On unique and almost unique factorization of complete ideals. II
Let (R,m) be a complete normal local domain of dimension \( 2.\) The set of complete m-primary ideals m(R) of R has a natural semigroup structure. This paper generalizes results obtained by the same author in part I of this paper [Am. J. Math. 111, No.3, 417-433 (1989; Zbl 0697.13007)] in the case where R/m is algebraically closed. - Consider the following properties of R: (a1) m(R) has unique factorization; (a2) R is a UFD; (b1) R satisfies condition (N); (b2) Cl(R) is a torsion group; (b3) m(R) is semifactorial. In the present paper (theorem 5) it is proved that \((a1)\Rightarrow(a2)\), and an example is given to show that the implication \((a2)\Rightarrow(a1)\) is false. (However Lipman has shown that \((a2)\Rightarrow(a1)\) is true if \(R/m\) is an algebraically closed field.) The implications \((b1)\Rightarrow(b3)\) were proved by \textit{H. Göhner} [J. Algebra 24, 403-429 (1975; Zbl 0308.13023)]. Theorem 4 of the present paper shows that \((b1)\Leftrightarrow(b2)\). Condition (N) was introduced by \textit{H. T. Muhly} and \textit{M. Sakuma} in J. Lond. Math. Soc. 38, 341-350; 494 (1963; Zbl 0142.288): R satisfies condition (N) if any proper birational morphism \(X\to Spec(R)\) factors as a sequence of blow-ups of height 2 ideals with irreducible exceptional divisors. The paper concludes with an example of a complete normal local domain of dimension 2 such that the normalization of the blow-up of any m-primary ideal has reducible exceptional divisor. This answers a question of J. Sally.
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