## 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}
```

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