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 $\mathbf{A}^ n_ R$.

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