INFINITESIMALS IN A RECURSIVELY ENUMERABLE PRIME MODEL

TRISTRAM DE PIRO

Abstract. Using methods developed by Robinson, we find a complete theory suitable for a first order description of infinitesimal neighborhoods. We use this to construct a specialisation having universal properties and to find a recursively enumerable model in which the algebraic version of Bezout’s theorem is provable by non-standard methods.

1. Specialisations and Valuations

Let L and K be fields with an imbedding i : L^{∗} → K^{∗}. In the case when L and K have the same characteristic, we will consider L as a subfield of K, otherwise we will by some abuse of notation refer to the embedded set i(L^{∗}) ∪ {0} as L. Let P (K) = ⋃_{n≥1} P^{n}(K) and P (L) = ⋃_{n≥1} P^{n}(L). By a closed algebraic subvariety of P^{n}(K), we mean a set W (K) where W is defined by homogeneous polynomial equations with coefficients in K. We say that W (K) is defined over L if we can take the coefficients to lie in L. Let W_{n}^{m}(K) denote the m′th Cartesian product of P^{n}(K). By a closed algebraic subvariety of W_{n}^{m}(K), we mean a set W (K) defined by multi-homogeneous polynomial equations with coefficients in K, similarly we can make sense of the notion of being defined over L. Note that if K is not algebraically closed, it is not necessarily true that the projection maps pr_{k,m} : W_{n}^{k}(K) → W_{n}^{m}(K) preserve closed algebraic subvarieties.

Definition 1.1. A specialisation is a map π = ⋃_{n≥1} π_{n} : P (K) → P (L), such that each π_{n} : P^{n}(K) → P^{n}(L) has the following property;

Let W_{n}^{m}(K) denote the m’th Cartesian product of P^{n}(K). Then, if V ⊂ W_{n}^{m}(K) is a closed algebraic subvariety defined over L and ā is an m-tuple of elements from W_{n}(K), such that V (ā) holds, then V (π_{n}(ā)) holds as well.

The author was supported by a Marie Curie research fellowship.

The following compatibility requirement must also hold between the π_{n};

Fix the following chain of embeddings i_{n} of P^{n}(K) and P^{n}(L) into P^{n+1}(K) and P^{n+1}(L) for n ≥ 1.

i_{n} : [x_{0} : …: x_{n}] ↦ [x_{0} : …: x_{n} : 0].

Then we require that π_{n+1} ∘ i_{n} = i_{n+1} ∘ π_{n}.

Definition 1.2. A Krull valuation v is a map v : K → Γ ∪ ∞ where Γ is an ordered abelian group with the following properties;

(i). v(x) = ∞ iff x = 0. (ii). v(xy) = v(x) + v(y) (iii). v(x + y) = min{v(x), v(y)}

Here, we adopt the convention that γ < ∞ for γ ∈ Γ and extend + naturally to Γ ∪ ∞.

We let O_{v} = {x ∈ K : v(x) ≥ 0} be the valuation ring of v and M_{v} = {x ∈ K : v(x) > 0} the unique maximal ideal. We also require;

(iv). The inclusion i : L^{∗} ∪ 0 → O_{v}^{∗} ∪ 0 maps L isomorphically onto O_{v}/M_{v}, the residue field of v.

Definition 1.3. We say that two Krull valuations v_{1} and v_{2} are equivalent, denoted by v_{1} ∼ v_{2} if O_{v_{1}} = O_{v_{2}}.

Lemma 1.4. v_{1} and v_{2} are equivalent iff there exists Θ : Γ_{1} → Γ_{2} such that Θ ∘ v_{1} = v_{2}.

In order to see this, define Θ(v_{1}(x)) = v_{2}(x), this is well defined as if v_{1}(x) = v_{1}(x′), then v_{1}(x/x′) = 0, hence x/x′ and x′/x belong to O_{v_{1}}. If v_{1} ∼ v_{2}, then x/x′ and x′/x belong to O_{v_{2}} as well, which gives that v_{2}(x) = v_{2}(x′). One can easily check that Θ is an isomorphism of ordered abelian groups as required.

Our main result in this section is the following;

Theorem 1.5. Let X := {π : P (K) → P (L)} be the set of specialisations and Y := {v/ ∼: v : K → Γ} be the set of equivalence classes of Krull valuations. Then there exists a natural bijection between X and Y. Specifically, there exists maps Φ and Ψ;

Φ : X → Y

Ψ : Y → X

with Ψ ∘ Φ = Id_{X} and Φ ∘ Ψ = Id_{Y}

We first show;

Theorem 1.6. There exists Ψ : Y → X

Proof. Let [v] denote a class of Krull valuations on K. We define a specialisation map π_{[v]} as follows;

Let (x_{0} : x_{1} : …: x_{n}) denote an element of P^{n}(K) written in homogeneous coordinates. For some λ ∈ K, the elements {λx_{0}, …, λx_{n}} will lie in O_{v} and not all of them will lie in M_{v}. Let π : O_{v} → L denote the unique ring morphism such that π ∘ i = Id_{L} where i is the inclusion map from L into O_{v}. Then (π(λx_{0}) : π(λx_{1}) : …: π(λx_{n})) defines an element of P^{n}(L). As is easily checked, the mapping is independent of the choice of λ and depends only on O_{v}, hence we obtain π_{n,[v]} : P^{n}(K) → P^{n}(L). We need to check that each π_{n,[v]} satisfies the property required of a specialisation. We will just verify this in the case when m ≤ 2 for each n ≥ 1, the other cases are straightforward generalisations;

For m = 1, let V ⊂ P^{n}(K) be a closed subvariety defined over L, then V is defined by a system of homogeneous equations in the variables {x_{0}, …, x_{n}} with coefficients in L. Taking a tuple ā belonging to V, we can assume that the elements {a_{0}, a_{1}, …, a_{n}} belong to O_{v}. Now, using the fact that the residue map π is a ring homomorphism fixing L, the reduced elements {π(a_{0}), π(a_{1}), …, π(a_{n})} also satisfy the same homogeneous equations as required.

For the case when m = 2, we use the Segre embedding which is defined by;

Segre : P^{n}(K) × P^{n}(K) → P^{n(n+2)}(K)

((x_{0} : …: x_{n}), (y_{0} : …: y_{n})) ↦ (x_{0}y_{0} : …: x_{0}y_{n} : x_{1}y_{0} : …: x_{n}y_{n})

The following diagram is easily checked to commute:

P^{n}(K) × P^{n}(K) ⟶^{Segre} P^{n(n+2)}(K)
↓^{π_{n,[v]}×π_{n,[v]}}  ↓^{π_{n(n+2),[v]}}
P^{n}(L) × P^{n}(L) ⟶^{Segre} P^{n(n+2)}(L) .

Therefore, it is sufficient to prove that the property holds for π_{n(n+2),[v]} : P^{n(n+2)}(K) → P^{n(n+2)}(L) when m = 1.This is the case covered above.

Finally, we need to check the compatibility requirement for the π_{n,[v]}, this is a trivial calculation.

Denote the specialisation map we have obtained by π_{[v]} and let Ψ([v]) = π_{[v]}.

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We now show;

Theorem 1.7. There exists Φ : X → Y

Proof. Suppose that we are given a specialisation π. In particular we have a map π_{1} : P^{1}(K) → P^{1}(L) satisfying the requirements above. We want to show how to recover a Krull valuation on K.

Let γ : K → P^{1}(K) be the map γ : k ↦ [k : 1], so π_{1} ∘ γ : K → P^{1}(L). Let U ⊂ P^{1}(L) be the open subset defined by P^{1} \ [1 : 0]. Let O_{K} = (π_{1} ∘ γ)^{−1}(U) and M_{K} = (π_{1} ∘ γ)^{−1}([0 : 1]). We now claim the following;

Lemma 1.8. O_{K} is a subring of K with F rac(O_{K}) = K and M_{K} is an ideal of O_{K}.

Proof. Suppose that {x, y} ⊂ O_{K}, then both π_{1}([x : 1]) and π_{1}([y : 1]) are in U. Let C ⊂ P^{1}(K) × P^{1}(K) × P^{1}(K) be the closed set defined in coordinates ([u : v], [w : x], [y : z]) by the equation uwz = yvx. As is easily checked, we have that C([x : 1], [y : 1], [xy : 1]). By the defining property of π_{1}, C(π_{1}([x : 1]), π_{1}([y : 1]), π_{1}([xy : 1])) also holds. Therefore, C([λ : 1], [μ : 1], [α, β]) where λ, μ, α, β are in L. By definition of C, we have λμβ = α which forces β ≠ 0. Hence, π_{1}([xy : 1]) ∈ U and therefore xy ∈ O_{K}. Let D ⊂ P^{1}(K) × P^{1}(K) × P^{1}(K) be defined using the same choice of coordinates by the equation uxz + wvz = yvx. Then we have that D([x : 1], [y : 1], [x + y : 1]) and therefore D(π_{1}([x : 1]), π_{1}([y : 1]), π_{1}([x + y : 1])). Again, we must have D([λ : 1], [μ, 1], [δ, ϵ]) where λ, μ, δ, ϵ are in L. This forces (λ + μ)ϵ = δ and therefore ϵ ≠ 0, so x + y ∈ O_{K}. Clearly, 1 ∈ O_{K} which shows that O_{K} is a subring of K as required. In order to see that M_{K} is an ideal of O_{K}, let x ∈ O_{K} and y ∈ M_{K}. We have that C([λ : 1], [0 : 1], [α, β]) where π_{1}([xy : 1]) = [α, β]. Then λ.0.β = 1.1.α forcing α = 0 and β = 1, so xy ∈ M_{K}. If x ∈ M_{K} and y ∈ M_{K} we obtain D([0 : 1], [0 : 1], [δ, ϵ]) where π_{1}([x + y : 1] = [δ, ϵ]. Then 0.1.β + 1.0.ϵ = 1.1.δ, so δ = 0 and ϵ = 1, hence x + y ∈ M_{K} as required. Finally, we show that F rac(O_{K}) = K. Suppose x ∉ O_{K}, then π_{1}([x : 1]) = [1 : 0]. We have that C([x : 1], [1/x : 1], [1 : 1]), hence C([1 : 0], [α, β], [1 : 1]) where π_{1}([1/x : 1]) = [α, β]. This forces 1.α.1 = 0.β.1, hence α = 0 and β = 1. Therefore 1/x ∈ O_{K} as required. □

We now further claim the following;

Lemma 1.9. If π_{1} is non-trivial, that is π_{1} is not a bijection between P^{1}(K) and P^{1}(L), then O_{K} is a proper subring of K

Proof. By the same argument as above we have that π_{1} ∘ γ(1/M_{K}) = [1 : 0], hence if O_{K} = K, using the previous lemma, we must have that M_{K} = 0. If π_{1} is non-trivial, we can find x ∈ K and y ∈ K distinct such that π_{1}([x : 1]) = π_{1}([y : 1]). By the usual arguments, we then have that π_{1}([x − y : 1]) = [0 : 1], so x − y ∈ M_{K} contradicting the fact that M_{K} = {0}.

