A class of quantum Zariski geometries

Boris Zilber Oxford University

17 November, 2006

1 Introduction

This paper is an attempt to understand the nature of non-classical Zariski geometries. Examples of such structures were first discovered in [HZ].

These examples showed that contrary to some expectations, one-dimensional Zariski geometries are not necessarily algebraic curves. Given a smooth algebraic curve C with a big enough group of regular automorphisms, one can produce a “smooth” Zariski curve C̃ along with a finite cover p : C̃ → C. C̃ cannot be identified with any algebraic curve because the construction produces an unusual subgroup of the group of regular automorphisms of C̃ ([HZ], section 10). The main theorem of [HZ] states that every Zariski curve has the form C̃, for some algebraic C. So, only in the limit case, when p is bijective, is the curve algebraic.

Typical example of an unusual subgroup of such a C̃ is the nilpotent group of two generators U and V with the central commutator ϵ = [U, V] of finite order N. So, the defining relations are

UV = ϵ VU, ϵ^{N} = 1.

This, of course, hints towards the known object of non-commutative geometry, the non-commutative (quantum) torus at the Nth root of unity. This observation encourages us to look for systematic links between noncommutative geometry and model theory. More specifically, we would like to give arguments towards the thesis that any non-classical Zariski geometry is in some way an object of non-commutative (quantum) geometry and the classical ones are just the limit cases of the general situation. Towards this end we carried out some analysis of the above examples in [Z2].

In this paper we attempt to give a general method which associates a “geometric object” to a typical quantum algebra. Note that this is in fact an open question. Yu. Manin mentions this foundational problem in [Man] I.1.4. Indeed, in general non-commutative geometry does not assume that one has (as is the case in commutative geometry) a procedure of getting a manifold-like structure from the algebra of “observables”, yet it is desirable both for technical and conceptual reasons. See also the survey paper [Sk]. The approach in [RVW] looks quite similar to what our paper suggests.

More specifically, we restrict ourselves with quantum algebras at roots of unity.

Strictly speaking the general notion of a quantum algebra does not exist, and we have to start our construction by introducing algebraic assumptions on A which make the desired theorem feasible.

The next step, after proving that the geometric object we obtain has the right properties, would be to check if our assumptions cover all interesting cases. If it were the case our assumptions would deserve the status of a definition of a quantum algebra.

Our construction always produces a Zariski geometry and when the algebra in question is big enough the structure is provably non-classical, that is not an object of (commutative) algebraic geometry. This might be seen as a good criterion for the adequacy of the construction. Among the structures which satisfy our assumptions is the quantum group U_{ϵ}(𝔰l_{2}), but we couldn’t check it for higher-dimensional objects because of algebraic difficulties.

In more detail, we consider F-algebras A over an algebraically closed field F. Our assumptions imply that a typical irreducible A-module is of finite dimension over F.

We introduce the structure associated with A as a two-sorted structure (Ṽ, F) where F is given with the usual field structure and Ṽ is the bundle over an affine variety V of A-modules of a fixed finite F-dimension N. Again by the assumptions the isomorphism types of N-dimensional A-modules are determined by points in V. “Inserting” a module M_{m} of the corresponding type in each point m of V we get

Ṽ = ∐_{m∈V} M_{m}.

In fact, for any m belonging to an open subset of V, the module M_{m} is irreducible.

Our language contains a function symbol U_{i} acting on each M_{m} (and so on the sort Ṽ) for each generator U_{i} of the algebra A. We also have the binary function symbol for the action of F by scalar multiplication on the modules. Since M_{m} may be considered an A/Ann M_{m}-module we have the bundle of finite-dimensional algebras A/Ann M_{m}, m ∈ V, represented in Ṽ. In typical cases the intersection of all such annihilators is 0. As a consequence of this, the algebra A is faithfully represented by its action on the bundle of modules. In fact the whole construction of the structure is aiming to present the category of all finite dimensional A-modules.

We write down our description of Ṽ as the set of first-order axioms Th(A-mod).

We prove two main theorems.

Theorem A (Sections 2.0.4 and 3.0.7) The theory Th(A-mod) is categorical in uncountable cardinals and model complete.

Theorem B (Section 4.0.13) Ṽ is a Zariski geometry in both sorts.

Theorem A is rather easy to prove, and in fact the proof uses not all of the assumptions on A we assumed. Yet despite the apparent simplicity of the construction, for certain A, Ṽ is not definable in an algebraically closed field, that is Ṽ(A) is not classical.

Theorem B requires much more work, mainly the analysis of definable sets. This is due to the fact that the theory of Ṽ, unlike the case of Zariski geometries coming from algebraic geometry, does not have quantifier elimination in the natural algebraic language. We hope that this technical analysis will be instrumental in practical applications to noncommutative geometry.

Acknowledgement I would like to thank Shahn Majid for very helpful discussions.

I started to work on the paper while I was a member of Model Theory and Applications to Algebra and Analysis programme at the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK. I am grateful to the organisers of the program, the staff of the Institute, and the participants.

2 From algebras to structures

2.0.1 We fix below until the end of the paper an F-algebra A, satisfying the following.

Assumptions.

1. We assume that F is an algebraically closed field and A is an associative unital affine F-algebra with generators U_{1}, …, U_{d} and defining relations with parameters in a finite C ⊂ F. We also assume that A is a finite dimensional module over its central subalgebra Z_{0}.

2. Z_{0} is a unital finitely generated commutative F-algebra without zero divisors, so Max Z_{0}, the space of maximal ideals of Z_{0}, can be identified with the F-points of an irreducible affine algebraic variety V over C.

3. There is a positive integer N such that to every m ∈ Max Z_{0} we can put in corresponds with m an A-module M_{m} of dimension N over F with the property that the maximal ideal m annihilates M_{m}.

The isomorphism type of the module M_{m} is determined uniformly by a solution to a system of polynomial equations P^{A} in variables t_{ijk} ∈ F and m ∈ V such that:

for every m ∈ V there exists t = {t_{ijk} : i ≤ d, j, k ≤ N} satisfying P^{A}(t, m) = 0 and for each such t there is a basis e(1), …, e(N) of the F-vector space on M_{m} with

⋀_{i≤d, j≤N} U_{i} e(j) = ∑_{k=1}^{N} t_{ijk}e(k).

We call any such basis e(1), …, e(N) canonical.

4. There is a finite group Γ and a map g : V × Γ → GL_{N}(F) such that, for each γ ∈ Γ, the map g(·, γ) : V → GL_{N}(F) is rational C-definable (defined on an open subset of V) and, for any m ∈ V,

Dom_{m}, the domain of definition of the map g(m, ·) : Γ → GL_{N}(F), is a subgroup of Γ,

g(m, ·) is an injective homomorphism on its domain, and for any two canonical bases e(1), …, e(N) and e′(1), …, e′(N) of

M_{m} there is λ ∈ F^{∗} and γ ∈ Dom_{m} such that

e′(i) = λ ∑_{1≤j≤N} g_{ij}(m, γ)e(j), i = 1, …, N.

We denote

Γ_{m} := g(m, Dom_{m}).

Remark The correspondence m ↦ M_{m} between points in V and the isomorphism types of modules is bijective by the assumption 2. Indeed, for distinct m_{1}, m_{2} ∈ Max Z_{0} the modules M_{m_{1}} and M_{m_{2}} are not isomorphic, for otherwise the module will be annihilated by Z_{0}.

2.0.2 The structure

Recall that V(A) or simply V stands for the F-points of the algebraic variety Max Z_{0}. By assumption 2.0.1.1 this can be viewed as the set of Amodules M_{m}, m ∈ Z_{0}.

Consider the set Ṽ as the disjoint union

Ṽ = ∐_{m∈V} M_{m}.

We also pick up arbitrarily for each m ∈ V a canonical basis e = {e(1), …, e(N)} in M_{m} and all the other canonical bases conjugated to e by Γ_{m}. We denote the set of bases for each m ∈ V as

E_{m} := Γ_{m}e = {(e′(1), …, e′(N)) : e′(i) = ∑_{1≤j≤N} γ_{ij}e(j), γ ∈ Γ_{m}}.

Consider, along with the sort Ṽ also the field sort F, the sort V identified with the corresponding affine subvariety V ⊆ F^{k}, some k, and the projection map

π : x ↦ m if x ∈ M_{m} from Ṽ to V.

We assume the full language of Ṽ contains:

1. the ternary relation S(x, y, z) which holds if and only if there is m ∈ V such that x, y, z ∈ M_{m} and x + y = z in the module;

2. the ternary relation a · x = y which for a ∈ F and x, y ∈ M_{m} is interpreted as the multiplication by the scalar a in the module M_{m}; 3. the binary relations U_{i}x = y, (i = 1, …, d) which for x, y ∈ M_{m} are

interpreted as the actions by the corresponding operators in the module M_{m};

4. the relations E ⊆ V × Ṽ^{N} with E(m, e) interpreted as e ∈ E_{m}.

The weak language is the sublanguage of the full one which includes 1-3 above only.

Finally, denote Ṽ the 3-sorted structure (Ṽ, V, F) described above, with V endowed with the usual Zariski language as the algebraic variety.

Remarks 1.Notice that the sorts V and F are bi-interpretable over C.

2. The map g : V × Γ → GL_{N}(F) being rational is definable in the weak language of Ṽ.

Now we introduce the first order theory Th(A-mod) describing (Ṽ, V, F). It consists of axioms:

Ax 1. F is an algebraically closed field of characteristic p and V is the Zariski structure on the F-points of the variety Max Z_{0}.

Ax 2. For each m ∈ V the action of scalars of F and operators U_{1}, …, U_{d} defines on π^{−1}(m) the structure of an A-module of dimension N.

Ax 3. Assumption 2.0.1.3 holds for the given P^{A}.

