1991 example by Hrushovski in the light of non-commutative geometry. A case study.

B.Zilber

13 September 2004

In 1991, after the main result of [HZ] identifying any ample Zariski geometry as a finite cover of an algebraic curve was obtained, Hrushovski produced an example of a non-algebraic Zariski geometry ([HZ], section 10). A more specific version of the example was discussed in an earlier version of [Z1] (with Y.Peterzil). There an interpretation of the example as a ’compactification of a non-compact Riemann surface’ was given. Here I come back to this example to give a more detailed construction of the Riemann surface on the one hand, and argue that the object in question is a curve in the context of non-commutative algebraic geometry, on the other. Moreover, we show that in a richer natural language, the structure is analytic Zariski. The really new material is in section 2 of the paper treating the Hrushovski structure D as a non-commutative curve. We first construct an infinite dimensional F-vector space H of locally Zariski functions D → F, where F is the algebraically closed field in the base of the construction. These functions are not definable in D in general but the construction is almost canonical. We note that making our choices in the course of the construction we endow the field F with a function sgn with values +1, −1 (in general can be values in the group of roots of unity of order n). Next we define a natural algebra A(D) of linear operators on H and consider its irreducible representations with the spectrum of a basic operator in the sector of F marked by sgn = +1. The set of polarised irreducible representations endowed with the natural Zariski topology given by ideals of A(D) gives us the Hrushovski D back. In the next section we extend this construction to a series of coordinate rings with sings and show that they give rise to non-algebraic Zariski structures, in fact included in examples of section 10 of [HZ]. This version of the article does not answer many natural questions, and should be treated as a draft of a prospective paper. In particular, we would like to show that the relations in the Zariski structures arising from the noncommutative rings are in correspondence with certain ideals of the rings.

1 The example

1.0.1 Let ℤ act nontrivially on ℤ/4ℤ, and let G be the semidirect product: f, h are the generators of ℤ/4ℤ, ℤ, respectively,

f^{4} = 1; fh = hf^{−1}  (1)

are the defining relations. Notice, that h^{2} commutes with f, and hence, the centre Z(G) of G is generated by h^{2} and f^{2}, so has index 4 in G. Fix a ∈ ℝ, a > 1. We claim that there is a strongly minimal double cover p : D → {ℂ − (0)} in the language containing the relations induced by the field structure on ℂ and unary operations f, h, defining a faithful action of G on D such that: f, h respect the kernel of p (that is p(x) = p(x′) implies p(f(x)) = p(f(x′)) and p(h(x)) = p(h(x′)) and induce the (unfaithful) action on {ℂ − (0)} :

p(h(x)) = a^{2}p(x), p(f(x)) = −p(x).  (2)

Notation For g ∈ G reduced to the form g = f^{k}h^{n}, n, k ∈ ℤ, let |g| = (−1)^{k}a^{2n}.

Notice that |g| ∈ ℝ \ (0) and the unfaithful action of G on ℂ \ (0) described above can be represented as

yg = |g|y.

The final example is a structure with the universe

D̄ = D ∪ {0, ∞}

and an equivalence relation E given by the kernel of p on D and 0 and ∞ are one-element equivalence classes, in particular, we can consider some p̂: D̄ → D̄/E = P^{1} ; f, h are operations, acting on D as described above and identical on 0 and ∞; for any Zariski-closed relation S on P^{1} we have p^{−1}(S) as a relation on D̄. Call the relations introduced above closed and introduce the dimension notion in D as induced from ℂ by p. Then using elimination of quantifiers following from the Lemma above one gets 1.0.2 Theorem C of [HZ] D̄ is an irreducible complete 1-dimensional pre-smooth Zariski structure Proof Suppose D̄ is interpretable in a pure field. We first prove that D̄ would then be birational with an irreducible curve C. Indeed, C is a double cover of the projective line P^{1}, so dim C = 1 and C has at most two irreducible components, say C_{1} and C_{2}. Supposing that they are distinct we get that p̂ : C_{i} → P^{1} is injective for both i = 1 and i = 2. Thus, the equivalence relation E links a generic element x of C_{1} to a generic element x′ of C_{2}. By construction x′ = f^{2}(x). Necessarily, the setwise stabiliser G^{0} of C_{1} (and C_{2}) in G is of index 2 in G, so f^{2} has to be in G^{0}, contradicting the construction. We have proven the irreducibility. Since every algebraic curve is birationally equivalent to a smooth one, G embeds into the subgroup of the group of birational transformations of a smooth curve C′. Now remember that any birational transformation of a smooth algebraic curve is biregular. Notice also that both f and h fix two different points of C′. But G is not such a subgroup of the automorphism group of any smooth (irreducible) algebraic curve: The automorphism group is finite in genus 2 or higher; in genus 0, it is PGL(2, ℂ), and any subgroup of the group fixing two different points can be embedded into the multiplicative group of the complex numbers, while G is non-commutative. In genus 1 (the elliptic curve) the group of automorphisms of the curve fixing a point is given by complex multiplication (when the elliptic curve is viewed as ℂ/L, L a lattice), and one sees again the contradiction in the non-commutativity of G.□

