Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 13 (Algebraic Geometry)

For a presheaf $\mathcal{F}$, first form its stalks ${\mathcal{F}}_x={\underrightarrow{\lim }}_{x\in U}\mathcal{F}(U)$. Define ${\mathcal{F}}^{+}(U)$ to consist of families $(s_x{)}_{x\in U}$, with $s_x\in {\mathcal{F}}_x$, which are locally represented by sections of $\mathcal{F}$: every $x\in U$ has a neighbourhood $V\subseteq U$ and $t\in \mathcal{F}(V)$ with $s_y=t_y$ for all $y\in V$. Addition is pointwise and restriction discards the components outside the smaller open set. The local representability condition is itself local, so compatible such families on an open cover glue uniquely. Thus ${\mathcal{F}}^{+}$ is a sheaf of abelian groups. This is sheafification by locally representable germs.

The canonical presheaf morphism isThe universal property of sheafification says that, for any sheaf $\mathcal{H}$, composition with $\theta$ gives a natural bijectionIndeed, locally representing a family by $t$ defines its image locally by the image of $t$ in $\mathcal{H}$. Equal germs give locally equal images, and the sheaf gluing axiom gives the unique global image. If $\mathcal{F}$ is already a sheaf, a section with all germs zero is zero, while every locally represented family glues to an actual section. Hence ${\theta }_U$ is both injective and surjective for every $U$, so sheafification leaves a sheaf unchanged.

For a continuous map $f:X\to Y$, the direct image sheaf isAn open cover pulls back to an open cover, so its sheaf axioms follow directly from those of $\mathcal{F}$. To construct the inverse image sheaf, first setRestriction uses the inclusion of these neighbourhood systems when $V$ shrinks. The use of sheafification is important: $P$ need not itself be a sheaf. Continuity and the stalk construction give $(f^{-1}\mathcal{G}{)}_x\cong {\mathcal{G}}_{f(x)}$.

Given an f-morphism of sheaves $\varphi$, define its map on stalks byIf two representatives have the same germ at $f(x)$, they agree on a neighbourhood there. Restriction compatibility makes their images agree on its inverse image, a neighbourhood of $x$, proving well-definedness. The map is a group homomorphism. We index it by $x$, since different points over the same $f(x)$ have different target stalks.

For $s\in \mathcal{G}(U)$, its class in $P(f^{-1}U)$ and then in its sheafification defines the canonical f-morphism of sheaves $\theta (U)$. Its germ at $x$ is simply $s_{f(x)}$. To factor any $\varphi$, take a section $t\in (f^{-1}\mathcal{G})(V)$. Locally on an open cover $V=\bigcup V_a$, it comes from a section $s_a\in \mathcal{G}(U_a)$ with $V_a\subseteq f^{-1}{U}_a$. Define the prospective image on $V_a$ byOn an overlap, the representatives have the same inverse-image germs, so their images have the same germs by the maps ${\varphi }_x$. Two sheaf sections with equal germs everywhere are equal. The local images therefore glue uniquely, independently of every representative and cover choice. This construction is additive and commutes with restrictions, giving a sheaf morphism $\psi :f^{-1}\mathcal{G}\to \mathcal{F}$ withConversely any factorization must have the prescribed image on these locally generating sections, proving uniqueness. This is the universal property of an inverse image sheaf.

An f-morphism of sheaves is exactly a sheaf morphism $\mathcal{G}\to f_{\ast }\mathcal{F}$. Thus the two constructions give the inverse-image direct-image adjunctionNaturality follows from composing the local representatives and their images with morphisms in either sheaf variable. This is an adjunction for sheaves of abelian groups; it is not the tensor-adjusted pullback of modules on a ringed space.

For an affine variety $V\subseteq {\mathbb{A}}_k^N$, one construction defines ${\mathcal{O}}_V(U)$ as the functions $U\to k$ which locally have the form $p/q$, where $p,q$ are elements of the coordinate ring $k[V]$ and $q$ does not vanish on the neighbourhood in question. Restrictions are restrictions of functions, and locality gives the sheaf axioms. In particularThe second formula identifies its local rings.

