Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 113 / 4

We prove the vanishing by an injective resolution of sheaves and the flasque-kernel section-lifting lemma.

For a short exact sequence of sheaves $0\to \mathcal{F}\to \mathcal{G}\to \mathcal{H}\to 0$ with flasque kernel, every section of $\mathcal{H}$ on an open set lifts to $\mathcal{G}$ there. Indeed choose a maximal partial lift by the Zorn lemma, using the sheaf gluing axiom on chains. At any point outside its domain choose a local lift. The difference on the overlap is a section of $\mathcal{F}$ and extends to the new open set by flasqueness; subtract that extension from the new lift and glue. Maximality forces the lift's domain to be the whole open set. Thus taking sections preserves this short exact sequence.

Every injective sheaf is flasque: for $U\subseteq V$, the monomorphism between the extension by zero sheaves ${\mathbb{Z}}_U\hookrightarrow {\mathbb{Z}}_V$ and the defining extension property of injectivity make the restriction $\mathcal{J}(V)\to \mathcal{J}(U)$ surjective. Embed $\mathcal{F}$ into an injective sheaf $\mathcal{J}$. The quotient $\mathcal{Q}$ is flasque as well: lift a section of $\mathcal{Q}$ on a smaller open to $\mathcal{J}$, extend it in $\mathcal{J}$, and project.

The long exact sequence in sheaf cohomology now gives $H^1(X,\mathcal{F})=0$, since $\Gamma (X,\mathcal{J})\to \Gamma (X,\mathcal{Q})$ is surjective, and for $i\ge 2$ gives $H^i(X,\mathcal{F})\cong H^{i-1}(X,\mathcal{Q})$, since injectives have zero positive-degree derived functors. Repeat the construction for the flasque quotient. After finitely many such shifts any specified positive degree reduces to a vanishing first group. HenceNo separation, compactness or scheme hypothesis is used.

Write $U_0=D_{+}(t_0)=\mathrm{Spec}\mathbb{C}[x]$, where $x=t_1/t_0$, and $U_1=D_{+}(t_1)=\mathrm{Spec}\mathbb{C}[y]$, with $y=1/x$ on the overlap. Since $\mathcal{G}$ is a quasi-coherent sheaf, it is associated to a $\mathbb{C}[x]$-module $M$ on $U_0$. The preimages under the open inclusion $f$ of $U_0,U_1,U_0\cap U_1$ are $U_0,D(x),D(x)$, all affine schemes. Thus $f$ is an affine morphism and its direct image sheaf $f_{\ast }\mathcal{G}$ is quasi-coherent by direct image of a quasi-coherent sheaf under an affine morphism.

The two-chart cover is an acyclic cover by vanishing of quasi-coherent cohomology on an affine scheme, so the acyclic cover theorem identifies its Čech cohomology with sheaf cohomology. Its only potentially nontrivial positive-degree differential isThis is surjective because of the second summand. There are no normalized Čech terms in degree two or higher for a two-open cover, soThis is the acyclic direct image from an affine chart of the projective line. The pushforward need not itself be a flasque sheaf; the acyclic affine cover is what justifies this computation.