The Stacks project

Lemma 75.5.2. Let $S$ be a scheme. Let $i : Z \to X$ be an immersion. Let $\varphi : U \to X$ be an ├ętale morphism where $U$ is a scheme. Set $Z_ U = U \times _ X Z$ which is a locally closed subscheme of $U$. Then

\[ \mathcal{C}_{Z/X}|_{Z_ U} = \mathcal{C}_{Z_ U/U} \]

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 just states 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 66.13.1. $\square$


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