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Tag 01EA

Chapter 20: Cohomology of Sheaves > Section 20.9: Mayer-Vietoris

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}

    Comments (3)

    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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