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Bleg: unramified and radicial morphisms

Posted by Akhil Mathew under

algebraic geometry,

blegs
[2] Comments
I’ve been trying to write up a bunch of notes on Zariski’s Main Theorem and its applications, which is the main reason I haven’t blogged in a while. They should be done (as in, in a rough but complete state) soon, after which I will post them. In writing them up, I ran into the following doubt. Can anyone clarify? I don’t really know whether this is MO level, so I’ll just ask it here.

*Is an unramified, radicial morphism an immersion?*

I know this is true for etale morphisms (SGA I, expose 1.5); in that case the map is even open.

**Edit: **I know this is also true if the morphism is unramified and proper (it follows easily from the fact that a proper and quasi-finite morphism is finite). If the properness hypothesis is unnecessary, then we have a very nice functorial description of what an immersion is.

**Edit 2: **Never mind, this is false.

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January 18, 2011 at 3:42 pm

Why not ask on math stackexchange?

January 18, 2011 at 3:45 pm

I might try that if I don’t get an answer here. As it is I’ve been asking quite a few questions in the recent past, though.