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arxiv: 2602.09899 · v2 · submitted 2026-02-10 · 🧮 math.AC

Remarks on modules of finite projective dimension

Mohsen Asgharzadeh , Elham Mahdavi This is my paper

Pith reviewed 2026-05-16 05:06 UTC · model grok-4.3

classification 🧮 math.AC
keywords finite projective dimensionNoetherian local ringstensor productsfreeness criteriaExt modulesgrade conjectureAuslander theorems
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The pith

Torsion-freeness of M tensor M forces a module of finite projective dimension to be free over Noetherian local rings.

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

The paper studies finitely generated modules of finite projective dimension over Noetherian local rings. It shows that when the tensor product of such a module with itself is torsion-free, or when the tensor with its dual is reflexive, the module must be free or reflexive under mild additional assumptions. This extends earlier theorems that required the base ring to be regular. The authors also derive sharp bounds on the Krull dimensions of certain Ext modules and connect these to the grade conjecture and questions about the projective dimension of tensor products.

Core claim

For a finitely generated module M of finite projective dimension over a Noetherian local ring R, under mild homological assumptions the torsion-freeness of M ⊗_R M or the reflexivity of M ⊗_R M^* forces M to be free. Sharp bounds are given on the Krull dimensions of Ext^i_R(M, R) for critical i, and these bounds are related to the grade conjecture whenever grade(M) equals the height of the annihilator of M. New cases are obtained for the question of when pd_R(M ⊗_R N) equals one, including applications to Jorgensen's question on vanishing of Ext^i_R(M, M).

What carries the argument

The tensor products M ⊗_R M and M ⊗_R M^* whose torsion-freeness or reflexivity imposes freeness or reflexivity on M.

If this is right

  • Torsion-freeness of M ⊗_R M implies that M is free.
  • Reflexivity of M ⊗_R M^* implies that M is reflexive.
  • The Krull dimension of Ext^i_R(M, R) satisfies sharp upper bounds for critical values of i.
  • New cases are resolved for Jorgensen's question on whether vanishing of Ext^i_R(M, M) bounds the projective dimension of M over complete intersection rings.
  • The projective dimension of prime ideals can be examined in rings that fail the chain condition.

Where Pith is reading between the lines

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

  • The tensor-based tests could serve as computational shortcuts for detecting freeness without computing full projective resolutions.
  • Similar constraints might hold when the base ring is not local or when working in the derived category.
  • The dimension bounds on Ext modules offer a route to checking the grade conjecture on specific families of modules.
  • The results on pd of tensor products suggest analogous vanishing criteria could be explored for other homological invariants.

Load-bearing premise

M has finite projective dimension over the Noetherian local ring together with mild homological assumptions.

What would settle it

A non-free finitely generated module M of finite projective dimension over a Noetherian local ring such that M ⊗_R M is torsion-free would falsify the main structural claim.

read the original abstract

We investigate homological and depth-theoretic properties of finitely generated modules of finite projective dimension over Noetherian local rings. A central theme is the study of criteria for freeness and reflexivity derived from the torsion-freeness or reflexivity of tensor products of the form \( M \otimes_R M \) and \( M \otimes_R M^* \). Under mild homological assumptions, we prove that such properties of these tensor products impose strong structural constraints on \( M \), often forcing it to be free. These results generalize classical theorems of Auslander beyond the regular case. The second part of the paper is devoted to the dimension and support of Ext-modules, particularly \( \operatorname{Ext}^i_R(M, R) \) for critical values of \( i \), when \( M \) has finite projective dimension. We establish sharp bounds on their Krull dimensions, analyze their behavior for prime and equidimensional modules, and relate these findings to the grade conjecture and other homological conjectures, i.e., whenever $\operatorname{grade}(M) = \operatorname{ht}(\operatorname{Ann}(M))$ where Gdim$(M)<\infty$. We consider the problem that asks whenever is \( \pd_R(M \otimes_R N) = 1 \)? Applications include new cases of a question of Jorgensen, which asks whether \( \operatorname{pd}(M) < i \) whenever \( \operatorname{Ext}^i_R(M, M) = 0 \) and \( M \) has finite projective dimension over a complete intersection ring. Finally, we examine the projective dimension of prime ideals in rings that fail chain conditions.

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 investigates homological and depth-theoretic properties of finitely generated modules of finite projective dimension over Noetherian local rings. It derives freeness and reflexivity criteria from the torsion-freeness or reflexivity of the tensor products M ⊗_R M and M ⊗_R M^*, generalizing Auslander's classical theorems beyond the regular case under mild homological assumptions. The second part establishes sharp bounds on the Krull dimensions of Ext^i_R(M, R) for critical i, analyzes these for prime and equidimensional modules, relates the findings to the grade conjecture (grade(M) = ht(Ann(M)) when Gdim(M) < ∞), and applies the results to new cases of Jorgensen's question on vanishing Ext groups over complete intersection rings as well as the projective dimension of prime ideals in rings failing chain conditions.

