Lemma 63.2.2 (0F6G)—The Stacks project

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Lemma 63.2.2. Let $X$ be a scheme. Let $Z \subset X$ be a locally closed subscheme. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. Given $U, U' \subset X$ open containing $Z$ as a closed subscheme, there is a canonical bijection

\[ \{ s \in \mathcal{F}(U) \mid \text{Supp}(s) \subset Z\} = \{ s \in \mathcal{F}(U') \mid \text{Supp}(s) \subset Z\} \]

which is given by restriction if $U' \subset U$.

Proof. Since $Z$ is a closed subscheme of $U \cap U'$, it suffices to prove the lemma when $U' \subset U$. Then it is a special case of Lemma 63.2.1. $\square$

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