finitely generated.

Now apply Theorem \[the:Approximation\_for\_finitely\_presented\_kG-modules\_for\_Eleks\_dimension\_function\] and Theorem \[the:Comparing\_Eleks\_dimension\_and\_the\_Ore\_dimension\].\
 \[the:dim\_approximation\_over\_fields:chain\_complexes\] We obtain from additivity, the exactness of the functor $S^{-1}kG \otimes_{kG}\_$ and the right exactness of the functor $k \otimes_{kG}\_$ that $$\begin{aligned}
\dim^{{\operatorname{Ore}}}_{kG}\bigl(H_i(C_*)\bigr)
&= 
\dim^{{\operatorname{Ore}}}_{kG} \bigl({\operatorname{coker}}(c_{i+1})\bigr)
+\dim^{{\operatorname{Ore}}}_{kG}\bigl({\operatorname{coker}}(c_i)\bigr)
- \dim^{{\operatorname{Ore}}}_{kG}\bigl(C_{i-1}\bigr),\\
\dim_{k}\bigl(H_i(k \otimes_{kG_n} C_*)\bigr)
&= 
\dim_{k}\bigl(k \otimes_{kG_n}   {\operatorname{coker}}(c_{i+1})\bigr)
+\dim_{k}\bigl(k \otimes_{kG_n}  {\operatorname{coker}}(c_i)\bigr)
\\  & \hspace{10mm} - \dim_{k}\bigl(k \otimes_{kG_n} C_{i-1}\bigr).\end{aligned}$$ Hence the claim follows from assertion \[the:dim\_approximation\_over\_fields:finitely\_presented\_modules\] applied to the finitely presented $kG$-modules ${\operatorname{coker}}(c_{i+1})$, ${\operatorname{coker}}(c_i)$ and $C_{i-1}$.\
 \[the:dim\_approximation\_over\_fields:CW-complexes\] This follows from assertion \[the:dim\_approximation\_over\_fields:chain\_complexes\] applied to the cellular chain complex of $X$.

Extension to the virtually torsionfree case {#sec:Extension_to_the_virtually_torsionfree_case}
===========================================

Next we explain how Theorem \[the:dim\_approximation\_over\_fields\] can be extended to the virtually torsionfree case.

For the remainder of this section let $k$ be a skew field, let $G$ be an amenable group which possesses a subgroup $H$ of finite index with $\Delta^+(H) = 1$, and let $k*G$ be a crossed product such that $k*H$ is a left Goldie ring. We define the *virtual Ore dimension* of a $k*G$-module $M$ by $$\label{eq:virtual_Ore_dimension}
  {\operatorname{vdim}}^{{\operatorname{Ore}}}_{k*G}(M) =
  \frac{\dim^{{\operatorname{Ore}}}_{k*H}\bigl({\operatorname{res}}_{k*G}^{k*H} M\bigr)}{[G:H]},
  $$ where ${\operatorname{res}}_{k*G}^{k*H} M$ is the $k*H$-module obtained from the $k*G$-module $M$ by restricting the $G$-action to $H$.

We have to show that this is independent of the choice of 