One of the standard facts in algebraic geometry is that a projective scheme is proper. In the language of varieties, one says that the image of a projective variety is closed. The precise statement one proves is that:

Theorem 1Let be any variety over the algebraically closed field . Let be a closed subset. Then the projection of to is closed.

This statement is sometimes phrased as saying that is “complete.” In many ways, it is a compactness statement. Recall that a compact space has the property that the map for any is a closed map. The converse is also true if there are reasonable assumptions (I think locally compact Hausdorff on will do it). Of course, these reasonable assumptions don’t apply to a variety with the Zariski topology, but they do if we are working with a variety (say, a quasiprojective variety) over and we can define the *complex topology*. And in fact, it turns out that projective varieties are indeed compact in the complex topology. The scheme-theoretic version is a little different. First, the theorem for varieties said that the object was “complete” in a certain sense. But the philosophy of Grothendieck is to consider not so much schemes but morphisms of schemes, and to do everything in a relative context. The idea is:

Definition 2A morphism isproperif it is separated, of finite type, and universally closed (i.e. any base change is a closed morphism).

I don’t really want to define separated in this post. On the other hand, I should explain what’s going on for quasiprojective varieties over an algebraically closed fields. In this case, the conditions of finite type and separated are redundant. The key condition is *universal closedness*, which won’t be satisfied for a general morphism. For instance, to say that is universally closed implies the first lemma about closedness of the projection.

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