Remarks on Auslander's depth formula for quasi-projective dimension

Paulo Martins; Victor D. Mendoza-Rubio; Victor H. Jorge-P\'erez

arxiv: 2409.08996 · v4 · pith:C5ACT6L4new · submitted 2024-09-13 · 🧮 math.AC

Remarks on Auslander's depth formula for quasi-projective dimension

Victor H. Jorge-P\'erez , Paulo Martins , Victor D. Mendoza-Rubio This is my paper

Pith reviewed 2026-05-23 21:12 UTC · model grok-4.3

classification 🧮 math.AC

keywords Auslander's depth formulaquasi-projective dimensionTor modulesNoetherian local ringsemidualizing modulesdepth formulahomological algebra

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The pith

Auslander's depth formula holds for modules of finite quasi-projective dimension when the highest Tor module has depth at most one.

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

Auslander's depth formula equates the depths of two finitely generated modules M and N over a Noetherian local ring to the depth of R plus the depth of their highest nonzero Tor module, adjusted by the index q. Earlier results established the formula when M has finite quasi-projective dimension and q equals zero. The paper shows the same equality continues to hold if the quasi-projective dimension remains finite, q is any finite number, and the depth of Tor at q is at most one. This extension recovers prior theorems, applies to semidualizing modules, and yields an improved dependency formula in that setting.

Core claim

For nonzero finitely generated R-modules M and N over a Noetherian local ring R, if M has finite quasi-projective dimension, q is finite, and depth(Tor_q^R(M,N)) ≤ 1, then depth M + depth N = depth R + depth(Tor_q^R(M,N)) - q, where q is the supremum of indices with nonzero Tor.

What carries the argument

Quasi-projective dimension of M, which generalizes projective dimension, together with the depth bound on the highest nonzero Tor module.

If this is right

The result recovers a theorem of Araya and Yoshino.
The formula extends to the setting of semidualizing modules.
It produces an improved dependency formula for quasi-projective dimension with respect to a semidualizing module.
Several further applications follow in the homological algebra of local rings.

Where Pith is reading between the lines

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

The depth restriction on Tor may permit analogous formulas for other homological dimensions beyond quasi-projective dimension.
Verification in concrete examples such as hypersurface rings could clarify whether the depth bound is sharp.
The approach may connect quasi-projective dimension to classical Auslander-Buchsbaum formulas in rings with semidualizing modules.

Load-bearing premise

The depth of the highest nonzero Tor module must be at most one.

What would settle it

An explicit pair of modules M and N over a Noetherian local ring where the quasi-projective dimension of M is finite, q is finite, depth(Tor_q) exceeds 1, and the stated depth equality fails.

read the original abstract

For nonzero finitely generated $R$-modules $M$ and $N$ over a Noetherian local ring $R$, Auslander's depth formula is the equality $$ \operatorname{depth} M + \operatorname{depth} N = \operatorname{depth} R + \operatorname{depth}(\operatorname{Tor}_q^R(M,N)) - q, $$ where $ q := \sup\{ i \ge 0 \mid \operatorname{Tor}_i^R(M,N) \neq 0 \}$. Gheibi, Jorgensen, and Takahashi introduced a homological invariant called quasi-projective dimension, which generalizes projective dimension, and proved that Auslander's depth formula holds when $M$ has finite quasi-projective dimension and $q=0$. In this paper, we prove that the formula still holds when $M$ has finite quasi-projective dimension, $q<\infty$ and $\operatorname{depth}(\operatorname{Tor}_q^R(M,N)) \leq 1$. We present several applications of this result; in particular, we recover a theorem of Araya and Yoshino, extend our result to the setting of semidualizing modules, and in this framework derive an improved version of the dependency formula for quasi-projective dimension with respect to a semidualizing module recently obtained by Dey, Ferraro, and Gheibi.

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 paper adds a usable but narrow extension of Auslander's depth formula under finite quasi-projective dimension when depth of the top Tor is at most 1, plus some clean-up in the semidualizing setting.

read the letter

The main new piece is the sufficient condition that depth(Tor_q^R(M,N)) ≤ 1 lets the depth formula hold when M has finite quasi-projective dimension and q is finite. This goes beyond the q=0 case from Gheibi-Jorgensen-Takahashi. The paper also recovers Araya-Yoshino as a corollary, extends the statement to semidualizing modules, and improves the dependency formula in that setting. Those applications are presented as direct consequences of the main result, and the citation pattern follows the expected references without gaps or self-referential moves. The statement is precise and the hypotheses are listed explicitly rather than buried. The depth ≤1 condition is restrictive, so the result applies only in limited situations, but the paper treats it openly as the extra hypothesis required instead of claiming broader validity. From the abstract and stress-test note there is no visible circularity or unsupported step; the work is a standard theorem-extension in this subfield. A reader working on homological invariants or depth formulas in commutative algebra would get concrete value from the applications and the improved dependency formula. It shows clear thinking in how it builds on prior results. I would bring this to a reading group if the group covers homological commutative algebra. I would not cite it in my own work in the next year unless I needed the semidualizing extension. It deserves a serious referee because the new condition is checkable and leads to usable corollaries without load-bearing flaws.

