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arxiv: 1907.05472 · v1 · pith:GGG6M7WRnew · submitted 2019-07-11 · 🧮 math.AC

Quasi-cyclic modules and coregular sequences

Robin Hartshorne , Claudia Polini This is my paper

Pith reviewed 2026-05-24 22:18 UTC · model grok-4.3

classification 🧮 math.AC
keywords quasi-cyclic modulescoregular sequencescodepthset-theoretic complete intersectionslocal cohomologycurves in projective spaceNoetherian rings
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The pith

Modules of codepth at least two over a Noetherian ring are increasing unions of cyclic modules.

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

The paper extends the definitions of coregular sequences and codepth to modules over Noetherian rings that need not be finitely generated or artinian. It proves that any module of codepth at least two must be quasi-cyclic. This property supplies a revised version of Hellus' theorem that detects set-theoretic complete intersections from the structure of local cohomology modules. The same machinery produces several concrete necessary conditions that any curve in projective three-space must satisfy if it is a set-theoretic complete intersection.

Core claim

We develop the theory of coregular sequences and codepth for arbitrary modules over a Noetherian ring. We show that modules of codepth at least two are quasi-cyclic, that is, increasing unions of cyclic modules. This yields a new version of Hellus' theorem characterizing set-theoretic complete intersections via local cohomology. We then derive a number of necessary conditions for a curve in projective three-space to be a set-theoretic complete intersection.

What carries the argument

Quasi-cyclic modules, defined as increasing unions of cyclic modules, which the paper proves contain every module of codepth at least two.

If this is right

  • A new local-cohomology criterion identifies set-theoretic complete intersections.
  • Any curve in P^3 that fails one of the listed conditions cannot be a set-theoretic complete intersection.
  • The open question whether every connected curve in P^3 is a set-theoretic complete intersection can now be tested against these necessary conditions.
  • The quasi-cyclic property applies directly to local cohomology modules arising from curves or other varieties.

Where Pith is reading between the lines

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

  • The same codepth condition might be used to simplify explicit calculations of local cohomology in examples that are not finitely generated.
  • Checking the listed conditions on explicit curves in P^3 could produce either supporting evidence or a counterexample to the open question.
  • The quasi-cyclic characterization may connect to questions about the structure of local cohomology in higher-dimensional projective varieties.

Load-bearing premise

The notions of coregular sequences and codepth extend in a useful way from finitely generated or artinian modules to arbitrary modules over a Noetherian ring.

What would settle it

Exhibit a module over a Noetherian ring that has codepth at least two yet cannot be written as an increasing union of cyclic submodules.

read the original abstract

We develop the theory of coregular sequences and codepth for modules that need not be finitely generated or artinian over a Noetherian ring. We use this theory to give a new version of a theorem of Hellus characterizing set-theoretic complete intersections in terms of local cohomology modules. We also define quasi-cyclic modules as increasing unions of cyclic modules, and show that modules of codepth at least two are quasi-cyclic. We then focus our attention on curves in projective three-space and give a number of necessary conditions for a curve to be a set-theoretic complete intersection. Thus an example of a curve for which any of these necessary conditions does not hold would provide a negative answer to the still open problem, whether every connected curve in projective three-space is a set-theoretic complete intersection

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.

Referee Report

1 major / 0 minor

Summary. The paper extends the theory of coregular sequences and codepth from the finitely generated or artinian setting to arbitrary modules over Noetherian rings. It defines quasi-cyclic modules as increasing unions of cyclic modules and proves that modules of codepth at least two are quasi-cyclic. It gives a revised version of Hellus' theorem characterizing set-theoretic complete intersections in terms of local cohomology modules, and derives several necessary conditions for a curve in projective 3-space to be a set-theoretic complete intersection.

Significance. If the extensions of the definitions are valid and the stated theorems follow, the work supplies new tools for analyzing local cohomology of non-finitely-generated modules and necessary conditions that could help address the open question of whether every connected curve in P^3 is a set-theoretic complete intersection. The quasi-cyclic property and the Hellus-type characterization are potentially useful if they hold without additional hidden hypotheses.

major comments (1)
  1. The central claims depend on the extension of coregular sequences and codepth to arbitrary (possibly non-f.g.) modules; the abstract invokes this extension when applying the theory to local cohomology modules and to ideal sheaves of curves, but the load-bearing definitions and their properties are not visible in the provided abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for highlighting this point about the abstract. We respond to the major comment below.

read point-by-point responses

  1. Referee: The central claims depend on the extension of coregular sequences and codepth to arbitrary (possibly non-f.g.) modules; the abstract invokes this extension when applying the theory to local cohomology modules and to ideal sheaves of curves, but the load-bearing definitions and their properties are not visible in the provided abstract.

    Authors: The abstract is a concise summary of the paper's contributions and does not contain technical definitions, which is standard in mathematical writing. The extensions of coregular sequences and codepth to arbitrary modules, together with all required properties, are defined and proved in detail in the body of the manuscript (beginning in Section 2) before any applications to local cohomology modules or ideal sheaves of curves are discussed. The abstract correctly states that these extensions are developed and used; readers interested in the precise statements are directed to the full text. We therefore see no need to alter the abstract. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper develops an extension of coregular sequences and codepth to arbitrary (not necessarily f.g. or artinian) modules over Noetherian rings, then uses the new notions to prove that codepth >=2 implies quasi-cyclic and to obtain a variant of Hellus' theorem on local cohomology. These steps are definitional and deductive from the introduced concepts plus standard local-cohomology machinery; they do not reduce by construction to prior fitted quantities or to a self-citation chain. The Hellus citation is external and supports only the comparison statement, not the new definitions or the quasi-cyclic claim. No load-bearing step matches any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 3 invented entities

The paper rests on the standard axioms of commutative algebra over Noetherian rings together with the new definitions of coregular sequences and codepth; no free parameters or invented physical entities appear.

axioms (1)
  • standard math Standard properties of local cohomology and Noetherian rings hold for the modules under consideration.
    Invoked when extending Hellus' theorem and when defining codepth.
invented entities (3)
  • coregular sequence no independent evidence
    purpose: Generalization of regular sequence to non-finitely-generated modules
    Newly defined object whose properties are developed in the paper.
  • codepth no independent evidence
    purpose: Measure of depth-like behavior for general modules
    Newly defined invariant used to state the quasi-cyclic theorem.
  • quasi-cyclic module no independent evidence
    purpose: Increasing union of cyclic modules
    Newly defined class shown to contain all modules of codepth >=2.

pith-pipeline@v0.9.0 · 5654 in / 1390 out tokens · 21700 ms · 2026-05-24T22:18:08.566457+00:00 · methodology

discussion (0)

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Reference graph

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