Properness and separatedness

Warning: this is way too long. I had to split in subsections, please consider using them.

Topology background

Recall (or believe me) that a topological space X is called T2, or Hausdorff, if any two points have disjoint neighborhoods, This is equivalent to the diagonal in XxX being closed. X is called compact if every open cover has a finite subcover. This is equivalent to saying that for every other topological space Y the projection XxY to Y is closed.

Note that for real and complex manifolds, the product as manifolds coincides with the product as topological spaces, so no confusion is possible when mentioning XxX and XxY.

T2 property says there are enough open sets on the topology; it remains true if you change topology on X to a finer one (with more open sets).

Compactness says there are not too many open sets; it remains true if you change topology on X to a coarser one (with fewer open sets).

Compact and Hausdorff is so to speak the perfect middle, and a theorem states that if a compact topology on X is finer than a Hausdorff one, they must coincide.

This has led to renaming, so that some parts of maths use compact for compact and Hausdorff, and quasicompact for compact. That's what Grothendieck learned.

Note that there is also a relative version, namely a continuous map of topological spaces f:X\to B is separated if the diagonal X to Xx_BX is a closed embedding (a homeomorphism with its image, which is closed) and it is proper if for every other continuous map Y\to B the projection Xx_BY\to Y is closed.

As usual, the absolute property for X corresponds to the relative property for the unique continuous map X to a point (i.e., a/the final object in the category of topological spaces). This is rarely mentioned for manifolds because fibered products do not always exist, but when they do they agree with those for  topological spaces.

Separatedness and properness for morphisms of schemes


For schemes, fiber products (finite limits) always exist, but the forgetful functor to topological spaces doesn't preserve fiber products.

Thus Grothendieck was led to define the following properties of a morphisms f:X \to Y of schemes: 
f is separated if the diagonal X\to Xx_YX is a closed embedding; f is universally closed if every base change of f is closed; f is proper if it is separated and universally closed.

Note that the naming is supposed to remind one of the corresponding properties of continuous maps of topological spaces, but is different so we can still use the topological properties if we need to. Note also the precise correspondences with standard [used by analysts] topology terminology are

separated : Hausdorff
universally closed: compact [quasicompact]
proper : Hausdorff and compact [compact]

Affine morphisms are always separated (exercise) so any morphism of schemes has locally closed diagonal (exercise).

Most schemes one encounters are separated over whatever base one is using, typically Spec k with k an algebraically closed field in classical algebraic geometry, so much so that at the beginning this property was made part of the definition of scheme, just as it was for the definition of topological space. In either case soon the more lax definition was found to be very useful.

A lot of the theorems one is used to in point set topology have an algebraic geometry counterpart, with the same proof (once one takes into account that fibered products aren't the same!):
  1. If X and Y are separated over S, then any S-morphism X \to Y is also separated;
  2. If X is proper over S, and Y is separated over S, then any S-morphism X \to Y is proper.
  3. A composition fo separated morphism is separated, a composition of proper morphisms is proper.
I suggest you do not try and prove this using the definition, but rather the very practical valuative criteria discussed in the next section. 

Example Let k be a field. X:=A^1_k is not proper over Spec k, because the projection Xx_kX \to X is not closed. Note that X\times_kX=A^2_k; the subset C:={xy=0} is closed, but its image via the projection (x,y)\mapsto y is {y≠0} which is open and dense, but not closed. 

Example Let k be a field. If we glue two copies of A^1_k along the identity isomorphism of the open subsets D(t) complement of the origin, we get an affine line with two origins O and O'. This is not separated, since the diagonal only contains (O,O) and (O',O') but its closure also contains (O,O') and (O',O).

These two examples are very important. They can be generalized, replacing A^1with an open subset of any affine smooth curve over an algebraically closed field k.

Example Let R be any scheme, n>0 an integer. Then if we define P:=P^n_R and X:=Spec R, the natural projection P^n_R\to R is proper.

Example Every open embedding is separated; every closed embedding is proper.

Example If X is proper over a field k, then every morphism X\to A^1_k has discrete image. In particular, if X is also reduced and connected, then O_X(X)=k. This may remind you that every holomorphic function on a compact connected complex manifold is constant. In particular an affine scheme of finite type over a field k is proper over k if and only if it is zero dimensional.


Valuative criteria for separatedness and properness

Our motivation are key criteria for separatedness and properness in the topological sense applying to very special spaces. A topological space X is metrizable if its topology topology can be induced by a metric; it is locally metrizable if it has an open cover by metrizable spaces.

A continuous map f:X \to Y of locally metrizable spaces is separated if and only for every sequence x_n in X such that f(x_n) converges to y in Y, there exist at most one x in f⁻¹(y) such that there is a convergent subsequence of x_n with limit x.

A continuous map f:X \to Y of locally metrizable spaces is topologically proper if and only if a sequence x_n in X admits a convergent subsequence iff f(x_n) admits a convergent subsequence in Y.

For an arbitrary morphism of schemes f:X\to Y we have the so called valuative criteria, which are based on the following commutative diagram.

Insert commutative square diagram
U X
C Y
as well as a red arrow from C to X

Valuative criteria A morphism f:X\to Y is separated/universally closed/proper iff for every diagram as above with C the spectrum of a valuation domain A and U the spectrum of its fraction field K(A) there is at most one/there is one/there is exactly red arrow keeping the diagram commutative.

You can find a proof online, and use the criterion to easily prove a number of properties:
  1. A morphism of schemes X\to Y is separated/universally closed/proper iff f_red:X_red \to Y_red is;
  2. Given morphisms f:X\to Y, g:Y\to Z and h:=g∘ f:X\to Z, if [add details]
What is unclear is how you should understand it, especially if like me you have no idea what a valuation domain is.

Valuative criteria: classical versions


We work now in classical algebraic geometry, that is all schemes are locally of finite type over a fixed algebraically closed field k. We use the same commutative diagram.

Geometrical valuative criteria A morphism f:X\to Y is separated/universally closed/proper iff for every diagram as above, with C a smooth irreducible curve over k and U an open nonempty subset, there is at most one/there is one/there is exactly red arrow keeping the diagram commutative.

Exercise Assuming the criteria show that it still works if you assume C = Spec A to be affine, and U to be C minus one closed point p.

Comment about this means that a curve with a double point is essentially the only example of non separatedness, and a curve minus one point the only example of non-universally closedness.

Connecting the general and classical case


If you are willing to go slightly beyond schemes of finite type for your criterion, show that you can further reformulate it by replacing C and U with Spec O_{C,p} and Spec K(C); this way it looks more similar to the valuative criteria, in nat O_{C,p} is a discrete valuation ring and K(C) its field of fractions. Note that restricting the schemes allowed let us also restrict the valuative domain allowed.

If you are familiar with formal completions, you can actually use exactly one C and one U, namely C=Spec k[[t]] and U=Spec k((t)).

I will have to add a lot to this page.





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