Ax 4. For the g : V × Γ → GL_{N}(F) given by the assumption 2.0.1.4, for any e, e′ ∈ E_{m} there exists γ ∈ Γ such that

e′(i) = ∑_{1≤j≤N} g_{ij}(m, γ)e(j), i = 1, …, N.

Moreover, E_{m} is an orbit under the action of Γ_{m}.

Remark The referee of the paper notes that if M_{m} is irreducible then associated to a particular collection of coefficients t_{kij} there is a unique (up to scalar multiplication) canonical base for M_{m} (as in 2.1.3). It follows that the only possible automorphisms of Ṽ which fix all of F are induced by multiplication by scalars in each module (the scalars do not have to be the same for each fibre, and typically are not). So the ’projective’ bundle ∐_{m∈V}(M_{m}/scalars) is internal to the field F, but the original Ṽ is not.

2.0.3 Examples We assume below that ϵ ∈ F is a primitive root of 1 of order ℓ, and ℓ is not divisible by the characteristic of F.

1. Let A be generated by U, V, U^{−1}, V^{−1} satisfying the relations

UU^{−1} = 1 = VV^{−1}, UV = ϵVU.

We denote this algebra T_{ϵ}^{2} (equivalent to O_{ϵ}((F^{×})^{2}) in the notations of [BG]).

The centre Z = Z_{0} of T_{ϵ}^{2} is the subalgebra generated by U^{ℓ}, U^{−ℓ}, V^{ℓ}, V^{−ℓ}. The variety Max Z is isomorphic to the 2-dimensional torus F^{∗} × F^{∗}.

Any irreducible T_{ϵ}^{2}-modules M is an F-vector space of dimension N = ℓ. It has a basis {e_{0}, …, e_{ℓ−1}} of the space consisting of U-eigenvectors and satisfying, for an eigenvalue μ of U and an eigenvalue ν of V,

\begin{matrix}Ue_{i} = μϵ^{i}e_{i} \\ Ve_{i} = \{ \begin{matrix}νe_{i+1}, i < ℓ − 1, \\ νe_{0}, i = ℓ − 1. \end{matrix} \end{matrix}

We also have a basis of V-eigenvectors {g_{0}, …, g_{ℓ−1}} satisfying

g_{i} = e_{0} + ϵ^{i}e_{1} + ⋯ + ϵ^{i(ℓ−1)}e_{ℓ−1}

and so

\begin{matrix}Vg_{i} = νϵ^{i}g_{i} \\ Ug_{i} = \{ \begin{matrix}μg_{i+1}, i < ℓ − 1, \\ μg_{0}, i = ℓ − 1. \end{matrix} \end{matrix}

For μ^{ℓ} = a ∈ F^{∗} and ν^{ℓ} = b ∈ F^{∗}, (U^{ℓ} − a), (V^{ℓ} − b) are generators of Ann(M). The module is determined uniquely once the values of a and b are given. So, V is isomorphic to the 2-dimensional torus F^{∗} × F^{∗}.

The coefficients t_{ijk} in this example are determined by μ and ν, which satisfy the polynomial equations μ^{ℓ} = a, ν^{ℓ} = b.

Γ_{m} = Γ is the fixed nilpotent group of order ℓ^{3} generated by the matrices

( \begin{matrix}0 1 0 … 0 \\ 0 0 1 … 0 \\ … & … \\ 1 0 … 0 \end{matrix} ) and ( \begin{matrix}1 0 0 … 0 \\ 0 ϵ 0 … 0 \\ … & … \\ 0 0 … ϵ^{ℓ−1} \end{matrix} )

2. Similarly, the d-dimensional quantum torus T_{ϵ,θ}^{d} generated by U_{1}, …, U_{d}, U_{1}^{−1} …, U_{d}^{−1} satisfying

U_{i}U_{i}^{−1} = 1, U_{i}U_{j} = ϵ^{θ_{ij}}U_{j}U_{i}, 1 ≤ i, j ≤ d,

where θ is an antisymmetric integer matrix, g.c.d.{θ_{ij} : 1 ≤ j ≤ d}) = 1 for some i ≤ d.

There is a simple description of the bundle of irreducible modules all of which are of the same dimension N = ℓ.

T_{ϵ,θ}^{d} satisfies all the assumptions.

3. A = U_{ϵ}(𝔰l_{2}), the quantum universal enveloping algebra of SL_{2}(F). It is given by generators K, K^{−1}, E, F satisfying the defining relations

KK^{−1} = 1, KEK^{−1} = ϵ^{2}E, KF K^{−1} = ϵ^{−2}F, EF − F E = \frac{K − K^{−1}}{ϵ − ϵ^{−1}} .

The centre Z of U_{ϵ}(𝔰l_{2}) is generated by K^{ℓ}, E^{ℓ}, F^{ℓ} and the element

C = F E + \frac{Kϵ + K^{−1}ϵ^{−1}}{(ϵ − ϵ^{−1})^{2}} .

We use [BG], Chapter III.2, to describe Ṽ. We assume ℓ ≥ 3 odd.

Let Z_{0} = Z and so V = Max Z is an algebraic extension of degree ℓ of the commutative affine algebra K^{ℓ}, K^{−ℓ}, E^{ℓ}, F^{ℓ}.

To every point m = (a, b, c, d) ∈ V corresponds the unique, up to isomorphism, module with a canonical basis e_{0}, …, e_{ℓ−1} satisfying

Ke_{i} = μϵ^{−2i}e_{i},

F e_{i} = \{ \begin{matrix}e_{i+1}, i < ℓ − 1, \\ be_{0}, i = ℓ − 1, \end{matrix}

Ee_{i} = \{ \begin{matrix}ρe_{ℓ−1}, i = 0, \\ (ρb + \frac{(ϵ^{i}−ϵ^{−i})(μϵ^{1−i}−μ^{−1}ϵ^{i−1})}{(ϵ−ϵ^{−1})^{2}})e_{i−1}, i > 0. \end{matrix}

where μ, ρ satisfy the polynomial equations

μ^{ℓ} = a, ρb + \frac{μϵ + μ^{−1}ϵ^{−1}}{(ϵ − ϵ^{−1})^{2}} = d  (1)

and

ρ ∏_{i=1}^{ℓ−1} (ρb + \frac{(ϵ^{i} − ϵ^{−i})(μϵ^{1−i} − μ^{−1}ϵ^{i−1})}{(ϵ − ϵ^{−1})^{2}}) = c.  (2)

We may characterise V as

V = {(a, b, c, d) ∈ F^{4} : ∃ ρ, μ (1) and (2) hold }.

In fact, the map (a, b, c, d) ↦ (a, b, c) is a cover of the affine variety A^{3} ∩{a ≠ 0} of order ℓ.

In almost all points of V, except for the points of the form (1, 0, 0, d_{+}) and (−1, 0, 0, d_{−}), the module is irreducible. In the exceptional cases, for each i ∈ {0, …, ℓ − 1} we have exactly one ℓ-dimensional module (denoted Z_{0}(ϵ^{i}) or Z_{0}(−ϵ^{i}) in [BG], depending on the sign) which satisfies the above description with μ = ϵ^{i} or −ϵ^{i}. The Casimir invariant is

d_{+} = \frac{ϵ^{i+1} + ϵ^{−i−1}}{(ϵ − ϵ^{−1})^{2}} or d_{−} = −\frac{ϵ^{i+1} + ϵ^{−i−1}}{(ϵ − ϵ^{−1})^{2}}

and the module, for i < ℓ − 1, has the unique proper irreducible submodule of dimension ℓ − i − 1 spanned by e(i + 1), …, e(ℓ − 1). For i = ℓ − 1 the module is irreducible. According to [BG],III.2 all the irreducible modules of A have been listed above, either as M_{m} or as submodules of M_{m} for the exceptional m ∈ V.

To describe Γ_{m} consider two canonical bases e and e′ in M_{m}. If e′ is not of the form λe, then necessarily e′_{0} = λe_{k}, for some k ≤ ℓ − 1, b ≠ 0 and

e′_{i} = \{ \begin{matrix}λe_{i+k}, 0 ≤ i < ℓ − k, \\ λbe_{i+k}, ℓ − 1 ≥ i ≥ ℓ − k, \end{matrix}

If we put λ = λ_{k} = ν^{−k}, for ν^{ℓ} = b, we get a finite order transformation. So we can take Γ_{(a,b,c,d)}, for b ≠ 0, to be the Abelian group of order ℓ^{2} generated by the matrices

( \begin{matrix}0 ν^{−1} 0 … 0 \\ 0 0 ν^{−1} … 0 \\ . . . . . . ν^{−1} \\ ν^{ℓ−1} 0 … 0 \end{matrix} ) and ( \begin{matrix}ϵ 0 0 … 0 \\ 0 ϵ 0 … 0 \\ … & … \\ 0 0 … ϵ \end{matrix} )

where ν is defined by

ν^{ℓ} = b.

When b = 0 the group Γ_{(a,0,c,d)} is just the cyclic group generated by the scalar matrix with ϵ on the diagonal.

The isomorphism type of the module depends on ⟨a, b, c, d⟩ only. This basis satisfies all the assumptions 1-4.

U_{ϵ}(𝔰l_{2}) is one of the simplest examples of a quantum group. Quantum groups, as all bi-algebras, have the following crucial property: the tensor product M_{1} ⊗ M_{2} of any two A-modules is well-defined and is an A-module. So, the tensor product of two modules in Ṽ produces a U_{ϵ}(𝔰l_{2})-module of dimension ℓ^{2}, definable in the structure, and which ’contains’ finitely many modules in Ṽ. This defines a multivalued operation on V (or on the open subset of V, in the second case).

More examples and the most general known cases U_{ϵ}(𝔤), for 𝔤 a semisimple complex Lie algebra, and O_{ϵ}(G), the quantised group G, for G a connected simply connected semisimple complex Lie group, are shown to have properties 1 and 2 for the central algebra Z_{0} generated by the corresponding U_{i}^{ℓ}, i = 1, …, d.

The rest of the assumptions are harder to check. We leave this open.

