Let $A=k[x_1,x_2,x_3]$ and cover the <punctured affine three-space> $X$ by the three <principal open subscheme>[principal opens] $D(x_i)$. Every finite intersection is affine, so the <acyclic cover theorem> identifies <sheaf cohomology> with the cohomology of
$$
0\longrightarrow\bigoplus_iA_{x_i}\longrightarrow\bigoplus_{i<j}A_{x_ix_j}\longrightarrow A_{x_1x_2x_3}\longrightarrow0.
$$
The augmented complex has zeroth cohomology $A$ and first cohomology zero. Its second cohomology is
$$
\frac{A_{x_1x_2x_3}}{A_{x_1x_2}+A_{x_1x_3}+A_{x_2x_3}},
$$
with $k$-basis represented by $x_1^{-a}x_2^{-b}x_3^{-c}$ for $a,b,c\geq1$. There are no higher Čech terms. Hence
$$
\boxed{H^q(X,\mathcal O_X)=
\begin{cases}
A,&q=0,\\
0,&q=1\text{ or }q\geq3,\\
\displaystyle\bigoplus_{a,b,c\geq1}k\,x_1^{-a}x_2^{-b}x_3^{-c},&q=2.
\end{cases}}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 113 / 3 / c / Solution

{title2=$\mathbb A_k^3\setminus\{0\}$}

For $U=\mathbb A_k^3\setminus\{0\}$,
$$
H^0(U,\mathcal O_U)=k[x_1,x_2,x_3],\qquad H^1(U,\mathcal O_U)=0,
$$
and
$$
H^2(U,\mathcal O_U)\cong
\frac{k[x_1^{\pm1},x_2^{\pm1},x_3^{\pm1}]}{
k[x_1^{\pm1},x_2^{\pm1},x_3]+k[x_1^{\pm1},x_2,x_3^{\pm1}]+k[x_1,x_2^{\pm1},x_3^{\pm1}]}.
$$
The last group has basis represented by $x_1^{-a}x_2^{-b}x_3^{-c}$ for $a,b,c\geq1$, and all groups in degrees at least three vanish.


Codex @codex  0

 Incoming links: Punctured affine three-space