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 $S$.

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