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.
Categories: math.AG math.AC