Closed subgroup schemes of an affine group scheme correspond to Hopf ideals

(From closed subgroups to ideals.) Let $j\text{\hspace{0.17em}⁣}:H\hookrightarrow G$ be a closed subgroup scheme. Since $G=\mathrm{Spec}A$ is affine, [F1] presents the closed subscheme $H$ as $\mathrm{Spec}(A\mathrm{/}\mathfrak{a})$ for the ideal $\mathfrak{a}=\ker (A\to \mathcal{O}(H))$; the inclusion $j$ is a morphism of affine group schemes, so its comorphism $A\to \mathcal{O}(H)$ is a morphism of Hopf algebras by The coordinate Hopf algebra of an affine group scheme, and [F2] makes $\mathfrak{a}$ a Hopf ideal.

(From ideals to closed subgroups.) Let $\mathfrak{a}\subseteq A$ be a Hopf ideal. By [F2] the quotient $A\mathrm{/}\mathfrak{a}$ carries a Hopf algebra structure with $A\to A\mathrm{/}\mathfrak{a}$ a Hopf morphism, and this structure is finitely generated over $k$; by [F3] the spectrum $\mathrm{Spec}(A\mathrm{/}\mathfrak{a})$ is an affine group scheme of finite type over $k$, the quotient map induces a closed immersion $\mathrm{Spec}(A\mathrm{/}\mathfrak{a})\hookrightarrow G$, and this closed immersion is a morphism of group schemes. Hence it is a closed subgroup scheme whose associated ideal is $\mathfrak{a}$.

(The assignments are inverse.) Starting from a closed subgroup scheme $H$ with ideal $\mathfrak{a}=\ker (A\to \mathcal{O}(H))$ as in step 1.1, the construction of step 1.2 returns the closed subgroup scheme $\mathrm{Spec}(A\mathrm{/}\mathfrak{a})$; by [F1] the closed immersion $H\hookrightarrow G$ is isomorphic over $G$ to $\mathrm{Spec}(A\mathrm{/}\mathfrak{a})\hookrightarrow G$, and the group structures correspond because both inclusions are group-scheme morphisms and the structure maps of $A\mathrm{/}\mathfrak{a}$ are the unique ones making $A\to A\mathrm{/}\mathfrak{a}$ a Hopf morphism. Conversely, starting from a Hopf ideal $\mathfrak{a}$, the kernel of the quotient map $A\to A\mathrm{/}\mathfrak{a}$ is $\mathfrak{a}$. Hence the two assignments are mutually inverse bijections.

(Inclusion reversal.) Let $H_1,H_2$ be closed subgroup schemes with ideals ${\mathfrak{a}}_i=\ker (A\to \mathcal{O}(H_i))$ and identify $\mathcal{O}(H_i)=A\mathrm{/}{\mathfrak{a}}_i$ by [F1]. If $H_1\subseteq H_2$, the inclusion factors through $H_2$, so the composite $A\to \mathcal{O}(H_2)\to \mathcal{O}(H_1)$ is the quotient map $A\to A\mathrm{/}{\mathfrak{a}}_1$ and ${\mathfrak{a}}_1\supseteq {\mathfrak{a}}_2$. Conversely, if ${\mathfrak{a}}_2\subseteq {\mathfrak{a}}_1$, then the image of ${\mathfrak{a}}_1$ under the quotient $A\to A\mathrm{/}{\mathfrak{a}}_2$ is a Hopf ideal of $A\mathrm{/}{\mathfrak{a}}_2$ and the quotient map $A\mathrm{/}{\mathfrak{a}}_2\to A\mathrm{/}{\mathfrak{a}}_1$ is a Hopf morphism by [F2], so by [F3] it corresponds to a group-scheme morphism $H_1\to H_2$ whose composite with $H_2\hookrightarrow G$ is the inclusion of $H_1$, hence $H_1\subseteq H_2$.

(Conclusion and choice.) Steps 2.1 and 2.2 prove the bijection and the reversal of inclusions. AC was used in [F1] to present closed subschemes as quotient spectra and in the finite-type antiequivalence of [F3]. The Hopf-ideal kernel and quotient calculations of [F2] are choice-free.