First neighborhood complete ideals in two-dimensional Muhly local domains
Let \((R, \mathfrak m)\) be a two-dimensional Muhly local domain [\textit{H. T. Muhly} and \textit{M. Sakuma}, Trans. Am. Math. Soc. 106, 210--221 (1963; Zbl 0123.03601); \textit{H. T. Muhly}, J. Lond. Math. Soc. 40, 99--107 (1965; Zbl 0123.03701)]. The author studies these rings from the point of view of first neighborhood complete ideals, also known as complete ideals adjacent to the maximal ideal. The study extends the ideas in \textit{S.~Noh} [Commun. Algebra 23, No. 8, 3127--3143 (1995; Zbl 0836.13013)] on two-dimensional regular local rings. The author shows a one-to-one correspondence between the set of immediate quadratic transforms of the ring and the set of first neighborhood complete ideals. He also gives a criterion for the regularity of a two-dimensional Muhly local domain in terms of the set of Rees valuations of the first neighborhood complete ideals. When \(R\) has embedding dimension three, the author characterizes those domains that are rational singularities in terms of their first neighborhood complete ideals. To be specific, he shows that \((R, \mathfrak m)\) is a rational singularity if and only if for each first neighborhood complete ideal \(I\) of \(R\) there exists a corresponding minimal reduction \((x, y)\) of \(\mathfrak m\) such that \(I \mathfrak m = I(x, y)\). Finally, a necessary and sufficient condition for an immediate prime divisor of the ring to possess an asymptotically irreducibly ideal is derived, and this ideal is described in the case that the ring is a rational singularity.
scientific article; zbMATH DE number 5534982