Generalized cover ideals and the persistence property

2013/01/21 by Bhat, Ashwini, Biermann, Jennifer, Van Tuyl, Adam
#05C25 #13A15 #13F20 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1301.5020

Abstract

Let I be a square-free monomial ideal in R = k[x₁,…,x_n], and consider the sets of associated primes \rm Ass(I^s) for all integers s ≥ 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 ≥ 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.

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