Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 14 (Algebraic Geometry)

Let $V\subseteq U$ be open subsets, and let $s\in {\mathcal{F}}_3(V)$. The assumed flasque-kernel section-lifting lemma lifts $s$ to $\tilde{s}\in {\mathcal{F}}_2(V)$. Since ${\mathcal{F}}_2$ is a flabby sheaf, extend $\tilde{s}$ to a section on $U$, then project to ${\mathcal{F}}_3(U)$. Its restriction is $s$. Thus every restriction map of ${\mathcal{F}}_3$ is surjective, proving that the quotient of two flabby sheaves in a short exact sequence is flabby.

For the construction of sheaf cohomology, embed a sheaf of abelian groups $\mathcal{F}$ into the sheafby sending a section to all its germs. This is injective because a section with every germ zero is zero. Arbitrary families of stalk elements glue point by point, and restriction maps are projections; hence $C(\mathcal{F})$ is a flabby sheaf. Set ${\mathcal{Q}}^0=\mathcal{F}$, ${\mathcal{I}}^j=C({\mathcal{Q}}^j)$ and ${\mathcal{Q}}^{j+1}={\mathcal{I}}^j/{\mathcal{Q}}^j$. Compose the quotient map with the next embedding to obtain the flasque resolutionThe sheaf cohomology groups are the cohomology groups of the resulting cochain complex of global sections:The usual comparison of resolutions identifies this construction with the right derived functors of global sections; the construction above is the Godement resolution.

If $\mathcal{F}$ is itself a flabby sheaf, the quotient argument shows inductively that every ${\mathcal{Q}}^j$ is flabby. The assumed lifting result therefore makesan exact sequence for every $j$. The global-section complex is consequently exact in every positive degree. In particular,This establishes the vanishing from the section-lifting property rather than assuming it in advance.

For the smooth algebraic curve $X$, a divisor on an algebraic curve is a finite formal sum $D={\sum }_P{n}_{P}P$ of closed points with integer coefficients. At each closed point, the local ring of a smooth algebraic curve is a discrete valuation ring; write its normalized discrete valuation as $v_P$. For $f\in k(X{)}^{\ast }$ the principal divisor is $\mathrm{div}(f)={\sum }_P{v}_P(f)P$. Only finitely many coefficients are nonzero: zeros and poles are proper closed subsets in the Noetherian Zariski topology of a curve. The divisor class group is the abelian groupEquivalently, its elements are divisors on an algebraic curve modulo linear equivalence of divisors.

Put $\mathcal{D}={\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$, the quotient sheaf of abelian groups. At a closed point $P$, the discrete valuation gives an isomorphismIndeed, a local parameter on a smooth algebraic curve has valuation one, and the valuation kernel is precisely the unit group of the discrete valuation ring. Thus a local section represented by $f$ determines its local principal divisor, and changing $f$ by a regular function which is a unit does not change those coefficients.

These local valuations identify $\Gamma (X,\mathcal{D})$ with $\mathrm{Div}(X)$. To see that they give finite support, represent a section locally by finitely many rational functions on an algebraic variety, using Noetherian Zariski topology to take a finite cover. Each representative has only finitely many zeros and poles, so their combined support is finite. Conversely, for $D=\sum n_{P}P$, choose at each support point a local parameter $t_P$ and shrink its neighbourhood to exclude every other zero or pole of $t_P$ and every other support point. The local rational equation $t_P^{n_P}$ represents the desired valuations there; on the complement of the support use $1$. On overlaps these equations differ by units, so their classes glue in the quotient sheaf. Finally, a section with every valuation zero has zero germ everywhere and is the identity section. Hence the identification is both surjective and injective.

The map on global sections sends $f\in k(X{)}^{\ast }$ to $\mathrm{div}(f)$. Therefore its cokernel is

For a divisor on an algebraic curve $D=\sum n_{P}P$, define its line bundle associated to a divisor inside the sheaf of rational functions byLocally choose a rational equation $f_i$ for $D$, as in the construction of the quotient sheaf above. ThenConsequently this is an invertible sheaf. Multiplication gives an isomorphismas is checked on the displayed local generators. If $D=\mathrm{div}(f)$, multiplication by $f$ identifies ${\mathcal{O}}_X(D)$ with ${\mathcal{O}}_X$. Thus $D\mapsto [{\mathcal{O}}_X(D)]$ induces a group homomorphism from the divisor class group to the Picard group.

