Associated points and integral closure of modules

Abstract

Let $X:=\mathrm{Spec}(R)$ be an affine Noetherian scheme, and $\mathcal{M} \subset \mathcal{N}$ be a pair of finitely generated $R$-modules. Denote their Rees algebras by $\mathcal{R}(\mathcal{M})$ and $\mathcal{R}(\mathcal{N})$. Let $\mathcal{N}^{n}$ be the $n$th homogeneous component of $\mathcal{R}(\mathcal{N})$ and let $\mathcal{M}^{n}$ be the image of the $n$th homegeneous component of $\mathcal{R}(\mathcal{M})$ in $\mathcal{N}^n$. Denote by $\overline{\mathcal{M}^{n}}$ be the integral closure of $\mathcal{M}^{n}$ in $\mathcal{N}^{n}$. We prove that $\mathrm{Ass}_{X}(\mathcal{N}^{n}/\overline{\mathcal{M}^{n}})$ and $\mathrm{Ass}_{X}(\mathcal{N}^{n}/\mathcal{M}^{n})$ are asymptotically stable, generalizing known results for the case where $\mathcal{M}$ is an ideal or where $\mathcal{N}$ is a free module. Suppose either that $\mathcal{M}$ and $\mathcal{N}$ are free at the generic point of each irreducible component of $X$ or $\mathcal{N}$ 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 $\mathrm{Ass}_{X}(\mathcal{N}^{n}/\overline{\mathcal{M}^{n}})$. Notably, we show that if $x \in \mathrm{Ass}_{X}(\mathcal{N}^{n}/\overline{\mathcal{M}^{n}})$ for some $n$, then $x$ is the generic point of a codimension-one component of the nonfree locus of $\mathcal{N}/\mathcal{M}$ or $x$ is a generic point of an irreducible set in $X$ where the fiber dimension $\mathrm{Proj}(\mathcal{R}(\mathcal{M})) \rightarrow X$ jumps. We prove a converse to this result without requiring $X$ to be universally catenary. Many of our results are stated and proved more generally for standard graded algebras. Also, we recover, strengthen, and prove a sort of converse of an important result of Kleiman and Thorup about integral dependence of modules.