What are your favorite properties of finite type k-schemes that can be checked at closed points (or possibly at “closed geometric points” — at closed points after you base change to the algebraic closure of k)? My reason for asking: this gives a connection to the classical theory of varieties.

Any open condition will work, but please list those here. I’m looking for some “variety-specific” facts. There seem to be remarkably few.

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

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