In the Shadows of a hypergraph: looking for associated primes of powers of squarefree monomial ideals

TL;DR

This paper explores the relationship between hypergraph shadows and the associated primes of powers of squarefree monomial ideals, revealing how smaller hypergraphs influence algebraic properties and persistence.

Contribution

It introduces the concept of hypergraph shadows and demonstrates their role in understanding associated primes of ideal powers.

Findings

Shadows preserve information about associated primes.

Hypergraph shadows relate to the persistence property.

New insights into the structure of associated primes.

Abstract

The aim of this paper is to study the associated primes of powers of squarefree monomial ideals. Hypergraphs and squarefree monomial ideals are strongly connected. The cover ideal $J(H)$ of a hypergraph $H$ is the intersection of the primes corresponding to the edges of $H$. We define the $shadow$ of $H$ as a certain set of smaller hypergraphs related to $H$. We then describe how the shadows of $H$ preserve information about the associated primes of the powers of $J(H)$. Some implications to the persistence property are studied.