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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

  • Am i correct to say that fibers of étale morphisms are always finite, i.e. étale implies quasi-finite?

  • 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 $k=\mathbb{C}$.

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What's your definition of étale? If you only require an étale morphism to be locally of finite presentation, then an infinite disjoint union of copies of $\operatorname{Spec} k$ is étale over $\operatorname{Spec} k$, for obvious reasons. – Zhen Lin Aug 15 '12 at 11:23
@Zhen, i'm following Hartshorne, III.10.2: $f:X \rightarrow Y$ is smooth of relative dimension $n$ if it is flat and for all $y \in Y$ the scheme $X_{\overline{y}} := X_y \times_{k(y)} \overline{k(y)}$ is regular and equidimensional of dimension $n$. Now étale means smooth of relative dimension 0. – Joachim Aug 15 '12 at 11:54
Hartshorne only defines ‘smooth of relative dimension $n$’ for morphisms of finite type. Being étale and of finite type is stable under base change, so the fibre over $x$ is étale and of finite type over $\operatorname{Spec} \kappa (x)$, where $\kappa (x)$ is the residue field at $x$ – and so the fibre must be a finite discrete set of points. – Zhen Lin Aug 15 '12 at 12:14
@ZhenLin Thanks! By the way, i defined $f$ as a morphism of varieties, and if i am correct those are always of finite type. – Joachim Aug 15 '12 at 12:25
up vote 5 down vote accepted

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 $(\mathbb{A}^1 \setminus \{0\}) \amalg (\mathbb{A}^1 \setminus \{1\}) \to \mathbb{A}^1$.

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Thanks. How about étale morphisms between projective irreducible varieties, are those always finite? (It seems to me that noncompleteness is the key-point of your examples) – Joachim Aug 16 '12 at 14:03
Yes, a morphism between proper varieties is proper. And a proper quasi-finite morphism is finite. – AFK Aug 16 '12 at 23:05
Thanks, that settles it. – Joachim Aug 17 '12 at 7:15

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