In the classical convention, an algebraic variety over $k$ is an irreducible ringed space with a sheaf of $k$-valued functions, admitting a finite open cover by spaces isomorphic to affine varieties, and satisfying the separated variety condition that its diagonal is closed in $X\times X$. Its topology is Noetherian. One can instead allow reducible reduced varieties; the arguments below still work with the denominator-clearing version of the localization argument. A morphism of varieties is a continuous map whose pullback takes local regular functions to regular functions, equivalently a morphism of locally ringed $k$-spaces. The maps on local rings are local because a function nonzero at the image point stays nonzero at the source point.

An affine variety supplies its own finite affine cover, is Noetherian, and is separated: its diagonal in $V\times V$ is cut out by the differences of corresponding coordinates. A regular map between affine varieties pulls coordinate functions back to regular functions. It is continuous because the inverse image of any polynomial zero set is the zero set of the pulled-back regular functions; on affine charts such zero sets are closed. Pulling back a locally represented fraction $p/q$ gives a regular fraction wherever its denominator is nonzero. Hence a regular map is a morphism of varieties in this definition.

For a morphism $\varphi :X\to Y$ which is an isomorphism over each member $U_i$ of an open cover of $Y$, every fibre contains exactly one point. Thus $\varphi$ is bijective. Its inverse is continuous and regular on every $U_i$, since there it is the given inverse of the local isomorphism. These inverses agree on overlaps, being inverses of the same map. They glue to a global inverse morphism of varieties. Therefore being an isomorphism is local on the target.

Let $A=\Gamma (X,{\mathcal{O}}_X)$, and let $Y$ be affine with coordinate ring $B$. Given a $k$-algebra homomorphism $\alpha :B\to A$, choose a presentation $B=k[y_1,\dots ,y_N]/I$. The maplands in $Y$ because all the equations in $I$ become zero functions. Its coordinates are global regular functions, so it is a morphism on every affine chart of $X$, and therefore globally. On a target neighbourhood where a fraction $b/c$ is defined its pullback is $\alpha (b)/\alpha (c)$, proving that the induced map on global sections is exactly $\alpha$. This also proves independence of the chosen generators and uniqueness. This is the affine-target adjunction for varieties.

For $f\in A$, the natural map $A_f\to \Gamma (X_f,{\mathcal{O}}_X)$ sends $a/f^N$ to that regular function. In the irreducible convention, if $f\ne 0$, $X_f$ is dense, so this map is injective: a global regular function vanishing there vanishes everywhere. Cover $X$ by finitely many affine charts $U_j$. A section $s$ on $X_f$ restricts on $U_j\cap X_f=D_{U_j}(f{|}_{U_j})$ to an element of $\Gamma (U_j,{\mathcal{O}}_X{)}_{f{|}_{U_j}}$. A common power $f^N$ clears all these finitely many denominators. The resulting regular sections on $U_j$ agree on the dense principal open in each overlap, hence agree there altogether and glue to a global section $a$. Thus $s=a/f^N$. If $f=0$, both sides are the zero ring of sections on the empty open. ConsequentlyThis is localization of global sections on a principal open. For reduced reducible varieties, equality on a principal open instead means that a sufficiently large power of $f$ annihilates the difference; finitely many charts and overlap refinements allow one common extra power. The same denominator-clearing proof then gives the stated localization isomorphism without a density assumption.

Now suppose ${\sum }_i{g}_i{f}_i=1$ in $A$ and every $X_{f_i}$ is affine. Each $A_{f_i}=\Gamma (X_{f_i},{\mathcal{O}}_X)$ is a finitely generated $k$-algebra. Select finitely many generators, writing them as $a_{ij}/f_i^{N_{ij}}$. Let $B\subseteq A$ be the $k$-subalgebra generated by all $f_i$, all $g_i$, and all $a_{ij}$. It is finitely generated, andThe inclusion from left to right is immediate, while the chosen generators give the reverse inclusion. This construction does not assume that $A$ was finitely generated in advance.