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We can now construct a Krull valuation on K by a standard method. Let Γ = K^{∗}/O_{K}^{∗} and define v : K → Γ by v(x) = x mod O_{K}^{∗} and v(0) = ∞. Define an ordering on the abelian group Γ by declaring v(x) ≤ v(y) iff y/x ∈ O_{K}. This is well defined as if v(x) = v(x′) and v(y) = v(y′), then y′/y, y/y′,x/x′ and x′/x are all in O_{K}. We have that y′/x′ = y/x.y′/y.x/x′ and y/x = y′/x′.y/y′.x′/x, therefore y′/x′ ∈ O_{K} iff y/x ∈ O_{K} as required. Transitivity of the ordering follows from the fact that O_{K} is a subring of K. ≤ is a linear ordering as if x ∈ K^{∗} and y ∈ K^{∗} then either x/y or y/x lies in O_{K}. Finally, we clearly have that if y/x ∈ O_{K} then yz/xz ∈ O_{K}, hence v(x) ≤ v(y) implies v(x) + v(z) ≤ v(y) + v(z). This turns Γ into an ordered abelian group. Properties (i) and (ii) of the axioms for a Krull valuation are trivial to check. Suppose property (iii) fails, then we can find x, y with v(x + y) < v(x) and v(x + y) < v(y). Therefore (x + y)/x ∉ O_{K} and (x + y)/y ∉ O_{K}. As 1 ∈ O_{K}, we have that x/y ∉ O_{K} and y/x ∉ O_{K} which is a contradiction. Finally, we check property (iv). By definition of π_{1}, we have that L^{∗} ⊂ O_{K}^{∗}, hence v|L is trivial. If k ∈ O_{K}^{∗}, we can find l ∈ L^{∗} such that π_{1}([k : 1] = [l : 1], then π_{1}([k −l : 1]) = [0 : 1] and k − l ∈ M_{K}. It follows that L maps onto O_{K}/M_{K}, and O_{K}/M_{K} ≅ L as required. Denote the valuation we have obtained by v_{π} and set Φ(π) = [v_{π}]. This ends the proof of Theorem 1.7

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We now complete the proof of Theorem 1.5;

Proof. Φ ∘ Ψ = Id_{Y}.

Let [v] be a class of Krull valuations on K with corresponding specialisation π_{[v]} provided by Ψ. Let π_{1,[v]} be the restriction to P^{1}(K). By definition, if k ∈ O_{v} then π_{1,[v]}([k : 1]) = [π(k), 1] where π is the residue map for v. If k ∉ O_{v}, then π_{1,[v]}([k : 1]) = [0, 1], so we see that O_{K} as defined above is exactly O_{v}. The valuation v_{π_{[v]}} constructed from π_{[v]} therefore has the same valuation ring O_{v}, so v ∼ v_{π_{[v]}} which gives the result.

Ψ ∘ Φ = Id_{X}.

Let π be a given specialisation and [v_{π}] the corresponding class of Krull valuations. Let π_{1} be the restriction of π to P^{1}(K) and π_{1,v_{π}} the specialisation constructed from v_{π} restricted to P^{1}(K). We have;

(i). π_{1,v_{π}}([k : 1]) = [0 : 1] iff v_{π}(k) > 0 iff k ∈ M_{v_{π}} iff k ∈ M_{K} as defined above iff π_{1}([k : 1]) = [0 : 1]

(ii). π_{1,v_{π}}([k : 1]) = [1 : 0] iff v_{π}(k) < 0 iff k ∉ O_{v_{π}} iff k ∉ O_{K} as defined above iff π_{1}([k : 1]) ∉ U iff π_{1}([k : 1]) = [1 : 0]

(iii). π_{1,v_{π}}([1 : 0]) = π_{1}([1 : 0]) = [1 : 0] trivially.

If k ∈ O_{v_{π}}, then π_{1,v_{π}}([k : 1]) = [α(k) : 1] where α is the residue mapping associated to v_{π}. We also have that π_{1}([k : 1]) ∈ U, hence as π_{1} is a specialisation that π_{1}([k : 1]) = [β(k) : 1] where β is a homomorphism from O_{v_{π}} to L. We thus obtain two homomorphisms α, β : O_{v_{π}} → L such that (by (i)) Ker(α) = Ker(β) = M_{v_{π}} and with the property that α∘i = β ∘i = Id_{L} where i is the natural inclusion of L in O_{v_{π}}. We thus obtain the splitting O_{v_{π}} = L ⊕ Ker(α) = L ⊕ Ker(β) = L ⊕ M with Ker(α) = Ker(β) = M. Now, using this fact, we can write any element of O_{v_{π}} uniquely in terms of L and M, hence the corresponding projections α and β are the same.

We have shown that π_{1} = π_{1,v_{π}}, it remains to check that π_{n} = π_{n,v_{π}} for all n ≥ 1. We prove this by induction on n, the case n = 1 having been established.

By the induction hypothesis and the compatibility requirement between the π_{n}, for {k_{0}, k_{1}, …, k_{n}} ⊂ O_{v_{π}};

π_{n+1}([k_{0} : k_{1} : …: k_{n} : 0]) = [π(k_{0}) : π(k_{1}) : …: π(k_{n}) : 0] (*)

where π is the residue map on O_{v_{π}}.

Let C ⊂ P^{n+1}(K) be the closed subvariety defined using coordinates [x_{0} : x_{1} : …: x_{n+1}] by the equations x_{0} = x_{1} = … = x_{n−1} = 0. Then by arguments as above and the fact that C is preserved by π_{n+1}, we can find a Krull valuation v′ on K with corresponding residue mapping π′ such that;

π_{n+1}([0 : …: 0 : 1 : k_{n+1}]) = [0 : …: 0 : 1 : π′(k_{n+1})] if v′(k_{n+1}) ≥ 0

= [0 : …: 0 : 0 : 1] otherwise (**)

Now let D be the closed subvariety of P^{n+1}(K) defined by the equations x_{1} = … = x_{n} and x_{0} = x_{n+1}. Again, we have that π_{n+1} preserves D, hence there exists a Krull valuation v′′ on K with corresponding residue mapping π′′ such that;

π_{n+1}([k : 1 : …: 1 : k]) = [π′′(k) : 1 …: 1 : π′′(k)] if v′′(k) ≥ 0

= [1 : 0 : …: 0 : 1] otherwise (***)

Let Sum be the closed subvariety of P^{n+1}(K)×P^{n+1}(K)×P^{n+1}(K) defined using coordinates [x_{0} : x_{1} : …: x_{n+1}], [y_{0} : y_{1} : …: y_{n+1}] and [z_{0} : z_{1} : …: z_{n+1}] by the equations x_{0}y_{1}z_{1} + y_{0}x_{n}z_{1} = z_{0}x_{n}y_{1} and x_{n+1}y_{1}z_{1} + y_{n+1}x_{n}z_{1} = z_{n+1}x_{n}y_{1}. Then, for k ∈ K, we have that Sum([0 : 0 : …: 1 : k], [k : 1 : …: 0 : 0], [k : 1 : …: 1 : k]), hence by the properties of a specialisation that Sum(π_{n+1}([0 : 0 : …: 1 : k]), π_{n+1}([k : 1 : …: 0 : 0]), π_{n+1}([k : 1 : …: 1 : k])).

In the generic case when v_{π}(k), v′(k), v′′(k) are all non-negative, we obtain Sum([0 : 0 : …: 1 : π′(k)], [π(k) : 1 : …: 0 : 0], [π′′(k) : 1 : …: 1 : π′′(k)]) which gives the relations 0.1.1 + π(k).1.1 = π′′(k).1.1 and π′(k).1.1 + 0.1.1 = π′′(k).1.1, so π(k) = π′(k) = π′′(k).

A simple calculation shows that v_{π}(k) < 0 iff v′(k) < 0 iff v′′(k) < 0, hence O_{v_{π}} = O_{v′} = O_{v′′}. We have now shown the following further compatibility between π_{1} and π_{n+1}. Namely;

If γ : P^{1}(K) → P^{n+1}(K) is given by γ : [x_{0}, x_{1}] ↦ [0 : 0 : …: x_{0} : x_{1}] then π_{n+1} ∘ γ = γ ∘ π_{1}. (†)

Finally, let Sum′ be the closed subvariety of P^{n+1}(K) × P^{n+1}(K) × P^{n+1}(K) defined in coordinates [x_{0} : …: x_{n+1}], [y_{0} : …: y_{n+1}], [z_{0} : …: z_{n+1}] by the (n + 1) equations x_{j}y_{1}z_{1} + y_{j}x_{n}z_{1} + z_{j}x_{n}y_{1} for j ≠ n. Let [k_{0} : …: k_{n+1}] be an arbitrary element of P^{n+1}(K). Without loss of generality, we may assume that {k_{0} : …: k_{n+1}} ⊂ O_{v_{π}} and that k_{n} ∈ O_{v_{π}}^{∗}. Hence, dividing by k_{n}, the element is of the form [k_{0} : …: k_{n−1} : 1 : k_{n+1}] with {k_{0}, …, k_{n−1}, k_{n+1}} ⊂ O_{v_{π}}. We have that Sum′([0 : …: 0 : 1 : k_{n+1}], [k_{0} : …: k_{n−1} : 1 : 0], [k_{0} : …: k_{n−1} : 1 : k_{n+1}]), hence by specialisation and (†), Sum′([0 : …: 0 : 1 : π(k_{n+1})], [π(k_{0}) : …: π(k_{n−1}) : 1 : 0], [l_{0} : …: l_{n} : l_{n+1}]). where {l_{0}, …, l_{n+1}} ⊂ L. As is easily checked, the case when l_{n} = 0 leads to a contradiction, hence we can assume that l_{n} = 1 (multiplying by 1/l_{n}). Now the equations give that l_{j} = π(k_{j}) for j ≠ n. We have therefore shown that π_{n+1} = π_{n+1,v_{π}} as required.

Theorem 1.5 is now proved.

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2. A Model Theoretic Language of Specialisations

We now introduce a model theoretic language which will enable us to describe specialisations in the context of algebraic geometry. In this section, we will assume that K and its residue field have the same characteristic. We will use a many sorted structure {⋃ S_{n} : n ∈ N }. Each sort will be the domain of P^{n}(K) for an algebraically closed field K. We fix an algebraically closed constant field L which we assume to be countable and let K be some non-trivial extension of L, having the same characteristic. In order to describe algebraic geometry, we introduce sets of predicates {V_{n}^{m}} on the Cartesian powers S_{n}^{m} to describe closed algebraic subvarieties of P^{n}(K) defined over L. In particular, we have constants to denote the individual elements of P^{n}(L) on each sort S_{n}. We introduce function symbols i_{n} : S_{n} → S_{n+1} to describe the imbeddings P^{n}(K) → P^{n+1}(K) defined above. Finally, we will have symbols {π_{n} : n ∈ N } to describe the specialisation map π = ∪_{n≥1}π_{n}. Strictly speaking, as P^{n}(L) is not definable, each π_{n} will be a union over l ∈ P^{n}(L) of unary predicates defined as {x ∈ P^{n}(K) : π_{n}(x) = l}. We denote the language < {V_{n}^{m}}, i_{n}, π_{n} > by L_{spec} and the theory of the structure < P (K), P (L), π > in this language by T_{spec}. We denote the theory of the structure < P (K), P (L) > in the language L_{spec} \ {π_{n}} by T_{alg}. Note that the structure < K, 0, 1, +, . > is interpretable in the structure < P (K), P (L) > in the language L_{spec} \ {π_{n}} (∗). This follows by noting that the points [1 : 0], [0 : 1] and [1 : 1] are named as elements in the sort S_{1} and the operations of +, . define algebraic subvarieties in the sorts S_{1}^{3}. The structure < L, 0, 1, +, . > is not interpretable but any model of T_{alg} will contain an isomorphic copy of P (L) as a substructure. It follows that the models of T_{alg} are exactly of the form < P (K), P (L) > for some algebraically closed field K properly extending L (use the fact that the axiomatisation of T h(< K, 0, 1, +, . >) can be interpreted in T_{alg} and the field structure can be related to the predicates {V_{n}^{m}} using the imbeddings i_{n}). We now claim the following;

Theorem 2.1. The theory T_{spec} is axiomatised by T_{axioms} = T_{alg} ∪ Σ where Σ is the set of sentences given by;

(i). The mappings {π_{n}} preserve the predicates {V_{n}^{m}}.

(ii). The compatibility requirement π_{n+1} ∘ i_{n} = i_{n+1} ∘ π_{n} holds.

(see definition 1.1). In particular, T_{axioms} is complete. Moreover, T_{axioms} is model complete.

The proof of this theorem will be based on Theorem 1.5 and the following result by Robinson, given in [6];

Theorem 2.2. Let K be an algebraically closed field with a non trivial Krull valuation v and residue field l. Then T_{K} is model complete in the language L_{val} and admits quantifier elimination in the language L_{rob}. Moreover, the completions of K are determined by the pair (char(l), char(K)), that is T_{K} ∪ Σ is complete where Σ is the possibly infinite set of sentences specifying the characteristic of K and l.

Here, by the language L_{rob} we mean the language of algebraically closed fields together with a binary predicate Div(x, y) denoting v(x) ≤ v(y). By the language L_{val}, we mean a 2-sorted language for the value group and the field, with the usual language for the field sort and the language of ordered groups on the group sort. T_{K} is the theory which asserts that K is an algebraically closed field, the value group Γ is linearly ordered and abelian, the valuation is non-trivial. For our purposes, we will require a slightly refined version of this result. Namely, we will fix a set of constants for an algebraically closed field L which we can assume to be countable, add to T_{K} the atomic diagram of L, relativized to the field sort, the requirement that v|L is trivial and π, the residue mapping, maps L injectively and homomorphically into the residue field. (Note, the condition that L maps onto the residue field is not definable and that the homomorphism requirement ensures that the residue field l and K have equal characteristic, hence the characteristic of K is already determined by the characteristic of L.) We will denote the corresponding theory by T_{K,L} and the expanded languages by L_{rob} and L_{val} again. It is no more difficult to prove that T_{K,L} is model complete, Robinson’s original proof in [6] requires the solution of certain valuation equations in the model K given that these equations have a solutions in an extension K′, it makes no difference if some of the elements from K are named. In order to show that T_{K,L} is complete, it is sufficient to exhibit a prime model of the theory;

Case 1. Char(K, L) = (p, p), with p ≠ 0. Take L(ϵ)^{alg} where ϵ is transcendental over L, define the valuation on L to be zero and extend it to L(ϵ) non-trivially using say v_{ord,ϵ}, the order valuation in ϵ. Take any extension to L(ϵ)^{alg}.

Case 2. Char(K, L) = (0, 0), define a similar valuation on L(ϵ)^{alg}.

We now show the following lemma;

Lemma 2.1. Amalgamation of Specialisations

Let (P (K_{1}), P (L), π_{1}) and (P (K_{2}), P (L), π_{2}) be models of T_{axioms}, then there exists a further model (P (K_{3}), P (L), π_{3}) such that;

(P (K_{1}), P (L), π_{1}) ≤ (P (K_{3}), P (L), π_{3})

and

(P (K_{2}), P (L), π_{2}) ≤ (P (K_{3}), P (L), π_{3})

Proof. By Theorem 1.5, we can find Krull valuations v_{1} and v_{2} on K_{1} and K_{2} such that π_{1} = π_{v_{1}} and π_{2} = π_{v_{2}}. Using the refined version of Robinson’s completeness result, we can jointly embed (K_{1}, v_{1}) and (K_{2}, v_{2}) over L into (K_{3}, v_{3}) (*). Let L′ be the residue field of v_{3}, then as K_{3} is algebraically closed, so is L′ and extends the residue field L of v_{1} and v_{2}. By standard results, we can construct a Krull valuation v on L′ with residue field L, for example use the construction given in [2]. Using Theorem 1.5 again, we can construct specialisations (P (K_{3}), P (L′), π_{v_{3}}) and (P (L′), P (L), π_{v}), the composition gives a specialisation (P (K_{3}), P (L), π_{3}). It remains to see that in fact π_{3} extends the specialisations π_{1} and π_{2}. This follows from the fact that if k ∈ K_{1} or k ∈ K_{2} and there exists l ∈ L such v_{1}(k − l) > 0 or v_{2}(k − l) > 0 then this relation is preserved in the embedding (∗). Hence the specialisation π_{v_{3}} already extends the specialisations π_{1} and π_{2} of P (K_{1}) and P (K_{2}) into P (L). As the specialisation π_{v} fixes L, this proves the lemma.

□

Lemma 2.3. Transfer of Formulas

Let (P (K), P (L), π) be a specialisation with corresponding (K, v), then there exists a map;

σ : P (K) → K^{eq}

σ : L_{spec}-formulae → L_{val}-formulae

such that for any ϕ(x_{1}, …, x_{n}) which is a L_{spec}-formula and (k_{1}, …, k_{n}) ⊂ P (K);

(P (K), P (L), π) ⊧ ϕ(k_{1}, …, k_{n}) iff (K, v) ⊧ σ(ϕ)(σ(k_{1}), …, σ(k_{n}))

. (†)

Moreover, the definition of the map is uniform in K.

Proof. The map σ is defined on the sorts P^{n}(K) by sending [k_{0}, …, k_{n}] to (k_{0}, …, k_{n})/ ∼_{n} where ∼_{n} is the equivalence relation defined on K^{n+1} from multiplication by K^{∗}. Similarly, σ maps a variable from the sort S_{n} to the corresponding variable from the sort in K^{eq} defined by ∼_{n}. A closed algebraic subvariety in {V_{n}^{m}} is defined by a multihomogeneous equation in the variables {(x_{01}, …, x_{n1}), …, (x_{0m}, …, x_{nm})}. Let C_{n}^{m} be the algebraic variety in K^{m(n+1)} defined by this equation. Then the corresponding formula in K^{eq} is given by;

(y_{1}, …, y_{m}) ∈ (∼_{n})^{m}[∃x_{1} … x_{m}(C_{n}^{m}(x_{1}, …, x_{m}) ∧ ⋀_{i=1}^{m} x_{i}/ ∼_{n}= y_{i})]

For the inclusion maps i_{n}, let us identify each i_{n} with its graph, then clearly we can define σ to map the formula i(x) = y to a corresponding formula relating the sorts ∼_{n} and ∼_{n+1} in K^{eq}.

Note that if l ∈ P^{n}(L) is a constant, then σ(l) = (l_{0}, …, l_{n})/ ∼_{n} where each l_{i} is a constant from the atomic diagram of L.

Finally, let π_{n} : P^{n}(K) → P^{n}(L) be a specialisation. Again, let us assume that we can identify π_{n} with its graph. We then have that;

π_{n}([x_{0} : …: x_{n}]) = [l_{0} : …: l_{n}]

iff

∃z∃z_{0} … ∃z_{n}((⋀_{i=0}^{n} x_{i}z = l_{i} + z_{i}) ∧ (⋀_{i=0}^{n} v(z_{i}) > 0)).

It is now clear how to define σ(π_{n}) as a union of formulas in the sort defined by ∼_{n}.

This completes the definition of σ, it is clear that the definition is uniform in K and a straightforward induction on the length of a formula from L_{spec} shows that it has the required property (†).

□

Theorem 2.1 is now a fairly straightforward consequence of the above lemmas. We first show model completeness. Suppose that we have models of T_{axioms}; (P (K_{1}), P (L), π_{1}) ≤ (P (K_{2}), P (L), π_{2})

By theorem 1.5, we can find Krull valuations v_{1} and v_{2} such that (K_{1}, v_{1}) ≤ (K_{2}, v_{2}) and (K_{1}, v_{1}), (K_{2}, v_{2}) ⊧ T_{K,L}. By the refined model completeness result after Theorem 2.2, we have (K_{1}, v_{1}) ≺ (K_{2}, v_{2}), hence using Lemma 2.3, we must have that;

(P (K_{1}), P (L), π_{1}) ≺ (P (K_{2}), P (L), π_{2})

as required. Completeness now follows directly from Lemma 2.1 and model completeness. Alternatively, one can exhibit a prime model of the theory, this is clearly possible by taking the specialisations corresponding to the prime models of T_{K,L} above.

3. A First Order Definition of Intersection Multiplicity and Bezout’s Theorem

We now formulate a non-standard definition of intersection multiplicity in the language L_{spec}. We will do this only in the case of projective curves inside P^{2}(L), the reader may wish to try formulating a corresponding definition in higher dimensions.

Let C_{1} and C_{2} be projective curves of degree d and degree e in P^{2}(K) defined over L. The parameter spaces for such curves are affine spaces of dimension (d + 1)(d + 2)/2 and (e + 1)(e + 2)/2 respectively. We can give them a projective realisation by noting that if (l̄) is a non-zero vector defining a curve of degree d, then multiplying it by a constant μ defines the same curve. Let P^{d(d+3)/2}(K) and P^{e(e+3)/2}(K) define these spaces which we will denote by P_{d} and P_{e} for ease of notation. Let Curve_{d} and Curve_{e} be the closed projective subvarieties of P_{d} ×P^{2}(K) and P_{e} × P^{2}(K), defined over the prime subfield of L, such that, for l ∈ P_{d}, the fibre Curve_{d}(l) defines the corresponding projective curve of degree d in P^{2}(K). For l in P^{n}(L), we denote its infinitesimal neighborhood V_{l} to be the inverse image under the specialisation π_{n}.

Now suppose that C_{1} and C_{2} (which may not be reduced or irreducible), of degrees d and e respectively, are defined by parameters l_{1} and l_{2} and intersect at an isolated point l in P^{2}(L). Then we define;

Mult(C_{1}, C_{2}, l) ≥ n iff

∃x_{1}, x_{2} ∈ V_{l_{1}}, V_{l_{2}}, ∃_{y_{1}≠...≠y_{n}} ∈ V_{l}({y_{1}, …, y_{n}} ⊂ Curve_{d}(x_{1})∩Curve_{e}(x_{2}))

Then define Mult(C_{1}, C_{2}, l) = n

iff

Mult(C_{1}, C_{2}, l) ≥ n and ¬Mult(C_{1}, C_{2}, l) ≥ n + 1.

Clearly, the statement that Mult(C_{1}, C_{2}, l) = n naturally defines a sentence in the language L_{spec}. One consequence of the completeness result given above is that the statement ”The curves C_{1} and C_{2} intersect with multiplicity n at l” depends only on the theory T_{axioms} and is independent of the particular structure (P (K), P (L), π). In the paper [3], we showed that this non-standard definition of multiplicity is equivalent to the algebraic definition of multiplicity when computed in the structure (P (K_{univ}), P (L), π_{univ}) (see the next section). It therefore follows that the non-standard definition of multiplicity is equivalent to the algebraic definition even when computed in a prime model of T_{axioms} which I will denote by (P (K_{prime}), P (L), π_{prime}).

We now turn to the statement of Bezout’s theorem. In algebraic language, this says that if projective algebraic curves C_{1} and C_{2} of degree d and degree e in P^{2}(L) intersect at finitely many points {l_{1}, …, l_{n}}, then;

∑_{i=1}^{n} I(C_{1}, C_{2}, l_{i}) = de

where I(C_{1}, C_{2}, l_{i}) is the algebraic intersection multiplicity. The nonstandard version of this result can be formulated in the language L_{spec} by the sentence;

Bezout(C_{1}, C_{2}) ≡ ∃_{m_{1},...,m_{n};m_{1}+...+m_{n}=de}(⋀_{i=1}^{n} Mult(C_{1}, C_{2}, l_{i}) = m_{i})

Again, in the paper [3], we proved the algebraic version of Bezout’s theorem by non-standard methods in the structure (P (K_{univ}), P (L), π_{univ}). It follows that the sentences Bezout(C_{1}, C_{2}) are all proved by the theory T_{axioms} and therefore hold in the structure (P (K_{prime}), P (L), π_{prime}) as well. This demonstrates the fact that we can prove an algebraic statement of Bezout’s theorem using only infinitesimals from a straightforward extension of L, namely L(ϵ)^{alg}, in particular in a structure such that the infinitesimal neighborhoods V_{l} are all recursively enumerable. This seems to provide some answer to a general objection concerning the use of infinitesimals, originating in [1]. It may also provide an effective alternative method to compute intersection multiplicities generally in algebraic geometry.

4. Constructing a Universal Specialisation

In the papers [2] and [3], we used the existence of a specialisation (P (K_{univ}), P (L), π_{univ}) having the following ”universal” property;

If L ⊂ L_{m} is an algebraically closed extension of L with transcendence degree m, and (P (L_{m}), P (L), π_{m}) is a specialisation, then there exists an L-embedding α_{L} : L_{m} → K_{univ} with the property that π_{univ} ∘ α_{L} = π_{m}. (*)

Unfortunately, the construction of K_{univ} was flawed. We correct this difficulty here;

Model theoretically, using theorem 2.1, it is easy to show the existence of such a structure. Namely, let (P (K_{univ}), P (L), π_{univ}) be a 2^{ω} saturated model of the theory T_{axioms}. Then, if L ⊂ L_{m} is an algebraically closed extension of L of transcendence degree m, clearly ⋃_{n≥1} Card(S^{n}(T h(M))) ≤ 2^{ω}, where M = (P (L_{m}), P (L), π_{m}). This follows as L was assumed to be countable. Hence, by elementary model theory, there exists an L-embedding α_{L} with the required properties. For the non-model theorist, we give a more algebraic construction, replacing the use of types by an explicit amalgamation of the possible valuations;

Proof. Suppose, inductively, we have already constructed a specialisation (P (K_{n}), P (L), π_{n}) which has the property (∗) for all extensions L ⊂ L_{m} with L_{m} algebraically closed of transcendence degree m ≤ n. We will construct K_{n+1} having this property for m ≤ n + 1. By theorem 1.5, we can find a Krull valuation v_{n} on K_{n} corresponding to the specialisation π_{n}. Let t be a new transcendental element. The extensions of v_{n} to K_{n}(t) are completely classifiable. In fact, we have the following result in [4] (Theorem 3.9), we refer the reader to the paper for the definition of each family of valuations; The extensions of v_{n} are of the form;

(i). v_{n,a,γ} where a ∈ K_{n} and γ is an element of some ordered group extension of v(K).

(ii). v_{n,A} where A is a pseudo Cauchy sequence in (K_{n}, v_{n}) of transcendental type.

Let I be a fixed enumeration of these valuations. Inductively, we assume that Card(K_{n}) ≤ 2^{ω} in which case the dimension of v(K_{n}) as a vector space over Q has dimension at most 2^{ω} as well. Clearly then the number of non-isomorphic (over K_{n}) valuations from (ii) is at most 2^{ω} and the same holds for the valuations obtained from (i) by noting that the number of order types of γ is at most 2^{ω} (it is easily checked that 2 new elements of the value group, γ_{1} and γ_{2}, having the same order type, define isomorphic valuations in the case of (i)). Hence, we can assume that I is well ordered and apply the method of transfinite induction to construct a series of specialisations (P (K_{n,i}), P (L), π_{n,i}) as follows;

For i = 0, set (P (K_{n,0}), P (L), π_{n,0}) = (P (K_{n}), P (L), π_{n})

Given i ∈ I with i not a limit ordinal, let v_{i+1} be the next valuation in the enumeration. Let (K_{n}\{t\}, \overline{v_{i+1}}) be the completion of (K_{n}(t), v_{i+1}) and let \overline{v_{i+1}} also denote the unique extension of this valuation to the algebraic closure K_{n}{t}^{alg}. This defines a Krull valuation and hence a specialisation (P (K_{n}{t}^{alg}), P (L′), π_{n,i+1}) where L′ is the algebraic closure of the residue field of v_{i+1}, having transcendence degree at most 1 over L. Using arguments as above, we can construct a specialisation (P (L′), P (L), π). Composing these specialisations, we obtain a specialisation (P (K_{n}{t}^{alg}), P (L), π_{n,i+1}). (One can omit this step by enumerating in I only those valuations which preserve the residue field L) Now, using Lemma 2.1 and Theorem 2.1, amalgamate the specialisations (P (K_{n}{t}^{alg}), P (L), π_{n,i+1}) and (P (K_{n,i}), P (L), π_{n,i}) to form a specialisation;

(P (K_{n,i}), P (L), π_{n,i}) ≺ (P (K_{n,i+1}), P (L), π_{n,i+1}).

For i a limit ordinal, we set;

(P (K_{n,i}), P (L), π_{n,i}) = ⋃_{j<i}(P (K_{n,j}), P (L), π_{n,j})

By the usual union of chains arguments we have that;

(P (K_{n,j}), P (L), π_{n,j}) ≺ (P (K_{n,i}), P (L), π_{n,i}) for j < i.

Repeating this process, we obtain a structure (P (K_{n+1}), P (L), π_{n+1}) such that;

(P (K_{n}), P (L), π_{n}) ≺ (P (K_{n+1}), P (L), π_{n+1}).

It remains to check that this structure has the universal property (∗) for m = n + 1. Let L_{n+1} be an algebraically closed extension of L with transcendence degree n + 1 and specialisation (P (L_{n+1}), P (L), π). Let v_{π} be the corresponding valuation and its restriction to L ⊂ L_{n} ⊂ L_{n+1}, a subfield of transcendence degree n. The corresponding specialisation (P (L_{n}), P (L), π) already factors through (P (K_{n}), P (L), π_{n}) (†) and the valuation v_{π} appears as v_{i} in the enumeration I when restricted to L_{n}(t). By a standard result in valuation theory, see [5], there exists an L_{n}(t)-embedding τ : L_{n}(t)^{alg} → L_{n}\{t\}^{alg} such that v_{π} = \overline{v_{i}} ∘ τ (††) (see notation above). Combining (†) and (††), we obtain an embedding α : (P (L_{n+1}), P (L)) → (P (K_{n,i}), P (L)) such that π = π_{n,i} ∘ α. This proves the result. It is now clear that the structure

(P (K_{univ}), P (L), π_{univ}) = ⋃_{i>0}(P (K_{i}), P (L), π_{i})

has the required universal property, is a model of T_{axioms} and;

(P (K_{i}), P (L), π_{i}) ≺ (P (K_{univ}), P (L), π_{univ}) for i > 0.

□

References

[1] ”The Analyst: a Discourse addressed to an Infidel Mathematician”, George Berkeley (1734)

[2] Zariski Structures and Algebraic Curves, Tristram de Piro, available on my website http://www.magneticstrix.net, (2010), old version ”Zariski Structures and Algebraic Geometry” on AG ArXiv (0402301) (2004)

[3] A Non-Standard Bezout Theorem for Curves, Tristram de Piro, available on my website http://www.magneticstrix.net, (2010), old version ”A Non Standard Bezout Theorem” on AG/LO ArXiv (0406176) (2004)

[4] Value groups, residue fields and bad places of function fields, Franz-Viktor Kuhlmann, (2003)

[5] Algebraic Number Theory, Jurgen Neukirch, Volume 322, Springer (1992)

[6] Complete Theories, Abraham Robinson, Studies in Logic and the Foundations of Mathematics, North-Holland, (1956).

Mathematics Department, The University of Camerino, Camerino, Italy

 E-mail address: tristam.depiro@unicam.it