Asymptotic behaviour of associated primes of monomial ideals with combinatorial applications

Let  R be a commutative Noetherian ring and I be an ideal of R. We say that I satisfies the persistence property if  \mathrm{Ass}_R(R/I^k)\subseteq \mathrm{Ass}_R(R/I^{k+۱}) for all positive integers k\geq ۱, which \mathrm{Ass}_R(R/I) denotes the set of associated prime ideals of I. In this paper, we introduce a class of square-free monomial ideals in the polynomial ring  R=K[x_۱,\ldots,x_n] over field K which are associated to  unrooted trees  such that if  G is a unrooted tree and I_t(G) is the ideal generated by the paths of G of length t, then J_t(G):=I_t(G)^\vee, where I^\vee denotes the Alexander dual of I, satisfies the persistence property. We also present a class of graphs such that the path ideals generated by paths of length two satisfy the persistence property. We conclude  this paper by giving a criterion for normally torsion-freeness of monomial ideals.

Keywords:

Monomial ideals , associated prime ideals , trees , paths