SEPARATED AND PROPER MORPHISMS

SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN The notions o separatedness and properness are the algebraic geometry analogues o the Hausdor condition and compactness in topology. For varieties over the complex numbers, it is possible to use the analytic topology inherited rom the usual topology on C in place o the Zariski topology, and the analytic topology corresponds well to our intuition, and allows us to use many tools rom topology and algebraic topology. An ongoing theme in algebraic geometry is that many such properties and tools can in act be translated into the context o varieties over an arbitrary ield, or more general schemes, but the translation may take some work. Oten one can proceed by inding a more abstract version o the usual topological deinition (or instance, involving products) and then translating this abstracted deinition into algebraic geometry. I 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 Hausdor property are requently called separation axioms, separatedness is the condition or (amilies o) schemes which plays the role o the Hausdor property in classical topology. The irst step is to observe that the usual deinition o Hausdor or 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 deine what it should mean or it to be a variety. We will see later that this does indeed recover the classical Hausdor condition when applied to complex prevarieties, so it as well behaved as one could hope or when translating classical topological conditions to (pre)varieties over an arbitrary ield. Even better, this idea generalizes immediately to schemes. We irst deine: Deinition 1.1. Given a morphism : X Y o schemes, the diagonal morphism : X X Y X is induced by the identity morphisms X X on both actors. We then have: Deinition 1.2. A morphism : X Y is separated i the image o : 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 i and only i its associated scheme is separated over Spec k (note that this is not completely trivial, because the underlying topological spaces o a prevariety considered as a prevariety or as a scheme dier due to the inclusion o generic points). In particular, the scheme associated to any quasiprojective variety is separated over Spec k. We have: Proposition 1.4. For any morphism : X Y, the associated diagonal morphism : X X Y X is an immersion. I X and Y are aine schemes, then is a closed immersion. Proo. We irst show that is a closed immersion when X = Spec R and Y = Spec S are aine. In this case, is induced by the ring homomorphism R S R R determined by r r rr. This is visibly a surjective map, so it induces a closed immersion o aine schemes. For the general case, i we cover Y by aine open subschemes {V i }, and then cover X by aine open subschemes {U j } with the property that or each j, we have (U j ) V ij or some i j, then we 1

have aine open subschemes U j Vij U j which cover the image o X under, with the preimage o each equal to U j. Restricting to each o these open sets we ind that is the diagonal morphism or U j V ij, so is a closed immersion. We conclude that is an immersion, as desired. We immediately conclude: Corollary 1.5. A morphism o aine schemes is separated. Corollary 1.6. A morphism : X Y is separated i and only i the diagonal morphism is a closed immersion. Proo. Clearly i the diagonal is a closed immersion, then is separated. Conversely, by Proposition 1.4 i has closed image it must be a closed immersion. 2. Proper morphisms Properness is the algebraic geometry analogue o compactness, generalizing the deinition o complete varieties. The deinition is very similar to that or complete varieties, although to ensure good behavior we throw in a technical condition which is automatic or varieties. Deinition 2.1. A morphism X S is universally closed i or all morphisms S S, we have that the projection X S S S is closed. A morphism is proper i it is o inite type, separated, and universally closed. Here we say a morphism is closed i the induced map on the underlying topological spaces is closed. Example 2.2. A variety is complete i and only i its associated scheme is proper over Spec k. Remark 2.3. It is in act not terribly hard to show that a universally closed morphism is necessarily quasi-compact. Thus, in the deinition o properness we could simply have imposed that the morphism be locally o inite type. 3. Valuative criteria: statements Intuitively, one can think o separatedness as (a relative version o) uniqueness o limits, and properness as (a relative version o) existence o (unique) limits. It is not immediately obvious how to ormalize these ideas in algebraic geometry, but it turns out to be doable, via valuative criteria. Aside rom providing some intuition or separatedness and properness in terms o uniqueness and existence o limits, the valuative criteria are extremely important when working with moduli spaces. In this case they amount to studying the behavior o amilies o objects over valuation rings. We include the valuative criterion or universal closedness separately, because most algebraic stacks are not separated, and it is nonetheless helpul to know whether they are universally closed. The most classical version o this sort o criterion applies to prevarieties over an algebraically closed ield k. Let X be a prevariety. Then one can think o the setup or limits as ollows: let C be a smooth curve over k, and P C a point; then consider a morphism : C {P } X. We could picture that the points (Q) or Q C {P } have a limit as Q approaches P i extends to a morphism on all o C. Uniqueness o the limit then corresponds to uniqueness o the extension. In act, this works with prevarieties: as we ve mentioned previously, X is a variety i and only i limits are unique in this sense, which is to say i and only i or all C, P and as above, there is at most one extension o to all o C. Similarly a variety X is complete i and only i limits exist, which is to say or all C, P and as above there exists a (necessarily unique) extension o to all o C. The above criterion works exactly the same i we replace C by the local scheme Spec O C,P and C P by Spec K(C), where K(C) is the unction ield o C (and the ield o ractions o O C,P ). 2

