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Criteria of Separatedness and Properness

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Criteria of Separatedness and Properness VALUATIVE CRITERIA OF SEPARATEDNESS AND PROPERNESS STERGIOS ANTONAKOUDIS 1. Introduction We discuss the valuative criteria of separatedness and properness for Noetherian rings using discrete valuation rings. We consider schemes of finite type over an algebraically closed field and see how to substitute DVR's with smooth one-dimensional k-algebras. These arguments easily generalize to a less restrictive setting [GD67, II.7.2 and II.7.3]. Our exposition follows [Har77] closely. Acknowledgments. We would like to thank Dennis Gaitsgory, Bao Le Hung and Thanos Papa¨ıoannoufor fruitful conversations and help. 2. Valuative criterion of separatedness 2.1. Motivation. Here is the rough idea: begin with a curve C missing a point P , and a scheme X over a field. We would like to say that X is separated if given any morphism from C − P to X, there is at most one way that we can extend it to a morphism from all of C into X. Since the above question is local, we should replace the curve by its local ring at P , which is a discrete valuation ring (we will see that in the case of schemes of finite type over a field this is not necessary). Theorem 1. Let f : X ! Y be a morphism of Noetherian schemes. Then f is separated if and only if the following condition holds. Given any DVR R with field of fractions K, with i : Spec(K) ! Spec(R) induced from the inclusion of R in K, and given any two morphisms Spec(K) ! X and Spec(R) ! Y such that the diagram Spec(K) / X ; i f Spec(R) / Y commutes, there is at most one morhism from Spec(R) to X making the whole diagram commutative. Before we prove the theorem we need to prove three lemmas. Date: December 3, 2009. 1 2 STERGIOS ANTONAKOUDIS Lemma 1. Let R be a valuation ring of a field K. Let T = Spec(R) and let U = Spec(K). To give a morphism of U to a scheme X is equivalent to giving a point x1 2 X and an inclusion of fields k(x1) ⊆ K. To give a morphism from T to X is equivalent to giving two points x0; x1 2 X, with x0 a specialization of x1, and an inclusion of fields k(x1) ⊆ K, such that R dominates the local ring O of x0 on the subscheme Z = fx1g of X with its reduced induced structure. Proof. [Har77, Section 2.4, Lemma 4.4]. Lemma 2. Let f : X ! Y be a quasi-compact morphism of schemes. Then the subset f(X) of Y is closed if and only if it is stable under specialization. Proof. [Har77, Section 2.4, Lemma 4.5.]. Lemma 3. Let O be a noetherian local domain with field of fractions K, and let L be a finitely-generated field extension of K, then there exists a discrete valuation ring R of L dominating O, i.e. the inclusion O ! R is a local ring homomorphism. Proof. Let y1; ··· ; yn be transcendental elements of L over K such that L is finite over K(y1; ··· ; yn). Then the extension of the maximal ideal m of O in O[y1; ··· ; yn] is not the unit ideal and therefore by localizing at a prime ideal above it, it suffices to find a DVR in L dominating that localization. Hence we can assume without loss of generality that L is a finite field extension of K. Next, let x1; ··· ; xn be a system of parameters for m, then x1; ··· ; xn are algebraically independent over K. Therefore the extension of m in 0 O = O[x2=x1; ··· ; xn=x1] is (x1), which is not the unit ideal. (An alternative way to see this is to choose x1 a non-nilpotent element of degree 1 in gr(m), which exists since the latter has the same dimension with O and hence it is not Artinian). Let p be a minimal 0 prime above (x1). Then p has height 1 by Krull's theorem. Let B be the localization of O at p. Then B is a noetherian local domain of dimension 1 and hence by the Krull-Akizuki theorem (See Appendix, Lemma 7) the integral closure of B in L, call it C, is noetherian of dimension 1. In particular if we localize C at a maximal ideal we get a DVR in L, which dominates O, as desired. Proof of theorem. First suppose that f is separated, and suppose given a diagram as above, with T = Spec(R) and U = Spec(K), where there are two morphisms h; h0 of T to