4. A = O_{ϵ}(F^{2}), Manin’s quantum plane is given by generators U and V and defining relations UV = ϵVU. The centre Z is again generated by U^{ℓ} and V^{ℓ} and the maximal ideals of Z in this case are of the form ⟨(U^{ℓ} − a), (V^{ℓ} − b)⟩ with ⟨a, b⟩ ∈ F^{2}.

This example, though very easy to understand algebraically, does not quite fit into our construction. Namely, the assumption 3 is satisfied only in generic points of V = Max Z. But the main statement still hold true for this case as well. We just have to construct Ṽ by glueing two Zariski spaces each corresponding to a localisation of the algebra A.

To each maximal ideal with a ≠ 0 we put in correspondence the module of dimension ℓ given in a basis e_{0}, …, e_{ℓ−1} by

\begin{matrix}Ue_{i} = μϵ^{i}e_{i} \\ Ve_{i} = \{ \begin{matrix}e_{i+1}, i < ℓ − 1, \\ be_{0}, i = ℓ − 1. \end{matrix} \end{matrix}

for μ satisfying μ^{ℓ} = a.

To each maximal ideal with b ≠ 0 we put in correspondence the module of dimension ℓ given in a basis g_{0}, …, g_{ℓ−1} by

\begin{matrix}Vg_{i} = νϵ^{i}g_{i} \\ Ug_{i} = \{ \begin{matrix}g_{i+1}, i < ℓ − 1, \\ ae_{0}, i = ℓ − 1. \end{matrix} \end{matrix}

for ν satisfying ν^{ℓ} = b.

When both a ≠ 0 and b ≠ 0 we identify the two representations of the same module by choosing g (given e and ν) so that

g_{i} = e_{0} + ν^{−1}ϵ^{i}e_{1} + ⋯ + ν^{−k}ϵ^{ik}e_{k} + ⋯ + ν^{−(ℓ−1)}ϵ^{i(ℓ−1)}e_{ℓ−1}.

This induces a definable isomorphism between modules and defines a glueing between Ṽ_{a≠0} and Ṽ_{b≠0}. In fact Ṽ_{a≠0} corresponds to the algebra given by three generators U, U^{−1} and V with relations UV = ϵVU and UU^{−1} = 1, a localisation of O_{ϵ}(F^{2}), and Ṽ_{b≠0} corresponds to the localisation by V^{−1}.

2.0.4 Categoricity

Lemma (i) Let Ṽ_{1} and Ṽ_{2} be two structures in the weak language satisfying 2.0.1.1-2.0.1.3 and 2.0.2.1-2.0.2.3 with the same P^{A} over the same algebraically closed field F. Then the natural isomorphism i : V_{1} ∪ F → V_{2} ∪ F over C can be lifted to an isomorphism

i : Ṽ_{1} → Ṽ_{2}.

(ii) Let Ṽ_{1} and Ṽ_{2} be two structures in the full language satisfying 2.0.1.1-2.0.1.4 and 2.0.2.1-2.0.2.4 with the same P^{A} over the same algebraically closed field F. Then the natural isomorphism i : V_{1} ∪ F → V_{2} ∪ F over C can be lifted to an isomorphism

i : Ṽ_{1} → Ṽ_{2}.

Proof We may assume that i is the identity on V and on the sort F. The assumptions 2.0.1 and the description 2.0.2 imply that in both struc-

tures π^{−1}(m), for m ∈ V, has the structure of a module. Denote these π_{1}^{−1}(m) and π_{2}^{−1}(m) in the first and second structure correspondingly.

For each m ∈ V the modules π_{1}^{−1}(m) and π_{2}^{−1}(m) are isomorphic.

Indeed, using 2.0.1.3 choose t_{ijk} satisfying P^{A} for m and find bases e in π_{1}^{−1}(m) and e′ in π_{2}^{−1}(m) with the U_{i}’s represented by the matrices {t_{ijk} : k, j = 1, …, N} in both modules. It follows that the map

i_{m} : ∑ z_{j}e(j) ↦ ∑ z_{j}e′(j), z_{1}, …, z_{N} ∈ F

is an isomorphism of the A-modules

i_{m} : π_{1}^{−1}(m) → π_{2}^{−1}(m).

Hence, the union

i = ⋃_{m∈V} i_{m}, i : Ṽ_{1} → Ṽ_{2},

is an isomorphism. This proves (i).

In order to prove (ii) choose, using 2.0.1.4, e and e′ in E_{m} in π_{1}^{−1}(m) and π_{2}^{−1}(m) correspondingly. Then the map i_{m} by the same assumption also preserves E_{m}, and so i is an isomorphism in the full language. □

As an immediate corollary we get

Theorem Th(A-mod) is categorical in uncountable cardinals both in the full and the weak languages.

Remark The above Lemma is a special case of Lemma 3.0.7.

2.0.5 We prove in this subsection that despite the simplicity of the construction and the proof of categoricity the structures obtained from algebras A in our list of examples are nonclassical.

Assume for simplicity that char F = 0. The statements in this subsection are in their strongest form when we choose the weak language for the structures.

Proposition Ṽ(T_{ϵ}^{n}) is not definable in an algebraically closed field, for n ≥ 2.

Proof We write A for T_{ϵ}^{2}. We consider the structure in the weak language.

Suppose towards the contradiction that Ṽ(A) is definable in some F′. Then F is also definable in this algebraically closed field. But, as is wellknown, the only infinite field definable in an algebraically closed field is the field itself. So, F′ = F and so we have to assume that Ṽ is definable in F.

Given W ∈ A, v ∈ Ṽ, x ∈ F and m ∈ V, denote Eig(W; v, x, m) the statement:

v is an eigenvector of W in π^{−1}(m) (or simply in M_{m}) with the eigenvalue x.

For any given W the ternary relation Eig(W; v, x, m) is definable in Ṽ by 2.0.2.

Let m ∈ V be such that μ is an U-eigenvalue and ν is a V-eigenvalue in the module M_{m}. ⟨μ^{ℓ}, ν^{ℓ}⟩ determines the isomorphism type of M_{m} (see 2.0.3), in fact m = ⟨μ^{ℓ}, ν^{ℓ}⟩.

Consider the definable set

Eig(U) = {v ∈ Ṽ : ∃μ, mEig(U; v, μ, m)}.

By our assumption and elimination of imaginaries in ACF this is in a definable bijection with an algebraic subset S of F^{n}, some n, defined over some finite C′. We may assume that C′ = C. Moreover the relations and functions induced from Ṽ on Eig(U) are algebraic relations definable in F over C.

Consider μ and ν as variables running in F and let F̃ = F{μ, ν} be the field of Puiseux series in variables μ, ν. Since S(F̃) as a structure is an elementary extension of Eig(U) there is a tuple, say e_{μ}, in S(F̃) which is an U-eigenvector with the eigenvalue μ.

By definition the coordinates of e_{μ} are Laurent series in the variables μ^{\frac{1}{k}} and ν^{\frac{1}{k}} , for some positive integer k. Let K be the subfield of F̃ consisting of all Laurent series in variables μ^{\frac{1}{k}} , ν^{\frac{1}{k}} , for the k above. Fix δ ∈ F such that

δ^{k} = ϵ.

The maps

ξ : t(μ^{\frac{1}{k}} , ν^{\frac{1}{k}}) ↦ t(δμ^{\frac{1}{k}} , ν^{\frac{1}{k}}) and ζ : t(μ^{\frac{1}{k}} , ν^{\frac{1}{k}}) ↦ t(μ^{\frac{1}{k}} , δν^{\frac{1}{k}}),

for t(μ^{\frac{1}{k}} , ν^{\frac{1}{k}}) Laurent series in the corresponding variables, obviously are automorphisms of K over F. In particular ξ maps μ to ϵμ and leaves ν fixed, and ζ maps ν to ϵν and leaves μ fixed. Also note that the two automorphisms commute and both are of order ℓk.

Since U is F-definable, ξ^{m}(e_{μ}) is a U-eigenvector with the eigenvalue ϵ^{m}μ, for any integer m.

By the properties of A-modules Ve_{μ} is an U-eigenvector with the eigenvalue ϵμ, so there is α ∈ F̃

Ve_{μ} = αξ(e_{μ}).  (3)

But α is definable in terms of e_{μ}, ξ(e_{μ}) and C, so by elimination of quantifiers α is a rational function of the coordinates of the elements, hence α ∈ K.

Since V is definable over F, we have for every automorphism γ of K,

γ(Ve) = Vγ(e).

So, (3) implies

Vξ^{i}e_{μ} = ξ^{i}(α)ξ^{i+1}(e_{μ}), i = 0, 1, 2, …

and, since

V^{kℓ}e_{μ} = ν^{kℓ}e_{μ},

applying V to both sides of (3) kℓ − 1 times we get

∏_{i=0}^{kℓ−1} ξ^{i}(α) = ν^{kℓ}.  (4)

Now remember that

α = a_{0}(ν^{\frac{1}{k}}) · μ^{\frac{d}{k}} · (1 + a_{1}(ν^{\frac{1}{k}})μ^{\frac{1}{k}} + a_{2}(ν^{\frac{1}{k}})μ^{\frac{2}{k}} + …)

where a_{0}(ν^{\frac{1}{k}}), a_{1}(ν^{\frac{1}{k}}), a_{2}(ν^{\frac{1}{k}}) … are Laurent series in ν^{\frac{1}{k}} and d an integer. Substituting this into (4) we get

ν^{kℓ} = a_{0}(ν^{\frac{1}{k}})^{kℓ}δ^{\frac{kℓ(kℓ−1)}{2}} μ^{dℓ} · (1 + a′_{1}(ν^{\frac{1}{k}})μ^{\frac{1}{k}} + a′_{2}(ν^{\frac{1}{k}})μ^{\frac{2}{k}} + …)

It follows that d = 0 and a_{0}(ν^{\frac{1}{k}}) = a_{0} · ν, for some constant a_{0} ∈ F. That is

α = a_{0} · ν · (1 + a_{1}(ν^{\frac{1}{k}})μ^{\frac{1}{k}} + a_{2}(ν^{\frac{1}{k}})μ^{\frac{2}{k}} + …)  (5)

Now we use the fact that ζ(e_{μ}) is an U eigenvector with the same eigenvalue μ, so by the same argument as above there is β ∈ K such that

ζ(e_{μ}) = βe_{μ}.  (6)

So,

ζ^{i+1}(e_{μ}) = ζ^{i}(β)ζ^{i}(e_{μ})

and taking into account that ζ^{kℓ} = 1 we get

∏_{i=0}^{kℓ−1} ζ^{i}(β) = 1.

