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.

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