## 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.
$\square$

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