Again we analyse β as a Laurent series and represent it in the form

β = b_{0}(μ^{\frac{1}{k}}) · ν^{\frac{d}{k}} · (1 + b_{1}(μ^{\frac{1}{k}})ν^{\frac{1}{k}} + b_{2}(μ^{\frac{1}{k}})ν^{\frac{2}{k}} + …)

where b_{0}(μ^{\frac{1}{k}}), b_{1}(μ^{\frac{1}{k}}), b_{2}(μ^{\frac{1}{k}}) … are Laurent series of μ^{\frac{1}{k}} and d is an integer. By an argument similar to the above using (7) we get

β = b_{0} · (1 + b_{1}(μ^{\frac{1}{k}})ν^{\frac{1}{k}} + b_{2}(μ^{\frac{1}{k}})ν^{\frac{2}{k}} + …)  (7)

for some b_{0} ∈ F.

Finally we use the fact that ξ and ζ commute. Applying ζ to (3) we get

Vζ(e_{μ}) = ζ(α)ζξ(e_{μ}) = ζ(α)ξζ(e_{μ}) = ξ(β)ζ(α)ξ(e_{μ}).

On the other hand

Vζ(e_{μ}) = βVe_{μ} = βαξ(e_{μ}).

That is

\frac{α}{ζ(α)} = \frac{ξ(β)}{β}.

Substituting (5) and (7) and dividing on both sides we get the equality

ϵ^{−1}(1 + a′_{1}(ν^{\frac{1}{k}})μ^{\frac{1}{k}} + a′_{2}(ν^{\frac{1}{k}})μ^{\frac{2}{k}} + …) = 1 + b′_{1}(μ^{\frac{1}{k}})ν^{\frac{1}{k}} + b′_{2}(μ^{\frac{1}{k}})ν^{\frac{2}{k}} + …

Compairing the constant terms on both sides we get the contradiction. This proves the proposition in the case n = 2.

To end the proof we just notice that the structure Ṽ(T_{ϵ}^{2}) is definable in any of the other Ṽ(T_{ϵ}^{n}), maybe with a different root of unity. This follows from the fact that the A-modules in all cases have similar description.□

Corollary The structure V(U_{ϵ}(𝔰l_{2})) (Example 2.0.3.3) is not definable in an algebraically closed field.

Indeed, consider

V_{0} = {(a, b, c, d) ∈ V : b ≠ 0, c = 0} and Ṽ_{0} = π^{−1}(V_{0})

with the relations induced from Ṽ.

Set U := K, V = F and consider the reduct of the structure Ṽ_{0} which ignores the operators E and C. This structure is isomorphic to Ṽ(T_{ϵ^{2}}^{2}) and is definable in V(U_{ϵ}(𝔰l_{2})), so the latter is not definable in an algebraically closed field.□

Remark Note that T_{ϵ}^{2} here does not have any immediate connection to the non-classical Zariski curve T_{N} in [Z2]. So the Proposition does not “explain” the earlier examples, though an attentive reader could spot similarities in the proof of the Proposition and that of the non-algebraicity of T_{N}. A possible connection remains an open question.

3 Definable sets

3.0.6 Given variables v_{1,1}, …, v_{1,r_{1}}, …, v_{s,1} …, v_{s,r_{s}} of the sort Ṽ, m_{1}, …, m_{s} of the sort V and variables x = {x_{1}, …, x_{p}} of the sort F, denote A_{0}(e, m, t) the formula

⋀_{i≤s, j≤N} E(e_{i}, m_{i}) & P^{A}({t_{iknℓ}}_{k≤d, ℓ,n≤N}; m_{i}) = 0 & ⋀_{k≤d,j≤N,i≤s} U_{k}e_{i}(j) = ∑_{ℓ≤N} t_{ikjℓ}e_{i}(ℓ).

Denote A(e, m, t, z, v) the formula

A_{0}(e, m, t) & ⋀_{i≤s; j≤r_{i}} v_{ij} = ∑_{ℓ≤N} z_{ijℓ}e_{i}(ℓ).

The formula of the form

∃ e_{1}, … e_{s}∃ m_{1}, …, m_{s}
∃ {t_{ikjl} : k ≤ d, i ≤ s, j, ℓ ≤ N} ⊆ F
∃ {z_{ijl} : i ≤ s, j ≤ r_{i}, ℓ ≤ N} ⊆ F :
A(e, m, t, z, v)& R(m, t, x, z),

where R is a boolean combination of Zariski closed predicates in the algebraic variety V^{s}×F^{q} over C, q = |t|+|x|+|z| (constructible predicate over C) will be called a core ∃-formula with kernel R(m, t, x, z) over C. The enumeration of variables v_{ij} will be referred to as the partitioning enumeration.

We also refer to this formula as ∃eR.

Comments (i) A core formula is determined by its kernel once the partition of variables (by enumeration) is fixed. The partition sets that π(e_{i}(j)) = π(e_{i}(k)), for every i, j, k, and fixes the components of the subformula A(e, m, t, z, v).

(ii) The relation A_{0}(e, m, t) defines the functions

e ↦ (m, t),

that is given a canonical basis {e_{i}(1), …, e_{i}(N)} in M_{m_{i}} we can uniquely determine m_{i} and t_{ikjℓ}.

For the same reason A(e, m, t, z, v) defines the functions

(e, v) ↦ (m, t, z).

3.0.7 Lemma Let

a = ⟨a_{1,1}, …, a_{1,r_{1}}, …, a_{s,1} …, a_{s,r_{s}}⟩ ∈ Ṽ × ⋯ × Ṽ, b = ⟨b_{1}, …, b_{n}⟩ ∈ F^{n}.

The complete type tp(a, b) of the tuple over C is determined by its subtype ctp(a, b) over C consisting of core ∃-formulas.

Proof We are going to prove that, given a′, b′ satisfying the same core type ctp(a, b) there is an automorphism of any ℵ_{0}-saturated model, α : (a, b) ↦ (a′, b′).

We assume that the enumeration of variables has been arranged so that π(a_{ij}) = π(a_{kn}) if and only if i = k. Denote m_{i} = π(a_{ij}).

Let e_{i} be bases of modules π^{−1}(m_{i}), i = 1, …, s, j = 1, …, N, such that ⊧ A_{0}(e, m, t) for some t = {t_{ikjℓ}} (see the notation in 3.0.6 and the assumption 2.0.1.3), in particular e_{i} ∈ E_{m_{i}}. By the assumption the correspondent systems span M_{m_{i}}, so there exist c_{ijℓ} such that

⋀_{i≤s; j≤r_{i}} a_{ij} = ∑_{ℓ≤N} c_{ijℓ}e_{i}(ℓ),

and let p = {P_{i} : i ∈ ℕ} be the complete algebraic type of (m, t, b, c).

The type ctp(a, b) contains core formulas with kernels P_{i}, i = 1, 2, … By assumptions and saturatedness we can find e′ m′, t′ and c′ satisfying the correspondent relations for (a′, b′). In particular, the algebraic types of (m, t, b, c) and (m′, t′, b′, c′) over C coincide and e′_{i} ∈ E_{m′_{i}}. It follows that there is an automorphism α : F → F over C such that α : (m, t, b, c) ↦ (m′, t′, b′, c′).

Extend α to π^{−1}(m_{1}) ∪ … ∪ π^{−1}(m_{s}) by setting

α(∑_{j} z_{j}e_{i}(j)) = ∑_{j} α(z_{j})e′_{i}(j)  (8)

for any z_{1}, …, z_{N} ∈ F and i ∈ {1, …, s}. In particular α(a_{ij}) = a′_{ij} and, since α(Γ_{m_{i}}) = Γ_{m′_{i}}, also α(E_{m_{i}}) = E_{m′_{i}}.

Now, for each m ∈ V \ {m_{1}, …, m_{s}} we construct the extension of α, α_{m}^{+} : π^{−1}(m) → π^{−1}(m′), for m′ = α(m), as in 2.0.4. Use 2.0.1.3 to choose t_{ijk} satisfying P^{A} for m and find bases e ∈ E_{m} and e′ ∈ E_{m′} with the U_{i}’s represented by the matrices {t_{ijk} : k, j = 1, …, N} in π^{−1}(m) and by {α(t_{ijk}) : k, j = 1, …, N} in π^{−1}(m′). It follows that the map

α_{m}^{+} : ∑ z_{j}e(j) ↦ ∑ α(z_{j})e′(j), z_{1}, …, z_{N} ∈ F

is an isomorphism of the A-modules

α_{m}^{+} : π^{−1}(m) → π^{−1}(m′).

Hence, the union

α^{+} = ⋃_{m∈V} α_{m}^{+}

is an automorphism of Ṽ.□

By the compactness theorem we immediately get from the lemma.

Corollary Every formula in Ṽ with parameters in C ⊆ F is equivalent to the disjunction of a finite collection of core formulas.

