Some Results on Normal Homogeneous Ideals

Abstract

In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an ℕ-graded ring Aof the form A ≥m ≔ ⊕ℓ≥m A ℓand monomial ideals in a polynomial ring over a field. For ideals of the form A ≥m we generalize a recent result of Faridi. We prove that a monomial ideal in a polynomial ring in nindeterminates over a field is normal if and only if the first n − 1 positive powers of the ideal are integrally closed. We then specialize to the case of ideals of the form I( λ ) ≔ , where J( λ ) = (,…, ) ⊆ K[x 1,…, x n ]. To state our main result in this setting, we let ℓ = lcm(λ1,…, ,…λ n ), for 1 ≤ i ≤ n, and set λ ′ = (λ1,…, λ i−1, λ i + ℓ, λ i+1,…, λ n ). We prove that if I( λ ′) is normal then I( λ ) is normal and that the converse holds with a small additional assumption.