Associated points and integral closure of modules

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.