## 78.7 Examples of group algebraic spaces

If $G \to S$ is a group scheme over the base scheme $S$, then the base change $G_ B$ to any algebraic space $B$ over $S$ is an group algebraic space over $B$ by Lemma 78.5.2. We will frequently use this in the examples below.

Example 78.7.1 (Multiplicative group algebraic space). Let $B \to S$ as in Section 78.3. Consider the functor which associates to any scheme $T$ over $B$ the group $\Gamma (T, \mathcal{O}_ T^*)$ of units in the global sections of the structure sheaf. This is representable by the group algebraic space

\[ \mathbf{G}_{m, B} = B \times _ S \mathbf{G}_{m, S} \]

over $B$. Here $\mathbf{G}_{m, S}$ is the multiplicative group scheme over $S$, see Groupoids, Example 39.5.1.

Example 78.7.2 (Roots of unity as a group algebraic space). Let $B \to S$ as in Section 78.3. Let $n \in \mathbf{N}$. Consider the functor which associates to any scheme $T$ over $B$ the subgroup of $\Gamma (T, \mathcal{O}_ T^*)$ consisting of $n$th roots of unity. This is representable by the group algebraic space

\[ \mu _{n, B} = B \times _ S \mu _{n, S} \]

over $B$. Here $\mu _{n, S}$ is the group scheme of $n$th roots of unity over $S$, see Groupoids, Example 39.5.2.

Example 78.7.3 (Additive group algebraic space). Let $B \to S$ as in Section 78.3. Consider the functor which associates to any scheme $T$ over $B$ the group $\Gamma (T, \mathcal{O}_ T)$ of global sections of the structure sheaf. This is representable by the group algebraic space

\[ \mathbf{G}_{a, B} = B \times _ S \mathbf{G}_{a, S} \]

over $B$. Here $\mathbf{G}_{a, S}$ is the additive group scheme over $S$, see Groupoids, Example 39.5.3.

Example 78.7.4 (General linear group algebraic space). Let $B \to S$ as in Section 78.3. Let $n \geq 1$. Consider the functor which associates to any scheme $T$ over $B$ the group

\[ \text{GL}_ n(\Gamma (T, \mathcal{O}_ T)) \]

of invertible $n \times n$ matrices over the global sections of the structure sheaf. This is representable by the group algebraic space

\[ \text{GL}_{n, B} = B \times _ S \text{GL}_{n, S} \]

over $B$. Here $\mathbf{G}_{m, S}$ is the general linear group scheme over $S$, see Groupoids, Example 39.5.4.

Example 78.7.5. Let $B \to S$ as in Section 78.3. Let $n \geq 1$. The determinant defines a morphism of group algebraic spaces

\[ \det : \text{GL}_{n, B} \longrightarrow \mathbf{G}_{m, B} \]

over $B$. It is the base change of the determinant morphism over $S$ from Groupoids, Example 39.5.5.

Example 78.7.6 (Constant group algebraic space). Let $B \to S$ as in Section 78.3. Let $G$ be an abstract group. Consider the functor which associates to any scheme $T$ over $B$ the group of locally constant maps $T \to G$ (where $T$ has the Zariski topology and $G$ the discrete topology). This is representable by the group algebraic space

\[ G_ B = B \times _ S G_ S \]

over $B$. Here $G_ S$ is the constant group scheme introduced in Groupoids, Example 39.5.6.

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