Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 113 (Algebraic Geometry)

For every homogeneous $s\in S_{+}$,Henceso $U$ is open in the Zariski topology.

If $\mathfrak{p}\in U$, then ${\phi }^{-1}(\mathfrak{p})$ is a homogeneous prime ideal that does not contain the irrelevant ideal of a graded ring $S_{+}$. Thusdefines a map $U\to \mathrm{Proj}S$. On every Standard affine open of Proj the graded homomorphism inducesand therefore an affine scheme morphismThese morphisms agree after localization on overlaps, so they glue to the required morphism of schemes $f:U\to \mathrm{Proj}S$.

Let $\mathfrak{p}\in \mathrm{Proj}T$. Some homogeneous $t\in T_{+}$ lies outside $\mathfrak{p}$. Choose $N$ with $N\mathrm{deg}t\ge d_0$. Since ${\phi }_{N\mathrm{deg}t}$ is surjective, $t^N=\phi (s)$ for some homogeneous $s\in S_{+}$; primality gives $t^N\notin \mathfrak{p}$. Thus $\mathfrak{p}\in U$, and $U=\mathrm{Proj}T$.

Fix a positive-degree homogeneous $s\in S$ and write $q=\phi (s)$. The mapis surjective: after multiplying the numerator and denominator of any degree-zero fraction by a sufficiently large power of $q$, its numerator has degree at least $d_0$ and therefore lifts through $\phi$. It is injective by the same device: if $a/s^n$ maps to zero, then $q^m\phi (a)=0$ for some $m$, and after increasing $m$ the injectivity of $\phi$ in degree $(m+n)\mathrm{deg}s$ gives $s^{m}a=0$. Hence the displayed map is a ring isomorphism.

The opens $D_{+}(q)$ obtained in this way cover $\mathrm{Proj}T$, and on every one of them $f$ is an isomorphism onto $D_{+}(s)$. The inverses agree on overlaps, so $f$ is an isomorphism of schemes. This proves the Invariance of Proj under an eventual graded isomorphism.

PutSinceis an integral domain, $(x,z)$ is a prime ideal. Moreover $\dim R=3$ and $\dim R/(x,z)=2$, so its height of an ideal is one. Therefore $D=V(x,z)$ is a prime Weil divisor.

Localizing at $x$ eliminates $y$ and givesa unique factorization domain. The Nagata theorem for divisor class groups says that $\mathrm{Cl}(R)$ is generated by the height-one primes containing $x$. Sincethese are $D=V(x,z)$ and $E=V(x,w)$. Both occur with multiplicity one in the principal divisorThe units of $R_x\cong k[z,w][x,x^{-1}]$ are exactly $c{x}^n$ with $c\in k^{\ast }$ and $n\in \mathbb{Z}$. Consequently the only relation supplied by localization is $[D]+[E]=0$, andThis is the divisor class group of the three-dimensional affine quadric cone.

Cover $Y$ byThese sets cover because a point of $D_Y$ has $(y,w)\ne (0,0)$. On $U_y$, the relation $x=zw/y$ shows that $D_Y$ has local equation $z$; on $U_w$, the relation $z=xy/w$ gives local equation $x$; and on $U_0$ its local equation is $1$. Thus $D_Y$ is a Cartier divisor.

For the line bundle associated to a divisor ${\mathcal{O}}_Y(D_Y)$, choose local framesThe transition functions areon the corresponding overlaps. Every displayed ratio is a unit in a ring in the corresponding ring of regular functions, and the Čech cocycle condition follows immediately.

The two pairs of regular functionsdefine maps to the projective line on the loci where their respective coordinates do not vanish simultaneously. Those loci cover $Y$, since simultaneous failure would force $x=y=z=w=0$. On their overlap the equation $xy=zw$ says that the two projective points are equal. They therefore glue to a morphism

Let $[A:B]$ be the homogeneous coordinates on ${\mathbb{P}}_k^1$. The pullback of the hyperplane divisor $V(A)$ has local equation $x$ on the first chart and $z$ on the second. It is therefore exactly $D_Y=V(x,z)$. Compatibility of the pullback of a sheaf of modules with the line bundle associated to a divisor givesThis is the ruling morphism of the punctured three-dimensional affine quadric cone associated with $D_Y$.

