Punctured affine plane
= Punctured affine plane
{title2=$\mathbb A_k^2\setminus\{0\}$}
The punctured affine plane is obtained by deleting the closed point $(x,y)$ from $\operatorname{Spec}k[x,y]$. Its regular functions still form $k[x,y]$, while
$$
H^1(\mathbb A_k^2\setminus\{0\},\mathcal O)\cong k[x^{\pm1},y^{\pm1}]/\bigl(k[x^{\pm1},y]+k[x,y^{\pm1}]\bigr),
$$
which has basis represented by $x^{-a}y^{-b}$ for $a,b\geq1$.