The strong Rees property of powers of the maximal ideal and Takahashi–Dao's question

Abstract

Dedicated to Craig Huneke on the occasion of his 65th Birthday.Abstract. In this paper, we introduce the notion of the strong Rees property (SRP) for m-primary ideals of a Noetherian local ring and prove that any power of the maximal ideal m has its property if the associated graded ring G of m satisfies depth G ≥ 2. As its application, we characterize two-dimensional excellent normal local domains so that m is a pg-ideal.Finally we ask what m-primary ideals have SRP and state a conjecture which characterizes the case when m n are the only ideals which have SRP.Question 1.2. Assume that (A, m) is a two-dimensional excellent normal local domain. If an inequality (µ(I) − 1) · ℓℓ(I) ≥ e(I) holds for any m-primary integrally closed ideal I, then is m a p g -ideal?It turns out that this question is related to the "strong Rees property" of powers of the maximal ideal. The main result in this paper is the following theorem. Theorem 3.2. Let (A, m) be a Noetherian local ring. Assume that depth A ≥ 2 and H 1 M (G) has finite length, where G = G(m) = ⊕ n≥0 m n /m n+1 . If m ℓ is Ratliff-Rush closed, then m ℓ has the strong Rees property.By Remark 2.5, any power m ℓ is Ratliff-Rush closed whenever depth G ≥ 1. Hence we have the following corollary.