3.0.8 We consider now a more general form of core formulas with parameters in both sorts Ṽ and F.

The general core formula of variables x = {x_{1}, …, x_{p}} and v = {v_{ij} : i ≤ s + u, j ≤ r_{i}} and parameters C ⊆ F, ê ⊆ Ṽ will be of the form

∃ e_{1}, … e_{s}∃ m_{1}, …, m_{s}
∃ {t_{ikjl} : k ≤ d, i ≤ s, j, ℓ ≤ N}
∃ {z_{ijl} : i ≤ s, j ≤ r_{i}, ℓ ≤ N}
∃ {y_{ijl} : i ≤ u, j ≤ r_{s+i}, ℓ ≤ N}
A(e, m, t, z, v)& B(ê, y, v)& R(m, t, x, y, z),

where ê = (ê_{s+1}, …, ê_{s+u}) are names of fixed canonical bases of some modules Mm̂_{s+1}, …, Mm̂_{s+u} in Ṽ, m, t, z and A are the same as in 3.0.6, y is {y_{ijl} : i ≤ u, j ≤ r_{s+i}, ℓ ≤ N}, R is a Boolean combination of Zariski closed predicates in variables m, t, x, y, z and B(ê, y, v) is the formula

⋀_{i≤u; j≤r_{s+i}} v_{s+i,j} = ∑_{ℓ≤N} y_{ijℓ} · ê_{i}(ℓ).

As before we call R appearing in the general core formula the kernel of the formula and write ∃e R for the general core formula with kernel R.

Remark Given the set in Ṽ defined by a general core formula ∃e R the values of parameters m̂_{s+1}, …, m̂_{s+u} are determined uniquely as π(v_{ij}) with i = s + 1, …, s + u, j ≤ r_{i}. Hence ê_{i}, s < i ≤ s + u, are determined up to a linear transformation inside Mm̂_{i}. So, choosing a different ê′ = γê one can still define the same set by using the formula ∃e R′ where R′(m, t, x, y′, z) is obtained from R(m, t, x, y, z) by substituting y′γ in place of y. In other words,

we may assume that two equivalent general core formulas have the same parameters ê.

Proposition Every formula with parameters in Ṽ is equivalent to the disjunction of a finite collection of general core formulas.

Proof By 3.0.6 it is enough to prove that there is such a form for the formula obtained from a core formula ∃e R in variables x = {x_{1}, …, x_{p}} and v = {v_{ij} : i ≤ s + u, j ≤ r_{i}} with parameters C ⊆ F,

∃ e_{1}, … e_{s}, … e_{s+u}∃ m_{1}, …, m_{s}, …, m_{s+u}

∃ {t_{ikjl} : i ≤ s + u, k ≤ d, j, ℓ ≤ N}  (9)
∃ {z_{ijl} : i ≤ s + u, j ≤ r_{i}, ℓ ≤ N}
A(e, m, t, z, v)& R(m, t, x, z)

by substituting

v_{s+1,1} := a_{s+1,1}, …, v_{s+1,q_{s+1}} := a_{s+1,q_{s+1}},
…
v_{s+u,1} := a_{s+u,1}, …, v_{s+u,q_{s+u}} := a_{s+u,q_{s+u}},

some a_{ij} ∈ Ṽ and 1 ≤ q_{i} ≤ r_{i}, i = s + 1, …, s + u.

Notice that once the substitution v_{s+i,1} := a_{s+i,1} occured the value of m_{s+i} will be fixed as m_{s+i} = π(a_{s+i}). Denote this m̂_{s+i}. Correspondingly there are finitely many possible values for e_{s+i} ∈ Em̂_{s+i}. Choosing any such canonical basis ê_{s+i}, the corresponding t_{s+i,kjl} described in A(e, m, t, z, v) will be fixed, denote the correspondent elements in F as t̂_{s+i,kjl}. For the same reason we have the z_{s+i,jl}, for j ≤ q_{s+i}, fixed as ẑ_{s+i,jl} by A(e, m, t, z, v).

So, ∃e R^{v_{I}:=a_{I}} is equivalent to

⋁_{ê_{s+1}∈Em̂_{s+1}...ê_{s+u}∈Em̂_{s+u}} ∃ e_{1}, … e_{s}∃ m_{1}, …, m_{s}

∃ {t_{ikjl} : i ≤ s, k ≤ d, j, ℓ ≤ N}
∃ {z_{ijl} : i ≤ s, j ≤ r_{i}, ℓ ≤ N}

(A(e, m, t, z, v)& R(m, t, x, z))^{v_{I}:=a_{I},m_{I}=m̂_{I},t_{I}=t̂_{I},z_{I}=ẑ_{I}}

Now we rename z_{ijl} with s < i ≤ s + u and q_{i} < j ≤ r_{i} as y_{i−s,j−q_{i},l}. R(m, t, x, z)^{v_{I}:=a_{I},m_{I}=m̂_{I},t_{I}=t̂_{I},z_{I}=ẑ_{I}} becomes then some constructible predicate in variables m, t, x, y, z and parameters C and m̂_{I}, t̂_{I}, ẑ_{I}.

We now want to reduce

A(e, m, t, z, v)^{v_{I}:=a_{I},m_{I}=m̂_{I},t_{I}=t̂_{I},z_{I}=ẑ_{I}}

to a suitable equivalent form. To this end we delete from the formula the conjuncts which are trivially true, namely E(ê_{i}, m̂_{i}) and the equalities of the form

P^{A}({t̂_{iknℓ}}_{k≤d, ℓ,n≤N}; m_{i}) = 0, a_{ij} = ∑_{l} ẑ_{ijl}ê_{i}(l) and U_{k}ê_{i}(j) = ∑ t̂_{ikjl}ê_{i}(l)

for i > s. The only equations with indices i > s remaining will have the form

v_{ij} = ∑_{l} y_{i−s,j,l}ê_{i}(l),

and the conjunction of all these will form our B(ê, y, v) (we rename ê_{i} as ê_{i−s} in the final form). The remaining part of A(e, m, t, z, v)^{v_{I}:=a_{I},m_{I}=m̂_{I},t_{I}=t̂_{I},z_{I}=ẑ_{I}} will be exactly A(e, m, t, z, v) where e, m, t, z, v are as in the definition of a general core formula. □

Remark We have also proved that the result ∃e R^{v_{I}:=a_{I}} of the substitution in a given core formula (9) with kernel R(m, t, x, z) is equivalent to a disjunction of general core formulas each with the kernel

R(m, t, x, z)^{m_{I}=m̂_{I},t_{I}=t̂_{I},z_{I}=ẑ_{I}y_{I}},

where the substitution z_{I} = ẑ_{I}y_{I} replaces z_{ijl} with s < i ≤ s + u by ẑ_{ijl}, for j ≤ q_{i}, or by y_{i−s,j−q_{i},l}, for q_{i} < j ≤ r_{i}.

Corollary Every formula in Ṽ with parameters in Ṽ is equivalent to the disjunction of a finite collection of general core formulas.

3.0.9 We assume from now on the stronger assumption 2.0.1.4 and prove in this section that the core formulas in Corollaries of 3.0.7 and 3.0.8 can have a form more suitable for technical purposes.

Let Γ be the group in 2.0.1.4. Given a Zariski closed predicate R := R(m, t, x, y, z) with m ranging in V^{s} and t, x, y, z tuples in F in accordance with the notation in 3.0.7, we define, for γ = (γ_{1}, …, γ_{s}) ∈ Γ^{s}, the predicate R^{γ}(m, t, x, y, z) of the same variables.

First we consider the case when R is irreducible. Set

V_{R} := {m ∈ V^{s} : ∃ t, x, y, z R(m, t, x, y, z)},

the projection of R on V^{s}. Let V_{R,γ} be the open subset of V_{R} equal to the domain of definition of the map (in variables m)

g(m, γ) := ⟨g(m_{1}, γ_{1}), …, g(m_{s}, γ_{s})⟩.

Set

Γ_{R}^{s} := {γ ∈ Γ^{s} : V_{R,γ} ≠ ∅}.

This is a subgroup of Γ^{s} since V_{R,γ} is a dense open subset when nonempty. In case γ ∈ Γ_{R}^{s} define R^{γ} to be the Zariski closure of the set

{⟨m, t, x, y, z⟩ : ∃t′, z′ m ∈ V_{R,γ} & t′ = g(m, γ)^{−1} · t · g(m, γ) &

& z′ = z · g(m, γ) & R(m, t′, x, y, z′)}.

Remember that t is a collection of N × N matrices and z is a list of Ntuples, coordinates of elements of M_{m_{i}} in the corresponding canonical bases e. So in the definition above z · g(m, γ) corresponds to the coordinates of the same elements in bases e′ = g(m, γ) · e, and g(m, γ)^{−1} · t · g(m, γ) is the result of the corresponding transformation of the matrices in t.

With an obvious abuse of notation we will often write R(m, t^{γ}, x, y, zγ) for R^{γ}(m, t, x, y, z), when γ ∈ Γ_{R}^{s}.

For a general Zariski closed R we first represent R = R_{1} ∪ ⋯ ∪ R_{k} as the union of its irreducible components and then set

R^{γ} = R_{1}^{γ} ∪ ⋯ ∪ R_{k}^{γ},
Γ_{R}^{s} = Γ_{R_{1}}^{s} ∩ ⋯ ∩ Γ_{R_{k}}^{s}.

Remarks (i) Obviously, R^{id} = R, for id the unit element of Γ^{s}; (ii) For γ ∈ Γ_{R}^{s} the set V_{R^{γ}} ∩ V_{R} is a dense open subset of V_{R} equal to

V_{R} ∩ V_{R,γ}; (iii) If R does not depend on t and z then R^{γ} = R for every γ ∈ Γ_{R}^{s}; (iv) Suppose P ⊆ R is a Zariski closed relation. Then P^{γ} ⊆ R^{γ} for every

γ ∈ Γ^{s} and Γ_{P}^{s} ⊆ Γ_{R}^{s}; (v) Let R^{∗} = ⋃_{γ∈Γ^{s}} R^{γ}. Then

R^{∗}^{γ} = R^{∗}

for every γ ∈ Γ_{R}^{s} such that

We will say that R is Γ-invariant if R^{γ} = R for every γ ∈ Γ_{R}^{s}.

