Download Separated and proper morphisms

SEPARATED AND PROPER MORPHISMS
BRIAN OSSERMAN
The notions of separatedness and properness are the algebraic geometry analogues of the Hausdorff condition and compactness in topology. For varieties over the complex numbers, it is possible
to use the “analytic topology” inherited from the usual topology on C in place of the Zariski topology, and the analytic topology corresponds well to our intuition, and allows us to use many tools
from topology and algebraic topology. An ongoing theme in algebraic geometry is that many such
properties and tools can in fact be translated into the context of varieties over an arbitrary field, or
more general schemes, but the translation may take some work. Often one can proceed by finding
a more abstract version of the usual topological definition (for instance, involving products) and
then translating this abstracted definition into algebraic geometry. If the translation works well, it
should recover the classical notions when we restrict our attention to complex varieties.
1. Separated morphisms
Recalling that topological conditions such as the Hausdorff property are frequently called “separation axioms,” separatedness is the condition for (families of) schemes which plays the role of the
Hausdorff property in classical topology. The first step is to observe that the usual definition of
Hausdorff for a topological space X is equivalent to the condition that the diagonal ∆(X) ⊆ X × X
be a closed subset. We then used this idea in the context that X is a prevariety to define what
it should mean for it to be a variety. We will see later that this does indeed recover the classical
Hausdorff condition when applied to complex prevarieties, so it as well behaved as one could hope
for when translating classical topological conditions to (pre)varieties over an arbitrary field. Even
better, this idea generalizes immediately to schemes. We first define:
Definition 1.1. Given a morphism f : X → Y of schemes, the diagonal morphism ∆ : X →
X ×Y X is induced by the identity morphisms X → X on both factors.
We then have:
Definition 1.2. A morphism f : X → Y is separated if the image of ∆ : X → X ×Y X is closed.
In this case, we also say that X is separated over Y .
Example 1.3. A prevariety is a variety if and only if its associated scheme is separated over Spec k
(note that this is not completely trivial, because the underlying topological spaces of a prevariety
considered as a prevariety or as a scheme differ due to the inclusion of generic points). In particular,
the scheme associated to any quasiprojective variety is separated over Spec k.
We have:
Proposition 1.4. For any morphism f : X → Y , the associated diagonal morphism ∆ : X →
X ×Y X is an immersion. If X and Y are affine schemes, then ∆ is a closed immersion.
Proof. We first show that ∆ is a closed immersion when X = Spec R and Y = Spec S are affine.
In this case, ∆] is induced by the ring homomorphism R ⊗S R → R determined by r ⊗ r0 7→ rr0 .
This is visibly a surjective map, so it induces a closed immersion of affine schemes.
For the general case, if we cover Y by affine open subschemes {Vi }, and then cover X by affine
open subschemes {Uj } with the property that for each j, we have f (Uj ) ⊆ Vij for some ij , then we
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have affine open subschemes Uj ×Vij Uj which cover the image of X under ∆, with the preimage of
each equal to Uj . Restricting to each of these open sets we find that ∆ is the diagonal morphism
for Uj → Vij , so is a closed immersion. We conclude that ∆ is an immersion, as desired.
We immediately conclude:
Corollary 1.5. A morphism of affine schemes is separated.
Corollary 1.6. A morphism f : X → Y is separated if and only if the diagonal morphism is a
closed immersion.
Proof. Clearly if the diagonal is a closed immersion, then f is separated. Conversely, by Proposition
1.4 if ∆ has closed image it must be a closed immersion.
2. Proper morphisms
Properness is the algebraic geometry analogue of compactness, generalizing the definition of
complete varieties. The definition is very similar to that for complete varieties, although to ensure
good behavior we throw in a technical condition which is automatic for varieties.
Definition 2.1. A morphism X → S is universally closed if for all morphisms S 0 → S, we have
that the projection X ×S S 0 → S 0 is closed. A morphism is proper if it is of finite type, separated,
and universally closed.