For surjectivity, let $\mathcal{L}$ be an invertible sheaf. Trivialize its one-dimensional rational fibre over the function field $k(X)$; equivalently, choose a nonzero rational section and identify $\mathcal{L}$ with a subsheaf of the rational-function sheaf. A finite trivializing cover givesOn overlaps $a_i/a_j$ is a regular unit. The integers $-v_P(a_i)$ therefore agree wherever two descriptions apply and define a finite divisor on an algebraic curve $D$. Its local description gives $\mathcal{L}={\mathcal{O}}_X(D)$. A different rational trivialization multiplies all $a_i$ by the same rational function, changing $D$ by a principal divisor, so the resulting divisor class is intrinsic.

For injectivity, an isomorphism ${\mathcal{O}}_X(D)\to {\mathcal{O}}_X(E)$ becomes multiplication by some $g\in k(X{)}^{\ast }$ on the rational fibre. At each closed point its effect on the local free generators givesHence $D-E=\mathrm{div}(g)$. Conversely this equality gives the required isomorphism by multiplication by $g$. We have proved the group isomorphism

Since $X$ is an irreducible variety, every nonempty open subset is irreducible and the sheaf of nonzero rational functions on an irreducible variety has value $k(X{)}^{\ast }$ there. All its restriction maps between nonempty opens are identities, and restriction to the empty set is surjective. Thus ${\mathcal{K}}_X^{\ast }$ is a flabby sheaf, and the introductory argument gives $H^1(X,{\mathcal{K}}_X^{\ast })=0$.

Apply the long exact sequence in sheaf cohomology to the short exact sequence of sheavesIts relevant portion isThe connecting homomorphism is therefore surjective, with kernel the image of $k(X{)}^{\ast }$. The computation in part (a) now givesConcretely, local rational equations $f_i$ of a divisor on an algebraic curve differ by regular units on overlaps. Their ratios give the gluing data of an invertible sheaf; changing the local equations by units changes those data by a coboundary. This is the transition-function interpretation of the identification with the Picard group. Reversing all transition ratios replaces a class by its inverse; either consistent convention gives the same group isomorphism.

Let $X$ be an affine variety over an algebraically closed field $k$, with reduced coordinate ring $A=k[X]$. On an open subset $U$, define ${\mathcal{O}}_X(U)$ to be the functions $h:U\to k$ which, near every point, have an expression $a/b$ with $a,b\in A$ and $b$ nowhere zero on that neighbourhood. Restriction is ordinary restriction of functions. Compatible local functions glue uniquely and remain locally quotients, so this defines the structure sheaf. On the principal open subset $D(f)$ it gives $A_f$, and at $P$ it gives the local ring $A_{{\mathfrak{m}}_P}$. The principal-open identification can also be seen by identifying $D(f)$ with the affine algebraic set obtained by adjoining a coordinate $t$ and the equation $tf=1$; its coordinate ring is $A[t]/(tf-1)\cong A_f$. This construction permits several irreducible components; no division by a function vanishing identically on a component is assumed.

An abstract algebraic variety is a locally ringed space locally isomorphic to such affine models, with a finite affine cover; we use the usual separated convention. Reducible varieties are allowed here. The separation condition says that the diagonal morphism is a closed immersion. Two varieties are related by a birational map with an inverse when some dense open subset of one is isomorphic as a locally ringed space to a dense open subset of the other.

Define the ring of rational functions on a reduced variety by pairs $(U,h)$, where $U\subseteq X$ is dense open and $h\in {\mathcal{O}}_X(U)$, identifying two pairs when their restrictions agree on a common dense open subset. Finite intersections of dense open subsets are dense, so addition and multiplication on these intersections are well defined. For an irreducible variety this gives the usual function field; for a reducible variety it need not be a field. Restriction to any dense open subset $V$ induces an isomorphismIndeed, intersecting domains with $V$ defines the restriction, and a dense open subset of $V$ is also dense open in $X$, which defines its inverse. An isomorphism between dense opens therefore induces an isomorphism of their rational rings. Thus birational varieties have isomorphic rings of rational functions, also in the reducible case.

