Affine plane (${\mathbb{A}}_k^2$)

The affine plane over a field $k$ is the affine scheme $\mathrm{Spec}k[x,y]$.

The two-dimensional scheme has an affine space of rational points over its base field; its scheme structure also records non-rational points and local rings.

The points of the real affine plane as a scheme are its prime ideals: the zero ideal, the principal ideals of irreducible real polynomials, and its maximal ideals. The latter have residue field $\mathbb{R}$ or $\mathbb{C}$, by the Zariski lemma and the real closed field property. They correspond respectively to real coordinate pairs and conjugate pairs of nonreal complex coordinate pairs. Thus the scheme contains both nonclosed generic points and closed points absent from the real locus. Its topology is the Zariski topology, and its structure sheaf has sections $\mathbb{R}[x,y{]}_h$ on each principal open subscheme $D(h)$.

The punctured affine plane is obtained by deleting the closed point $(x,y)$ from $\mathrm{Spec}k[x,y]$. Its regular functions still form $k[x,y]$, whilewhich has basis represented by $x^{-a}{y}^{-b}$ for $a,b\ge 1$.

The punctured affine plane $U={\mathbb{A}}_k^2\setminus \{(0,0)\}$ is not an affine scheme. On its cover $D(x)\cup D(y)$, global sections are $k[x,y{]}_x\cap k[x,y{]}_y=k[x,y]$ inside the field of fractions. Coprimality of $x$ and $y$ proves this equality. If $U$ were affine, its inclusion into ${\mathbb{A}}_k^2$ would be an isomorphism by affine scheme reconstruction from global sections, contradicting the omitted point.