Hidden constructions in abstract algebra (3) Krull Dimension of distributive lattices and commutative

rings

Thierry Coquand (^{∗}) Henri Lombardi (^{†}),

may 2002

Abstract

We present constructive versions of Krull’s dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Español and the authors. We show that the notion of Krull dimension has an explicit computational content in the form of existence (or lack of existence) of some algebraic identities. We can then get an explicit computational content where abstract results about dimensions are used to show the existence of concrete elements. This can be seen as a partial realisation of Hilbert’s program for classical abstract commutative algebra.

MSC 2000: 13C15, 03F65, 13A15, 13E05

Key words: Krull dimension, distributive lattices, Constructive Mathematics.

∗ Chalmers, University of Göteborg, Sweden, email: coquand@cs.chalmers.se † Equipe de Mathématiques, CNRS UMR 6623, UFR des Sciences et Techniques, Université de Franche-Comté, 25 030 BESANCON cedex, FRANCE, email: lombardi@math.univ-fcomte.fr

Contents

Introduction 3

1 Distributive lattice, Entailment relations 4 1.1 Distributive lattices, filters and spectrum . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Distributive lattices and entailment relations . . . . . . . . . . . . . . . . . . . 8 1.3 Spectrum and completeness theorem . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Krull dimension of distributive lattices 10 2.1 Definition of Kr_{ℓ}(L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Partially specified chains of prime ideals . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Krull dimension of a distributive lattice . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Implicative lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Decidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.6 Dimension of Spectral Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.7 Connections with Joyal’s definition . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Zariski and Krull lattice 15 3.1 Zariski lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Krull lattices of a commutative ring . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Krull dimension of a polynomial ring over a discrete field . . . . . . . . . . . . . 18

Bibliographie 20

Annex: The completeness theorem and LLPO 22 A.1 Theories and models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 A.2 Completeness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 A.3 Compactness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 A.4 LPO and LLPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 A.5 Geometric formulae and theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Introduction

We present constructive versions of Krull’s dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Español and the authors. We show that the notion of Krull dimension has an explicit computational content in the form of existence (or lack of existence) of some algebraic identities. This confirms the feeling that commutative algebra can be seen computationally as a machine that produces algebraic identities (the most famous of which being called Nullstellensatz). This can be seen as a partial realisation of Hilbert’s program for classical abstract commutative algebra.

Our presentation follows Bishop’s style (cf. in algebra [19]). As much as possible, we kept minimum any explicit mention to logical notions. When we say that we have a constructive version of an abstract algebraic theorem, this means that we have a theorem the proof of which is constructive, which has a clear computational content, and from which we can recover the usual version of the abstract theorem by an immediate application of a well classified non-constructive principle. An abstract classical theorem can have several distinct interesting constructive versions.

In the case of abstract theorem in commutative algebra, such a non-constructive principle is the completeness theorem, which claims the existence of a model of a formally consistent propositional theory. We recall the exact formulation of this theorem in the appendix, as well as its derivation from the compactness theorem When this is used for algebraic structures of enumerable presentation (in a suitable sense) the compactness and completeness theorem can be seen as a reformulation of Bishop LLPO (a real number is ≥ 0 or ≤ 0).

To avoid the use of completeness theorem is not motivated by philosophical but by practical considerations. The use of this principle leads indeed to replace quite direct (but usually hidden) arguments by indirect ones which are nothing else than a double contraposition of the direct proofs, with a corresponding lack of computational content. For instance [2] the abstract proof of 17^{th} Hilbert’s problem claims : if the polynomial P is not a sum of rational fractions there is a field K in which one can find an absurdity by reading the (constructive) proof that the polynomial is everywhere positive or zero. The direct version of this abstract proof is: from the (constructive) proof that the polynomial is everywhere positive or zero, one can show (using arguments of the abstract proofs) that any attempt to build K will fail. This gives explicitly the sum of squares we are looking for. In the meantime, one has to replace the abstract result: “any real field can be ordered” by the constructive theorem: “in a field in which any attempt to build an ordering fails −1 is a sum of squares”. One can go from this explicit version to the abstract one by completeness theorem, while the proof of the explicit version is hidden in the algebraic manipulations that appear in the usual classical proof of the abstract version.

Here is the content of the paper.

Distributive lattices In this section, we present basic theorems on distributive lattices. An important simplification of proofs and computations is obtained via the systematic use of the notion of entailment relation, which has its origin in the cut rule in Gentzen’s sequent calculus, with the fundamental theorem 1.7.

Dimension of distributive lattices In this section, we develop the theory of Krull dimension of distributive lattices, explaining briefly the connection with Español’s developments of Joyal’ s theory. We show that the property to have a Krull dimension ≤ ℓ can be formulated as the existence of concrete equalities in the distributive lattice.

Zariski and Krull lattice In section 3 we define the Zariski lattice of a commutative ring (whose elements are radicals of finitely generated ideals), which is the constructive counterpart of Zariski spectrum : the points of Zariski spectrum are the prime ideals of Zariski lattice, and the constructible subsets of Zariski spectrum are the elements of the Boolean algebra generated by the Zariski lattice. Joyal’s idea is to define Krull dimension of a commutative ring as the dimension of its Zariski lattice. This avoids any mention of prime ideals. We show the equivalence between this (constructive) point of view and the (constructive) presentation given in [14], showing that the property to have a Krull dimension ≤ ℓ can be formulated as the existence of concrete equalities in the ring.

Conclusion This article confirms the actual realisation of Hilbert’s program for a large part of abstract commutative algebra. (cf. [2, 4, 10, 11, 12, 13, 14, 15, 16, 17]). The general idea is to replace ideal abstract structures by partial specifications of these structures. The very short elegant abstract proof which uses these ideal objects has then a corresponding computational version at the level of the partial specifications of these objects. Most of classical results in abstract commutative algebra, the proof of which seem to require in an essential way excluded middle and Zorn’s lemma, seem to have in this way a corresponding constructive version. Most importantly, the abstract proof of the classical theorem always contains, more or less implicitly, the constructive proof of the corresponding constructive version. Finally, we should note that the explicit characterisations of Krull dimension of distributive lattices, Theorem 2.9, of spectral spaces, Theorem 2.14, and of rings, Corollary 3.6, are new.

1 Distributive lattice, Entailment relations

Elementary though it has become after successive presentations and simplifications, the theory of distributive lattices is the ideal instance of a mathematical theory, where a syntax is specified together with a complete description of all models, and what is more, a table of semantic concepts and syntactic concepts is given, together with a translation algorithm between the two kinds of concepts. Such an algorithm is a “completeness theorem” (G. C. Rota [20]).

1.1 Distributive lattices, filters and spectrum

As indicated by the quotation above, the structure of distributive lattices is fundamental in mathematics, and G.C. Rota has pointed out repeatedly its potential relevance to commutative algebra and algebraic geometry. A distributive lattice is an ordered set with finite sups and infs, a minimum element (written 0) and a maximum element (written 1). The operations sup and inf are supposed to be distributive w.r.t. the other. We write these operations ∨ and ∧. The relation a ≤ b can then be defined by a ∨ b = b or, equivalently, a ∧ b = a. The theory of distributive lattices is then purely equational. It makes sense then to talk of distributive lattices defined by generators and relations.

A quite important rule, the cut rule, is the following

(((x ∧ a) ≤ b) & (a ≤ (x ∨ b))) ⟹ a ≤ b.

In order to prove this, write x ∧ a ∧ b = x ∧ a and a = a ∧ (x ∨ b) hence

a = (a ∧ x) ∨ (a ∧ b) = (a ∧ x ∧ b) ∨ (a ∧ b) = a ∧ b.

A totally ordered set is a distributive lattice as soon as it has a maximum and a minimum element. We write n for the totally ordered set with n elements (this is a distributive lattice for n ≠ 0.) A product of distributive lattices is a distributive lattice. Natural numbers with the divisibility relation form a distributive lattice (with minimum element 1 and maximum element 0). If L and L′ are two distributive lattices, the set Hom(L, L′) of all morphisms (i.e., maps preserving sup, inf, 0 and 1) from L to L′ has a natural order given by

φ ≤ ψ ⟺^{def} ∀x ∈ L φ(x) ≤ ψ(x).

A map between two totally ordered distributive lattices L and S is a morphism if, and only if, it is nondecreasing and 0_{L} and 1_{L} are mapped into 0_{S} and 1_{S}.

The following proposition is direct.

Proposition 1.1 Let L be a distributive lattice and J a subset of L. We consider the distributive lattice L′ generated by L and the relations x = 0 for x ∈ J (L′ is a quotient of L). Then

• the equivalence class of 0 is the set of a such that for some finite subset J_{0} of J:

a ≤ ⋁_{x∈J_{0}} x in L

• the equivalence class of 1 is the set of b such that for some finite subset J_{0} of J:

1 = (b ∨ ⋁_{x∈J_{0}} x) in L

• More generally a ≤_{L′} b if, and only if, for some finite subset J_{0} of J:

a ≤ (b ∨ ⋁_{x∈J_{0}} x)

In the previous proposition, the equivalence class of 0 is called an ideal of the lattice; it is the ideal generated by J. We write it ⟨J⟩_{L}. We can easily check that an ideal I is a subset such that:

0 ∈ I
x, y ∈ I ⟹ x ∨ y ∈ I
x ∈ I, z ∈ L ⟹ x ∧ z ∈ I

(the last condition can be written (x ∈ I, y ≤ x) ⇒ y ∈ I).

Furthermore, for any morphim φ : L_{1} → L_{2}, φ^{−1}(0) is an ideal of L_{1}.

A principal ideal is an ideal generated by one element a. We have ⟨a⟩_{L} = {x ∈ L ; x ≤ a}. Any finitely generated ideal is principal.

The dual notion of ideal is the one of filter. A filter F is the inverse image of 1 by a morphism. This is a subset such that:

1 ∈ F
x, y ∈ F ⟹ x ∧ y ∈ F
x ∈ F, z ∈ T ⟹ x ∨ z ∈ F

Notation 1.2 We write P_{f}(X) for the set of all finite subsets of the set X. If A is a finite subset of a distributive lattice L we define

⋁ A := ⋁_{x∈A} x and ⋀ A := ⋀_{x∈A} x

We write A ⊢ B or A ⊢_{L} B for the relation defined on the set P_{f}(L):

A ⊢ B ⟺^{def} ⋀ A ≤ ⋁ B

Note the relation A ⊢ B is well defined on finite subsets because of associativity commutativity and idempotence of the operations ∧ and ∨. Note also ∅ ⊢ {x} ⇒ x = 1 and {y} ⊢ ∅ ⇒ y = 0. This relation satisfies the following axioms, where we write x for {x} and A, B for A ∪ B.

a ⊢ a  (R)
(A ⊢ B) & (A ⊆ A′) & (B ⊆ B′) ⟹ A′ ⊢ B′ (M)
(A, x ⊢ B) & (A ⊢ B, x) ⟹ A ⊢ B  (T)

we say that the relation is reflexive, monotone and transitive. The last rule is also called cut rule. Let us also mention the two following rules of “distributivity”:

\begin{aligned}(A, x ⊢ B) & (A, y ⊢ B) & ⟺ & A, x ∨ y ⊢ B \\ (A ⊢ B, x) & (A ⊢ B, y) & ⟺ & A ⊢ B, x ∧ y \end{aligned}

The following is proved in the same way as Proposition 1.1.

Proposition 1.3 Let L be a distributive lattice and (J, U) a pair of subsets of L. We consider the distributive lattice L′ generated by L and by the relations x = 0 for x ∈ J and y = 1 for y ∈ U (L′ is a quotient of L). We have that:

• the equivalence class of 0 is the set of elements a such that:

∃J_{0} ∈ P_{f}(J), U_{0} ∈ P_{f}(U) a, U_{0} ⊢_{L} J_{0}

• the equivalence class of 1 is the set of elements b such that: vérifient:

∃J_{0} ∈ P_{f}(J), U_{0} ∈ P_{f}(U) U_{0} ⊢_{L} b, J_{0}

• More generally a ≤_{L′} b if, and only if, there exists a finite subset J_{0} of J and a finite

subset U_{0} of U such that, in L:

a, U_{0} ⊢_{L} b, J_{0}

We shall write L/(J = 0, U = 1) for the quotient lattice L′ described in Proposition 1.3. Let ψ : L → L′ be the canonical surjection. If I is the ideal ψ^{−1}(0) and F the filter ψ^{−1}(1), we say that the ideal I and the filter F are conjugate. By the previous proposition, an ideal I and a filter F are conjugate if, and only if, we have:

[I_{0} ∈ P_{f}(I), F_{0} ∈ P_{f}(F), (x, F_{0} ⊢ I_{0})] ⟹ x ∈ I  and
[I_{0} ∈ P_{f}(I), F_{0} ∈ P_{f}(F), (F_{0} ⊢ x, I_{0})] ⟹ x ∈ F.

This can also be formulated as follows:

(f ∈ F, x ∧ f ∈ I) ⟹ x ∈ I and (j ∈ I, x ∨ j ∈ F) ⟹ x ∈ F.

When an ideal I and a filter F are conjugate, we have

1 ∈ I ⟺ 0 ∈ F ⟺ (I, F) = (L, L).

We shall also write L/(I, F) for L′ = L/(J = 0, U = 1) . By Proposition 1.3, a homomorphism φ from L to another lattice L_{1} satisfying φ(J) = {0} and φ(U) = {1} can be factorised in an unique way through the quotient L′.

As shown by the example of totally ordered sets a quotient of distributive lattices is not in general characterised by the equivalence classes of 0 and 1.

Classically a prime ideal I of a lattice is an ideal whose complement F is a filter (which is then a prime filter). This can be expressed by

1 ∉ I  and (x ∧ y) ∈ I ⟹ (x ∈ I or y ∈ I)  (∗)

which can also be expressed by saying that I is the kernel of a morphism from L into the lattice with two elements written 2. Constructively, at least in the case where L is discrete, it seems natural to take the definition (∗), where “or” is used constructively. The notion of prime filter is then defined in a dual way.

Definition 1.4 Let L be a distributive lattice.

• An idealistic prime in L is given by a pair (J, U) of finite subsets of L. We consider this as an incomplete specification for a prime ideal P satisfying J ⊆ P and U ∩ P = ∅.

• To any idealistic prime (J, U) we can associate a pair (I, F) as described in Proposition 1.3 where I is an ideal, F is a filter and I, F are conjugate.

• We say that the idealistic prime (J, U) collapses iff we have I = F = L. This means that the quotient lattice L′ = T/(J = 0, U = 1) is a singleton i.e., 1 ≤_{L′} 0, which means also U ⊢ J.

Theorem 1.5 (Simultaneous collapse for idealistic primes) Let (J, U) be an idealistic prime for a lattice L and x be an element of L. If the idealistic primes (J ∪ {x}, U) and (J, U ∪ {x}) collapse, then so does (J, U).

Proof. We have two finite subsets J_{0}, J_{1} of J and two finite subsets U_{0}, U_{1} of U such that

x, U_{0} ⊢ J_{0} and U_{1} ⊢ x, J_{1}

hence

x, U_{0}, U_{1} ⊢ J_{0}, J_{1} and U_{0}, U_{1} ⊢ x, J_{0}, J_{1}

By the cut rule

U_{0}, U_{1} ⊢ J_{0}, J_{1}
□

Notice the crucial role of the cut rule.

1.2 Distributive lattices and entailment relations

An interesting way to analyse the description of distributive lattices defined by generators and relations is to consider the relation A ⊢ B defined on the set P_{f}(L) of finite subsets of a lattice L. Indeed if S ⊆ L generates the lattice L, then the relation ⊢ on P_{f}(S) is enough to characterise the lattice L, because any formula on S can be rewritten, in normal conjunctive form (inf of sups in S) and normal disjunctive form (sup of infs in S). Hence if we want to compare two elements of the lattice generated by S we write the first in normal disjunctive form, the second in normal conjunctive form, and we notice that

⋁_{i∈I} (⋀ A_{i}) ≤ ⋀_{j∈J} (⋁ B_{j})  ⟺  &_{(i,j)∈I×J} (A_{i} ⊢ B_{j})

Definition 1.6 For an arbitrary set S, a relation over P_{f}(S) which is reflexive, monotone and transitive (see page 6) is called an entailment relation.

The notion of entailment relations goes back to Gentzen sequent calculus, where the rule (T) (the cut rule) is first explicitly stated, and plays a key role. The connection with distributive lattices has been emphasized in [3, 4]. The following result (cf. [3]) is fundamental. It says that the three properties of entailment relations are exactly the ones needed in order to have a faithfull interpretation in distributive lattices.

Theorem 1.7 (fundamental theorem of entailment relations) Let S be a set with an entailment relation ⊢_{S} over P_{f}(S). Let L be the lattice defined by generators and relations as follows: the generators are the elements of S and the relations are

⋀ A ≤ ⋁ B

whenever A ⊢_{S} B. For any finite subsets A and B of S we have

A ⊢_{L} B ⟺ A ⊢_{S} B.

Proof. We give an explicit possible description of the lattice L. The elements of L are represented by finite sets of finite sets of elements of S

X = {A_{1}, …, A_{n}}

(intuitively X represents ⋀ A_{1} ∨ ⋯ ∨ ⋀ A_{n}). We define then inductively the relation A ≺ Y with A ∈ P_{f}(S) and Y ∈ L (intuitively ⋀ A ≤ ⋁_{C∈Y} (⋀ C))

• if B ∈ Y and B ⊆ A then A ≺ Y

• if A ⊢_{S} y_{1}, …, y_{m} and A, y_{j} ≺ Y for j = 1, …, m then A ≺ Y

It is easy to show that if A ≺ Y and A ⊆ A′ then we have also A′ ≺ Y. It follows that A ≺ Z holds whenever A ≺ Y and B ≺ Z for all B ∈ Y. We can then define X ≤ Y by A ≺ Y for all A ∈ X and one can then check that L is a distributive lattice^{1} for the operations

0 = ∅, 1 = {∅}, X ∨ Y = X ∪ Y, X ∧ Y = {A ∪ B | A ∈ X, B ∈ Y }.

For establishing this one first show that if C ≺ X and C ≺ Y we have C ≺ X ∧ Y by induction on the proofs of C ≺ X and C ≺ Y. We notice then that if A ⊢_{S} y_{1}, …, y_{m} and A, y_{j} ⊢_{S} B for all j then A ⊢_{S} B using m times the cut rule. It follows that if we have A ⊢_{L} B, i.e., A ≺ {{b} | b ∈ B}, then we have also A ⊢_{S} B. □

1 L is actually the quotient of P_{f}(P_{f}(S)) by the equivalence relation: X ≤ Y and Y ≤ X.

As a first application, we give the description of the Boolean algebra generated by a distributive lattice. A Boolean algebra can be seen as a distributive lattice with a complement operation x ↦ \overline{x} such that x ∧ \overline{x} = 0 and x ∨ \overline{x} = 1. The application x ↦ \overline{x} is then a map from the lattice to its dual.

Proposition 1.8 Let L be a distributive lattice. There exists a free Boolean algebra generated by L. It can be described as the distributive lattice generated by the set L_{1} = L ∪ \overline{L} (^{2}) with the entailment relation ⊢_{L_{1}} defined as follows: if A, B, A′, B′ are finite subsets of L we have

A, \overline{B} ⊢_{L_{1}} A′, \overline{B′} ⟺^{def} A, B′ ⊢ A′, B in L

If we write L_{Bool} for this lattice (which is a Boolean algebra), there is a natural embedding of L_{1} in L_{Bool} and the entailment relation of L_{Bool} induces on L_{1} the relation ⊢_{L_{1}} .

Proof. See [3]. □

Notice that by Theorem 1.7 we have x ⊢_{L} y if, and only if, x ⊢_{L_{1}} y hence the canonical map L → L_{1} is one-to-one and L can be identified to a subset of L_{1}.

1.3 Spectrum and completeness theorem

The spectrum of the lattice L, written Spec(L) is defined as the set Hom(L, 2). It is isomorphic to the ordered set of all detachable prime ideals. The order relation is then reverse inclusion. We have Spec(2) ≃ 1, Spec(3) ≃ 2, Spec(4) ≃ 3, etc…

Proposition 1.9 The completeness theorem implies the following result. If (J, U) is an idealistic prime which does not collapse then there exists φ ∈ Spec(L) such that J ⊆ φ^{−1}(0) and U ⊆ φ^{−1}(1). In particular if a ≰ b, there exists φ ∈ Spec(L) such that φ(a) = 1 and φ(b) = 0. Also, if L ≠ 1, Spec(L) is nonempty.

Proof. This follows from the completeness theorem for geometric theories (see Appendix). □

A corollary is the following representation theorem (Birkhoff theorem)

Theorem 1.10 (Representation theorem) The completeness theorem implies the following result. The map θ_{L} : L → P(Spec(L)) defined by a ↦ {φ ∈ Spec(L) ; φ(a) = 1} is an injective map of distributive lattice. This means that any distributive lattice can be represented as a lattice of subsets of a set.

Another corollary is the following proposition.

Proposition 1.11 The completeness theorem implies the following result. Let φ : L → L′ a map of distributive lattices; φ is injective if, and only if, Spec(φ) : Spec(L′) → Spec(L) is surjective.

2 \overline{L} is a disjoint copy of L.

Proof. We have the equivalence

a ≠ b ⟺ a ∧ b ≠ a ∨ b ⟺ a ∨ b ≰ a ∧ b

Assume that Spec(φ) is surjective. If a ≠ b in L, take a′ = φ(a), b′ = φ(b) and let ψ ∈ Spec(L) be such that ψ(a ∨ b) = 1 and ψ(a ∧ b) = 0. Since Spec(φ) is surjective there exists ψ′ ∈ Spec(L′) such that ψ = ψ′φ hence ψ′(a′ ∨ b′) = 1 and ψ′(a′ ∧ b′) = 0, hence a′ ∨ b′ ≰ a′ ∧ b′ and a′ ≠ b′. Suppose that φ is injective. We identify L to a sublattice of L′. If ψ ∈ Spec(L), take I = ψ^{−1}(0) and F = ψ^{−1}(1). By the compactness theorem (see appendix), there exists ψ′ ∈ Spec(L′) such that ψ′(I) = 0 and ψ′(F) = 1, which means ψ = ψ′ ∘ φ. □

Of course, these three last results are hard to interpret in a computational way. An intuitive interpretation is that we can proceed “as if” any distributive lattice is a lattice of subsets of a set. The goal of Hilbert’s program is to give a precise meaning to this sentence, and explain what is meant by “as if” there.

2 Krull dimension of distributive lattices

2.1 Definition of Kr_{ℓ}(L)

To develop a suitable constructive theory of the Krull dimension of a distributive lattice we have to find a constructive counterpart of the notion of increasing chains of prime ideals.

Definition 2.1 To any distributive lattice L and ℓ ∈ ℕ we associate a distributive lattice Kr_{ℓ}(L) which is the lattice defined by the generators φ_{i}(x) for i ≤ ℓ and x ∈ L (thus we have ℓ+1 disjoint copies of L and we let φ_{i} be the bijection between L and the ith copy) and relations

• ⊢ φ_{i}(1)

• φ_{i}(0) ⊢

• φ_{i}(a), φ_{i}(b) ⊢ φ_{i}(a ∧ b)

• φ_{i}(a ∨ b) ⊢ φ_{i}(a), φ_{i}(b)

• φ_{i}(a) ⊢ φ_{i}(b) whenever a ≤ b in L

• φ_{i+1}(a) ⊢ φ_{i}(a) for i < ℓ

Let S be the disjoint union ⋃ φ_{i}(L) and ⊢_{S} the entailment relation generated by these relations.

From this definition, we get directly the following theorem.

Theorem 2.2 The maps φ_{i} are morphisms from the lattice L to the lattice Kr_{ℓ}(L). Furthermore the lattice Kr_{ℓ}(L) with the maps φ_{i} is then a solution of the following universal problem: to find a distributive lattice K and ℓ + 1 homomorphisms φ_{0} ≥ φ_{1} ≥ ⋯ ≥ φ_{ℓ} from L to K such that, for any lattice L′ and any morphism ψ_{0} ≥ ψ_{1} ≥ ⋯ ≥ ψ_{ℓ} ∈ Hom(L, L′) we have one and only one morphism η : K → L′ such that ηφ_{0} = ψ_{0}, ηφ_{1} = ψ_{1}, …, ηφ_{ℓ} = ψ_{ℓ}.

The next theorem is the main result of this paper, and uses crucially the notion of entailment relation.

Theorem 2.3 If U_{i} and J_{i} (i = 0, …, ℓ) are finite subsets of L we have in Kr_{ℓ}(L)

φ_{0}(U_{0}) ∧ … ∧ φ_{ℓ}(U_{ℓ}) ≤ φ_{0}(J_{0}) ∨ … ∨ φ_{ℓ}(J_{ℓ})

if, and only if,

φ_{0}(U_{0}), …, φ_{ℓ}(U_{ℓ}) ⊢_{S} φ_{0}(J_{0}), …, φ_{ℓ}(J_{ℓ})

if, and only if, there exist x_{1}, …, x_{ℓ} ∈ L such that (where ⊢ is the entailment relation of L):

\begin{aligned}x_{1}, U_{0} & ⊢ & J_{0} \\ x_{2}, U_{1} & ⊢ & J_{1}, x_{1} \\ ⋮ & ⋮ & ⋮ \\ x_{ℓ}, U_{ℓ−1} & ⊢ & J_{ℓ−1}, x_{ℓ−1} \\ U_{ℓ} & ⊢ & J_{ℓ}, x_{ℓ} \end{aligned}

Proof. The equivalence between the first and the second statement follows from Theorem 1.7.

We show next that the relation on P_{f}(S) described in the statement of the theorem is indeed an entailment relation. The only point that needs explanation is the cut rule. To simplify notations, we take ℓ = 3. We have then 3 possible cases, and we analyse only one case, where X, φ_{1}(z) ⊢_{S} Y and X ⊢_{S} Y, φ_{1}(z), the other cases being similar. By hypothesis we have x_{1}, x_{2}, x_{3}, y_{1}, y_{2}, y_{3} such that

\begin{aligned}x_{1}, U_{0} & ⊢ & J_{0} & y_{1}, U_{0} & ⊢ & J_{0} \\ x_{2}, U_{1}, z & ⊢ & J_{1}, x_{1} & y_{2}, U_{1} & ⊢ & J_{1}, y_{1}, z \\ x_{3}, U_{2} & ⊢ & J_{2}, x_{2} & y_{3}, U_{2} & ⊢ & J_{2}, y_{2} \\ U_{3} & ⊢ & J_{3}, x_{3} & U_{3} & ⊢ & J_{3}, y_{3} \end{aligned}

The two entailment relations on the second line give

x_{2}, y_{2}, U_{1}, z ⊢ J_{1}, x_{1}, y_{1}  x_{2}, y_{2}, U_{1} ⊢ J_{1}, x_{1}, y_{1}, z

hence by cut

x_{2}, y_{2}, U_{1} ⊢ J_{1}, x_{1}, y_{1}

i.e.,

x_{2} ∧ y_{2}, U_{1} ⊢ J_{1}, x_{1} ∨ y_{1}

Finally, using distributivity

\begin{aligned}(x_{1} ∨ y_{1}), U_{0} & ⊢ & J_{0} \\ (x_{2} ∧ y_{2}), U_{1} & ⊢ & J_{1}, (x_{1} ∨ y_{1}) \\ (x_{3} ∧ y_{3}), U_{2} & ⊢ & J_{2}, (x_{2} ∧ y_{2}) \\ U_{3} & ⊢ & J_{3}, (x_{3} ∧ y_{3}) \end{aligned}

and hence φ_{0}(U_{0}), …, φ_{3}(U_{3}) ⊢_{S} φ_{0}(J_{0}), …, φ_{3}(J_{3}). Finally it is left to notice that the entailment relation we have defined is clearly the least possible relation ensuring the φ_{i} to form a non-increasing chain of morphisms. □

Notice that the morphisms φ_{i} are injective: it is easily seen that for a, b ∈ L the relation φ_{i}(a) ⊢_{S} φ_{i}(b) implies a ⊢ b, and hence that φ_{i}(a) = φ_{i}(b) implies a = b.

2.2 Partially specified chains of prime ideals

Definition 2.4 In a distributive lattice L, a partial specification for a chain of prime ideals (that we shall call idealistic chain) is defined as follows. An idealistic chain of length ℓ is a list of ℓ + 1 idealistic primes of L: C = ((J_{0}, U_{0}), …, (J_{ℓ}, U_{ℓ})). An idealistic chain of length 0 is nothing but an idealistic prime.

We think of an idealistic chain of length ℓ as a partial specification of an increasing chains of prime ideals P_{0}, …, P_{ℓ} such that J_{i} ⊆ P_{i}, U_{i} ∩ P_{i} = ∅, (i = 0, …, ℓ).

Definition 2.5 We say that an idealist chain ((J_{0}, U_{0}), …, (J_{ℓ}, U_{ℓ})) collapses if, and only if, we have in Kr_{ℓ}(L)

φ_{0}(U_{0}), …, φ_{ℓ}(U_{ℓ}) ⊢_{S} φ_{0}(J_{0}), …, φ_{ℓ}(J_{ℓ})

Thus an idealistic chain ((J_{0}, U_{0}), …, (J_{ℓ}, U_{ℓ})) collapses in L if, and only if, the idealistic prime P = (φ_{0}(J_{0}), …, φ_{ℓ}(J_{ℓ}); φ_{0}(U_{0}), …, φ_{ℓ}(U_{ℓ})) collapses in Kr_{ℓ}(L). From the completeness theorem we deduce the following result which justifies this idea of partial specification.

Theorem 2.6 (formal Nullstellensatz for chains of prime ideals) The completeness theorem implies the following result. Let L be a distributive lattice and ((J_{0}, U_{0}), …, (J_{ℓ}, U_{ℓ})) be an idealistic chain in L. The following properties are equivalent:

(a) There exist ℓ + 1 prime ideals P_{0} ⊆ ⋯ ⊆ P_{ℓ} such that J_{i} ⊆ P_{i}, U_{i} ∩ P_{i} = ∅, (i = 0, …, ℓ).

(b) The idealistic chain does not collapse.

Proof. If (b) holds then the idealistic prime P = (φ_{0}(J_{0}), …, φ_{ℓ}(J_{ℓ}); φ_{0}(U_{0}), …, φ_{ℓ}(U_{ℓ})) does not collapse in Kr_{ℓ}(L). It follows then from Proposition 1.9 that there exists σ ∈ Spec( Kr_{ℓ}(L)) such that σ is 0 on φ_{0}(J_{0}), …, φ_{ℓ}(J_{ℓ}) and 1 on φ_{0}(U_{0}), …, φ_{ℓ}(U_{ℓ})). We can then take P_{i} = (σ ∘ φ_{i})^{−1}(0). That (a) implies (b) is direct. □

2.3 Krull dimension of a distributive lattice

Definition 2.7

1) An elementary idealistic chain in a distributive lattice L is an idealistic chain of the form

((0, x_{1}), (x_{1}, x_{2}), …, (x_{ℓ}, 1))

(with x_{i} in L).

2) A distributive lattice L is of dimension ≤ ℓ−1 iff it satisfies one of the equivalent conditions

– Any elementary idealistic chain of length ℓ collapses.

– For any sequence x_{1}, …, x_{ℓ} ∈ L we have

φ_{0}(x_{1}), …, φ_{ℓ−1}(x_{ℓ}) ⊢ φ_{1}(x_{1}), …, φ_{ℓ}(x_{ℓ})

in Kr_{ℓ}(L),

The following result shows that this definition coincides with the classical definition of Krull dimension for lattices.

Theorem 2.8 The completeness theorem implies that the Krull dimension of a lattice L is ≤ ℓ − 1 if, and only if, there is no strictly increasing chains of prime ideals of length ℓ.

Using Theorem 2.3, we get the following characterisation.

Theorem 2.9 A distributive lattice L is of Krull dimension ≤ ℓ − 1 if, and only if, for all x_{1}, …, x_{ℓ} ∈ L there exist a_{1}, …, a_{ℓ} ∈ L such that

a_{1} ∧ x_{1} = 0, a_{2} ∧ x_{2} ≤ a_{1} ∨ x_{1}, , …, a_{ℓ} ∧ x_{ℓ} ≤ a_{ℓ−1} ∨ x_{ℓ−1}, 1 = a_{ℓ} ∨ x_{ℓ}

In this way we have given a concrete form of the statement that the distributive lattice L has a dimension ≤ ℓ − 1 in the form of an existence of a sequence of inequalities.

In particular the distributive lattice L is of dimension ≤ −1 if, and only if, 1 = 0 in L, and it is of dimension ≤ 0 if, and only if, L is a Boolean algebra (any element has a complement).

We have furthermore.

Lemma 2.10 A distributive lattice L generated by a set G is of dimension ≤ ℓ − 1 if, and only if, for any sequence x_{1}, …, x_{ℓ} ∈ G

φ_{0}(x_{1}), …, φ_{ℓ−1}(x_{ℓ}) ⊢ φ_{1}(x_{1}), …, φ_{ℓ}(x_{ℓ})

in Kr_{ℓ}(L).

Indeed using distributivity, one can deduce

a ∨ a′, A ⊢ b ∨ b′, B a ∧ a′, A ⊢ b ∧ b′, B

from a, A ⊢ b, B and a′, A ⊢ b′, B. Furthermore any element of L is an inf of sups of elements of G.

2.4 Implicative lattice

A lattice L is said to be an implicative lattice [5] or Heyting algebra [8] if, and only if, there is a binary operation → such that

a ∧ b ≤ c ⟺ a ≤ b → c

Theorem 2.11 If L is an implcative lattice, we have in Kr_{ℓ}(L)

φ_{0}(U_{0}), …, φ_{ℓ}(U_{ℓ}) ⊢_{S} φ_{0}(J_{0}), …, φ_{ℓ}(J_{ℓ})

if, and only if,

1 = u_{ℓ} → (j_{ℓ} ∨ (u_{ℓ−1} → (j_{ℓ−1} ∨ … (u_{0} → j_{0}))))

where u_{j} = ∧U_{j} and j_{k} = ∨J_{k}.

In the case where L is an implicative lattice, we can write explicitely that L is of dimension ≤ ℓ − 1 as an identity. For instance that L is of dimension ≤ 0 is equivalent to the identity

1 = x ∨ ¬x

where ¬x = x → 0 and that L is of dimension ≤ 1 is equivalent to the identity

1 = x_{2} ∨ (x_{2} → (x_{1} ∨ ¬x_{1}))

and so on.

Corollary 2.12 An implicative lattice L is of dimension ≤ ℓ−1 if, and only if, for any sequence x_{1}, …, x_{ℓ}

1 = x_{ℓ} ∨ (x_{ℓ} → … (x_{2} ∨ (x_{2} → (x_{1} ∨ ¬x_{1}))) …)

2.5 Decidability

To any distributive lattice L we have associated a family of distributive lattices Kr_{ℓ}(L) with a complete description of their ordering. A lattice is discrete if, and only if, its ordering is decidable, which means intuitively that there is an algorithm to decide the ordering (or, equivalently, the equality) in this lattice. It should be intuitively clear that we could find a discrete lattice L such that Kr_{1}(L) is not discrete since, by 2.3, the ordering on Kr_{1}(L) involves an existential quantification on the set L, that may be infinite (this point is discussed in [1], with another argument). However we can use the characterisation of Theorem 2.3 to give a general sufficient condition ensuring that all Kr_{ℓ}(L) are discrete.

Theorem 2.13 Suppose that the lattice L is a discrete implicative lattice then each Kr_{ℓ}(L) is discrete.

Proof. This is direct from Theorem 2.11. □

2.6 Dimension of Spectral Spaces

This subsection is written from a classical point of view. Following [7], a topological space X is called a spectral space if it satisfies the following conditions: (a) X is a compact T_{0}- space; (b) X has a compact open basis which is closed under finite intersections; (c) each irreducible closed subspace of X has a generic point. Spec(R), with the Zariski topology, is spectral for any commutative ring R with identity. Similary, if we take for basic open the sets U_{a} = {ϕ ∈ Spec(L) | ϕ(a) = 1} then Spec(L) is spectral for any distributive lattice. The compact open subsets of a spectral space form a distributive lattice, and it is well-known [21, 8] that, if L is an arbitrary distributive lattice, then L is isomorphic to the lattice of compact open subsets of the space Spec(L).

If U, V are open subsets of a topological space X we define U → V to be the largest open W such that W ∩ U ⊆ V and ¬ U = U → ∅. In a classical setting a spectral space X is said to be of dimension ≤ ℓ − 1 if, and only if, there is no strictly increasing chains of length ℓ of irreducible closed subsets of X. We can reformulate Theorem 2.9 as follows.

Theorem 2.14 A spectral space X is of dimension ≤ ℓ − 1 if, and only if, for any compact open subsets x_{1}, …, x_{ℓ} of X

X = x_{ℓ} ∨ (x_{ℓ} → … (x_{2} ∨ (x_{2} → (x_{1} ∨ ¬x_{1}))) …)

2.7 Connections with Joyal’s definition

Let L be a distributive lattice, Joyal [6] gives the following definition of dim(L) ≤ ℓ. Let φ_{i}^{ℓ} : L → Kr_{ℓ}(L) be the ℓ + 1 universal morphisms. By universality of Kr_{ℓ+1}(L), we have ℓ + 1 morphisms σ_{i} : Kr_{ℓ+1}(L) → Kr_{ℓ}(L) such that σ_{i} ∘ φ_{j}^{ℓ+1} = φ_{j}^{ℓ} if j ≤ i and σ_{i} ∘ φ_{j}^{ℓ+1} = φ_{j−1}^{ℓ} if j > i. Joyal defines then dim(L) ≤ ℓ to mean that (σ_{0}, …, σ_{ℓ}) : Kr_{ℓ+1}(L) → Kr_{ℓ}(L)^{ℓ+1} is injective. This definition can be motivated by Proposition 1.11: the elements in the image of Sp(σ_{i}) are the chains of prime ideals (α_{0}, …, α_{ℓ}) with α_{i} = α_{i+1}, and Sp(σ_{0}, …, σ_{ℓ}) is surjective if, and only if, for any chain (α_{0}, …, α_{ℓ}) there exists i < ℓ such that α_{i} = α_{i+1}. This means exactly that there is no nontrivial chain of prime ideals of length ℓ + 1. Using the completeness theorem, one can then see the equivalence with Definition 2.7. One could check directly this equivalence using a constructive metalanguage, but for lack of space, we shall not present here this argument. Similarly, it would be possible to establish the equivalence of our definition with the one of Español [6] (here also, this connection is clear via the completeness theorem).

3 Zariski and Krull lattice

3.1 Zariski lattice

Let R be a commutative ring. We write ⟨J⟩ or explicitly ⟨J⟩_{R} for the ideal of R generated by the subset J ⊆ R. We write M(U) for the monoid (^{3}) generated by the subset U ⊆ R. Given a commutative ring R the Zariski lattice Zar(R) has for elements the radicals of finitely generated ideals (the order relation being inclusion). It is well defined as a lattice. Indeed \sqrt{I_{1}} = \sqrt{J_{1}} and \sqrt{I_{2}} = \sqrt{J_{2}} imply \sqrt{I_{1}I_{2}} = \sqrt{J_{1}J_{2}} (which defines \sqrt{I_{1}} ∧ \sqrt{I_{2}}) and \sqrt{I_{1} + I_{2}} = \sqrt{J_{1} + J_{2}} (which defines \sqrt{I_{1}} ∨ \sqrt{I_{2}}). The Zariski lattice of R is always distributive, but may not be discrete, even if R is discrete. Nevertheless an inclusion \sqrt{I_{1}} ⊆ \sqrt{I_{2}} can always be certified in a finite way if the ring R is discrete. This lattice contains all the informations necessary for a constructive development of the abstract theory of the Zariski spectrum. We shall write ã for \sqrt{⟨a⟩}. Given a subset S of R we write S̃ for the subset of Zar(R) the elements of which are s̃ for s ∈ S. We have ã_{1} ∨ ⋯ ∨ ã_{m} = \sqrt{⟨a_{1}, …, a_{m}⟩} and ã_{1} ∧ ⋯ ∧ ã_{m} =

{ \atop a_{1} ⋯ a_{m}}.

Let U and J be two finite subsets of R, we have

Ũ ⊢_{Zar(R)} J̃ ⟺ ∏_{u∈U} u ∈ \sqrt{⟨J⟩} ⟺ M(U) ∩ ⟨J⟩ ≠ ∅

This describes completely the lattice Zar(R). More precisely we have:

Proposition 3.1 The lattice Zar(R) of a commutative ring R is (up to isomorphism) the lattice generated by (R, ⊢) where ⊢ is the least entailment relation over R such that

\begin{aligned}0 & ⊢ & & x, y & ⊢ & xy \\ & ⊢ & 1 & xy & ⊢ & x & x + y & ⊢ & x, y \end{aligned}

Proof. It is clear that the relation U ⊢ J defined by “M(U) meets ⟨J⟩” satisfies these axioms. It is also clear that the entailment relation generated by these axioms contains this relation. Let us show that this relation is an entailment relation. Only the cut rule is not obvious. Assume that M(U, a) meets ⟨J⟩ and that M(U) meets ⟨J, a⟩. There exist then m_{1}, m_{2} ∈ M(U) and k ∈ ℕ, x ∈ R such that a^{k}m_{1} ∈ ⟨J⟩ , m_{2} + ax ∈ ⟨J⟩. Eliminating a this implies that M(U) intersects ⟨J⟩ . □

We have ã = b̃ if, and only if, a divides a power of b and b divides a power of a.

Proposition 3.2 In a commutative ring R to give an ideal of the lattice Zar(R) is the same as to give a radical ideal of R. If I is a radical ideal of R one associates the ideal

I = {J ∈ Zar(R) | J ⊆ I}

of Zar(R). Conversely if I is an ideal of Zar(R) one can associate the ideal

I = ⋃_{J∈I} J = {x ∈ R | x̃ ∈ I},

which is a radical ideal of R. In this bijection the prime ideals of the ring correspond to the prime ideals of the Zariski lattice.

3 A monoid will always be multiplicative.

Proof. We only prove the last assertion. If I is a prime ideal of R, if J, J′ ∈ Zar(R) and J ∧ J′ ∈ I, let a_{1}, …, a_{n} ∈ R be some “generators” of J (i.e., J = \sqrt{⟨a_{1}, …, a_{n}⟩}) and let b_{1}, …, b_{m} ∈ R be some generators of J′. We have a_{i}b_{j} ∈ I and hence a_{i} ∈ I or b_{j} ∈ I for all i, j. It follows from this (constructively) that we have a_{i} ∈ I for all i or b_{j} ∈ I for all j. Hence J ∈ I or J′ ∈ I and I is a prime ideal of Zar(R). Conversely if I is a prime ideal of Zar(R) and if we have x̃y ∈ I then x̃ ∧ ỹ ∈ I and hence x̃ ∈ I or ỹ ∈ I. This shows that {x ∈ R | x̃ ∈ I} is a prime ideal of R. □

3.2 Krull lattices of a commutative ring

Definition 3.3 We define Kru_{ℓ}(R) := Kr_{ℓ}( Zar(R)). This is called the Krull lattice of order ℓ of the ring R. We say also that R is of Krull dimension ≤ ℓ iff the distributive lattice Zar(R) is of dimension ≤ ℓ.

Theorem 3.4 The ring R is of dimension ≤ ℓ − 1 if, and only if, for any x_{1}, …, x_{n} ∈ R we have in Kru_{ℓ}(R)

φ_{0}(x̃_{1}), …, φ_{ℓ−1}(x̃_{ℓ}) ⊢ φ_{1}(x̃_{1}), …, φ_{ℓ}(x̃_{ℓ})

Proof. This is a direct consequence of Lemma 2.10 and the fact that the elements x̃ generates Zar(R). □

Theorem 3.5 Let C = ((J_{0}, U_{0}), …, (J_{ℓ}, U_{ℓ})) be a list of ℓ + 1 pairs of finite subsets of R, the following properties are equivalent:

1. there exist j_{i} ∈ ⟨J_{i}⟩, u_{i} ∈ M(U_{i}), (i = 0, …, ℓ), such that

u_{0} · (u_{1} · (⋯ (u_{ℓ} + j_{ℓ}) + ⋯) + j_{1}) + j_{0} = 0

2. there exist L_{1}, …, L_{ℓ} ∈ Zar(R) such that in Zar(R):

\begin{aligned}L_{1}, Ũ_{0} & ⊢ & J̃_{0} \\ L_{2}, Ũ_{1} & ⊢ & J̃_{1}, L_{1} \\ ⋮ & ⋮ & ⋮ \\ L_{ℓ}, Ũ_{ℓ−1} & ⊢ & J̃_{ℓ−1}, L_{ℓ−1} \\ Ũ_{ℓ} & ⊢ & J̃_{ℓ}, L_{ℓ} \end{aligned}

3. there exist x_{1}, …, x_{ℓ} ∈ R such that (for the entailment relation described in Proposition 3.1):

\begin{aligned}x_{1}, U_{0} & ⊢ & J_{0} \\ x_{2}, U_{1} & ⊢ & J_{1}, x_{1} \\ ⋮ & ⋮ & ⋮ \\ x_{ℓ}, U_{ℓ−1} & ⊢ & J_{ℓ−1}, x_{ℓ−1} \\ U_{ℓ} & ⊢ & J_{ℓ}, x_{ℓ} \end{aligned}

Proof. It is clear that 1 entails 3: simply take

x_{ℓ} = u_{ℓ} + j_{ℓ}, x_{ℓ−1} = x_{ℓ}u_{ℓ−1} + j_{ℓ−1}, …, x_{0} = x_{1}u_{0} + j_{0}

and that 3 entails 2.

Let us prove that 2 implies 3. We assume:

\begin{aligned}L_{1}, Ũ_{0} & ⊢ & I_{0} \\ L_{2}, Ũ_{1} & ⊢ & I_{1}, L_{1} \\ Ũ_{2} & ⊢ & I_{2}, L_{2} \end{aligned}

The last line means that M(U_{2}) intersects I_{2} + L_{2} and hence I_{2} + ⟨x_{2}⟩ for some element x_{2} of L_{2}. Hence we have Ũ_{2} ⊢ I_{2}, x̃_{2}. Since x̃_{2} ≤ L_{2} in Zar(R) we have x̃_{2}, Ũ_{1} ⊢ I_{1}, L_{1}. We have then replaced L_{2} by x̃_{2}. Reasoning as previously one sees that one can replace as well L_{1} by a suitable x̃_{1}. One gets then 3.

Finally, let us show that 3 entails 1: if we have for instance

\begin{aligned}x_{1}, U_{0} & ⊢ & I_{0} \\ x_{2}, U_{1} & ⊢ & I_{1}, x_{1} \\ U_{2} & ⊢ & I_{2}, x_{2} \end{aligned}

by the last line we know that we can find y_{2} both in the monoid M_{2} = M(U_{2}) + ⟨I_{2}⟩ and in ⟨x_{2}⟩ . Since y_{2} ⊢ x_{1}

y_{2}, U_{1} ⊢ I_{1}, x_{1}

and since y_{2} ∈ M_{2} we can find y_{1} both in the monoid M_{1} = M_{2}M(U_{1}) + ⟨I_{1}⟩ and in ⟨x_{1}⟩. We have y_{1} ⊢ x_{1} and hence

y_{1}, U_{0} ⊢ I_{0}

and since y_{1} ∈ M_{1} this implies 0 ∈ M_{1}M(U_{0}) + ⟨I_{0}⟩ as desired. □

Corollary 3.6 A ring R is of Krull dimension ≤ ℓ − 1 iff for any sequence x_{1}, …, x_{ℓ} there exist a_{1}, …, a_{ℓ} ∈ R and m_{1}, …, m_{ℓ} ∈ ℕ such that

x_{1}^{m_{1}}(⋯ (x_{ℓ}^{m_{ℓ}}(1 + a_{ℓ}x_{ℓ}) + ⋯) + a_{1}x_{1}) = 0

Proof. By Theorem 3.4, we have in Kru_{ℓ}(R)

φ_{0}(x̃_{1}), …, φ_{ℓ−1}(x̃_{ℓ}) ⊢ φ_{1}(x̃_{1}), …, φ_{ℓ}(x̃_{ℓ})

we can then apply Theorem 3.5 to the elementary idealistic chain

((0, x̃_{1}), (x̃_{1}, x̃_{2}), …, (x̃_{ℓ}, 1))

and we get in this way j_{i} ∈ ⟨x_{i}⟩ , j_{0} = 0 and u_{i} ∈ M(x_{i+1}), u_{ℓ} = 1 such that

u_{0} · (u_{1} · (⋯ (u_{ℓ} + j_{ℓ}) + ⋯) + j_{1}) + j_{0} = 0

as desired. □

This concrete characterisation of the Krull dimension of a ring can be found in [14], where it is derived using dynamical methods [2].

Lemma 3.7 If R is coherent and noetherian then Zar(R) is an implicative lattice.

Proof. Let L ∈ Zar(R), radical of an ideal generated by elements y_{1}, …, y_{n} and x ∈ R, we show how to define an element x̃ → L ∈ Zar(R) such that, for any M ∈ Zar(R)

M ∧ x̃ ≤ L ⟺ M ≤ x̃ → L

For this, we consider the sequence of ideals

I_{k} = {z ∈ R | zx^{k} ∈ ⟨y_{1}, …, y_{n}⟩}

Since R is coherent, each I_{k} is finitely generated. Since furthermore R is noetherian and I_{k} ⊆ I_{k+1} the sequence I_{k} is stationary and ⋃_{k} I_{k} is finitely generated. We take for x̃ → L the radical of this ideal.

If M ∈ Zar(R) then M is the radical of an ideal generated by finitely many elements x_{1}, …, x_{m} and we can take M → L = (x̃_{1} → L) ∧ … ∧ (x̃_{m} → L). □

Corollary 3.8 If R is coherent, noetherian and strongly discrete then each lattice Kr_{n}(R) is discrete.

Proof. Using Theorem 2.13 and Lemma 3.7 we are left to show that Zar(R) is discrete. We have M ≤ L if, and only if, 1 = M → L. But to test if an element of Zar(R) is equal to the ideal ⟨1⟩ is decidable since R is strongly discrete. □

The hypotheses of this corollary are satisfied if R is a polynomial ring K[X_{1}, …, X_{n}] over a discrete field K [19].

3.3 Krull dimension of a polynomial ring over a discrete field

Let R be a commutative ring, let us say that a sequence x_{1}, …, x_{ℓ} is singular if, and only if, there exists a_{1}, …, a_{ℓ} ∈ R and m_{1}, …, m_{ℓ} ∈ ℕ such that

x_{1}^{m_{1}}(⋯ (x_{ℓ}^{m_{ℓ}}(1 + a_{ℓ}x_{ℓ}) + ⋯) + a_{1}x_{1}) = 0

A sequence is pseudo regular if, and only if, it is not singular. Corollary 3.6 can be reformulated as: a ring R is of Krull dimension ≤ ℓ − 1 if, and only if, any sequence in R of length ℓ is singular.

Proposition 3.9 Let K be a discrete field, R a commutative K-algebra, and x_{1}, …, x_{ℓ} in R algebraically dependent over K. The sequence x_{1}, …, x_{ℓ} is singular.

Proof. Let Q(x_{1}, …, x_{ℓ}) = 0 be a algebraic dependence relation over K. Let us order the nonzero monomials of Q along the lexicographic ordering. We can suppose that the coefficient of the first monomial is 1. Let x_{1}^{m_{1}}x_{2}^{m_{2}} ⋯ x_{ℓ}^{m_{ℓ}} be this momial, it is clear that Q can be written on the form

Q = x_{1}^{m_{1}} ⋯ x_{ℓ}^{m_{ℓ}} + x_{1}^{m_{1}} ⋯ x_{ℓ}^{1+m_{ℓ}}R_{ℓ} + x_{1}^{m_{1}} ⋯ x_{ℓ−1}^{1+m_{ℓ−1}}R_{ℓ−1} + ⋯ + x_{1}^{m_{1}}x_{2}^{1+m_{2}}R_{2} + x_{1}^{1+m_{1}}R_{1}

and this is the desired collapsus. □

Let us say that a ring is of dimension ℓ if it is of dimension ≤ ℓ but not of dimension ≤ ℓ−1. It follows that we have:

Theorem 3.10 Let K be a discrete field. The Krull dimension of the ring K[X_{1}, …, X_{ℓ}] is equal to ℓ.

Proof. Given Proposition 3.9 it is enough to check that the sequence (X_{1}, …, X_{ℓ}) is pseudo regular, which is direct. □

Notice that we got this basic result quite directly from the characterisation of Corollary 3.6, and that our argument is of course also valid classically (with the usual definition of Krull dimension). This contradicts the current opinion that constructive arguments are necessarily more involved than classical proofs.

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Annex: Completeness, compactness theorem, LLPO and geometric theories

A.1 Theories and models

We fix a set V of atomic propositions or propositional letters. A proposition ϕ, ψ, … is a syntactical object built from the atoms p, q, r ∈ V with the usual logical connectives

0, 1, ϕ ∧ ψ, ϕ ∨ ψ, ϕ → ψ, ¬ϕ

We let P_{V} be the set of all propositions. Let F_{2} be the Boolean algebra with two elements. A valuation is a function v ∈ F_{2}^{V} that assigns a truth value to any of the atomic propositions. Such a valuation can be extended to a map P_{V} → {0, 1}, ϕ ⟼ v(ϕ) in the expected way. A theory T is a subset of P_{V}. A model of T is a valuation v such that v(ϕ) = 1 for all ϕ ∈ T.