For the finite-cover assertion, write the closed complement of $U$ as $V(I)$, where $I=(f_1,\dots ,f_r)$ is an ideal of the Noetherian ring $A$. ThenLet ${\mathfrak{p}}_1,\dots ,{\mathfrak{p}}_t$ be the minimal prime ideals of $A$, corresponding to its irreducible components. Since $U$ is dense, it meets every component, so $I$ is contained in none of these primes. Prime avoidance supplies $s\in I$ outside their union. Consequently $D(s)\subseteq U$ meets a dense open subset of every component and is dense in $X$. Add $D(s)$ to the finite cover if necessary. This proves that a finite principal-open cover can be chosen with a dense member. It does not say that every principal-open cover already has such a member: for $X=V(xy)$, the dense open $X\setminus \{(0,0)\}=D(x)\cup D(y)$ has neither displayed member dense, although adding $D(x+y)$ supplies a dense member.

For clarity, the zero divisors in this reduced Noetherian ring are exactly ${\bigcup }_i{\mathfrak{p}}_i$. If $ab=0$ and $a$ avoids every ${\mathfrak{p}}_i$, reduction in the domain $A/{\mathfrak{p}}_i$ forces $b\in {\mathfrak{p}}_i$ for all $i$, hence $b=0$. Conversely, if $a\in {\mathfrak{p}}_i$, choose for every $j\ne i$ an element $b_j\in {\mathfrak{p}}_j\setminus {\mathfrak{p}}_i$, possible because distinct minimal primes are incomparable. Their product $b$ is not in ${\mathfrak{p}}_i$ but belongs to all the other primes. Then $b\ne 0$ and $ab\in {\bigcap }_j{\mathfrak{p}}_j=0$. For a single minimal prime, reducedness makes that prime zero and this argument uses $b=1$. It follows that

The total quotient ring $S^{-1}A$, where $S$ consists of these non-zero-divisors, maps to $\mathrm{Rat}(X)$ byThe equivalence relation for localization makes this a well-defined $k$-algebra homomorphism. For surjectivity, a rational class represented on $U$ restricts to the dense principal open $D(s)\subseteq U$ constructed above. Its section there belongs to $A_s$, so it is $a/s^r$ and lies in the image. For injectivity, if such a fraction is zero on a dense open subset, its numerator vanishes on that dense subset. The numerator's zero set is closed, so it vanishes everywhere on $X$ and is zero in the reduced coordinate ring. ThusIn particular $A\to S^{-1}A$ is canonically injective: $a/1=0$ implies $sa=0$ for some non-zero-divisor $s$, hence $a=0$.

Finally let $Y$ be the reduced closed union of exactly those irreducible components of $X$ which contain $P$. Removing all the other components gives an open neighbourhood of $P$ on which $X$ and $Y$ coincide, including their structure sheaves. Hence ${\mathcal{O}}_{X,P}\cong {\mathcal{O}}_{Y,P}$. Write $B=k[Y]$ and let ${\mathfrak{n}}_P$ be its maximal ideal at $P$. Every minimal prime of $B$ is contained in ${\mathfrak{n}}_P$, because every component of $Y$ contains $P$. If $b\notin {\mathfrak{n}}_P$, it avoids each such prime and is a non-zero-divisor. Therefore the map of localizationsis defined and injective: if its value is zero, a non-zero-divisor annihilates $a$, so $a=0$. We obtain the desired local ring embeds in rational functions on incident components:At an intersection point one must retain all incident components. For example, the local ring of $V(xy)$ at the origin has two nonzero germs $x,y$ with $xy=0$; it cannot embed into the function field of just one of its axes.

A quasi-coherent sheaf is a sheaf of modules which locally has a presentation by free module sheaves:where the index sets may be infinite. Equivalently, on affine neighbourhoods it is an affine module sheaf. A coherent sheaf is locally finitely generated, with the additional requirement that the kernel of every map from a finite free module sheaf is locally finitely generated. On a Noetherian scheme, these conditions are equivalent to being locally associated with a finitely generated module.