X making the whole diagram commutative. Spec(K) ; / X www; h wwww i wwww f wwwwh0 www Spec(R) / Y 00 Then we obtain a morphism h : T ! X ×Y X. We observe that the restrictions of h and h0 to U are the same and the closure of U is all of T . Since ∆(X) is closed, the image of T lies in the diagonal (in the set-theoric sense). But since T is reduced it factors through the reduced scheme structure of the diagonal. But that implies that h and h0 are equal, by the universal property of h00. VALUATIVE CRITERIA OF SEPARATEDNESS AND PROPERNESS 3 Conversely, let us suppose that the condition of the theorem is satisfied. We need to show that f is separated and hence it will be enough to show that ∆(X) is closed. Since X is Noetherian, the diagonal morphism is quasi-compact and hence by the previous lemma it suffices to show that ∆(X) is stable under specialization. Assume that there exists p1 in ∆(X) with p0 2 fp1g. Let's assume that p0 is not in the diagonal. Let K = k(p1) and O be the local ring at p0 of fp1g, with its reduced induced structure. Then O is a local noetherian domain with field of fractions K. Hence by the lemma we proved above there is a DVR R of K that dominates O. The field of fractions of R is K and if we let T = Spec(R) and U = Spec(K) then by the lemma above we get a morphism T ! X ×Y X, sending the generic point t1 of T to p1 and the closed point t0 of T to p0. Now, we are almost done, since composing with the two projections π1 X ×Y X //X π2 gives two morphisms of T to X which give the same morhism to Y, and moreover whose restrictions to U are the same. So by the conditions of the theorem these two morphisms are the same. That means that the morphim T ! X ×Y X factors through the diagonal morhism and hence the point p0 belongs to the diagonal ∆(X). That finishes the proof. 3. Valuative criterion of properness 3.1. Motivation. The rough idea is similar to that of the valuative criterion of separat- edness in Section 2.1, namely given a curve C missing a point P , and a scheme X over a field, we would like to say that X is proper if given any morphism from C − P to X, there is precisely one way to we extend it to a morphism from all of C into X. Since the above question is local, again, we should replace the curve by its local ring at P , which is a discrete valuation ring. Theorem 2. Let f : X ! Y be a morphism of finite type of Noetherian schemes. Then f is proper if and only if the following condition holds. For any DVR R with field of fractions K, with i : Spec(K) ! Spec(R) induced from the inclusion of R in K, for any two morphisms Spec(K) ! X and Spec(R) ! Y such that the following diagram commutes. U / X > i f T / Y Then there is a unique morhism of Spec(R) to X making the whole diagram commutative. Proof. First assume that f is proper. Then f is also separated and by the previous theorem if such a morphism T ! X exists, it has to be unique. So we only need to prove that it exists. Consider the base change T ! Y and let XT be X ×Y T . Then we get a map 4 STERGIOS ANTONAKOUDIS U ! XT by the universality of fiber product, such that the following diagram commutes. U / XT / X f 0 f T / Y Let p1 2 XT be the image of the unique point t1 of U. Let Z = fp1g. Then Z is a closed 0 subset of XT . Since f is proper, it is also universally closed and hence f is a closed map. In particular the image of Z under f 0 is a closed subset of T . But by the commutativity 0 of the diagram above we have that f (p1) = t1, which is the generic point of T , hence 0 0 f (Z) = T . In particular there is a point p0 2 Z such that f (p0) = t0, the closed point of T . That gives us a homomorphism of local rings R ! OZ;p0 corresponding to the morphism 0 f . The function field of Z is k(p1), which is contained in K. But R is a discrete valuation ring, dominated by a local ring inside K, therefore R is isomorphic to OZ;p0 . Here I use the fact that valuation rings of a field K are maximal subrings of K with respect to the ordering given by domination. Therefore by the lemma in the previous section we get a morphism from T to XT sending t0; t1 to p0; p1, respectively.
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