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 $\mathbf{A}^ n_ R$.

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

\[ \mathbf{A}^ n_ S = \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{O}_ S[T_1, \ldots , T_ n]) \]

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

\[ \mathbf{A}^ n_ S = S \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} \mathbf{A}^ n_{\mathbf{Z}}. \]

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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