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 ∈ 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 ∈ 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 = {x1} 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 ∈ {p1}. Let’s assume that p0 is not in the diagonal. Let K = k(p1) and O be the local ring at p0 of {p1}, 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 separatedness 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 ∈ XT be the image of the unique point t1 of U. Let Z = {p1}. 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 ∈ 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. Composing with XT → X gives the existence of the desired morphism T → X. Conversely, suppose that the condition of the theorem holds. We need to show that f is proper. By the previous theorem it only suffices to show that f is universally closed. Let Y 0 → Y be a base change and let f 0 : X0 → Y 0 be the corresponding morphism obtained from f. The morphism f 0 has the same property with f (i.e. for any commutative square as in the statement of the theorem, the bottom horizontal map factors through X0), which follows immediately using the universality of the fiber product. But in order to prove that a map of Noetherian schemes is universally closed it suffices to consider base changes by Noetherian schemes. Hence the problem reduces to showing that f is a closed map. Let Z be a closed subset of X, and give it the reduced induced structure.

Z ⊆ X0 / X

f 0 f Y 0 / Y

We need to show that f(Z) is closed in Y . Since f is of finite type so is the restriction of f to Z. Therefore the morphism f : Z → Y is quasi-compact and hence by the previous lemma, we only need to show that f(Z) is stable under specialization. So let z1 ∈ Z be a point, let y1 = f(z1), and let y0 ∈ {y1}. Let O be the local ring of y0 on {y1}, with its reduced induced structure. Then the field of fractions of O is k(y1), which is a subfield of K = k(z1). By the lemma in the previous section we can find a discrete valuation ring of K VALUATIVE CRITERIA OF SEPARATEDNESS AND PROPERNESS 5 which dominates O. From this and the first lemma above we get a commutative diagram U / Z / X ~~ i ~~ ~~ f ~ ~ T / Y Hence by the conditions of the theorem we can find a morphism T → X. Since Z is closed and the generic point of T goes to z1 ∈ Z, this morphism factors to give a morphism from T to Z, this is true as a map of schemes as T is reduced. Now let z0 be the image of t0, then f(z0) = y0, so y0 ∈ f(Z). This finishes the proof.

4. The case of schemes of finite type over an algebraically closed field We will prove the following two theorems. Theorem 3. Let f : X → Y be a morphism of schemes of finite type over k, where k is an algebraically closed field. Then f is separated if and only if the following condition holds. For any finitely-generated smooth k-algebra A which is of dimension 1, p a closed point of Spec(A) and i : Spec(A) \{p} → Spec(A) the inclusion map, for any two morphisms Spec(A) \{p} → X and Spec(A) → Y such that the following diagram commutes. Spec(A) \{p} 9/ X i f Spec(A) / Y Then there is at most one morhism of Spec(A) to X making the whole diagram commutative. Theorem 4. Let f : X → Y be a morphism of schemes of finite type over k, where k is an algebraically closed field. Then f is proper if and only if the following condition holds. For any finitely-generated smooth k-algebra A which is of dimension 1, p a closed point of Spec(A) and i : Spec(A) \{p} → Spec(A) the inclusion map, for any two morphisms Spec(A) \{p} → X and Spec(A) → Y such that the following diagram commutes. Spec(A) \{p} 9/ X i f Spec(A) / Y Then there is one morhism of Spec(A) to X making the whole diagram commutative. We call an integral separated scheme of finite type over an algebraically closed field a variety over k. If it is proper we call it a complete variety. If it is of dimension 1 we call it a curve, e.g. Spec(A) in the theorem above is a curve and any non-complete curve is of that form. 6 STERGIOS ANTONAKOUDIS

