Local cohomology of the affine plane supported at the origin
= Local cohomology of the affine plane supported at the origin
{title2=$H_{(x,y)}^i(k[x,y])$}
For $X=\operatorname{Spec}k[x,y]$ and $Z=V(x,y)$, the only nonzero local cohomology group of the structure sheaf is
$$
H_Z^2(X,\mathcal O_X)
\cong\frac{k[x^{\pm1},y^{\pm1}]}{k[x^{\pm1},y]+k[x,y^{\pm1}]}.
$$
It has basis $x^{-a}y^{-b}$ for $a,b\geq1$. The two-standard-open <Čech cochain complex> of the <punctured affine plane> proves the formula.