Write $A=k[x,y]$, $X=\operatorname{Spec}A$, and $U=X\setminus Z$. Since $A$ is an <integral domain>, no nonzero global section is supported only at the origin, so
$$
H_Z^0(X,\mathcal O_X)=0.
$$
The <punctured affine plane> has the affine cover $U=D(x)\cup D(y)$. Its <Čech cochain complex> for the structure sheaf is
$$
0\longrightarrow A_x\oplus A_y
\xrightarrow{(a,b)\mapsto a-b}A_{xy}\longrightarrow0.
$$
Consequently
$$
H^0(U,\mathcal O_U)=A,
\qquad
H^1(U,\mathcal O_U)=\frac{A_{xy}}{A_x+A_y},
\qquad
H^i(U,\mathcal O_U)=0\quad(i\geq2).
$$

Because $X$ is affine, the higher <sheaf cohomology> of $\mathcal O_X$ vanishes. The <long exact sequence for local cohomology> therefore gives
$$
H_Z^1(X,\mathcal O_X)=0,
\qquad
H_Z^2(X,\mathcal O_X)
\cong\frac{k[x^{\pm1},y^{\pm1}]}{k[x^{\pm1},y]+k[x,y^{\pm1}]},
$$
and $H_Z^i(X,\mathcal O_X)=0$ for $i\geq3$. The nonzero group has the $k$-basis
$$
\{x^{-a}y^{-b}:a,b\geq1\}.
$$
This computes the <local cohomology of the affine plane supported at the origin> in every degree.

Solution

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

For $U=X\setminus Z$, local and ordinary <sheaf cohomology> fit into the natural sequence
$$
\cdots\to H^{i-1}(U,\mathcal F|_U)
\to H_Z^i(X,\mathcal F)
\to H^i(X,\mathcal F)
\to H^i(U,\mathcal F|_U)
\to H_Z^{i+1}(X,\mathcal F)\to\cdots.
$$

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 Incoming links: Long exact sequence for local cohomology