Adjacent integrally closed ideals in 2-dimensional regular local rings

Abstract

Let ( A, m ) be a 2-dimensional regular local ring with algebraically closed residue field. Zariski's Unique Factorization Theorem asserts that every integrally closed (complete) m-primary ideal I is uniquely factored into a product of powers of simple complete ideals I = P1 a 1 P2 a 2 ⋯ Pn a n, where Pi is a simple complete ideal for ai {greater than or slanted equal to} 1 and n {greater than or slanted equal to} 1. In this paper, we give a new characterization for a simple complete ideal in terms of adjacent complete ideals. We also give a characterization for a complete ideal I to have finitely many adjacent complete m-primary over-ideals. Namely, we show that I is simple if and only if it has a unique adjacent over-ideal and that I = P1 a 1 P2 a 2 ⋯ Pn a n has only finitely many complete adjacent over-ideals if and only if ai = 1 for every i and there are no proximity relations among Pi. © 2005 Elsevier Inc. All rights reserved.