The reduced algebra $B$ is the coordinate ring of an affine variety $Y$ (irreducible when $X$ is). The map $B\hookrightarrow A$ produces $X\to Y$ by the preceding construction. The sets $D_Y(f_i)$ cover $Y$, since their functions generate the unit ideal already in $B$. Their inverse images are $X_{f_i}$, and on them the map is the isomorphism corresponding to $B_{f_i}=A_{f_i}$. The target-local argument above now provesThis is affineness from a unit-ideal principal affine cover. The printed sets are $X_{f_i}$; the missing index in the TeX aid is not a different hypothesis.

A sheaf of modules $\mathcal{F}$ over ${\mathcal{O}}_X$ is quasi-coherent when locally it admits a presentationwith arbitrary indexing sets $I,J$. On an affine chart, sheafifying the corresponding module presentation and using exactness of localization identifies its cokernel with a sheaf $\tilde{N}$. Thus one may equivalently require that locally on affine charts $\mathcal{F}$ is associated with a module. This does not impose finite generation, which belongs to the stronger coherent condition.

On the affine variety $X$ with coordinate ring $A$, put $M=\Gamma (X,\mathcal{F})$. Refine local module charts to a finite principal open cover $X={\bigcup }_{j}D(g_j)$ such that $\mathcal{F}{|}_{D(g_j)}\cong {\tilde{N}}_j$ for an $A_{g_j}$-module $N_j$. This is possible because principal opens form an open basis and $X$ is quasi-compact. On the overlap $D(g_j{g}_k)$ the sections are the corresponding further localization of $N_j$ or $N_k$. The sheaf gluing axiom gives an equalizerwhere the last map is the difference of restrictions. For any $f\in A$, localize this sequence. Exactness of localization and commutation with finite products give exactly the equalizer for the cover $D(f)={\bigcup }_{j}D(f{g}_j)$. HenceThe isomorphisms are canonical and commute with all basic-open restrictions. Since the sections of $\tilde{M}$ on these opens are $M_f$, they identify the two sheaves:Conversely $\tilde{M}$ is quasi-coherent for every $A$-module $M$, since a free presentation of $M$ gives the required sheaf presentation. Also $\Gamma (X,\tilde{M})=M$, and module homomorphisms sheafify, while a sheaf morphism is determined on every $D(f)$ by the localization of its map on global sections. This proves the affine module-sheaf equivalence.

For a short exact sequence of quasi-coherent sheaves, the sequence of their stalks is exact. Under this equivalence the stalk at $x$ is the localization of the global-section module at ${\mathfrak{m}}_x$. Exactness of module sequences can be checked at all maximal ideals, by localization detects zero elements applied to their homology, so the global-section modules also form a short exact sequence. Therefore global sections are exact on quasi-coherent sheaves over an affine variety. This is stronger than the left exactness of the global section functor on arbitrary sheaves, and follows from localization, not from assuming the cohomology vanishing still to be proved.

For an open inclusion $j:U\hookrightarrow X$, the requested sheaf isRestriction gives $\mathcal{F}\to {}_U\mathcal{F}$. This direct image from an open restriction is not extension by zero: its stalks outside $U$ may be nonzero.

The locally vanishing principle for sheaf cohomology says that a class $\xi \in H^i(X,\mathcal{F})$, $i>0$, is killed on a suitable neighbourhood of every point. Under the stated hypothesis that basis opens and their finite intersections have zero cohomology in degrees $1,\dots ,i-1$, these neighbourhoods can be chosen in that basis so that the image of $\xi$ underis zero. In particular its restriction in $H^i(U,\mathcal{F}{|}_U)$ is zero. This is a statement about individual classes; it does not assert that every locally vanishing class is already globally zero.

Here is a noncircular proof of vanishing of quasi-coherent cohomology on an affine scheme, applied to the variety. Induct on $i>0$, simultaneously for every affine variety and quasi-coherent sheaf. Assume all lower positive degrees vanish. The principal-open basis is closed under finite intersections, so it satisfies the principle's hypothesis. Given $\xi \in H^i(X,\mathcal{F})$, choose a finite principal cover $\mathcal{U}=(D(f_j))$ on which it restricts to zero.

