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Tag 01AK

Chapter 17: Sheaves of Modules > Section 17.3: The abelian category of sheaves of modules

Lemma 17.3.4. Let $j : U \to X$ be an open immersion of topological spaces. The functor $j_! : \textit{Ab}(U) \to \textit{Ab}(X)$ is exact.

Proof. Follows from the description of stalks given in Sheaves, Lemma 6.31.6. $\square$

    The code snippet corresponding to this tag is a part of the file modules.tex and is located in lines 280–285 (see updates for more information).

    \begin{lemma}
    \label{lemma-j-shriek-exact}
    Let $j : U \to X$ be an open immersion of topological spaces.
    The functor $j_! : \textit{Ab}(U) \to \textit{Ab}(X)$
    is exact.
    \end{lemma}
    
    \begin{proof}
    Follows from the description of stalks
    given in Sheaves, Lemma \ref{sheaves-lemma-j-shriek-abelian}.
    \end{proof}

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