Every finite-type characteristic-zero group scheme is smooth

By [F1] and [F3] extend to an algebraic closure and write $R={\mathcal{O}}_{G,e}$, with maximal ideal $\mathfrak{m}$ and residue field $k$. Its reduction is regular by [F1]. Multiplication induces a map $R\to R{\otimes }_k(R\mathrm{/}{\mathfrak{m}}^2)$: restrict multiplication to $\mathrm{Spec}R$ in the first factor and the second infinitesimal neighbourhood of $e$ in the second. Its underlying image lies in every open neighbourhood of $e$ containing $\mathrm{Spec}R$, so pullbacks of local functions are defined; a denominator invertible at $e$ remains invertible because its image modulo the nilpotent second-factor ideal is that denominator in $R$. The identity restrictions imply that the image of $a\in \mathfrak{m}$ has the form $a\otimes 1+1\otimes \bar{a}+y$, where $y\in \mathfrak{m}{\otimes }_k(\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2)$. If $a$ is nonzero nilpotent, choose its least nilpotence exponent $n\ge 2$. Expanding the $n$th power of this image, with the second-factor ideal square zero, gives $n{a}^{n-1}\otimes \bar{a}\in a^{n-1}\mathfrak{m}{\otimes }_k(R\mathrm{/}{\mathfrak{m}}^2)$. The class of $a^{n-1}$ modulo $a^{n-1}\mathfrak{m}$ is nonzero: otherwise $a^{n-1}=t{a}^{n-1}$ for $t\in \mathfrak{m}$, and the unit $1-t$ would annihilate a nonzero element. Since $n$ is invertible, projection onto that nonzero class forces $\bar{a}=0$. Hence every nilpotent in $R$ belongs to ${\mathfrak{m}}^2$.

Reduction preserves Krull dimension, and the inclusion of the nilradical in ${\mathfrak{m}}^2$ shows that it also preserves cotangent dimension. The regularity of $R_{\mathrm{r}\mathrm{e}\mathrm{d}}$ from [F1] therefore makes $R$ regular by [F2]. Over the algebraically closed characteristic-zero field regularity at the rational identity is smoothness there. Translation carries the identity to every closed point; the open smooth locus consequently contains every closed point. Its complement, if nonempty, would contain a closed point, so it is empty. Finally [F1] descends smoothness to the original field. The argument used only the local multiplication near $(e,e)$, and thus applies to nonaffine groups. AC is used in [F3] and inherited from [F1].