arxiv: 2605.11943 · v1 · submitted 2026-05-12 · 🧮 math.AC

Recognition: 1 theorem link

· Lean Theorem

Bass numbers of graded components of local cohomology modules

Maryam Jahangiri

Pith reviewed 2026-05-13 04:01 UTC · model grok-4.3

classification 🧮 math.AC

keywords bass numberslocal cohomologygraded modulesasymptotic behaviorcohomological dimensionfiniteness dimensionrelative Cohen-Macaulayirrelevant ideal

0 comments

The pith

In standard graded rings, the Bass numbers of graded pieces of local cohomology modules follow specific asymptotic patterns in three cases.

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

The paper examines sequences of Bass numbers μ^i(p0, H^j_{R+}(M)_n) as n runs over all integers, for local cohomology of a finitely generated graded module M with respect to the irrelevant ideal of a standard graded ring R. It establishes that these sequences display predictable long-term behavior when i is 0 or 1 and j is at most the finiteness dimension, when the base ring R0 is regular and i sits at or just below the height of p0 with j equal to the cohomological dimension, and when M itself is relative Cohen-Macaulay with respect to R+. Readers care because these patterns clarify how injective resolutions of local cohomology modules behave degree by degree, which in turn informs vanishing, support, and structure questions in graded commutative algebra.

Core claim

The sequence {μ^i(p0, H^j_{R+}(M)_n)}n∈Z exhibits specific asymptotic behavior in the cases (1) i=0 or 1 and j≤f_{R+}(M), (2) R0 regular with i=hei(p0) or hei(p0)-1 and j=cd_{R+}(M), (3) M relative Cohen-Macaulay w.r.t. R+.

What carries the argument

Bass numbers μ^i(p0, N) counting the multiplicity of the injective hull of R0/p0 in the minimal injective resolution of N, applied to each graded piece N = H^j_{R+}(M)_n.

Load-bearing premise

R is a standard graded ring and M is a finitely generated graded R-module, with local cohomology taken relative to the irrelevant ideal R+.

What would settle it

A concrete standard graded ring R, finitely generated graded M, and prime p0 where, for i=0 and j equal to the finiteness dimension, the sequence μ^0(p0, H^j_{R+}(M)_n) keeps changing for arbitrarily negative n instead of stabilizing.

read the original abstract

Let $R=\bigoplus_{n\in \NN_0}R_n$ be a standard graded ring, $R_+=\bigoplus_{n\in \NN}R_n$ its irrelevant ideal, and $M$ a finitely generated graded $R$-module. In this paper, we study the asymptotic behavior of the sequence $\{\mu^i(\p_0, H^j_{R_+}(M)_n)\}_{n\in \Z}$ of Bass numbers of graded components of local cohomology modules with respect to an ideal $\p_0\in \Spec(R_0)$ in each of the following cases: (1) $i=0$ or $i= 1$ and $j\leq f_{R_+}(M)$, (2) $R_0$ is regular, $i= \hei(\p_0)$ or $i= \hei(\p_0)- 1$ and $j= \cd_{R_+}(M)$, (3) $M$ is relative Cohen-Macaulay with respect to $R_+$. Here, $\cd_{R_+}(M)$ and $f_{R_+}(M)$ denote the cohomological dimension and finiteness dimension of $M$ with respect to $R_+$, respectively.

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 gives targeted asymptotic descriptions for Bass numbers of graded local cohomology pieces in three standard cases, but the value hinges on the proofs which aren't visible here.

read the letter

The paper studies the asymptotic behavior of the sequence of Bass numbers μ^i(p0, H^j_{R+}(M)_n) as n varies over the integers, for a standard graded ring R, finitely generated graded module M, and prime p0 in the degree-zero part. It focuses on three regimes: small i (0 or 1) with j at most the finiteness dimension, the case where the base ring R0 is regular and i sits at or just below the height of p0 when j equals the cohomological dimension, and the relative Cohen-Macaulay case. These are natural extensions of existing work on local cohomology invariants rather than a broad new framework. The setup uses the usual definitions of cd and f, which keeps everything grounded in the literature. The paper does well by isolating concrete situations where one can actually say something definite about the sequence, avoiding vague general claims. That kind of specificity can be helpful for later calculations or bounds. The main soft spot is that only the abstract is in front of me, so I cannot check the actual arguments or see whether the proofs handle the graded pieces cleanly or introduce any extra vanishing assumptions. If the derivations are direct and the examples are convincing, the results should hold up; otherwise the claims stay formal. This is aimed at commutative algebraists who already work with local cohomology and Bass numbers in graded settings. Someone computing these invariants or looking for stabilization results in specific degrees would get the most out of it. The work is incremental but precise enough that it deserves a serious referee rather than a desk rejection, provided the full proofs check out.

