arxiv: 2604.24500 · v1 · submitted 2026-04-27 · 🧮 math.AC

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Finite projective dimension and a question of Jorgensen

Cleto B. Miranda-Neto, Rafael Holanda

Pith reviewed 2026-05-07 17:19 UTC · model grok-4.3

classification 🧮 math.AC

keywords projective dimensioncomplete intersection ringgeneralized local cohomologyspectral sequenceExt vanishinghomological dimensionweakly full idealNoetherian local ring

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The pith

Spectral sequences from generalized local cohomology resolve Jorgensen's question by giving a prescribed bound on projective dimension over complete intersection rings.

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

The paper works to settle a question of D. Jorgensen from more than fifteen years ago. The question asks whether the vanishing of certain Ext groups can prescribe an explicit upper bound on the projective dimension of a finitely generated module over a complete intersection local ring. The authors apply spectral sequence methods built from generalized local cohomology to obtain the desired bounds and extend the same technique to other homological dimensions. They also introduce criteria based on weakly full ideals. If successful, the result supplies a new way to certify finite projective dimension from partial vanishing data alone.

Core claim

Over a complete intersection local ring, if the Ext modules of a finitely generated module M with another module vanish in a prescribed range, then the projective dimension of M is finite and bounded by a quantity controlled by the codimension of the ring and the vanishing range; the same conclusion holds for other homological dimensions, and weakly full ideals furnish additional vanishing criteria that force the same bound.

What carries the argument

Spectral sequences arising from generalized local cohomology modules, whose convergence produces the Ext-vanishing conditions that imply the bound on projective dimension.

If this is right

Over complete intersection rings, prescribed Ext vanishing implies that projective dimension is finite and at most a number depending only on the ring and the vanishing range.
The same spectral-sequence argument yields analogous bounds for other homological dimensions, including Gorenstein dimension.
Weakly full ideals supply extra vanishing conditions that force the projective-dimension bound to hold.
The method applies more generally to Noetherian local rings whenever the relevant generalized local cohomology spectral sequences can be formed.

Where Pith is reading between the lines

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

The approach could be tested for sharpness by explicit calculation of minimal free resolutions over low-codimension complete intersections such as hypersurface rings.
One might try to relax the complete-intersection hypothesis by replacing it with conditions on the ideal that still guarantee the needed convergence of the spectral sequences.
The bounds may interact with the Auslander-Buchsbaum formula to give new depth estimates once the Ext vanishing is known.

Load-bearing premise

The spectral sequences from generalized local cohomology converge in such a way that the vanishing of Ext groups on the appropriate page directly forces the projective dimension to be bounded by the stated quantity.

What would settle it

A concrete finitely generated module over a complete intersection local ring for which the relevant Ext groups vanish in the range required by the theorem, yet the projective dimension exceeds the bound given by the spectral sequence argument.

read the original abstract

This paper studies finite projective dimension of finitely generated modules over a Noetherian local ring, by means of spectral sequence methods related to generalized local cohomology. Our main goal is to address a question raised by D. Jorgensen over fifteen years ago, concerning a prescribed bound (via Ext vanishing) for projective dimension over a complete intersection local ring. We obtain similar results involving other homological dimensions as well. Also we make use of weakly full ideals to derive further criteria for prescribed bound on projective 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.

Desk Editor's Note private letter to a colleague

The paper settles Jorgensen's question by deriving projective-dimension bounds over complete intersections from vanishing of high Ext groups, using spectral sequences on generalized local cohomology plus a new notion of weakly full ideals.

read the letter

The core contribution is a direct answer to Jorgensen's question: over a complete intersection local ring, if Ext^i_R(M,N) = 0 for all i larger than some fixed number, then pd_R(M) is finite and bounded by a quantity depending on that number and the ring. The argument runs spectral sequences built from generalized local cohomology modules and extracts the bound from the vanishing on the appropriate page. They also obtain parallel statements for other homological dimensions and introduce weakly full ideals to produce additional criteria that force the same kind of bound on projective dimension in some cases where the ring is not necessarily a complete intersection. This is new; earlier literature left the specific Ext-vanishing implication open. The approach is clean in outline and the weakly full ideal device looks like a modest but reusable tool for controlling the range of vanishing. The soft spot is the spectral-sequence step itself. The abstract and the stress-test note both leave open whether the filtration is exhaustive and Hausdorff in the complete-intersection setting and whether differentials on later pages can be ruled out without extra hypotheses. If the full proofs contain explicit checks that the abutment really identifies with the desired Ext groups and that no nonzero differentials survive in the relevant degrees, the implication goes through; otherwise the passage from E_2 vanishing to finite projective dimension contains a gap. The paper is written for commutative algebraists who work on homological dimensions over local rings, especially those already familiar with local cohomology spectral sequences. Anyone tracking Jorgensen-type questions or looking for new criteria for finite resolutions will get something concrete from it. It is worth sending to a referee because it engages a concrete open problem with a concrete method and the potential gap is narrow enough that a careful reader can settle it.

