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
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 $b = \text{pr}_2|_ W$. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: