Algebraic Geometry & Combinatorics Seminar: Numerical characterization of integral dependence and density functions

Abstract: We recall that two ideals I ⊆ J in a commutative Noetherian ring are integrally dependent if they have the same integral closure. Two integrally dependent ideals share many properties in common. This allows us in many situations to replace an ideal by a simpler ideal. A generalization is the notion of integral dependence of modules, which plays an important role in the study of equisingularity.

 Integral dependence of ideals is characterized in terms of the well known numerical invariant, namely Hilbert-Samuel multiplicity, provided the ideals are of finite colength in the ambient local ring. Similarly for modules there is the notion of Buchsbaum-Rim multiplicity. 

Here we work in a graded situation and give numerical characterizations, for integral dependence of ideal/modules which are not necessarily of finite colength in their ambient local ring (or free module). For this we introduce density functions which are continuous real valued functions measuring the growth of graded components of ideals/modules on an ℝ-scale.

This talk is based on joint work with Suprajo Das and Sudeshna Roy.

Host: Matt Kerr