More generally given a Boolean algebra B we can define B-valuation to be a function v ∈ B^{V}. This can be extended as well to a map P_{V} → B, ϕ ⟼ v(ϕ). A B-model of T is a valuation v such that v(ϕ) = 1 for all ϕ ∈ T. The usual notion of model is a direct special case, taking for B the Boolean algebra F_{2}. For any theory there exists always a free Boolean algebra over which T is a model, the Lindenbaum algebra of T, which can be also be defined as the Boolean algebra generated by T, thinking of the elements of V as generators and the elements of T as relations. The theory T is formally consistent if, and only if, its Lindenbaum algebra is not trivial.

A.2 Completeness theorem

Theorem A.1 (Completeness theorem) Let T be a theory. If T is formally consistent then T has a model.

This theorem is the completeness theorem for propositional logic. Such a theorem is strongly related to Hilbert’s program, which can be seen as an attempt to replace the question of existence of model of a theory by the formal fact that this theory is not contradictory.

Let B the Lindenbaum algebra of T. To prove completeness, it is enough to find a morphism B → F_{2} assuming that B is not trivial, wich is the same as finding a prime ideal (which is then automatically maximal) in B. Thus the completeness theorem is a consequence of the existence of prime ideal in nontrivial Boolean algebra. Notice that this existence is clear in the case where B is finite, hence that the completeness theorem is direct for finite theories.

A.3 Compactness theorem

The completeness theorem for an arbitrary theory can be seen as a corollary of the following fundamental result.

Theorem A.2 (Compactness theorem) Let T be a theory. If all finite subsets of T have a model then so does T.

Suppose indeed that the compactness theorem holds, and let T be a formally consistent theory. Then an arbitrary finite subset T_{0} of T is also formally consistent. Furthermore, we have seen that this implies the existence of a model for T_{0}. It follows then from the compactness theorem that T itself has a model.

Conversely, it is clear that the compactness theorem follows from the completeness theorem, since a theory is formally consistent as soon as all its finite subsets are.

A simple general proof of the compactness theorem is to consider the product topology on {0, 1}^{V} and to notice that the set of models of a given subset of T is a closed subset. The theorem is then a corollary of the compactness of the space W := {0, 1}^{V} when compactness is expressed (in classical mathematics) as: if a family of closed subsets of W has non-void finite intersections, then its intersection is non-void.

A.4 LPO and LLPO

If V is countable (i.e., discrete and enumerable) we have the following alternative argument. One writes V = {p_{0}, p_{1}, …} and builds by induction a partial valuation v_{n} on {p_{i} | i < n} such that any finite subset of T has a model which extends v_{n}, and v_{n+1} extends v_{n}. To define v_{n+1} one first tries v_{n+1}(p_{n}) = 0. If this does not work, there is a finite subset of T such that any of its model v that extends v_{n} satisfies v(p_{n}) = 1 and one can take v_{n+1}(p_{n}) = 1.

The non-effective part of this argument is contained in the choice of v_{n+1}(p_{n}), which demands to give a gobal answer to an infinite set of (elementary) questions.

Now let us assume also that we can enumerate the infinite set T. We can then build a sequence of finite subsets of T in a nondecreasing way K_{0} ⊆ K_{1} ⊆ … such that any finite subset of T is a subset of some K_{n}. Assuming we have construct v_{n} such that all K_{j}’s have a model extending v_{n}, in order to define v_{n+1}(p_{n}) we have to give a global answer to the questions: do all K_{j}’s have a model extending v_{n+1} when we choose v_{n+1}(p_{n}) = 1 ? For each j this is an elementary question, having a clear answer. More precisely let us define g_{n} : ℕ → {0, 1} in the following way: g_{n}(j) = 1 if there is a model v_{n,j} of K_{j} extending v_{n} with v_{n,j}(p_{n}) = 1, else g_{n}(j) = 0. By induction hypothesis if g_{n}(j) = 0 then all K_{ℓ} have a model v_{n,ℓ} extending v_{n} with v_{n,ℓ}(p_{n}) = 1, and all models v_{n,ℓ} of K_{ℓ} extending v_{n} satisfy v_{n,ℓ}(p_{n}) = 1 if ℓ ≥ j. So we can “construct” inductively the infinite sequence of partial models v_{n} by using at each step the non-constructive Bishop’s principle LPO (Least Principle of Omniscience): given a function f : ℕ → {0, 1}, either f = 1 or ∃j ∈ ℕ f(j) ≠ 1. This principle is applied at step n to the function g_{n}.

In fact we can slightly modify the argument and use only a combination of Dependant Choice and of Bishop’s principle LLPO (Lesser Limited Principle of Omniscience), which is known to be strictly weaker than LPO: given two non-increasing functions g, h : ℕ → {0, 1} such that, for all j

g(j) = 1 ∨ h(j) = 1

then we have g = 1 or h = 1. Indeed let us define h_{n} : ℕ → {0, 1} in a symmetric way: h_{n}(j) = 1 if there is a model v_{n,j} of K_{j} extending v_{n} with v_{n,j}(p_{n}) = 0, else h_{n}(j) = 0. Cleraly g_{n} and h_{n} are non-increasing functions. By induction hypothesis, we have for all j g_{n}(j) = 1 ∨ h_{n}(j) = 1. So, applying LLPO, we can define v_{n+1}(p_{n}) = 1 if g_{n} = 1 and v_{n+1}(p_{n}) = 1 if h_{n} = 1. Nevertheless, we have to use dependant choice in order to make this choice inifnitely often since the answer “g = 1 or h = 1” given by the oracle LLPO may be ambiguous.

In a reverse way it is easy to see that the completeness theorem restricted to the countable case implies LLPO.

A.5 Geometric formulae and theories

What would have happened if topologies without points had been discovered before topologies with points, or if Grothendieck had known the theory of distributive lattices? (G. C. Rota [20]).

A formula is geometric if, and only if, it is built only with the connectives 0, 1, ϕ ∧ ψ, ϕ ∨ ψ from the propositional letters in V. A theory if a (propositional) geometric theory iff all the formula in T are of the form ϕ → ψ where ϕ and ψ are geometric formulae.

It is clear that the formulae of a geometric theory T can be seen as relations for generating a distributive lattice L_{T} and that the Lindenbaum algebra of T is nothing else but the free Boolean algebra generated by the lattice L_{T}. It follows from Proposition 1.8 that T is formally consistent if, and only if, L_{T} is nontrivial. Also, a model of T is nothing else but an element of Spec(L_{T}).

Theorem A.3 (Completeness theorem for geometric theories) Let T be a geometric theory. If T generates a nontrivial distributive lattice, then T has a model.

The general notion of geometric formula allows also existential quantification, but we restrict ourselves here to the propositional case. Even in this restricted form, the notion of geometric theory is fundamental. For instance, if R is a commutative ring, we can consider the theory with atomic propositions D(x) for each x ∈ R and with axioms

• D(0_{R}) → 0

• 1 → D(1_{R})

• D(x) ∧ D(y) → D(xy)

• D(xy) → D(x)

• D(x + y) → D(x) ∨ D(y)

This is a geometric theory T. The model of this theory are clearly the complement of the prime ideals. What is remarkable is that, while the existence of models of this theory is a nontrivial fact which may be dependent on set theoretic axioms (such as dependent axiom of choices) its formal consistency is completely elementary (as explained in the beginning of the section 3). This geometric theory, or the distributive lattice it generates, can be seen as a point-free description of the Zariski spectrum of the ring. The distributive lattice generated by this theory (called in this paper the Zariski lattice of R) is isomorphic to the lattice of compact open of the Zariski spectrum of R, while the Boolean algebra generated by this theory is isomorphic to the algebra of the constructible sets.