First we observe that proving one of the directions of the above two theorems is easy. It is identical to the case of valuation rings above since a curve is a reduced scheme. Before we continue we need to prove the following lemmas. Lemma 4. Let f : X → Y be a morphism of schemes which are both of finite type over k and k is an algebraically closed field. If the morphism f is surjective on closed points then it is surjective on topological spaces. Proof. The closed points of a scheme over k, with k algebraically closed, are naturally in bijection with the k-points of the scheme. By Chevalley’s theorem the image of the above morphism is a constructible set. If f is not surjective, then the complement of its image is a non-empty constructible set which contains no closed points, a contradiction to the Nullstellensatz. Lemma 5. Let X be scheme which is of finite type over k and k is an algebraically closed field, U an open dense subset of X and p a closed point of X. Then there is an affine curve in X intersecting U and passing through p. Proof. We can assume that p is not in U. Further by restricting to a neighborhood around p we could assume that X = Spec(A) is affine and p corresponds to a maximal ideal m of A. By Noether normalization lemma we have an inclusion k[x1, ··· , xn] ,→ A that is finite and integral. Using the going-up theorem we see that the restriction of m is a maximal ideal and hence by Hilbert Nullstellensatz it is pn = (x1 − a1, ··· , xn − an), where (a1, ··· , an) is a k-point of the affine scheme Spec(k[x1, ··· , xn]). Let q be a k-point of U whose image in Spec(k[x1, ··· , xn]) corresponds to a maximal ideal (x1 −b1, ··· , xn −bn). Then we can find an affine line that passes through the two points that corresponds to a prime ideal pn−1 of Spec(k[x1, ··· , xn]) such that we have a maximal chain of prime ideals 0 ⊂ p1 ⊂ · · · ⊂ pn. Then by the going-up theorem the prime ideal lying above pn−1 is of height n − 1 and hence cutting along it defines a curve in Spec(A) that intersects U and passes through p. Hence we constructed an affine curve locally-closed embedded in X. Hence the result. Lemma 6. Let X be a variety over k and X → C a dominant morphism into a curve over k and k is an algebraically closed field. Then there is a curve in X such that the restriction of that morphism is dominant. Proof. We have a morphism X(k) → C(k) and hence we see that there is a closed point p ∈ C such that the fiber of p is non-empty. Let U be the complement of p in C. Then U in a open subset of C whose closure is everything. Also the pre-image of U in a non- empty open subset of X; hence it is dense in X. Therefore by the previous lemma we get a curve C0 in X that intersects both the pre-image of U and the fiber of p. Hence we get a non-constant morphism C0 → C. Since C0 is irreducible the closure of the image of that morphism is an irreducible closed subset of C that is not a closed point. Hence the morphism in dominant. Proof of properness. All we have to prove is that f is universally closed. For that to be true it suffices to prove that we get a closed morphism under any base change by schemes that are of finite type over k. Also the condition of extending curves would hold true for VALUATIVE CRITERIA OF SEPARATEDNESS AND PROPERNESS 7 any base change of f (the proof of that claim is identical to the proof for DVRs). Hence without loss of generality it would be enough to show that f is closed. Next, for any closed subset Z of X, with its reduced induced structure we get a morphism Z,→X→ Y . We need to show that it has closed image. But since any curve is a reduced scheme it follows that the morphism Z → Y has same extension property as f. Hence again without loss of generality it is sufficient to prove that the image of X under f is closed in Y . By considering the decomposition of X into irreducible components and the induced decomposition of Y by the closure of the image of these irreducible components, we can further assume that X and Y are both integral schemes, Y is affine and the morphism f is dominant. Hence by the lemma above we are reduced to showing that the image of f contains all closed points of Y . Let p be a closed point of Y . Since the image of f is constructible we can find open sets F F U1, ··· ,Un and closed sets C1, ··· ,Cn in Y such that f(X) = (U1 ∩ C1) ··· (Un ∩ Cn). Hence f(X) ⊆ C1 ∪ · · · ∪ Cn ⊆ Y and by irreducibility of Y we see that some Ci = Y and therefore we conclude that there is an open dense subset of Y , call it U, that is contained in the image of f. Now we can apply the lemma above to get a curve C = Spec(A) which is a closed subscheme of Y that passes through p and intersects U. In particular all but finitely many points of C are in U. By considering the base change we get a closed embedding of X0 into X, X0 / X

f C / Y and since C is irreducible and the image of X0 contains an open subset of C it follows that the ﬁrst vertical morphism is dominant. Moreover we can safely replace X0 by one of it’s irreducible components and hence we can assume without loss of generality that X0 is a variety over k. Hence by the lemma above we can ﬁnd a curve C0 = Spec(A0) in X0 that dominates C. Hence we have constructed a commutative diagram

C0 / X

f 0 f C / Y where the two vertical morphisms are dominant and the two horizontal ones are locally closed embeddings. The two curves that we constructed are both affine. Let D = Spec(Ae) and D0 = Spec(Af0), where Ae and Af0 are the integral closures of A and A0 in their field of fractions K and K0, respectively. By Noether’s theorem they are finitely-generated k- algebras [Eis95, Section 13.3]. The morphisms D → C and D0 → C0 are finite and dominant and hence in particular surjective. It is also clear that we get a dominant morphism D0 → D 8 STERGIOS ANTONAKOUDIS such that the following square commutes.

