Generalized cover ideals and the persistence property

Abstract

Let $I$ be a square-free monomial ideal in $R=k[x_1,\dots ,x_n]$, and consider the sets of associated primes $\mathrm{Ass}(I^s)$ for all integers $s\ge 1$. 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 $\mathrm{Ass}(I^s)$ for all $s\ge 1$. As consequences, not only can we compute the index of stability, we can also show that this family of ideals has the persistence property.

15 pages; revised version has a new introduction; references updated; to appear in J. Pure. Appl. Algebra