Here we say a morphism is closed if the induced map on the underlying topological spaces is
closed.
Example 2.2. A variety is complete if and only if its associated scheme is proper over Spec k.
Remark 2.3. It is in fact not terribly hard to show that a universally closed morphism is necessarily
quasi-compact. Thus, in the definition of properness we could simply have imposed that the
morphism be locally of finite type.
3. Valuative criteria: statements
Intuitively, one can think of separatedness as (a relative version of) uniqueness of limits, and
properness as (a relative version of) existence of (unique) limits. It is not immediately obvious how
to formalize these ideas in algebraic geometry, but it turns out to be doable, via valuative criteria.
Aside from providing some intuition for separatedness and properness in terms of uniqueness and
existence of limits, the valuative criteria are extremely important when working with moduli spaces.
In this case they amount to studying the behavior of families of objects over valuation rings. We
include the valuative criterion for universal closedness separately, because most algebraic stacks are
not separated, and it is nonetheless helpful to know whether they are universally closed.
The most classical version of this sort of criterion applies to prevarieties over an algebraically
closed field k. Let X be a prevariety. Then one can think of the setup for limits as follows: let C
be a smooth curve over k, and P ∈ C a point; then consider a morphism f : C r {P } → X. We
could picture that the points f (Q) for Q ∈ C r {P } have a limit as Q approaches P if f extends to
a morphism on all of C. Uniqueness of the limit then corresponds to uniqueness of the extension.
In fact, this works with prevarieties: as we’ve mentioned previously, X is a variety if and only if
“limits are unique” in this sense, which is to say if and only if for all C, P and f as above, there
is at most one extension of f to all of C. Similarly a variety X is complete if and only if “limits
exist,” which is to say for all C, P and f as above there exists a (necessarily unique) extension of
f to all of C.
The above criterion works exactly the same if we replace C by the local scheme Spec OC,P and
C r P by Spec K(C), where K(C) is the function field of C (and the field of fractions of OC,P ).
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Now, OC,P is a discrete valuation ring, and if we want to work with schemes not necessarily of finite
type over an algebraically closed field, we should consider arbitrary discrete valuation rings. In fact,
if we want to work with non-Noetherian schemes, we should consider not just discrete valuation
rings, but arbitrary valuation rings. This is precisely what the valuative criteria do. We begin by
recalling the basic definitions and properties for valuation rings.
Definition 3.1. An integral domain R with fraction field K is a valuation ring if for all x ∈ K ∗ ,
either x ∈ R or x−1 ∈ R.
Remark 3.2. The reason for the terminology is that we obtain a homomorphism ν from K ∗ to an
ordered abelian group, with the property that x ∈ R if and only if ν(x) > 0. This homomorphism
is simply obtained by setting the abelian group equal to K ∗ /R∗ , with ordering determined by the
above condition. Moreover, given x, x0 ∈ K ∗ , with x + x0 6= 0, we have ν(x + x0 ) > min{ν(x), ν(x0 )}.
Indeed, suppose without loss of generality that ν(x) 6 ν(x0 ), so that by definition x0 /x ∈ R. Then
(x + x0 )/x = 1 + x0 /x ∈ R, so ν(x + x0 ) > ν(x), as desired. Such a homomorphism is called a
valuation, and as in §I.6 of Hartshorne [1], valuation rings may be defined equivalently in terms of
existence of a valuation.
The topological space underlying a valuation ring may have arbitrary dimension, but we nonetheless have the following fact, which follows easily from Remark 3.2:
Proposition 3.3. A valuation ring is a local ring.
In particular, if R is a valuation ring, then Spec R has unique generic and closed points, corresponding to the zero ideal and the maximal ideal, respectively. Throughout our discussion, a
morphism Spec K → Spec R is always assumed to be the canonical inclusion of the generic point.
We can now state the valuative criteria.
