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Suppose I have a morphism of Noetherian schemes over a field $k$ (if one needs this then assume $k$ is algebraically closed) $f:C'\rightarrow S$ which is finite with geometrically connected and reduced fibers. Is $f$ an isomorphism?

flag	asked Nov 24 at 23:08 HNuer 915●2●7

	Note that this is true if $S$ is normal, by Zariski's main theorem. – Damian Rössler Nov 26 at 8:07
	Also note that if you do not assume that $S$ is surjective, then any closed immersion gives a counterexample (Jason Starr's answer is a closed immersion but you can see that he assumed that you wanted the morphism to be surjective). – Damian Rössler Nov 26 at 8:10

Jason Starr · Accepted Answer · 2012-11-24 23:39:15Z UTC

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No: $\text{Spec} k[\epsilon]/\langle \epsilon \rangle$ mapping to $\text{Spec} k[\epsilon]/\langle \epsilon^2 \rangle$ .

answered Nov 24 at 23:39

Jason Starr

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Ray Hoobler · Answer 2 · 2012-11-25 21:39:02Z UTC

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Or, perhaps more to the point, the map from the normalization to a cusp on a curve. Note that such a map is an isomorphism on points without being an isomorphism of schemes.

answered Nov 25 at 21:39

Ray Hoobler
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Fiber above the cusp is not reduced. – René Pannekoek Nov 26 at 9:39

Yes. You are right. I was thinking only about the isomorphism on points, not the fiber! – Ray Hoobler Nov 28 at 17:24

Mohan · Answer 3 · 2012-11-25 16:36:09Z UTC

Though any closed embedding will give a counter example, it is true if you ask whether it is an isomorphism to the (scheme theoretic) image. In other words, the map is an isomorphism from $C'\to f(C')$ where $f(C')$ is thought of as the scheme theoretic image (which makes sense since the map is assumed to be finite), with $k$ algebraically closed.

Does finite+reduced fibers+connected fibers imply isomorphism?

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