VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS

VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Recall that or prevarieties, we had criteria or being a variety or or being complete in terms o existence and uniqueness o limits, where limits are described in terms o extending morphisms rom punctured nonsingular curves over the missing point. There are parallel criteria or separatedness and properness, which we now discuss. Due to the generality o the schemes considered, we have to replace curves by the spectra o valuation rings (recalling that the local ring o a curve is a discrete valuation ring, this is not such a leap), so these criteria are reerred to as valuative criteria. We will irst state the criteria, then give the proos assuming certain statements relating to valuation rings, and inally we will prove those statements. 1. Statements We irst need to develop basic properties o valuation rings, and we start by recalling the deinition: Deinition 1.1. An integral domain A with raction ield K is a valuation ring i or all x K, either x A or x 1 A. Remark 1.2. The reason or the terminology is that we obtain a homomorphism ν rom K to an ordered abelian group, with the property that x A i and only i ν(x) 0. This homomorphism is simply obtained by setting the abelian group equal to K /A, 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 A. Then (x + x )/x = 1 + x /x A, 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 1.2: Proposition 1.3. A valuation ring is a local ring, with maximal ideal A A. In particular, i A is a valuation ring, then Spec A has unique generic and closed points, corresponding to the zero ideal and the maximal ideal, respectively. Throughout our discussion, a morphism Spec K Spec A is always assumed to be the canonical inclusion o the generic point. We can now state the valuative criteria. Theorem 1.4. Let : X Y be a morphism o schemes, and assume that is quasicompact. Then is separated i and only i or every commutative diagram Spec A Y with A a valuation ring and K its raction ield, there is at most one way o illing in the dashed arrow so that the diagram remains commutative. 1

Theorem 1.5. Let : X Y be a morphism o inite type, with quasicompact. Then is proper i and only i or every commutative diagram Spec A Y with A a valuation ring and K its raction ield, there is exactly one way o illing in the dashed arrow so that the diagram remains commutative. Remark 1.6. Note that in both 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. However, it is typically no harder to work with arbitrary valuation rings. Also, 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. Also, quasiseparatedness is automatically satisied i X is locally Noetherian (which is or instance true i Y is locally Noetherian and is locally o inite type), so under mild Noetherian hypotheses we do not need to worry about checking this condition separately. 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. 2. Proos Each valuative criterion is o course two statements: irst, that separatedness (respectively, properness) implies the stated criterion, and second, that the criterion implies separatedness (respectively, properness). 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. The irst concept we need is that o specialization: Deinition 2.1. 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. Because schemes have 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 make this more precise in an important special case: Proposition 2.2. 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 2.3. To see that quasicompactness is necessary in the proposition, 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 1.5 does not imply that a morphism is closed without a quasicompactness hypothesis. 2

Proposition 2.2 will be enough or checking separatedness, but or properness it is convenient to develop the statement into one on closed morphisms: Corollary 2.4. 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. 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 2.2. In order to apply our criteria, the main use o valuation rings is the ollowing result, generalizing our main lemma on existence o morphisms rom nonsingular curves: Proposition 2.5. Let X be a scheme, and x, x X with x specializing to x. Then there exists a valuation ring A and a morphism Spec A X with the generic point o Spec A mapping to x, and the closed point o Spec A 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 A, with raction ield K, and a commutative diagram Spec K Spec A X Y such that the image o Spec K is x, and the generic and closed points o Spec A map to (x) and y respectively. We deer the proo to the next section. From this, it is not hard to prove that the stated valuative criteria imply separatedness and properness. Proo o i direction o Theorem 1.4. Suppose that the stated criterion is satisied, so we wish to show that is separated. Since we have assumed quasi-compact, by Proposition 2.2 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 2.5 there exists a valuation ring A with raction ield K, and a morphism ψ : Spec A X Y X such that the generic point o Spec A maps to z, and the closed point o Spec A maps to z. Taking irst and second projection yields two morphisms p 1 ψ and p 2 ψ rom Spec A to X, which give the same morphism Spec A Y ater composition with. We claim that p 1 ψ agrees with p 2 ψ i we precompose with ι : Spec K Spec A. 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 2.6. Suppose a morphism : X Y satisies the valuative criterion stated in Theorem 1.5. Then or every morphism Y Y, the base change X Y Y Y o satisies the same valuative criterion. 3

