isms to graded $R_{\mathbb{Z}}$-module homomorphisms, $\Phi$ is a functor.

Conversely, suppose that $\widetilde{N}\in R_{\mathbb{Z}}{\text{-}{\textsf}{qgr}}$ and pick a preimage $N\in
R_{\mathbb{Z}}{{\textsf}{\text{-}grmod}}$. Then $N$ is generated by $\bigoplus_{i=0}^a N_i$, for some $a$, and so $N_j=R_{ja}N_a$, for all $j\geq a$. For $j\geq i\geq a$ we have natural maps of $R_a$-modules $$\theta_{ji}: R^*_{ia}\otimes N_i \cong
 R^*_{ia}\otimes R^*_{ji}\otimes R_{ji}\otimes N_i \cong
 R^*_{ja}\otimes (R_{ji}\otimes N_i )
 \twoheadrightarrow R^*_{ja}\otimes N_j,$$ where the tensor products are over the appropriate $R_k$. By the associativity of tensor products, $\theta_{ki}=\theta_{kj}\theta_{ji}$, for all $k\geq j\geq
 i\geq a$. Since each $N_i$ is a noetherian $R_i$-module, each $R^*_{ia}\otimes N_i$ is a noetherian $R_a$-module and so $\theta_{ji}$ is an isomorphism for all $j\geq i\gg 0$. Equivalently, $N_j\cong R_{ji}\otimes N_i$ for all such $j\geq i$.

Set $\Theta(\widetilde{N})
 =R^*_{j0}\otimes N_j\in R_0{\text{-}{\textsf}{mod}}$ for some $j\gg 0$. Since any two preimages of $\widetilde{N}$ in $R_{\mathbb{Z}}{{\textsf}{\text{-}grmod}}$ agree in high degree, $\Theta(\widetilde{N})$ is independent of the choice of $N$. Moreover, as $R^*_{j0}=R^*_{k0}R_{kj}$, $$\phi(\Theta(\widetilde{N}))_{\geq j}
 \cong \bigoplus_{k\geq j} R_{k0}\otimes R^*_{j0}\otimes N_j
 \cong \bigoplus_{k\geq j} R_{kj}\otimes N_j = \bigoplus_{k\geq j}N_k,$$ and so $\Phi\Theta(\widetilde{N}) = \widetilde{N}.$ Checking that $\Theta$ and $\Phi$ are inverse equivalences is now routine.

 {#section-1}

We remark that many of the standard techniques and results concerned with associated graded modules for unital algebras extend routinely to ${\mathbb{Z}}$-algebras. These only appear in peripheral ways in this paper and so we refer the reader to [@GS2] for a discussion of these results.

The main theorem {#sect-filt}
================

 {#sect601}

In this section we prove the main theorem of the paper by proving Theorem \[mainthm-intro\] fr