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Favorite properties of varieties (finite type k-schemes) checkable at closed points?

Posted by ravivakil under

Big lists
[4] Comments
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.

June 13, 2011 at 9:36 am

Open sets are determined by their intersection with the subset of closed points.

In particular, closed points are dense; and also constructible sets are determined by their intersection with the subset of closed points.

June 13, 2011 at 9:36 am

Maps of reduced finite type kschemes over algebraically closed fields are determined by their maps on closed points. (Variant: maps of finite type schemes are determined on the level of sets by their maps of closed points)

June 13, 2011 at 9:36 am

Universal injectivity of a map of varieties can be checked on the level of maps of closed points.

June 13, 2011 at 9:37 am

(Universal) surjectivity of a map of finite type k-schemes can be checked on the level of maps of closed points. Reason: the first comment.