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
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
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
Comments (2)
Comment #6695 by old friend on
Comment #6901 by Johan on