Some characterizations of first neighborhood complete ideals in dimension two
The author continues his study of first neighborhood complete ideals in a two-dimensional Muhly local domain begun in [J. Pure Appl. Algebra 213, No. 6, 1140--1151 (2009; Zbl 1163.13004); Proc. Am. Math. Soc. 137, No. 5, 1649--1656 (2009; Zbl 1166.13011); J. Algebra 344, No. 1, 14--46 (2011; Zbl 1271.13019)]. Specifically, when \((R, \mathfrak m)\) is a two-dimensional Muhly local domain and \(I\) is a complete quasi-one-fibered \(\mathfrak m\)-primary ideal of \(R\), it is shown that \(I\) is a first neighborhood complete ideal if and only if the multiplicity of \(I\) is exactly one more than that of \(\mathfrak m\). Two consequences follow: \((R, \mathfrak m)\) is a rational singularity if and only if a, equivalently every, first neighborhood complete ideal has minimal multiplicity; and if \((R, \mathfrak m)\) has minimal multiplicity, then for any first neighborhood complete ideal \(I\) of \(R\), \(\mathfrak m I \subseteq\) core(\(I\)). Moreover, assuming that \((R, \mathfrak m)\) has minimal multiplicity, the author shows that \(I\) is a first neighborhood complete ideal if and only if certain numerical invariants associated with \(I\) are minimal: the number of elements in a minimal ideal basis of \(I\), the degree function coefficients of \(I\), the rank of \(I\), and more. Finally, the author gives a simple proof that in a two-dimensional Muhly local domain with minimal multiplicity, any first neighborhood complete ideal is projectively full.
scientific article; zbMATH DE number 6156052
scientific article