Now, O C,P is a discrete valuation ring, and i we want to work with schemes not necessarily o inite type over an algebraically closed ield, we should consider arbitrary discrete valuation rings. In act, i 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 deinitions and properties or valuation rings. Deinition 3.1. An integral domain R with raction ield K is a valuation ring i or all x K, either x R or x 1 R. Remark 3.2. The reason or the terminology is that we obtain a homomorphism ν rom K to an ordered abelian group, with the property that x R i and only i 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, x K, with x+x 0, we have ν(x+x ) min{ν(x), ν(x )}. Indeed, suppose without loss o generality that ν(x) ν(x ), so that by deinition x /x R. Then (x + x )/x = 1 + x /x R, so ν(x + x ) ν(x), as desired. Such a homomorphism is called a valuation, and as in I.6 o Hartshorne [1], valuation rings may be deined equivalently in terms o existence o a valuation. The topological space underlying a valuation ring may have arbitrary dimension, but we nonetheless have the ollowing act, which ollows easily rom Remark 3.2: Proposition 3.3. A valuation ring is a local ring. In particular, i 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 o the generic point. We can now state the valuative criteria. Deinition 3.4. We say a morphism : X Y satisies the existence (respectively, uniqueness) part o the valuative criterion i or every commutative diagram Spec K X Spec R Y with R a valuation ring and K its raction ield, there exists (respectively, there is at most one) one way o illing in the dashed arrow so that the diagram remains commutative. Theorem 3.5. Let : X Y be a morphism o schemes, and assume that is quasicompact. Then is separated i and only i it satisies the uniqueness part o the valuative criterion. Theorem 3.6. Let : X Y be a morphism o schemes, and assume that is quasicompact. Then is universally closed i and only i it satisies the existence part o the valuative criterion. From the above two theorems, we immediately conclude the valuative criterion or properness. Theorem 3.7. Let : X Y be a morphism o inite type, with quasicompact. Then is proper i and only i it satisies both the existence and uniqueness parts o the valuative criterion. Remark 3.8. Note that in the valuative criteria, i we have Y locally Noetherian and locally o inite type, then it is enough to only consider discrete valuation rings instead o arbitrary ones. In addition, in this case (and more generally i X is locally Noetherian) the condition that be quasi-comapct is automatically satisied. Thus, under mild Noetherian hypotheses we do not need to worry about checking this condition separately. 3

More generally, the condition that is quasi-compact is an annoying one, but it comes up naturally in a number o 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 o 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 satisying the valuative condition. Aside rom providing some intuition or separatedness and properness in terms o uniqueness and existence o limits, the valuative criteria are extremely important when working with moduli spaces. In this case they amount to studying the behavior o amilies o objects over valuation rings. We include the valuative criterion or universal closedness separately, because most algebraic stacks are not separated, and it is nonetheless helpul to know whether they are universally closed. 4. Valuative criteria: proos Each valuative criterion is o course two statements: irst, that separatedness (respectively, universal closedness) implies the stated criterion, and second, that the criterion implies separatedness (respectively, universal closedness). We thus have our statements to prove, and the proos are rather independent o one another. There is however substantial commonality between the proos that the two criteria imply separatedness and properness, so we will begin with the proos o these statements. Recall the ollowing concept: Deinition 4.1. Given a scheme X, and points x, x X, we say that x specializes to x i x is in the closure o {x}. A subset Z X is closed under specialization i or all points x, x X with x Z and x specializing to x, we also have x Z. Closed sets are visibly closed under specialization. Because schemes have (unique) generic points or all their irreducible closed subsets, the idea is that a subset o 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 rom the spectra o valuation rings. Putting these statements together will quickly yield the relevant direction o 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 : X Y is a quasi-compact morphism. I (X) is closed under specialization, then (X) is closed. For the proo, see Lemma 4.5 o Chapter II o Hartshorne [1]. Example 4.3. To see that quasicompactness is necessary, let Y = A 1 k, and let X be an ininite disjoint union o closed points o A 1 k, with the inclusion. Then (X) contains only closed points, so is closed under specialization, but is not a closed set. Note that this morphism is even locally o inite type, so quasicompactness is really the crucial hypothesis. Because is not closed, this is also an example that the criterion o Theorem 3.6 does not imply that a morphism is closed without a quasicompactness hypothesis. Proposition 4.2 will be enough or checking separatedness, but or universal closedness it is convenient to develop the statement into one on closed morphisms: Corollary 4.4. I : X Y is a quasicompact morphism, and or all x X, and y Y such that (x) specializes to y, we have some x X with x specializing to x, and (x ) = y, then is closed. 4