For an affine variety $X$ with coordinate ring $A$ and an $A$-module $M$, setRestriction on principal open subsets is the natural module localization map. On an arbitrary open subset, sections are families of these stalk values locally represented by a single fraction from a localization. This constructs the affine module sheaf; its global sections are $M$. Choose a free-module presentation $A^{(J)}\to A^{(I)}\to M\to 0$. Exactness of localization and its compatibility with direct sums yieldThus $\tilde{M}$ is a quasi-coherent sheaf, without a finite-generation assumption.

We now prove coherence of an affine module sheaf. If $M$ is finitely generated, take a surjection $A^r\to M$. Since $A$ is a Noetherian ring, its kernel is finitely generated, so $M$ has a finite presentation of a module. More generally, on every principal affine open, the kernel of a map $A_a^s\to M_a$ is a submodule of a finite free module over the Noetherian ring $A_a$, and is finitely generated. On sheafifying, this verifies both parts of the definition of a coherent sheaf.

Conversely, if $\tilde{M}$ is coherent, choose a finite principal-open cover $X={\bigcup }_{i}D(a_i)$ on which it has finitely many generators. Write those local generators as $m_{ij}/a_i^{r_{ij}}$. Their numerators generate a finite submodule $N={\sum }_{i,j}A{m}_{ij}$ of $M$, and $N_{a_i}=M_{a_i}$ for every $i$. To check that $M/N=0$, let $q\in M/N$. Each equality after localization gives an exponent $r_i$ such that $a_i^{r_i}q=0$. The ideal generated by the $a_i^{r_i}$ is the unit ideal: otherwise a maximal ideal containing all of them would give a point outside the principal-open cover. A linear combination of these powers equal to $1$ consequently annihilates $q$. Every $q$ is zero, proving

Next consider a morphism $\varphi :{\mathcal{G}}_1\to {\mathcal{G}}_2$ between quasi-coherent sheaves. On an affine open $V$ of the target variety, use the assumed affine module-sheaf equivalence to write ${\mathcal{G}}_j{|}_V=\widetilde{M_j}$, with $M_j=\Gamma (V,{\mathcal{G}}_j)$. The sheaf morphism is the sheafification of the module map $\psi =\Gamma (V,\varphi )$. Indeed, its action on a section $m/a^r$ of a principal open is forced by module-linearity and restriction to be $\psi (m)/a^r$. Therefore exactness of localization givesThese identities are compatible with restriction, so $(\ker \varphi ){|}_V=\widetilde{\ker \psi }$. Since this holds on an affine cover, the kernel sheaf is quasi-coherent on every variety. This proof of kernels of quasi-coherent sheaf morphisms does not assert that taking global sections is right exact.

For direct image along a morphism of affine varieties, write $Y$ with coordinate ring $A$, $X$ with coordinate ring $B$, and let $\alpha :A\to B$ be the induced ring homomorphism. If $\mathcal{F}=\tilde{M}$ on $X$, regard $M$ as an $A$-module by restriction of scalars. For $a\in A$,The last equality identifies localization with the scalar action of $a$. It identifies the direct image sheaf with the affine module sheaf of the restricted module, so $f_{\ast }\mathcal{F}$ is quasi-coherent.

Suppose $Y$ is affine and $X={\bigcup }_{i=1}^r{U}_i$ is a finite affine cover. Since a variety is separated, every $U_i\cap U_j$ is affine: it is identified with the inverse image of the closed diagonal morphism inside the affine product $U_i{\times }_k{U}_j$. Write $g_i=f{|}_{U_i}$ and $g_{ij}=f{|}_{U_i\cap U_j}$. Form the morphism of sheaves of modulesBoth sums are finite, and every summand is quasi-coherent by the affine calculation. For every open $V\subseteq Y$, the sheaf gluing axiom on $f^{-1}(V)$ identifies the kernel of this map on sections with $\mathcal{F}(f^{-1}(V))$. HenceThe kernel result proved above establishes quasi-coherence.

