Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 16 (Algebraic Geometry)

Construct the tensor product of sheaves by first forming the presheaf $U\mapsto \mathcal{F}(U){\otimes }_{{\mathcal{O}}_X(U)}\mathcal{G}(U)$, with restriction maps induced by those of the two sheaves of modules, and then applying sheafification. The local module actions are compatible with restrictions and therefore give the sheaf an ${\mathcal{O}}_X$-module structure. Its stalks areIndeed, a finite collection of germs of sheaf sections can be represented on a common neighbourhood, and every finite tensor relation holds on a sufficiently small neighbourhood. Equivalently, this construction represents bilinear maps of sheaves of modules that are balanced over the structure sheaf. Sections of the resulting sheaf need not themselves be tensors of global sections: that is why the sheafification step matters.

The direct image sheaf is defined on an open set $V\subseteq Y$ byRestrictions are those of $\mathcal{F}$, and the sheaf gluing axiom follows by taking inverse images of an open cover. The morphism of ringed spaces supplies ${\varphi }^{\#}:{\mathcal{O}}_Y\to {\varphi }_{\ast }{\mathcal{O}}_X$. Thus $a\in {\mathcal{O}}_Y(V)$ acts on $s\in \mathcal{F}({\varphi }^{-1}V)$ by ${\varphi }^{\#}(a)s$, making ${\varphi }_{\ast }\mathcal{F}$ a sheaf of modules over ${\mathcal{O}}_Y$. This definition uses no quasi-coherence.

First form the inverse image sheaf ${\varphi }^{-1}\mathcal{H}$ by applying sheafification toIt is a sheaf of modules over ${\varphi }^{-1}{\mathcal{O}}_Y$. The morphism of ringed spaces gives a ring map ${\varphi }^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$, so the pullback of a sheaf of modules isAt $x\in X$, writing $y=\varphi (x)$, its stalk is ${\mathcal{O}}_{X,x}{\otimes }_{{\mathcal{O}}_{Y,y}}{\mathcal{H}}_y$. The tensor construction makes the action of the structure sheaf explicit; taking the inverse image alone would not give the requested ${\mathcal{O}}_X$-module.

The opening change-of-rings tensor quotient sends $m{\otimes }_{R}n$ to $m{\otimes }_{A}n$. The target pairing is bilinear and $R$-balanced because $\theta (r)m{\otimes }_{A}n=m{\otimes }_A\theta (r)n$. On the source, let $A$ act through the first factor; commutativity makes that action compatible with the $R$-balancing relations, and the map is $A$-linear. It is surjective: the target is the quotient imposing the additional relations $am{\otimes }_{R}n=m{\otimes }_{R}an$ for every $a\in A$. The construction commutes with module homomorphisms in both variables.

The pullback-direct-image adjunction unit is obtained locally by pulling a section $h\in \mathcal{H}(V)$ to ${\varphi }^{-1}V$ and sending it to $1\otimes h$ in ${\varphi }^{\ast }\mathcal{H}({\varphi }^{-1}V)$. These maps respect restrictions and the ${\mathcal{O}}_Y$-actions, hence defineFor $\mathcal{H}={\mathcal{O}}_Y$, multiplication identifies ${\varphi }^{\ast }{\mathcal{O}}_Y$ with ${\mathcal{O}}_X$, and ${\eta }_{{\mathcal{O}}_Y}$ is precisely the structure morphism ${\varphi }^{\#}:{\mathcal{O}}_Y\to {\varphi }_{\ast }{\mathcal{O}}_X$.

To construct the direct-image tensor comparison, on $V\subseteq Y$ send $s\otimes t$, with $s\in \mathcal{F}({\varphi }^{-1}V)$ and $t\in \mathcal{G}({\varphi }^{-1}V)$, to its tensor section of $\mathcal{F}{\otimes }_{{\mathcal{O}}_X}\mathcal{G}$. This pairing is ${\mathcal{O}}_Y(V)$-balanced through ${\varphi }^{\#}$ and is compatible with restrictions. The universal property of the tensor product of modules and sheafification therefore giveApply this with $\mathcal{G}={\varphi }^{\ast }\mathcal{H}$ after tensoring the unit ${\eta }_{\mathcal{H}}$ with ${\varphi }_{\ast }\mathcal{F}$. The composite is the map in the projection formula for sheaves. If $\mathcal{H}{|}_V\cong {\mathcal{O}}_V^r$, then ${\varphi }^{\ast }\mathcal{H}{|}_{{\varphi }^{-1}V}\cong {\mathcal{O}}_{{\varphi }^{-1}V}^r$, and the composite identifies withthe identity on the $r$ components. Thus it is an isomorphism for a locally free sheaf of finite rank. Neither $\mathcal{F}$ nor $\mathcal{G}$ was assumed quasi-coherent.

