Nonsingular Hasse principle for rings I

L. Darnière

July 11, 1998

1 Introduction

There have been numerous model theoretic studies in the last twenty years on fields with families of topologies. Most of them involved various local-global principles (LGP) analogous to the geometric Hasse principle.

In 1986, Rumely [7] proved that Hilbert’s tenth problem had a positive answer for the ring Z̃ of all algebraic integers. His approach involved a new kind of LGP, adapted to the context of rings. This paved the way to new results on the model theory of rings, and indeed several important theorems concerning decidability, quantifier elimination or model-completeness for various domains were obtained independently by van den Dries, Macintyre, Prestel and Schmid ([8], [9] and [5]) by using apparently different LGP’s, related to Rumely’s one.

In this paper we introduce another LGP for rings we call LGPH. It turns out to contain all the preceding ones as particular cases, and to first order (in the ordinary language of rings). On the other hand, the proof we give in this article of the axiomatisability of the LGPH requires new material, wich was not needed by the above LGP’s.

A motivation to introduce the LGPH is the remark that a ring A satisfying any of the other above mentioned LGP’s has the property that the henselisation of any valuation ring containing A must have a separably closed fraction field, a condition wich is very restrictive (but it is precisely what makes the axiomatisation of those LGP’s much easier than for the LGPH). On the other hand, Moret-Bailly extended widely the theorem of Rumely in [3], giving a lot of exemples of rings satisfying the LGPH but not this additionnal condition. The integral closure of Z (the ring of integers) in Q_{p}, wich we note Z̃ ∩ Q_{p}, is an example of such a ring.

Prestel and Schmid proposed in [6] an axiomatisation of Z̃ ∩ R and Z̃ ∩ Q_{p} based on a variant of their LGP in [5]. They succeeded for Z̃ ∩ R, but for the reason we have just mentionned Z̃∩Q_{p} cannot satisfy this LGP as the p-adic valuation ring is henselian and has a fraction field wich is not separably closed. This is actually the only obstruction, and it appears that the methods used in [6] work quite well if we replace their LGP by the LGPH.

We present this approach in the last section of this paper, where we obtain exactly the results announced in [6], theorems 3.2 to 3.4 and [6], corollary 3.5, namely a recursive axiomatisation of Z̃ ∩ Q_{p} as one of the completions of a modele-complete theory. In particular we prove that this ring is decidable. Similar results can be obtained on the integral closure of Z or F_{p}[T] in various local or semi-local fields (see [1] or some forthcoming papers).

2 Notations

Let us introduce now a few facts and conventions we are going to use throughout this paper.

All our rings are commutative with 1. For any element a of a ring A we note M_{A}(a) (resp. M_{A}(a)^{c}) the set of maximal ideals of A containing a (resp. not containing a), the subscript A is omitted whenever the context makes it superfluous. If A is a local ring then m(A) denotes its unique maximal ideal. If A is a domain then Frac(A) denotes its quotient field, and W_{A} is the set of valuation rings R such that A ⊆ R ⊆ Frac(A) and m(R) ∩ A is a maximal ideal of A.

Given a field K we denote by K^{s} (resp. K^{a}) any separable (resp. algebraic) closure of K. For any valuation ring R with fraction field K we note R^{h} the henselisation of R, and R^{s} (resp. R^{a}) the (unique) extension of R^{h} to K^{s} (resp. K^{a}).

For any x, y in K we note:

|x|_{R} ≤ |y|_{R} ⟺ ∃u ∈ R, x = uy.

For any x = (x_{1}, …, x_{m}) ∈ K^{m} and any y = (y_{1}, …, y_{n}) ∈ K^{n} we note:

|x|_{R} ≤ |y|_{R} ⟺ max_{i≤m} |x_{i}|_{R} ≤ max_{j≤n} |y_{j}|_{R}.

One can easily deduce from the Taylor formula for polynomials that for any x, y ∈ R^{n} and any p ∈ R[X_{1}, …, X_{n}]:

|p(x) − p(y)|_{R} ≤ |x − y|_{R}.

In order to make the typing easier, if R = A_{p}, A_{p}^{h}, A_{p}^{s} or A_{p}^{a} for some subring A and some prime ideal p of A, we write |x|_{p} ≤ |ε|_{p} in place of |x|_{R} ≤ |ε|_{R}.

Our varieties are affine, geometrically reduced and geometrically irreducible. When we give a system of equations f ∈ K[X]^{m} (with X = (X_{1}, …, X_{n})) for a variety V defined over a field K, we always mean that f generates a prime ideal of K^{a}[X], but the proof will always be left to the reader, who may also look at [1]. A point of V in a K-algebra A is always an affine point, that is a zero of f in A^{n}, and V (A) (resp. V^{∘}(A)) denotes the set of points (resp. smooth points) of V with coordinates in A. A generic point of V over K is a point x ∈ V (A) such that K[x] is isomorphic to the coordinate ring K[V] of V.

We will make an extensive use of the following notion, introduced by Prestel and Schmid [5]. A r-ring is a ring A together with a radical relation on A, that is a binary relation ≤ such that for every a, b, c in A:

1. a ≤ a;

2. a ≤ b, b ≤ c ⇒ a ≤ c;

3. a ≤ b ⇒ ac ≤ bc;

4. 1 ≤ a;

5. a ≤ b^{2} ⇒ a ≤ b.

A r-domain is an r-ring whose underlying ring is a domain. It is easy to check that if P is a nonempty set of prime ideals of A, the relation defined by:

a ≤_{P} b ⟺ (∀p ∈ P, a ∈ p ⇒ b ∈ p)

is a radical relation. Prestel and Schmid ([5], theorem 2.5) proved that every radical relation is of that kind. More precisely, they denoted by Spec(A, ≤) the set of prime ideals p of a such that:

a ≤ b, a ∈ p ⇒ b ∈ p.

They proved that Spec(A, ≤) is nonempty, and:

a ≤ b ⟺ (∀p ∈ Spec(A, ≤), a ∈ p ⇒ b ∈ p).

Remark 2.1 Prestel and Schmid required their r-rings to satisfy the additional axiom 0 ≰ 1. It is easy to check that this condition is not necessary to have this result, the case 0 ≤ 1 corresponding to the radical relation defined by the empty set, so we will allow our radical relations to satisfy 0 ≤ 1 if necessary.

The main radical relation we will consider in a ring A is the relation ≤_{Specmax(A)} defined by the maximal spectrum of A. This radical relation will simply be noted ≤_{A}:

a ≤_{A} b ⟺ M_{A}(a) ⊆ M_{A}(b).

It is important to notice that this relation is definable in the language of rings:

a ≤_{A} b ⟺ ∀c ∈ A, ((b, c) = 1 ⇒ (a, c) = 1)

where (x, y) = 1 means that the ideal generated by x, y in A is A himself, in other words A ⊧ ∃u, ∃v(ux + vy = 1).

The residual upper semi-lattice L(A, ≤) associated to any radical relation on A is the set of all P (a) := {p ∈ Spec(A, ≤) | a ∈ p} with a ranging over A. Ordered by inclusion, L(A, ≤) has a largest element P (0), a least element P (1) and is stable by finite union: P (ab) = P (a) ∪ P (b). If in addition A is Bezout (that is, any finitely generated ideal is principal) then L(A, ≤) is also stable by finite intersection, hence a distributive lattice (we refer to [5] for this observation: if A is Bezout then for any a, b in A there is d in A such that aA+bA = dA, and for any such d, P (d) = P (a)∩P (b)). The quotient ordered set canonically associated to the preorder ≤ is isomorphic to L(A, ≤), hence L(A, ≤) is interpretable in (A, ≤) and L(A, ≤_{A}) is interpretable in A. We note L(A) for L(A, ≤_{A}).

We note L_{rad} the language of r-rings and r-domains, that is the language of rings with an additional binary relation symbol for the radical relation.

3 The local-global principle

We say that a ring A satisfies the local-global principle with respect to henselisations (LGPH) if and only if it is a domain and the following two conditions hold:

1. ∀a ∈ A \ {0}, M(a)^{c} is compact.

2. Every V defined over the fraction field of A satisfies:

V^{∘}(A) ≠ ∅ ⟺ ∀R ∈ W_{A}, V^{∘}(R^{h}) ≠ ∅.

This last condition is called the geometric condition of the LGPH. Notice that any field satisfies the LGPH.

Proposition 3.1 A domain wich satisfies LGPH is Bezout.

Proof: Let A be a domain satisfying the LGPH, with quotient field K. For any a, b ∈ A \ {0} the variety V defined over K by (X(aU + bV) = a, Y (aU + bV) = b) is smooth, and has a point in every R ∈ W_{A} (because R is Bezout). Hence the LGPH gives (u, v, x, y) ∈ V (A) and we have then d := au + bv wich divides both a and b, hence aA + bA = dA.

It follows that in the geometric condition of the LGPH we can replace W_{A} by the set of all the A_{m}’s, with m ranging over the maximal spectrum of A.

Van den Dries and Macintyre [9] say that a ring A has the block factorisation property if for any nonzero a and any b in A, there exists a′, a′′ in A such that a = a′a′′, M(a′) = M(a)∩M(b) and M(a′′) = M(a)\M(b). We are going to prove that the LGPH implies the block factorisation property.

