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arxiv: 2601.15403 · v2 · submitted 2026-01-21 · 🧮 math.AC

F-Purity of Binomial Edge Ideals

Adam LaClair , Jason McCullough This is my paper

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classification 🧮 math.AC
keywords binomial edge idealsF-purityweakly closed graphsasteroidal tripleschordal graphsF-injective algebraspositive characteristic
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The pith

Binomial edge ideals are F-pure in characteristic 2 if and only if the graph is weakly closed.

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

The paper resolves both of Matsuda's conjectures on F-purity of binomial edge ideals. It proves that over a field of characteristic two, the quotient by the binomial edge ideal is F-pure precisely when the underlying graph is weakly closed. It shows that the second conjecture fails strongly: any graph containing an asteroidal triple, such as the net, yields a binomial edge ideal that is not F-pure in any positive characteristic. These results also classify F-pure binomial edge ideals completely for chordal graphs and produce infinite families of standard graded algebras that are F-injective yet neither F-pure nor F-rational in every characteristic.

Core claim

The binomial edge ideal of a graph G defines an F-pure quotient in characteristic 2 if and only if G is weakly closed, and any graph containing an asteroidal triple defines a binomial edge ideal that fails to be F-pure in every positive characteristic.

What carries the argument

The binomial edge ideal J_G of a graph G, with F-purity detected by whether the Frobenius endomorphism splits on the quotient ring.

Load-bearing premise

The standard combinatorial definitions of weakly closed graphs and asteroidal triples correctly identify the algebraic condition of F-purity via the Frobenius map on finite simple graphs.

What would settle it

A single graph that is not weakly closed yet has an F-pure binomial edge ideal in characteristic 2, or a graph containing an asteroidal triple whose binomial edge ideal is F-pure in some positive characteristic, would disprove the claims.

Figures

Figures reproduced from arXiv: 2601.15403 by Adam LaClair, Jason McCullough.

Figure 1
Figure 1. Figure 1: The finite graphs appearing in Theorem 4.11 [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Infinite families appearing in Theorem 4.11 Theorem 5.3 ([41, Theorem 2.3]). Let G be a graph, and let k be any field of positive characteristic. If G is a weakly closed graph, then R/JG is F-pure. The forward implication of Conjecture A follows from Theorem 5.3. However, the reverse implication has remained open. Matsuda provided an analogue of Theorem 5.1, which had previously been observed by Kratsch an… view at source ↗
Figure 3
Figure 3. Figure 3: The graph XF2n+3 1 , n ≥ 0 We establish common notation that we will use in the remainder of this subsection [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The graph XF2n+3 5 , n ≥ 0 9.4. The Graph co−XF2n+2 6 . We define the graph co−XF2n+2 6 . Definition 9.21. For n ≥ 0, XF2n+2 6 denotes the graph on 2n + 7 vertices having edge set {{1, i} 2n+5 i=2 , {2, i} 2n+5 i=3 , {i, i + 1} 2n+4 i=2 , {2, 2n + 6}, {2n + 5, 2n + 6}, {1, 2n + 7}, {3, 2n + 7}}. Then, co−XF2n+2 6 := XF2n+2 6 1 2 3 2n + 4 4 2n + 5 2n + 6 2n + 7 · · · [PITH_FULL_IMAGE:figures/full_fig_p034_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The graph XF2n+2 6 , n ≥ 0 Remark 9.22. The labeling of the vertices in defining co−XF2n+2 6 in Definition 9.21 is different than the labeling of the vertices as defined in Trotter [52]. We establish common notation that we will use in the remainder of this subsection. Setup 9.23. By n we denote a positive integer greater than or equal to 2.3 We denote by G the graph co−XF2n+2 6 . For 3 ≤ i ≤ 2n + 3, we de… view at source ↗
read the original abstract

In 2012, K. Matsuda introduced the class of weakly closed graphs and investigated when binomial edge ideals are F-pure. He proved that weakly closed binomial edge ideals are F-pure whenever the base field has positive characteristic. He conjectured that: (i) when the base field has characteristic two, every F-pure binomial edge ideal comes from a weakly closed graph; and (ii) that every binomial edge ideal is F-pure provided that the characteristic of the residue field is sufficiently large. In this paper, we resolve both of Matsuda's conjectures. We confirm Matsuda's first conjecture, showing that the binomial edge ideal of a graph defines an F-pure quotient in characteristic 2 if and only if the graph is weakly closed. We also show that Matsuda's second conjecture is false in a very strong way by showing that graphs containing asteroidal triples, such as the net, define non-F-pure binomial edge ideals in any positive characteristic. Our results yield a complete classification of F-pure binomial edge ideals of chordal graphs as well as large families of standard graded algebras that are F-injective but neither F-pure nor F-rational in all characteristics.

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 / 1 minor

Summary. The paper resolves two conjectures of Matsuda on F-purity of binomial edge ideals. It proves that in characteristic 2 the binomial edge ideal of a graph is F-pure if and only if the graph is weakly closed, and disproves the second conjecture by exhibiting graphs with asteroidal triples (e.g., the net) whose binomial edge ideals fail to be F-pure in every positive characteristic. The work also gives a complete classification of F-pure binomial edge ideals of chordal graphs and constructs infinite families of standard graded algebras that are F-injective but neither F-pure nor F-rational in all positive characteristics.

Significance. If the results hold, the paper supplies a definitive combinatorial resolution of the conjectures together with explicit counterexamples and new families of algebras with controlled F-properties. These contributions clarify the boundary between F-pure and merely F-injective rings in the binomial-edge setting and furnish concrete test cases for broader questions in positive-characteristic commutative algebra.

minor comments (1)
  1. [Introduction] A brief recall of the definition of an asteroidal triple (or a reference to a standard source) would improve readability in the introduction and in the counterexample section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending acceptance. The report accurately reflects the resolution of both of Matsuda's conjectures and the additional contributions on chordal graphs and F-injective but non-F-pure algebras.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper resolves both of Matsuda's conjectures via direct combinatorial arguments: the 'if' direction applies the standard Frobenius purity criterion to the binomial generators of weakly closed graphs, while the 'only if' direction in characteristic 2 and the counterexamples for asteroidal triples (such as the net) rely on explicit verification that no test element satisfies the purity condition. These steps use only the standard definitions of binomial edge ideals, weakly closed graphs, and asteroidal triples together with the usual properties of the Frobenius map on finite simple graphs; no parameter fitting, self-definitional reductions, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies entirely on standard definitions and theorems from commutative algebra and graph theory; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)
  • standard math Standard definitions and basic properties of binomial edge ideals, F-purity, F-injectivity, and F-rationality in positive characteristic
    Invoked throughout to state the conjectures and theorems.
  • standard math Standard combinatorial definitions of weakly closed graphs and asteroidal triples
    Used to formulate the classification and counterexamples.

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