3.0.10 Lemma 1. We may assume that the kernels in core formulas in Corollary 3.0.7 are of the form R(m, t, x, z) & ¬S(m, t, x, z), where R, S are given by systems of equations and S is Γ-invariant.

Proof We go back to the proof of Lemma 3.0.7 and consider the deductively complete type p in the language of fields, p = {P_{i}}, with conjunctions of P_{i} appearing in the end as the kernels of core formulas. We may assume that each P_{i} is either a system of equations R(m, t, x, z) in variables m, t, x, z or the negation ¬S(m, t, x, z) of the system of equations S. We are going to prove that, for a given ¬S ∈ p there is a system of equations R ∈ p, and a negation ¬S̄ ∈ p, with ¬S̄^{γ} = ¬S̄, for all γ ∈ Γ_{S}^{s}, such that R & ¬S̄ ⊧ ¬S. This implies that we can replace all P_{i} by R & ¬S̄ and would prove the Lemma.

If ⋀_{γ∈Γ^{s}} ¬S^{γ} ∈ p then this formula, being equivalent to a negation ¬S̄ of a system of equations, is invariant under Γ^{s} and satisfies ¬S̄ ⊧ ¬S.

So we assume the opposite, ⋀_{γ∈Γ^{s}} ¬S^{γ} ∉ p. Hence, for some nonempty proper subset Δ ⊊ Γ^{s}, with 1 ∈ Δ,

¬T = ⋀_{γ∈Δ} ¬S^{γ} ∈ p.

We assume Δ to be maximal with this property.

Obviously

⋁_{γ∈Γ^{s}} ¬T^{γ} ∈ p.

Denote

Stab(Δ) = {γ ∈ Γ^{s} : γΔ = Δ}.

Since by maximality for any γ ∈ Γ^{s} \ Stab(Δ) we have ¬T^{γ} & ¬T ∉ p, necessarily T^{γ} ∈ p and so

⋀_{γ∈Γ^{s}\StabΔ} T^{γ} ∈ p.

But

⋁_{γ∈Γ^{s}} ¬T^{γ} & ⋀_{γ∈Γ^{s}\StabΔ} T^{γ} ⊧ ⋁_{γ∈StabΔ} ¬T^{γ}

The latter is equivalent to ¬T, and ¬T ⊧ ¬S. So we can take ⋀_{γ∈Γ^{s}\StabΔ} T^{γ} for R and ⋁_{γ∈Γ^{s}} ¬T^{γ} for ¬S̄.□

Lemma 2. We may assume that in Corollary 3.0.7 the kernels of core formulas are of the form R&¬S with R, S given by systems of polynomial equations and both are Γ-invariant.

Proof We use Lemma 1. Observe first that in general

Claim ∃e, t, zA(e, m, t, v, z) & P (m, t, z, x) is equivalent to ∃e, t, zA(e, m, t, v, z) & P^{∗}(m, t, z, x), where

P^{∗} = ⋁_{γ∈Γ^{s}} P^{γ}.

Indeed, ∃e P obviously implies ∃e P^{∗}. To see the converse note that, given v and x, if for some e and γ ∈ Γ^{s} we have ⊧ A(e, m, t, v, z) & P (m, t^{γ}, zγ, x) then, letting e′ = γe, we will have ⊧ A(e′, m, t^{γ}, v, zγ) and so, ⊧ A(e′, m, t′, v, z′) & P (m, t′, z′, x) for t′ = t^{γ} and z′ = zγ.

Applying the Claim to our ∃e R & ¬S we will get the equivalent formula ∃e R^{∗} & ¬S since S^{∗} = S. □

Combining Lemma 2 with the Remark in 3.0.8 we get.

Corollary We may assume that in Corollary 3.0.8 the kernels of general core formulas are of the form R&¬S with R, S given by systems of polynomial equations and both are Γ-invariant.

Now we discuss general core formulas with Γ-invariant kernels.

Lemma 3. Assuming that R_{2} is Γ-invariant we have

(i) ∃e (R_{1} & R_{2}) ≡ (∃e R_{1}) & (∃e R_{2});

(ii) ∃e ¬R_{2} ≡ ¬∃e R_{2}.

Proof (i) The left-hand-side obviously implies the formula on the right. Assume for converse that the right-hand-side is true. That is for given v, x and y there is e and e′ such that ⊧ A(e, m, t, v, z) & B(ê, y, v)& R_{1}(m, t, x, y, z) and ⊧ A(e′, m, t′, v, z′) & B(ê, y, v)& R_{2}(m, t′, x, y, z′). Since e′ = γe for some γ ∈ Γ^{s}, we have ⊧ A(e, m, t, v, z) & B(ê, y, v)& R_{2}(m, t^{γ}, x, y, zγ). But R_{2} is Γ-invariant, hence we get ⊧ A(e, m, t, v, z) & B(ê, y, v)& R_{1}(m, t, z, x) & R_{2}(m, t, z, x), as requred.

(ii) We need only prove the implication from left to right. Assume that ⊧ A(e, m, t, v, z) & B(ê, y, v)& ¬R_{2}(m, t, x, y, z). We need to check that for no e′ it is possible ⊧ A(e′, m, t′, v, z′) & B(ê, y, v)& R_{2}(m, t′, x, y, z′). Indeed, as above by Γ-invariance the latter is equivalent to ⊧ A(e, m, t, v, z) & B(ê, y, v)& R_{2}(m, t, x, y, z), which would contradict the former. □

Lemma 4 Suppose ∃eR_{1} ≡ ∃eR_{2}, both sides are general core formulas with the same partition of v-variables, u and ê_{1}, …, ê_{u} are same in both formulas, and R_{1}, R_{2} are Γ-invariant. Then R_{1} ≡ R_{2}.

Proof By Lemma 3

∃e (R_{1} & ¬R_{2}) ≡ ∃e R_{1} & ∃e ¬R_{2} ≡ ∃e R_{1} & ¬∃e R_{2}

and so R_{1} & ¬R_{2} is inconsistent, that is ⊧ R_{1} → R_{2}. By symmetry R_{1} ≡ R_{2}.□

4 Zariski geometry

In this section we introduce on Ṽ and its finite cartesian powers a topology which is naturally coming from the coordinate algebra A. To see that this is a Noetherean topology satisfying also the definition of a pre-smooth Zariski geometry (see [Z1] for this) we have to have more than just a quantifier elimination to existential formulas. To this end we carry out a more detailed analysis of general core formulas and their behavior under Boolean operations and projections.

4.0.11 We introduce the A-topology declaring basic closed subsets of Ṽ^{n} × F^{p} the subsets defined by general core formulas ∃eR with kernels R given by Γ-invariant systems of polynomial equations with coefficients in F.

We also assume that R contains the equation P^{A}(t, m) = 0 (see 2.0.1.3), which is in the A-part of ∃eR.

We often denote R̂ the closed set defined by the formula ∃eR.

The closed subsets of the topology are given by applying finite unions and arbitrary intersections to basic closed subsets.

Claim 1 Intersection of an infinite family of basic closed subsets of a Cartesian power of Ṽ is equal to the intersection of its finite subfamily.

Indeed, since for a given set of variables there are finite number of ways to partition (enumerate) the variables as {v_{ij} : i ≤ s, j ≤ r_{i}}, we may assume that all core formulas defining the members of the family have the same partition of variables. Now by Lemma 2 and Lemma 3(i) of 3.0.10 the intersection of sets defined by ∃e R_{α}, α ∈ I, reduces to the intersection of Zariski closed sets defined by R_{α}, α ∈ I, which obviously stabilises.

Using Koenig’s Lemma we get

Claim 2 The A-topology is Noetherian.

Since for s + u = 0 a general core formula ∃e R takes the form R(x, y) the following is obvious.

Claim 3 The restriction of the A-topology to the sort F is the classical Zariski topology.

Claim 4 Any definable subset of a Cartesian power of Ṽ is equal to the Boolean combination of closed subsets, that is, is constructible.

Indeed, by the Corollary in 3.0.8 it is sufficient to prove the statement for subsets defined by general core formulas. The Corollary in 3.0.10 together with Lemmas 3(ii) provide the rest.

We will also need a more detailed presentation of sets obtained by projecting closed sets onto coordinate subspaces, as well as fibers of these projections.

Lemma 1 Let ∃e R be the general core formula in the notation of 3.0.8 and a ∈ {1, …, s + u}, b ∈ {1 …, r_{a}} some indices. Then the formula ∃v_{ab}∃e R is equivalent to a general core formula ∃e′ R′ with the kernel R′ equivalent to one of the following

(i) ∃y_{a−s,b1} … y_{a−s,bN}R,
(ii) ∃z_{ab1} … z_{abN}R or
(iii) ∃m_{a}∃{t_{akjl} : k ≤ d, j, l ≤ N}∃z_{ab1} … z_{abN}R.

Proof (i) Suppose s < a ≤ s + u. Since v_{ab} does not occur in A(e, m, t, z, v) and R(m, t, x, y, z), the formula ∃v_{ab}∃e R is equivalent to

∃ e_{1}, … e_{s}∃ …
A(e, m, t, z, v)& (∃v_{ab}B(ê, y, v)) & R(m, t, x, y, z),

with the quantifier prefix the same as in ∃e R. Looking at the form of B(ê, y, v) one sees that (∃v_{ab}B(ê, y, v)) is equivalent to some B(ê, y′, v′) with y′ = y \ {y_{a−s,b1} … y_{a−s,bN}} and v′ = v \ {v_{ab}}. Now we can equivalently rewrite the formula as

∃ e_{1}, … e_{s}∃ …
A(e, m, t, z, v)& B(ê, y′, v′)& ∃y_{a−s,b1} … y_{a−s,bN}R(m, t, x, y, z),

where ∃y_{a−s,b1} … y_{a−s,bN} moved from the quantifier prefix to the end of the formula. Of course, by quantifier elimination in algebraically closed fields, ∃y_{a−s,b1} … y_{a−s,bN}R is a constructible predicate.