Lemma 3.2 Let A be a domain satisfying the LGPH, a ∈ A \ {0} and p_{1}, p_{2} two prime ideals of A. Suppose that none of p_{1}, p_{2} contains the other. Then there exists a_{1}, a_{2} ∈ A such that a = a_{1}a_{2}, |a_{1}|_{p_{1}} = |a_{2}|_{p_{2}} = 1.

Proof: Fix c_{1} ∈ p_{1} \ p_{2}, c_{2} ∈ p_{2} \ p_{1} and d a generator of c_{1}A + c_{2}A. Replacing c_{1}, c_{2} by c_{1}/d, c_{2}/d if necessary one can suppose c_{1}, c_{2} relatively prime. We consider the variety V defined over Frac(A) by the equations (A_{1}A_{2} = a, c_{1}Y_{1} + A_{1}B_{1} = 1, c_{2}Y_{2} + A_{2}B_{2} = 1). It is smooth, and has a point (a_{1,R}, a_{2,R}, b_{1,R}, b_{2,R}, y_{1,R}, y_{2,R}) in every R ∈ W_{A}: if c_{1} ∉ m(R), (a, 1, 0, 1, c_{1}^{−1}, 0) ∈ V (R), if not then as c_{1} and c_{2} are relatively prime c_{2} ∉ m(R), hence (1, a, 1, 0, 0, c_{2}^{−1}) ∈ V (R). The LGPH then gives a point (a_{1}, a_{2}, b_{1}, b_{2}, y_{1}, y_{2}) ∈ V (A) and we have a = a_{1}a_{2}, a_{1} ∉ p_{1} because c_{1} ∈ p_{1} and c_{1}y_{1} + a_{1}b_{1} = 1, and a_{2} ∉ p_{2} by the same argument hence |a_{1}|_{p_{1}} = |a_{2}|_{p_{2}} = 1.

Proposition 3.3 A domain satisfying the LGPH has the block factorisation property.

Remark 3.4 It has to be noticed that until now we have used only the geometric condition of the LGPH. On the other hand, it is straightforward that a domain with the block factorisation property satisfies the non geometric condition of the LGPH. Hence the LGPH is equivalent to the block factorisation property plus the geometric condition.

Proof: Let A be a domain satisfying the LGPH, and a, α ∈ A \ {0}. We are looking for a factorisation a = bc with M(b) = M(a) ∩ M(α) and M(c) = M(a) \ M(α). Let M := M(a) \ M(α) and N := M(a) ∩ M(α). As a closed subset of M(α)^{c} (resp. of Specmax(A)), M (resp. N) is compact. (Note this is the first (and last) time we use the non geometric condition of the LGPH, in order to guarantee that M(α)^{c} is compact.) For any x in A, we note M(x) = M ∩ M(x) and similarly for N (x).

Let n be a fixed element of N. For every m ∈ M, 3.2 gives a non zero b_{n,m} in A such that b_{n,m} divides a, |b_{n,m}|_{m} = 1 et |b_{n,m}|_{n} = |a|_{n}. The intersection of all M(b_{n,m}) for m ranging over M is empty, hence there exist m_{1}, …, m_{m} in M such that

M(b_{n,m_{1}}) ∩ ⋯ ∩ M(b_{n,m_{m}}) = ∅.

As A is Bezout, let b_{n} = ∑_{i=1}^{m} λ_{i}b_{n,m_{i}} be a generator of the ideal generated by the

b_{n,m_{i}}’s for i ≤ m. We have a/b_{n} = (a/b_{n,m_{1}})(b_{n,m_{1}}/b_{n}) ∈ A, hence |b_{n}|_{n} ≥ |a|_{n}. On the other hand:

|b_{n}|_{n} ≤ max_{1≤i≤m} |λ_{i}b_{n,m_{i}}|_{n} ≤ max_{1≤i≤m} |b_{n,m_{i}}|_{n} = |a|_{n}.

It follows that |b_{n}|_{n} = |a|_{n}. In addition for every m in M there is some i ≤ m such that b_{n,m_{i}} ∉ m, and as b_{n} divides b_{n,m_{i}} it follows that b_{n} ∉ m, that is |b_{n}|_{m} = 1.

For each n in N we set c_{n} := a/b_{n}. As |c_{n}|_{n} = |a/b_{n}|_{n} = 1, that is c_{n} ∉ n, the N (c_{n})’s have an empty intersection, hence there are elements n_{1}, …, n_{n} in N such that:

N (c_{n_{1}}) ∩ ⋯ ∩ N (c_{n_{n}}) = ∅.

Let c := ∑_{i=1}^{n} μ_{i}c_{n_{i}} be a generator of the ideal generated by the c_{n_{i}}’s. Let b := a/c; as c

divides c_{n_{1}} wich divides a, b belongs to A.

For each n in N there is an index i ≤ n such that c_{n_{i}} ∉ n; and as c divides c_{n_{i}} it follows that c ∉ n, hence M(c)∩M(a)∩M(α) is empty. For every m in M, |c|_{m} ≥ |a|_{m} because c divides a, but on the other hand:

|c|_{m} ≤ max_{1≤i≤n} |μ_{i}c_{n_{i}}|_{m} ≤ max_{1≤i≤n} |c_{n_{i}}|_{m} = max_{1≤i≤n} |a/b_{n_{i}}|_{m} = |a|_{m}

hence |c|_{m} = |a|_{m}. It follows that |b|_{m} = 1, that is b ∉ m, hence M(b) ∩ M(a) \ M(α) is empty. Since M(b) ∪ M(c) = M(bc) = M(a), we conclude that M(b) = M(a) ∩ M(α) and M(c) = M(a) \ M(α).

Recall that a constructible subset of the maximal spectrum of a ring A is any (finite) Boolean combination of susbets M(a) with a ∈ A.

Lemma 3.5 Let A be a ring having the block factorisation property. Then any constuctible subset of Specmax(A) equals M(a) or M(a)^{c} for some nonzero a in A.

Proof: For any a, b in A we have M(a) ∪ M(b) = M(ab). If in addition A has the block factorisation property, then for any non zero a there are elements c, d such that M(a) \ M(b) = M(c) and M(a) ∩ M(b) = M(d), hence an immediate induction shows that any constructible subset of Specmax(A) of the form M(a) or M(a)^{c} for some a in A, and we can suppose a non zero since M(0) = M(1)^{c}.

Let us say that a Prüfer domain A has the block approximation property (BAP) if for any disjoint constructible subsets M_{1}, …, M_{n} of Specmax(A), for any a_{1}, …, a_{n} in A and any nonzero ε_{1}, …, ε_{n} in A there is a ∈ A such that |a − a_{i}|_{m} ≤ |ε_{i}|_{m} for every m ∈ M_{i} and every i ≤ n.

Proposition 3.6 A domain satisfying the LGPH has the block approximation property.

Proof: Let A be a domain satisfying the LGPH, M_{1}, …, M_{n} be a family of disjoint constructible subsets of Specmax(A), a_{1}, …, a_{n} be any elements of A, and ε_{1}, …, ε_{n} be nonzero elements of A. Using the block factorisation property and 3.5, one can find nonzero α_{1}, …, α_{n} in A such that M(α_{i}) = M_{i} ∩ M(ε_{i}) for i ≤ n.

Let V be the smooth variety defined over Frac(A) by the following equations:

\{ \begin{matrix}(X − a_{i})V_{i} = ε_{i}U_{i} & i ≤ n \\ V_{i} + α_{i}T_{i} = 1 & i ≤ n \end{matrix}

For any j ≤ n and each maximal ideal m ∈ M(α_{j}) take: x_{m} = a_{j}, t_{j,m} = u_{j,m} = 0, v_{j,m} = 1 and for every i ≠ j, t_{i,m} = α_{i}^{−1} and u_{i,m} = v_{i,m} = 0. As α_{i} ∉ m for every i ≠ j, (x_{m}, t_{m}, u_{m}, v_{m}) belongs to V (A_{m}). This point also belongs to V (A_{n}) for any n out of M(α_{1}) ∪ ⋯ ∪ M(α_{n}). The LGPH then gives (x, t, u, v) ∈ V (A) and for every i ≤ n and every m ∈ M_{i}, we have: |x − a_{i}|_{m} ≤ 1 = |ε_{i}|_{m} if m ∉ M(α_{i}) since M(ε_{i}) ∩ M_{i} = M(α_{i}), |x − a_{i}|_{m} = |ε_{i}u_{i}/v_{i}|_{m} ≤ |ε_{i}|_{m} if m ∈ M(α_{i}) since v_{i} + α_{i}t_{i} = 1 ⇒ M(v_{i}) ∩ M(α_{i}) = ∅.

Corollary 3.7 If a domain A satisfies the LGPH, then any two R_{1} ≠ R_{2} ∈ W_{A} are independent (that is R_{1}R_{2} = Frac(A)).

Proof: Take α_{1} in m(R_{1}) ∩ A \ m(R_{2}), and similarly for α_{2}.The block approximation property with M_{1} = M(α_{1}) \ M(α_{2}) and M_{2} = M(α_{2}) \ M(α_{1}) proves that R_{1} and R_{2} satisfy the usual strong approximation property, hence are independant.

4 Axiomatisation

Domains having the block approximation property have in common a lot of nice properties, wich have been extensively studied in [1]. In this paper we focus on those wich allow us to prove that the LGPH is a first order property (in the language of rings).

We already noticed that the relation defined by a ≤ b ⟺ M(a) ⊆ M(b) is (uniformally) definable, hence we can use here the formula a ≤ b as an abbreviation for the corresponding formula in the language of rings.

