39.4 Group schemes

Let us recall that a group is a pair $(G, m)$ where $G$ is a set, and $m : G \times G \to G$ is a map of sets with the following properties:

1. (associativity) $m(g, m(g', g'')) = m(m(g, g'), g'')$ for all $g, g', g'' \in G$,

2. (identity) there exists a unique element $e \in G$ (called the identity, unit, or $1$ of $G$) such that $m(g, e) = m(e, g) = g$ for all $g \in G$, and

3. (inverse) for all $g \in G$ there exists a $i(g) \in G$ such that $m(g, i(g)) = m(i(g), g) = e$, where $e$ is the identity.

Thus we obtain a map $e : \{ *\} \to G$ and a map $i : G \to G$ so that the quadruple $(G, m, e, i)$ satisfies the axioms listed above.

A homomorphism of groups $\psi : (G, m) \to (G', m')$ is a map of sets $\psi : G \to G'$ such that $m'(\psi (g), \psi (g')) = \psi (m(g, g'))$. This automatically insures that $\psi (e) = e'$ and $i'(\psi (g)) = \psi (i(g))$. (Obvious notation.) We will use this below.

Definition 39.4.1. Let $S$ be a scheme.

1. A group scheme over $S$ is a pair $(G, m)$, where $G$ is a scheme over $S$ and $m : G \times _ S G \to G$ is a morphism of schemes over $S$ with the following property: For every scheme $T$ over $S$ the pair $(G(T), m)$ is a group.

2. A morphism $\psi : (G, m) \to (G', m')$ of group schemes over $S$ is a morphism $\psi : G \to G'$ of schemes over $S$ such that for every $T/S$ the induced map $\psi : G(T) \to G'(T)$ is a homomorphism of groups.

Let $(G, m)$ be a group scheme over the scheme $S$. By the discussion above (and the discussion in Section 39.2) we obtain morphisms of schemes over $S$: (identity) $e : S \to G$ and (inverse) $i : G \to G$ such that for every $T$ the quadruple $(G(T), m, e, i)$ satisfies the axioms of a group listed above.

Let $(G, m)$, $(G', m')$ be group schemes over $S$. Let $f : G \to G'$ be a morphism of schemes over $S$. It follows from the definition that $f$ is a morphism of group schemes over $S$ if and only if the following diagram is commutative:

$\xymatrix{ G \times _ S G \ar[r]_-{f \times f} \ar[d]_ m & G' \times _ S G' \ar[d]^ m \\ G \ar[r]^ f & G' }$

Lemma 39.4.2. Let $(G, m)$ be a group scheme over $S$. Let $S' \to S$ be a morphism of schemes. The pullback $(G_{S'}, m_{S'})$ is a group scheme over $S'$.

Proof. Omitted. $\square$

Definition 39.4.3. Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$.

1. A closed subgroup scheme of $G$ is a closed subscheme $H \subset G$ such that $m|_{H \times _ S H}$ factors through $H$ and induces a group scheme structure on $H$ over $S$.

2. An open subgroup scheme of $G$ is an open subscheme $G' \subset G$ such that $m|_{G' \times _ S G'}$ factors through $G'$ and induces a group scheme structure on $G'$ over $S$.

Alternatively, we could say that $H$ is a closed subgroup scheme of $G$ if it is a group scheme over $S$ endowed with a morphism of group schemes $i : H \to G$ over $S$ which identifies $H$ with a closed subscheme of $G$.

Lemma 39.4.4. Let $S$ be a scheme. Let $(G, m, e, i)$ be a group scheme over $S$.

1. A closed subscheme $H \subset G$ is a closed subgroup scheme if and only if $e : S \to G$, $m|_{H \times _ S H} : H \times _ S H \to G$, and $i|_ H : H \to G$ factor through $H$.

2. An open subscheme $H \subset G$ is an open subgroup scheme if and only if $e : S \to G$, $m|_{H \times _ S H} : H \times _ S H \to G$, and $i|_ H : H \to G$ factor through $H$.

Proof. Looking at $T$-valued points this translates into the well known conditions characterizing subsets of groups as subgroups. $\square$

Definition 39.4.5. Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$.

1. We say $G$ is a smooth group scheme if the structure morphism $G \to S$ is smooth.

2. We say $G$ is a flat group scheme if the structure morphism $G \to S$ is flat.

3. We say $G$ is a separated group scheme if the structure morphism $G \to S$ is separated.

Add more as needed.

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