Some Results on Normal Homogeneous Ideals

Abstract

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 Ngraded ring A of the form A ≥m := ℓ≥m A ℓ and 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 n indeterminates 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) := J(λ), where J(λ) = (x λ1

Canonical modules of Rees algebras

“…Let I be the integral closure of the ideal (X 2 , Y 3 , Z 5 ) ⊂ K[X, Y, Z], K a field. It has been noticed by Reid, Roberts, and Vitulli [25] (and can easily be checked by normaliz [9]) that R = R(I ) is normal. A K-basis of R is given by all monomials Remark 2.6.…”

Serre's Condition Rℓfor Affine Semigroup Rings

“…see the discussion in [19]). By Proposition 3.7 of [19] to prove that a is normal, it suffices to show that a 2 = S ≥2L . For this we proceed as in the proof of Theorem III.2.2 of [5].…”

“…Observe that the homogeneous ideal a = S ≥L is integrally closed and that the integral closure of a t is S ≥tL for t ≥ 1 (e.g. see the discussion in [19]). By Proposition 3.7 of [19] to prove that a is normal, it suffices to show that a 2 = S ≥2L .…”

Clôture intégrale et opérations de tores algébriques de complexité un dans les variétés affines

“…La prochaine assertion est une traduction combinatoire de [RRV,Proposition 3.1] via la correspondance du théorème 3.5.…”

“…For the case of non-elliptic affine k ⋆ -surfaces, we obtain a combinatorial proof for the normality of any integrally closed invariant ideals of such surfaces. As another application, we obtain the following new criterion of normality which generalizes Reid-Roberts-Vitulli's Theorem [RRV,3.1] in the case of complexity 0 (see 5.5).…”