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

Let $A=k[x_1,x_2,x_3]$ and cover the punctured affine three-space $X$ by the three principal opens $D(x_i)$. Every finite intersection is affine, so the acyclic cover theorem identifies sheaf cohomology with the cohomology ofThe augmented complex has zeroth cohomology $A$ and first cohomology zero. Its second cohomology iswith $k$-basis represented by $x_1^{-a}{x}_2^{-b}{x}_3^{-c}$ for $a,b,c\ge 1$. There are no higher Čech terms. Hence