Integrally Closed Ideals and Rees Valuation

ISSN： 0092-7872

Source： Communications in Algebra, Vol.40, Iss.9, 2012-09, pp. : 3397-3413

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Abstract

Let (R, m, k) be a normal Noetherian analytically irreducible universally catenary local domain of dimension d ≥ 1 with infinite residue field k and field of fractions k, and let I be an m-primary ideal. For P ∈ Min(m R[It]) there exists at least one such that P = Q ∩ R[It]. Let v be the Rees valuation of . We obtain necessary and sufficient conditions for the quotient ring R[It]/P to be a polynomial ring in d variables over k in terms of the v-values of a suitably chosen minimal generating set for I. Let J: = (a ₁,…, a _d)R be a minimal reduction of I. If I is a normal ideal and is a polynomial ring in d variables over k, we show that the residue field k(v) of v, is generated over k by the images of . If (R, m, k) is a two-dimensional regular local ring with algebraically closed residue field k and I is a product of distinct simple integrally closed m-primary ideals, we show for each positive integer n and each Q ∈ Min(m R[I ⁿ t]) that the ring is a two-dimensional normal Cohen-Macaulay graded domain with minimal multiplicity at its maximal homogeneous ideal, with this multiplicity being n.