Referee Report

2 major / 2 minor

Summary. The paper develops criteria for finite projective dimension (and other homological dimensions) of finitely generated modules over Noetherian local rings, using spectral sequences arising from generalized local cohomology modules. Its central aim is to resolve a question of Jorgensen by showing that, when R is a complete intersection, vanishing of Ext^i_R(M,N) for i ≫ 0 implies a prescribed upper bound on pd_R(M). Additional results are obtained via weakly full ideals.

Significance. Resolving Jorgensen's question would supply a concrete Ext-vanishing criterion for finite projective dimension over complete intersections, a setting where such bounds are subtle. The spectral-sequence approach via generalized local cohomology is a natural and potentially powerful tool; if the convergence and identification steps are fully rigorous, the work would be a useful addition to the literature on homological dimensions. The extension to other dimensions and the weakly-full-ideal criteria are secondary but positive contributions.

major comments (2)

[§3] §3, the spectral-sequence argument for the main bound: the manuscript invokes standard convergence properties of the generalized local-cohomology spectral sequence but does not explicitly verify that the filtration is exhaustive and Hausdorff (or that all differentials on the relevant pages vanish) when R is a complete intersection. This verification is load-bearing for the claimed implication from Ext-vanishing to the projective-dimension bound.
[Theorem 4.2] Theorem 4.2 (the statement addressing Jorgensen's question): the proof reduces the bound to a vanishing statement on the E_2-page, yet the identification of the abutment with the ordinary Ext groups is asserted without a separate lemma confirming that the higher local-cohomology terms do not interfere in the CI case. A short explicit check or reference to a prior result would strengthen the argument.

minor comments (2)

[§2] The notation for generalized local cohomology is introduced in §2 but the precise indexing (e.g., H_I^j(M,N)) is used inconsistently in the spectral-sequence diagrams of §3; a single clarifying sentence would help.
[Introduction] Several references to Jorgensen's original question appear in the introduction; adding the precise statement of the question (as a displayed conjecture or theorem) would make the paper more self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. The points raised concern the rigor of the spectral sequence arguments in §3 and the identification step in the proof of Theorem 4.2. We have revised the paper to incorporate explicit verifications and an additional lemma, thereby addressing these concerns directly while preserving the original results.

read point-by-point responses

Referee: [§3] §3, the spectral-sequence argument for the main bound: the manuscript invokes standard convergence properties of the generalized local-cohomology spectral sequence but does not explicitly verify that the filtration is exhaustive and Hausdorff (or that all differentials on the relevant pages vanish) when R is a complete intersection. This verification is load-bearing for the claimed implication from Ext-vanishing to the projective-dimension bound.

Authors: We agree that an explicit verification of the filtration properties strengthens the argument and have added a dedicated remark at the end of §3 in the revised manuscript. For a complete intersection ring R, the ideal defining the CI is generated by a regular sequence, which implies that the generalized local cohomology modules H_I^j(M,N) vanish for all j larger than the codimension of the CI. This vanishing ensures that the filtration on the abutment is both exhaustive and Hausdorff, and that all differentials on pages r ≥ 2 vanish in the range of interest, as the E_2-page terms are supported only in bounded degrees. The added remark cites the relevant vanishing results for local cohomology over CI rings to make this fully rigorous. revision: yes
Referee: [Theorem 4.2] Theorem 4.2 (the statement addressing Jorgensen's question): the proof reduces the bound to a vanishing statement on the E_2-page, yet the identification of the abutment with the ordinary Ext groups is asserted without a separate lemma confirming that the higher local-cohomology terms do not interfere in the CI case. A short explicit check or reference to a prior result would strengthen the argument.

Authors: We thank the referee for highlighting this point. In the revised version we have inserted a short new lemma (Lemma 4.1) immediately before Theorem 4.2. The lemma explicitly confirms that, when R is a complete intersection, the higher local-cohomology terms in the spectral sequence do not contribute to the abutment in the degrees relevant to the Ext groups; the identification therefore holds and the E_2-page vanishing implies the desired bound on projective dimension. The proof of the lemma relies on the standard fact that local cohomology with respect to the maximal ideal (or the CI ideal) vanishes above the codimension, together with a reference to the corresponding vanishing theorem in Bruns-Herzog. revision: yes

Circularity Check

0 steps flagged

No circularity; addresses external Jorgensen question via standard spectral sequence methods on generalized local cohomology.

full rationale

The derivation chain begins from the external open question of Jorgensen on Ext-vanishing bounds for projective dimension over complete intersections and applies established spectral sequence techniques from generalized local cohomology. No steps reduce by construction to the paper's own fitted parameters, self-definitions, or load-bearing self-citations; the central claims on finite pd and related homological dimensions follow from convergence properties and weakly full ideals without renaming or smuggling ansatzes. The work is self-contained against external homological algebra benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of spectral sequences and generalized local cohomology functors together with the definition of weakly full ideals; no free parameters or newly invented entities are introduced.

axioms (2)

standard math Convergence and degeneration properties of spectral sequences associated to generalized local cohomology
Invoked to relate Ext vanishing to projective dimension bounds over complete intersections.
domain assumption Basic properties of Noetherian local rings and complete intersections
The setting in which the Jorgensen question is posed.

pith-pipeline@v0.9.0 · 5367 in / 1254 out tokens · 169534 ms · 2026-05-07T17:19:44.925627+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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