In the same way, we will write (a, b) = d for the formula:

∃u, v, a′, b′ (a = a′d ∧ b = b′d ∧ au + bv = d)

wich says that the ideal generated by a and b is principal, with generator d.

Any element of the fraction field of a Bezout domain A can be written u/v with (u, v) = 1. It follows that given any a, b and any non zero α in A, a divides b in every A_{m} such that m ∈ M(α) if and only if there are u, v in A such that (v, α) = 1 and av = bu, that is if and only if A models the following formula:

DivLoc(a, b, α) ≡ ∃u, v [(v, α) = 1 ∧ av = bu].

Proposition 4.1 The class of Bezout domains having the block factorisation and the block approximation properties is elementary.

Proof: A domain is Bezout if and only if it models ∀a, b∃d (a, b) = d.

It has the block factorisation property if and only if it models:

∀a, b, [a ≠ 0 → ∃a_{1}, a_{2} (a = a_{1}a_{2} ∧ (a_{1}, b) = 1 ∧ a_{2} ≤ b)].

An immediate induction shows that a Bezout domain has the block approximation property if and only if it satisfies this property with respect to any two disjoint constructible sets. We can even suppose each M_{i} ⊆ M(ε_{i}). If in addition the block factorisation property holds, by 3.5 those two constructible sets can be taken as M(α_{1}) and M(α_{2}) with α_{1}A+α_{2}A = A. Hence a Bezout domain having the block factorisation property has the block approximation property if and only if it models:

∀α_{1}, α_{2}, a_{1}, a_{2}, ε_{1}, ε_{2} (ε_{1}ε_{2} ≠ 0 ∧ (α_{1}, α_{2}) = 1) →
∃a, DivLoc(ε_{1}, a − a_{1}, α_{1}) ∧ DivLoc(ε_{2}, a − a_{2}, α_{2}).

Lemma 4.2 Let α be a nonzero element of a Bezout domain A having the block factorisation property. Let M_{1}, …, M_{q} be constructible subsets of Specmax(A) such that M(α) ⊆ ⋃_{k=1}^{q} M_{k}. Let m_{1}, …, m_{q} be distinct maximal ideals of A such that m_{k} ∈ M_{k} for every k ≤ q. Then there exist non zero α_{1}, …, α_{q} such that for every k ≤ q:

m_{k} ∈ M(α_{k}) ⊆ M_{k}

and (M(α_{k}))_{k≤q} is a partition of M(α).

Proof: For each k ≤ q, by 3.5 there is β_{k} in A such that M(β_{k}) = M(α) ∩ M_{k}.

We prove the result by induction on q. If q = 1, take α_{1} = α and the assertion is proved. Suppose q > 1 and it has been proved for q − 1.

The block factorisation property gives a nonzero α′ such that M(α′) = M(α)∩M(β) where β is the product of β_{1}, …, β_{q−1}. Applying the induction hypothesis to α′ and β_{1}, …, β_{q−1} gives α′_{1}, …, α′_{q−1} wich define a partition of M(α′). The block factorisation property gives α′_{q} such that M(α′_{q}) = M(α)\M(α′). Then M(α′_{q}) ⊆ M(β_{q}), (M(α′_{k}))_{k≤q} is a partition of M(α), and for every k < q we have m_{k} ∈ M(α′_{k}) ⊆ M(β_{k}).

If m_{q} ∈ M(α′_{q}) it suffices to set α_{k} = α′_{k} for every k ≤ q to finish the proof. If not, then m_{q} ∈ M(α′) hence there is l < q such that m_{q} ∈ M(α′_{l}). Take γ ∈ m_{l} \ m_{q}, by 3.5 there is γ_{q} in A such that M(γ_{q}) = M(β_{q}) ∩ M(α′_{l}) \ M(γ). The reader may easily check that m_{q} belongs to M(α′_{q}γ_{q}) which is contained in M(β_{q}), that m_{l} belongs to M(α′_{l}) \ M(γ_{q}) which is contained in M(β_{l}), and that those two constructible sets form a partition of M(α′_{l}α′_{q}). Hence setting α_{q} = α′_{q}γ_{q}, α_{l} = some element of A such that M(α_{l}) = M(α′_{l}) \ M(γ_{q}), and α_{k} = α′_{k} for every k ≠ l, q yields the conclusion (note that every α_{k} can be supposed non zero, as if α_{k} = 0 then k = q = 1 and one can replace α_{k} by α).

Lemma 4.3 Let A be a Bezout domain having the block approximation and the block factorisation properties. Fix a non zero α in A and p_{1}, q_{1}, …, p_{r}, q_{r} in A[X] with X = (X_{1}, …, X_{n}). The following are equivalent:

1. For every m ∈ M(α) there is a_{m} ∈ A_{m}^{n} such that |p_{i}(a_{m})|_{m} ≤ |q_{i}(a_{m})| ≠ 0 for every i ≤ r.

2. There is a ∈ A^{n} such that |p_{i}(a)|_{m} ≤ |q_{i}(a)| ≠ 0 for every m ∈ M(α) and every i ≤ r.

Proof: We prove only (1) ⇒ (2), since the other direction is trivial.

For every m in M(α), since each p_{i}(a_{m})/q_{i}(a_{m}) belongs to A_{m} there is a non zero v_{m} in A such that v_{m} ∉ m and v_{m}p_{i}(a_{m})/q_{i}(a_{m}) ∈ A for every i ≤ m. The family of all M(v_{m})^{c}’s covers M(α), hence by compactness of M(α) there are m_{1}, …, m_{q} in M(α) such that M(α) ⊆ ⋃_{k≤q} M(v_{m_{k}})^{c}. By 4.2 we can find α_{1}, …, α_{q} in A \ {0} such that M(α_{k}) ⊆ M(v_{m_{k}})^{c} for each k ≤ q, and the M(α_{k})’s form a partition of M(α). The block approximation property then gives a in A^{n} such that for every k ≤ q and every m in M(α_{k}) we have |a − a_{m_{k}}|_{m} ≤ |αh(a_{m_{k}})|_{m} < |h(a_{m_{k}})|_{m} with h := ∏_{i≤r} q_{i}.

Fix any m in M(α), there is k ≤ q such that m belongs to M(α_{k}). For every i ≤ r, since q_{i} ∈ A[X] the Taylor formula for polynomials yields:

|q_{i}(a) − q_{i}(a_{m_{k}})|_{m} ≤ |a − a_{m_{k}}|_{m} < |q_{i}(a_{m_{k}})|_{m}

hence |q_{i}(a)|_{m} = |q_{i}(a_{m_{k}})|_{m} ≠ 0. The same argument yields to:

|p_{i}(a) − p_{i}(a_{m_{k}})|_{m} ≤ |a − a_{m_{k}}|_{m} ≤ |q_{i}(a_{m_{k}})|_{m} = |q_{i}(a)|_{m}.

Since m ∈ M(α_{k} ⊆ M(v_{m_{k}})^{c} we have p_{i}(a_{m_{k}})/q_{i}(a_{m_{k}}) ∈ v_{m_{k}}^{−1}A ⊆ A_{m_{k}}, hence |p_{i}(a_{m_{k}})|_{m} ≤ |q_{i}(a_{m_{k}})|_{m} and the conclusion follows:

|p_{i}(a)|_{m} ≤ max(|p_{i}(a) − p_{i}(a_{m_{k}})|_{m}, |p_{i}(a_{m_{k}})|_{m}) ≤ |q_{i}(a)|_{m}.

Let us recall this well known fact: a valuation ring R is dense in its henselisation if and only if for every f ∈ R[X], every non zero ε in R and every x in R such that f(x) ∈ m(R) and f′(x) ∉ m(R), there is an element y in R such that |f(y)|_{R} ≤ |ε|_{R}. This is always true if the valuation group can be embedded in R_{+}^{∗}, but not in the general case.

Proposition 4.4 Let A be a Bezout domain having block factorisation and block approximation properties. Then the following are equivalent:

1. Every R in W_{A} is dense in its henselisation.

2. For every f ∈ A[X], every a and every nonzero ε in A there is an element b in A such that |f(b)|_{m} ≤ |ε|_{m} for every m in M(f(a)) \ M(f′(a)).

Remark 4.5 The reader may notice that a Prüfer domain A having the block factorisation is dense in each R ∈ W_{A}. Indeed, if x = u/v ∈ R with R = A_{m}, u, v ∈ A and v ∉ m, for any nonzero ε the block approximation property gives a in A such that:

\{ \begin{matrix}|a − u|_{n} ≤ |ε|_{n} & for every n ∈ M(v)^{c}, \\ |a|_{n} ≤ |v|_{n} & for every n ∈ M(v). \end{matrix}

Hence v divides a in A (because A = ⋂ W_{A} and |a|_{n} ≤ |v|_{n} for every maximal ideal of A, hence a/v belongs to every R ∈ W_{A}) and as m ∈ M(v)^{c} we have |b − x|_{m} ≤ |ε|_{m} with b := a/v ∈ A.

Proof: Suppose that (2) above holds, and fix R = A_{m} in W_{A}. Let g be any polynomial in R[X], suppose there is x in R such that g(x) ∈ m(R) and g′(x) ∉ m(R), and fix an arbitrary δ in R \ {0}. Since A is dense in R there exists a ∈ A and f ∈ A[X] such that f − g ∈ δR[X], f(a) ∈ m and f′(a) ∉ m. Write δ = ε/u with ε, u ∈ A \ {0} and u ∉ m. By (2) there is b in A such that |f(b)|_{m} ≤ |ε|_{m} = |δ|_{m} hence |g(b)|_{m} = |f(b) + (g(b) − f(b))|_{m} ≤ |δ|_{m} wich proves that R is dense in R^{h}.

