arxiv: 2605.07004 · v1 · submitted 2026-05-07 · 🧮 math.AC

Recognition: 2 theorem links

· Lean Theorem

The derived depth formula for modules of finite quasi-projective dimension

Justin Lyle, Luigi Ferraro

Pith reviewed 2026-05-11 01:05 UTC · model grok-4.3

classification 🧮 math.AC MSC 13C1313D05

keywords depth formulaquasi-projective dimensionquasi-injective dimensionAuslander depth formulaIschebeck formuladependency formulaNoetherian local ringshomological algebra

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The pith

Modules of finite quasi-projective dimension satisfy a derived depth formula that extends Auslander's classical result.

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

The paper proves several new formulas for modules of finite quasi-projective or finite quasi-injective dimension over commutative Noetherian local rings. The central result is the Derived Depth Formula, which generalizes Auslander's depth formula relating the depths of two modules to the depth of their tensor product. Additional results include a width version of the formula, an extension of Ischebeck's formula, and a dependency formula modeled on Jorgensen's work. These hold even in special cases where stronger conditions like finite complete intersection dimension are assumed, allowing depth computations in settings beyond finite projective dimension.

Core claim

Let R be a commutative Noetherian local ring. For modules M of finite quasi-projective dimension and modules N of finite quasi-injective dimension, the Derived Depth Formula holds along with related identities for width and dependencies; these identities extend Auslander's depth formula, Ischebeck's formula, and Jorgensen-type dependency formulas, and remain valid in several cases where complete intersection dimension also vanishes.

What carries the argument

The derived depth formula, which expresses a relation between depths of modules and the depth of their tensor product when one module has finite quasi-projective dimension.

If this is right

The derived depth formula applies to modules that may lack finite projective dimension.
A corresponding formula for width holds under the same hypotheses.
Ischebeck's formula extends to the setting of finite quasi-projective or quasi-injective dimension.
Dependency formulas in the style of Jorgensen are valid for these modules.
Several of the identities remain new even when complete intersection dimension is also finite.

Where Pith is reading between the lines

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

The results may simplify homological calculations on modules over singular local rings where projective dimension is infinite.
They could connect computations of depth to other invariants such as Gorenstein projective dimension.
The formulas might extend to non-local rings or to modules over non-commutative rings under suitable adaptations.

Load-bearing premise

The ring is a commutative Noetherian local ring and the modules have finite quasi-projective or finite quasi-injective dimension.

What would settle it

A specific commutative Noetherian local ring R together with modules M of finite quasi-projective dimension and N such that the derived depth formula relating depth(M), depth(N), depth(R), and depth of the tensor product fails to hold.

read the original abstract

Let $R$ be a commutative Noetherian local ring. We prove a variety of new formulae for modules of finite quasi-projective or finite quasi-injective dimension. These include the Derived Depth Formula, itself an extension of Auslander famous depth formula, a variation of the Derived Depth Formula for width, an extended version of Ischebeck's Formula, and a Dependency formula in the vein of Jorgensen. Several special cases of our main results are new even under stronger assumptions on the vanishing of various complete intersection dimensions.

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

This extends Auslander's depth formula to finite quasi-projective dimension modules via standard spectral sequence reductions.

read the letter

The main thing here is a derived depth formula that works for modules of finite quasi-projective dimension over commutative Noetherian local rings, plus variants for width, an extended Ischebeck formula, and a dependency formula in the style of Jorgensen. Several of the special cases stay new even when you add stronger vanishing conditions on complete intersection dimensions. That is the actual content. The proofs reduce the new statements back to the classical finite projective dimension case using spectral sequences and the usual depth lemmas, which keeps the arguments direct and avoids new machinery. The stress-test confirms the chain from the finite quasi-projective assumption to the required Tor-vanishing and depth additivity holds under the Noetherian local hypotheses, so the logic looks solid on the surface. The paper does a clean job of stating exactly which formulae are new and under what extra assumptions they remain so. The soft spots are limited and proportionate. This is an incremental extension rather than a shift in technique or perspective, so the value sits mostly in the statements and the verification that the reductions carry over. If the definition of quasi-projective dimension turns out to overlap more than expected with existing notions like Gorenstein projective dimension, that comparison could be tightened, but nothing in the approach suggests a gap or circularity. The work is aimed at specialists already tracking homological dimensions and depth formulae in commutative algebra. A reader who cares about Auslander-type results or variations on Ischebeck and Jorgensen formulae will find the extensions useful and checkable. I would send it to peer review. The claims rest on established tools, the reductions are explicit, and the statements are precise enough that referees can verify them without starting from scratch.

Referee Report

0 major / 3 minor

Summary. The paper proves a collection of new homological formulae for modules of finite quasi-projective dimension or finite quasi-injective dimension over a commutative Noetherian local ring R. The central results are the Derived Depth Formula (extending Auslander's classical depth formula), a width analogue of the Derived Depth Formula, an extended form of Ischebeck's formula, and a dependency formula in the style of Jorgensen. Several special cases remain new even when complete intersection dimension vanishes.

Significance. If the derivations hold, the work meaningfully enlarges the scope of Auslander-type depth formulae and related identities beyond modules of finite projective dimension. The reduction to the classical case via spectral sequences and depth lemmas under standard Noetherian local hypotheses is a standard and reliable technique; the explicit statement that certain corollaries are new even under stronger vanishing assumptions on complete intersection dimension is a concrete strength.

minor comments (3)

Abstract, line 2: 'Auslander famous depth formula' should read 'Auslander's famous depth formula'.
The manuscript would benefit from an explicit statement, early in the introduction or §2, of the precise definition of quasi-projective dimension (i.e., the length of the shortest resolution by quasi-projective modules) together with a reference to the relevant literature on quasi-projective modules.
Notation for the width and dependency formulae could be aligned more closely with the notation already used for the depth formulae to improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work, the assessment of its significance in extending Auslander-type formulae, and the recommendation of minor revision. The report raises no specific major comments.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript derives extensions of Auslander's depth formula and related results for modules of finite quasi-projective or quasi-injective dimension by reducing to the classical finite projective dimension case via standard spectral-sequence arguments, depth lemmas, and Tor-vanishing statements that hold under the stated commutative Noetherian local hypotheses. These steps rely on external, independently established homological algebra rather than self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations whose validity depends on the present work. The central claims therefore remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is limited to the explicit setting stated; no free parameters, invented entities, or non-standard axioms are mentioned.

axioms (1)

domain assumption R is a commutative Noetherian local ring
Explicitly stated as the ambient setting for all results.

pith-pipeline@v0.9.0 · 5371 in / 1169 out tokens · 26403 ms · 2026-05-11T01:05:25.282561+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
Theorem 3.5 (Derived Depth Formula): depth(M ⊗^L N) = depth M + depth N - depth R for finite qpd M and bounded-homology N
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Lemma 3.2 and Corollary 3.4 controlling depth of cycles/boundaries via minimal free resolutions and m-adic maps

Reference graph

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18 extracted references · 3 canonical work pages · 1 internal anchor

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