(ii) and (iii). Suppose a ≤ s. Then the formula ∃v_{ab}∃e R is equivalent to

∃ e_{1}, … e_{s}∃ …
(∃v_{ab}A(e, m, t, z, v)) & B(ê, y, v)& R(m, t, x, y, z).

One can obviously eliminate the quantifier from ∃v_{ab}A(e, m, t, z, v) by substituting v_{ab} everywhere in A(e, m, t, z, v) by the term ∑_{ℓ≤N} z_{abℓ}e_{a}(ℓ). This makes the conjunct v_{ab} = ∑_{ℓ≤N} z_{abℓ}e_{a}(ℓ) in A(e, m, t, z, v) a tautology and after removing it we get a formula without v_{ab} which in the case r_{a} ≠ 1 is again of the form A(e, m, t, z′, v′), where v′ = v \{v_{ab}}, z′ = z \{z_{ab1} … z_{abN}}. This gives us a general core formula of the form (ii) for ∃v_{ab}∃e R.

In case r_{a} = 1 the conjunct v_{ab} = ∑_{ℓ≤N} z_{abℓ}e_{a}(ℓ) is the only one that uses the variables e_{a}. By eliminating this conjunct we made other subformulas containing e_{a} redundant. So we eliminate

⋀_{j≤N} E(e_{a}, m_{a}) & P^{A}({t_{aknℓ}}_{k≤d, ℓ,n≤N}; m_{a}) = 0 & ⋀_{k≤d,j≤N} U_{k}e_{a}(j) = ∑_{ℓ≤N} t_{akjℓ}e_{a}(ℓ)

from A(e, m, t, z, v) as well (notice that by our assumptions P^{A}({t_{aknℓ}}_{k≤d, ℓ,n≤N}; m_{a}) = 0 is also copied in R). The resulting formula is again of the form A(e′, m′, t′, z′, v′), with v′ = v \ {v_{ab}}, m′ = m \ {m_{a}}, z′ = z \ {z_{ab1} … z_{abN}} and t′ = t \ {t_{akjl} : k ≤ d, j, ℓ ≤ N}. Now we may push the quantifiers ∃m_{a}, ∃{z_{ab1} … z_{abN}} and ∃{t_{akjℓ} : k ≤ d, j, ℓ ≤ N} to the end of the formula and get the general core formula of the form (iii).□

Lemma 2 Suppose ∃e R does not contain x, free variables of the sort F. Let ∃e′R′ be the general core formula equivalent to ∃v_{ab}∃e R as in the above Lemma. More precisely R′ = ∃uR(u, w), some u depending on the case. Let v′ = v \ {v_{a}b} and v̂′ is a tuple in Ṽ satisfying ∃e′R′. Then (∃eR)^{v′:=v̂′} has kernel of the form R(u, ŵ), for some ŵ.

Proof Follow the analysis in the proof of Proposition in 3.0.8. In case (i) the substituition v′ := v̂′ fixes the whole of e, m, t, z and y\{y_{a−s,b,1} …, y_{a−s,b,N}}, so the kernel is R^{m=m̂,t=t̂,z=ẑ,y′=ŷ′}. In other words, in this case we satisfied the requirement of the Lemma with u = (y_{a−s,b,1} …, y_{a−s,b,N}) and w = (m, t, z, y′).

In case (ii) again v̂′ fixes the whole of e, m, t, y and z \ {z_{ab1} … z_{abN}}. In case (iii) v̂′ fixes e \ e_{a} m \ m_{a}, t \ t_{a}, y and z \ {z_{ab1} … z_{abN}}.□

4.0.12 For further purposes we need a more detailed understanding of intersections of closed sets.

Let {v_{1}, …, v_{n}} be a linear reenumeration of variables {v_{ij} : i ≤ s, j ≤ r_{i}}, n = r_{1} + ⋯ + r_{s}, of sort Ṽ in a general core formula ∃eR (the variables {v_{ij} : s < i ≤ s + u, j ≤ r_{i}} remain unchanged). We write k ∼_{R} k′ for k, k′ ∈ {1, …, n} if k and k′ correspond to some (i, j) and (i, j′) in the old enumeration. This is an equivalence relation. We denote I_{R} the subset {1, …, n} corresponding to {(i, 1) : i = 1, …, s} in the partitioning enumeration, the set of representatives of ∼_{R}-classes.

We use the abbreviation e_{i} for {e_{i}(1), …, e_{i}(N)}, t_{i} for {t_{ikjℓ} : k ≤ d; j, l ≤ N} and z_{i} for {z_{ijℓ} : j ≤ r_{i}; ℓ ≤ N}, i = 1, …, n, along with other obvious abbreviations. In particular, Ue_{i} = t_{i}e_{i} stands for

⋀_{k≤d,j≤N,i≤s} U_{k}e_{i}(j) = ∑_{ℓ≤N} t_{ikjℓ}e_{i}(ℓ).

We rewrite equivalently the general core formula ∃e R of 3.0.8 as R̃(v, x) :

∃ e_{1}, … e_{n} ∃ t_{1}, …, t_{n}, ∃ z_{1}, …, z_{n} y ∃ m_{1}, … m_{n}

⋀_{i≤n} E(e_{i}, m_{i}) & Ue_{i} = t_{i}e_{i} & v_{i} = z_{i}e_{i} & ⋀_{i∼_{R}j} e_{i} = e_{j} & B(ê, y, v)

& R(m, t, z, x, y) & ⋀_{i∼_{R}j} m_{i} = m_{j} & t_{i} = t_{j}.

This is not a core formula because of the component ⋀_{i∼_{R}j} e_{i} = e_{j}.

Remark 1 In R only m_{i}, t_{i} with i ∈ I_{R} as well as z_{i}, i ≤ n, y and x occur explicitly. Let d_{R} = dim R, the dimension of the variety defined by R in the space given by these variables. Obviously we may assume that R depends on all variables m_{i}, t_{i}, z_{i}, i ≤ n, y and x. Then in the bigger ambient space we still have

d_{R} = dim[R& ⋀_{i∼j} m_{i} = m_{j} & t_{i} = t_{j}].

Let ∃e S be another general core formula of the same variables with possibly different partitioning enumeration {v_{ij} : i ≤ s′ + u, j ≤ r′_{i}}. We assume that variables {v_{ij} : s′ < i ≤ s′ + u, j ≤ r′_{i}} and parameters ê_{s′+1}, …, ê_{s′+u} are the same in both ∃e R and ∃e S.

We re-enumerate the variables {v_{ij} : i ≤ s′, j ≤ r′_{i}} linearly as v_{1}, …, v_{n}. We have the corresponding equivalence relation ∼_{S} on {1, …, n} and a set of its representatives I_{S}. As above ∃eS can be equivalently rewritten as the formula S̃(v, x) :

∃ e′_{1}, … e′_{n} ∃ t′_{1}, …, t′_{n}, ∃ z′_{1}, …, z′_{n} y ∃ m_{1}, … m_{n}

⋀_{i≤n} E(e′_{i}, m_{i}) & Ue′_{i} = t′_{i}e′_{i} & v_{i} = z′_{i}e′_{i} & ⋀_{i∼_{S}j} e′_{i} = e′_{j} & B(ê, y, v)

S(m′, t′, x, y, z′) & ⋀_{i∼_{S}j} m_{i} = m_{j} & t′_{i} = t′_{j}.

Lemma The formula R̃(v, x) & S̃(v, x) is equivalent to the formula T̃(v, x) :

∃ e_{1}, … e_{n} ∃ t_{1}, …, t_{n}, ∃ z_{1}, …, z_{n} y ∃ m_{1}, … m_{n}

⋀_{i≤n} E(e_{i}, m_{i}) & Ue_{i} = t_{i}e_{i} & v_{i} = z_{i}e_{i} & ⋀_{i∼_{RS}j} e_{i} = e_{j} & B(ê, y, v)

R(m, t, x, y, z)&S(m, t, x, y, z) & ⋀_{i∼_{RS}j} m_{i} = m_{j} & t_{i} = t_{j},

where ∼_{RS} is the transitive closure of the composition of the two equivalence relations ∼_{R} and ∼_{S} .

Proof The implication T̃(v, x) → R̃(v, x) & S̃(v, x) is obvious.

For converse suppose R̃(v, x) & S̃(v, x) holds. This implies the existence of e_{i}, e′_{i}, t_{i}, t′_{i}, z_{i}, z′_{i}, m_{i} (i = 1, …, n) and y which satisfy

⋀_{i≤n} E(e_{i}, m_{i}) & Ue_{i} = t_{i}e_{i} & v_{i} = z_{i}e_{i} & ⋀_{i∼_{R}j} e_{i} = e_{j} & B(ê, y, v)

R(m, t, x, y, z) & ⋀_{i∼_{R}j} m_{i} = m_{j} & t_{i} = t_{j}

and

⋀_{i≤n} E(e′_{i}, m_{i}) & Ue′_{i} = t′_{i}e′_{i} & v_{i} = z′_{i}e′_{i} & ⋀_{i∼_{S}j} e′_{i} = e′_{j} & B(ê, y, v)

S(m, t′, x, y, z′) & ⋀_{i∼_{S}j} m_{i} = m_{j} & t′_{i} = t′_{j}.

m_{i} must be the same in both formulas since m_{i} = π(v_{i}). It follows from the assumption 2.0.1.4 that for some γ_{i} ∈ Γ_{m}, e_{i} = γ_{i}e′_{i}. Since R is Γ-invariant we can exchange, for i ∈ I_{R}, e_{i} by γ_{i}e_{i}, t_{i} by t_{i}^{γ_{i}} and z_{i} by z_{i}γ_{i} without changing the validity of R and so may assume that γ_{i} = 1 and e_{i} = e′_{i} for i ∈ I_{R}.

