What are your favorite open and closed and locally closed conditions (under reasonable hypotheses). (My motivation for doing all three of these at once: to keep people saying “the reduced locus is open” and “the nonreduced locus is closed”.)

For simplicity, please put one per comment, so people can respond.

Related question: I was intending to have one list on properties that can be checked at closed points, but I realized that all examples I had in mind were of two sorts, which I am asking separately. (i) Some are true because they are open conditions, and thus for a quasicompact scheme it suffices to check them at closed points (see the comment after Important Exercise 6.1.E). (ii) Some are true because they a variety-specific result, usually involving the Nullstellensatz. Are there any things you like to check at closed points that aren’t of either of these two sorts?

Advertisements