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arxiv: 1907.06532 · v1 · pith:GMN4ZZSGnew · submitted 2019-07-15 · 🧮 math.AC

Almost normally torsionfree ideals

Claudia Andrei-Ciobanu This is my paper

Pith reviewed 2026-05-24 21:02 UTC · model grok-4.3

classification 🧮 math.AC
keywords edge idealsfacet idealsalmost normally torsionfreeconnected graphsodd cyclesBorel idealst-spread ideals
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The pith

All connected graphs whose edge ideals are almost normally torsionfree are described, along with qualifying odd cycles and degree-3 t-spread Borel ideals.

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

The paper establishes which connected graphs produce edge ideals that are almost normally torsionfree by giving a complete description of them. It proves that the facet ideal associated to a special odd cycle also satisfies the almost normally torsionfree condition. It then determines exactly which t-spread principal Borel ideals generated in degree 3 have this property. These results connect the structure of graphs and hypergraphs directly to the behavior of associated ideals in a polynomial ring. The classification matters because the torsion properties control how the powers of the ideal interact with associated primes.

Core claim

The paper describes all connected graphs whose edge ideals are almost normally torsionfree. It proves that the facet ideal of a special odd cycle is almost normally torsionfree. It determines the t-spread principal Borel ideals generated in degree 3 which are almost normally torsionfree.

What carries the argument

Edge ideals of graphs and facet ideals of cycles, together with the algebraic condition of being almost normally torsionfree, that translate combinatorial features into statements about ideal powers and torsion.

If this is right

  • The edge ideal of any connected graph is almost normally torsionfree precisely when the graph belongs to the described family.
  • The facet ideal of any special odd cycle is almost normally torsionfree.
  • Only certain explicitly determined t-spread principal Borel ideals generated in degree 3 are almost normally torsionfree.
  • The listed objects provide concrete families where powers of the ideal exhibit controlled torsion behavior.

Where Pith is reading between the lines

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

  • The classification supplies explicit test cases for checking whether almost normally torsionfree coincides with other ideal properties such as being normal or having linear resolution.
  • One could test the boundary cases by computing associated primes of powers for small graphs just outside the described class.
  • The results for degree 3 may suggest a pattern that extends to higher degrees or other Borel-type constructions, though that extension is not addressed here.

Load-bearing premise

The standard definitions of edge ideals, facet ideals, t-spread Borel ideals, and almost normally torsionfree allow graph and cycle properties to translate directly into algebraic statements about the ideals.

What would settle it

A connected graph whose edge ideal is almost normally torsionfree but falls outside the paper's described class, or a degree-3 t-spread principal Borel ideal claimed to be almost normally torsionfree that fails the property under direct computation.

Figures

Figures reproduced from arXiv: 1907.06532 by Claudia Andrei-Ciobanu.

Figure 1
Figure 1. Figure 1: G Let G be a graph on the vertex set [n] and I = I(G) ⊂ S its edge ideal. The symbolic Rees algebra of I is Rs(I) = ⊕k≥0I (k) t k , where I (k) is the k-th symbolic power of I. This algebra is actually the vertex cover algebra of G which is generated as an S-algebra by all the monomials x a t b where a is an indecomposable vertex cover of G of order b. Recall from [9] that a vector a = (a1, . . . , an) ∈ N… view at source ↗
read the original abstract

We describe all connected graphs whose edge ideals are almost normally torsionfree. We also prove that the facet ideal of a special odd cycle is almost normally torsionfree. Finally, we determine the t-spread principal Borel ideals generated in degree 3 which are almost normally torsionfree.

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

0 major / 3 minor

Summary. The paper classifies all connected graphs whose edge ideals are almost normally torsionfree. It proves that the facet ideal of a special odd cycle is almost normally torsionfree. It also determines the t-spread principal Borel ideals generated in degree 3 which are almost normally torsionfree.

Significance. If the classifications hold, the results give explicit combinatorial characterizations of monomial ideals satisfying the almost normally torsionfree condition. This is useful for understanding the associated primes of powers and the structure of Rees algebras in the setting of edge ideals, facet ideals, and Borel ideals. The direct translation from graph/hypergraph properties to algebraic conditions is a strength.

minor comments (3)
  1. The abstract states that descriptions and proofs exist but provides no indication of the combinatorial criteria or proof techniques used; expanding the abstract slightly would improve accessibility without altering length substantially.
  2. Notation for 'almost normally torsionfree' and related notions (e.g., normal torsionfreeness, Rees algebra) should be recalled or referenced explicitly in the introduction for readers outside the immediate subfield.
  3. In the classification of graphs, ensure that the forbidden subgraphs or minimal non-examples are illustrated with figures or explicit small examples to make the combinatorial translation easier to verify.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review and positive assessment of the manuscript. The referee recommends minor revision but has not listed any specific major comments. Accordingly, we have no points to address in a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript is a classification result: it enumerates connected graphs whose edge ideals satisfy the almost normally torsionfree property, proves the property for facet ideals of special odd cycles, and identifies the degree-3 t-spread principal Borel ideals with the property. All arguments are direct combinatorial translations of the algebraic condition into forbidden subgraphs or explicit generator lists, relying only on standard facts about monomial ideals and Rees algebras. No equations, fitted parameters, predictions from data subsets, or load-bearing self-citations appear; the derivations are self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated.

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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