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 AssR(R/Ik) ? AssR(R/Ik+1) for all positive integers k ? 1, which AssR(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[x1, . . . , xn] over field K which are associated to unrooted trees such that if G is a unrooted tree and It(G) is the ideal generated by the paths of G of length t, then Jt(G) := It(G)_, where I_ denotes the Alexander dual of I, satis- fies 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.