we obtain that $$\dim_k({\operatorname{coker}}(  f\vert_{Q_n\setminus
\partial_RQ_n}))-\dim_k({\operatorname{coker}}(M[Q_n]))\le r\cdot  \bigl|\partial_R Q_n\bigr|.$$ Since $$\frac{|\partial_R Q_n|}{[G/G_n]}=\frac{ |\partial_R Q_n|}{
    |Q_n|}\xrightarrow{n\to\infty} 0$$ we get that $$\lim_{n\to\infty}\frac{\dim_k( k[G/G_n]\otimes_{kG}M)}{[G:G_n]}$$ exists if and only if $$\lim_{n\to\infty}\frac{\dim_k( M[Q_n])}{|Q_n|}$$ exists, and in this case they are equal. Now the assertion follows from Theorem \[the:equality\_elek\_and\_approximative\_dimension\_of\_finitely\_presented\_modules\].

Comparing dimensions {#sec:Comparing_dimensions}
====================

The main result of this section is:

\[the:Comparing\_dimensions\] Let $G$ be a group, let $k$ be a skew field, and let $k*G$ be a crossed product which is prime left Goldie. Let $\dim$ be any dimension function which assigns to a finitely generated left $k*G$-module a nonnegative real number and satisfies

1.  $\dim(k*G) = 1$.

2.  For every short exact sequence $0 \to M_0 \to M_1 \to M_2 \to 0$ of finitely generated left $k*G$-modules, we get $$\dim(M_1) = \dim(M_0) + \dim(M_2).$$

3.  If the finitely generated left $k*G$-module $M$ satisfies $\dim(M) =
    0$, then every quotient module $Q$ of $M$ satisfies $\dim(Q) = 0$.

Then for every finitely presented left $k*G$-module $M$, we get $\dim(M) = \dim_{k*G}^{{\operatorname{Ore}}}(M)$.

Let $S$ denote the non-zero-divisors of $k*G$. We have to show that for all $r,s \in \mathbb{N}$ and every $r\times
s$ matrix $A$ with entries in $k*G$ $$\dim_{k*G}^{{\operatorname{Ore}}}\bigl({\operatorname{coker}}\big(r_A \colon S^{-1}k*G^r \to
S^{-1}k*G^s\bigr) \bigr)=
\dim\bigl({\operatorname{coker}}\big(r_A \colon 