Asymptotic behaviour of associated primes of monomial ideals with combinatorial applications

Document Type : Research Paper

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University of Payame Noor

Abstract

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, 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.

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