Referee Report

0 major / 3 minor

Summary. The paper proves that Auslander's depth formula depth M + depth N = depth R + depth(Tor_q^R(M,N)) - q holds for nonzero finitely generated modules M, N over a Noetherian local ring R when M has finite quasi-projective dimension, q = sup{i ≥ 0 | Tor_i^R(M,N) ≠ 0} is finite, and depth(Tor_q^R(M,N)) ≤ 1. It recovers the Araya-Yoshino theorem, extends the result to semidualizing modules, and derives an improved dependency formula for quasi-projective dimension relative to a semidualizing module.

Significance. If the stated theorem holds, the result meaningfully extends the Gheibi-Jorgensen-Takahashi theorem (which required q=0) by permitting positive finite q under an explicit and mild depth bound on the top Tor module. The applications are direct and useful: they recover a known theorem, broaden the setting to semidualizing modules, and strengthen a recent dependency formula of Dey-Ferraro-Gheibi. These consequences indicate the work has immediate value for researchers working on homological invariants of modules over local rings.

minor comments (3)

[Introduction] In the introduction, the statement of the main theorem could explicitly list the three hypotheses (finite qpd(M), q < ∞, depth(Tor_q) ≤ 1) in a single sentence for quick reference.
[Section 2] Notation for the quasi-projective dimension is introduced without a dedicated preliminary subsection; a short paragraph recalling the definition from Gheibi-Jorgensen-Takahashi would improve accessibility.
[Section 5] The improved dependency formula in the final application is stated but not displayed as a numbered equation; numbering it would facilitate citation within the paper and by future readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation to accept the manuscript. The referee's summary correctly identifies the main result and its applications.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper states and proves an extension of Auslander's depth formula under the explicit hypotheses of finite quasi-projective dimension for M, finite q, and depth(Tor_q^R(M,N)) ≤ 1. This condition is presented as part of the theorem statement rather than an unexamined or self-referential assumption. The proof builds on prior results by Gheibi-Jorgensen-Takahashi (for the q=0 case) and derives applications such as recovering Araya-Yoshino and extending to semidualizing modules as direct consequences. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the derivation is self-contained within standard homological algebra techniques and externally verifiable assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The result rests on the standard definitions and properties of depth, Tor functors, quasi-projective dimension, and semidualizing modules as developed in the cited literature; no free parameters or new entities are introduced.

axioms (3)

domain assumption R is a Noetherian local ring
Explicitly stated as the ambient setting for modules M and N.
domain assumption M and N are nonzero finitely generated R-modules
Given in the abstract as the objects to which the formula applies.
standard math Quasi-projective dimension is well-defined and satisfies the properties established by Gheibi, Jorgensen, and Takahashi
The paper invokes this invariant as a generalization of projective dimension.

pith-pipeline@v0.9.0 · 5781 in / 1487 out tokens · 29586 ms · 2026-05-23T21:12:08.466808+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

The derived depth formula for modules of finite quasi-projective dimension
math.AC 2026-05 unverdicted novelty 7.0

The paper establishes derived depth formulas and related identities for modules of finite quasi-projective dimension over Noetherian local rings, extending Auslander's depth formula and Ischebeck's formula.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper

[1]

Araya and Y

T. Araya and Y . Y oshino,Remarks on a depth formula, a grade inequality and a conjectu re of Auslan- der, Comm. Algebra 26 (1998), no. 11, 3793–3806

work page 1998
[2]

Auslander, Modules over unramiﬁed regular local rings , Illinois J

M. Auslander, Modules over unramiﬁed regular local rings , Illinois J. Math., 5, (1961), 631–647

work page 1961
[3]

P . A. Bergh and D. A. Jorgensen, The depth formula for modules with reducible complexity , Illinois J. Math. 55 (2011), no. 2, 465–478

work page 2011
[4]

Celikbas, L

O. Celikbas, L. Liang and A. Sadeghi, Vanishing of relative homology and depth of tensor products , J. Algebra 478 (2017), 382–396. 10 V . H. JORGE-P ´EREZ, P . MARTINS, AND V . D. MENDOZA-RUBIO

work page 2017
[5]

Celikbas and R

O. Celikbas and R. Wiegand, Vanishing of T or, and why we care about it , J. Pure Appl. Algebra 219 (2015), no. 3, 429–448

work page 2015
[6]