Proo. 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 o Z is closed under specialization, so (Z) Y is closed by Proposition 4.2. The main use o valuation rings will be the ollowing result, stating that specializations (in a relative and absolute setting) can be detected via morphisms rom the spectra o valuation rings. Proposition 4.5. Let X be a scheme, and x, x X with x specializing to x. Then there exists a valuation ring R and a morphism Spec R X with the generic point o Spec R mapping to x, and the closed point o Spec R mapping to x. More generally, i : X Y is any morphism, and we have x X, and y Y a specialization o (x), then there exists a valuation ring R, with raction ield K, and a commutative diagram Spec K Spec R X Y such that the image o Spec K is x, and the generic and closed points o Spec R map to (x) and y respectively. We omit the proo. From this, it is not hard to prove that the stated valuative criteria imply separatedness and universal closedness. Proo o i direction o Theorem 3.5. Suppose that the stated criterion is satisied, so we wish to show that is separated. Since we have assumed quasi-compact, by Proposition 4.2 it is enough to show that the image o is closed under specialization. Accordingly, suppose we have z specializing to z in X Y X, with z = (x) or some x X. By Proposition 4.5 there exists a valuation ring R with raction ield K, and a morphism ψ : Spec R X Y X such that the generic point o Spec R maps to z, and the closed point o Spec R maps to z. Taking irst and second projection yields two morphisms p 1 ψ and p 2 ψ rom Spec R to X, which give the same morphism Spec R Y ater composition with. We claim that p 1 ψ agrees with p 2 ψ i we precompose with ι : Spec K Spec R. It suices to check that ψ ι actors through, but since this is a morphism rom Spec K, it is enough to observe that Spec K maps to z, which by hypothesis is a point o. We thus obtain the claim, and then by hypothesis we conclude that p 1 ψ = p 2 ψ, and thus that ψ actors through. It ollows inally that z (X), so (X) is closed under specialization, as desired. The ollowing lemma is used in checking properness. We leave the proo, which uses only the universal property o ibered products, to the reader. Lemma 4.6. Suppose a morphism : X Y satisies the existence part o the valuative criterion. Then or every morphism Y Y, the base change X Y Y Y o satisies the existence part o the valuative criterion. Proo o i direction o Theorem 3.6. Suppose our criterion is satisied. Let Y Y be any morphism, and X = X Y Y ;. We thus wish to show that X Y is closed. Since quasicompactness is preserved under base change, by Corollary 4.4 it is enough to show that or any x X, and y Y with (x) specializing to y, there exists x X with x specializing to x and (x ) = y. 5

By Proposition 4.5 there exists a valuation ring R with raction ield K and a diagram Spec K Spec R with the image o Spec K being x, and the image o the generic and closed points o Spec R being (x) and y, respectively. By Lemma 4.6, our criterion holds also or X Y, so we conclude that we can ill in the dashed arrow so that the diagram still commutes, and setting x to be the image o the closed point o 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 ormer directly, while the latter requires an additional result on morphisms and spectra o valuation rings. Proo o only i direction o Theorem 3.5. Suppose that is separated, and we have a diagram as in the statement, and morphisms g 1, g 2 : Spec X making the diagram commute. We thus obtain a morphism g : Spec R X Y X such that g ι actors through, where ι : Spec K Spec R is the canonical inclusion. Because (X) is closed by hypothesis, we conclude that g(spec R) (X), and thus that g actors through, since Spec R is reduced. It thus ollows that g 1 = g 2. Proposition 4.7. Let R be a valuation ring with raction ield K. Suppose that : X Spec R is a closed morphism. Then given a morphism Spec K X such that the diagram X Y Spec K X Spec R id Spec R commutes, there exists a morphism illing in the dashed arrow so that the diagram still commutes. One can rephrase the proposition as saying that or a closed morphism to Spec R, every generic section extends to a section. We omit the proo. Proo o only i direction o Theorem 3.6. Suppose that is universally closed. Given a diagram as in the valuative criterion, we wish to prove existence o the dashed arrow. Consider the base change X := X Y Spec R Spec R, which is closed by hypothesis. The morphism Spec K X then induces a morphism Spec K X, and by Proposition 4.7 we obtain a morphism Spec R X which when composed with the projection morphism X X, gives us what we want. 5. Properties o properties o morphisms Properties o morphisms tend to have certain standard properties. One o the most important is that they should be stable under base change: that is, i : X Y is a morphism, and Y Y any other morphism, i has the property in question, then so should the induced morphism X Y Y Y. Another crucial property is that the property in question be closed under composition. We will assume (see Exercise II.3.11 (a) o Hartshorne [1], but note that it s easier to do (b) irst and then deduce (a)), the act that closed immersions are stable under base change; rom this one deduces easily that arbitrary immersions are likewise stable under base change. Beore moving on, we describe a consequence o this basic act: the notion o graph o a morphism. 6

