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

Write $A=k[x,y]$. If $\dim Z=1$, its prime ideal has height one and is generated by an irreducible polynomial $f$, because $A$ is a unique factorization domain. Then $U=D(f)$ is a principal open subscheme andIf $\dim Z=0$, its complement has codimension two. Since ${\mathbb{A}}_k^2$ is normal, regular functions extend across that codimension-two subset, givingThis includes the calculation for the punctured affine plane. If $Z={\mathbb{A}}_k^2$ has dimension two, then $U$ is empty and its ring of sections is the zero ring.

For an open cover $\mathcal{U}=(U_i)$ and a sheaf $\mathcal{F}$, the Čech cochain complex iswith the alternating sum of restriction maps as differential. Its cohomology is the Čech cohomology ${\check{H}}^p(\mathcal{U},\mathcal{F})$.

Cover ${\mathbb{P}}_k^1$ by $U_0=D_{+}(x_0)$ and $U_1=D_{+}(x_1)$, and write $t=x_1/x_0$. This is an acyclic affine cover for the twisting sheaf on projective space $\mathcal{O}(-1)$. In compatible trivializations, its degree-one Čech quotient isThe first summand contains every nonnegative power of $t$ and the second every negative power, so

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

Multiplication by the homogeneous equation $f_d$ and restriction to its zero scheme give the structure-sheaf sequence of a hypersurfaceBy cohomology under a closed immersion, $H^q({\mathbb{P}}^3,i_{\ast }{\mathcal{O}}_X)\cong H^q(X,{\mathcal{O}}_X)$. The associated long exact sequence in cohomology containsBoth outer groups vanish by the cohomology of twisting sheaves on projective space, because they are intermediate cohomology groups on ${\mathbb{P}}^3$. Therefore