Conversely, suppose (1) holds, and let f, a be as in (2). The block factorisation property gives a non zero α in A such that M(α) = M(f(a)) \ M(f′(a)). Fix some non zero ε in A, for each m ∈ M(α) (1) gives a_{m} ∈ A_{m} such that |f(a_{m})|_{m} ≤ |ε|_{m}, hence 4.3 proves (2).

Corollary 4.6 The class of Bezout domains A having both the block approximation and block factorisation properties, and such that every R in W_{A} is dense in its henselisation, is an elementary class. It is recursively axiomatisable.

Proof: By 4.1 it is sufficient to axiomatise this class modulo the theory of Bezout domains having the block factorisation and block approximation properties, but by 4.4 it is axiomatised by the following family of axioms, with d ∈ N^{∗}:

∀c_{0}, …, c_{d}∀ε∀a [ε ≠ 0 → ∃b, u, v, d (f(b)v = εu ∧ (v, f(a)) = d ∧ d ≤ f′(a))]

Indeed, observe that (v, f(a)) = d ∧ d ≤ f′(a) is equivalent to saying that M(v) is disjoint from M(f(a)) \ M(f′(a)).

Proposition 4.7 If a domain A satisfies the LGPH, then A is dense in A_{m}^{h} for every m ∈ Specmax(A).

This result will be strengthened in the next section. Proof: By remark 4.5 we know that A is dense in A_{m} hence it is sufficient to prove that A_{m} is dense in A_{m}^{h}, and for this we are going to use criterion 4.4. Let f be a polynomial with one variable and coefficients in A, fix an element a of A and a non zero ε in A. If f(a) = 0 there is nothing to prove, if not the block factorisation property gives a non zero α in A such that M(α) = M(f(a)) \ M(f′(a)).

Let us considere the variety V defined over Frac(A) by the equations εY = Zf(X) and αU + Z = 1. For every m ∈ M(α) since by construction f(a) ∈ m and f′(a) ∉ m, Hensel’s lemma gives x_{m} ∈ A_{m}^{h} such that f(x_{m}) = 0. Taking y_{m} = u_{m} = 0 and z_{m} = 1 then gives a point of V with coordinates in A_{m}^{h}. For every maximal ideal m of A not in M(α), x_{m} = a, y_{m} = z_{m} = 0, u_{m} = α^{−1} defines a point of V with coordinates in A_{m}.

Since V is smooth the LGPH provides a point (x, y, z, u) ∈ V (A) such that for every m ∈ M(α), z ∉ m since αu + z = 1 hence:

|f(x)|_{m} = |εyz^{−1}|_{m} ≤ |ε|_{m}

and the conclusion follows by 4.4.

The following result is supposed to be well known, although we never heard about any published reference on it, except in the case of absolute values wich has been proved by Lang. The interested reader will find in [1], I.5, a simple proof valid for any valuation ring and based on non standard arguments.

Lemma 4.8 Let R be a henselian valuation ring, and f = (f_{1}, …, f_{m}) ∈ R[X]^{m} with X = (X_{1}, …, X_{n}) be a system of equations for a variety V defined over F := Frac(R). Let Δ be a (n − dim V)-subdeterminant of the Jacobian matrix of f, and let μ be a nonzero element of R. Suppose that for any nonzero ε in R there is a_{ε} ∈ R^{n} such that |Δ(a_{ε})|_{R} ≥ |μ|_{R} and |f_{i}(a_{ε})|_{R} ≤ |ε|_{R} for every i ≤ m. Then V has a smooth point with coordinates in R.

Lemma 4.9 (accessibility lemma) Let A be a Prüfer domain with fraction field K, and for every maximal ideal n of A let K_{n} be the fraction field of A_{n}^{h}. Fix a maximal ideal m of A and x in K_{m}^{n}. Let us suppose that A_{m} is dense in A_{m}^{h}. Then for every nonzero ε in A there is a constructible subset M of Specmax(A) and y in K^{n} such that:

1. m ∈ M and |x − y|_{m} < |ε|_{m}.

2. For every n ∈ M there is a K-conjugate x_{n} of x in K_{n}^{n} such that |x_{n} − y|_{n} < |ε|_{n}.

If in addition A has the block factorisation property one can choose M = M(α) for some non zero α ∈ m. Moreover, if x belongs to (A_{m}^{h})^{n} and A is dense in A_{m}, then y and x_{n} can be taken in (A_{n}^{h})^{n} for every n ∈ M.

Proof: We prove it first with n = 1. Fix some nonzero ε in K. Let f be the minimal polynomial of x over K, let u in A be a non zero element such that g := uf ∈ A[X] . By hypothesis K is dense in K_{m} hence there exists y in K such that:

|x − y|_{m} < |ε|_{m}.  (1)

As K_{m} is a separable extension of K, g′(x) is non zero hence by continuity of polynomial functions we can choose y so that:

|g(y)|_{m} < min (|g′(y)^{2}|_{m}, |εg′(y)|_{m}) .  (2)

Let a, b, u, v in A such that u, v ∉ m, g(y)/g′(y)^{2} = a/u and g(y)/εg′(y) = c/v. By (2) we have a, b ∈ m hence:

m ∈ M := M(a) ∩ M(b) \ (M(u) ∪ M(v)).  (3)

For every n in M we have |g(y)|_{n} < |g′(y)^{2}|_{n} and |g(y)|_{n} < |εg′(y)|_{n} Hensel-Rychlik’s theorem then gives x_{n} ∈ K_{n} such that g(x_{n}) = 0, hence x_{n} is a K-conjugate of x over K, and:

|x_{n} − y|_{n} = |\frac{g(y)}{g′(y)}|_{n} < |ε|_{n}

wich proves this case.

Let now n take any value in N^{∗}. The primitive element theorem gives θ ∈ K_{m} such that K[x] = K[θ]. Let p ∈ K[T]^{n} be such that x = p(θ), and u ∈ A \ {0} such that q := up ∈ A[T]^{n}. Finally fix any non zero ε in K. Let t ∈ K, M a construtible subset of Specmax(A) given by the above case. Let y := p(t) and for any n ∈ M let θ_{n} ∈ K_{n} be a K-conjugate of θ such that |θ_{n} − t|_{n} < |uε|_{n}, and x_{n} := p(θ_{n}). By construction x_{n} is K-conjugate of x and we have:

|x_{n} − y|_{n} = |u^{−1}|_{n}|q(θ_{n}) − q(t)|_{n} ≤ |u^{−1}|_{n}|θ_{n} − t|_{n} < |ε|_{n}.

This ends the proof of the main statement. For the two additional remarks, let us observe that 3.5 and (3) above give a non zero α in A such that M = M(α). If we suppose that A is dense in A_{m} and x ∈ (A_{m}^{h})^{n} then t can be taken so that y ∈ A_{m}^{n} hence there is s ∈ A\m such that sy ∈ A^{n}. If we take ε in A and replace M by N = M\M(s), for every n ∈ N we have y ∈ A_{n} and |x_{n} − y| < |ε|_{n} ≤ 1 hence x_{n} ∈ (A_{m}^{h})^{n}.

Proposition 4.10 Let A be a Bezout domain having the block factorisation and block approximation properties. Suppose that every R in W_{A} is dense in its henselisation. Let V be a variety defined over K := Frac(A) and let f = (f_{1}, …, f_{m}) ∈ A[X]^{m} with X = (X_{1}, …, X_{n}) be a system of equations for V. For any non zero α in A the following are equivalent:

1. V^{∘}(A_{m}^{h}) ≠ ∅ for every m in M(α).

2. There is a (n−dim V)-subdeterminant of the Jacobian matrix of f and a non zero element μ of A with the property that for every non zero ε in A there exists a ∈ A^{n} such that for every m in M(α):

\{ \begin{matrix}|Δ(a)|_{m} ≥ |μ|_{m} \\ |f(a)|_{m} ≤ |ε|_{m} \end{matrix}

Proof: (2) ⇒ (1) Suppose (2) holds and fix m in M(α). For any non zero δ in A_{m}^{h} by density there is ε in A such that |ε|_{m} = |δ|_{m} hence (2) implies V^{∘}(A_{m}^{h}) ≠ ∅ by 4.8.

(1) ⇒ (2) Conversely, suppose (1) holds: for each m ∈ M(α) there is x_{m} in V (A_{m}^{h}) and a submatrix of size n − dim V of the Jacobian matrix of f whose determinant Δ_{m}(x_{m}) is non zero. Let Δ be any of those determinants, and fix m in M(α). Since Δ ∉ I(V) the implicit function theorem gives a point x′_{m} in V (A_{m}^{h}) such that Δ(x′_{m}) ≠ 0. Replacing x_{m} by x′_{m} if necessary, we can suppose that Δ_{m} = Δ, in other words Δ_{m} does not depend on m. The next step is to reduce to a finite number of local points, in order to construct something like a common lower bound for the Δ(x_{m})’s.

