The Stacks project

Lemma 59.46.1. Let $i : Z \to X$ be a closed immersion of schemes. Let $U, U'$ be schemes étale over $X$. Let $h : U_ Z \to U'_ Z$ be a morphism over $Z$. Then there exists a diagram

\[ \xymatrix{ U & W \ar[l]_ a \ar[r]^ b & U' } \]

such that $a_ Z : W_ Z \to U_ Z$ is an isomorphism and $h = b_ Z \circ (a_ Z)^{-1}$.

Proof. Consider the scheme $M = U \times _ X U'$. The graph $\Gamma _ h \subset M_ Z$ of $h$ is open. This is true for example as $\Gamma _ h$ is the image of a section of the étale morphism $\text{pr}_{1, Z} : M_ Z \to U_ Z$, see Étale Morphisms, Proposition 41.6.1. Hence there exists an open subscheme $W \subset M$ whose intersection with the closed subset $M_ Z$ is $\Gamma _ h$. Set $a = \text{pr}_1|_ W$ and $b = \text{pr}_2|_ W$. $\square$

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