Referee Report

0 major / 3 minor

Summary. The paper studies the asymptotic behavior of the sequence of Bass numbers {μ^i(p0, H^j_{R+}(M)_n)}n∈Z for a standard graded ring R, finitely generated graded R-module M, and prime p0 ∈ Spec(R0). It establishes results on this behavior in three regimes: (1) i=0 or 1 with j ≤ f_{R+}(M); (2) R0 regular, i=ht(p0) or ht(p0)−1 with j=cd_{R+}(M); (3) M relative Cohen-Macaulay w.r.t. R+.

Significance. If the claimed asymptotic statements hold, the results provide concrete information on the eventual vanishing or constancy of Bass numbers of graded pieces of local cohomology, which is useful for understanding the homological structure of H^j_{R+}(M) in graded settings. The cases cover standard situations involving finiteness dimension, cohomological dimension, and relative Cohen-Macaulayness, building on existing literature on local cohomology.

minor comments (3)

The abstract and introduction should explicitly state whether the asymptotic behavior means eventual constancy, eventual vanishing, or a specific formula for large |n|, with a reference to the precise statement in the main theorems.
Notation: ensure that the subscript n on H^j_{R+}(M)_n is consistently defined as the degree-n component throughout, and clarify the range of n (positive or all integers) in each case.
The paper would benefit from a brief comparison table or remark contrasting the three cases with previously known results on Bass numbers of local cohomology (e.g., works on asymptotic stability of Ext or Tor).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work on the asymptotic behavior of Bass numbers of graded components of local cohomology modules and for recommending minor revision. No specific major comments were provided in the report, so there are no individual points requiring detailed rebuttal or defense. We will address any minor editorial or presentational issues in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained on standard definitions

full rationale

The paper examines asymptotic behavior of Bass numbers μ^i(p0, H^j_{R+}(M)_n) in three conventional regimes using the standard definitions of cohomological dimension cd_{R+}(M) and finiteness dimension f_{R+}(M) from the local cohomology literature. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or claimed results. The setup (standard graded ring, fg graded module, Spec(R0) prime) is external and non-circular. All listed cases rely on prior independent results about local cohomology rather than reducing to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract introduces no explicit free parameters, new axioms, or invented entities; it relies on standard notions from local cohomology theory.

pith-pipeline@v0.9.0 · 5526 in / 1072 out tokens · 100542 ms · 2026-05-13T04:01:09.714697+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
We study the asymptotic behavior of the sequence {μ^i(p0, H^j_{R+}(M)_n)}n∈Z of Bass numbers of graded components of local cohomology modules... in cases (1) i=0 or 1 and j≤f_{R+}(M), (2) R0 regular with i=ht(p0) or ht(p0)-1 and j=cd_{R+}(M), (3) M relative Cohen-Macaulay w.r.t. R+.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

˜Alvarez Montaner, F

J. ˜Alvarez Montaner, F. Sohrabi,Bass numbers of local cohomology of cover ideals of graphs, J. Alg. Comb.53(2021) 263–297, https://doi.org/10.1007/s10801-019-00928-0

work page doi:10.1007/s10801-019-00928-0 2021
[2]

Brodmann,Asymptotic behaviour of cohomology: tameness, supports and associated primes, S

M. Brodmann,Asymptotic behaviour of cohomology: tameness, supports and associated primes, S. Ghorpade, H. Srinivasan, J. Verma (Eds.), ”Commutative Algebra and Algebraic Geometry” Proceedings, Joint International Meeting of the AMS and the IMS on Commutative Algebra and Algebraic Geometry, Bangalore/India, December 17-20, 2003, Contemporary Mathematics,39...

work page 2003
[3]

Brodmann, R.Y

M. Brodmann, R.Y. Sharp,Local Cohomology: an algebric introduction with geometric applications, 2nd ed. Cambridge University Press, (2012)

work page 2012
[4]

Bruns and J

W. Bruns and J. Herzog,Cohen–Macaulay rings, 2nd ed. Cambridge University Press, (1998)

work page 1998
[5]

