The smooth locus of a normal completion of a group has only constant functions

By [F1], place $G$ as a locally closed subvariety of projective space. Its reduced closure $X$ is integral, and $G$ is open in $X$. Normalize each affine chart of $X$ in $k(G)$. By [F2] the resulting affine maps are finite and agree on principal-overlap charts, so they glue to a finite normal variety $\bar{G}\to X$. A finite map is proper by its integral affine ring description and lying-over after arbitrary base change. Thus $\bar{G}$ is proper by [F2]. Above the normal open $G$ the integral closures equal the original rings, so $G$ embeds as a dense open of $\bar{G}$.

Let $U$ be the smooth locus of $\bar{G}$. It contains $G$. By [F3] every height-zero or height-one point is smooth, since normal height-one local rings are DVRs and $k$ is perfect; hence $\bar{G}\setminus U$ has codimension at least two. A global function on $U$ is a rational function on $\bar{G}$, regular in each height-one local ring. On each normal affine chart it therefore belongs to its coordinate ring by the intersection assertion of [F3]. These extensions agree in the function field and glue. Thus $\mathrm{\Gamma }(U,\mathcal{O})=\mathrm{\Gamma }(\bar{G},\mathcal{O})=k$ by [F4]. AC is inherited from [F1]–[F4]. No general compactification theorem is used: the ample-sheaf construction supplied the projective completion.