Consequences of Properness in Algebraic Geometry

Question

Let's call a variety $X$ proper if the projection $Y\times X\rightarrow X$ is a closed map (where $Y$ is any variety). I read in Vakil's notes that properness is a version compactness in algebraic geometry (AG) (so it is supposed to compensate for the failure of Hausdorff condition). I want to understand the consequences of this fact. For instance, what kind of operations/properties do we have in the projective space (due to properness) that do not exist in the affine case?

Answer 1

In topology we have concepts of compactness and a proper map which is one where the preimage of any compact set is compact. First let's see what makes these concepts useful in topology other than being able to always work with finite covers. Compact spaces satisfy some nice properties. For example, continuous images of compact spaces are compact, closed subsets of compact spaces are compact, and compact subspaces of a Hausdorff space are closed.

Let $f: X \to Y$ be a continuous map of spaces. If $X$ is compact and $Y$ Hausdorff then $f$ is closed and proper. If $f$ is proper and $Y$ is locally compact and Hausdorff then $f$ is closed. If $X$ is Hausdorff and $Y$ is locally compact and Hausdorff then $f$ is proper if and only if $f \times \operatorname{id} : X \times Z \to Y \times Z$ is closed for any $Z$. In particular, $X$ is compact if and only if $X \to \operatorname{pt}$ is proper so if $X$ is Hausdorff, it is compact if and only if the projection $X \times Z \to Z$ is closed or any $Z$.

An argument can be made that in topology, most of the time when you are using a compactness or properness assumption you are probably using one of these properties above. So in some sense, the important thing is not compactness or properness itself, but these particular properties that proper maps and continuous maps of compact Hausdorff spaces have. Now, in algebraic geometry, because of the Hausdorff assumption, we can't expect these properties to be true in general, but since they're such useful properties, we instead make them definitions. This follows a general philosophy of algebraic geometry that the important thing isn't the properties that a space satisfies but rather the properties morphisms satisfy.

We call a morphism $f: X \to Y$ separated if the diagonal morphism $X \to X \times_Y X$ is a closed immersion. If you're just working with varieties you probably won't have to worry about this. This takes the place of Hausdorff and in fact is a condition that maps of Hausdorff spaces satisfy. We call a morphism proper if it is separated and if for any $Z$, $f \times \operatorname{id} : X \times Z \to Y \times Z$ is a closed map. As you can see this is exactly the equivalent definition of proper in the topological sense when $X$ and $Y$ were Hausdorff and $Y$ locally compact. Finally, we say $X$ is proper (respectively separated) if the morphism $X \to \operatorname{pt}$ is proper (respectively separated), again this is the analogue of the fact that a space is compact if and only if the map to a point is proper.

Now, with these definitions, we get analogues of the properties of continuous maps of compact spaces and of proper maps stated above but for proper morphism taking the place of proper continuous map and separated taking the place of Hausdorff. Namely, any proper morphism is closed, if $f:X \to Y$ is a morphism with $X$ proper and $Y$ separated then $f$ is a proper morphism and in this case $f(X)$ is also proper (see here). This is the analogue of the fact that continuous images of compact spaces are compact. This last statement illustrates one of the usefulnesses of properness: we couldn't even talk about $f(X)$ as a variety if $f$ weren't closed. Properness gives us a good grasp of the image of a morphism. Finally, when we are working over the complex numbers, we have the complex topology that we can give $X$ instead of the Zariski one. Then we have that $X$ is proper as a variety if and only if it is compact in the complex topology and a morphism $f:X \to Y$ of varieties is a proper morphism if and only if it is a proper continuous map in the complex topology. So over the complex numbers the concepts really do coincide.

Now I'll give some more algebraic applications of proper morphisms and why we want them. First, like I said, properness gives us a good way to talk about images of morphisms since they will be closed subvarieties. In particular, we need proper to have a well defined pushforward of divisors (or more generally algebraic cycles). So the Picard group, or more generally the Chow groups, a sort of homology theory for varieties, are only functorial for proper morphisms.

Similarly, for any morphism $f:X \to Y$ and coherent sheaf $\mathcal{F}$ on $X$ we can define the pushfoward sheaf $f_*\mathcal{F}$. In general this sheaf is not coherent but if $f$ is proper then it is. In fact, for varieties $f$ is proper if and only if the pushfoward $f_*$ always preserves coherent sheaves.

A more down to earth application is the following well known theorem about projective varieties which holds for any proper varieties. If $X$ is proper, then the only regular functions on $X$ are constants.

Finally, one very useful but technical theorem that proper morphisms satisfy is the proper base change theorem. This gives you, for example, a very useful criterion for a pushforward of a sheaf to be locally free: under certain conditions, if $f:X \to Y$ is proper and $\mathcal{F}$ is a coherent sheaf on $X$, then $f_*\mathcal{F}$ is locally free if and only if the dimension of the global sections of $\mathcal{F}$ are constant on the fibers of $f$.