Affine three-space (${\mathbb{A}}_k^3$)

Affine three-space is $\mathrm{Spec}k[x_1,x_2,x_3]$.

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

For $U={\mathbb{A}}_k^3\setminus \{0\}$,andThe last group has basis represented by $x_1^{-a}{x}_2^{-b}{x}_3^{-c}$ for $a,b,c\ge 1$, and all groups in degrees at least three vanish.