The Stacks project

Lemma 20.20.1. Let $i : Z \to X$ be a closed immersion of topological spaces. For any abelian sheaf $\mathcal{F}$ on $Z$ we have $H^ p(Z, \mathcal{F}) = H^ p(X, i_*\mathcal{F})$.

Proof. This is true because $i_*$ is exact (see Modules, Lemma 17.6.1), and hence $R^ pi_* = 0$ as a functor (Derived Categories, Lemma 13.16.9). Thus we may apply Lemma 20.13.6. $\square$

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