is $kG$-isomorphic to the cokernel of $r_A \colon kG^r \to kG^s$. Since $A$ is a finite matrix and each element in $kG$ has finite support, we can find a finitely generated subgroup $H \subseteq G$ such that $A \in
M_{r,s}(kH)$. Both $kG$ and $kH$ are prime left Goldie by Lemma \[lem:delta\_implies\_prime\] and Theorem \[thm:elementary\_amenable\_goldie\]. Consider the finitely presented $kH$-module $N := {\operatorname{coker}}\bigl(r_A \colon kH^r \to kH^s\bigr)$. Then $M = kG \otimes_{kH} N$. We can also consider the Ore localization $T^{-1}kH$ for $T$ the set of non-zero-divisors of $kH$. Put $H_n = H \cap G_n$. We obtain a residual chain $(H_n)_{n\ge 0}$ of $H$ and have: $$\begin{aligned}
\dim_{kG}^{{\operatorname{Ore}}}(M)
&= 
\dim_{S^{-1}kG} \bigl(S^{-1}kG \otimes_{kG} M\bigr)\\
&= 
\dim_{S^{-1}kG} \bigl(S^{-1}kG \otimes_{kG} kG \otimes_{kH} N\bigr)
\\
&= 
\dim_{S^{-1}kG}\bigl(S^{-1}kG \otimes_{T^{-1}kH} T^{-1}kH \otimes_{kH} N\bigr)
\\
&= 
\dim_{T^{-1}kH}\bigl(T^{-1}kH \otimes_{kH}  N)
\\
&= 
\dim_{kH}^{{\operatorname{Ore}}}(N).\end{aligned}$$ We compute $$\begin{aligned}
\frac{\dim_k\bigl(k\otimes_{kG_n} M\bigr)}{[G:G_n]} 
&=  
\frac{\dim_k\bigl(k[G/G_n]\otimes_{kG} M\bigr)}{[G:G_n]}\\
&=  
\frac{\dim_k\bigl(k[G/G_n]\otimes_{kG} kG \otimes_{kH} N\bigr)}{[G:G_n]}
\\
&=  
\frac{\dim_k\bigl(k[G/G_n]\otimes_{k[H/H_n]} k[H/H_n] \otimes_{kH} N\bigr)}{[G:G_n]}
\\
&=  
\frac{[G/G_n:H/H_n] \cdot \dim_k\bigl(k[H/H_n] \otimes_{kH} N\bigr)}{[G:G_n]}
\\
&=  
\frac{[G/G_n:H/H_n] \cdot \dim_k\bigl(k \otimes_{kH_n} N\bigr)}{[G/G_n:H/H_n] \cdot [H:H_n]}
\\
&=  
\frac{\dim_k\bigl(k \otimes_{kH_n} N\bigr)}{[H:H_n]}.\end{aligned}$$ Therefore the claim holds for $M$ over $kG$ if it holds for $N$ over $kH$. Hence we can assume without loss of generality that $G$ is 