Morphisms and Closed Subgroups
Morphisms of algebraic groups
A morphism of algebraic groups \(\varphi : G \to G'\) over \(S\) is a morphism of schemes over \(S\) that respects the group-object structure on each side. Equivalently:
- in the typeclass form, a morphism in \(\mathsf{Grp}(\mathsf{Over}\, S)\) — a scheme morphism intertwining the multiplication, identity, and inverse;
- in the coordinate-ring form (over an affine base), a homomorphism of \(R\)-Hopf algebras \(A' \to A\) (note the contravariant direction).
The categorical group-object axiom (\(\varphi \circ \mu_G = \mu_{G'} \circ (\varphi \times \varphi)\)) is a property of a scheme morphism — once a candidate \(\varphi\) is fixed, the group-hom property is checked, not extra data.
Examples
- Determinant \(\det : \operatorname{GL}_n \to \mathbb{G}_m\).
- Inclusion \(\operatorname{SL}_n \hookrightarrow \operatorname{GL}_n\).
- \(n\)-th power \((-)^n : \mathbb{G}_m \to \mathbb{G}_m\) (kernel = \(\mu_n\)).
- Frobenius \(F : G \to G^{(p)}\) in characteristic \(p\).
Closed subgroups
A closed subgroup of \(G\) is a closed subscheme \(H \subseteq G\) such that the group operations of \(G\) restrict to \(H\) (equivalently, the inclusion is a morphism in \(\mathsf{Grp}\)). Closed subgroups are the kernels of group-scheme morphisms.
Examples.
- \(\operatorname{SL}_n \subseteq \operatorname{GL}_n\) (kernel of det).
- \(\mu_n \subseteq \mathbb{G}_m\) (kernel of \(n\)-th power).
- Diagonal torus \(T \subseteq \operatorname{GL}_n\).
- Borel subgroup \(B \subseteq \operatorname{GL}_n\) (upper-triangular).
- Unipotent radical \(U \subseteq B\) (strictly upper-triangular).
Closed-image construction
Given any morphism \(\varphi : G \to G'\), the scheme-theoretic image of \(\varphi\) is a closed subscheme of \(G'\) (when \(\varphi\) is quasi-compact and quasi-separated, which always holds for finite-type morphisms over a Noetherian base). Carrying the group-object structure across, this gives the image subgroup \(\operatorname{Im}(\varphi) \subseteq G'\), a closed subgroup scheme.
Kernels
Dually, the kernel of a morphism \(\varphi : G \to G'\) is the fibre product \(G \times_{G'} \{e\}\), a closed subgroup of \(G\). See [[node:linear_algebraic_groups.kernels_images_and_quotients|kernels, images, and quotients]] for the exact sequence formalism.
Faithfully flat / faithful descent
For surjective morphisms, the right notion is faithfully flat: a morphism is faithfully flat iff it is flat and surjective. Over a field, every morphism of finite-type group schemes is automatically flat once it is surjective on geometric points.
Closed subgroups via Hopf-algebra quotients
Under the coordinate-ring equivalence, a closed subgroup \(H \subseteq G\) corresponds to a quotient Hopf algebra \(\Gamma(G, \mathcal{O}_G) \twoheadrightarrow \Gamma(H, \mathcal{O}_H)\): the ideal of \(H\) is a Hopf ideal (an ideal closed under \(\Delta\) and \(S\) in a suitable sense, equivalently the kernel of a Hopf algebra map).
This is the algebraic shadow of the embedding theorem ([[node:linear_algebraic_groups.embedding_theorem|every affine algebraic group over a field embeds as a closed subgroup of \(\operatorname{GL}_n\)]]).