We note that the full subcategory of $\operatorname{FI}_V$ whose objects are the sets $[n] = \{1,\ldots,n\}$ is naturally equivalent to $\operatorname{FI}_V$. For this reason we will often restrict ourselves to working with this subcategory. Also observe that if $|V| = 1$, then $\operatorname{FI}_V$ is naturally equivalent to the category $\operatorname{FI}$ of finite sets and injections. $\operatorname{FI}$-modules were introduced by Church, Ellenberg, and Farb in their seminal work [@CEF], and can be seen as one of the primary driving forces behind recent interest in so-called asymptotic algebra.\
In this work we will be primarily interested in studying finitely generated $\operatorname{FI}_V$-modules. To this end, we have the following critical theorem.\

\[noeth\] Assume that $|V| = d$. If $W$ is a finitely generated $\operatorname{FI}_V$-module over a Noetherian ring $k$, then all submodules of $W$ are also finitely generated. Moreover, if $k$ is a field, then there exists polynomials $p_1^W,\ldots,p_d^W \in {\mathbb{Q}}[x]$ such that $$\dim_k(W_n) = p_1^W(n) + p_2^W(n)2^n + \ldots + p_d^W(n)d^n$$ for all $n\gg 0$.

Prior to the provided source, the Noetherian property in the above theorem was proven for $\operatorname{FI}_d$-modules over a field of characteristic 0 in [@Sn Theorem 2.3]. The statement on dimensional stability was proven in the case where 