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.