Associated points and integral closure of modules (Q1643544)

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 homogeneous component of $\mathcal{R}(\mathcal{M})$ in $\mathcal{N}^n$. Denote by $\overline{\mathcal{M}^{n}}$ the integral closure of $\mathcal{M}^{n}$ in $\mathcal{N}^{n}$. One of the main goals of this paper is to describe the sets $\mathrm{Ass}_{X}(\mathcal{N}^{n}/\overline{\mathcal{M}^{n}})$ and $\mathrm{Ass}_{X}(\mathcal{N}^{n}/\mathcal{M}^{n}).$ \par The author obtains a complete classification of the points $x$ of $X$ that appear in the set $\bigcup_{n=1}^\infty \mathrm{Ass}_{X}(\mathcal{N}^{n}/\overline{\mathcal{M}^{n}})$ when $X$ is universally catenary. In particular, he proves that this set is finite. More generally, without assuming $X$ to be universally catenary and any additional hypothesis on the pair $\mathcal{M} \subset \mathcal{N},$ the author proves 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. \par Suppose that either $\mathcal{M}$ 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. If $X$ is universally catenary, the author proves a generalization of a classical result due to \textit{S. McAdam} [Proc. Am. Math. Soc. 80, 555--559 (1980; Zbl 0445.13002)] and obtain a geometric classification of the points appearing in $\mathrm{Ass}_{X}(\mathcal{N}^{n}/\overline{\mathcal{M}^{n}})$. More precisely, he shows 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. He proves a converse of this result without requiring $X$ to be universally catenary. \par Finally, he recovers, strengthens, and proves a sort of converse of an important result of Kleiman and Thorup about integral dependence of modules.