Group Schemes of Finite Type over a Field

Let $k$ be a field. A group scheme of finite type over $k$ is a finite-type $k$-scheme $G$ with $k$-morphisms $m:G{\times }_{k}G\to G,e:\mathrm{Spec}k\to G,i:G\to G$ satisfying the following identities of scheme morphisms. Multiplication is associative, $m\circ (m\times \mathrm{id})=m\circ (\mathrm{id}\times m)$ on $G^3$; $m\circ (e\times \mathrm{id})=\mathrm{id}=m\circ (\mathrm{id}\times e)$ under the canonical identifications; and $m\circ (i,\mathrm{id})=e\circ p=m\circ (\mathrm{id},i)$, where $p:G\to \mathrm{Spec}k$ is the structure map. The products exist by Existence of all scheme fibre products; the base and finite-type conventions are Schemes and morphisms over a base and Locally finite type and finite type morphisms.

For every $k$-scheme $T$, put $G(T)={\mathrm{Hom}}_k(T,G)$. The three structure morphisms give a group law on $G(T)$, naturally under precomposition in $T$. In particular for every commutative unital $k$-algebra $R$, $G(R)$ means $G(\mathrm{Spec}R)$ and is a group, including for algebras with nilpotents. The definition imposes neither reducedness nor smoothness, and allows finite nonreduced group schemes. A group scheme is called commutative if $m$ agrees with its composition with the factor-exchange map.

Let $G,H$ be group schemes of finite type over a field $k$, as in Group schemes of finite type over a field. A morphism of $k$-group schemes is a $k$-morphism $f:G\to H$ satisfying $f\circ m_G=m_H\circ (f\times f),f\circ e_G=e_H,i_H\circ f=f\circ i_G\mathrm{.}$ Consequently $G(T)\to H(T)$ is a group homomorphism for every $k$-scheme $T$, naturally in $T$.

A closed subgroup scheme of $G$ is a closed immersion $j:H\hookrightarrow G$ in the sense of Closed immersions of schemes, where $H$ is a group scheme and $j$ is a morphism of group schemes. Its multiplication, identity, and inverse are the restrictions of those of $G$. A closed immersion is a monomorphism: a factorization through its subscheme, when it exists, is unique, since the map of sheaves onto the subscheme's structure sheaf is surjective. Therefore these restricted structure morphisms are uniquely determined. A closed subscheme of $G$ is not assumed to be a subgroup merely because its $k$-rational points form one; all algebra-valued points, including points over nonreduced algebras, are relevant.

Let $k$ be a field, $G$ a group scheme of finite type over $k$, and $j:H\hookrightarrow G$ a closed subscheme. Then $H$ has the unique induced structure of a closed subgroup scheme if and only if $H(R)\subseteq G(R)$ is a subgroup for every commutative unital $k$-algebra $R$. Equivalently, the identity $e_G$, multiplication restricted to $H{\times }_{k}H$, and inverse restricted to $H$ all factor through $H$. No reducedness, smoothness, or algebraic closedness hypothesis is imposed.