1.0.3 Interpretation in the reals. Denote K_{1} = {x ∈ ℂ : Re(x) ≥ 0 & Im(x) > 0}, K_{2} = iK_{1}, K_{3} = −K_{1}, K_{4} = −iK_{1}, D = K_{1} ∪K_{2} ∪K_{3} ∪K_{4}, D̄ = D ∪{0, ∞}. p(x) = x^{2}, f(x) = ix,

h(x) = \{ \begin{matrix}ax, & if x ∈ K_{1} ∪ K_{3} \\ −ax, & if x ∈ K_{2} ∪ K_{4} \end{matrix}  (3)

Notice that f^{4}(x) = x, f(h(x)) = h(f^{−1}(x)) for any x ∈ ℂ and p(h(x)) = a^{2}p(x), p(f(x)) = −p(x) and, more generally, p(g(x)) = |g|p(x). It follows, by (1) and (2) that the construction is isomorphic to the Example above.

1.0.4 A Riemann surface and its “compactification” Now we construct an open, in the real topology, subset D^{0} of D. Let

A = {r ∈ ℝ : a^{2n} < r < a^{2n+1}, n ∈ ℤ}, B = {r ∈ ℝ : a^{2n−1} < r < a^{2n}, n ∈ ℤ},

⟨a⟩ = {a^{m} : m ∈ ℤ},
Γ = ⟨a⟩ ∪ i⟨a⟩ ∪ −⟨a⟩ ∪ −i⟨a⟩,
D^{0} = D \ Γ.

We claim that D^{0} can be interpreted as a non-compact Riemann manifold. Fix a positive real ϵ and let

U_{1} = {u + iv : −ϵ < u < 0, v ∈ A},
V_{1} = {u + iv : −ϵ < u < 0, v ∈ B}.

These are open subsets of the complex plane and so are

K_{1}^{ϵ} = K_{1} ∪ U_{1} ∪ V_{1} \ Γ,
K_{2}^{ϵ} = iK_{1}^{ϵ}, K_{3}^{ϵ} = −K_{1}^{ϵ}, K_{4}^{ϵ} = −iK_{1}^{ϵ}.

Now we construct a Riemann manifold patching the four open charts K_{i}^{ϵ} by biholomorphic maps. First let

ϕ_{12} : x ∈ V_{1} ↦ x ∈ K_{2}^{ϵ},
ϕ_{14} : x ∈ U_{1} ↦ −x ∈ K_{4}^{ϵ},
ϕ_{23} : x ∈ iV_{1} ↦ x ∈ K_{3}^{ϵ},
ϕ_{21} : x ∈ iU_{1} ↦ −x ∈ K_{1}^{ϵ},
ϕ_{34} : x ∈ −V_{1} ↦ x ∈ K_{4}^{ϵ},
ϕ_{32} : x ∈ −U_{1} ↦ −x ∈ K_{2}^{ϵ},
ϕ_{41} : x ∈ −iV_{1} ↦ x ∈ K_{1}^{ϵ},
ϕ_{43} : x ∈ −iU_{1} ↦ −x ∈ K_{3}^{ϵ}.

