Lemma 76.6.2. Let S be a scheme. Let i : Z \to X be an immersion of algebraic spaces over S. Let \varphi : U \to X be an étale morphism where U is a scheme. Set Z_ U = U \times _ X Z which is a locally closed subscheme of U. Then
canonically and functorially in U.
Proof. Let T \subset X be a closed subspace such that i defines a closed immersion into X \setminus T. Let \mathcal{I} be the quasi-coherent sheaf of ideals on X \setminus T defining Z. Then the lemma follows from the fact that \mathcal{I}|_{U \setminus \varphi ^{-1}(T)} is the sheaf of ideals of the immersion Z_ U \to U \setminus \varphi ^{-1}(T). This is clear from the construction of \mathcal{I} in Morphisms of Spaces, Lemma 67.13.1. \square
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