Finally, for arbitrary $Y$, take an affine neighbourhood $V\subseteq Y$ and apply the preceding argument to $f^{-1}(V)\to V$. The open subset $f^{-1}(V)$ is again a variety and has a finite affine cover, because varieties have Noetherian Zariski topology. The restriction of $f_{\ast }\mathcal{F}$ to $V$ is exactly the direct image for this restricted morphism. ThusThe finiteness and separation conditions are the relevant hypotheses, not affineness of the original source or target. More generally, quasi-coherence of direct image under a quasi-compact quasi-separated morphism is valid. If one permits nonseparated finite-type prevarieties, each overlap in the argument has a finite affine cover; replacing its summand by the sum of direct images from these charts gives the same kernel, since agreement can be tested on a cover. This also proves the assertion in that convention. It does not extend by this argument to arbitrary spaces with an unrestricted infinite cover.

Let $S=k[X_0,\dots ,X_n]$ with its usual grading, and put $S(m{)}_d=S_{m+d}$. Define the twisting sheaf on projective space byMore explicitly, on $U_i=D_{+}(X_i)$ its sections are the homogeneous degree-$m$ part of $S_{X_i}$. They are a free rank-one module over $(S_{X_i}{)}_0=k[X_j/X_i:j\ne i]$, with generator $e_i=X_i^m$. These generators make sense for every integer $m$, since $X_i$ is invertible on this chart. Their transition rule isThe ratios are regular units on overlaps and satisfy the cocycle identity, so they glue to an invertible sheaf. In particular $\mathcal{O}(m)\otimes \mathcal{O}(r)\cong \mathcal{O}(m+r)$.

For $n\ge 1$, a global section is a compatible homogeneous degree-$m$ rational expression lying in every $S_{X_i}$. Inside the fraction field of $S$,To justify this, write a fraction in lowest terms in the unique factorization domain $S$. Membership in $S_{X_i}$ forces every irreducible divisor of its denominator to be associated with $X_i$. Since there are at least two nonassociated variables, the denominator must be a unit. ThereforeThe degree-$m$ monomials are indexed by nonnegative integers $a_0,\dots ,a_n$ with sum $m$. The stars and bars count givesHere and below, an all-integer binomial shorthand must use the counting convention $\left(\genfrac{}{}{0ex}{}{a}{n}\right)=0$ when $a<n$, including negative $a$. The polynomial extension of binomial coefficients to negative upper arguments does not give the stated dimensions. In dimension zero, ${\mathbb{P}}^0$ is one point and every ${\mathcal{O}}_{{\mathbb{P}}^0}(m)$ is trivial, so its space of global sections is $k$ for every $m$; this case is treated separately.

For a closed inclusion $i:H\hookrightarrow {\mathbb{P}}^n$, direct image sheaf formation is exact: its stalk is the original stalk at a point of $H$ and zero outside $H$. It preserves flabby sheaves, since restrictions on the target are restrictions between the corresponding open subsets of $H$. Push forward a flasque resolution of $\mathcal{F}$; exactness gives a flasque resolution of $i_{\ast }\mathcal{F}$, and its complex of global sections is the original one becauseTaking cohomology of these identical complexes proves sheaf cohomology under a closed inclusion:This argument applies even without quasi-coherence.

We derive the remaining sheaf cohomology dimensions using the given eventual vanishing, the long exact sequence in sheaf cohomology, and induction on dimension. Multiplication by the linear equation of a hyperplane $H\cong {\mathbb{P}}^{n-1}$ gives the hyperplane exact sequence for twisting sheavesExactness can be checked in each chart by quotienting its polynomial ring by the equation of $H$; multiplication is injective because that equation is a non-zero-divisor. We use the closed-inclusion cohomology identification to replace the cohomology of the final sheaf by that on $H$.

The dimension-zero base is a point, whose global-section functor is exact. Thus all its higher sheaf cohomology is zero. For $n=1$, the relevant long exact sequence isWhen $m\ge 0$, restriction of homogeneous polynomials to the hyperplane point is surjective: after choosing that point as $[1:0]$, the section $X_0^m$ restricts to a generator. Therefore the consecutive $H^1$ groups are isomorphic for $m\ge 0$. The given vanishing for large $m$ propagates down to $m=-1$. For $m<0$, both displayed $H^0$ groups are zero, and the sequence becomesStarting with $H^1(\mathcal{O}(-1))=0$, induction downward gives $h^1(\mathcal{O}(m))=-m-1$ for $m\le -2$. For degrees $i>1$, the point contributes no cohomology to the exact sequence, so consecutive $H^i$ groups are isomorphic for every $m$; the given eventual vanishing makes them all zero. Hence the complete positive-degree formula holds for ${\mathbb{P}}^1$.

Now let $n\ge 2$ and assume the formulas for $H\cong {\mathbb{P}}^{n-1}$. For every $m$, the restrictionis surjective: for $m\ge 0$ extend a polynomial in the hyperplane coordinates to the same polynomial on ${\mathbb{P}}^n$, and for $m<0$ both groups vanish because $H$ has positive dimension. Consequently the exact sequence gives an injectionFor $2\le i\le n-1$, the induction hypothesis says $H^{i-1}(H,{\mathcal{O}}_H(m))=0$, and gives the same injection in degree $i$. For each fixed $m$, composing finitely many injections reaches a sufficiently large twist, whose higher cohomology is zero by the given hypothesis. ThereforeFor $i>n$, the two neighbouring hyperplane groups $H^{i-1}(H,{\mathcal{O}}_H(m))$ and $H^i(H,{\mathcal{O}}_H(m))$ are zero by induction. Consecutive $H^i$ groups on ${\mathbb{P}}^n$ are then isomorphic, and eventual vanishing again makes them zero. This derives vanishing above the dimension too, without needing a special projective-space cohomology theorem.

For top cohomology the exact sequence now reduces toWrite $b_n(m)=h^n({\mathcal{O}}_{{\mathbb{P}}^n}(m))$. The induction hypothesis and eventual vanishing give the recurrenceConsecutive top groups agree for $m\ge -n+1$, so descending from a large twist gives $b_n(m)=0$ for $m\ge -n$. Below this range, iterating the recurrence gives a finite sum:The last equality follows by summing Pascal's identity. These short exact sequences also prove that every group whose dimension is being counted is finite-dimensional. Thus, for $n\ge 1$,With the stated counting convention, the latter is the requested single binomial formula. The base ${\mathbb{P}}^0$ has $h^0=1$ for all twists and no higher cohomology; it is not governed by the negative-twist $H^0$ rule for positive-dimensional projective space.

Finally calculate the canonical line bundle of a smooth variety from its transition functions. On $U_0$ write $t_j=X_j/X_0$ for $1\le j\le n$. On $U_i$, $i>0$, take the coordinates $u_j=X_j/X_i$ in increasing order of $j\ne i$. ThusIn their wedge product, $d{u}_0=-t_i^{-2}d{t}_i$ and
$d{u}_j=t_i^{-1}d{t}_j-t_j{t}_i^{-2}d{t}_i$. Every term involving a second $d{t}_i$ vanishes, so reordering the remaining differentials givesThis computation is valid in every characteristic. Multiply the ordered local generator on $U_i$ by $(-1{)}^i$. The resulting generators ${\eta }_i$ satisfy ${\eta }_i=(X_i/X_0{)}^{-n-1}{\eta }_0$ on $U_i\cap U_0$, and hence the corresponding transition rule on every overlap. They are exactly the transitions of $\mathcal{O}(-n-1)$. Therefore the canonical bundle of projective space is

For a line bundle, Serre duality statesThe intermediate groups on both sides vanish. In the two outer degrees, the top group at twist $m$ has the same dimension as the global-section group at twist $-m-n-1$:The polynomial degree on the right is nonnegative precisely when $m\le -n-1$, and its dimension is $\left(\genfrac{}{}{0ex}{}{-m-1}{n}\right)$. Interchanging $m$ and $-m-n-1$ checks the other outer degree. On ${\mathbb{P}}^0$ both sides are one-dimensional. Thus all the formulas are consistent with Serre duality, which was used only for this final consistency check, not to prove the cohomology formulas.