Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 16 / 3

An $A$-module homomorphism $h:M\to N$ induces, by localization of a module, compatible mapson all principal open subsets $D(a)$. These glue to an ${\mathcal{O}}_X$-linear morphism of sheaves $\tilde{h}:\tilde{M}\to \tilde{N}$.

Conversely, taking global sections of an ${\mathcal{O}}_X$-linear morphism of sheaves gives an $A$-module homomorphism $h:M\to N$, since $\Gamma (X,\tilde{M})=M$ and $\Gamma (X,\tilde{N})=N$. On $D(a)$, compatibility with restriction determines the image of $m/1$, and ${\mathcal{O}}_X$-linearity determines the image of $a^{-r}m$. It must be $h(m)/a^r$. Thus the morphism of sheaves is uniquely recovered from $h$. Both constructions preserve addition and composition, proving the affine module-sheaf equivalence on morphisms:No finite-generation assumption is used.

Use the local-presentation definition of a quasi-coherent sheaf: locally it admits a presentation ${\mathcal{O}}^{(I)}\to {\mathcal{O}}^{(J)}\to \mathcal{F}\to 0$ by direct sums of copies of the structure sheaf, with no finiteness requirement on the index sets. If $M$ is an $A$-module, choose a free module presentation $A^{(I)}\to A^{(J)}\to M\to 0$. Exactness of localization of a module gives the corresponding local presentation of $\tilde{M}$. Hence a sheaf which is associated with a module on every affine open subset is a quasi-coherent sheaf.

For the converse, begin with a quasi-coherent sheaf and a local presentation on an affine open subset $V=\mathrm{Spec}R$. The free sheaves have global sections $R^{(I)}$ and $R^{(J)}$: a section has locally finite support, and quasi-compactness of $V$ makes its support finite globally. Part (i) identifies the presentation map with an $R$-module homomorphism. Exactness of localization then identifies its cokernel sheaf with ${\tilde{M}}_V$ for the module cokernel $M_V$. Thus there is an affine open cover on which $\mathcal{F}$ is associated with modules.

It remains to show this on an arbitrary affine open subset $U=\mathrm{Spec}A$, rather than only the selected cover. Around each point of $U$, choose an affine neighborhood contained in $U$ with such a local presentation. Choose a finite cover $U={\bigcup }_{i=1}^{r}D(s_i)$ refining these neighborhoods. On each $D(s_i)$ the restriction is associated with a module. Its restriction to $D(s_i{s}_j)$ is therefore the corresponding localization, and the same is true after intersecting with $D(a)$. Put $M=\Gamma (U,\mathcal{F})$. The sheaf gluing axiom giveswhere the arrow is the difference of the two restriction maps. Localization of a module commutes with kernels and finite products. Localizing this formula at $a$ therefore gives exactly the gluing formula for $D(a)$, covered by the $D(a{s}_i)$. Consequentlycompatibly with restriction. These isomorphisms on the principal open subsets identify $\tilde{M}$ with $\mathcal{F}{|}_U$. We have proved the global-section localization for quasi-coherent sheaves and, in particular,

Work on an affine open subset $U=\mathrm{Spec}A$ and write $\mathcal{F}{|}_U=\tilde{M}$ and $\mathcal{E}{|}_U=\tilde{N}$ using part (ii). We first show that any section $e\in \Gamma (U,\mathcal{E})$ lifts to $\Gamma (U,\mathcal{G})$.

A surjection of sheaves supplies local lifts $g_i$ on a finite cover by principal open subsets $D(s_i)$. Their differencessatisfy $c_{ij}+c_{jk}=c_{ik}$ on triple intersections. We spell out why this Čech cocycle is a coboundary. Choose a common sufficiently large power $n$ so that $s_j^n{c}_{ij}$ can be represented in $M_{s_i}$ for every pair. Increasing $n$ also kills the finitely many localization errors in the cocycle identities: after extending $s_k^n{c}_{ik}$ to $M_{s_i}$ and $s_k^n{c}_{jk}$ to $M_{s_j}$, their difference in $M_{s_i{s}_j}$ is $s_k^n{c}_{ij}$. This follows because any error which vanishes after localizing at $s_k$ is killed by a power of $s_k$; there are only finitely many indices.

The $D(s_i^n)$ still cover $U$, so the ideal generated by the $s_i^n$ is $A$. Choose $a_i\in A$ with ${\sum }_i{a}_i{s}_i^n=1$ and putusing the extended representatives just chosen. On an overlap, the cocycle identity givesThus $g_i-b_i$ agree and glue to a lift of $e$. This is the degree-one argument behind the exactness of the unit-ideal localization Čech complex.

The same reasoning applies to every $D(a)\subseteq U$. Hence we have short exact sequences of modules of sections on $U$ and on $D(a)$. Localize the first sequence at $a$ and compare them. The maps on the outer modules are the isomorphisms from part (ii):An elementary diagram chase makes the middle map an isomorphism: subtract a lift from the right to reduce surjectivity to the left, and use injectivity on the left to prove injectivity. These isomorphisms identify $\mathcal{G}{|}_U$ with the quasi-coherent sheaf associated with $\Gamma (U,\mathcal{G})$. Doing this on every affine open subset provesThe proof even works without the Noetherian assumption.

Take the projective line $X={\mathbb{P}}_k^1$ and the coherent sheaves $\mathcal{F}={\mathcal{O}}_X$ and $\mathcal{G}={\mathcal{O}}_X(-1)$, where the latter is a twisting sheaf on projective space. Both are line bundles, so each stalk is a free rank-one ${\mathcal{O}}_{X,x}$-module and therefore ${\mathcal{F}}_x\simeq {\mathcal{G}}_x$ at every point.

They are nevertheless not isomorphic. On the standard charts with coordinates $t$ and $t^{-1}$, choose frames $e_0,e_1$ of $\mathcal{O}(-1)$ with $e_1=t^{-1}{e}_0$. A global section would give polynomials $a(t)$ and $b(t^{-1})$ satisfyingThe left side has only nonnegative powers of $t$, and the right side only strictly negative powers. Thus both vanish, giving $\Gamma (X,\mathcal{O}(-1))=0$. For the structure sheaf, the corresponding equality is $a(t)=b(t^{-1})$, whose solutions are precisely the constants. HenceA morphism of sheaves which is an isomorphism on all stalks would be an isomorphism; this example only asserts the existence of separate stalk isomorphisms, not that they come from one morphism of sheaves.