The Stacks project

41.18 Étale local structure of étale morphisms

This is a bit silly, but perhaps helps form intuition about étale morphisms. We simply copy over the results of Section 41.17 and change “closed immersion” into “isomorphism”.

Lemma 41.18.1. Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots , x_ n \in X$ be points having the same image $s$ in $S$. Assume $f$ is étale at each $x_ i$. Then there exists an étale neighbourhood $(U, u) \to (S, s)$ and opens $V_{i, j} \subset X_ U$, $i = 1, \ldots , n$, $j = 1, \ldots , m_ i$ such that

  1. $V_{i, j} \to U$ is an isomorphism,

  2. $u$ is not in the image of $V_{i, j} \cap V_{i', j'}$ unless $i = i'$ and $j = j'$, and

  3. any point of $(X_ U)_ u$ mapping to $x_ i$ is in some $V_{i, j}$.

Proof. An étale morphism is unramified, hence we may apply Lemma 41.17.1. Now $V_{i, j} \to U$ is a closed immersion and étale. Hence it is an open immersion, for example by Theorem 41.14.1. Replace $U$ by the intersection of the images of $V_{i, j} \to U$ to get the lemma. $\square$

Lemma 41.18.2. Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots , x_ n \in X$ be points having the same image $s$ in $S$. Assume $f$ is separated and $f$ is étale at each $x_ i$. Then there exists an étale neighbourhood $(U, u) \to (S, s)$ and a disjoint union decomposition

\[ X_ U = W \amalg \coprod \nolimits _{i, j} V_{i, j} \]

such that

  1. $V_{i, j} \to U$ is an isomorphism,

  2. the fibre $W_ u$ contains no point mapping to any $x_ i$.

In particular, if $f^{-1}(\{ s\} ) = \{ x_1, \ldots , x_ n\} $, then the fibre $W_ u$ is empty.

Proof. An étale morphism is unramified, hence we may apply Lemma 41.17.2. As in the proof of Lemma 41.18.1 the morphisms $V_{i, j} \to U$ are open immersions and we win after replacing $U$ by the intersection of their images. $\square$

The following lemma is in some sense much weaker than the preceding one but it may be useful to state it explicitly here. It says that a finite étale morphism is étale locally on the base a “topological covering space”, i.e., a finite product of copies of the base.

Lemma 41.18.3. Let $f : X \to S$ be a finite étale morphism of schemes. Let $s \in S$. There exists an étale neighbourhood $(U, u) \to (S, s)$ and a disjoint union decomposition

\[ X_ U = \coprod \nolimits _ j V_ j \]

such that each $V_ j \to U$ is an isomorphism.

Proof. An étale morphism is unramified, hence we may apply Lemma 41.17.3. As in the proof of Lemma 41.18.1 we see that $V_{i, j} \to U$ is an open immersion and we win after replacing $U$ by the intersection of their images. $\square$


Comments (0)


Post a comment

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04HK. Beware of the difference between the letter 'O' and the digit '0'.