Let K denote the fraction field of A. For each m in M(α) let γ_{m} be a non zero element of m given by the accessibility lemma 4.9: for any n in M(γ_{m}) there is a Kconjugate of x_{m} in (A_{n}^{h})^{n}. Replacing A_{n}^{h} by some K-isomorphic image we can suppose that x_{m} itself belongs to (A_{n}^{h})^{n}. Let β_{m} ∈ A \ m such that M(β_{m}) = M(α) \ M(γ_{m}). The family of all M(β_{m})^{c}’s covers M(α) as m ranges over M(α). By compactness there are m_{1}, …, m_{q} in M(α) such that M(α) ⊆ ⋃_{k=1}^{q} M(β_{m_{k}})^{c}. By 4.2 there is a partition (M(α_{1}), …, M(α_{q})) of M(α) such that:

M(α_{k}) ⊆ M(α) \ M(β_{m_{k}}) ⊆ M(γ_{m_{k}})

We may observe that we have already reduced to a finite number of local points. Indeed for every m in M(α) there is k ≤ q such that m ∈ M(α_{k}) hence x_{m_{k}} ∈ V (A_{m}^{h}).

For each k ≤ q let d_{k}, e_{k} be nonzero elements integral over A such that Δ(x_{m_{k}}) = d_{k}/e_{k}, and μ_{k} be the norm of d_{k} over A (that is the product of all K-conjugates of d_{k}). Eventually let μ′ := ∏_{k≤q} μ_{k} ∈ A \ {0} and use the block factorisation property to write μ′ = μ′′μ with M(μ) = M(μ′) ∩ M(α) and M(μ′′) = M(μ′) \ M(α), hence |μ|_{m} = |μ′|_{m} for every m ∈ M(α). It is immediate that |Δ(x_{m_{k}})|_{m} ≥ |d_{k}|_{m} ≥ |μ|_{m} for every k ≤ q and every m in M(α_{k}).

It remains to check that (2) holds with this μ. Fix any non zero ε in A, for each m in M(α) there is k ≤ q such that m belongs to M(β_{k}). The density of A in A_{m}^{h} gives a_{m} ∈ A^{n} such that |a_{m}−x_{m_{k}}|_{m} ≤ |ε|_{m}. We can even choose a_{m} so that |Δ(a_{m})|_{m} ≥ |μ|_{m}. Moreover since f ∈ A[X]^{m} we have:

|f(a_{m})|_{m} ≤ |a_{m} − x_{m_{k}}|_{m} ≤ |ε|_{m}.

Then 4.3 gives a ∈ A^{n} such that |Δ(a)|_{m} ≥ |μ|_{m} and |f(a)|_{m} ≤ |ε|_{m} for every m ∈ M(α).

We can now set the result wich was the aim of this section.

Theorem 4.11 LGPH is a recursively axiomatisable property (in the language of rings).

Proof: By 4.6 it it sufficient to axiomatise the LGPH modulo the theory, noted T for this proof, of Bezout domains A having the block factorisation and the block approximation properties, and such that each R in W_{A} is dense in its henselisation. By 3.4 we only have to deal with the geometric condition.

Since the theory of algebraically closed fields has the elimination of quantifiers, t is well known that there is a formula V ar_{m,n,e,d}(x) in the language of rings such that for any domain A and any tuple c (whose length N is a function of m, n and e) of coefficients in A defining a system f_{c} = (f_{1}, …, f_{m}) of polynomials with n indeterminates and degree less than or equal to e, we have A ⊧ V ar_{m,n,e,d}(c) if and only if f is a system of equations of a variety V_{c} defined over Frac(A) with dim V_{c} = d. The set of (n−d)-subdeterminants of the jacobian matrix of f_{c} is defined uniformly in c, we denote it by J.

For any model A of T and any c ∈ A^{N} such that A ⊧ V ar_{m,n,e,d}(c), 4.10 shows that V_{c} has a smooth point in R^{h} for every R ∈ W_{A} if and only if A models the following formula (with parameters c):

ZH(c) ≡ ∀α, α ≠ 0 → ∃μ, [μ ≠ 0 ∧ ∨∨_{Δ_{c}∈J}∀ε, (0 ≠ ε →

∃a = (a_{1}, …, a_{n}) DivLoc(Δ_{c}(a), μ, α) ∧ ∧∧_{i=1}^{m} DivLoc(ε, f_{i}(a), α))].

It follows that a domain A satisfies the LGPH if and only if it is a model of T and of the following family of formulas:

∀c = (c_{1}, …, c_{N}) V ar_{m,n,e,d}(c) ∧ ZH(c) →

∃a = (a_{1}, …, a_{n}) f_{c}(a) = 0 ∧ ∨∨_{Δ_{c}∈J} Δ_{c}(a) ≠ 0.

5 LGPH with additionnal conditions

Proposition 5.1 (density property) Let A be a domain satisfying the LGPH and m a fixed maximal ideal of A. Let V be a variety defined over Frac(A), and for any maximal ideal n of A suppose that V has a smooth point x_{n} with coordinates in A_{n}^{h}. Then V^{∘}(A) is dense in V^{∘}(A_{m}^{h}) (with respect to the topology induced by the valuation). In particular given any polynomials h_{0}, h_{1}, …, h_{r} with coefficients in A such that h_{0} ∉ I(V), the LGPH provides smooth points a of V in A such that:

\{ \begin{matrix}h_{0}(a) ≠ 0 \\ h_{i}(a) ∈ m ⟺ h_{i}(x_{m}) ∈ m(A_{m}^{h}), & i ≤ r \end{matrix}

Proof: Fix any non zero ε in A, we can assume ε ∈ m. Let α be a non zero element of m given by the accessibility lemma 4.9: there is y in K^{n} such that |x_{m} − y|_{m} ≤ |ε|_{m} and for every n in M(α) there is a K-conjugate y_{m,n} of x_{m} in (A_{n}^{h})^{n} such that |y_{m,n} − y|_{n} ≤ |ε|_{n}. Replacing if necessary A_{n}^{h} by some K-isomorphic image we can assume y_{m,n} = x_{m} ∈ V^{∘}(A_{n}^{h}). The block factorisation property gives ε′, ε′′ in A such that ε = ε′ε′′, M(ε′) = M(ε) ∩ M(α) and M(ε′′) = M(ε) \ M(α). Since |ε|_{m} = |ε′|_{m}, by replacing if necessary ε by ε′, we can suppose M(ε) ⊆ M(α).

Now let f = (f_{1}, …, f_{m}) ∈ K[X] with X = (X_{1}, …, X_{n}) be a system of equations of V, and consider the variety V′ defined over K by:

\{ \begin{matrix}f(X) = 0 \\ X − y = εU \end{matrix}

with U = (U_{1}, …, U_{n}) a set of new variables. A point (x, u) of V′ is easily seen to be smooth if and only if x is a smooth point of V. For every maximal ideal n, if n ∈ M(α) then (x_{m}, ε^{−1}(x_{m} − y)) is a smooth point of V′ whose coordinates belong to A_{n}^{h}, as x_{m} ∈ V^{∘}(A_{n}^{h}) and |x_{m} − y|_{n} ≤ |ε|_{n}. If not, since M(ε) ⊆ M(α) we have ε^{−1} ∈ A_{n} hence (x_{n}, ε^{−1}(x_{n} − y)) ∈ V^{′∘}(A_{n}^{h}). The LGPH then gives (x, u) ∈ V^{′∘}(A), hence x ∈ V^{∘}(A) and |x − y|_{m} = |εu|_{m} ≤ |ε|_{m}. The conclusion then follows:

|x − x_{m}|_{m} ≤ max (|x − y|_{m}, |x_{m} − y|_{m}) ≤ |ε|_{m}.

The additional assertion on the h_{i}’s is then an easy consequence, by continuity of polynomial fonctions and the fact that m(A_{m}^{h}) is closed open in A_{m}^{h}. The reader has only to remember that as h_{0} does not belong to I(V), the implicit function theorem in A_{m}^{h} gives points x′_{m} in V^{∘}(A_{m}^{h}) arbitrarily close to x_{m} such that h_{0}(x′_{m}) ≠ 0. Replacing x_{m} by such an x′_{m} if necessary, we can suppose that h_{0}(x_{m}) ≠ 0, and the conclusion follows by density.

Ershov introduced in [2] a very general notion of regular closure, wich contains all the ordinary local-global principles for fields. When W is a family of valuation rings with fraction field K, Ershov says that K is regularly closed with respect to F_{W} := {Frac(R^{h}) | R ∈ W } if and only if for every variety V defined over K:

V^{∘}(K) ≠ ∅ ⟺ ∀F ∈ F, V^{∘}(F) ≠ ∅.

The following corollary makes clearer the relation between the LGPH and the relative regular closure.

Corollary 5.2 Let A be a domain with fraction field K. If A satisfies the LGPH, then K is regularly closed with respect to F_{W_{A}}.

Proof: Let V be a variety defined over K having a smooth point in Frac(R^{h}) for every R ∈ W_{A}. Then the homogenisation V^{∗} of V has a smooth point in R^{h} for every R ∈ W_{A}, hence in A by the LGPH. Moreover the density property allows us to choose a smooth point (a_{1}, …, a_{n}, u) in V^{∗}(A) such that u ≠ 0 and (a_{1}/u, …, a_{n}/u) belongs to V^{∘}(K) and the conclusion follows.

The converse is true in some cases, in particular when W_{A} is the family of all pvaluation rings of K. However it is false in general: let R be the valuation ring of rational fonctions in one variable over Q with no pole at 0. Let A be the integral closure of R in the algebraic closure of Frac(R). As Frac(A) is algebraically closed, trivially Frac(A) is regularly closed with respect to F_{W_{A}} = {Frac(A)}. But it has been proved that A does not satisfy the geometric condition of the LGPH (see [9], 5.5).

