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Tag 02US

17.7. A canonical exact sequence

We give this exact sequence its own section.

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 656–709 (see updates for more information).

    \section{A canonical exact sequence}
    \label{section-canonical-exact-sequence}
    
    \noindent
    We give this exact sequence its own section.
    
    \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 (2)

    Comment #3209 by denis lieberman (site) on February 23, 2018 a 1:51 am UTC

    Should j _1 j ^* be written as j _* j^ ! as the right adjoint [SGA4 V.6].

    In other words, should the j's be switched?

    Comment #3210 by denis lieberman (site) on February 23, 2018 a 3:55 am UTC

    Should j _1 j ^* be written as j _* j^ ! as the right adjoint [SGA4 V.6].

    In other words, should the j's be switched?

    Also, is it traditional to write i before j, for example, 0-->i_i^! F --> -->j_ j^* F-->0.

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