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)?
2026-03-26 07:58:18.1774511898
Is a proper morphism of schemes open?
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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.