E0 o ? _D0 / C0

f 0 EDo ? _ / C Both D and D0 are smooth aﬃne curves, because in dimension 1 being normal and smooth is the same. Next, there aresmooth projective curves E and E0 over k with open subsets isomorphic to D and D0, respectively [Har77, Chapter 1, Corollary 6.10]. In particular E and E0 are proper schemes over k and the inclusion of function ﬁelds K0 ,→ K induces a morphism E → E0 such that the above diagram commutes. Moreover this morphism is proper and dominant (since it is non-constant) and hence surjective. The proof of the existence of the above morphism is the same as in the language of varieties [Har77, Chapter 1, Section 6]. Therefore there is a closed point q in E0 whose image in C along the above diagram is p; the closed point in Y that we started with. That gives a smooth curve D00 = D0 ∪ {q} such that the following diagram commutes

0 / X D _ f D0 ∪ {q} / Y By restricting to an open affine neighborhood of q we can assume that the conditions of the theorem are satisfied. Therefore we get a morphism D00 → X. Then the image of q in X is a closed point that maps to p under f. Hence we proved that f is surjective on closed points. Hence we are done. Proof of separatedness. The proof of separatedness will be similar to what we did earlier. We want to prove that the diagonal morphism X → X ×Y X has closed image. Let p be a closed point in the closure of the image of the diagonal. Then as above we get a smooth affine curve D00 with a closed point q and D00 = D0 ∪ {q} with D0 also an affine smooth curve (here we use the same notation as in the previous proof) such that we have a commutative square 0 / X D _ ∆

0 D ∪ {q} / X ×Y X with the property that the point q maps to the point p. Composing with the two projections

π1 X ×Y X /X π2 VALUATIVE CRITERIA OF SEPARATEDNESS AND PROPERNESS 9 gives two morphisms of D00 to X which give the same morphism to Y, and moreover whose restrictions to D0 are the same. So by the conditions of the theorem there two morphisms 00 are the same. That means that the morphism D → X ×Y X factors through the diagonal morphism and hence the point q belongs to the diagonal ∆(X). Hence we are done.

5. Appendix Lemma 7 (Krull-Akizuki Theorem). n Let O be a noetherian domain of dimension 1 with field of fractions K. Let L be a finite field extension of K then the integral closure of O in L is a Dedekind domain; in particular it is Noetherian of dimension 1 and hence its localization at non-zero prime ideal is a DVR.

Proof. [Neu99] We will reduce it to the case L = K. Let S be the integral closure of R in L. Then S is clearly integrally closed and of dimension 1 by going-up. We want to show that S in Noetherian. Let a1, ··· , an be a basis of L over K which is contained in S. Let 0 0 R = R[a1, ··· , an], a finitely-generated R-module and hence Noetherian. Moreover R is again one-dimensional since the extension R,→ R0 is integral. But the field of fractions of R0 is L and hence it suffices to prove the theorem for L = K, as claimed. Let S be the integral closure of R in K and let I be a non-zero ideal of S. We will prove that I is finitely-generated and therefore that S in Noetherian. The fact that the dimension of S is 1 follows from the going-up theorem. There is a non-zero element a ∈R∩I. The n images of the set of ideals {a S∩R}n in the Artinian ring R/aR (Artinian because it is Noetherian of dimension 0) form a descending chain of ideals, so there exists an integer l such that for all n ≥ l,(anS∩R) + aR = (an+1S∩R) + aR. Claim : alS⊆ al+1S+R. S ∼ alS al+1S+R ∼ R The claim proves the result because then we have aS = al+1S ⊆ al+1S = al+1S∩R , so that the image of I in S/aS is finitely-generated over R, hence I is finitely-generated, as claimed. Proof of claim: It suffices to prove the claim after localizing at each maximal ideal m of R, hence we can assume without loss of generality that R is a local ring with maximal ideal m. If a is a unit then there is nothing to show, hence we can assume that a ∈ m. b Let x be a non-zero element in S. Write x = c for some non-zero b, c ∈ R. Then n+1 1 n+1 there exists an integer n ≥ l such that m ⊆ cR ⊆ x R. In particular, a x ∈ R. Thus an+1x ∈ an+1S∩R⊆ an+2S∩R+aR, so that an ∈ an+1S + R. Now let n be the smallest integer greater than or equal to l such that anx ∈ an+1S + R. If n > l, then anx ∈ (an+1S + R) ∩ anS= an+1S + anS∩R⊆ an+1S + an+1S∩R+aR = an+1S + aR, so that an−1x ∈ anS+R, contradicting the minimality of n. This proves that n = l and hence the claim.

References [Eis95] D. Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, 1995, With a view toward algebraic geometry. 10 STERGIOS ANTONAKOUDIS

[GD67] A. Grothendieck and J. Dieudonné, Elémentsde géométriealgébrique, Publications mathématiques de l’I.H.E.S., vol. 4, 8, 11, 17, 20, 24, 28, 32, Institut des Hautes Etudes Scientifiques, 1960, 1961, 1963–1967. [Har77] R. Hartshorne, Algebraic geometry, 2nd ed., Graduate Texts in Mathematics, vol. 52, Springer- Verlag, 1977. [Neu99] J. Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften, vol. 322, Springer-Verlag, 1999.