A new characterization of rational surface singularities (Q2639118)

The rational surface singularities have a lot of characterizations. See for example: \textit{J. Lipman}, Publ. Math., Inst. Hautes Étud. Sci. 36, 195-279 (1970; Zbl 0181.489). The main new theorem is the converse of a result of J. Lipman. Let (R,\({\mathfrak m})\) be an excellent normal local domain of dimension 2 such that \(k=R/{\mathfrak m}\) is algebraically closed. Then R has a rational singularity if and only if, for any \({\mathfrak m}\)- primary complete ideal I in R, \(I^ 2\) is a complete ideal. The author also proves that if k is not algebraically closed, the theorem fails (the example is a simple elliptic singularity over \({\mathbb{Q}})\).