F-Purity of Binomial Edge Ideals

TL;DR

This work classifies the F-purity of binomial edge ideals $R_G=R/J_G$ across characteristics, proving Matsuda's conjecture in characteristic $2$ by showing $R_G$ is F-pure iff $G$ is weakly closed, and disproving the conjecture that large characteristics force eventual F-purity by exhibiting graphs with asteroidal triples (e.g., the net) that yield non-F-pure $R_G$ in every positive characteristic. It develops a robust toolkit—Fedder’s criterion, colon-ideal computations, and a cycle-based parity analysis—to reduce F-purity questions to induced subgraphs and finite obstruction lists via Gallai’s theorem, then applies these to co-regular and AT-containing graphs. A complete chordal-graph classification is obtained: $R_G$ is F-pure in all characteristics exactly when $G$ is AT-free (equivalently weakly closed). The paper also derives practical consequences: unmixed binomial edge ideals of König type are weakly closed and thus F-pure in positive characteristic, and weakly closed graphs are stable under vertex deletion and completion, enriching the interplay between combinatorial graph structure and F-singularities in commutative algebra.

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.

Figures (5)

• Figure 1: The finite graphs appearing in Theorem \ref{['thm:forbidden_subgraph_AT_free']}
• Figure 2: Infinite families appearing in Theorem \ref{['thm:forbidden_subgraph_AT_free']}
• Figure 3: The graph $\mathrm{XF}_1^{2n+3}$, $n\geq 0$
• Figure 4: The graph $\mathrm{XF}_5^{2n+3}$, $n\geq 0$
• Figure 5: The graph $\mathrm{XF}_6^{2n+2}$, $n\geq 0$

Theorems & Definitions (149)

• Conjecture A: Matsuda matsuda2018weakly
• Theorem A
• Conjecture B: Matsuda matsuda2018weakly
• Theorem B: Theorem \ref{['thm:non_F_purity']}, Corollary \ref{['cor:at_triple_implies_non_f_purity']}
• Definition 2.1
• Theorem 2.2: fedder1983
• Definition 2.3
• Proposition 2.4: herzog2010binomial
• Definition 2.5
• Lemma 3.1