Theorem 5.3 Let A be a domain satisfying the LGPH, with fraction field K. Every R^{h} (R ∈ W_{A}) is endowed with the radical relation induced by its maximal spectrum, and so is A. Let f = (f_{1}, …, f_{m}) ∈ A[X]^{m} with X = (X_{1}, …, X_{n}) be a system of equations of a variety V defined over K, and g a fixed element of A[X]\I(V). For every i ∈ {1, …, u} and every j ∈ {1, …, v} fix some p_{i}, p′_{i}, q_{j}, q′_{j} in A[X] with the p_{i}’s not belonging to I(V). Suppose that for every maximal ideal m of A the variety V has a smooth point x_{m} with coordinates in A_{m}^{h} such that for every i ≤ u:

A_{m}^{h} ⊧ 0 ≠ p_{i}(x_{m}) ≤ p′_{i}(x_{m}) (i ≤ u).

Suppose in addition that for each j ≤ v there is a maximal ideal m(j) such that:

A_{m(j)}^{h} |̸= q_{j}(x_{m(j)}) ≤ q′_{j}(x_{m(j)}).

Then V has a smooth point a with coordinates in A such that g(a) ≠ 0, A ⊧ p_{i}(x) ≤ p′_{i}(x) for every i ≤ u and A |̸= q_{j}(x) ≤ q′_{j}(x) for every j ≤ v.

Proof: We are going to use several times the following simple argument:

(∗) For any valuation ring R and any a, b ∈ R^{n} such that |a−b|_{R} < 1, for any h ∈ R[X] we have |h(a) − h(b)|_{R} < 1 hence h(a) ∈ m(R) if and only if h(b) ∈ m(R). In particular for any h, h′ ∈ R[X] we have h(a) ≤_{R} h′(a) if and only if h(b) ≤_{R} h′(b).

Since none of the p_{i}’s belong to I(V), for every maximal ideal m of A the implicit function theorem gives a smooth point x′_{m} of V arbitrarily close to x_{m} such that p_{i}(x_{m}) ≠ 0 for every i ≤ u. Moreover the above argument (∗) shows that if x′_{m} is close enough to x_{m} it satisfies the radical conditions of the hypothesis, as x_{m} does. Replacing x_{m} by x′_{m} if necessary, we can then suppose that p_{i}(x_{m}) ≠ 0 for every i ≤ u.

Let p be the product of the p_{i}’s for i ≤ u and let a_{0} ∈ V^{∘}(A) be given by the density property so that α := p(a_{0}) ≠ 0. By hypothesis there are distinct maximal ideals m_{1}, …, m_{r} of A and a partition J_{1}, …, J_{r} of {1, …, v} such that for every k ∈ {1, …, r} and every j ∈ J_{k}:

A_{m_{k}}^{h} |̸= q_{j}(x_{m_{k}}) ≤ q′_{j}(x_{m_{k}}).

Muliplying α if necessary by a non zero element of m_{k} for each k ≤ r, we can suppose that M(α) contains {m_{1}, …, m_{r}}.

For every m ∈ M(α) let γ_{m} be a non zero element of m given by the accessibility lemma 4.9: there is b_{m} in A^{n} such that |x_{m} − b_{m}|_{m} ≤ |α|_{m} and for every n in M(γ_{m}) there is a K-conjugate x′_{m,n} of x_{m} in (A_{n}^{h})^{n} such that |x′_{m,n} − b_{m}|_{n} ≤ |α|_{n}. Replacing if necessary A_{n}^{h} by some K-isomorphic image we can assume x_{m} = x′_{m,n} ∈ V^{∘}(A_{n}^{h}).

By 3.5 there is for each m ∈ M(α) a non zero β_{m} in A such that:

M(β_{m}) = M(α) \ [M(γ_{m}) ∩ ⋂_{i≤u} M(p′_{i}(b_{m})) ∩ M(p_{i}(b_{m}))^{c}] .

Since A_{m}^{h} ⊧ ∧∧_{i≤u} p_{i}(x_{m}) ≤ p′_{i}(x_{m}) we have m ∈ M(α) ∩ M(β_{m})^{c}. Moreover for every n ∈ M(α) ∩ M(β_{m})^{c} by construction A_{n}^{h} ⊧ ∧∧_{i≤u} p_{i}(b_{m}) ≤ p′_{i}(b_{m}), and since |x_{m} − b_{m}|_{n} < 1 the argument (∗) applies:

A_{n}^{h} ⊧ ∧∧_{i≤u} p_{i}(x_{m}) ≤ p′_{i}(x_{m}).

The M(β_{m})^{c}’s cover M(α) hence a finite subfamily of them, wich we can suppose to include M(β_{m_{k}})^{c} for every k ≤ r, covers M(α). Say M(α) ⊆ ⋃_{k≤s} M(β_{m_{k}})^{c} with s ≥ r. We can suppose all those m_{k}’s to be distinct, hence 4.2 provides a partition (M(α_{k}))_{k≤s} of M(α) such that m_{k} ∈ M(α_{k}) ⊆ M(α) ∩ M(β_{m_{k}})^{c} for every k ≤ s.

The block approximation property gives a point b in A^{n} such that for every k ≤ s and every m ∈ M(α_{k}) we have |b−b_{m_{k}}|_{m} ≤ |α|_{m}. Since by construction |b_{m_{k}} −x_{m_{k}}|_{m} ≤ |α|_{m} it follows that:

|b − x_{m_{k}}|_{m} ≤ |α|_{m}.  (1)

Since p(x_{m_{k}}) ≠ 0 the block approximation property also gives non zero δ_{i} in A such that for i ≤ u, every k ≤ s and every m ∈ M(α_{k}):

|δ_{i}|_{m} ≤ |p_{i}(x_{m_{k}})|_{m}.  (2)

Use the block factorisation property to write δ_{i} = δ′_{i}δ′′_{i}, with M(δ′_{i}) = M(δ_{i}) ∩ M(α) and M(δ′′_{i}) = M(δ_{i}) \ M(α). For every k ≤ s and every m ∈ M(α_{k}) we then have |δ′_{i}|_{m} = |δ_{i}|_{m}. Replacing δ_{i} by δ′_{i} if necessary, we can suppose M(δ_{i}) ⊆ M(α).

Let V′ be the variety defined over K by the following equations, with Y a n-tuple of new variables:

\{ \begin{matrix}f(X) = 0 \\ αY = X − b, \\ p_{i}(X)Z_{i} = δ_{i}, & i ≤ u \end{matrix}

A point (x, y, z) of V′ is smooth if and only if x is a smooth point of V.

For every m in M(α) there is k ≤ s such that m ∈ M(α_{k}), hence x_{m_{k}} is a smooth point of V (A_{m}^{h}) such that |b − x_{m_{k}}|_{m} ≤ |α|_{m} by (1) and |δ_{i}|_{m} ≤ |p_{i}(x_{m_{k}})|_{m} by (2). Taking x = x_{m_{k}}, y = α^{−1}(x_{m_{k}} − b), and z_{i} = δ_{i}/p_{i}(x_{m_{k}}) for i ≤ u then gives a smooth point of V′ with coordinates in A_{m}^{h}.

For every m in M(α)^{c} by construction α and the p_{i}(a_{0})’s are invertible in A_{m} hence taking x = a_{0}, y = α^{−1}(a_{0} − b) and z_{i} = δ_{i}/p_{i}(a_{0}) gives a smooth point of V′ with coordinates in A_{m}^{h}.

The LGPH then gives a smooth point (x, y, z) of V′ with coordinates in A such that g(x) ≠ 0. Hence x is a smooth point of V with coordinates in A such that g(x) ≠ 0.

Fix any i ≤ u. Since p_{i}(x) divides δ_{i} it follows that M(p_{i}(x)) ⊆ M(α). For every m ∈ M(p_{i}(a)) there is then some k ≤ s such that m ∈ M(α_{k}).

|x − x_{m_{k}}|_{m} ≤ max (|x − b|_{m}, |b − x_{m_{k}}|_{m}) ≤ |α|_{m} < 1.

Since A_{m} ⊧ p_{i}(x_{m}) ≤ p′_{i}(x_{m}) the argument (∗) then implies that A_{m} ⊧ p_{i}(x) ≤ p′_{i}(x) hence p′_{i}(x) ∈ m. This proves that p_{i}(x) ≤_{A} p′_{i}(x) for every i ≤ u.

Fix now any j ≤ v; there is k ≤ r such that j belongs to J_{k} hence A_{m_{k}}^{h} |̸= q_{j}(x_{m_{k}}) ≤ q′_{j}(x_{m_{k}}). Since |x − x_{m_{k}}|_{m} < 1 the argument (∗) implies that A_{m_{k}}^{h} |̸= q_{j}(x) ≤ q′_{j}(x). It follows that q_{j}(x) ∈ m_{k} and q′_{j}(x) ∉ m_{k} hence A |̸= q_{j}(x) ≤ q′_{j}(x).