Corollary 5.1. Suppose : X Y is a morphism o schemes over S. Then the morphism Γ : X X S Y induced by the identity and is a base change o Y/S. In particular Γ is an immersion, and i Y is separated over S, then Γ is a closed immersion. This construction induces a bijection between S-morphisms : X Y and subschemes Γ X S Y with the property that p 1 induces an isomorphism Γ X. Proo. For the irst assertion, one simply checks that the square X Γ Y X S Y id Y S Y satisies the universal property o the ibered product. It is clear then that given a morphism, we obtain a subscheme which maps isomorphically to X under p 1. Conversely, given such a subscheme, the composition p 2 p 1 1 yields an S-morphism X Y. It is straightorward to check that these constructions are maturally inverse. Because the diagonal is always an immersion, there are many properties o morphisms which are always, or requently, satisied by. For instance, is always locally o inite type. I is locally o inite type, then is locally o inite presentation. By deinition, i is separated then is a closed immersion, and hence proper. Again by deinition, i is quasiseparated, then is quasicompact, hence o inite type. These sorts o statements motivate the ollowing proposition. Proposition 5.2. Suppose that P is a property o morphisms such that: (i) P is closed under composition; (ii) P is stable under base change. Then it ollows that (iii) A product o morphisms having P has P (that is, i X Y and X Y are morphisms o S-schemes which each have P, then the induced morphism X S X Y S Y also has P ); (iv) i : X Y and g : Y Z are morphisms such that g has P and g has P, then has P ; (v) i : X Y has P and X red X has P, then red : X red Y red has P. The proo is the same as that o Exercise II.4.8 o Hartshorne [1]. To illustrate the technique, i : X Y and g : Y Z are morphisms such that g has P and g has P, then is the composition X Γ X Z Y p 2 Y, and Γ is a base change o g, while p 2 is a base change o g, so by (i) and (ii) we conclude that has P, as desired. Corollary 5.3. Suppose that P is a property o morphisms such that: (i) closed immersions have P ; (ii) P is closed under composition; (iii) P is stable under base change. Then it ollows that (iv) A product o morphisms having P has P ; (v) i : X Y and g : Y Z are morphisms such that g has P and g is separated, then has P ; (vi) i : X Y has P, then red : X red Y red has P. 7

I urther quasicompact immersions have P, then it is enough in (v) or g to be quasiseparated, and i all immersions have P, then no condition on g is necessary in (v). Proposition 5.4. The ollowing properties o morphisms satisy (i)-(iii) o Corollary 5.3 (and hence (iv)-(vi) as well): closed immersion; universally closed; proper. The ollowing properties satisy (i)-(iii) in the stronger orm that quasicompact immersions have the given property, and hence quasiseparatedness is enough in (v): quasi-compact; inite type. The ollowing properties satisy (i)-(iii) in the stronger orm that all immersions have the given property, and hence separatedness is unnecessary in (v): immersion; locally o inite type; separated. Finally, the property o being locally o inite presentation satisies (ii)-(vi) o Corollary 5.3, except that in (v) g is required to be locally o inite type rather than separated. We omit the proos, which are mostly straightorward. Note that one should not check these properties or separatedness or properness using the valuative criterion as in Hartshorne, as it is not signiicantly harder to check them directly, and one avoids unnecessary hypotheses that way. Example 5.5. A good example o a property which satisies (i)-(vi) o Corollary 5.3 but not their stronger version is closed immersions: or instance, i we take to be an inclusion morphism rom the aine line to the line with the doubled origin, and g the morphism rom the line with the doubled origin to the aine line which sends both origins to the origin, then g is the identity, hence a closed immersion, but is not a closed immersion. Thus the separatedness o g is necessary or closed immersions to satisy (v). On the other hand, open immersions don t satisy any orm o (v). Example 5.6. Note that a consequence o this is that i X, Y are locally o inite type over S (as is or instance the case or prevarieties), then any morphism between X and Y over S is automatically locally o inite type as well. (We can say the same or inite type i we assume also that Y is separated, or i we impose a locally Noetherian condition on S, which ensures that immersions are quasi-compact) As an immediate consequence o Corollary 5.3 (v) and Proposition 5.4 we obtain the ollowing analogue o the classical act that the continuous image o a compact set is compact, and hence closed in any Hausdor space: Corollary 5.7. Let : X Y be a morphism o schemes over S, with X universally closed over S, and Y separated over S. Then (X) is closed in Y, and more generally, is a (universally) closed morphism. Reerences [1] Robin Hartshorne, Algebraic geometry, Springer-Verlag, 1977. [2] Hideyuki Matsumura, Commutative ring theory, Cambridge University Press, 1986. 8