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Tag 00AC

Chapter 6: Sheaves on Spaces > Section 6.31: Open immersions and (pre)sheaves

Remark 6.31.13. Let $j : U \to X$ be an open immersion of topological spaces as above. Let $x \in X$, $x \not \in U$. Let $\mathcal{F}$ be a sheaf of sets on $U$. Then $\mathcal{F}_x = \emptyset$ by Lemma 6.31.4. Hence $j_!$ does not transform a final object of $\mathop{\textit{Sh}}\nolimits(U)$ into a final object of $\mathop{\textit{Sh}}\nolimits(X)$ unless $U = X$. According to our conventions in Categories, Section 4.23 this means that the functor $j_!$ is not left exact as a functor between the categories of sheaves of sets. It will be shown later that $j_!$ on abelian sheaves is exact, see Modules, Lemma 17.3.4.

    The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 4831–4844 (see updates for more information).

    \begin{remark}
    \label{remark-j-shriek-not-exact}
    Let $j : U \to X$ be an open immersion of topological spaces as above.
    Let $x \in X$, $x \not \in U$. Let $\mathcal{F}$ be a sheaf of sets
    on $U$. Then $\mathcal{F}_x = \emptyset$ by Lemma \ref{lemma-j-shriek}.
    Hence $j_!$ does not transform a final object of $\Sh(U)$
    into a final object of $\Sh(X)$ unless $U = X$.
    According to our conventions in
    Categories, Section \ref{categories-section-exact-functor}
    this means that the functor $j_!$ is not left exact
    as a functor between the categories of sheaves of sets.
    It will be shown later that $j_!$ on abelian sheaves is exact,
    see Modules, Lemma \ref{modules-lemma-j-shriek-exact}.
    \end{remark}

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