Lemma 59.79.3 (0A45)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.79: Cohomology with support in a closed subscheme
Lemma 59.79.3 (cite)

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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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