Generalized cover ideals and the persistence property

TL;DR

This paper introduces a new family of square-free monomial ideals related to graphs, explicitly determines associated primes for all powers when the graph is a tree, and proves the persistence property for these ideals.

Contribution

It generalizes cover ideals to a broader class associated with graphs, explicitly computes associated primes for all powers in the case of trees, and establishes the persistence property.

Findings

Explicit determination of associated primes for all powers when the graph is a tree.

Computation of the index of stability for the introduced ideals.

Proof that this family of ideals satisfies the persistence property.

Abstract

Let $I$ be a square-free monomial ideal in $R = k[x_1,\ldots,x_n]$, and consider the sets of associated primes ${\rm Ass}(I^s)$ for all integers $s \geq 1$. Although it is known that the sets of associated primes of powers of $I$ eventually stabilize, there are few results about the power at which this stabilization occurs (known as the index of stability). We introduce a family of square-free monomial ideals that can be associated to a finite simple graph $G$ that generalizes the cover ideal construction. When $G$ is a tree, we explicitly determine ${\rm Ass}(I^s)$ for all $s \geq 1$. As consequences, not only can we compute the index of stability, we can also show that this family of ideals has the persistence property.