Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 113 / 4 / i

A flasque sheaf has surjective restriction maps $\mathcal{F}(V)\to \mathcal{F}(U)$ for every pair of opens $U\subseteq V$. To prove the claim for an injective sheaf of modules, let $j_U:U\hookrightarrow X$ and $j_V:V\hookrightarrow X$. The natural mapis a monomorphism: its stalks are either the identity on ${\mathcal{O}}_{X,x}$, the map from zero to that stalk, or the zero-to-zero map. Here $j_{!}$ is extension by zero for module sheaves.

The extension-by-zero adjunction identifiesThe injective object property extends every morphism from $(j_U{)}_{!}{\mathcal{O}}_U$ to one from $(j_V{)}_{!}{\mathcal{O}}_V$. Under the displayed identification this is exactly surjectivity of $I(V)\to I(U)$. Injective module sheaves are therefore flasque. This argument works on an arbitrary ringed space, without Noetherian or separation assumptions.