Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 113 (Algebraic Geometry)

A morphism of schemes is proper when it is a finite type morphism, a separated morphism, and a universally closed morphism. The valuative criterion for properness says, under the usual finite-type and Noetherian hypotheses, that $f:X\to Y$ is proper exactly when every commutative squarewith $R$ a discrete valuation ring and $K=\mathrm{Frac}R$ has a unique diagonal lift $\mathrm{Spec}R\to X$.

For ${\mathbb{P}}_k^2\to \mathrm{Spec}k$, a $K$-point is $[x_0:x_1:x_2]$ with not all $x_i$ zero. If $\pi$ is a uniformizer, multiply all coordinates by one power of $\pi$ so that ${\min }_{i}v(x_i)=0$. The resulting coordinates lie in $R$ and at least one is a unit, so they define an $R$-point extending the given $K$-point. If two extensions exist, on a chart where one coordinate is a unit their affine coordinate ratios agree in $K$ and therefore in the integral domain $R$; hence the extensions agree. This verifies existence and uniqueness directly.

Take the projective line with a doubled point over $\mathbb{C}$, obtained by gluing two copies of ${\mathbb{P}}_{\mathbb{C}}^1$ along the complement of one point. After every base change of a morphism of schemes, a closed subset has closed image from each of the two projective-line charts, so its total image, the union of those two images, is closed. The structure morphism is therefore universally closed. The two doubled points have no disjoint neighborhoods, so the scheme is not separated and hence is not proper.

Because ${\mathcal{A}}_{\bullet }$ is coherent and locally generated by ${\mathcal{A}}_1$, there is locally a finite-rank coherent sheaf $\mathcal{E}$ and a surjection of graded algebrasThe Relative Proj construction turns this into a closed immersionConsequently $Y\to X$ is a projective morphism and therefore a proper morphism. Since $X\to \mathrm{Spec}k$ is proper and proper morphisms are closed under composition, $Y\to \mathrm{Spec}k$ is proper.

The coincidence locus of two scheme morphisms is the fibre productThe diagonal morphism ${\Delta }_Y$ is a locally closed immersion, and this property is stable under base change of a morphism of schemes, so $E\hookrightarrow X$ is locally closed. The universal property of a fibre product says that a morphism $T\to X$ factors through $E$ exactly when $f{|}_T=g{|}_T$. Thus $E$ is the largest locally closed subscheme on which they coincide. If $Y$ is separated, ${\Delta }_Y$ is a closed immersion, and its base change $E\hookrightarrow X$ is closed.

A Weil divisor on $X$ is a finite formal sumover integral codimension-one closed subschemes $Z$. Since $X$ is regular in codimension one, the local ring at the generic point of each $Z$ is a discrete valuation ring. A nonzero rational function $f\in k(X{)}^{\times }$ therefore defines the principal divisor $\mathrm{div}(f)={\sum }_Z{v}_Z(f)[Z]$. The divisor class group is

The scheme ${\mathbb{P}}_k^3$ is Noetherian, integral, separated, and regular. Every open subscheme inherits these properties, so $U={\mathbb{P}}_k^3\setminus C$ satisfies $(\star )$. Because $C$ has dimension one, it has codimension two in ${\mathbb{P}}_k^3$ and contains no prime Weil divisor. The localization sequence for the divisor class group therefore makes restriction an isomorphismThe hyperplane divisor generates the class group of projective space, so

The assertion is false. For $n\ge 2$, take the affine hypersurfaceIt is an integral scheme, and its only possible singular point is the origin, which has codimension two. Hence it is regular in codimension one; as a hypersurface it satisfies Serre's condition $(S_2)$, so the Serre criterion for normality also makes it a normal scheme. The Divisor class group of an A-type surface singularity isgenerated by $V(x,z)$. Thus a closed affine subscheme satisfying $(\star )$ can have nonzero torsion in its class group.

The standard affine charts of ${\mathbb{P}}^3$ are copies of ${\mathbb{A}}^3$, and the product charts of ${\mathbb{P}}^1\times {\mathbb{P}}^1$ are copies of ${\mathbb{A}}^2$. Their local rings are localizations of polynomial rings over $k$ and hence are regular local rings. Both schemes are therefore regular.

On a regular integral scheme every Weil divisor is Cartier, so the divisor class group is naturally the Picard group. Pullback of line bundles along the Segre embedding therefore definesThe two groups areThe Segre coordinates are bihomogeneous of bidegree $(1,1)$, so ${\iota }^{\ast }{\mathcal{O}}_{{\mathbb{P}}^3}(1)=\mathcal{O}(1,1)$. In these bases the map is $m\mapsto (m,m)$ and

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