The Stacks project

Lemma 59.79.3. Let $i : Z \to X$ be a closed immersion of schemes. Let $j : U \to X$ be the inclusion of the complement of $Z$. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. There is a distinguished triangle

\[ i_*R\mathcal{H}_ Z(\mathcal{F}) \to \mathcal{F} \to Rj_*(\mathcal{F}|_ U) \to i_*R\mathcal{H}_ Z(\mathcal{F})[1] \]

in $D(X_{\acute{e}tale})$. This produces an exact sequence

\[ 0 \to i_*\mathcal{H}_ Z(\mathcal{F}) \to \mathcal{F} \to j_*(\mathcal{F}|_ U) \to i_*\mathcal{H}^1_ Z(\mathcal{F}) \to 0 \]

and isomorphisms $R^ pj_*(\mathcal{F}|_ U) \cong i_*\mathcal{H}^{p + 1}_ Z(\mathcal{F})$ for $p \geq 1$.

Proof. To get the distinguished triangle, choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $. Then we obtain a short exact sequence of complexes

\[ 0 \to i_*\mathcal{H}_ Z(\mathcal{I}^\bullet ) \to \mathcal{I}^\bullet \to j_*(\mathcal{I}^\bullet |_ U) \to 0 \]

by the discussion above. Thus the distinguished triangle by Derived Categories, Section 13.12. $\square$

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