A mixed multiplicity formula for complete ideals in 2-dimensional rational singularities

Abstract

Let ( R , m ) (R,m) be a 2-dimensional rational singularity with algebraically closed residue field and for which the associated graded ring is an integrally closed domain. We use the work of Göhner on condition ( N ) (N) and the theory of degree functions developed by Rees and Sharp to give a very short and elementary proof of a formula for the (mixed) multiplicity of complete m m -primary ideals in ( R , m ) (R,m) .