On Zero-Dimensional Glicci Monomial Ideals

Benjamin Mudrak

arxiv: 2602.03703 · v2 · pith:UIH6ZEY6new · submitted 2026-02-03 · 🧮 math.AC

On Zero-Dimensional Glicci Monomial Ideals

Benjamin Mudrak This is my paper

Pith reviewed 2026-05-21 14:17 UTC · model grok-4.3

classification 🧮 math.AC

keywords monomial idealsGorenstein liaisongliccicomplete intersectionm-primary idealszero-dimensional idealsliaison class

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The pith

All m-primary monomial ideals in k[x,y,z] with at most eight generators are homogeneously glicci.

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

The paper proves that every m-primary monomial ideal in the polynomial ring over three variables that requires at most eight minimal generators belongs to the Gorenstein liaison class of a complete intersection. The result follows from explicit constructions of homogeneous Gorenstein links that reach a complete intersection while keeping every second ideal in the chain an m-primary monomial ideal. A separate construction produces monomial ideals in any number of variables that are glicci yet lie outside the complete intersection liaison class of a complete intersection. These statements classify a concrete family of zero-dimensional monomial ideals and show that the glicci property holds for all members of this family under the stated bound on generators.

Core claim

We prove that all m-primary monomial ideals in k[x,y,z] with at most eight generators are homogeneously glicci. We also construct a large class of m-primary monomial ideals in R_n for any n with any number of minimal generators that are homogeneously glicci but not in the complete intersection liaison class of a complete intersection (licci). All Gorenstein links used are constructed explicitly and every second step links to another m-primary monomial ideal.

What carries the argument

Explicit homogeneous Gorenstein links that connect an m-primary monomial ideal to a complete intersection while ensuring every second ideal in the liaison chain remains an m-primary monomial ideal.

If this is right

All m-primary monomial ideals in three variables with at most eight generators lie in the Gorenstein liaison class of a complete intersection.
There exist m-primary monomial ideals in any number of variables that are homogeneously glicci but not licci.
The Gorenstein links for these ideals can be written down explicitly while preserving the monomial and m-primary conditions at alternate steps.

Where Pith is reading between the lines

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

The generator bound of eight may be tied to the three-variable case; the same explicit-link method might succeed for some ideals with more generators.
The technique of alternating between monomial m-primary ideals could be tested on zero-dimensional ideals that are not monomial.
If the pattern continues, the glicci property might hold for all zero-dimensional monomial ideals regardless of generator count.

Load-bearing premise

Suitable Gorenstein links can always be chosen so that every second ideal in the liaison chain stays an m-primary monomial ideal.

What would settle it

An explicit m-primary monomial ideal in k[x,y,z] with eight or fewer generators that cannot be connected to a complete intersection by any sequence of homogeneous Gorenstein links in which every second ideal remains monomial and m-primary.

read the original abstract

Consider the polynomial ring $R_n = k[x_1,...,x_n]$, where $k$ is a field. Let $m = (x_1,...,x_n)$ and $I$ be an $m$-primary monomial ideal in $R$. We consider the problem of determining whether such ideals are in the Gorenstein liasion class of a complete intersection (glicci). We prove that all $m$-primary monomial ideals in $k[x,y,z]$ with at most eight generators are homogeneously glicci. We also construct a large class of $m$-primary monomial ideals in $R_n$ for any $n$ with any number of minimal generators that are homogeneously glicci but not in the complete intersection liaison class of a complete intersection (licci). All Gorenstein links used are constructed explicitly and every second step links to another $m$-primary monomial ideal.

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

Explicit constructions classify all m-primary monomial ideals in three variables with at most eight generators as homogeneously glicci and give glicci-but-not-licci families in higher dimensions.

read the letter

The punchline for this paper is that it classifies a chunk of the glicci monomial ideals in three variables: every m-primary monomial ideal in k[x,y,z] with no more than eight generators is homogeneously glicci. The proof relies on explicit Gorenstein links, and the same technique yields a large class of examples in n variables that are glicci but not licci. What the paper does well is the hands-on construction. Instead of proving existence, it gives specific sets of monomial generators for each link and checks that the resulting ideals stay m-primary and monomial. The stress-test note confirms that these steps check out without hidden assumptions or unverified parts. That level of detail is helpful for anyone who wants to see how the liaison works in practice. On the soft spots, the bound of eight generators feels specific to the cases they checked. It works for those, but the paper does not indicate whether nine or ten generators would still work or where the method breaks. This is not a flaw in the stated results, just a limit on how far the classification goes. Everything stays within monomial ideals, so general non-monomial cases are left aside, which matches the paper's focus. This work is for people in commutative algebra who study Gorenstein liaison and monomial ideals. Someone trying to understand the difference between glicci and licci classes will find the explicit examples useful. It deserves a serious referee. The claims are grounded in concrete constructions that can be inspected directly.

Referee Report

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Summary. The manuscript proves that every m-primary monomial ideal in k[x,y,z] with at most eight minimal generators is homogeneously glicci, achieved via explicit constructions of Gorenstein links where every second step remains an m-primary monomial ideal. It additionally constructs a large family of m-primary monomial ideals in R_n for arbitrary n that are homogeneously glicci but not licci, again using explicit links preserving the required properties.

Significance. If the explicit constructions hold, this advances liaison theory for monomial ideals by completely settling the homogeneous glicci question for all cases with up to eight generators in three variables and by supplying concrete examples that distinguish glicci from licci classes in higher dimensions. The direct, case-by-case verification of the links and the preservation of the m-primary monomial property at each step is a clear strength, as it renders the claims verifiable without hidden parameters or reductions.

Simulated Author's Rebuttal

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We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We appreciate the recognition that the explicit constructions and direct verification of the links, including preservation of the m-primary monomial property, constitute a clear strength of the work.

Circularity Check

0 steps flagged

No significant circularity; results rest on explicit constructions

full rationale

The paper proves its claims by supplying explicit, case-by-case constructions of Gorenstein links for every m-primary monomial ideal in k[x,y,z] with at most eight minimal generators, verifying directly that each second link remains an m-primary monomial ideal. A similar explicit-link technique produces the larger family in n variables that are glicci but not licci. These steps are self-contained constructive verifications with concrete monomial generators; no fitted parameters, self-definitional reductions, or load-bearing self-citations appear in the argument chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard commutative algebra without introducing fitted parameters or new entities.

axioms (1)

standard math Polynomial rings over a field admit monomial ideals and Gorenstein links in the usual way
Basic setup for R_n and m-primary ideals.

pith-pipeline@v0.9.0 · 5681 in / 1109 out tokens · 73964 ms · 2026-05-21T14:17:14.219621+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/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that all m-primary monomial ideals in k[x,y,z] with at most eight generators are homogeneously glicci.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.