Chardin, S

M. Chardin, S. D. Cutkosky, J. Herzog, H. Srinivasan,Duality and tameness,Michigan Math. J., 57(2008) 137-155, DOI: 10.1307/mmj/1220879401

work page doi:10.1307/mmj/1220879401 2008
[6]

L. Chu, Y. Gu,A Problem of Local Cohomology Modules,Communications in Algebra,36:4 (2008) 1603-1607, DOI: 10.1080/00927870701869253

work page doi:10.1080/00927870701869253 2008
[7]

Divaani-Aazar, R

K. Divaani-Aazar, R. Naghipour and M. Tousi,Cohomological dimension of certain algebraic varieties, Proc. Amer. Math. Soc.130(12) (2002) 3537-3544

work page 2002
[8]

D. Helm, E. Miller,Bass numbers of semigroup-graded local cohomology, Pacific Journal of Mathematics, 209:1 (2003) 41-66

work page 2003
[9]

Jahangiri, R

M. Jahangiri, R. Ahangari Maleki,Bass numbers of local cohomology modules at the first and last non-vanishing levels, https://doi.org/10.48550/arXiv.2603.18724. BASS NUMBERS OF GRADED COMPONENTS OF... 19

work page doi:10.48550/arxiv.2603.18724
[10]

Jahangiri, A

M. Jahangiri, A. Rahimi,Relative Cohen-Macaulauness and relative unmixedness of bi-graded modules, Journal of Commutative Algebra,4:4 (2012) 551-575

work page 2012
[11]

Jahangiri and H

M. Jahangiri and H. Zakeri,Local cohomology modules with respect to an ideal containing the irrelevant ideal, J. Pure and Appl. Algebra213(2009) 573- 581

work page 2009
[12]

Herzog,Komplex aufl¨asungen und dualit¨at in der lokalen algebra, Habilitationsschrift, Universit¨at Regensburg, (1974)

J. Herzog,Komplex aufl¨asungen und dualit¨at in der lokalen algebra, Habilitationsschrift, Universit¨at Regensburg, (1974)

work page 1974
[13]

Huneke and R

C. Huneke and R. Y. Sharp,Bass numbers of local cohomology modules,Trans. Amer. Math. Soc. 339 (1993) 765-779

work page 1993
[14]

Kirby,Artinian modules and hilbert polynomials, Q

D. Kirby,Artinian modules and hilbert polynomials, Q. J. Math.,24:1 (1973) 47-57

work page 1973
[15]

Lyubeznik,Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra), Invent

G. Lyubeznik,Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra), Invent. Math., 113 (1993) 41-55

work page 1993
[16]

Lyubeznik,On the vanishing of local cohomology in characteristicp >0, Compositio Math.,142 (2006) 207-221

G. Lyubeznik,On the vanishing of local cohomology in characteristicp >0, Compositio Math.,142 (2006) 207-221

work page 2006
[17]

Ogus,Local cohomological dimensional of algebraic varieties, Ann

A. Ogus,Local cohomological dimensional of algebraic varieties, Ann. Math.,98:2 (1973) 327-365

work page 1973
[18]

J. J. Rotman,An introduction to homological algebra, 2nd ed. Academic press (2008)

work page 2008
[19]

R. Y. Sharp,Bass numbers in the graded case, a-invariant formula, and an analogue of Faltings annihilator theorem, J. Algebra222(1999) 246-270

work page 1999
[20]

Puthenpurakal,Graded components of local cohomology modules, Collectanea Mathematica,73:2 (2022) 135-171

Tony J. Puthenpurakal,Graded components of local cohomology modules, Collectanea Mathematica,73:2 (2022) 135-171

work page 2022
[21]

T. J. Puthenpurakal,Graded components of local cohomology modules over polynomial rings, https://doi.org/10.48550/arXiv.2411.13090

work page doi:10.48550/arxiv.2411.13090
[22]

Wisbauer,Founations of Module and Ring Theory, Algebra, Logic and Applications, 3, Amsterdam: Gordon and Breach (1991)

R. Wisbauer,Founations of Module and Ring Theory, Algebra, Logic and Applications, 3, Amsterdam: Gordon and Breach (1991)

work page 1991
[23]

Yanagawa,Bass numbers of local cohomology modules with supports in monomial ideals, Math

K. Yanagawa,Bass numbers of local cohomology modules with supports in monomial ideals, Math. Proc. Cambridge Philos. Soc.,131(2001) 45-60. Department of Mathematics, F aculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran. Email address:jahangiri@khu.ac.ir, jahangiri.maryam@gmail.com

work page 2001