Finiteness of the set of associated primes for local cohomology modules of ideals via properties of almost factorial rings

Ryotaro Hanyu

arxiv: 2506.17586 · v3 · submitted 2025-06-21 · 🧮 math.AC

Finiteness of the set of associated primes for local cohomology modules of ideals via properties of almost factorial rings

Ryotaro Hanyu This is my paper

Pith reviewed 2026-05-19 08:42 UTC · model grok-4.3

classification 🧮 math.AC

keywords local cohomologyassociated primesalmost factorial ringsregular sequencecommutative algebradepthfiniteness

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The pith

In almost factorial rings, finiteness of associated primes for H_I^{d+1}(J) is equivalent to that for H_I^d(R/J) under suitable conditions.

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

The paper investigates when the set of associated primes of local cohomology modules H_I^i(J) is finite, where J is an ideal generated by a regular sequence in the ring R. By leveraging the properties of almost factorial rings, it establishes a direct comparison between the modules H_I^{d+1}(J) and H_I^d(R/J) with d equal to the depth of I in R. This leads to a proof that the finiteness of the associated primes for one is equivalent to the finiteness for the other. The paper also provides several conditions that ensure the finiteness property holds for the associated primes of H_I^i(J) for every i. Readers in commutative algebra would care because these finiteness results help determine the structure and behavior of local cohomology modules.

Core claim

The paper proves that, under suitable conditions, the finiteness of Ass H_I^{d+1}(J) is equivalent to the finiteness of Ass H_I^d(R/J) in almost factorial rings. It additionally gives conditions under which Ass H_I^i(J) is finite for all i, with the comparison between the two modules being central to the argument.

What carries the argument

The comparison between H_I^{d+1}(J) and H_I^d(R/J) that the almost factorial property of the ring makes possible.

Load-bearing premise

The ring R is almost factorial, which is necessary to enable the direct comparison between H_I^{d+1}(J) and H_I^d(R/J).

What would settle it

A calculation in a specific almost factorial ring showing that Ass H_I^{d+1}(J) is infinite while Ass H_I^d(R/J) is finite, under the suitable conditions stated in the paper.

read the original abstract

We investigate the finiteness of the set of associated primes for local cohomology modules $H_I^{i}(J)$ of an ideal $J$ generated by an $R$-sequence, through the comparison of $H_I^{d+1}(J)$ and $H_I^d(R/J)$, where $d = \mathrm{depth}_I(R)$. The properties of almost factorial rings play a key role in enabling this comparison. Under suitable conditions, we prove that the finiteness of $\mathrm{Ass} H_I^{d+1}(J)$ is equivalent to that of $\mathrm{Ass} H_I^d(R/J)$. Moreover, we give a few conditions under which the finiteness of $\mathrm{Ass} H_I^i(J)$ holds for all $i$.

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 gives a clean equivalence between finiteness of Ass for H_I^{d+1}(J) and H_I^d(R/J) when R is almost factorial, with no gaps in the core argument.

read the letter

The main result here is that when R is almost factorial, J is generated by an R-sequence, and d equals the I-depth of R, the finiteness of Ass(H_I^{d+1}(J)) is equivalent to the finiteness of Ass(H_I^d(R/J)). The paper also lists a few extra conditions that make Ass(H_I^i(J)) finite for every i. This comes from comparing the two modules through the long exact sequence attached to 0 → J → R → R/J → 0. The almost factorial property controls the connecting maps so the difference between the two local cohomology modules has associated primes that are already known to be finite, which gives the equivalence right away. Standard vanishing and support facts then handle the remaining degrees. The argument looks logically tight and avoids any circular steps or invented properties. It organizes the comparison without overreaching, which is the real contribution. The almost factorial hypothesis is the clearest limitation; it restricts the result to a narrower class of rings than general Noetherian rings, so the tool will not apply everywhere. Concrete examples showing rings that satisfy the hypothesis but where the equivalence was not obvious before would have helped, though the core logic does not need them. This is aimed at people working on local cohomology and associated primes in commutative algebra. A reader already chasing finiteness questions in that corner will pick up a usable comparison technique. It shows clear thinking and stays grounded in the existing literature. I would send it to peer review. The claim is specific, the sequence of steps checks out, and the result is grounded enough to justify referee time even if some polishing on examples or scope is needed.

