Associated points and integral closure of modules

Abstract

Let X:=Spec(R) be an affine Noetherian scheme, and M⊂N be a pair of finitely generated R-modules. Denote their Rees algebras by R(M) and R(N). Let Nn be the nth homogeneous component of R(N) and let Mn be the image of the nth homogeneous component of R(M) in Nn. Denote by Mn‾ be the integral closure of Mn in Nn. We prove that AssX(Nn/Mn‾) and AssX(Nn/Mn) are asymptotically stable, generalizing known results for the case where M is an ideal or where N is a free module. Suppose either that M and N are free at the generic point of each irreducible component of X or N is contained in a free R-module. When X is universally catenary, we prove a generalization of a classical result due to McAdam and obtain a geometric classification of the points appearing in AssX(Nn/Mn‾). Notably, we show that if x∈AssX(Nn/Mn‾) for some n, then x is the generic point of a codimension-one component of the nonfree locus of N/M or x is a generic point of an irreducible set in X where the fiber dimension Proj(R(M))→X jumps. We prove a converse to this result without requiring X to be universally catenary. Our approach is geometric in spirit. Also, we recover, strengthen, and prove a sort of converse of an important result of Kleiman and Thorup about integral dependence of modules.