# The Stacks Project

## Tag 01EA

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 $$\mathop{\rm Hom}\nolimits_{\mathcal{O}_X}(j_!\mathcal{O}_U, \mathcal{F}) = \mathop{\rm Hom}\nolimits_{\mathcal{O}_U}(\mathcal{O}_U, \mathcal{F}|_U) = \mathcal{F}(U)$$ 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 $$\mathop{\rm Hom}\nolimits_{\mathcal{O}_X}(j_!\mathcal{O}_U, \mathcal{I}) \longrightarrow \mathop{\rm Hom}\nolimits_{\mathcal{O}_X}(j'_!\mathcal{O}_{U'}, \mathcal{I})$$ is surjective, see Homology, Lemma 12.24.2. Putting everything together we obtain the lemma. $\square$

The code snippet corresponding to this tag is a part of the file cohomology.tex and is located in lines 716–726 (see updates for more information).

\begin{lemma}
\label{lemma-injective-restriction-surjective}
\begin{slogan}
Local sections in injective sheaves can be extended globally.
\end{slogan}
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.
\end{lemma}

\begin{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 \ref{sheaves-section-open-immersions}.
Since $j_!$ is a left adjoint to restriction we see that
for any sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules
$$\Hom_{\mathcal{O}_X}(j_!\mathcal{O}_U, \mathcal{F}) = \Hom_{\mathcal{O}_U}(\mathcal{O}_U, \mathcal{F}|_U) = \mathcal{F}(U)$$
see Sheaves, Lemma \ref{sheaves-lemma-j-shriek-modules}.
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 \ref{categories-lemma-yoneda}). 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
$$\Hom_{\mathcal{O}_X}(j_!\mathcal{O}_U, \mathcal{I}) \longrightarrow \Hom_{\mathcal{O}_X}(j'_!\mathcal{O}_{U'}, \mathcal{I})$$
is surjective, see
Homology, Lemma \ref{homology-lemma-characterize-injectives}.
Putting everything together we obtain the lemma.
\end{proof}

Comment #653 by Fan on June 3, 2014 a 6:36 pm UTC

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

Comment #664 by Johan (site) on June 4, 2014 a 8:51 pm UTC

Thanks! This is fixed here.

Comment #2597 by Rogier Brussee on June 5, 2017 a 10:21 am UTC

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

There are also 2 comments on Section 20.9: Cohomology of Sheaves.

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