Definition 3.4. We say a morphism f : X → Y satisfies the existence (respectively, uniqueness)
part of the valuative criterion if for every commutative diagram
Spec K
w
w
w
w
/X
w;
f
/Y
Spec R
with R a valuation ring and K its fraction field, there exists (respectively, there is at most one)
one way of filling in the dashed arrow so that the diagram remains commutative.
Theorem 3.5. Let f : X → Y be a morphism of schemes, and assume that ∆f is quasicompact.
Then f is separated if and only if it satisfies the uniqueness part of the valuative criterion.
Theorem 3.6. Let f : X → Y be a morphism of schemes, and assume that f is quasicompact.
Then f is universally closed if and only if it satisfies the existence part of the valuative criterion.
From the above two theorems, we immediately conclude the valuative criterion for properness.
Theorem 3.7. Let f : X → Y be a morphism of finite type, with ∆f quasicompact. Then f is
proper if and only if it satisfies both the existence and uniqueness parts of the valuative criterion.
Remark 3.8. Note that in the valuative criteria, if we have Y locally Noetherian and f locally of
finite type, then it is enough to only consider discrete valuation rings instead of arbitrary ones.
In addition, in this case (and more generally if X is locally Noetherian) the condition that ∆f be
quasi-comapct is automatically satisfied. Thus, under mild Noetherian hypotheses we do not need
to worry about checking this condition separately.
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More generally, the condition that ∆f is quasi-compact is an annoying one, but it comes up
naturally in a number of settings, enough so that it has a name – quasi-separated. Note that
a separated morphism is necessarily quasi-separated, since closed immersions are quasi-compact.
Thus, what the valuative criterion is really saying is that separatedness is the union of two properties: quasi-separatedness, and the valuative condition. Similarly, as we mentioned earlier, every
universally closed morphism is quasi-compact, so the valuative criterion says that being universally
closed is equivalent to being both quasi-compact and satisfying the valuative condition.
Aside from providing some intuition for separatedness and properness in terms of uniqueness
and existence of limits, the valuative criteria are extremely important when working with moduli
spaces. In this case they amount to studying the behavior of families of objects over valuation
rings. We include the valuative criterion for universal closedness separately, because most algebraic
stacks are not separated, and it is nonetheless helpful to know whether they are universally closed.
4. Valuative criteria: proofs
Each valuative criterion is of course two statements: first, that separatedness (respectively, universal closedness) implies the stated criterion, and second, that the criterion implies separatedness
(respectively, universal closedness). We thus have four statements to prove, and the proofs are
rather independent of one another. There is however substantial commonality between the proofs
that the two criteria imply separatedness and properness, so we will begin with the proofs of these
statements.
Recall the following concept:
Definition 4.1. Given a scheme X, and points x, x0 ∈ X, we say that x specializes to x0 if x0 is
in the closure of {x}. A subset Z ⊆ X is closed under specialization if for all points x, x0 ∈ X
with x ∈ Z and x specializing to x0 , we also have x0 ∈ Z.
Closed sets are visibly closed under specialization. Because schemes have (unique) generic points
for all their irreducible closed subsets, the idea is that a subset of a scheme which is reasonably
well behaved and closed under specialization should be closed. We then show that specializations
(in both the relative and absolute settings) can be detected via morphisms from the spectra of
valuation rings. Putting these statements together will quickly yield the relevant direction of the
valuative criteria.
We begin by making the statement on closed sets and specialization more precise in an important
special case:
Proposition 4.2. Suppose f : X → Y is a quasi-compact morphism. If f (X) is closed under
specialization, then f (X) is closed.
For the proof, see Lemma 4.5 of Chapter II of Hartshorne [1].
Example 4.3. To see that quasicompactness is necessary, let Y = A1k , and let X be an infinite
disjoint union of closed points of A1k , with f the inclusion. Then f (X) contains only closed points,
so is closed under specialization, but is not a closed set. Note that this morphism is even locally of
finite type, so quasicompactness is really the crucial hypothesis.