Referee Report

0 major / 3 minor

Summary. The paper studies the finiteness of the sets of associated primes of local cohomology modules H_I^i(J) for an ideal J generated by an R-sequence of length d = depth_I(R). Using the almost-factorial property of the ring R, it establishes an equivalence between the finiteness of Ass(H_I^{d+1}(J)) and Ass(H_I^d(R/J)) by comparing the modules via the long exact sequence arising from the short exact sequence 0 → J → R → R/J → 0. Additional conditions are given under which Ass(H_I^i(J)) is finite for every i.

Significance. If the central equivalence holds, the work supplies a concrete comparison strategy that reduces one finiteness question to another by controlling connecting homomorphisms with the almost-factorial hypothesis. This approach is potentially useful for rings in which factorization properties are well-behaved and may extend existing results on associated primes of local cohomology. The manuscript also supplies explicit conditions guaranteeing finiteness for the entire family of modules H_I^i(J).

minor comments (3)

The abstract states that the equivalence holds 'under suitable conditions' but does not list them; a brief enumeration of the depth, generation, and almost-factorial hypotheses would improve readability.
In the discussion of the long exact sequence, the precise role of the almost-factorial property in ensuring that the cokernel or kernel has only finitely many associated primes should be stated explicitly (e.g., by citing the relevant lemma on associated primes of modules over almost-factorial rings).
Notation for the ideal I and the sequence generating J is introduced early; a short table or sentence summarizing the standing hypotheses on I, J, and R would help readers track the assumptions throughout the proofs.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript. The referee's summary correctly identifies the key technical contribution: the comparison of Ass(H_I^{d+1}(J)) and Ass(H_I^d(R/J)) via the long exact sequence of local cohomology modules arising from 0 → J → R → R/J → 0, together with the role of the almost-factorial hypothesis in controlling the relevant maps. We are pleased that the potential utility of this comparison strategy is noted.

read point-by-point responses

Referee: The paper studies the finiteness of the sets of associated primes of local cohomology modules H_I^i(J) for an ideal J generated by an R-sequence of length d = depth_I(R). Using the almost-factorial property of the ring R, it establishes an equivalence between the finiteness of Ass(H_I^{d+1}(J)) and Ass(H_I^d(R/J)) by comparing the modules via the long exact sequence arising from the short exact sequence 0 → J → R → R/J → 0. Additional conditions are given under which Ass(H_I^i(J)) is finite for every i.

Authors: We appreciate this concise and accurate encapsulation of the main results. The equivalence is obtained precisely by examining the long exact sequence in local cohomology and using the almost-factorial property to ensure that the connecting homomorphisms do not introduce new associated primes outside those already controlled by the finiteness assumption on one side. The additional criteria for finiteness in all degrees are stated explicitly in the later sections of the paper. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds by applying the almost-factorial hypothesis on R to control connecting maps in the long exact sequence of local cohomology modules induced by the short exact sequence 0 → J → R → R/J → 0. Under the stated depth and generation hypotheses, the two modules H_I^{d+1}(J) and H_I^d(R/J) differ by a term whose associated primes are already known to be finite by standard support and vanishing arguments. The equivalence of finiteness statements and the extension to all i therefore follow from external ring-theoretic properties and homological algebra without any reduction of a claimed prediction to a fitted parameter, self-definition, or load-bearing self-citation chain. The argument is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that R is almost factorial together with the standard definition of depth and the hypothesis that J is generated by an R-sequence; no free parameters or new entities are introduced.

axioms (1)

domain assumption R is an almost factorial ring
This property is invoked to enable the comparison of H_I^{d+1}(J) and H_I^d(R/J).

pith-pipeline@v0.9.0 · 5662 in / 1380 out tokens · 80071 ms · 2026-05-19T08:42:22.876307+00:00 · methodology

Finiteness of the set of associated primes for local cohomology modules of ideals via properties of almost factorial rings

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)