A convenient category to study asymptotic primes and related questions

Abstract

Let $A$ be a Noetherian ring and let $\mathcal{R} = \bigoplus_{n \geq 0}\mathcal{R}_n$ be a standard graded ring with $\mathcal{R}_0 = A$. We define a category $\mathfrak{A}(\mathcal{R})$ of graded $\mathcal{R}$-modules (not necessarily finitely generated) with the following properties: if $X = \bigoplus_{n \in \mathbb{Z}} X_n \in \mathfrak{A}(\mathcal{R}) $ then (1) $X_i$ is finitely generated $A$-module for all $i \in \mathbb{Z}$ and $X_i = 0$ for $i \ll 0$. (2) There exists $n_0$ such that $\text{Ass}_A X_n = \text{Ass}_A X_{n_0}$ for all $n \geq n_0$. (3) If $X_n$ has finite length as an $A$-module for all $n$ then there exists $P_X(z) \in \mathbb{Q}[z]$ such that $P_X(n) = \ell_A(X_n)$ for all $n \gg 0$. (4) If $F$ is a coherent functor on the category of finitely generated $A$-modules then $F(X) = \bigoplus_{n \in \mathbb{Z}} F(X_n) \in \mathfrak{A}(\mathcal{R})$. (5) For an ideal $J$ in $A$, there exists $c_J^X$ such that $\text{grade}(J, X_n) = \text{grade}(J, X_{c_J^X})$ for all $n \geq c_J^X$. We give a unified proof of several results in theory of associate primes and related areas.