Integrally Closed Ideals and Rees Valuation

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.