Because f is not closed, this is also an example that the criterion of Theorem 3.6 does not imply
that a morphism is closed without a quasicompactness hypothesis.
Proposition 4.2 will be enough for checking separatedness, but for universal closedness it is
convenient to develop the statement into one on closed morphisms:
Corollary 4.4. If f : X → Y is a quasicompact morphism, and for all x ∈ X, and y ∈ Y such
that f (x) specializes to y, we have some x0 ∈ X with x specializing to x0 , and f (x0 ) = y, then f is
closed.
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Proof. Given Z ⊆ X closed, give Z the reduced induced structure. Then since closed immersions
are quasicompact, Z → Y is quasicompact, by the hypotheses we have the image of Z is closed
under specialization, so f (Z) ⊆ Y is closed by Proposition 4.2.
The main use of valuation rings will be the following result, stating that specializations (in a
relative and absolute setting) can be detected via morphisms from the spectra of valuation rings.
Proposition 4.5. Let X be a scheme, and x, x0 ∈ X with x specializing to x0 . Then there exists a
valuation ring R and a morphism Spec R → X with the generic point of Spec R mapping to x, and
the closed point of Spec R mapping to x0 .
More generally, if f : X → Y is any morphism, and we have x ∈ X, and y 0 ∈ Y a specialization
of f (x), then there exists a valuation ring R, with fraction field K, and a commutative diagram
Spec K
/X
f
/Y
Spec R
such that the image of Spec K is x, and the generic and closed points of Spec R map to f (x) and
y respectively.
We omit the proof.
From this, it is not hard to prove that the stated valuative criteria imply separatedness and
universal closedness.
Proof of “if ” direction of Theorem 3.5. Suppose that the stated criterion is satisfied, so we wish
to show that f is separated. Since we have assumed ∆f quasi-compact, by Proposition 4.2 it is
enough to show that the image of ∆f is closed under specialization. Accordingly, suppose we have
z specializing to z 0 in X ×Y X, with z = ∆f (x) for some x ∈ X. By Proposition 4.5 there exists
a valuation ring R with fraction field K, and a morphism ψ : Spec R → X ×Y X such that the
generic point of Spec R maps to z, and the closed point of Spec R maps to z 0 .
Taking first and second projection yields two morphisms p1 ◦ ψ and p2 ◦ ψ from Spec R to X,
which give the same morphism Spec R → Y after composition with f . We claim that p1 ◦ ψ agrees
with p2 ◦ψ if we precompose with ι : Spec K → Spec R. It suffices to check that ψ ◦ι factors through
∆f , but since this is a morphism from Spec K, it is enough to observe that Spec K maps to z, which
by hypothesis is a point of ∆f . We thus obtain the claim, and then by hypothesis we conclude that
p1 ◦ ψ = p2 ◦ ψ, and thus that ψ factors through ∆f . It follows finally that z 0 ∈ ∆f (X), so ∆f (X)
is closed under specialization, as desired.
The following lemma is used in checking properness. We leave the proof, which uses only the
universal property of fibered products, to the reader.
Lemma 4.6. Suppose a morphism f : X → Y satisfies the existence part of the valuative criterion.
Then for every morphism Y 0 → Y , the base change X ×Y Y 0 → Y 0 of f satisfies the existence part
of the valuative criterion.
Proof of “if ” direction of Theorem 3.6. Suppose our criterion is satisfied. Let Y 0 → Y be any morphism, and X 0 = X ×Y Y 0 ;. We thus wish to show that X 0 → Y 0 is closed. Since quasicompactness
is preserved under base change, by Corollary 4.4 it is enough to show that for any x ∈ X 0 , and
y ∈ Y with f (x) specializing to y, there exists x0 ∈ X 0 with x specializing to x0 and f (x0 ) = y.
