As an application of the relative spectrum we define affine n-space over a base scheme S as follows. For any integer n \geq 0 we can consider the quasi-coherent sheaf of \mathcal{O}_ S-algebras \mathcal{O}_ S[T_1, \ldots , T_ n]. It is quasi-coherent because as a sheaf of \mathcal{O}_ S-modules it is just the direct sum of copies of \mathcal{O}_ S indexed by multi-indices.
Definition 27.5.1. Let S be a scheme and n \geq 0. The scheme
over S is called affine n-space over S. If S = \mathop{\mathrm{Spec}}(R) is affine then we also call this affine n-space over R and we denote it \mathbf{A}^ n_ R.
Note that \mathbf{A}^ n_ R = \mathop{\mathrm{Spec}}(R[T_1, \ldots , T_ n]). For any morphism g : S' \to S of schemes we have g^*\mathcal{O}_ S[T_1, \ldots , T_ n] = \mathcal{O}_{S'}[T_1, \ldots , T_ n] and hence \mathbf{A}^ n_{S'} = S' \times _ S \mathbf{A}^ n_ S is the base change. Therefore an alternative definition of affine n-space is the formula
Also, a morphism from an S-scheme f : X \to S to \mathbf{A}^ n_ S is given by a homomorphism of \mathcal{O}_ S-algebras \mathcal{O}_ S[T_1, \ldots , T_ n] \to f_*\mathcal{O}_ X. This is clearly the same thing as giving the images of the T_ i. In other words, a morphism from X to \mathbf{A}^ n_ S over S is the same as giving n elements h_1, \ldots , h_ n \in \Gamma (X, \mathcal{O}_ X).
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