Finally, the patching maps are

ψ_{ij} = ϕ_{ij} ∪ ϕ_{ji}^{−1}, 1 ≤ i < j ≤ 4

with the natural domains. This is well-defined since V_{1} ∩ jU_{1} = ∅, for j ∈ {1, −1, i, −i}, as well as V_{1} ∩ jV_{1} = ∅, for j ∈ {−1, i, −i}. Notice that h is holomorphic on D^{0}, which can be seen by identifying h on each chart with a holomorphic function, namely with ax on K_{1}^{ϵ} ∪ K_{3}^{ϵ}, and −ax on K_{2}^{ϵ} ∪ K_{4}^{ϵ} : near boundary of a chart, say when x ∈ V_{1} ⊆ K_{2}, we have, by the definition (3), h(x) = −ax ∈ K_{4}. On the other hand

x ∈ V_{1} ⊆ K_{1}^{ϵ} →^{a} ax ∈ U_{1} →^{ϕ_{14}} −ax ∈ K_{4},

and analogously with the other charts. So, the Zariski structure D̄ can be considered a compactification of the Riemann manifold D^{0}.

1.0.5 The relationship between the structure on the Riemann manifold D^{0} and the Zariski curve D̄ is even more interesting. Let D^{♯} be the Zariski structure D̄ enriched by unary predicate for Γ. Then D^{♯} is an analytic Zariski structure with the quasi-compact topology generated by the Zariski topology of D̄ plus Γ̄ = Γ ∪ {∞, 0}, as a closed set. The theory of D^{♯} is superstable with U-rank of Γ equal to 1 and U-rank of D equals to ω. D^{0} is definable in D^{♯}. Proof The structure on D^{♯}/E, the compactification P^{1} of the complex torus ℂ^{×} with the distinguished multiplicative subgroup |G| = ⟨a^{2}⟩∪−⟨a^{2}⟩ is wellknown, see e.g. section 5 of [Z11] for the case when a is a transcendental number. It is superstable, allows elimination of quantifiers to the level of existential formulas of a simple form and as a result, one can use the specialisation criterion to prove that the topology is quasi-compact. Now we can use the fact that the finite cover does not change essential properties (Proposition 10.1 of [HZ]) to prove the same properties for D^{♯} itself. The reader is asked to check the analytic Zariski axioms using elimination of quantifiers himself. Obviously, Γ is a 0-dimensional analytic subset of the open set D^{0} and h is holomorphic on D^{0} ( the graph of the mapping is analytic) in the sense of [PZ].

2 A coordinate algebra A(D) of D.

2.1 Coordinate functions on D.

We consider an abstract algebraically closed field F of characteristic zero rather than ℂ.

2.1.1 Given a c ∈ F^{×} let ce be one of the two points in D such that p(ce) = c^{2}. Let

√ : V_{c^{2}} → V_{c}

be the unique inverse of the map v ↦ v^{2} between the two local neighborhoods in ^{∗}F, and similarly

√ : V_{−c^{2}} → V_{ic}.

Since any point c ∈ F^{×} is regular for p, function p maps V_{ce} bijectively on V_{c^{2}} and V_{f(ce)} bijectively on V_{−c^{2}}. Hence we get the unique bijection

\sqrt{p} : \begin{matrix}V_{ce} ↦ V_{c} \\ V_{f(ce)} ↦ V_{ic} \end{matrix} .

Since f : D → D induces via p function y ↦ −y on F, local Zariskicontinuous bijection \sqrt{pf} satisfies

\sqrt{p(f(t))} = i\sqrt{p(f(t))},

for t ∈ V_{c}. We let

D_{|c|} = V_{ce} ∪ V_{f(ce)} ∪ V_{f^{2}(ce)} ∪ V_{f^{3}(ce)}

and denote x_{|c|} the unique function

x_{|c|} : D_{|c|} → V_{c} ∪ V_{ic} ∪ V_{−c} ∪ V_{−ic} ⊂ ^{∗}F

satisfying

x_{|c|}(f^{k}(t)) = i^{k}\sqrt{p}(t) for t ∈ V_{ce}, k = 0, 1, 2, 3.  (4)

This identifies D_{|c|} with the subset V_{c} ∪ V_{ic} ∪ V_{−c} ∪ V_{−ic} = \sqrt{V_{c^{2}} ∪ V_{−c^{2}}} of ^{∗}F, so x_{|c|} is a local coordinate function. The construction depends on the choice of c and ce and so we need to satisfy

D_{|c|} = D_{|ic|} = D_{|−c|} = D_{|−ic|}.  (5)

For each c ∈ F^{×} we arrange our correspondence

{c, ic, −c, −ic} ↦ {ce, (ic)e, (−c)e, (−i)c} ⊂ D

so that (ic)e = f(ce), (−c)e = f^{2}(ce), (−ic)e = f^{3}(ce). This is coherent with the definition of ce. Moreover, it implies (5) and also guarantees that the definition of x_{|c|} does not depend on the choice of the representation of D_{|c|}.

