Lemma 58.10.4. Let $X$ be a scheme. Let $U \subset X$ be a dense open. Assume
$U \to X$ is quasi-compact,
every point of $X \setminus U$ is closed, and
for every $x \in X \setminus U$ the punctured spectrum of the strict henselization of $\mathcal{O}_{X, x}$ is connected.
Then $\textit{FÉt}_ X \to \textit{Fét}_ U$ is fully faithful.
Proof. Let $Y_1, Y_2$ be finite étale over $X$ and let $\varphi : (Y_1)_ U \to (Y_2)_ U$ be a morphism over $U$. We have to show that $\varphi $ lifts uniquely to a morphism $Y_1 \to Y_2$ over $X$. Uniqueness follows from Lemma 58.10.1.
Let $x \in X \setminus U$. Set $V = U \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$. Since every point of $X \setminus U$ is closed $V$ is the punctured spectrum of $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$. By Lemma 58.10.2 we can extend the morphism $\varphi _ V : (Y_1)_ V \to (Y_2)_ V$ uniquely to a morphism $(Y_1)_{\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})} \to (Y_2)_{\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})}$. By Limits, Lemma 32.20.3 (this uses that $U$ is retrocompact in $X$) we find an open $U \subset U'_ x$ containing $x$ and an extension $\varphi '_ x : (Y_1)_{U'_ x} \to (Y_2)_{U'_ x}$ of $\varphi $. Note that given two points $x, x' \in X \setminus U$ the morphisms $\varphi '_ x$ and $\varphi '_{x'}$ agree over $U'_ x \cap U'_{x'}$ as $U$ is dense in that open (Lemma 58.10.1). Thus we can extend $\varphi $ to $\bigcup U'_ x = X$ as desired. $\square$
Your email address will not be published.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: