1 DEFORMATIONS VIA DIMENSION THEORY BRIAN OSSERMAN Abstract. We show that standard arguments for deformations based on dimension counts can also be applied over a (not necessarily Noetherian) valuation ring A of rank 1. Key intermediate results are a principal ideal theorem for schemes of finite type over A, and a theorem on subadditivity of intersection codimension for schemes smooth over A. 1. Introduction In , Nagata began a process of extending certain dimension-related statements to a non-noetherian setting. Motivated by applications to tropical geometry, where one typically works over a non-noetherian valuation ring of rank 1, we continue in the same spirit as Nagata, proving several results involving dimensions of schemes over such valuation rings. We say that an irreducible catenary scheme S is intersection-regular if it has the property that codimension is subadditive under intersection: that is, for any irreducible closed subschemes Z 1, Z 2 of S, and any irreducible component Z of Z 1 Z 2, we have codim S Z codim S Z 1 + codim S Z 2. We say that S is universally intersection-regular if every irreducible scheme smooth over S is intersection-regular. It is a theorem of Serre (Theorem V.3 of ) that if S is regular, it is (universally) intersection-regular. If S is universally intersection-regular, one may use dimension-counting arguments to deform points in a family over S. Such techniques are important for instance in the theory of limit linear series developed by Eisenbud and Harris. The primary purpose of the present paper is to show that valuation rings of rank 1 are universally intersection-regular, allowing us to conclude the following theorem: Theorem 1.1. Suppose S is the spectrum of a valuation ring of rank 1, and suppose Y S is smooth, of relative dimension d. Suppose that X S is constructed by intersecting closed subschemes Z 1,..., Z m Y, and for each i = 1,..., m we have that every irreducible component of Z i has codimension less than or equal to d i in Y. Choose x in the special fiber of X over S, and suppose that the dimension of the special fiber at x is less than or equal to d m i=1 d i. Then there is a point x in the generic fiber of X over S specializing to x. If x is closed in the special fiber, x may be chosen to be closed in the generic fiber. We can apply this result in tropical geometry (*) to study the relationship between tropicalization and intersection. In the tropical setting, the additional statement in the case that x is closed is very important one works with valuation rings having algebraically closed residue and fraction fields, so if we know x is closed, we conclude it is in fact rational. 1 cite sam-me
2 2 BRIAN OSSERMAN In 2, we use Noetherian approximation to prove a result which could be interpreted as a relative form of the principal ideal theorem, which we then apply in two ways: we first conclude that schemes of finite type over a valuation ring of rank 1 satisfy the usual principal ideal theorem, and then prove a statement on the existence of closed points in generic fibers specializing to chosen closed points in special fibers. In 3, we use the principal ideal theorem to prove that the spectrum of a valuation ring of rank 1 is universally intersection-ring, adapting the usual argument for smooth varieties over a field. Finally, in 4 we adapt to the non- Noetherian setting the standard argument allowing us to deduce Theorem 1.1, and in fact prove a sharper result in a more general setting. Acknowledgements. I would like to thank Brian Conrad, David Rydh, David Eisenbud, William Heinzer, Bernd Ulrich, and Sam Payne for helpful conversations. 2. A relative principal ideal theorem In this section, we prove a sort of relative version of the Krull principal ideal theorem (Proposition 2.1) for morphisms of finite type of not necessarily Noetherian schemes, and apply it to conclude that the usual principal ideal theorem holds for schemes of finite type over a valuation ring of rank 1. We also prove a statement on the existence of closed points in generic fibers specializing to given closed points in special fibers. Proposition 2.1. Suppose f : X S is a dominant morphism between irreducible schemes, and assume that f is either flat and locally of finite type, or locally of finite presentation with equidimensional fibers. Suppose D is an effective Cartier divisor in X. If D does not meet the generic fiber of f, then for every s S, every irreducible component of D f 1 (s) is an irreducible component of f 1 (s). Proof. First, we observe that the hypotheses that f is flat and locally of finite type imply that f is necessary locally of finite presentation, by Corollary of , and with equidimensional fibers by Theorem (ii) of . The question is clearly local on both X and S, so we may assume that X is of finite presentation over S, and that both are affine, so that D is the vanishing locus of some function h on X. Say X = Spec A, S = Spec B. By finite presentation, there is a subring B of B which is finitely generated over Z, and hence Noetherian, such that X and h are defined over B that is, such that we have a finitely presented B -algebra A, and h A, with X = X := Spec A S S and h being the pullback of h to X. If X is not irreducible, we may replace it with its closed image; since X X is an affine map, this coincides with the closure of the image of the generic point of X, so is still irreducible. of its generic fiber, which will still be of finite type. Let f : X S be the structure morphism, and let D be the vanishing locus of h. If D does not meet the generic fiber of f, then the same is true for D and f. Choose s S with D f 1 (s) non-empty, and let s be the image of s in S. By equidimensionality of fibers of f, we have that the generic fiber of f and f 1 (s) are both pure of some fixed dimension d, and the same then necessarily holds for the generic fiber of f and for f 1 (s ) Corollary of . Similarly, D f 1 (s ) is pure dimension d if and only if D f 1 (s) is pure of dimension d, so it is enough to show that every irreducible component of D f 1 (s ) has dimension d. We claim that this follows from the Krull principal ideal theorem. Indeed, let D 1 be an irreducible component of D, meeting f 1 (s ). Then D 1 can have height
3 DEFORMATIONS VIA DIMENSION THEORY 3 at most 1 in X by the principal ideal theorem. Let s be the image of the generic point of D 1 in S. Since the generic point of D 1 is a specialization of the generic point of X and specializes to some point of f 1 (s ), we have that f 1 (s ) is pure of dimension d at the generic point of D 1, by upper semicontinuity of fiber dimension (Theorem of ). Since f is dominant and s is not the generic point of S, any component of f 1 (s ) has height at least 1 in X ; since D 1 has height at most 1, we conclude that D 1 f 1 (s ) is a component of f 1 (s ), and thus has dimension d. Applying Lemma of  to the closed image of D 1 in S, it follows that every irreducible component of D 1 f 1 (s ) has dimension at least d, as desired. Remark 2.2. Note that a hypothesis along the lines of flatness is in fact necessary for the proposition; otherwise, one obtains a counterexample from the blowup of the affine plane at the origin, with the divisor being the strict transform of a line through the origin. The first consequence of the proposition is that the Krull principal ideal theorem holds for schemes of finite type over valuation rings of rank 1. Corollary 2.3. Let f : X S be a morphism of finite type, with S = Spec A for A a valuation ring of rank 1, and let Z be a locally principal closed subscheme of X. Then every irreducible component of Z has codimension at most 1 in each component of X containing it. Proof. First note every closed subscheme of X can have only finitely many irreducible components, since each must have generic point in either the generic or closed fiber of f. The question is also local on X, and it suffices to work in a neighborhood of each generic point of Z, so we can reduce immediately to the case that Z is irreducible, by restricting to the complement of all but one of its components. We may further restrict to any given irreducible component of X containing Z, and we may further replace X by X red. We have thus reduced to the situation that X is integral and Z is irreducible. In this case, Z must either be equal to X, or a Cartier divisor in X, so we may assume we are in the latter situation. If X is supported in the special fiber of f, we immediately conclude the desired statement from the classical principal ideal theorem. On the other hand, if X is not supported in the special fiber, it is necessarily dominant, and hence flat. If Z meets the generic fiber, we may apply the classical principal ideal theorem to the generic fiber to conclude the desired statement. If Z is contained in the special fiber, we conclude from Proposition 2.1 that it is in fact a component of the special fiber. It thus follows that Z has height 1 in X. We can also conclude the following corollary on the topology of morphisms of finite type, using an argument in the spirit of Katz Lemma 4.15 of : Corollary 2.4. Let f : X S be a morphism of finite type, and suppose we have s, s S with s specializing to s. Given x a closed point of f 1 (s), suppose there exists x f 1 (s ) specializing to x. Then there exists a point x closed in f 1 (s ), with x specializing to x and x specializing to x. Proof. We may replace X by the closure of x, and then S by the closure of s, so we may assume X and S are integral, and s, s are the generic and closed points of S respectively. We claim it is enough to treat the case that S = Spec B, with
4 4 BRIAN OSSERMAN B a valuation ring. Indeed, let A be a valuation ring with a dominant morphism Spec A X mapping the closed point to x (Proposition of ). Then set B = A K(S); the inclusion B A is then a local homomorphism of valuation rings, and we have a commutative diagram Spec A Spec B X. Then set S = Spec B, X = X S Spec B. The maps from Spec A induce a map Spec A X which gives us a pair of points x, x X mapping to x, x in X, and with x specializing to x. We see moreover that x must be closed in the special fiber over S, since it maps to a closed point of f 1 (s). Thus, if we have the statement of the corollary over a valuation ring, we conclude that we have x closed in the generic fiber and with x specializing to x specializing to x. Let x be the image of x in X; we necessarily have that x specializes to x, which specializes to x. However, since there was no extension of residue fields at the generic point under the map Spec B S, we have that the generic fibers are isomorphic, so we conclude that x is closed in the generic fiber, giving the desired statement. We thus treat the case that S = Spec B is a valuation ring. We may further continue with the hypotheses that s, s are the generic and closed point of s, and X is integral. It then follows that X is flat over S. Moreover, because the morphism is of finite type, we may pass to Zariski open subsets of X and thus assume X = Spec A is also affine. The proof is by induction on the dimension of the generic fiber of X over S. If the dimension is 0, then x is already necessarily closed. Suppose now that the dimension of the generic fiber is d > 0, and we know the desired statement for fiber dimension d 1. By semicontinuity of fiber dimension, the fiber dimension at x is necessarily at least d, and in particular strictly positive. We may thus choose f A vanishing at x, but whose vanishing set D does not contain any component of the special fiber passing through x. D can have at most finitely many irreducible components which meet the generic fiber; we may thus replace X by a Zariski open subset and assume that there is no irreducible component of D which meets the generic fiber but does not contain x. By Proposition 2.1, we conclude that D must meet the generic fiber, so if we let Z be any irreducible component of D meeting the generic fiber, applying the induction hypothesis to Z we obtain a closed point x in the generic fiber of Z over S which specializes to x, giving us the desired statement. Remark 2.5. Corollary 2.4 is broadly useful in the context of tropical geometry. Indeed, in addition to its application to tropical intersection theory, it fixes a gap in the original proof of a foundational result of Speyer and Sturmfels, Theorem 2.1 of ; see also Payne  for an alternative argument. 3. Serre s theorem over non-noetherian valuation rings We now use the principal ideal theorem over valuation rings of rank 1 to prove that schemes smooth over a valuation ring of rank 1 are intersection-regular, giving a partial generalization of Serre s theorem to the non-noetherian case. As compared to Serre s theorem, our proof follows the much easier argument for varieties smooth over a field. S
5 DEFORMATIONS VIA DIMENSION THEORY 5 Theorem 3.1. Let S = Spec B, with B a valuation ring of rank 1. Then S is universally intersection-regular. Note that the rank condition is necessary not only in our argument, but for also for the validity of the theorem, as demonstrated by the following example. Example 3.2. Let B be a valuation ring of rank greater than 1, and b B an element generating an ideal of height greater than 1. Let A = B[x], and consider the ideals of A generated by x and by x b. These each have height 1, but the ideal (x, x b) = (x, b) has height strictly greater than 2. We will repeatedly use the following altitude formula of Nagata (Theorem 2 of ): Theorem 3.3 (Nagata). Let S = Spec A with A a valuation ring of finite rank, and let X be an irreducible scheme, with f : X S dominant and of finite type. Given x X with closure x, we have: codim X x + trdeg f(x) x = codim S f(x) + trdeg S X, and moreover, any maximal chain of irreducible closed subschemes containing x has length equal to codim X x. In particular, X is catenary. Proof. Since the statement is in terms of codimensions and transcendance degrees, it is clearly local, so we may assume X is affine. It is likewise topological, so we may assume X is reduced. The first statement in is then precisely what is proved by Nagata. The catenary assertion following by extending any two maximal chains between x x to maximal chains containing x. The main proposition necessary for the proof of Theorem 3.1 is the following. Proposition 3.4. Let S be the spectrum of a valuation ring of rank 1, and suppose we have an irreducible scheme X flat and of finite type over S, and Z 1 and Z 2 two irreducible closed subschemes of X. Then for every irreducible component Z of Z 1 S Z 2, and any irreducible components X of X S X, we have the inequality codim X Z codim X Z 1 + codim X Z 2. Proof. Since Z 1 red S Z 2 red is homeomorphic to Z 1 S Z 2 (Proposition of ), we may assume both Z 1 and Z 2 are reduced. Let Z 1, Z 2 and Z be the closures of the images in S of Z 1, Z 2 and Z respectively. Z is contained in, and hence still an irreducible component of, Z 1 Z Z Z 2 Z. Let Z i be irreducible components of Z i Z containing the closure of the image of Z, which is irreducible. Considering the fibers over the generic points of Z i and Z, and applying Corollary of , we have trdeg(z i/ Z) trdeg Z i / Z i for i = 1, 2. Next, we claim that we cannot have Z equal to the closed point of S unless at least one of Z 1, Z 2 is also equal to the closed point. Indeed, suppose that both Z 1 and Z 2 dominate S. Then they are both flat over S, and so too is Z 1 S Z 2, and it follows that Z must dominate S (Proposition of ). We thus have codim S Z codims Z1 + codim S Z2.
6 6 BRIAN OSSERMAN Finally, we have by Proposition (iii) of  that trdeg Z/ Z = trdeg Z 1/ Z + trdeg Z 2/ Z, and X S X is still flat over S and must thus have any irreducible component X flat over S, with trdeg X /S = 2 trdeg X/S, by Corollary of . Thus, by Theorem 3.3 we conclude as desired. codim X Z = codim S Z + trdeg X S X/S trdeg Z/ Z = codim S Z + 2 trdeg X/S trdeg Z 1 / Z trdeg Z 2/ Z codim S Z1 + trdeg X/S trdeg Z 1/ Z + codim S Z2 + trdeg X/S trdeg Z 2/ Z = codim X Z 1 + codim X Z 2, In order to conclude Theorem 3.1, we make the following naive definition in general, without assuming that the principal ideal theorem holds: Definition 3.5. A locally closed subscheme Z X is a locally complete intersection subscheme if, for every point z Z, there is an affine neighborhood U of z in X in which Z is the vanishing set of ideal I, generated by c sections of O X (U), and every irreducible component of Z passing through z has codimension c in every irreducible component of X containing it. We then recall the following well-known theorem on smooth morphisms: Theorem 3.6. Let f : X Y be an immersion, locally of finite presentation, of one smooth S-scheme into another. Then X is a local complete intersection subscheme of Y. In particular, if f : X S is smooth, then the diagonal : X X S X is a locally complete intersection subscheme. Proof. This is essentially the implication (a) implies (d) in Proposition 7 of 2.2 of . We note that the assertion is local, so we may assume that f is in fact a closed immersion. If (at given points x and f(x)) we denote by r and n the relative dimensions of X and Y over S, then given loc. cit. it is enough to verify that the codimension of X inside Y is n r. Let Z be any irreducible component of X containing x, and Z an irreducible component of Y containing Z; we claim that the codimension of Z in Z is n r. Since X and Y are smooth, we have that Z and Z must dominate an irreducible component Z of S, and we can compute the codimension after passing to the generic fiber. But in the generic fiber, we are simply of finite type over a field, so we can naively compute the codimension as n r, as desired. For the second assertion, note that the diagonal is locally of finite presentation by Corollary of , so we can apply the first assertion. We may now easily prove that the spectrum of a valuation ring of rank 1 is universally intersection-regular. Proof of Theorem 3.1. Suppose X is smooth over S, and Z 1 and Z 2 are two irreducible closed subschemes of X. By Proposition 3.4, if X is the connected (hence
7 DEFORMATIONS VIA DIMENSION THEORY 7 irreducible, by smoothness) component of X S X containing the diagonal, we have that every irreducible component Z of Z 1 S Z 2 satisfies codim X Z codim X Z 1 + codim X Z 2. By Theorem 3.3, all schemes in question are catenary, so we can inductively apply Corollary 2.3 to see that codimension can only decrease when intersecting with any locally complete intersection subscheme. Since Z 1 Z 2 can be realized as Z 1 S Z 2, we conclude the desired statement from Theorem Deformations via dimension theory We conclude by arguing that if we have that S is universally intersection-regular, then we can deform via dimension theory as in Theorem 1.1. Although the argument is standard, it is typically made for varieties over a field, where concepts of dimension are far more robust, and it requires somewhat more care in this setting. Even in the Noetherian setting, dimensions can be badly behaved globally, so for instance in  we state our comparable results in terms of dimensions of local rings at closed points of fibers and their images. Here, we instead propose the following definition, which applies in principal even to settings where local rings are infinite-dimensional. Definition 4.1. Let f : X S be a morphism of finite type, with S an irreducible scheme. We say that f has strong relative dimension at least ρ over S if the following conditions hold: (i) for every irreducible component Z of X, every component of every fiber of Z has dimension at least ρ; (ii) for every x X, every integral closed subscheme S S, and every irreducible component Z of f 1 (S ), if the dimension of f 1 (f(x)) at x is ρ + δ, then the codimension f(z ) in S is at most δ. Note that in the definition, we allow ρ to be negative, and can still draw conclusions in this situation. Proposition 4.2. Let S be universally intersection-regular, and Y smooth over S of relative dimension d. Suppose that f : X S is constructed by intersecting closed subschemes Z 1,..., Z m Y, and for each i = 1,..., m we have that every irreducible component of Z i has codimension less than or equal to d i in Y. Then X has strong relative dimension at least d m i=1 d i over S. Proof. By hypothesis, Y is intersection-regular, so we conclude by induction on m that every irreducible component Z of X has codimension at most m i=1 d i in Y. Now let S S be an irreducible closed subscheme; note that S necessarily has finite codimension c since we assume S to be catenary. Then because Y is smooth and catenary, any irreducible component of the preimage S of S in Y also has codimension c by Proposition of , so we conclude that an irreducible component Z of S X has codimension at most c + m i=1 d i. For any s S, if we choose S to be the closure of s, then every component of f 1 (s) Z has closure in Z equal to a component of S Z, so in order for the latter to have codimension less than or equal to c + m i=1 d i in Y, we see that the former must have codimension less than or equal to m i=1 d i in the fiber of Y over s, giving condition (i).
8 8 BRIAN OSSERMAN For condition (ii), if the dimension of f 1 (f(x)) at x is d m i=1 d i + δ, and η is the generic point of Z, the dimension of f 1 (f(η)) at η is less than or equal to d m i=1 d i + δ. It follows that the codimension of Z in the fiber of Y over f(η) is at least m i=1 d i δ. Let c be the codimension of f(z ) in S. Then the codimension in S of the fiber of Y over f(η) is also c as before, so we have that the codimension of Z in Y is at least c + c + m i=1 d i δ. We conclude that we have m m c + c + d i δ codim Y Z c + d i, so c δ, as desired. i=1 Proposition 4.3. Suppose f : X S has strong relative dimension at least ρ, and suppose we have x X such that the dimension of f 1 (f(x)) at x is equal to ρ. Then: (i) for all s S specializing to f(x), and for all irreducible components Z of X containing x, there is a point x of Z specializing to x with f(x ) = s. (ii) if x is closed in its fiber, x as above may be chosen to be closed in its fiber as well; (iii) if X is of finite presention over S, then f is open at x. Proof. Condition (i) is an immediate application of the definition of strong relative dimension, with δ = 0. We conclude (ii) from Corollary 2.4. Finally, we conclude (iii) from (i) by Chevalley s theorem (see Proposition of ). Putting the two propositions together, we immediately conclude: Corollary 4.4. Let S be universally intersection-regular, and Y smooth over S of relative dimension d. Suppose that X S is constructed by intersecting closed subschemes Z 1,..., Z m Y, and for each i = 1,..., m we have that every irreducible component of Z i has codimension less than or equal to d i in Y. Choose x X, and suppose that the dimension of f 1 (f(x)) at x is less than or equal to d m i=1 d i. Then for every irreducible component Z of X, and every point s S specializing to f(x), there is a point x in Z specializing to x with f(x ) = s. If x is closed in its fiber, x may be chosen to be closed in its fiber as well. We conclude Theorem 1.1 immediately from Corollary 4.4 and Theorem 3.1. Note however that we have in fact shown that the hypotheses of Corollary 4.4 are satisfied more generally: in particular, it follows from Theorem 3.1 that our base scheme S could be any irreducible scheme smooth over the spectrum of a valuation ring of rank 1. References 1. Siegfried Bosch, Werner Lutkebohmert, and Michel Raynaud, Neron models, Springer-Verlag, Alexander Grothendieck and Jean Dieudonné, Éléments de géométrie algébrique: I. Le langage des schémas, Publications mathématiques de l I.H.É.S., vol. 4, Institut des Hautes Études Scientifiques, , Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes, Publications mathématiques de l I.H.É.S., vol. 8, Institut des Hautes Études Scientifiques, i=1
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