On unique and almost unique factorization of complete ideals II

Abstract

The theory of complete ideals was begun by Zariski in 1938 [Z 1], and further developed in Appendix 5 of [ZS]. Zariski developed a theory of factorization of complete ideals in regular two-dimensional local rings. In this paper we develop the theory of factorization of complete ideals in complete normal two-dimensional local rings. Let (R, m) be a complete normal two-dimensional local ring, and let re(R) be the semi-group of complete m-primary ideals of R. We show how the divisor class group of R, CI(R), is related to factorization in re(R). We generalize results obtained in [C2] where it is assumed that R/m is algebraically closed. Assume that R is any regular local ring of dimension 2. Zariski proved that any complete ideal I of R has unique factorization into products of simple complete ideals. The distinct simple complete ideals occuring in the factorization of I are in 1 1 correspondence with the irreducible components of the exceptional divisor of the blowing up of I. This shows that any proper birational morphism X ~spec(R) with X normal can be decomposed into a sequence of blowups with irreducible exceptional divisors. Now assume that R is a complete normal local domain of dimension 2. It is natural to ask if the condition that m(R) has unique factorization is equivalent to R being a UFD. We show (Theorem 5) that if re(R) has unique factorization then R is a UFD. We also give an example (Example 1) to show that the converse does not hold. However, Lipman has shown in [L] that if R is a UFD, and if R/m is algebraically closed, then re(R) has unique factorization. Recall that R is a UFD if and only if CI(R)=0. G6hner [G] introduced the notion of a semigroup being semi-factorial. This definition is reproduced in definition 3 of this paper. We prove G6hner's conjecture (2 3 of Theorem 4) that re(R) is semifactorial if and only if CI(R) is a torsion group. As a corollary (Corollary to Theorem 4) we obtain that if Rim is algebraically closed of characteristic zero, then re(R) is semi-factorial if and only if R has a rational singularity. Another corollary is that CI(R) is torsion if and only if R satisfies the condition (N) of Muhly and Sakuma [-MS] (1r of Theorem 4). R satisfies condition (N) if any proper birational morphism X ~ spec(R) factors as a sequence of blowups of height 2 ideals with irreducible exceptional divisors.