39.5 Examples of group schemes
Example 39.5.1 (Multiplicative group scheme). Consider the functor which associates to any scheme $T$ the group $\Gamma (T, \mathcal{O}_ T^*)$ of units in the global sections of the structure sheaf. This is representable by the scheme
\[ \mathbf{G}_ m = \mathop{\mathrm{Spec}}(\mathbf{Z}[x, x^{-1}]) \]
The morphism giving the group structure is the morphism
\begin{eqnarray*} \mathbf{G}_ m \times \mathbf{G}_ m & \to & \mathbf{G}_ m \\ \mathop{\mathrm{Spec}}(\mathbf{Z}[x, x^{-1}] \otimes _{\mathbf{Z}} \mathbf{Z}[x, x^{-1}]) & \to & \mathop{\mathrm{Spec}}(\mathbf{Z}[x, x^{-1}]) \\ \mathbf{Z}[x, x^{-1}] \otimes _{\mathbf{Z}} \mathbf{Z}[x, x^{-1}] & \leftarrow & \mathbf{Z}[x, x^{-1}] \\ x \otimes x & \leftarrow & x \end{eqnarray*}
Hence we see that $\mathbf{G}_ m$ is a group scheme over $\mathbf{Z}$. For any scheme $S$ the base change $\mathbf{G}_{m, S}$ is a group scheme over $S$ whose functor of points is
\[ T/S \longmapsto \mathbf{G}_{m, S}(T) = \mathbf{G}_ m(T) = \Gamma (T, \mathcal{O}_ T^*) \]
as before.
Example 39.5.2 (Roots of unity). Let $n \in \mathbf{N}$. Consider the functor which associates to any scheme $T$ the subgroup of $\Gamma (T, \mathcal{O}_ T^*)$ consisting of $n$th roots of unity. This is representable by the scheme
\[ \mu _ n = \mathop{\mathrm{Spec}}(\mathbf{Z}[x]/(x^ n - 1)). \]
The morphism giving the group structure is the morphism
\begin{eqnarray*} \mu _ n \times \mu _ n & \to & \mu _ n \\ \mathop{\mathrm{Spec}}( \mathbf{Z}[x]/(x^ n - 1) \otimes _{\mathbf{Z}} \mathbf{Z}[x]/(x^ n - 1)) & \to & \mathop{\mathrm{Spec}}(\mathbf{Z}[x]/(x^ n - 1)) \\ \mathbf{Z}[x]/(x^ n - 1) \otimes _{\mathbf{Z}} \mathbf{Z}[x]/(x^ n - 1) & \leftarrow & \mathbf{Z}[x]/(x^ n - 1) \\ x \otimes x & \leftarrow & x \end{eqnarray*}
Hence we see that $\mu _ n$ is a group scheme over $\mathbf{Z}$. For any scheme $S$ the base change $\mu _{n, S}$ is a group scheme over $S$ whose functor of points is
\[ T/S \longmapsto \mu _{n, S}(T) = \mu _ n(T) = \{ f \in \Gamma (T, \mathcal{O}_ T^*) \mid f^ n = 1\} \]
as before.
Example 39.5.3 (Additive group scheme). Consider the functor which associates to any scheme $T$ the group $\Gamma (T, \mathcal{O}_ T)$ of global sections of the structure sheaf. This is representable by the scheme
\[ \mathbf{G}_ a = \mathop{\mathrm{Spec}}(\mathbf{Z}[x]) \]
The morphism giving the group structure is the morphism
\begin{eqnarray*} \mathbf{G}_ a \times \mathbf{G}_ a & \to & \mathbf{G}_ a \\ \mathop{\mathrm{Spec}}(\mathbf{Z}[x] \otimes _{\mathbf{Z}} \mathbf{Z}[x]) & \to & \mathop{\mathrm{Spec}}(\mathbf{Z}[x]) \\ \mathbf{Z}[x] \otimes _{\mathbf{Z}} \mathbf{Z}[x] & \leftarrow & \mathbf{Z}[x] \\ x \otimes 1 + 1 \otimes x & \leftarrow & x \end{eqnarray*}
Hence we see that $\mathbf{G}_ a$ is a group scheme over $\mathbf{Z}$. For any scheme $S$ the base change $\mathbf{G}_{a, S}$ is a group scheme over $S$ whose functor of points is
\[ T/S \longmapsto \mathbf{G}_{a, S}(T) = \mathbf{G}_ a(T) = \Gamma (T, \mathcal{O}_ T) \]
as before.
Example 39.5.4 (General linear group scheme). Let $n \geq 1$. Consider the functor which associates to any scheme $T$ 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 scheme
\[ \text{GL}_ n = \mathop{\mathrm{Spec}}(\mathbf{Z}[\{ x_{ij}\} _{1 \leq i, j \leq n}][1/d]) \]
where $d = \det ((x_{ij}))$ with $(x_{ij})$ the $n \times n$ matrix with entry $x_{ij}$ in the $(i, j)$-spot. The morphism giving the group structure is the morphism
\begin{eqnarray*} \text{GL}_ n \times \text{GL}_ n & \to & \text{GL}_ n \\ \mathop{\mathrm{Spec}}(\mathbf{Z}[x_{ij}, 1/d] \otimes _{\mathbf{Z}} \mathbf{Z}[x_{ij}, 1/d]) & \to & \mathop{\mathrm{Spec}}(\mathbf{Z}[x_{ij}, 1/d]) \\ \mathbf{Z}[x_{ij}, 1/d] \otimes _{\mathbf{Z}} \mathbf{Z}[x_{ij}, 1/d] & \leftarrow & \mathbf{Z}[x_{ij}, 1/d] \\ \sum x_{ik} \otimes x_{kj} & \leftarrow & x_{ij} \end{eqnarray*}
Hence we see that $\text{GL}_ n$ is a group scheme over $\mathbf{Z}$. For any scheme $S$ the base change $\text{GL}_{n, S}$ is a group scheme over $S$ whose functor of points is
\[ T/S \longmapsto \text{GL}_{n, S}(T) = \text{GL}_ n(T) =\text{GL}_ n(\Gamma (T, \mathcal{O}_ T)) \]
as before.
Example 39.5.5. The determinant defines a morphism of group schemes
\[ \det : \text{GL}_ n \longrightarrow \mathbf{G}_ m \]
over $\mathbf{Z}$. By base change it gives a morphism of group schemes $\text{GL}_{n, S} \to \mathbf{G}_{m, S}$ over any base scheme $S$.
Example 39.5.6 (Constant group). Let $G$ be an abstract group. Consider the functor which associates to any scheme $T$ 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 scheme
\[ G_{\mathop{\mathrm{Spec}}(\mathbf{Z})} = \coprod \nolimits _{g \in G} \mathop{\mathrm{Spec}}(\mathbf{Z}). \]
The morphism giving the group structure is the morphism
\[ G_{\mathop{\mathrm{Spec}}(\mathbf{Z})} \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} G_{\mathop{\mathrm{Spec}}(\mathbf{Z})} \longrightarrow G_{\mathop{\mathrm{Spec}}(\mathbf{Z})} \]
which maps the component corresponding to the pair $(g, g')$ to the component corresponding to $gg'$. For any scheme $S$ the base change $G_ S$ is a group scheme over $S$ whose functor of points is
\[ T/S \longmapsto G_ S(T) = \{ f : T \to G \text{ locally constant}\} \]
as before.
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