Applying the Čech cochain complex to a flasque resolution gives the Čech lifting below the first possible local cohomology degree comparison segment, under lower-degree vanishing on all intersectionsFor completeness, the double complex has terms ${\check{C}}^p(\mathcal{U},{\mathcal{I}}^q)$. Its augmented rows are exact because each ${\mathcal{I}}^q$ is flasque, so its total cohomology is the cohomology of global sections of the resolution. The vertical cohomology on intersections in degrees $0<q<i$ is zero. Equivalently, start with local primitives of a cocycle representing $\xi$, take their differences on pairwise overlaps, and solve successively for primitives of those differences in degrees $i-1,i-2,\dots ,1$. The last difference is a Čech $i$-cocycle with values in $\mathcal{F}$. This identifies the kernel of the restriction map with the displayed Čech group. No vanishing in degree $i$ on the intersections has been assumed.

It remains a module calculation. Write $\mathcal{F}=\tilde{M}$. The augmented Čech complex isSince the $D(f_j)$ cover $X$, the $f_j$ generate the unit ideal. This complex is exact. To see it algebraically, use alternating cochains, with repeated indices giving zero. For a cocycle $c$, choose $N$ large enough to clear every $f_j$-denominator in $f_j^N{c}_{jI}$, as well as the finitely many cocycle relations; vanishing criterion in a module localization allows a further power to clear relations which initially hold only after localization. Since the ideal $(f_j^N)$ is still the unit ideal, choose $a_j$ with ${\sum }_j{a}_j{f}_j^N=1$. Defineusing those cleared representatives in $M_{f_I}$. The cocycle identity gives $\delta b=({\sum }_j{a}_j{f}_j^N)c=c$. The same argument with the augmentation gives gluing and uniqueness in degree zero. This is exactness of the unit-ideal localization Čech complex.

Thus ${\check{H}}^i(\mathcal{U},\mathcal{F})=0$, so the class $\xi$, whose restrictions were zero, is zero. The induction starts at $i=1$, when the lower-degree condition is empty. We have proved

Order the index set of the open cover $\mathcal{U}=(U_a)$. The Čech cochain groups areand their differential is the alternating sum of restrictions:Terms obtained by deleting two indices cancel in pairs, so ${\delta }^2=0$. The Čech cohomology is ${\check{H}}^p=\ker \delta /\mathrm{im}\delta$, with zero incoming differential in degree zero. Its zeroth group is the group of global sections by sheaf gluing.

The acyclic cover theorem applies if every nonempty finite intersection is acyclic for $\mathcal{F}$. For a quasi-coherent sheaf on a variety it is sufficient that every such intersection is affine; on a separated variety, any affine open cover has this property. On ${\mathbb{P}}^n$, take the $n+1$ standard charts $U_i=D_{+}(X_i)$. Every intersection is a principal open in an affine chart, hence affine. The Čech complex has no terms above degree $n$, provingThis is the cohomological dimension bound from an affine cover and requires only quasi-coherence, not finite generation.

For the remaining projective calculations assume $n\ge 1$. There is a necessary zero-dimensional exception to the negative-twist assertion: ${\mathbb{P}}^0$ is a point, every twist is trivial there, and $H^0({\mathbb{P}}^0,\mathcal{O}(m))\cong k$ even for $m<0$.

Put $S=k[X_0,\dots ,X_n]$ with its usual grading. The twisting sheaf on projective space is the sheaf associated with the shifted graded module $S(m)$; on an intersection $U_I={\bigcap }_{i\in I}{U}_i$ its sections areOn $U_i$ a generator is $e_i=X_i^m$, with transition $e_j=(X_j/X_i{)}^m{e}_i$. These are regular units on overlaps and satisfy the cocycle relation, so they glue an invertible sheaf for every integer $m$, including negative $m$.

A global section is a compatible family of these homogeneous fractions, hence a single element of degree $m$ in ${\bigcap }_i{S}_{X_i}\subseteq \mathrm{Frac}S$. For $n\ge 1$, $X_0$ and $X_1$ are relatively prime in the unique factorization domain $S$, so $S_{X_0}\cap S_{X_1}=S$; therefore the full intersection is $S$. It follows thatFor $m\ge 0$ its dimension is $\left(\genfrac{}{}{0ex}{}{n+m}{n}\right)$. The proof uses all-chart compatibility; regularity on one chart alone would permit poles on its complement.

