localization in Subsection \[sub:ore\_localization\]. The Ore localization $S^{-1}kG$ is a skew field containing $kG$. Therefore the following definition makes sense:

\[def:Ore\_dimension\] Let $G$ be a torsionfree amenable group such that $kG$ contains no zero-divisors. The *Ore dimension* of a $kG$-module $M$ is defined by $$\dim^{{\operatorname{Ore}}}_{kG}(M) = \dim_{S^{-1}kG}\bigl(S^{-1}kG \otimes_{kG} M\bigr).$$

The following theorem is our main result; we will prove a more general version, including virtually torsionfree groups, in Section \[sec:Extension\_to\_the\_virtually\_torsionfree\_case\].

\[the:dim\_approximation\_over\_fields\] Let $k$ be a field. Let $G$ be a torsionfree amenable group for which $kG$ has no zero-divisors[^1]. Let $(G_n)_{n\ge 0}$ be a residual chain of $G$. Then:

1.  \[the:dim\_approximation\_over\_fields:finitely\_presented\_modules\] Consider a finitely presented $kG$-module $M$. Then $$\dim_{kG}^{{\operatorname{Ore}}}(M)
     = 
    \lim_{n \to \infty} \frac{\dim_k\bigl(k \otimes_{kG_n}  M\bigr)}{[G:G_n]};$$

2.  \[the:dim\_approximation\_over\_fields:chain\_complexes\] Consider a finite free $kG$-chain complex $C_*$. Then we get for all $i \ge 0$ $$\dim_{kG}^{{\operatorname{Ore}}}\bigl(H_i(C_*)\bigr)  
      = 
    \lim_{n \to \infty} \frac{\dim_k\bigl(H_i(k \otimes_{kG_n} C_*)\bigr)}{[G:G_n]};$$

3.  \[the:dim\_approximation\_over\_fields:CW-complexes\] Let $X$ be a finite free $G$-$CW$-complex. Then we get for all $i \ge 0$ $$\dim_{kG}^{{\operatorname{Ore}}} \bigl(H_i(X)\bigr)  
      = 
    \lim_{n \to \infty} \frac{\dim_k\bigl(H_i(G_n \backslash X;k)\bigr)}{[G:G_n]}.$$

\[rem:fields\_of\_characteristic\_zero\] Let $G$ be a group with a residual chain $(G_n)_{n \ge 0}$, and let $M$ be a finitely presented $kG$-module. Then the Approximation Theorem for $L^2$-Betti numbers says that $$\label{eq:limit_for_modules}
\dim_{{{\mathcal N}}(G)}\bigl({{\mathcal 