Now we assume that D_{|c|} and x_{|c|} are defined by the above construction (non canonically) for every c ∈ F^{×}. Let

D = ⋃{D_{|c|} : c ∈ F^{×}}

and

x(t) = x_{|c|}(t) for each c ∈ F^{×}, t ∈ D_{|c|},

a bijective function D → ^{∗}F. Denote x̌ the function D_{|c|} → F^{∗} defined as

x̌(t) = a^{−1}x(h(t)).

This is the dual coordinate function. It is bijective as well since h is bijective. In particular, it follows that h̄(t) = x^{−1}(x̌(t)) is a bijection on D and on each D_{|c|}. By definitions the identity

x(t) · x(t) = p(s) = x̌(t) · x̌(t)  (6)

holds on D. It follows that in each neighborhood V_{c} the restriction of the functions satisfy

x_{c} = x̌_{c} or x_{c} = −x̌_{c}.

But (1) forces that exactly for two out of four neighborhoods in D_{|c|} the second happens, that is we have x̌_{c} = −x_{c} = x_{−c}. It follows that applying a^{−1}x(h(t)) twice we get

ˇx̌(t) = a^{−2}x(h^{2}(t)) = x(t), any t ∈ D  (7)

and

h̄(h̄(t)) = t.  (8)

2.1.2 We now note that our construction of x and x̌ have some effect on F. Define the sign function

sgn(t) := \frac{x̌(t)}{x(t)}.

By (6) the value of sgn(t) is either 1 or −1. We then have

sgn(f(t)) = \frac{x̌(ft)}{x(ft)} = \frac{−ix̌(t)}{ix(t)} = −sgn(t).

In other words, the sign function of p(f^{k}t) changes the sign when k is odd and has the same sign as p(t) when k is even. So, it indeed depends on the value of p(t) rather than on t. We let sgn p(t) = sgn(t). Hence, we obtained a sign function on F^{×}

sgn : F^{×} → {1, −1}.

Using (2) we get the properties, for all λ ∈ F^{×} :

sgn(−λ) = −sgn(λ),
sgn(a^{2}λ) = sgn(λ).

By (6)

sgn x^{2}(t) = sgn p(t) = sgn(t).

Hence

sgnx(t)x̌(t) = sgn(x^{2}(t)sgn(t)) = sgn(x^{2}(t))sgn(t) = sgn(t)sgn(t)) = 1,

that is

sgn x(t)x̌(t) = 1, all t ∈ D.  (9)

2.1.3 The space of locally Zariski functions. A function ψ : D → ^{∗}F will be called locally Zariski if for any s ∈ D the function

ψ_{|V_{s}} : V_{s} → ^{∗}F

is a local Zariski continuous function (in particular, definable). By construction coordinate functions x and x̌ are locally Zariski. The sum and and the product of two locally Zariski functions are locally Zariski.

Let H be the F-algebra of locally continuous functions on D and H_{0} its subalgebra generated by x and x̌. We may also regard both as an F-vector space and its subspace with some linear operators on them. We define linear operators X, X̌ and F on H as follows:

X : ψ(t) ↦ x(t)ψ(t);
X̌ : ψ(t) ↦ x̌(t)ψ(t);
F : ψ(t) ↦ ψ(f(t)).

We thus have the following relations between the operators:

XX^{−1} = X^{−1}X = 1,  (10)
X^{2} = X̌^{2},  (11)
F X = iXF, F X̌ = −iX̌F  (12)
F^{4} = 1  (13)

So, we define the algebra A(D) = A

A = ⟨X, X^{−1}, F, H⟩

the F-algebra with given generators subject to relations (10)-(13). Identity (11) follows from (6). Identity (12) follows from (5). and (13) follow from (1).

