Definitions:
Let $k$ be a field, by an affine algebraic group we will mean an affine group scheme $G$ over $Spec(k)$ with the property that the coordinate ring of the underlying affine scheme $|G_{\bar{k}}|:=G \otimes_k Spec(\bar{k})$ is a coordinate ring of an algebraic $\bar{k}$-group (in the classical (variety) sense).
Question:
Is there something wrong with this proof of $|G|$'s smoothness (something is bothering me about it):
The "Proof":
Let $G$ be an affine algebraic $k$-group and denote by $\bar{G}$ the affine scheme \ $G\times_{Spec(k)} Spec(\bar{k})$. The universal property of the fibre product ensures that $\bar{G}$ is also an affine algebraic $\bar{k}$-group . Moreover, the inclusion of $k$ into $\bar{k}$ ensures $\bar{k}$ is a $k$-algebra, whence there is an isomorphism of $\bar{k}[G]$-modules : \begin{equation}
\Omega_{\bar{k}[G]|\bar{k}} \cong \Omega_{k[G]|k}\otimes_{k[G]} \bar{k}[G].
\end{equation} From which it is deduced that: $\Omega_{\bar{k}[G]|\bar{k}}$ is $\bar{k}[G]$-free of rank $n$ if and only if $\Omega_{k[G]|k}$ is $k[G]$-free of rank $n$. Therefore, it is enough to show the theorem for $\bar{G}$.
\item The algebraic variety $V(\bar{k}[\bar{G}])$ \textit{(in the classical sense)} contains a non-singular point \textit{p}, hence $\Omega_{\bar{k}[\bar{G}]_{\mathscr{I}(p)}|k}$ is a free $\bar{k}[\bar{G}]_{\mathscr{I}(p)}$-module.
$\bar{G}$ is a group, therefore the left multiplication $\lambda_g$ on $\bar{G}$ is an automorphism . Whence, for any point $p$ in $\bar{G}$, there is an isomorphism of $k$ algebras: $\psi: k[G]_g \rightarrow k[G]_x$. This induces a short exact sequence of $k[G]_p$-modules :
\begin{equation} \Omega_{k[G]_g|k[G]_g} \rightarrow \Omega_{k[G]_p|k[G]_g} \rightarrow 0 \end{equation}
$\Omega_{k[G]_g|k[G]_g}$ is a free $k[G]_g$-module on no generators, hence $\Omega_{k[G]_g|k[G]_g}$ is the $0$ $k[G]_g$-module.
The exactness of the above, therefore implies that $\Omega_{k[G]_|k[G]_g}$ is also trivial and in particular it is a projective $k[G]_p$-module.
Since the point $g$ was chosen to be nonsingular, the serre-swan theorem implies $\Omega_{k[G]_g|k}$ is an $k[G]_g$-projective module; whence: $\Omega_{k[G]_p|k}$ is $k[G]_p$-projective.
Since $p$ was chosen arbitrarily and $\Omega_{k[G]_p|k}\cong (\Omega_{k[G]|k})_p$, then $\Omega_{k[G]|k}$ is locally free since; by the serre-swan theorem $\bar{G}$, is smoothness is then deduced.