Remark 5.4 We already mentionned that previous local-global principles for rings boyl down to particular cases of the LGPH. More precisely, the experienced reader can infere from the above properties of rings satisfing the LGPH, that the good Rumely domains introduced by van den Dries and Macintyre in [9] are exactly the domains satisfying the LGPH wich have an algebraically closed fraction field, a Jacobson radical reduced to the zero ideal, and whose residual lattice is dense. Seemingly, it can be shown as a corollary of 5.3 that a domain A satisfies the local-global principle introduced by Prestel and Schmid in [5] if and only if it satisfies the LGPH and if in addition the henselisation of any R in W_{A} has a separably closed fraction field. As we won’t use it here, we refere the reader to [1], I.3 and II.5, for all of this.

6 The p-adic case

Let us recall from the literature some facts about formally p-adic fields (see [4]). Let p be a fixed prime number, a p-adically closed field is a field elementary equivalent to some finite extension of the p-adic field Q_{p}. Such a field K is the fraction field of a unique henselian valuation ring R, definable in K, whose residue field is a finite extension of the field F_{p} with p elements, and whose maximal ideal is generated by an element π such that |p|_{R} = |π^{e}|_{R} for some e ∈ N^{∗}. The ring R is called the holomorphy ring of K. The complete theory of any p-adically closed field is recursively axiomatisable and model-complete. It follows that if K ⊆ K′ are elementary equivalent p-adically closed fields with holomorphy ring R, R′ then R′ ∩ K = R, a remark that we will often use in what follows.

Until the end of this section let F be a fixed finite extension of Q_{p}, and denote by T_{F} its complete theory. Similarly we let T H_{F} (resp. T H_{F,∀}) denote the complete (resp. universal) theory of the holomorphy ring of F (in the language of rings). It is well known that given any model K of T_{F}, a subfield K_{0} of K is a model of T_{F} if and only of K is a regular extension of K_{0}. In particular Alg(K) the subfield of algebraic elements of K is a model of T_{F}.

Prestel and Schmid [5] called r_{∘}-domains those r-domains (A, ≤) for wich (0) ∈ Spec(A, ≤), that is (A, ≤) ⊧ ∀a, 0 ≤ a → a = 0. Let T (F) denote the theory of r_{∘}-domains (A, ≤) such that A ⊧ T H_{F,∀}. It is a universal theory and a fortiori an inductive theory, hence any model of T (F) embeds in an existentially closed model.

Theorem 6.1 Existentially closed models (A, ≤) of T (F) are exactly those which satisfy the following conditions:

1. A satisfies the LGPH;

2. Frac(A) models T_{F};

3. L(A) is dense;

4. The radical relation of (A, ≤) is induced by Specmax(A).

They form an elementary class, whose theory T (F)^{∗} is model-complete: it is the model companion of T (F).

This theorem, like all the results wich are proved in this section, has almost been proved by Prestel and Schmid in [6]. Here “almost” means that their local-global principle was not the LGPH, but if we replace it by the LGPH all the arguments used in [6] (wich were actually taken from [5]) carry over with only little change. In what follows we refer the reader to [5] for the easiest steps and rewrite the main steps.

The reader may have noticed that the second assertion of 6.1 (T (F)^{∗} is well defined) has already been proved: as Frac(A) and L(A) are interpretable in A, and the radical relation ≤_{A} is definable in A, it is clear that conditions 2 to 4 are first order, and we have proved in 4.11 that the LGPH (condition 1) is recursively axiomatisable.

The arguments used in [5], theorem 3.3, apply here to prove that any existentially closed model of T (F) is a model of T (F)^{∗}. For example, let us check the following:

Lemma 6.2 Any existentially closed model of T (F) satisfies the LGPH.

Proof: Let (A, ≤) be an existentially closed model of T (F), and K its fraction field. Let V be a variety defined over K having a smooth point x_{R} with coordinates in R^{h} for every R ∈ W_{A}. Let x_{V} be a generic point of V over K, and A′ = A[x_{V}]. For any non zero p ∈ Spec(A, ≤), let m be a maximal ideal of A containing p, and let R in W_{A} be such that m(R) ∩ A = m. Choose an elementary extension R′ of R^{h} with transcendence degree over K greater than or equal to dim V. Then the implicit function theorem applied to the smooth point x_{R} in R′ gives a point x′_{R} ∈ V (R′) such that the transcendence degree of K(x′_{R}) over K is dim V, hence x′_{R} is a generic point of V over K. Since A[x′_{R}] and A[x_{V}] are isomorphic over A, we can identify them and consider that A′ ⊆ R′. There is a prime ideal q′ in R′ such that q′ ∩ A = p (q′ is the ideal generated by p in R′). Let p′ := q′ ∩ A′, endow A′ with the radical relation defined by P′ := {p′ | p ∈ Spec(A, ≤) \ {(0)}} ∪ {(0)}. As by construction the restriction map p′ ↦ p′ ∩ A sends P′ onto Spec(A, ≤), it follows that (A′, ≤) is an extension of (A, ≤) (see [5], lemma 2.7) hence (A, ≤) is existentially closed in (A′, ≤). Since V has a smooth point with coordinates in A′ (namely x_{V}) it then follows that V^{∘}(A) ≠ ∅, proving that the LGPH holds in A.

By the above arguments we have reduced the proof of 6.1 to proving that T (F)^{∗} is model-complete. Let us recall the following basic properties of r-domains.

Lemma 6.3 Let (A, ≤) ⊆ (A′, ≤) with A, A′ Bezout.

1. For every p′ ∈ Spec(A′, ≤), p′ ∩ A belongs to Spec(A, ≤).

2. There is a natural lattice-embedding of L(A, ≤) into L(A′, ≤). We will use it to identify L(A, ≤) as a sub-lattice of L(A′, ≤).

3. Given another extension (A′′, ≤) of (A, ≤) with A′′ Bezout and an orderisomorphism σ between L(A′, ≤) and L(A′′, ≤) whose restriction to L(A, ≤) is identity, define:

σ̂ : p′′ ↦ {a′ ∈ A′ | p′′ ∈ σ(P (a′))}.

Then σ̂ is a bijection between Spec(A′′, ≤) and Spec(A′, ≤), and p′′∩A = σ̂(p′′)∩A for every p′′ ∈ Spec(A′′, ≤).

Proof: (1) is straightforward, (2) is [5], lemma 2.10 (e), and (3) is essentially [5], proposition 2.8.

Since a ring A satisfying the LGPH has the block factorisation property, it follows that for any non zero a and any b in A there is c in A such that M(a) \ M(b) = M(c). The argument used in the proof of [5], lemma 3.5 (viii), then proves that:

Spec(A, ≤_{A}) ⊆ Specmax(A) ∪ {(0)}.

It follows that for any a, b in A with a ≠ 0 there is c in A such that P (a) \ P (b) = P (c) hence L(A, ≤_{A}) is a relatively complemented lattice (that is a distributive lattice L such that for any element α of L except maybe the maximal element, the sublattice of elements of of L smaller than α is a Boolean algebra).

The following result can be proved along the lines of [5], page 194. For a complete proof see [1], Annexe A,.

Lemma 6.4 The theory of dense relatively complemented lattices is complete and modelcomplete (it is actually ℵ_{0}-categorical).

We come now to the main step of the proof of 6.1: proving that T (F)^{∗} is modelcomplete. Notice that for any model (A, ≤) of T (F)^{∗}, from A ⊧ T H_{F,∀} and Frac(A) ⊧ T_{F} we infer that the holomorphy ring R of Frac(A) contains A. Moreover R ∈ W_{A} since A/(m(R) ∩ A) is a subring of the finite field R/m(R), hence A/(m(R) ∩ A) is equal to the field itself, that is m(R) ∩ A is a maximal ideal of A. Proof: (finishing 6.1) Let (A, ≤) ⊆ (A′, ≤) be two models of T (F)^{∗}. Let K, K′ be their fraction fields, and R_{∘}, R′_{∘} their holomorphy rings. Since T_{F} is model-complete and R′_{∘} (resp. R_{∘}) is definable in K (resp. K′) we have R′_{∘} ∩ K = R_{∘}, and R′_{∘} is an elementary extension of R_{∘}. Let φ(x) be a quantifier free formula in L_{rad} with parameters in A and n free variables, such that (A′, ≤) ⊧ ∃xφ(x). We have to find a in A^{n} such that φ(a) holds in (A, ≤).

As (A, ≤) and (A′, ≤) are both r_{∘}-domains, they have the following property:

a ≤ b ⟺ b = 0 or 0 ≠ a ≤ b.

We can therefore suppose φ(x) of the following kind:

φ(x) ≡ ∧∧_{i=1}^{m} f_{i}(x) = 0 ∧ g(x) ≠ 0 ∧ ∧∧_{i=1}^{u} 0 ≠ p_{i}(x) ≤ p′_{i}(x) ∧ ∧∧_{i=1}^{v} q_{i}(x) ≰ q′_{i}(x)

with f_{i}, g, p_{i}, p′_{i}, q_{i}, q′_{i} in A[X] (X = (X_{1}, …, X_{n})).

Let a′ ∈ A^{′n} be such that (A′, ≤) ⊧ φ(a′). As K′ is a regular extension of K, a′ is the generic point of a variety V defined over K. As for any f ∈ I(V) and any point x of V we have f(x) = 0, it is sufficient to find a point a ∈ V (A) such that g(a) ≠ 0, p_{i}(a) ≤ p′_{i}(a) for i ≤ u and q_{j}(a) ≰ q′_{j}(a) for j ≤ v. In order to achieve this goal we are going to use 5.3, hence it is worthwhile to notice that since g(a′) and the p_{i}(a′)’s are non zero, neither g nor the p_{i}’s belong to the ideal of V.