Remark. The operator identities (10)-(13) give enough information on f and h we want to recover later. Indeed, we are going to show that even “smaller” algebra is sufficient for this purpose, since it has equivalent category of irreducible modules.

2.1.4 Remark. Consider the structure (D, x), that is D in the language expanded by the locally Zariski functions of 2.1.3 (x̌ is definable using x). Obviously, its substructure (D, x) gives rise, by the procedure of 2.1.3 to the same algebra A. We claim (without giving a proof here) that (D, x) ≼ (D, x). Moreover, the result of the construction in 2.1.3 is independent of the choices we had to make in the course of defining x(t). This brings the following

Question Consider a structure (D, x) which is existentially closed in the class of structures satisfying (4) and (5). What is the model-theoretic status of the theory of this structure? Is it simple?

Since (F, sgn) is definable in (D, x) we ask the same question about existentially closed theories of the field endowed by a sign function.

2.1.5 Natural irreducible representations Given t ∈ D, consider the subspace of H

H_{t}^{0} = {ψ ∈ H : ψ(gt) = 0 for all g ∈ ⟨f, h̄⟩},

where ⟨f, h̄⟩ is the 8-element group generated by f and h̄ (see subsection 2.1.1 and (7)). Notice that H_{t}^{0} is invariant under the action of X, X̌ and F and so is an Asubmodule of H. Checking definitions we see that the function xx̌ − x(t)x̌(t) is in H_{t}^{0}. Hence the quotient-space is the 4-dimensional A module

M_{t} = H/H_{t}^{0} = ⟨1, x, x̌, x^{2}⟩

where 1, x, x̌, x^{2} are the elements of the quotient corresponding to 1, x, x̌.x^{2}. Denote λ(t) := x(t)x̌(t). The natural representation determined by t is given by the following action on the basis of M_{t} :

X : \begin{matrix}1 ↦ x \\ x ↦ x^{2} \\ x̌ ↦ λ(t)1 \\ x^{2} ↦ λ(t)x̌. \end{matrix}

Similarly,

X̌ : \begin{matrix}1 ↦ x̌ \\ x ↦ λ(t)1 \\ x̌ ↦ x^{2} \\ x^{2} ↦ λ(t)x, \end{matrix}

F : \begin{matrix}1 ↦ 1 \\ x ↦ ix \\ x̌ ↦ −ix̌ \\ x^{2} ↦ −x^{2}. \end{matrix}

One can easily find that the following are independent eigenvectors for X

ψ_{0}(t) = x(t)^{2}1 + x(t)x + x̌(t)x̌ + x^{2},

with eigenvalues x(t),

ψ_{1}(t) = x(t)^{2}1 + ix(t)x − ix̌(t)x̌ − x^{2},

with eigenvalues ix(t),

ψ_{2}(t) = x(t)^{2}1 − x(t)x − x̌(t)x̌ + x^{2},

with eigenvalues −x(t), and

ψ_{3}(t) = x(t)^{2}1 − ix(t)x + ix̌(t)x̌ − x^{2},

with eigenvalues −ix(t).

The same vectors are eigenvectors of X̌ with eigenvalues x̌(t), −ix̌(t), −x̌(t), ix̌(t), correspondingly.

The module M_{t} is annihilated by XX̌ − x(t)x̌(t).

In other words, all the eigenvalues of the operator XX̌ obtained by natural representations in correspondence with (9) are of the sign +1. We formulate it as the extra assumption on algebra A(D). We assume that F is endowed with a sign function

sgn : F^{×} → {1, −1}

satisfying the properties, for all λ ∈ F^{×} :

sgn(−λ) = −sgn(λ),
sgn(a^{2}λ) = sgn(λ).

We now declare that we restrict ourselves to those representations of A only for which

the spectrum of operator XX̌ is of sign +1.  (14)

2.1.6 Comments The spectrum of an operator Y in a Banach algebra B is defined in terms more general than eigenvalues. More precisely, spec Y is the set of λ ∈ ℂ such that Y − λ is not invertible. This notion strongly depends on the norm of B. Thus, our assumtion (14) is a very weak version of a norm on A(D).

