Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 113 / 4 / c / Solution

Write $A=k[x,y]$, $X=\mathrm{Spec}A$, and $U=X\setminus Z$. Since $A$ is an integral domain, no nonzero global section is supported only at the origin, soThe punctured affine plane has the affine cover $U=D(x)\cup D(y)$. Its Čech cochain complex for the structure sheaf isConsequently

Because $X$ is affine, the higher sheaf cohomology of ${\mathcal{O}}_X$ vanishes. The long exact sequence for local cohomology therefore givesand $H_Z^i(X,{\mathcal{O}}_X)=0$ for $i\ge 3$. The nonzero group has the $k$-basisThis computes the local cohomology of the affine plane supported at the origin in every degree.