Since the theory of dense relatively complemented lattices is model complete by 6.4, L(A) ⪯ L(A′) hence they have some ultrapowers wich are isomorphic over L(A). As L(A) is interpretable in A, the residual lattice of an ultrapower of A is itself an ultrapower of L(A), and the same holds for A′. Hence the residual lattices of some elementary extensions of (A, ≤) and (A′, ≤) are isomorphic over L(A). Let (A′′, ≤) and (A′′′, ≤) be such elementary extensions of (A, ≤) and (A′, ≤) respectively. Replacing (A′, ≤) by (A′′′, ≤) if necessary we can suppose there is an order-isomorphism σ from L(A′) to L(A′′) whose restriction to L(A) is identity.

Let K′′ be the fraction field of A′′, and R′′_{∘} its holomorphy ring: recall that since (A′′, ≤) is an elementary extension of (A, ≤) it is a model of T (F)^{∗}, hence K′′ ⊧ T_{F} and R′′_{∘} ∩ K = R_{∘} = R′_{∘}. The bijection σ̂ between Spec(A′′, ≤) and Spec(A′, ≤) given by 3 of 6.3 sends obviously the zero ideal of A′′ to the zero ideal of A′, hence it induces a bijection between Specmax(A′′) and Specmax(A′). Moreover m′′ ∩ A = σ̂(m′′) ∩ A for every maximal ideal m′′ of A′′. We can compose σ̂ with the permutation of of Specmax(A′) wich exchanges σ̂(m(R′′_{∘}) ∩ A′′) with m(R′_{∘}) ∩ A′, and fixes every other maximal ideals of A′. Since we have:

σ̂(m(R′′_{∘}) ∩ A′′) ∩ A = m(R′′_{∘}) ∩ A = m(R_{∘}) ∩ A = m(R′_{∘}) ∩ A,

it follows that m′′ ∩ A = (τ ∘ σ̂(m′′)) ∩ A for every maximal ideal m′′ of A′′. Replacing σ̂ by τ ∘ σ̂ if necessary, we can then suppose that σ̂ sends m(R′′_{∘}) ∩ A to m(R′_{∘}) ∩ A′.

Fix any m′′ ∈ Specmax(A′′) and let R′′ := A_{m′′}, R′ := A′_{σ̂(m′′)} and R := R′′ ∩ K = R′ ∩ A. We can consider R^{h} as a common subring of R^{′h} and R^{′′h}.

If m′′ ≠ m(R′′_{∘}) ∩ A′′ then R′′ and R′′_{∘} are independent by 3.7, hence R^{′′h} has an algebraically closed fraction field (it is a well-known property of henselian fields that the henselisation of any valuation ring independent from the henselian one has a separably closed fraction field). The same holds for R^{′h} and since the theory of non trivial valuation rings with algebraically closed fraction fields has quantifier elimination it follows that R^{′′h} and R^{′h} are elementary equivalent over R^{h} (that is in the language of rings enriched with a constant symbol for each element of R^{h}).

If m′′ = m(R′′_{∘}) ∩ A, by construction R^{h} ⪯ R^{′h} and R^{h} ⪯ R^{′′h} as all three are models of T H_{F}, hence R^{′h} and R^{′′h} are elementary equivalent over R^{h} (in this case R = R^{h} but we prefer to write R^{h} in order to set a statement uniform in R, and similarly for R′, R′′).

Let {h_{1}, …, h_{r}} be an enumeration of all the p_{i}, p′_{i}, q_{j}, q′_{j}. Let Ω_{R′} be the set of indices i ≤ r such that h_{i}(a′) belongs to m(R′), and denote by Ω_{R′}^{c} its complement in {1, …, r}. As a′ is a generic point of V it is a smooth point, hence there is a determinant Δ with an appropriate size to caracterise smooth points of V such that Δ(a′) ≠ 0. The henselisation R^{′h} models ∃xφ_{R′}(x) where φ_{R′}(x) is the following formula, with parameters in R^{h}:

∧∧_{i=1}^{r} f_{i}(x) = 0 ∧ Δ(x) ≠ 0 ∧ ∧∧_{i∈Ω_{R′}} ∀y_{i}, y_{i}h_{i}(x) ≠ 1 ∧ ∧∧_{i∈Ω_{R′}^{c}} ∃y_{i}, y_{i}h_{i}(x) = 1.

Since R^{′′h} is elementary equivalent to R^{′h} over R^{h}, the same holds in R^{′′h} hence there is x_{m′′} ∈ V^{∘}(R^{′′h}) such that for every i ≤ r:

h_{i}(x_{m′′}) ∈ m(R^{′′h}) ⟺ h_{i}(a′) ∈ m(R^{′h}).

In particular, A_{m′′}^{′′h} ⊧ p_{i}(x_{m′′}) ≤ p′_{i}(x_{m′′}) for every i ≤ u, and for every j ≤ v there is n′′ ∈ Specmax(A′′) such that A_{n′′}^{′′h} ⊧ q_{i}(x_{n′′}) ≰ q′_{i}(x_{m′′}). Theorem 5.3 then gives a point a′′ ∈ V (A′′) such that g(a′′) ≠ 0, p_{i}(a′′) ≤ p′_{i}(a′′) for i ≤ u and q_{j}(a′′) ≰ q′_{j}(a′′) for j ≤ v, hence (A′′, ≤) ⊧ ∃xφ(x). As (A, ≤) ⪯ (A′′, ≤) the same holds in (A, ≤) wich ends the proof.

In [5], page 195 Prestel and Schmid defined the algebraic part Alg(A) of a domain A to be the subring of elements of A wich are algebraic over the prime field of Frac(A). Along the lines of [5], theorem 4.5, one can deduce from 6.1:

Corollary 6.5 Two models of T (F)^{∗} are elementary equivalent if and only if their algebraic parts are isomorphic.

We can now conclude that the ring Z̃ ∩ Q_{p} is decidable by setting F = Q_{p} in the following statement.

Corollary 6.6 The ring Z̃ ∩ F is decidable. Moreover the complete theory of (Z̃ ∩ F, ≤_{Z^{}˜ ∩F}) is decidable and model complete, it is axiomatised by T (F)^{∗} and the following family of axioms:

∀x, f(x) ≠ 0, for every nonmonic irreducible f in Z[X].

Proof: Let A := Z̃ ∩ F, since the radical relation defined by the maximal spectrum is definable in the ring, the decidability of A follows from the decidability of (A, ≤_{A}). Let T denote the above theory, that is T (F)^{∗} plus the condition on nonmonic polynomials.

Moret-Bailly has proved in [3] that A models the LGPH. The arguments given in [6], proposition 2.1, apply here and prove that L(A) is dense, hence (A, ≤_{A}) models T (F)^{∗}. The additional conditions on nonmonic poynomials are obviously satisfied by A hence (A, ≤_{A}) models T.

Conversely any model (A′, ≤) of T has a fraction field K′ wich models T_{F} hence Alg(K′) ≃ Alg(F) ≃ Frac(A). From the conditions on monic polynomials we infer that any element of Alg(A′) is integral over Z, and conversely since A′ is integrally closed Alg(A′) contains a copy of the integral closure of Z in Alg(K′), that is a copy of A, hence Alg(A′) ≃ A = Alg(A). By 6.5 we conclude that (A′, ≤) ≡ (A, ≤).

Since T (F)^{∗} is model-complete so is T, and since T is given by a recursive family of axioms the conclusion follows.

Remark 6.7 We have tried in this section to follow closely the lines of Prestel and Schmid, because it is the shorter way to prove 6.6. From 6.6 it is immediate that for any non zero integer n the ring (Z̃ ∩ Q_{p})[1/n] is decidable. A more general approach such as in [1], chapitre 5 gives necessary and sufficient condition of decidability for a larger class of rings between Z̃ ∩ Q_{p} and its fraction field. Let us mention here which result is actually proved with the local-global principle and the method of [5] in place of the LGPH: namely, that (Z̃ ∩ Q_{p})[1/n] is decidable for any non zero integer n such that p divides n.

References

[1] L. Darnière. Étude modèle-théorique d’anneaux satisfaisant un principe de Hasse non singulier. Doctoral thesis, Université de Rennes 1, january 1998.

[2] Y. L. Ershov. Relative regular closeness and π-valuations. Algebra and Logic, 31:170– 181, 1992.

[3] L. Moret-Bailly. Groupes de picard et problèmes de skolem II. Ann. scien. Ec. Norm. Sup., 22(4):181–194, 1989.

[4] A. Prestel and P. Roquette. Formally p-adic fields, volume 1050 of LNM. Springer-Verlag, 1984.

[5] A. Prestel and J. Schmid. Existentially closed domains with radical relations. J. reine und angew. Math., 407:178–201, 1990.

[6] A. Prestel and J. Schmid. Decidability of the rings of real algebraic and p-adic algebraic integers. J. reine und angew. Math., 414:141–148, 1991.

[7] R. Rumely. Arithmetic over the ring of all algebraic integers. J. reine und angew. Math., 368:127–133, 1986.

[8] L. van den Dries. Elimination theory for the ring of algebraic integers. J. reine und angew. Math., 388:189–205, 1988.

[9] L. van den Dries and A. Macintyre. The logic of Rumely’s local-global principle. J. reine und angew. Math., 407:33–56, 1990.

Luck Darnière Institut de recherche mathématique de Rennes Université de Rennes 1 35042 Rennes Cedex France e-mail: darniere@univ-rennes1.fr