By symmetry we can reduce to the situation that also e_{j} = e′_{j} for j ∈ I_{S}.

Claim We can choose e_{i} = e′_{i} for all i ≤ n.

Proof. By induction on n. We have already e_{i} = e′_{i} for all i ∈ I_{R} ∪ I_{S}, so we assume that I_{R} ∪ I_{S} ⊆ {1, …, n − 1} and we can choose e_{i} = e′_{i} for all i ≤ n−1. We have e_{n} = e_{ℓ} = e′_{ℓ} for some ℓ ∈ I_{R}, ℓ ∼_{R} n, and e′_{n} = e′_{k} = e_{k}, for some k ∈ I_{S}, k ∼_{S} n. From the equivalences it follows that m_{i} = m_{n} = m_{k}, i.e. the modules coincide. So, e_{n} = γe′_{n} for some γ ∈ Γ_{m}.

Let J_{k} = {i ≤ n : e_{i} = e_{k}}, J′_{k} = {i ≤ n : e′_{i} = e_{k}}. Note that n ∉ J_{k}. Apply the substitution e_{i} ↦ γe_{i} and e′_{j} ↦ γe′_{j} for all i ∈ J_{k}, j ∈ J′_{k}, leaving e_{i} and e′_{j} for i ∉ J_{k}, j ∉ J′_{k} unchanged in

⋀_{i≤n} E(e_{i}, m_{i}) & Ue_{i} = t_{i}e_{i} & v_{i} = z_{i}e_{i} & ⋀_{i∼_{R}j} e_{i} = e_{j}

and

⋀_{i≤n} E(e′_{i}, m_{i}) & Ue′_{i} = t′_{i}e′_{i} & v_{i} = z′_{i}e′_{i} & ⋀_{i∼_{S}j} e′_{i} = e′_{j}.

This induces the correspondent transformation of t, t′, z, z′ which by Γ-invariance does not change the validity of R(m, t, x, y, z) and S(m, t′, x, y, z′).

This preserves all the existing equalities and gives e′_{n} = e_{n} for the new value of e′_{n}. Claim proved.

This brings us to the situation with e_{i} = e′_{i}, t_{i} = t′_{i} and z_{i} = z′_{i} in the formulas above. Thus

⋀_{i≤n} E(e_{i}, m_{i}) & Ue_{i} = t_{i}e_{i} & v_{i} = z_{i}e_{i} & ⋀_{i∼_{RS}j} e_{i} = e_{j}

R(m, t, z, x)&S(m, t, z, x) & ⋀_{i∼_{RS}j} m_{i} = m_{j} & t_{i} = t_{j},

hold. This proves the converse implication.□

Corollary 1 The intersection of two basic closed sets given by general core formulas ∃e R and ∃e S with arbitrary partitioning enumerations and the same parameters ê_{s+1}, …, ê_{s+u} is a basic closed set given by a core formula ∃e T, with T equivalent to

R(m, t, x, y, z)&S(m, t, x, y, z) & ⋀_{i∼_{RS}j} m_{i} = m_{j} & t_{i} = t_{j}.

Indeed, T̃(v, x, y) can be transformed into a core formula by the following process.

Let I_{RS} ⊆ {1, …, n} be a set of representatives of ∼_{RS}-classes. Assuming I_{RS} ⊆ {1, …, u} re-enumerate v_{1}, …, v_{n} as v_{ij} : i ≤ u, j ≤ r_{i}}, v_{i1} is the v_{i} in the linear enumeration and indices (ij) correspond to indices equivalent to i by ∼_{RS} .

Using the equalities e_{i} = e_{j}, m_{i} = m_{j} and t_{i} = t_{j}, for i ∼_{RS} j, we delete e_{j}, m_{j} and t_{j}, with j > u, everywhere from T̃ along with the subformulas stating the equalities.

We re-enumerate z-variables in accordance with enumeration v_{ij}, so that now the formula T̃ says now that v_{ij} = z_{ij}e_{i} for every i ≤ u and j ≤ r_{i}.

After that T̃ transforms to

∃e_{1}, …, e_{u} ∃t_{1}, …, t_{u} ∃m_{1}, …, m_{u} ∃z_{11}, …, z_{ur_{u}}

⋀_{i≤u} E(e_{i}, m_{i}) & Ue_{i} = t_{i}e_{i} & ⋀_{j≤r_{u}} v_{ij} = z_{ij}e_{i} & B(ê, y, v)

R′(m, t, x, y, z)&S′(m, t, x, y, z),

where R′ and S′ obtained by substituting t_{i} and m_{i} instead of t_{j} and m_{j}, for j ∼_{RS} i, j > u. This is a general core formula.

Remark 2 By the Remark in 3.0.8 we can always assume that parameters ê in both formulas are the same.

Combining Corollary 1 with (i) and (ii) of the definition 4.0.11 we get.

Corollary 2 Every closed set in the A-topology is equal to the union of a finite family of closed sets each of the form P̂, for P a Zariski closed predicate.

Corollary 3 Given a basic closed set P ⊆ Ṽ^{n}×F^{p}, there is a core formula ∃e P defining P with the finest partition of the v-variables. That is, for every ∃e R defining the same set the partition ∼_{P} is refining ∼_{R} .

Fixing a choice of parameters ê (one of the finitely many), the Zariski closed relation P above is determined uniquely by the set P. Any other choice ê′ of parameters for P determines a Zariski closed relation P′ obtained from P by a linear transformation in variables y.

Indeed, take for ∃e P the formula obtained by taking the conjunction of all possible representations ∃e R of P, R Zariski closed, using Corollary 1.

Lemma 4 in 3.0.10 implies the uniqueness of P.□

We will say that the algebraic constructible set P (F) for P and P as above is associated with P. If P (F) is Zariski closed we call P (F) the variety associated with the closed set P.

4.0.13 From now on when we write a basic closed set in the form P̂ (equivalently, use the core formula ∃e P) the kernel P is canonical, that is is uniquely determined by the set P̂.

We define

dim P̂ := dim P (F),

where dim on the right is the dimension of the algebraic variety.

For a constructible set S we define

dim S := dim S̄, where S̄ is the closure of S.

Suppose v = v_{1}^{⌢}v_{2}, |v_{1}| = n_{1}, |v_{2}| = n_{2}, x = x_{1}^{⌢}x_{2}, |x_{1}| = k_{1}, |x_{2}| = k_{2} and let

pr : Ṽ^{n_{1}+n_{2}} × F^{k_{1}+k_{2}} → Ṽ^{n_{1}} × F^{k_{1}}

be the projection pr : v_{1}^{⌢}v_{2}^{⌢}x_{1}^{⌢}x_{2} ↦ v_{1}^{⌢}x_{1}.

Proposition Let S ⊆ Ṽ^{n_{1}+n_{2}} × F^{p_{1}+p_{2}} be a closed set. Then

(i) pr(S) is a constructible set; (ii) for each a ∈ pr(S), the set S ∩ pr^{−1}(a) is closed; (iii) for each nonnegative integer ℓ the set

{a ∈ pr(S) : dim S ∩ pr^{−1}(a) ≥ ℓ}

is constructible. If ℓ > min_{a∈pr(S)} dim S ∩ pr^{−1}(a) then the set is contained in a proper subset closed in pr(S).

(iv) assuming S is irreducible, we have

dim S = dim pr(S) + min_{a∈pr S} dim S ∩ pr^{−1}(a).

(v) for any two irreducible S_{1}, S_{2} ⊆ Ṽ^{n} × F^{p}, for every irreducible component S_{0} of S_{1} ∩ S_{2},

dim S_{0} ≥ dim S_{1} + dim S_{2} − dim Ṽ^{n} × F^{p}.

Proof (i) Follows from Claim 4 in 4.0.11. (ii) Just notice that

S ∩ pr^{−1}(a) = S ∩ {v_{1}^{⌢}x_{1} = a}

and notice that {v_{1}^{⌢}x_{1} = a} is a basic closed set.

(iii) By our definition of dimension and the two Lemmas in 4.0.11 this is equivalent to the same statement for S an affine algebraic variety. This is a well-known theorem for algebraic varieties used as an axiom (FC) for Zariski geometries in [Z1].

(iv) First observe Claim If S is irreducible then S = P̂ with the associated variety P of

the form

⋃_{γ∈Γ^{s}} R^{γ}, R irreducible.

Indeed, by definition S is a union of sets of the form P̂. Since it is irreducible there is just one such in the union. Let R(F) be an irreducible component of the variety P (F). By Γ-invariance R^{γ}(F) is also a component of P, for all γ ∈ Γ_{P} . By irreduciblity of S the union of all R^{γ} is equal to P. Claim proved.

Again as in (iii), by 4.0.11, (iv) is equivalent to

dim P = dim pr P + min_{b∈pr P} dim P ∩ pr^{−1}(b)

for an appropriate projection pr. But this is the known addition formula for algebraic varieties and more generally Zariski structures, see [Z1].

(v) First observe that by the Claim above irreducible S_{i} (i = 0, 1, 2) have to be of the form P̂_{i}, for P_{i} of the form ⋁_{γ∈Γ^{s}} R_{i}^{γ}, R_{i} irreducible.

The rest follows from Corollary 1 of 4.0.12 (see also Remark 1 in the same section). In the present notation we get by the Corollary that the kernel in P̂_{1} ∩ P̂_{2} corresponds to the intersection of two algebraic subvarieties of dimensions dim P_{1} and dim P_{2} in the ambient affine space of dimension dim Ṽ^{n} + dim F^{p}. By the Dimension Theorem for affine spaces we get the required inequality. □

Theorem For any algebra A satisfying the assumptions 2.0.1(1-4) the structure Ṽ is a Zariski geometry, satisfying the presmoothness condition provided the affine algebraic variety V is smooth.

Proof The Proposition above together with the topological subsection 4.0.11 proves all the assumptions defining Zariski geometries, see [Z1].□

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