Affine n-space - Knowlpedia

Let $k$ be a field and $n\ge 0$ . The affine $n$ -space over $k$ is the affine scheme

\mathbb A_k^n:=\operatorname{Spec}k[x_1,\ldots,x_n].

The variables $x_1,\ldots,x_n$ are coordinate functions. A $k$ -valued point $(a_1,\ldots,a_n)\in k^n$ determines the maximal ideal $(x_1-a_1,\ldots,x_n-a_n)$ . If $k$ is algebraically closed, every closed point arises this way, though the spectrum also contains nonclosed points corresponding to irreducible subvarieties.

The case $n=1$ is the affine line. The standard open charts of projective $n$ -space are each isomorphic to $\mathbb A_k^n$ .

Examples

For example, in $\mathbb A_k^2$ , the prime ideal $(y-x^2)\subseteq k[x,y]$ represents the whole parabola, not a single coordinate pair. This illustrates how scheme points encode both ordinary points and generic behavior of algebraic loci.