If some $d_i=0$, the ideal $(X_i^{d_i})$ contains $1$, so the requested containment is immediate. Otherwise every $d_i\ge 1$. A monomial $X_0^{a_0}\cdots X_n^{a_n}$ of degree ${\sum }_i{d}_i$ must have $a_i\ge d_i$ for some $i$: if not, its total degree would be at most ${\sum }_i(d_i-1)<{\sum }_i{d}_i$. Thus every degree-${\sum }_i{d}_i$ monomial, and hence every homogeneous polynomial of that degree, belongs to $(X_0^{d_0},\dots ,X_n^{d_n})$. This is monomial containment in an ideal of coordinate powers.

For $n>0$, a top-degree Čech cochain for $\mathcal{O}$ is a degree-zero Laurent polynomial on the full intersection. Write it with a common denominator asThe containment just proved gives $P={\sum }_i{Q}_i{X}_i^{d_i}$, with homogeneous $Q_i$ of degree ${\sum }_j{d}_j-d_i$. ConsequentlyThe $i$th term is regular on the intersection omitting $U_i$, and has degree zero. Give it the sign $(-1{)}^i$ in that component of the preceding Čech cochain. Its coboundary is the original fraction. Every top cochain is thus a coboundary, andThis is top Čech cohomology from missing-denominator monomials.

To obtain the negative-twist bound by induction on dimension, let $H\cong {\mathbb{P}}^{n-1}$ be a hyperplane, with inclusion $j$. Its equation gives the hyperplane exact sequence for twisting sheavesThe associated long exact sequence in sheaf cohomology and sheaf cohomology under a closed inclusion giveFor $n=1$, restriction $H^0({\mathbb{P}}^1,\mathcal{O})\to H^0(H,{\mathcal{O}}_H)$ is the surjection $k\to k$. Exactness and the already proved $H^1({\mathbb{P}}^1,\mathcal{O})=0$ give $H^1({\mathbb{P}}^1,\mathcal{O}(-1))=0$. The displayed surjections then give the same vanishing for every $m\ge 0$. This handles the point hyperplane without making a false negative-twist claim on ${\mathbb{P}}^0$.

For $n\ge 2$, the induction hypothesis in dimension $n-1$ gives $H^{n-1}(H,{\mathcal{O}}_H(m))=0$ whenever $m\ge -(n-1)$. Therefore consecutive top-degree groups are isomorphic for $m\ge 1-n$. Starting with $H^n({\mathbb{P}}^n,\mathcal{O})=0$ and stepping down through $m=0,-1,\dots ,1-n$ reaches $H^n({\mathbb{P}}^n,\mathcal{O}(-n))=0$. Stepping upward proves all positive twists too. We concludeThis is top-twist vanishing by hyperplane induction.

Finally, on $U_0$ write $x_a=X_a/X_0$ and ${\omega }_0=d{x}_1\wedge \cdots \wedge d{x}_n$. On $U_i$, let ${\omega }_i$ be the wedge of $d(X_a/X_i)$ for $a\ne i$, taken in increasing index order. For $i>0$, differentiating $X_0/X_i=x_i^{-1}$ and $X_a/X_i=x_a/x_i$ givesIn the wedge, all terms involving a second copy of $d{x}_i$ disappear; the surviving powers are $x_i^{-2}$ from $d(x_i^{-1})$ and $x_i^{-1}$ from the other $n-1$ differentials. Moving $d{x}_i$ into its original position produces the stated sign. Rescale each local generator by ${\eta }_i=(-1{)}^i{\omega }_i$. Then ${\eta }_j=(X_j/X_i{)}^{-n-1}{\eta }_i$ on every overlap, exactly the transition of the twist with $m=-n-1$. This canonical-form transition on projective space proves the canonical bundle of projective space:For line bundles the Serre duality pairing becomesIn particular $H^n(\mathcal{O}(-r))$ is dual to $H^0(\mathcal{O}(r-n-1))$, which is zero for $r\le n$, precisely the independently obtained bound. The case $r=0$ recovers $H^n(\mathcal{O})=0$, and $H^n(\mathcal{O}(-n-1))\cong k$ matches $H^0(\mathcal{O})\cong k$. Thus the calculated transition functions, global sections and top-degree vanishing agree with Serre duality.