= [G:H] \cdot [H : H \cap G_n]$ for $n \ge N$. This implies for every finitely presented $kG$-module $M$ $$\begin{aligned}
{\operatorname{vdim}}_{kG}^{{\operatorname{Ore}}}(M)
  = 
\frac{\dim_{kH}^{{\operatorname{Ore}}}\bigl({\operatorname{res}}_{kG}^{kH} M\bigr)}{[G:H]}
& =  
\lim_{n \to \infty} \frac{\dim_k\bigl(k \otimes_{k[H\cap G_n]}
{\operatorname{res}}_{kG}^{kH}  M\bigr)}{[G:H] \cdot [H:H \cap G_n]}
\\
& =  
\lim_{n \to \infty} \frac{\dim_k\bigl(k \otimes_{kG_n}
M\bigr)}{[G:G_n]}.\qedhere\end{aligned}$$

Because of Theorem \[thm:elementary\_amenable\_goldie\], Theorem \[the:dim\_approximation\_over\_fields\] is true in the case $k$ is a skew field and $G$ is an elementary amenable group in which the orders of the finite subgroups are bounded (clearly $\Delta^+(G_n) = 1$ for sufficiently large $n$). In particular Theorem \[the:dim\_approximation\_over\_fields\] is true for any virtually torsionfree elementary amenable group.

Examples {#sec:Examples}
========

\[rem:connection\_to\_L2\_for\_group\_C\] Let $(G_n)_{n\ge 0}$ be a residual chain of a group $G$. Let $X$ be a finite free $G$-$CW$-complex. Let $k$ be a field of characteristic ${\operatorname{char}}(k)$. For a prime $p$ denote by ${{\mathbb F}}_p$ the field of $p$ elements. Then we conclude from the universal coefficient theorem $$\begin{aligned}
\dim_{k}\bigl(H_i(G_n\backslash X;k)\bigr) & =  \dim_{{{\mathbb Q}}}\bigl(H_i(G_n\backslash X;{{\mathbb Q}})\bigr)  &  {\operatorname{char}}(k) = 0;
\\
\dim_{k}\bigl(H_i(G_n\backslash X;k)\bigr) & =  \dim_{{{\mathbb F}}_p}\bigl(H_i(G_n\backslash X;{{\mathbb F}}_p)\bigr)  &  p = {\operatorname{char}}(k) \not= 0;
\\
\dim_{{{\mathbb F}}_p}\bigl(H_i(G_n\backslash X;{{\mathbb F}}_p)\bigr) & \ge  \dim_{{{\mathbb Q}}}\bigl(H_i(G_n\backslash X;{{\mathbb Q}})\bigr).& \end{aligned}$$ In particular we conclude from Remark \[rem:fields\_of\_characteristic\_zero\] 