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:
(associativity) m(g, m(g', g'')) = m(m(g, g'), g'') for all g, g', g'' \in G,
(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
(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.
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.
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:
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.
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.
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.
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.
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.
We say G is a smooth group scheme if the structure morphism G \to S is smooth.
We say G is a flat group scheme if the structure morphism G \to S is flat.
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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