I was writing a question, it became too long, and i decided to split it into two parts. I hope posting two questions at the same time is not a problem.
First question: Checking flat- and smoothness: enough to check on closed points?
Now let $f: X \rightarrow Y$ be a morphism of varieties.
If $f$ is smooth of relative dimension 0, i.e. étale, its preimage of a point should be a 0-dimensional regular scheme, i.e. a collection of points. But a zero dimensional union of varieties (i.e. a zero-dimensional scheme that is a variety except that integral is replaced by reduced) always has a finite set as space, right? So
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Am i correct to say that fibers of étale morphisms are always finite, i.e. étale implies quasi-finite?
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Moreover, i understood that étale morphisms are not always finite, so to finish the picture could you give an example of a non-finite étale morphism?
Thanks a lot!
PS tag "complex-geometry" is included since i'm happy to assume $k=\mathbb{C}$.
Best Answer
For an exemple of a non-finite étale morphism, simply consider an open immersion like $\mathbb{A}^1 \setminus \{0\} \to \mathbb{A}^1$. You can even give an example of a surjective étale non-finite morphism by considering an open covering like $(\mathbb{A}^1 \setminus \{0\}) \amalg (\mathbb{A}^1 \setminus \{1\}) \to \mathbb{A}^1$.