Local sections in injective sheaves can be extended globally.

Lemma 20.9.1. Let $X$ be a ringed space. Let $U' \subset U \subset X$ be open subspaces. For any injective $\mathcal{O}_ X$-module $\mathcal{I}$ the restriction mapping $\mathcal{I}(U) \to \mathcal{I}(U')$ is surjective.

Proof. Let $j : U \to X$ and $j' : U' \to X$ be the open immersions. Recall that $j_!\mathcal{O}_ U$ is the extension by zero of $\mathcal{O}_ U = \mathcal{O}_ X|_ U$, see Sheaves, Section 6.31. Since $j_!$ is a left adjoint to restriction we see that for any sheaf $\mathcal{F}$ of $\mathcal{O}_ X$-modules

$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(j_!\mathcal{O}_ U, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{O}_ U, \mathcal{F}|_ U) = \mathcal{F}(U)$

see Sheaves, Lemma 6.31.8. Similarly, the sheaf $j'_!\mathcal{O}_{U'}$ represents the functor $\mathcal{F} \mapsto \mathcal{F}(U')$. Moreover there is an obvious canonical map of $\mathcal{O}_ X$-modules

$j'_!\mathcal{O}_{U'} \longrightarrow j_!\mathcal{O}_ U$

which corresponds to the restriction mapping $\mathcal{F}(U) \to \mathcal{F}(U')$ via Yoneda's lemma (Categories, Lemma 4.3.5). By the description of the stalks of the sheaves $j'_!\mathcal{O}_{U'}$, $j_!\mathcal{O}_ U$ we see that the displayed map above is injective (see lemma cited above). Hence if $\mathcal{I}$ is an injective $\mathcal{O}_ X$-module, then the map

$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(j_!\mathcal{O}_ U, \mathcal{I}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(j'_!\mathcal{O}_{U'}, \mathcal{I})$

is surjective, see Homology, Lemma 12.24.2. Putting everything together we obtain the lemma. $\square$

Comment #653 by Fan on

In the proof, $\cal F$ should be $\cal I$.

Comment #2597 by Rogier Brussee on

Suggested slogan: Local sections in injective sheaves can be extended globally.

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