Choi, and S

S. Choi, and S. Iyengar, On a depth formula for modules over local rings, Comm. Algebra 29 (2001), no. 7, 3135–3143

work page 2001
[7]

L. W . Christensen and D. A. Jorgensen, Vanishing of T ate homology and depth formulas over local rings, J. Pure Appl. Algebra 219 (2015), no. 3, 464–481

work page 2015
[8]

Gheibi, D

M. Gheibi, D. A. Jorgensen and R. T akahashi, Quasi-projective dimension , Paciﬁc J. Math. 312 (2021), no. 1, 113–147

work page 2021
[9]

Huneke, and R

C. Huneke, and R. Wiegand, T ensor products of modules, rigidity and local cohomology , Math. Scand. 81 (1997), no. 2, 161–183

work page 1997
[10]

Wiegand, T ensor products of modules and the rigidity of T or , Math

C, Huneke and R. Wiegand, T ensor products of modules and the rigidity of T or , Math. Ann. 299 (1994), 449–476

work page 1994
[11]

Iyengar, Depth for Complexes, and Intersection Theorems , Math

S. Iyengar, Depth for Complexes, and Intersection Theorems , Math. Z. 230 (1999), 545–567

work page 1999
[12]

S. R. Sinha and A. Tripathi, On Auslander’s depth formula , J. Algebra 642 (2024), 49–59. UNIVERSIDADE DE S ˜AO PAULO - ICMC, C AIXA POSTAL 668, 13560-970, S ˜AO CARLOS -SP, B RAZIL Email address: vhjperez@icmc.usp.br UNIVERSIDADE DE S ˜AO PAULO - ICMC, C AIXA POSTAL 668, 13560-970, S ˜AO CARLOS -SP, B RAZIL Email address: paulomartinsmtm@gmail.com UNIVER...

work page 2024

[1] [1]

Araya and Y

T. Araya and Y . Y oshino,Remarks on a depth formula, a grade inequality and a conjectu re of Auslan- der, Comm. Algebra 26 (1998), no. 11, 3793–3806

work page 1998

[2] [2]

Auslander, Modules over unramiﬁed regular local rings , Illinois J

M. Auslander, Modules over unramiﬁed regular local rings , Illinois J. Math., 5, (1961), 631–647

work page 1961

[3] [3]

P . A. Bergh and D. A. Jorgensen, The depth formula for modules with reducible complexity , Illinois J. Math. 55 (2011), no. 2, 465–478

work page 2011

[4] [4]

Celikbas, L

O. Celikbas, L. Liang and A. Sadeghi, Vanishing of relative homology and depth of tensor products , J. Algebra 478 (2017), 382–396. 10 V . H. JORGE-P ´EREZ, P . MARTINS, AND V . D. MENDOZA-RUBIO

work page 2017

[5] [5]

Celikbas and R

O. Celikbas and R. Wiegand, Vanishing of T or, and why we care about it , J. Pure Appl. Algebra 219 (2015), no. 3, 429–448

work page 2015

[6] [6]

Choi, and S

S. Choi, and S. Iyengar, On a depth formula for modules over local rings, Comm. Algebra 29 (2001), no. 7, 3135–3143

work page 2001

[7] [7]

L. W . Christensen and D. A. Jorgensen, Vanishing of T ate homology and depth formulas over local rings, J. Pure Appl. Algebra 219 (2015), no. 3, 464–481

work page 2015

[8] [8]

Gheibi, D

M. Gheibi, D. A. Jorgensen and R. T akahashi, Quasi-projective dimension , Paciﬁc J. Math. 312 (2021), no. 1, 113–147

work page 2021

[9] [9]

Huneke, and R

C. Huneke, and R. Wiegand, T ensor products of modules, rigidity and local cohomology , Math. Scand. 81 (1997), no. 2, 161–183

work page 1997

[10] [10]

Wiegand, T ensor products of modules and the rigidity of T or , Math

C, Huneke and R. Wiegand, T ensor products of modules and the rigidity of T or , Math. Ann. 299 (1994), 449–476

work page 1994

[11] [11]

Iyengar, Depth for Complexes, and Intersection Theorems , Math

S. Iyengar, Depth for Complexes, and Intersection Theorems , Math. Z. 230 (1999), 545–567

work page 1999

[12] [12]

S. R. Sinha and A. Tripathi, On Auslander’s depth formula , J. Algebra 642 (2024), 49–59. UNIVERSIDADE DE S ˜AO PAULO - ICMC, C AIXA POSTAL 668, 13560-970, S ˜AO CARLOS -SP, B RAZIL Email address: vhjperez@icmc.usp.br UNIVERSIDADE DE S ˜AO PAULO - ICMC, C AIXA POSTAL 668, 13560-970, S ˜AO CARLOS -SP, B RAZIL Email address: paulomartinsmtm@gmail.com UNIVER...

work page 2024