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 = P 1 a 1 P 2 a 2 ⋯ P n a n , where P i is a simple complete ideal for a i ⩾ 1 and n ⩾ 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 = P 1 a 1 P 2 a 2 ⋯ P n a n has only finitely many complete adjacent over-ideals if and only if a i = 1 for every i and there are no proximity relations among P i .