On Adjacent Ideals in Two-Dimensional Rational Singularities

Abstract

Let (R, ℳ) be a two-dimensional rational singularity with algebraically closed residue field. We investigate, for complete ℳ-primary ideals J ⊂ I in R, when length =length for all natural numbers n ≥ 1. This enables us to obtain some new results concerning adjacent and almost adjacent ideals in R. For example, in the special case where the associated graded ring of R is an integrally closed domain, we are able to show that e(I) = e(ℳ) +1 where I is a complete ideal adjacent to ℳ and e denotes the multiplicity. This result suggests the more general question of how much the multiplicities can differ for two adjacent ℳ-primary ideals in case there are no restrictions on the associated graded ring of R. We are able to answer this question using the theory of degree functions developed by Rees and Sharp, thereby generalizing the work of Noh.