A convenient category to study asymptotic primes and related questions

Let $A$ be a Noetherian ring and let $\mathcal{R}={⨁}_{n\ge 0}{\mathcal{R}}_n$ 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 $X={⨁}_{n\in \mathbb{Z}}{X}_n\in \mathfrak{A}(\mathcal{R})$ 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 $n\ge n_0$. (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 $P_X(n)={\mathrm{\ell }}_A(X_n)$ for all $n\gg 0$. (4) If $F$ is a coherent functor on the category of finitely generated $A$-modules then $F(X)={⨁}_{n\in \mathbb{Z}}F(X_n)\in \mathfrak{A}(\mathcal{R})$. (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 $n\ge c_J^X$. We give a unified proof of several results in theory of associate primes and related areas.