I am wondering whether a proper morphism of schemes, which is closed by definition, has to be open. Since nobody told me this I suspect it to be false, but I can't find a counter-example. Could anyone help me (just hints are very welcome too)?
Is a proper morphism of schemes open?
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$\begingroup$
algebraic-geometry
schemes
open-map
1 Answers
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Any closed immersion is finite, hence proper, and is never open unless it surjects onto a connected component of the target. From this you can get a plethora of counterexamples.
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1It's not quite true that it's "never open unless it's surjective"--what if the base scheme is disconnected? – 2017-01-04
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0@EricWofsey Thanks, fixed! – 2017-01-04