F-Purity of Binomial Edge Ideals

TL;DR

This paper proves two conjectures about when binomial edge ideals are F-pure, linking F-purity to graph properties and providing a complete classification for certain graph classes.

Contribution

It confirms Matsuda's conjecture for characteristic two and disproves his large characteristic conjecture, classifying F-pure binomial edge ideals based on graph structure.

Findings

F-purity in characteristic 2 corresponds exactly to weakly closed graphs.

Graphs with asteroidal triples do not produce F-pure binomial edge ideals in any characteristic.

Complete classification of F-pure binomial edge ideals for chordal graphs.

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.