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Tag 02UT

Chapter 17: Sheaves of Modules > Section 17.7: A canonical exact sequence

Lemma 17.7.1. Let $X$ be a topological space. Let $U \subset X$ be an open subset with complement $Z \subset X$. Denote $j : U \to X$ the open immersion and $i : Z \to X$ the closed immersion. For any sheaf of abelian groups $\mathcal{F}$ on $X$ the adjunction mappings $j_{!}j^*\mathcal{F} \to \mathcal{F}$ and $\mathcal{F} \to i_*i^*\mathcal{F}$ give a short exact sequence $$ 0 \to j_{!}j^*\mathcal{F} \to \mathcal{F} \to i_*i^*\mathcal{F} \to 0 $$ of sheaves of abelian groups. For any morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of abelian sheaves on $X$ we obtain a morphism of short exact sequences $$ \xymatrix{ 0 \ar[r] & j_{!}j^*\mathcal{F} \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & i_*i^*\mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & j_{!}j^*\mathcal{G} \ar[r] & \mathcal{G} \ar[r] & i_*i^*\mathcal{G} \ar[r] & 0 } $$

Proof. The functoriality of the short exact sequence is immediate from the naturality of the adjunction mappings. We may check exactness on stalks (Lemma 17.3.1). For a description of the stalks in question see Sheaves, Lemmas 6.31.6 and 6.32.1. $\square$

    The code snippet corresponding to this tag is a part of the file modules.tex and is located in lines 662–692 (see updates for more information).

    \begin{lemma}
    \label{lemma-canonical-exact-sequence}
    Let $X$ be a topological space.
    Let $U \subset X$ be an open subset with complement $Z \subset X$.
    Denote $j : U \to X$ the open immersion and
    $i : Z \to X$ the closed immersion.
    For any sheaf of abelian groups $\mathcal{F}$ on $X$
    the adjunction mappings $j_{!}j^*\mathcal{F} \to \mathcal{F}$ and
    $\mathcal{F} \to i_*i^*\mathcal{F}$ give a short exact
    sequence
    $$
    0 \to j_{!}j^*\mathcal{F} \to \mathcal{F} \to i_*i^*\mathcal{F} \to 0
    $$
    of sheaves of abelian groups. For any morphism
    $\varphi : \mathcal{F} \to \mathcal{G}$ of abelian sheaves on $X$
    we obtain a morphism of short exact sequences
    $$
    \xymatrix{
    0 \ar[r] &
    j_{!}j^*\mathcal{F} \ar[r] \ar[d] &
    \mathcal{F} \ar[r] \ar[d] &
    i_*i^*\mathcal{F} \ar[r] \ar[d] &
    0 \\
    0 \ar[r] &
    j_{!}j^*\mathcal{G} \ar[r] &
    \mathcal{G} \ar[r] &
    i_*i^*\mathcal{G} \ar[r] &
    0
    }
    $$
    \end{lemma}
    
    \begin{proof}
    The functoriality of the short exact sequence is
    immediate from the naturality of the adjunction mappings.
    We may check exactness on stalks (Lemma \ref{lemma-abelian}).
    For a description of the stalks in question see
    Sheaves, Lemmas \ref{sheaves-lemma-j-shriek-abelian}
    and \ref{sheaves-lemma-stalks-closed-pushforward}.
    \end{proof}

    Comments (1)

    Comment #1798 by Keenan Kidwell on January 19, 2016 a 11:40 pm UTC

    Can't one give a one-sentence proof of the naturality by invoking the naturality of the counit and unit of the adjunctions being used? Something like "the functoriality of the short exact sequence is immediate from the naturality of the adjunction mappings?"

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