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
$V_{i, j} \to U$ is an isomorphism,
$u$ is not in the image of $V_{i, j} \cap V_{i', j'}$ unless $i = i'$ and $j = j'$, and
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 finite disjoint union decomposition
\[ X_ U = W \amalg \coprod \nolimits _{i, j} V_{i, j} \]
of schemes such that
$V_{i, j} \to U$ is an isomorphism,
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 finite disjoint union decomposition
\[ X_ U = \coprod \nolimits _ j V_ j \]
of schemes 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.
$\square$
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