For $r=1$, the assertion is the assumed vanishing for coherent ideal sheaves. For $r>1$, project $\mathcal{F}\subseteq {\mathcal{O}}_X^r$ onto the last component. Its image $\mathcal{I}\subseteq {\mathcal{O}}_X$ is a coherent ideal sheaf, and its kernel ${\mathcal{F}}^{\prime }$ is a coherent sheaf contained in ${\mathcal{O}}_X^{r-1}$. Here images and kernels are coherent because a variety is Noetherian. The short exact sequencegives an exact segment $H^1(X,{\mathcal{F}}^{\prime })\to H^1(X,\mathcal{F})\to H^1(X,\mathcal{I})$ in the long exact sequence in sheaf cohomology. The outer terms vanish by mathematical induction and the hypothesis, so the middle term vanishes. This is ideal-sheaf vanishing for a coherent submodule of a trivial bundle.

Let ${\mathcal{I}}_Y$ be the coherent ideal sheaf of $Y$, and let ${\mathfrak{m}}_P$ be the ideal sheaf of a closed point. Because $P\notin Y$, the stalk $({\mathcal{I}}_Y{)}_P$ is ${\mathcal{O}}_{X,P}$. Evaluation at $P$ therefore gives a surjective morphism of sheaves ${\mathcal{I}}_Y\to k_P$ to the skyscraper sheaf at $P$. Its kernel $\mathcal{J}={\mathcal{I}}_Y\cap {\mathfrak{m}}_P$ is again a coherent ideal sheaf. Fromand $H^1(X,\mathcal{J})=0$, the long exact sequence in sheaf cohomology shows that $\Gamma (X,{\mathcal{I}}_Y)\to k$ is onto. Choose $f$ mapping to $1$. It vanishes on $Y$ and satisfies $f(P)=1$. Hence $X_f\subseteq U$, and inside the affine variety $U$ it is the principal open subset defined by $f{|}_U$. A principal open of an affine variety is affine. This gives affine principal neighbourhoods from ideal-sheaf vanishing.

The quasi-compactness of $X$ gives a finite cover by the affine principal neighbourhoods constructed above. Because at every point some $f_i$ is a unit in the local ring, the map of coherent sheavesis surjective. Its kernel $\mathcal{F}$ is a coherent submodule of the trivial bundle, so $H^1(X,\mathcal{F})=0$ by part (a). The long exact sequence in sheaf cohomology makes the map on global sections surjective. Lifting $1$ suppliesThis is the unit-ideal certificate from a principal affine cover.

The localization of global sections on a principal open gives $A_{f_i}=\Gamma (X_{f_i},{\mathcal{O}}_X)$. Each of these rings is a finitely generated algebra because $X_{f_i}$ is an affine variety. Choose finite generators of every $A_{f_i}$ and write them as $a_{ij}/f_i^{e_{ij}}$ with $a_{ij}\in A$. Let $B$ be the $k$-subalgebra of $A$ generated by all $f_i,g_i,a_{ij}$. It is a finitely generated algebra, and $B_{f_i}=A_{f_i}$ since these localizations contain the chosen generators and $f_i^{-1}$.

For any integer $N\ge 1$, raise ${\sum }_i{g}_i{f}_i=1$ to the power $r(N-1)+1$. Every resulting monomial contains some $f_i^N$, so it gives a unit-ideal identity for powersFor an arbitrary $a\in A$, equality $A_{f_i}=B_{f_i}$ implies $f_i^{N_i}a\in B$ for some $N_i$: multiply by an additional power if equality of localized fractions requires it. Choose a common $N$ and the identity above. Then $a={\sum }_i{c}_i(f_i^{N}a)\in B$. Consequently $A=B$, so $A$ is a finitely generated algebra. It is a reduced ring, since a nilpotent element global regular function has zero germ everywhere on the reduced variety $X$. This is finite generation from finitely many principal localizations.

Let $Y$ be the affine variety with coordinate ring $A$. Its principal opens $D(f_i)$ cover $Y$ because ${\sum }_i{g}_i{f}_i=1$. On $X_{f_i}$, the localization of global sections on a principal open identifies its coordinate ring with $A_{f_i}$, giving an isomorphismOn an overlap the two maps are induced by the same elements of $A$, or equivalently by the same identification with $A_{f_i{f}_j}$, so they agree. Glue them to $\varphi :X\to Y$. The inverses agree on the overlaps as well and glue to its inverse. At a closed point $P$, this is the map corresponding to the evaluation ring homomorphism $A\to k$, $a\mapsto a(P)$. This proves the cohomological criterion for affineness by an explicit global isomorphism.

For an affine variety, every coherent ideal sheaf is quasi-coherent; vanishing of quasi-coherent cohomology on an affine scheme therefore gives $H^1(X,\mathcal{I})=0$.

For the projective-space complement, assume $n\ge 1$ and choose two distinct closed points $P,Q\in X$. Take the ideal sheaf of two closed points $\mathcal{I}={\mathfrak{m}}_P\cap {\mathfrak{m}}_Q$ on $X$. The codimension-two extension of regular functions on a normal variety givesOne can see this directly: on every standard affine chart of ${\mathbb{P}}^n$, a rational function written in lowest terms cannot have a nonconstant denominator, because an irreducible polynomial factor of the denominator would define a pole along a codimension-one hypersurface, and such a hypersurface is not removed by $Z$. The extended function is constant because every global regular function on projective space is constant. Now the short exact sequencesends $k$ diagonally into $k^2$ on global sections. Its cokernel is $k$, and the long exact sequence in sheaf cohomology injects that cokernel into $H^1(X,\mathcal{I})$. ThusThe assumption $n\ge 1$ is necessary: ${\mathbb{P}}^0$ is already affine and has no such example.

For $U_{i_0\dots i_q}=U_{i_0}\cap \cdots \cap U_{i_q}$, the Čech cochain complex has groupsTerms in ${\delta }^2$ cancel in pairs, and Čech cohomology is $\ker \delta /\mathrm{im}\delta$. A degree-zero cocycle is exactly a family of compatible local sections. The sheaf gluing axiom gives their unique global section, proving ${\check{H}}^0(\mathcal{U},\mathcal{F})\cong \mathcal{F}(X)$. If the cover has affine finite intersections, a quasi-coherent sheaf has no higher cohomology on those intersections by vanishing of quasi-coherent cohomology on an affine scheme. The acyclic cover theorem then identifies all Čech groups with sheaf cohomology. In particular, a finite affine open cover of a separated variety has this property.

For the sheaf of units of the structure sheaf, a multiplicative degree-one cocycle is a family $g_{ij}\in {\mathcal{O}}_X^{\ast }(U_i\cap U_j)$ satisfying $g_{ij}{g}_{jk}=g_{ik}$, with $g_{ji}=g_{ij}^{-1}$. It glues trivial rank-one free modules into an invertible sheaf. Changing the local frames multiplies $g_{ij}$ by a coboundary, and two sets of transition data give isomorphic line bundles precisely when their cocycles differ this way. Tensoring line bundles multiplies their cocycles. Thus line bundles trivialized by an open cover give the group isomorphismFor the remaining arguments, work over the algebraically closed ground field. On the irreducible variety $V$, put $\mathcal{Q}={\mathcal{K}}^{\ast }/{\mathcal{O}}_V^{\ast }$, a quotient of sheaves of abelian groups. A global section of $\mathcal{Q}$ is locally represented by rational functions $f_i$ whose ratios are regular units. The corresponding unit cocycle $f_j/f_i$ defines an invertible sheaf. A single global rational function has trivial cocycle. Conversely, every line bundle has a nonzero rational section: choose a nonzero vector in its one-dimensional fibre at the generic point and express it in local frames. This supplies such local $f_i$. If the associated line bundle is trivial, changing frames makes all $f_i$ restrictions of one rational function. Thereforeis exact. This is the Cartier-divisor description of the Picard group. The sheaf of nonzero rational functions on an irreducible variety is flasque, since all restrictions between nonempty open sets are the identity on $k(V{)}^{\ast }$. Apply the long exact sequence in sheaf cohomology to $1\to {\mathcal{O}}_V^{\ast }\to {\mathcal{K}}^{\ast }\to \mathcal{Q}\to 1$. Since $H^1(V,{\mathcal{K}}^{\ast })=0$, its connecting map has exactly the cokernel just computed, provingFinally the Segre description of a smooth quadric surface identifies $V$ with ${\mathbb{P}}^1\times {\mathbb{P}}^1$. The two rulings of a smooth quadric surface have classes $F_1,F_2$ generating $\mathrm{Pic}(V)\cong {\mathbb{Z}}^2$. For completeness, remove one line in each ruling: the remaining chart is the affine plane, with factorial coordinate ring $k[s,t]$ and trivial divisor class group. The localization sequence for the divisor class group makes $F_1,F_2$ generators, and their degrees on the two ruling lines prove independence. The hyperplane class, and hence the conic $C$, has bidegree $(1,1)$. By Picard-group localization on a smooth variety, the Picard group of a smooth affine quadric surface isExplicitly, the restriction of ${\mathcal{O}}_V(1,0)$ is nontrivial: if it were trivial on $U$, its rational trivialization would have divisor supported on $C$, forcing $(1,0)$ to be an integer multiple of $(1,1)$, which is impossible.

On an affine chart $\mathrm{Spec}A\subseteq V$, the module of Kähler differentials ${\Omega }_{A/k}$ is generated by symbols $da$ subject to $k$-linearity and $d(ab)=adb+bda$. It represents $k$-derivations. These modules commute with localization, so their associated quasi-coherent sheaves glue to the Kähler differential sheaf ${\Omega }_{V/k}^1$. It is coherent: if $A=k[x_1,\dots ,x_N]/I$ and $I=(f_1,\dots ,f_s)$, then it has the finite presentation of a moduleFor a closed point $P$, put $R={\mathcal{O}}_{V,P}$ with maximal ideal $\mathfrak{m}$ and residue field $k$. The Zariski tangent space is ${\mathrm{Hom}}_k(\mathfrak{m}/{\mathfrak{m}}^2,k)$, equivalently ${\mathrm{Der}}_k(R,k)$. A derivation kills constants and ${\mathfrak{m}}^2$, so it factors through $a\mapsto a-a(P)$; conversely every linear functional on $\mathfrak{m}/{\mathfrak{m}}^2$ defines such a derivation by the product rule. The universal property of Kähler differentials therefore identifies this space withIn particular, ${\Omega }_{V,P}^1/{\mathfrak{m}}_P{\Omega }_{V,P}^1\cong {\mathfrak{m}}_P/{\mathfrak{m}}_P^2$, the algebraic cotangent space.

A point is a smooth point of a variety when its local ring is a regular local ring; over this algebraically closed field this says ${\dim }_k{T}_{V,P}=n=\dim V$. Tensor the finite presentation of a module above with $k(P)$. The tangent space is the kernel of the Jacobian matrix $J(P)$, hence has dimension $N-\mathrm{rank}J(P)$. This proves the Jacobian criterionFor a smooth irreducible variety $V$, choose at each point an invertible $(N-n)$-minor and shrink the affine chart so that it remains invertible. Its relations eliminate $N-n$ of the generators of ${\Omega }_{A/k}$, yielding a surjection $A^n\to {\Omega }_{A/k}$. At the generic point, its target has dimension $n$: over a perfect ground field, a separating transcendence basis of the function field has differentials forming a basis. Equivalently this follows from the assumed density of the smooth locus. The kernel therefore becomes zero over the fraction field of the integral domain $A$. As a submodule of $A^n$, it is a torsion-free module, so it is already zero. Thus these maps give local isomorphisms with ${\mathcal{O}}_V^n$, proving local freeness of differentials on a smooth variety with rank $n$.

For an affine chart $\mathrm{Spec}A\subseteq V$, write $W\cap \mathrm{Spec}A=\mathrm{Spec}(A/I)$ and $\mathcal{M}=\tilde{M}$. The restriction of a module sheaf to a closed subvariety is $\widetilde{M{\otimes }_{A}A/I}=\widetilde{M/IM}$. The Conormal exact sequence for Kähler differentials isIt follows from the generators and relations: passing to $A/I$ imposes precisely the additional relations $df=0$ for $f\in I$. The first map is well defined because $d(I^2)$ lies in $I{\Omega }_{A/k}$. Glue these exact module sequences, using exactness of localization, to obtainIn the final assertion, interpret locally principal subvariety as a proper local hypersurface. Its ideal on each chart is $(f)$ with $f\ne 0$. Since $V$ is an irreducible variety and reduced, $f$ is a non-zero-divisor, and $A/(f)\to (f)/(f^2)$, $\bar{a}\mapsto af$, is an isomorphism. These local rank-one descriptions make the conormal sheaf ${\mathcal{I}}_W/{\mathcal{I}}_W^2$ invertible.

Because $W$ is not contained in the singular locus of $V$, there is a dense open subset of $W$ where both $V$ and $W$ are smooth varieties. At a closed point there, $\dim W=\dim V-1$ by the Krull principal ideal theorem. The tangent description $T_{W,P}=\ker (df:T_{V,P}\to k)$ then forces $df\ne 0$. Hence the conormal map is injective at the generic point of $W$. Its kernel is a subsheaf of a line bundle on the integral variety $W$, so it is a torsion-free sheaf; a torsion-free sheaf with zero generic fibre is zero. This proves conormal injectivity for a generically smooth Cartier divisor. If zero equations were allowed in the phrase locally principal, $W=V$ would be a counterexample to invertibility; the proper-hypersurface convention is essential.