A scheme-faithful action fixing a point has a faithful finite jet representation

Put $X_n=\mathrm{Spec}({\mathcal{O}}_{X,P}\mathrm{/}{\mathfrak{m}}_P^{n+1})$. Its product with $G$ has underlying space $G\times \{P\}$ and hence lies in the regular domain. The fixed-point identity makes the point ideal stable and therefore makes its powers stable; the action restricts to $G\times X_n\to X_n$. Group identities restrict as well, yielding a representation ${\rho }_n:G\to \mathrm{GL}(V_n)$ with $V_n={\mathcal{O}}_{X,P}\mathrm{/}{\mathfrak{m}}_P^{n+1}$. Indeed after any affine base change the automorphism is a linear automorphism of the free module with basis $V_n$, so its matrix entries are regular and its determinant invertible. The spaces are finite dimensional because the local ring is Noetherian and its residue field is $k$. The kernels $H_n$ are closed, descend with $n$, and stabilize to a closed subgroup $H$ by the ascending chain condition on their ideal sheaves and a finite affine cover of $G$.

The subgroup $H$ acts as the identity on every $X_n$. Cover $H$ by affine charts $\mathrm{Spec}A$ and choose an affine neighbourhood $B$ of $P$ in $X$. In the regular-action case the equalizer ideal of action and projection restricts to zero in $A{\otimes }_k({\mathcal{O}}_{X,P}\mathrm{/}{\mathfrak{m}}_P^{n+1})$ for all $n$. In the rational case, cover the fixed-point slice by principal open neighbourhoods $D(d)\subset \mathrm{Spec}A\times B$ in the regular domain. The specialization $d(P)\in A$ is invertible after localizing at it, and these localizations cover $H$. On each such chart the equalizer ideal is generated by fractions with powers of $d$ as denominators. This denominator is invertible in every finite jet ring, since its constant specialization is a unit. Thus identity on every jet forces each numerator to have zero image in $A{\otimes }_k({\mathcal{O}}_{X,P}\mathrm{/}{\mathfrak{m}}_P^{n+1})$ for all $n$, with $A$ now that localized chart ring. Expand a numerator using finitely many $k$-linearly independent coefficients in $A$; its local-ring coefficients lie in every power of ${\mathfrak{m}}_P$ and are zero by [F2]. Since $B$ is integral, its coordinate ring injects into ${\mathcal{O}}_{X,P}$, and tensoring over $k$ preserves injectivity. The numerator is therefore zero already. Hence the action and projection agree on these nonempty source neighbourhoods. The integral $X$ makes such a neighbourhood schematically dense after tensoring with any $A$: restriction of its coordinate rings embeds into $A{\otimes }_{k}k(X)$. Consequently $H$ acts identically as a scheme-valued birational transformation, and identically everywhere in the regular-action case by the same schematic density. Scheme faithfulness gives $H=e$.

For a stabilizing $n$, ${\rho }_n$ has trivial scheme kernel and is a closed immersion by [F1]. The target is affine, so $G$ is affine. The proof retains infinitesimal kernels and does not infer faithfulness from ordinary rational points. AC is inherited from [F1]–[F2].