The Stacks project

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.

Comments (2)

Comment #7816 by Verroq on

Typo in the Diagram under Def 022S: On the right arrow, it should be m', not m.

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 022R. Beware of the difference between the letter 'O' and the digit '0'.



View Section 39.4 as pdf