In bigger generality we would like to use an n-sign function

sgn_{n} : F^{×} → {1, ϵ, …, ϵ^{n−1}},

where ϵ is a primitive root of unity of power n, with the properties

sgn_{n}(ϵλ) = ϵ · sgn_{n}(λ),
sgn_{n}(a^{n}λ) = sgn_{n}(λ).

Example 1. The partition of ℂ^{×} in 4 sectors in subsection 1.0.3 gives rise to a sign function

sgn_{4}(t) = i^{m}, if t ∈ K_{m}, m = 1, 2, 3, 4.

Similarly we can use the partition in (3) to define sgn_{2}.

There are natural operators acting on spaces of functions satisfying the assumption on the spectrum above or similar to it.

Example 2. Consider the space G_{′} of bounded C^{∞} functions ψ(t) : ℝ_{+} → ℂ on the positive reals with values in ℂ. Consider the differentiation operator \frac{d}{dt}. Then the eigenvectors of the operators can be described as multiples of exp(a + bi)t for a, b ∈ ℝ, a ≤ 0. And the spectrum consists of all such a + bi, that is equals to the closed left half plane

\{z ∈ ℂ : \frac{π}{2} ≤ arg z ≤ \frac{3π}{2} \}.

Example 3. Given a real 0 ≤ α < 2π, let

ℂ_{α} = {z ∈ ℂ : 0 ≤ arg z ≤ α}.

Let G_{>α} be the space of C^{∞} functions ψ(t) : ℂ_{α} → ℂ which have a bounded extension to ψ(t) : ℂ_{β} → ℂ for some real β > α. The differentiation operator \frac{d}{dt} again has the eigenvectors of the form exp(ct) for c ∈ ℂ, π/2 ≤ arg c + α < 3π/2. So the spectrum equals to this sector. Choosing α = (n − 2)π/n we get the spectrum

S_{n} = \{c ∈ ℂ^{×} : \frac{π}{2} ≤ arg c < \frac{π}{2} + \frac{2π}{n} \}.

Let ϵ be a primitive root of unity of power n. Then

ℂ^{×} = S_{n} ˙∪ϵS_{n} ˙∪ … ˙∪ϵ^{n−1}S_{n}

is a partition of ℂ^{×} and so determines the function

sgn_{n} : ℂ^{×} → {1, ϵ, …, ϵ^{n−1}}.

2.1.7 Irreducible representations of A. By definitions (see e.g. [BG, III]) A is a PI-algebra with center Z = F[Y, Y^{−1}], where Y = XX̌, and is a finite extension over the center. The action of A on H is an infinite-dimensional representation. All the irreducible representations of A are 4-dimensional over F and can be described as follows. By [BG], section III.1 (a consequence of Kaplansky’s theorem and Artin-Tate’s lemma) every irreducible A-module M is of finite-dimension over F and is annulated by a maximal ideal of the center Z. But Z ≅ F[Y, Y^{−1}] and so a maximal ideal I of Z is of the form ⟨Y − λ⟩, for some λ ∈ F^{×}. In other words Z/I ≅ F and Y = XX̌ acts on M as the multiplication by λ. It follows that the action of A on M reduces to the a action of the F-algebra

Ā_{λ} = ⟨X, X̌, F : XX̌ = λ, F^{4} = 1, XF = iFX, X̌F = −iFX̌⟩,

where X, X̌, F correspond to X, X̌ and F. Let e ∈ M be a common eigenvector of commuting operators X and X̌, and let X · e = μ · e. Then by (12) F e, F^{2}e, and F^{3}e are eigenvectors of X with eigenvalues iμ, −μ and −iμ correspondingly. We now want to determine the corresponding X̌-eigenvalues, in fact it suffices to determine the eigenvalue μ̌ for e_{0} only, since then the X̌-eigenvalues for e_{1}, e_{2} and e_{3} will be determined by (12) as −iμ̌, −μ̌, iμ̌ correspondingly. So let

X̌e_{0} = μ̌e_{0}.

Since XX̌e_{0} = μμ̌e_{0} = λe_{0} and μ^{4} = λ^{2}, we have two cases here:

μ^{2} = λ, μ̌ = μ  (15)

and

μ^{2} = −λ, μ̌ = −μ.  (16)

Both cases are consistent with the action of A, hence we have two nonisomorphic polarised A-modules annihilated by XX̌ − λ. Here polarised means that the module is given with its basis e_{0}, e_{1}, e_{2}, e_{3} and two polarised modules are considered isomorphic if the correspondence between the bases is an isomorphism of modules. We denote the modules above M^{σ}(μ) where σ = +1, in case (15), and σ = −1, in case (16). We denote their canonical bases as

{e_{0}^{σ}(μ), …, e_{3}^{σ}(μ)}.

Remark M^{+}(μ) and M^{−}(μ) are isomorphic as modules without polarisations.

2.1.8 Note that for each module M^{σ}(μ) constructed in 2.1.7

I = I^{σ}(μ) = {L ∈ A : Le^{σ}(μ)_{0} = 0},  (17)

is a maximal left ideal containing XX̌ − μμ̌. Moreover, with each such ideal we associate uniquely the polarised A-module

M(I) = A/I

with the choice of eigenvectors for X,

e_{0} = 1 + I, e_{1} = F + I, e_{2} = F^{2} + I, e_{3} = F^{3} + I

with eigenvalues μ, iμ, −μ, −iμ, correspondingly. There is one-to-one correspondence between polarised modules M(I) and maximal ideals of the form (17).

The ideals I corresponding to M^{+}(μ) and M^{−}(μ) are uniquely defined by their generators

I^{+}(μ) = ⟨(X − μ), (X̌ − μ)⟩

and

I^{−}(μ) = ⟨(X − μ), (X̌ + μ)⟩.

2.1.9 Let 2D = 2D(A) be the set of all the left modules above, that is

2D(A) = {I^{+}(μ), I^{−}(μ) : μ ∈ F^{×}} = {I^{σ}(μ) : μ ∈ F^{×}, σ = 1, −1}.

We consider the following functions on 2D :

f : I^{σ}(μ) ↦ I^{−σ}(iμ)

and

h : I^{σ}(μ) ↦ I^{σ}(σaμ).

We then have

f^{2} : I^{σ}(μ) ↦ I^{σ}(−μ),

fh : I^{σ}(μ) ↦^{h} I^{σ}(σaμ) ↦^{f} I^{−σ}(iaμ),
hf^{3} : I^{σ}(μ) ↦^{f} I^{−σ}(−iμ) ↦^{h} I^{−σ}(iaμ).

We may conclude that f^{4} is identity and fh = hf^{3} in accordance with (1).

We also consider the left ideals of A of the form

J(ν) = ⟨X^{2} − ν⟩, ν ∈ F^{×},

that is generated by elements of the form X^{2} − ν. We may identify F̃^{×} with F^{×} and introduce the usual (algebraic) Zariski relations and functions on it. In particular, we write cJ(ν) for J(cν) and so on. Let

F̃^{×} = {J(ν) : ν ∈ F^{×}}

and consider the map p : D → F̃^{×} given as

p : I^{σ}(μ) ↦ J(μ^{2}).

This is a 4-1 map satisfying

pf : I^{σ}(μ) ↦^{f} I^{−σ}(iμ) ↦^{p} J(−μ^{2})

and

ph : I^{σ}(μ) ↦^{h} I^{σ}(σaμ) ↦^{p} J(a^{2}μ^{2}).

That is we have

pf = −p, ph = a^{2}p

in correspondence with (2).

2.1.10 Representations with a given sign. We finally come to use the last part of the definition of A(D) given by assumption (14). This corresponds to the requirement that in the definition of A_{λ} in 2.1.7 we assume sgn(λ) = +1. By (15) and (16),

λ = σμ^{2} iff σ = sgn(μ^{2}).

Consequently, we let

D(A) = {I^{σ}(μ) ∈ 2D(A) : sgn(μ^{2}) = σ}.

Note that D(A) is closed under f and h, by the definitions in 2.1.9. Also, for λ ∈ F^{×} with sgn(λ) = σ,

p^{−1}(λ) ∩ D(A) = \{I^{σ}(\sqrt{λ}), I^{σ}(−\sqrt{λ})\}

that is the restriction of p to D(A) is a 2-1 map satisfying (2). This proves

Proposition D(A) is a model of Hrushovski’s example.

