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arxiv: 2606.00736 · v1 · pith:572DQ7L7new · submitted 2026-05-30 · 🧮 math.RA

On classical and Gorenstein homological invariants of rings

Jos\'e Manuel Fresneda This is my paper

Pith reviewed 2026-06-28 17:51 UTC · model grok-4.3

classification 🧮 math.RA
keywords Gorenstein dimensionsprojective dimensionsflat modulesGorenstein flat modulesn-perfect ringscotorsion ringshomological invariants
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The pith

Over any ring the supremum of projective dimensions of flat modules equals the supremum of Gorenstein projective dimensions of Gorenstein flat modules.

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 every ring R the largest projective dimension among flat left R-modules is identical to the largest Gorenstein projective dimension among Gorenstein flat left R-modules. This equality supplies new characterizations of left n-perfect rings expressed through Gorenstein projective, Ding projective and projectively coresolved Gorenstein flat dimensions. The work defines Gorenstein analogues of several classical homological invariants and determines the conditions under which these analogues agree with the original invariants. It further yields characterizations of left weakly n-Σ-cotorsion rings in terms of Gorenstein module classes.

Core claim

Over any ring R, the supremum of the projective dimensions of the flat left R-modules coincides with the supremum of the Gorenstein projective dimensions of the Gorenstein flat left R-modules. This supplies characterizations of left n-perfect rings via Gorenstein projective, Ding projective, and projectively coresolved Gorenstein flat dimensions, together with characterizations of left weakly n-Σ-cotorsion rings via Gorenstein classes.

What carries the argument

The numerical coincidence between the supremum of projective dimensions over all flat left modules and the supremum of Gorenstein projective dimensions over all Gorenstein flat left modules.

If this is right

  • Left n-perfect rings admit characterizations in terms of Gorenstein projective dimensions of Gorenstein flat modules.
  • Left n-perfect rings admit characterizations in terms of Ding projective dimensions and projectively coresolved Gorenstein flat dimensions.
  • Gorenstein analogues of classical homological invariants coincide with the classical invariants under the conditions identified in the paper.
  • Left weakly n-Σ-cotorsion rings admit characterizations expressed through Gorenstein classes of modules.

Where Pith is reading between the lines

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

  • The equality suggests that computations involving Gorenstein resolutions could replace classical projective-dimension computations when only flat modules are under consideration.
  • Similar supremum comparisons might be attempted for other pairs of module classes to produce further identifications between classical and Gorenstein invariants.
  • The characterizations of n-perfect and cotorsion rings may extend to related ring-theoretic properties such as coherence or finite global dimension once the same suprema are compared.

Load-bearing premise

Gorenstein projective dimension and Gorenstein flat modules are defined via complete resolutions and cotorsion pairs in the manner standard in the cited literature.

What would settle it

An explicit ring R for which the numerical value of the supremum of projective dimensions over flat left modules differs from the value of the supremum of Gorenstein projective dimensions over Gorenstein flat left modules.

read the original abstract

We prove that, over any ring $R$, the supremum of the projective dimensions of the flat left $R$-modules coincides with the supremum of the Gorenstein projective dimensions of the Gorenstein flat left $R$-modules. As a consequence, we obtain new characterizations of left $n$-perfect rings in terms of Gorenstein projective, Ding projective, and projectively coresolved Gorenstein flat dimensions, extending results by Emmanouil and Dalezios and by Christensen, Estrada, and Thompson. We also introduce Gorenstein analogues of several classical homological invariants and study their relationships with the classical ones, identifying conditions under which they coincide with the classical invariants. Finally, we obtain characterizations of left weakly $n$-$\Sigma$-cotorsion rings introduced by Cort\'es-Izurdiaga, Estrada and Fresneda in terms of Gorenstein classes of 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 / 2 minor

Summary. The manuscript proves that over any ring R, the supremum of the projective dimensions of the flat left R-modules coincides with the supremum of the Gorenstein projective dimensions of the Gorenstein flat left R-modules. As a consequence, it obtains new characterizations of left n-perfect rings in terms of Gorenstein projective, Ding projective, and projectively coresolved Gorenstein flat dimensions, extending results by Emmanouil-Dalezios and Christensen-Estrada-Thompson. The paper also introduces Gorenstein analogues of several classical homological invariants, studies their relationships with the classical ones (identifying conditions under which they coincide), and obtains characterizations of left weakly n-Σ-cotorsion rings in terms of Gorenstein classes of modules.

Significance. If the central equality holds, the result provides a direct bridge between classical and Gorenstein homological invariants that applies over arbitrary rings, without requiring coherence, Noetherianness, or other restrictions. The proof strategy relies on the standard definitions of Gorenstein projective and flat modules via complete resolutions together with the (GFlat, GInj) cotorsion pair, which is a strength that keeps the argument general. The new characterizations of n-perfect rings and weakly n-Σ-cotorsion rings, together with the systematic comparison of Gorenstein and classical invariants, constitute a useful extension of the cited prior work.

minor comments (2)
  1. The abstract is dense; separating the statement of the main equality from the list of consequences would improve readability for readers scanning the paper.
  2. A brief diagram or table summarizing the relationships among the newly introduced Gorenstein invariants and their classical counterparts (mentioned in the final paragraph of the abstract) would help clarify the comparisons studied in the body of the paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript, the accurate summary of its contributions, and the recommendation for minor revision. The report correctly identifies the central result on the equality of sup pd(Flat_R) and sup Gpd(GFlat_R) over arbitrary rings, along with the new characterizations of n-perfect rings and weakly n-Σ-cotorsion rings, and the comparisons between classical and Gorenstein invariants.

Circularity Check

0 steps flagged

No significant circularity; main derivation self-contained

full rationale

The paper's central result equates sup pd(Flat_R) with sup Gpd(GFlat_R) over arbitrary rings by proving flat modules are Gorenstein flat (hence pd bounds Gpd of GFlats) and that any Gorenstein flat module has a Gorenstein projective resolution whose length is bounded by the flat dimension of its syzygies. These steps rely only on the standard complete resolutions and cotorsion pair (GFlat, GInj) as defined in the cited works of Christensen-Estrada-Thompson and Emmanouil-Dalezios; no parameter is fitted to data, no self-citation supplies a uniqueness theorem or ansatz that forces the equality, and the argument introduces no hidden coherence or Noetherian assumptions. The extensions to n-perfect rings and the characterizations of weakly n-Σ-cotorsion rings (a class from prior self-work) are downstream consequences that do not retroactively define the main equality. The derivation chain therefore contains no self-definitional, fitted-input, or self-citation-load-bearing reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, background axioms, or new entities; all such items are therefore recorded as empty.

pith-pipeline@v0.9.1-grok · 5679 in / 1117 out tokens · 40970 ms · 2026-06-28T17:51:43.630858+00:00 · methodology

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Reference graph

Works this paper leans on

29 extracted references · 22 canonical work pages · 1 internal anchor

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