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It is well known that topology and algebraic geometry assign different meanings to the word "proper". Let us recall the relevant definitions from topology (and we work in the context of topological spaces):

  • A map $f:X\to Y$ is called separated iff the diagonal $X\hookrightarrow X\times_YX$ is a closed inclusion.

  • A map $f:X\to Y$ is called proper iff for every net $(x_\alpha)_\alpha$ in $X$ such that $(f(x_\alpha))_\alpha$ converges to $y\in Y$, there exists a subnet converging to $x\in X$ with $f(x)=y$. Equivalently, for every compact subspace $K\subseteq Y$ its inverse image $f^{-1}(K)\subseteq X$ is compact (compact means every open cover admits a finite subcover) EDIT: The condition of compact sets having compact inverse images is strictly weaker than properness. Here is a counterexample: take the identity map $\{0,1\}\to\{0,1\}$ where the domain has the discrete topology and the target has the topology whose open sets are $\{\},\{0\},\{0,1\}$. The inverse image of a compact set under this map is compact (every subspace of $\{0,1\}$ with the discrete topology is compact), however this map is not proper in the sense of the definition above.

The above are the definitions from topology. In algebraic geometry, a map is called "proper" iff it is proper and separated in the sense defined above.

Is there a good reason to prefer one definition of the word "proper" over the other? Does the answer depend on whether one is doing topology or algebraic geometry?

Note: I am not asking for which meaning was first historically, nor for whether someone should be blamed for changing the meaning of an existing mathematical term, nor for people's personal preferences between the two meanings. Rather I am looking for a mathematical reason why one definition might be preferred over the other.

For example, both "separated" and "proper" are preserved under pullback, thus so is "separated and proper". But:

Is there another important formal property of "separated and proper" which doesn't immediately follow from corresponding formal properties of "separated" and "proper" individually?

Or, in the other direction:

Is there some important property of "proper" maps which cannot be formally derived from results about "separated" and "separated and proper" maps? Is there any argument using the notion of a "proper" map which would become inconvenient if one only had access to the notions of "separated" and "separated and proper"?

For clarity, let me suggest that answers should specify whether they are using the word "proper" as I have defined it above or in the algebro-geometric sense (called "separated and proper" above).

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    $\begingroup$ It is useful to know that the diagonal of a proper morphism is proper. This makes proper maps formally analogous to étale maps in some sense. $\endgroup$ – crystalline May 4 '19 at 20:44
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    $\begingroup$ Transferring from the Zariski topology to the analytic topology of the $\mathbb C$-points, this is just the relative version of the usual disagreement about whether 'compact' requires 'Hausdorff'. $\endgroup$ – Kevin Casto May 4 '19 at 21:21
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    $\begingroup$ @KevinCasto: Yes, but so what? I don't see how this helps answer the question. $\endgroup$ – John Pardon May 5 '19 at 12:49
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I think the separated condition should be included, at least for purposes of sheaf cohomology. First let's consider the case where Y is a point. if X is a compact Hausdorff space, then sheaf cohomology over X commutes with filtered colimits. This is a nice property which fails if X is non-Hausdorff. For example, take X to be two copies of [0,1] glued along U=[0,1/2) via the identity map. For any sheaf F on U, we can extend it by zero to X, and the cohomology will sit in a long exact sequence with two copies of the compactly supported cohomology of F and one copy of the ordinary cohomology of F. By this construction we reduce to the fact that cohomology on U doesn't commute with filtered colimits.

Now, in the relative situation, there's another important property of separated proper maps f: X --> Y: the "proper base change theorem" holds, so the derived pushforward of sheaves commutes with pullback. (In particular, one can calculate the stalks of the pushforward in terms of cohomology of the fibers.) Similar to the above you can make a counterexample to this without the separated hypothesis.

The reason I think these facts about sheaf cohomology are important is that they form the basis for Grothendieck's "six functor formalism" and Grothendieck-Verdier duality. Those give a powerful and compact way of extracting the algebraic essence of a given topological situation. Actually, the six functor formalism also exists for at least mildly non-separated proper maps (e.g., those which are locally separated), but the "exceptional pushforward" f_! (which always satisfies the analog of "proper base change") and the ordinary pushforward f_* need no longer agree. This is just like for maps f which are separated, but not proper. (Actually, the proper but not separated situation is even worse somehow, because there is not even a natural transformation f_! -> f_* which one can ask to be an isomorphism. What's going on is that a condition on the diagonal of a morphism is more basic and fundamental than a condition on the morphism itself.)

To summarize, I would say that a crucial property of proper maps is that f_! = f_*, and for this one needs the separatedness. Philosophically, the issue is the one raised in the comments: for many purposes, given a class of maps, one wants a stronger closure than just closure under pullbacks: one also wants closure under passing to diagonals. This stronger closure is in essence used to build up a six functor formalism, surely among other things.

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In topology, separated and proper maps satisfy a certain lifting property which allows to show that separated and proper maps are contained in a certain orthogonal of several closed maps of finite topological spaces. See https://ncatlab.org/nlab/show/Tychonoff+theorem#ProofViaTaimanovTheoremAndLiftingProperties , Remark 2.4 and Question 2.5 for details.

In view of the remark in the previous answer

one wants a stronger closure than just closure under pullbacks: one also wants closure under passing to diagonals. This stronger closure is in essence used to build up a six functor formalism, surely among other things.

the class of separated and proper maps can probably be defined as the orthogonal wrt unique lifting property, and such orthogonals (wrt unique lifting property) are necessarily closed under passing to diagonals.

Well, I have not checked that being separated is enough, but it shouldn't be hard.

When the separated and proper maps is a map to a single point, this lifting property is known as Taimanov theorem and says that under some obviously necessary conditions a map to a compact Hausdorff space can be extended from a dense subspace. https://ncatlab.org/nlab/show/Taimanov+theorem

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