SOME CHARACTERIZATIONS OF FIRST NEIGHBORHOOD COMPLETE IDEALS IN DIMENSION TWO

作者：Debremaeker Raymond^*

来源：Journal of Commutative Algebra, 2012, 4(4): 505-524.

DOI：10.1216/JCA-2012-4-4-505

• First neighborhood complete ideal
• quasi-one-fibered ideal
• Muhly local domain

In this paper we study first neighborhood complete ideals in a two-dimensional normal Noetherian local domain (R, M) with algebraically closed residue field and the associated graded ring an integrally closed domain. It is shown that a complete quasi-one-fibered M-primary ideal I of R is a first neighborhood complete ideal if and only if e(I) = e(M) + 1. This implies that (R, M) is a rational singularity if and only if a (every) first neighborhood complete ideal has minimal multiplicity. Moreover, if (R, M) has minimal multiplicity, then a complete quasi-one-fibered M-primary ideal I of order one is a first neighborhood complete ideal if and only if certain numerical data associated with I are minimal. This yields a simple proof of the fact that first neighborhood complete ideals in such a local ring R are projectively full.

• 出版日期2012