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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