Group schemes over a base scheme

Let $S$ be a scheme. An $S$-group scheme is a group object in the category of $S$-schemes: an $S$-scheme $G$, with structure morphism $p:G\to S$, together with $S$-morphisms $m:G{\times }_{S}G\to G,e:S\to G,i:G\to G$ called multiplication, unit section and inverse, such that the following identities of $S$-morphisms hold. The fibre product is the one of Fibre product of schemes and exists by Existence of all scheme fibre products; the canonical identifications $S{\times }_{S}G\cong G\cong G{\times }_{S}S$ used below are those of Uniqueness of the fibre product.

(Associativity) $m\circ (m{\times }_S{\mathrm{id}}_G)=m\circ ({\mathrm{id}}_G{\times }_{S}m)$ as morphisms $G{\times }_{S}G{\times }_{S}G\to G$.

(Unit laws) $m\circ (e{\times }_S{\mathrm{id}}_G)={\mathrm{id}}_G$ and $m\circ ({\mathrm{id}}_G{\times }_{S}e)={\mathrm{id}}_G$ under the canonical identifications above.

(Inverse laws) $m\circ ({\mathrm{id}}_G,i)=e\circ p$ and $m\circ (i,{\mathrm{id}}_G)=e\circ p$ as morphisms $G\to G$, where $({\mathrm{id}}_G,i):G\to G{\times }_{S}G$ and $(i,{\mathrm{id}}_G):G\to G{\times }_{S}G$ are induced by the universal property of the fibre product.

A morphism of $S$-group schemes $f:G\to H$ is an $S$-morphism with $f\circ m_G=m_H\circ (f{\times }_{S}f)$, $f\circ e_G=e_H$ and $f\circ i_G=i_H\circ f$, where $f{\times }_{S}f:G{\times }_{S}G\to H{\times }_{S}H$ is the induced morphism.

Functor of points. For an $S$-scheme $T$ put $G(T)={\mathrm{Hom}}_S(T,G)$. The composites $T\stackrel{\mathrm{\Delta }}{\to }T{\times }_{S}T\stackrel{\alpha {\times }_S\beta }{\to }G{\times }_{S}G\stackrel{m}{\to }G$, $T\to S\stackrel{e}{\to }G$ and $i\circ \alpha$ make $G(T)$ a group, and this structure is natural in $T$ by the universal property of the fibre product. A morphism $f:G\to H$ of $S$-group schemes induces homomorphisms $G(T)\to H(T)$ compatible with the transition maps of $T$. For $S=\mathrm{Spec}k$ and $G$ of finite type this is the notion of Group schemes of finite type over a field, and the group object diagrams above are equivalent to the requirement that each $G(T)$ be a group naturally in $T$. The definition is a condition on given data and quotes no new existence statement.

Commutativity. $G$ is commutative when $m=m\circ \sigma$, where $\sigma :G{\times }_{S}G\to G{\times }_{S}G$ is the exchange isomorphism; equivalently, each $G(T)$ is abelian.

Closed subgroup schemes. A closed subgroup scheme of $G$ is a closed immersion $j:H\to G$ for which there exist $S$-morphisms $m_H:H{\times }_{S}H\to H$, $e_H:S\to H$, $i_H:H\to H$ with $j\circ m_H=m\circ (j{\times }_{S}j)$, $j\circ e_H=e$ and $j\circ i_H=i\circ j$; since $j$ is a monomorphism these morphisms are unique if they exist, $H$ becomes an $S$-group scheme and $j$ a morphism of $S$-group schemes. A closed subgroup scheme $H$ is normal when $H(T)$ is a normal subgroup of $G(T)$ for every $S$-scheme $T$; equivalently, the conjugation morphism $G{\times }_{S}H\to G$, $(\alpha ,\beta )\mapsto m_G(m_G(\alpha ,\beta ),i_G(\alpha ))$, factors through $j$.

Kernels. For a morphism $f:G\to H$ of $S$-group schemes, the kernel is the fibre product $K=G{\times }_{H}S$ formed with $f$ and the unit section $e_H:S\to H$, so that $K\to G$ is the base change of $e_H$ along $f$ and is a closed immersion whenever $e_H$ is one, in particular whenever $H$ is separated over $S$; the induced $S$-morphisms make $K$ an $S$-group scheme, and for every $S$-scheme $T$ the sequence $0\to K(T)\to G(T)\to H(T)$ is exact.

Base change. For any morphism $S^{\mathrm{\prime }}\to S$, the base change $G_{S^{\mathrm{\prime }}}=G{\times }_S{S}^{\mathrm{\prime }}$ of Base change of objects, morphisms and properties carries an induced $S^{\mathrm{\prime }}$-group structure with $m_{S^{\mathrm{\prime }}},e_{S^{\mathrm{\prime }}},i_{S^{\mathrm{\prime }}}$ obtained from $m,e,i$ under the canonical isomorphism $G_{S^{\mathrm{\prime }}}{\times }_{S^{\mathrm{\prime }}}{G}_{S^{\mathrm{\prime }}}\cong (G{\times }_{S}G){\times }_S{S}^{\mathrm{\prime }}$. It satisfies $G_{S^{\mathrm{\prime }}}(T)=G(T)$ for every $S^{\mathrm{\prime }}$-scheme $T$, regarded as an $S$-scheme via $T\to S^{\mathrm{\prime }}\to S$; this is the base-change convention used for all group schemes on this page. In particular a morphism $f:G\to H$ of $S$-group schemes base changes to a morphism $f_{S^{\mathrm{\prime }}}$ of $S^{\mathrm{\prime }}$-group schemes.