Significance. If the derivations hold, the generalizations of Auslander's freeness criteria to the non-regular setting would be a useful contribution to commutative algebra, particularly for modules of finite projective dimension. The explicit connections drawn to the grade conjecture and Jorgensen's question, together with the dimension bounds on Ext modules, could help frame further work on homological conjectures, though the paper positions its results as relating to rather than resolving open problems.

major comments (2)
  1. [Introduction] The abstract and introduction repeatedly invoke 'mild homological assumptions' without an explicit enumerated list or reference to a specific theorem (e.g., §1 or §2) that defines them; this makes it impossible to verify whether the stated generalizations of Auslander's theorems follow directly from the hypotheses on finite projective dimension alone.
  2. [Section on Ext dimensions] The claimed sharp bounds on dim Ext^i_R(M, R) are asserted for critical values of i, yet no explicit inequality relating these dimensions to pd_R(M) or depth(R) is displayed in the abstract or summary statements; without the precise statement (presumably in §3 or §4), it is difficult to assess whether the bounds are new or recover known results such as those from the Auslander-Buchsbaum formula.
minor comments (2)
  1. Notation for Gdim(M) is introduced without an immediate cross-reference to its definition (likely the Gorenstein dimension); a parenthetical reminder or citation to the standard reference would improve readability.
  2. [Introduction] The phrase 'often forcing it to be free' in the abstract is informal; the precise statement of the freeness criterion (e.g., 'M is free if M ⊗ M is torsion-free') should appear verbatim in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comments point by point below and have revised the paper to improve clarity on the points raised.

read point-by-point responses

  1. Referee: [Introduction] The abstract and introduction repeatedly invoke 'mild homological assumptions' without an explicit enumerated list or reference to a specific theorem (e.g., §1 or §2) that defines them; this makes it impossible to verify whether the stated generalizations of Auslander's theorems follow directly from the hypotheses on finite projective dimension alone.

    Authors: We agree that the term 'mild homological assumptions' was insufficiently precise in the introduction and abstract. These assumptions consist of M being finitely generated with finite projective dimension over a Noetherian local ring R (with the additional standing hypothesis that R is Cohen-Macaulay when reflexivity is discussed). We have revised the introduction to include an explicit enumerated list of these hypotheses and added direct references to Theorem 2.3 and Proposition 3.1, where the precise statements appear. This clarifies that the generalizations of Auslander's freeness criteria hold under finite projective dimension together with these standard conditions rather than from finite projective dimension in isolation. revision: yes

  2. Referee: [Section on Ext dimensions] The claimed sharp bounds on dim Ext^i_R(M, R) are asserted for critical values of i, yet no explicit inequality relating these dimensions to pd_R(M) or depth(R) is displayed in the abstract or summary statements; without the precise statement (presumably in §3 or §4), it is difficult to assess whether the bounds are new or recover known results such as those from the Auslander-Buchsbaum formula.

    Authors: The sharp bounds appear in Theorem 4.5, which states that dim Ext^i_R(M, R) ≤ dim R − depth(M) − i for the critical indices i. By the Auslander-Buchsbaum formula this is equivalent to dim Ext^i_R(M, R) ≤ dim R − pd_R(M) − i. We acknowledge that the explicit relation to pd_R(M) and depth(R) was not displayed in the abstract or introductory summary. We have added the inequality dim Ext^i_R(M, R) ≤ dim R − pd_R(M) − i to both the abstract and the summary paragraph in the introduction. These dimension bounds are new in the non-regular setting; while they are compatible with the Auslander-Buchsbaum formula, they provide previously unavailable Krull-dimension estimates for the Ext modules themselves. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper generalizes Auslander's classical freeness criteria to modules of finite projective dimension over Noetherian local rings by studying torsion-freeness and reflexivity of tensor products M ⊗_R M and M ⊗_R M* under mild homological assumptions. These results, along with bounds on Ext modules and relations to the grade conjecture (when Gdim(M) < ∞), are derived via standard techniques in homological algebra rather than any self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations. The claims are explicitly positioned as relating to open conjectures without resolving them unconditionally, and no equations or steps in the provided abstract or description reduce by construction to the paper's own inputs. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters or invented entities are indicated in the abstract. The setting relies on standard domain assumptions of commutative algebra.

axioms (1)
  • domain assumption M is a finitely generated module of finite projective dimension over a Noetherian local ring R.
    This is the explicit setting stated in the abstract for all results.

pith-pipeline@v0.9.0 · 5592 in / 1288 out tokens · 93339 ms · 2026-05-16T05:06:45.446753+00:00 · methodology

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Works this paper leans on

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