Associated Primes of the Square of the Alexander Dual of Hypergraphs

TL;DR

This paper develops combinatorial methods to determine associated primes of the square of the Alexander dual of hypergraph edge ideals, revealing new behaviors in higher dimensions and providing constructions for specific prime heights.

Contribution

It introduces a general combinatorial approach for detecting associated primes and extends known results from 2-hypergraphs to higher dimensions, including constructions of hypergraphs with prescribed prime heights.

Findings

Associated primes of 2-hypergraphs are either of height 2 or odd height > 2.

In 3-hypergraphs, no restrictions on the heights of associated primes.

Any prime height can be realized in hypergraphs of dimension ≥ 3.

Abstract

The purpose of this paper is to provide methods for determining the associated primes of the square of the Alexander dual of the edge ideal for an m-hypergraph H. We prove a general method for detecting associated primes of the square of the Alexander dual of the edge ideal based on combinatorial conditions on the m-hypergraph. Also, we demonstrate a more efficient combinatorial criterion for detecting the non-existence of non-minimal associated primes. In investigating 3-hypergraphs, we prove a surprising extension of the previously discovered results for 2-hypergraphs (simple graphs). For 2-hypergraphs, associated primes of the square of the Alexander dual of the edge ideal are either of height 2 or of odd height greater than 2. However, we prove that in the 3-hypergraph case, there is no such restriction - or indeed any restriction - on the heights of the associated primes. Further, we generalize this result to any dimension greater than 3. Specifically, given any integers m, q, and n with 3\leq m\leq q\leq n, we construct a m-hypergraph of size n with an associated prime of height q. We further prove that it is possible to construct connected m-hypergraphs under the same conditions.