of the residue field.

Core claim

If a commutative Noetherian ring admits a nonzero finitely generated module of finite injective dimension, then the ring is Cohen-Macaulay and additionally reduced, normal, and an integral domain; under further hypotheses it is a complete intersection or Gorenstein. Reflexivity of such a module forces the ring to be quasi-normal. The injective dimensions of tensor products and endomorphism rings are investigated, as is the case in which a high syzygy of the residue field surjects onto a nonzero module of finite injective dimension.

What carries the argument

Finitely generated modules of finite injective dimension, whose existence is used to derive forcing conditions on the ambient Noetherian ring via homological arguments.

If this is right

  • The ambient ring must be reduced.
  • The ambient ring must be normal.
  • The ambient ring must be an integral domain.
  • Under additional conditions the ambient ring is a complete intersection or Gorenstein.
  • Reflexive or torsionless modules of finite injective dimension force the ring to be quasi-normal.

Where Pith is reading between the lines

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

  • These forcing statements could be used to classify which Noetherian rings admit modules of finite injective dimension.
  • The results on reflexivity suggest a possible route to detecting quasi-normality via homological data alone.
  • The analysis of syzygies of the residue field may connect to questions about the minimal number of generators needed for such modules.

Load-bearing premise

The ring is commutative and Noetherian and the modules are finitely generated.

What would settle it

A non-reduced commutative Noetherian ring that admits a nonzero finitely generated module of finite injective dimension would serve as a counterexample.

read the original abstract

Motivated by Bass' conjecture, we study finitely generated modules of finite injective dimension and the additional constraints they impose on the ambient ring. Beyond ensuring the Cohen--Macaulay property, the presence of such modules enforces further conditions on the ring, including reducedness, normality, being an integral domain, and various singularity conditions such as complete intersection, Gorenstein, and beyond. This continues to detect non-singularity as well. We also address the reflexivity (and also torsionlessness) of modules with finite injective dimension and show that this forces the ring to be quasi-normal. In the same vein, we investigate the injective dimension of tensor products and endomorphism rings. Finally, we study the behavior of R when high syzygies of ksurject onto a non-zero R-module of finite injective dimension.

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

2 major / 2 minor

Summary. The manuscript presents notes motivated by the Bass conjecture on finitely generated modules M of finite injective dimension over a commutative Noetherian ring R. It claims that the existence of a nonzero such M forces R to be reduced, normal, an integral domain, complete intersection or Gorenstein (beyond the known Cohen-Macaulay property). Additional results address reflexivity and torsionlessness of such modules (forcing quasi-normality), injective dimension of tensor products and endomorphism rings, and the case where a high syzygy of the residue field k surjects onto a nonzero module of finite injective dimension.

Significance. If the derivations hold, the notes supply concrete additional constraints on rings admitting nonzero f.g. modules of finite injective dimension, extending the classical homological implications in the Noetherian setting where minimal injective resolutions and associated-prime arguments are available. The work stays within standard assumptions and does not introduce new ad-hoc axioms or entities.

major comments (2)
  1. [Abstract] Abstract and opening paragraphs: the forcing statements (reducedness, normality, domain, CI, Gorenstein) are asserted without any displayed theorem statements, proof sketches, or references to specific lemmas in the provided text; this prevents verification that the arguments are free of gaps or post-hoc choices and makes the central claims unverifiable from the manuscript as presented.
  2. The proofs are stated to rely on Noetherianness for the existence of minimal injective resolutions, structure of associated primes, and behavior of Ext/Tor; if any section claims the results extend beyond the Noetherian + f.g. setting without additional hypotheses or counter-example discussion, that would undermine the scope of the forcing statements.
minor comments (2)
  1. Notation such as id_R(M) and the precise definition of 'finite injective dimension' should be introduced explicitly in the first section rather than assumed from context.
  2. The manuscript would benefit from a short table or list summarizing which ring properties are forced by which module conditions, for quick reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed report. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and opening paragraphs: the forcing statements (reducedness, normality, domain, CI, Gorenstein) are asserted without any displayed theorem statements, proof sketches, or references to specific lemmas in the provided text; this prevents verification that the arguments are free of gaps or post-hoc choices and makes the central claims unverifiable from the manuscript as presented.

    Authors: We agree that the abstract and introductory paragraphs would benefit from explicit cross-references. In the revision we will add numbered theorem statements (or lemma references) immediately after each forcing claim, together with one-sentence indications of the key arguments (associated-prime analysis or Ext-vanishing) used in the proofs. revision: yes

  2. Referee: [—] The proofs are stated to rely on Noetherianness for the existence of minimal injective resolutions, structure of associated primes, and behavior of Ext/Tor; if any section claims the results extend beyond the Noetherian + f.g. setting without additional hypotheses or counter-example discussion, that would undermine the scope of the forcing statements.

    Authors: No section of the manuscript asserts or suggests that the results hold outside the commutative Noetherian, finitely generated setting. All statements and proofs explicitly invoke Noetherianness for minimal injective resolutions and associated primes; the scope is therefore unchanged. revision: no

Circularity Check

0 steps flagged

No circularity: standard algebraic derivations from Noetherian assumptions

full rationale

The paper derives implications for rings admitting finitely generated modules of finite injective dimension using classical Noetherian commutative algebra (minimal injective resolutions, associated primes, Ext/Tor behavior). These are proved from standard domain assumptions without self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations. The Bass conjecture motivation is external and the results remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard results from commutative algebra concerning injective dimension and the Bass conjecture; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math Standard properties of finitely generated modules and injective dimension over commutative Noetherian rings
    Invoked throughout the motivation and statements about ring constraints.

pith-pipeline@v0.9.0 · 5646 in / 1264 out tokens · 19102 ms · 2026-05-24T10:09:31.467082+00:00 · methodology

discussion (0)

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