On unique and almost unique factorization of complete ideals. (English) Zbl 0697.13007
Let \((R,m)\) be a complete normal local domain of dimension 2. The set of complete m-primary ideals of R, m(R), has a natural semigroup structure. Assume that \(k=R/m\) is algebraically closed. Then we have the following results:
(1) m(R) has unique factorization \(\Leftrightarrow\) R is a UFD;
(2) R satisfies condition (N) \(\Leftrightarrow\) the class group Cl(R) is torsion.
The implication \(\Leftarrow\) of (1) was proved by J. Lipman in Publ. Math., Inst. Hautes Étud. Sci. 36 (1969), 195-279 (1970; Zbl 0181.489). The implication \(\Leftarrow\) of (2) was proved by H. Göhner [J. Algebra 34, 403-429 (1975; Zbl 0308.13023)]. Condition (N) was introduced by H. T. Muhly and M. Sakuma [J. Lond. Math. Soc. 38, 341-350; 494 (1963; Zbl 0142.288)]. The other two implications are proved in the present paper. (These results are extended to the case where k is not algebraically closed in the part II of the paper under review [Invent. Math. 98, No.1, 59-74 (1989)].
When k is algebraically closed of characteristic zero, it is proved that R has a rational singularity if and only if R satisfies condition (N). An example is given of a nonrational singularity R defined over an algebraically closd field of characteristic \(p>0\) such that R satisfies condition (N). Finally several cases in dimension greater than 2 are considered.
(1) m(R) has unique factorization \(\Leftrightarrow\) R is a UFD;
(2) R satisfies condition (N) \(\Leftrightarrow\) the class group Cl(R) is torsion.
The implication \(\Leftarrow\) of (1) was proved by J. Lipman in Publ. Math., Inst. Hautes Étud. Sci. 36 (1969), 195-279 (1970; Zbl 0181.489). The implication \(\Leftarrow\) of (2) was proved by H. Göhner [J. Algebra 34, 403-429 (1975; Zbl 0308.13023)]. Condition (N) was introduced by H. T. Muhly and M. Sakuma [J. Lond. Math. Soc. 38, 341-350; 494 (1963; Zbl 0142.288)]. The other two implications are proved in the present paper. (These results are extended to the case where k is not algebraically closed in the part II of the paper under review [Invent. Math. 98, No.1, 59-74 (1989)].
When k is algebraically closed of characteristic zero, it is proved that R has a rational singularity if and only if R satisfies condition (N). An example is given of a nonrational singularity R defined over an algebraically closd field of characteristic \(p>0\) such that R satisfies condition (N). Finally several cases in dimension greater than 2 are considered.
Reviewer: L.G.Roberts
MSC:
| 13F15 | Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial) |
| 14J17 | Singularities of surfaces or higher-dimensional varieties |