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By Proposition 4.5 there exists a valuation ring R with fraction field K and a diagram
Spec K
w
w
w
w
/ X0
w;
f
/ Y0
Spec R
with the image of Spec K being x, and the image of the generic and closed points of Spec R being
f (x) and y, respectively. By Lemma 4.6, our criterion holds also for X 0 → Y 0 , so we conclude that
we can fill in the dashed arrow so that the diagram still commutes, and setting x0 to be the image
of the closed point of Spec R completes the argument.
We now move on to checking that conversely, separatedness and universal closedness also imply
the stated criteria. We can prove the former directly, while the latter requires an additional result
on morphisms and spectra of valuation rings.
Proof of “only if ” direction of Theorem 3.5. Suppose that f is separated, and we have a diagram as
in the statement, and morphisms g1 , g2 : Spec → X making the diagram commute. We thus obtain a
morphism g : Spec R → X ×Y X such that g◦ι factors through ∆f , where ι : Spec K → Spec R is the
canonical inclusion. Because ∆f (X) is closed by hypothesis, we conclude that g(Spec R) ⊆ ∆f (X),
and thus that g factors through ∆f , since Spec R is reduced. It thus follows that g1 = g2 .
Proposition 4.7. Let R be a valuation ring with fraction field K. Suppose that f : X → Spec R
is a closed morphism. Then given a morphism Spec K → X such that the diagram
Spec K
s
s s id
Spec R
/
s9 X
s
s
f
/ Spec R
commutes, there exists a morphism filling in the dashed arrow so that the diagram still commutes.
One can rephrase the proposition as saying that for a closed morphism to Spec R, every generic
section extends to a section. We omit the proof.
Proof of “only if ” direction of Theorem 3.6. Suppose that f is universally closed. Given a diagram
as in the valuative criterion, we wish to prove existence of the dashed arrow. Consider the base
change X 0 := X ×Y Spec R → Spec R, which is closed by hypothesis. The morphism Spec K → X
then induces a morphism Spec K → X 0 , and by Proposition 4.7 we obtain a morphism Spec R → X 0
which when composed with the projection morphism X 0 → X, gives us what we want.
5. Properties of properties of morphisms
Properties of morphisms tend to have certain standard properties. One of the most important
is that they should be stable under base change: that is, if f : X → Y is a morphism, and
Y 0 → Y any other morphism, if f has the property in question, then so should the induced
morphism X ×Y Y 0 → Y 0 . Another crucial property is that the property in question be closed
under composition. We will assume (see Exercise II.3.11 (a) of Hartshorne [1], but note that
it’s easier to do (b) first and then deduce (a)), the fact that closed immersions are stable under
base change; from this one deduces easily that arbitrary immersions are likewise stable under base
change. Before moving on, we describe a consequence of this basic fact: the notion of graph of a
morphism.
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Corollary 5.1. Suppose f : X → Y is a morphism of schemes over S. Then the morphism
Γf : X → X ×S Y induced by the identity and f is a base change of ∆Y /S . In particular Γf is an
immersion, and if Y is separated over S, then Γf is a closed immersion.
This construction induces a bijection between S-morphisms f : X → Y and subschemes Γf ⊆
∼
X ×S Y with the property that p1 induces an isomorphism Γf → X.
Proof. For the first assertion, one simply checks that the square
f
X
Γf
X ×S Y
f ×id
/Y
∆
/ Y ×S Y
satisfies the universal property of the fibered product.
It is clear then that given a morphism, we obtain a subscheme which maps isomorphically to
X under p1 . Conversely, given such a subscheme, the composition p2 ◦ p−1
1 yields an S-morphism
X → Y . It is straightforward to check that these constructions are maturally inverse.
Because the diagonal ∆f is always an immersion, there are many properties of morphisms which
are always, or frequently, satisfied by ∆f . For instance, ∆f is always locally of finite type. If f is
locally of finite type, then ∆f is locally of finite presentation. By definition, if f is separated then
∆f is a closed immersion, and hence proper. Again by definition, if f is quasiseparated, then ∆f is
quasicompact, hence of finite type. These sorts of statements motivate the following proposition.