Proo o i direction o Theorem 1.5. Suppose our criterion is satisied, and we wish to show that is proper. We know that is separated by Theorem 1.5, so it is enough to check that is universally closed. 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 2.4 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. By Proposition 2.5 there exists a valuation ring A with raction ield K and a diagram Spec K Spec A with the image o Spec K being x, and the image o the generic and closed points o Spec A being (x) and y, respectively. By Lemma 2.6, 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 A completes the argument. We now move on to checking that conversely, separatedness and properness 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 1.4. 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 A X Y X such that g ι actors through, where ι : Spec K Spec A is the canonical inclusion. Because (X) is closed by hypothesis, we conclude that g(spec A) (X), and thus that g actors through, since Spec A is reduced. It thus ollows that g 1 = g 2. Proposition 2.7. Let A be a valuation ring with raction ield K. Suppose that : X Spec A is a closed morphism. Then given a morphism Spec K X such that the diagram X Y Spec A id Spec A 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 A, every generic section extends to a section. The proo o the proposition is deerred to the next section, but we can now easily conclude the proo o Theorem 1.5. Proo o only i direction o Theorem 1.5. Suppose that is proper. Given a diagram as in the statement, we know any extension is unique i it exists, by Theorem 1.4. We thus wish to prove existence. Consider the base change X := X Y Spec A Spec A, which is closed by hypothesis. The morphism Spec K X then induces a morphism Spec K X, and by Proposition 2.7 we obtain a morphism Spec A X which when composed with the projection morphism X X, gives us what we want. 3. Proos o statements on valuation rings Recall that i K is a ield, we have a partial order on the local rings strictly contained in K given by A B i A B and m B A = m A, or equivalently i the inclusion A B is a local homomorphism. This partial order is called dominance. The basic algebra act we need is the ollowing: 4

Proposition 3.1. The valuation rings A with given raction ield K are precisely the maximal elements under dominance among local rings strictly contained in K. The proo o this is not diicult; see Theorem 10.2 o Matsumura [2]. Proo o Proposition 2.5. Note that the irst statement ollows rom the second, by taking to be the identity map. For the second statement, by replacing Y with the closure o (x) (with the reduced induced subscheme structure), it is clear that we may assume Y is integral, and (x) is the generic point o Y. Then, by Proposition 3.1, there exists a valuation ring A o K(Y ) dominating O Y,y, so we have a morphism Spec A Y sending the generic point to (x) and the closed point to y. Since K(Y ) = k((x)), we also have an inclusion K(Y ) k(x), so we can let A be a valuation ring o k(x) which dominates A. Then we obtain a morphism Spec A Y which sends the generic point to (x) and the closed point to y, and by construction the raction ield o A is k(x), so there is a canonical morphism rom Spec K := Spec k(x) X with image x commuting with the given morphisms, as desired. Note that the proo in act shows that we can always assume Spec A X induces an isomorphism K = k(x). Proo o Proposition 2.7. Let x be the image o Spec K; then by hypothesis, (x) is the generic point o Spec A. I Z X is the closure o x, the hypothesis that is closed implies that (Z) = Spec A, so there is some x X with x specializing to x, and (x ) being the closed point o Spec A. The morphisms Spec K X Spec A having composition equal to the canonical inclusion must then yield an isomorphism k(x) = K. Thus, i we put the reduced subscheme structure on Z, we have Z Spec A inducing an isomorphism on unction ields. Then since x maps to the closed point, we have that O Z,x dominates A, and must thereore be equal to A by Proposition 3.1. We conclude that the canonical morphism Spec O Z,x Z X yields the desired morphism Spec A X. Reerences 1. Robin Hartshorne, Algebraic geometry, Springer-Verlag, 1977. 2. Hideyuki Matsumura, Commutative ring theory, Cambridge University Press, 1986. 5