Affine n-space

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

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.

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.

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