Proposition 5.2. Suppose that P is a property of morphisms such that:
(i) P is closed under composition;
(ii) P is stable under base change.
Then it follows that
(iii) A product of morphisms having P has P (that is, if X → Y and X 0 → Y 0 are morphisms
of S-schemes which each have P , then the induced morphism X ×S X 0 → Y ×S Y 0 also has
P );
(iv) if f : X → Y and g : Y → Z are morphisms such that g ◦ f has P and ∆g has P , then f
has P ;
(v) if f : X → Y has P and Xred → X has P , then fred : Xred → Yred has P .
The proof is the same as that of Exercise II.4.8 of Hartshorne [1]. To illustrate the technique,
if f : X → Y and g : Y → Z are morphisms such that g ◦ f has P and ∆g has P , then f is the
composition
p2
Γf
X → X ×Z Y Y ,
and Γf is a base change of ∆g , while p2 is a base change of g ◦ f , so by (i) and (ii) we conclude
that f has P , as desired.
Corollary 5.3. Suppose that P is a property of morphisms such that:
(i) closed immersions have P ;
(ii) P is closed under composition;
(iii) P is stable under base change.
Then it follows that
(iv) A product of morphisms having P has P ;
(v) if f : X → Y and g : Y → Z are morphisms such that g ◦ f has P and g is separated, then
f has P ;
(vi) if f : X → Y has P , then fred : Xred → Yred has P .
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If further quasicompact immersions have P , then it is enough in (v) for g to be quasiseparated,
and if all immersions have P , then no condition on g is necessary in (v).
Proposition 5.4. The following properties of morphisms satisfy (i)-(iii) of Corollary 5.3 (and
hence (iv)-(vi) as well):
• closed immersion;
• universally closed;
• proper.
The following properties satisfy (i)-(iii) in the stronger form that quasicompact immersions have
the given property, and hence quasiseparatedness is enough in (v):
• quasi-compact;
• finite type.
The following properties satisfy (i)-(iii) in the stronger form that all immersions have the given
property, and hence separatedness is unnecessary in (v):
• immersion;
• locally of finite type;
• separated.
Finally, the property of being locally of finite presentation satisfies (ii)-(vi) of Corollary 5.3,
except that in (v) g is required to be locally of finite type rather than separated.
We omit the proofs, which are mostly straightforward. Note that one should not check these
properties for separatedness or properness using the valuative criterion as in Hartshorne, as it is
not significantly harder to check them directly, and one avoids unnecessary hypotheses that way.
Example 5.5. A good example of a property which satisfies (i)-(vi) of Corollary 5.3 but not their
stronger version is closed immersions: for instance, if we take f to be an inclusion morphism from
the affine line to the line with the doubled origin, and g the morphism from the line with the
doubled origin to the affine line which sends both origins to the origin, then g ◦ f is the identity,
hence a closed immersion, but f is not a closed immersion. Thus the separatedness of g is necessary
for closed immersions to satisfy (v).
On the other hand, open immersions don’t satisfy any form of (v).
Example 5.6. Note that a consequence of this is that if X, Y are locally of finite type over S
(as is for instance the case for prevarieties), then any morphism between X and Y over S is
automatically locally of finite type as well. (We can say the same for finite type if we assume
also that Y is separated, or if we impose a locally Noetherian condition on S, which ensures that
immersions are quasi-compact)
As an immediate consequence of Corollary 5.3 (v) and Proposition 5.4 we obtain the following
analogue of the classical fact that the continuous image of a compact set is compact, and hence
closed in any Hausdorff space:
Corollary 5.7. Let f : X → Y be a morphism of schemes over S, with X universally closed over
S, and Y separated over S. Then f (X) is closed in Y , and more generally, f is a (universally)
closed morphism.
References
[1] Robin Hartshorne, Algebraic geometry, Springer-Verlag, 1977.
[2] Hideyuki Matsumura, Commutative ring theory, Cambridge University Press, 1986.
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