The direct image sheaf is determined on the basis consisting of principal open subschemes $D(b)\subseteq \mathrm{Spec}B$. Their inverse images areand henceOn the other hand, localization of the restricted-scalar $B$-module givesThese isomorphisms commute with restriction to smaller principal opens, so they define the equality of quasi-coherent sheaves

Because $X$ is a Noetherian scheme, choose a finite affine cover $X={\bigcup }_i{U}_i$. Every intersection $U_i\cap U_j$ is quasi-compact, so choose a finite affine cover $U_i\cap U_j={\bigcup }_k{U}_{ijk}$. If $j_i:U_i\hookrightarrow X$ and $j_{ijk}:U_{ijk}\hookrightarrow X$ are the inclusions, the sheaf axiom gives an exact sequencewhere the last arrow is the difference of the two restrictions to each overlap chart.

Applying the left-exact direct image sheaf functor identifies $f_{\ast }\mathcal{F}$ with the kernel ofEvery map $U_i\to Y$ and $U_{ijk}\to Y$ is a morphism between affine schemes. Part (a) shows that all sheaves in the two finite sums are quasi-coherent sheaves. Kernels of morphisms of quasi-coherent sheaves on the affine scheme $Y$ are quasi-coherent, because they correspond to kernels of module homomorphisms. Therefore $f_{\ast }\mathcal{F}$ is quasi-coherent. This is the quasi-coherence of direct image under a quasi-compact quasi-separated morphism in the present Noetherian case.

If the germ of a sheaf section $s_p$ is zero, then by the definition of a stalk of a sheaf there is a neighbourhood $V$ of $p$ on which $s{|}_V=0$. Every point of $V$ also has zero germ. Thus the complement of $\mathrm{Supp}s$ is open, so the support of a sheaf section is closed.

The kernel of the restriction mapconsists exactly of sections whose germs vanish outside $Z$, namely ${\Gamma }_Z(X,\mathcal{F})$. This proves exactness. If $\mathcal{F}$ is a flasque sheaf, the restriction map is surjective by definition.

Now let $0\to {\mathcal{F}}_1\to {\mathcal{F}}_2\to {\mathcal{F}}_3\to 0$ be exact. The global section functor is left exact, so a section of ${\Gamma }_Z(X,{\mathcal{F}}_2)$ mapping to zero lifts uniquely to a global section of ${\mathcal{F}}_1$. Its germs outside $Z$ vanish because ${\mathcal{F}}_1\to {\mathcal{F}}_2$ is injective at each stalk of a sheaf. It therefore lies in ${\Gamma }_Z(X,{\mathcal{F}}_1)$, proving exactness of the supported-section sequence.

Suppose in addition that ${\mathcal{F}}_1$ is flasque and take $s_3\in {\Gamma }_Z(X,{\mathcal{F}}_3)$. Surjectivity on global sections gives a lift $s_2\in \Gamma (X,{\mathcal{F}}_2)$. On $X\setminus Z$, its restriction comes from some $t_1\in \Gamma (X\setminus Z,{\mathcal{F}}_1)$ by left exactness. Extend $t_1$ to ${\tilde{t}}_1\in \Gamma (X,{\mathcal{F}}_1)$ using flasqueness. Then $s_2-{\tilde{t}}_1$ maps to $s_3$ and vanishes outside $Z$, proving surjectivity on the right.

Write $A=k[x,y]$, $X=\mathrm{Spec}A$, and $U=X\setminus Z$. Since $A$ is an integral domain, no nonzero global section is supported only at the origin, soThe punctured affine plane has the affine cover $U=D(x)\cup D(y)$. Its Čech cochain complex for the structure sheaf isConsequently

Because $X$ is affine, the higher sheaf cohomology of ${\mathcal{O}}_X$ vanishes. The long exact sequence for local cohomology therefore givesand $H_Z^i(X,{\mathcal{O}}_X)=0$ for $i\ge 3$. The nonzero group has the $k$-basisThis computes the local cohomology of the affine plane supported at the origin in every degree.