3 A generalisation

3.0.11 A coordinate algebra B_{ϵ}. Let ϵ be a primitive root of unity of order n. We define an F-algebra B_{ϵ} = B

B_{ϵ} = ⟨U, U^{−1}, V, V^{−1}⟩

as the algebra given by the generators and relations (18)-(20).

UU^{−1} = 1, V V^{−1} = 1,  (18)
U^{n} = V^{n},  (19)
UV = ϵV U.  (20)

Comment. Consider U = XF and V = X̌F in algebra A(D) of section 2. Then (18)-(20) hold for ϵ = −1, n = 2. That is B_{−1} is a subalgebra of A.

3.0.12 Irreducible representations of B_{ϵ} The centre of B = B_{ϵ} obviously contains U^{n} = V^{n}, so the algebra is a finite over its centre Z. Hence any irreducible B-module M is finite dimensional and is annihilated by a maximal ideal of the centre. It is easy to check that Z = F[Y, Y^{−1}], Y = U^{n} = V^{n}, so a maximal ideal of Z is of the form ⟨U^{n}−λ⟩, λ ∈ F^{×}. Hence M is a B̄_{λ}-module, where

B̄_{λ} = ⟨Ū, V̄ : Ū^{n} = V̄^{n} = λ⟩.

Let e ∈ M be an eigenvector of U with eigenvalue μ. Then μ^{n} = λ. Let μ̌ satisfy μ̌^{n} = λ and define

e_{k} = μ̌^{−k}V^{k}e.

Since

UV^{k}e = ϵ^{k}V^{k}Ue = μϵ^{k}V^{k}e,

the e_{k} are U-eigenvectors for k = 0, …, n − 1 with distinct eigenvalues ϵ^{k}μ. It follows M is spanned by {e_{0}, …, e_{n−1}} and its isomorphism type as a B-module with a polarisation is determined by μ and μ̌. Let σ = μ̌/μ. This can be any root of unity of power n. We denote this polarised module M^{σ}(μ).

Remark M^{σ}(μ) and M^{τ}(ν) are isomorphic as B_{ϵ}-modules without polarisations, for any σ, τ and μ, ν satisfying σ^{n} = 1 = τ^{n} and μ^{n} = ν^{n}.

3.0.13 Let nD = nD(B_{ϵ}) be the set of all the polarised modules above, that is

nD = {M^{σ}(μ) : σ^{n} = 1, μ ∈ F^{×}}.

Let δ^{n} = ϵ, that is δ is a root of ϵ of order n. We consider the following functions on nD :

f_{n} : M^{σ}(μ) ↦ M^{ϵσ}(δμ)

and

h_{n} : M^{σ}(μ) ↦ M^{σ}(σaμ).

We then have,

f_{n}^{n} : M^{σ}(μ) ↦ M^{σ}(ϵμ),
f_{n}^{n^{2}} : M^{σ}(μ) ↦ M^{σ}(μ),

that is the order of f_{n} is n^{2}.

f_{n}h_{n} : M^{σ}(μ) ↦^{h} M^{σ}(σaμ) ↦^{f} M^{ϵσ}(δσaμ),
h_{n}f_{n}^{n^{2}−n+1} : M^{σ}(μ) ↦^{f^{n^{2}−n+1}} M^{ϵσ}(ϵ^{−1}δμ) ↦^{h} M^{ϵσ}(δσaμ).

Hence

f_{n}h_{n} = h_{n}f^{n^{2}−n+1},

that is the action by conjugation by h_{n} on the finite cyclic group of order n^{2} is determined as f_{n} ↦ f_{n}^{n^{2}−n+1}.

We define

D(B_{ϵ}) = {M^{σ}(μ) ∈ nD : σ = sgn_{n}μ^{n}}.

It is immediate from definitions that D(B_{ϵ}) is closed under f_{n} and h_{n} and that the map

D(B_{ϵ}) → F^{×} : M^{σ}(μ) ↦ μ^{n}

is n-to-1. The same arguments as in 1.0.2 (more details in [HZ] section 10, especially Proposition 10.1) show that D(B_{ϵ}) is an irreducible Zariski curve, not definable in an algebraically closed field.

References

[HZ] E.Hrushovski and B.Zilber, Zariski Geometries. Journal of AMS, 9(1996), 1-56

[Z1] B.Zilber, Lecture notes on Zariski structures, 1994-2004, web-page