On asymptotic depth of integral closure filtration and an application

2017/09/19 by Puthenpurakal, Tony J.
#13D45 #13H15 #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13A30 #Secondary 13H10

paper · doi:10.48550/arxiv.1709.06244

Abstract

Let (A,\mathfrakm) be an analytically unramified formally equidimensional Noetherian local ring with depth A ≥ 2. Let I be an \mathfrakm-primary ideal and set I^* to be the integral closure of I. Set G^*(I) = \bigoplus_{n≥ 0} (Iⁿ)^*/(Iⁿ⁺¹)^* be the associated graded ring of the integral closure filtration of I. We prove that depth G^*(Iⁿ) ≥ 2 for all n ≫ 0. As an application we prove that if A is also an excellent normal domain containing an algebraically closed field isomorphic to A/\m then there exists s₀ such that for all s ≥ s₀ and J is an integrally closed ideal strictly containing (\mathfrakm^s)^* then we have a strict inequality μ(J) < μ((\mathfrakm^s)^*) (here μ(J) is the number of minimal generators of J).

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