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For opens $U\subseteq V\subseteq X$, there is a monomorphism between the extensions by zero of ${\mathcal{O}}_U$ and ${\mathcal{O}}_V$. Morphisms from these sheaves into an injective sheaf of modules $I$ identify with sections on $U$ and $V$, respectively. The injective object extension property therefore makes the restriction $I(V)\to I(U)$ surjective, proving that $I$ is flasque. The argument applies to any ringed space.

On a ringed space $(X,{\mathcal{O}}_X)$, an injective module sheaf is an injective object in the category of sheaves of modules over ${\mathcal{O}}_X$. Its defining extension property concerns module-sheaf morphisms; it is distinct from injectivity solely in the category of sheaves of abelian groups.

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.