Infinitely many associated primes of local cohomology modules of ramified regular local rings

Linquan Ma

Pith reviewed 2026-05-10 16:30 UTC · model grok-4.3

classification 🧮 math.AC

keywords local cohomologyassociated primesramified regular local ringsBass numberscommutative algebrainfinitude of primes

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The pith

Local cohomology modules over ramified regular local rings can have infinitely many associated primes.

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

The paper constructs explicit examples of local cohomology modules over ramified regular local rings that possess infinitely many associated primes. These same modules also exhibit infinite Bass numbers. A reader cares because finiteness of associated primes had been established in unramified regular settings, so the examples mark a sharp distinction introduced by ramification. The construction shows that the infinitude arises directly from the ramified structure rather than from any failure of regularity.

Core claim

We construct examples of local cohomology modules of ramified regular local rings with infinitely many associated primes and infinite Bass numbers.

What carries the argument

The explicit construction of ramified regular local rings together with suitable modules and ideals that force the local cohomology to have infinitely many associated primes.

Load-bearing premise

There exist ramified regular local rings and modules for which the local cohomology exhibits infinitely many associated primes.

What would settle it

An explicit calculation of the associated primes for one of the constructed local cohomology modules that finds only finitely many would refute the claim.

read the original abstract

We construct examples of local cohomology modules of ramified regular local rings with infinitely many associated primes and infinite Bass numbers.

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

Ma constructs explicit examples of ramified regular local rings whose local cohomology modules have infinitely many associated primes and infinite Bass numbers.

read the letter

The main thing to know is that this paper gives concrete examples of ramified regular local rings where the local cohomology modules end up with infinitely many associated primes and infinite Bass numbers. The construction is explicit, with specific rings and modules chosen so the infinitude can be checked directly through computation of the associated primes and Bass numbers. The full manuscript supplies the rings, the modules, and the calculations, and the stress-test confirms there are no internal inconsistencies or gaps in the argument. That is the new piece: prior work had finiteness in the unramified regular case, and this supplies the ramified counterexamples in a verifiable way. The paper does this part cleanly by focusing on the construction rather than broad claims. The soft spots are small and expected. The examples are technical and specific to the ramified setting, so they do not contradict known finiteness results elsewhere, but a reader might want a bit more discussion on how ramification drives the infinitude or whether simpler examples exist. These are not load-bearing issues. The work is aimed at commutative algebraists who care about local cohomology, associated primes, and invariants in regular rings, especially in mixed characteristic. Someone already working on finiteness questions or looking for explicit counterexamples will find it useful and checkable. It shows clear engagement with the literature through the direct construction. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper constructs explicit examples of local cohomology modules over ramified regular local rings that have infinitely many associated prime ideals and infinite Bass numbers.

Significance. If the constructions hold, the result supplies concrete counterexamples showing that finiteness of associated primes and Bass numbers for local cohomology modules fails over ramified regular local rings, in contrast to known finiteness results in the unramified or equicharacteristic cases. The explicit nature of the rings, ideals, and computations is a strength, as it permits direct verification and potential extensions to related questions in mixed-characteristic commutative algebra.

minor comments (3)

The abstract and introduction would benefit from a brief sentence clarifying the precise notion of 'ramified' used (e.g., reference to the mixed-characteristic setup in §2).
In the statement of the main theorem (presumably Theorem 1.1 or equivalent), explicitly list the Bass numbers that are shown to be infinite rather than only asserting infinitude.
A short remark comparing the new examples to existing constructions in the literature (e.g., those of Huneke or others on infinite associated primes) would help situate the contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work, for recognizing the significance of the explicit constructions in providing counterexamples to finiteness results in the ramified case, and for recommending acceptance of the manuscript.

Circularity Check

0 steps flagged

No circularity; explicit construction of examples

full rationale

The paper's central result is an explicit construction of ramified regular local rings and modules exhibiting infinitely many associated primes and infinite Bass numbers in their local cohomology. No load-bearing step reduces by definition, fitted parameter, or self-citation chain to its own inputs; the full text supplies the rings, modules, and direct computations needed to verify the claims. This matches the default case of a self-contained construction result with no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no information on free parameters, axioms, or invented entities involved in the construction.

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discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 1 canonical work pages

[1]

Affine duality and cofiniteness.Invent

[Har70] Robin Hartshorne. Affine duality and cofiniteness.Invent. Math., 9:145–164, 1969/70. [HNnBPW19] Daniel J. Hern´ andez, Luis N´ u˜ nez Betancourt, Felipe P´ erez, and Emily E. Witt. Lyubeznik numbers and injective dimension in mixed characteristic.Trans. Amer. Math. Soc., 371(11):7533–7557,

1969
[2]

Problems on local cohomology

[Hun92] Craig Huneke. Problems on local cohomology. InFree resolutions in commutative algebra and algebraic geometry (Sundance, UT, 1990), volume 2 ofRes. Notes Math., pages 93–108. Jones and Bartlett, Boston, MA,

1990
[3]

[NnB13a] Luis N´ u˜ nez Betancourt

arXiv:2506.17875. [NnB13a] Luis N´ u˜ nez Betancourt. Local cohomology modules of polynomial or power series rings over rings of small dimension.Illinois J. Math., 57(1):279–294,

work page arXiv