Dec 19, 2002 - observation is that definable sets of finite Morley rank in differentially ... ule for (R, â) is an R-module V together with a sequence D = (D0,D1, .
Two remarks on differential fields Anand Pillay∗ University of Illinois at Urbana-Champaign December 19, 2002
Abstract We make two observations concerning differential fields. The first is a model-theoretic proof of the existence and uniqueness of a PicardVessiot extension for an iterative linear differential equation (in positive characteristic) answering a question of Hrushovski. The second observation is that definable sets of finite Morley rank in differentially closed fields of characteristic zero are (possibly incomplete) Zariskitype structures, answering a question of Zilber.
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Introduction
The common theme in this paper is the structure of “finite-dimensional” sets definable or type-definable in differential fields. We first consider iterative differential fields of positive characteristic. Such an object is a field K equipped with an iterative Hasse derivation ∂ = (∂i : i = 0, 1, 2, ..). An iterative linear differential equation over K is a system (*) ∂i (y) = Ai y, for i = 1, 2, .., (where y is a 1 × n column vector of unknowns and each Ai is an n × n matrix over K) which comes from a ∂-module M over K. A Picard Vessiot extension of K is an iterative differential field extension of K generated by a fundamental matrix of solutions for (*) and which has no new constants. Such things ∗
Partially supported by NSF grants and a Humboldt Foundation Award
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were considered by Okugawa [10] and more recently by Matzat and van der Put ([7], [8]). Matzat and van der Put proved existence and uniqueness of the Picard-Vessiot extension (which were left open by Okugawa). We will give a proof of this using the model theory of separably closed strict Hasse fields of Ersov invariant 1. This was more or less asked for in [3]. Everything generalizes easily to the case of finitely many commuting Hasse derivations. In the characteristc zero case, Poizat [14] used the existence of prime models (in the theory of differentially closed fields of characteristic zero) to get a nice account of the Picard-Vessiot theory. This does not help in the positive characteristic case; one reason being that we are dealing with a really infinite system of equations, another that there do not exist prime models over arbitrary sets in the theory of separably closed fields. We deal with the problem by considering an associated type-definable structure which has finite Morley rank. Our second observation concerns definable sets and/or types of finite Morley rank in a differentially closed field of characteristic zero. Zilber asked whether such a definable set can be viewed as a “Zariski-type structure” as in [19]. The answer is yes, and I am sure is known to several other people. In any case as this was not made explicit earlier, we supply some details. The interest here is connected to whether there is a kind of fine structure theory common to all higher-dimensional Zariski structures. The paper [5] gives such a fine structure theorem in the strongly minimal case. In particular any strongly minimal Zariski structure can be “compactified”. Compactifiability of arbitrary Zariski structures, especially those arizing from finite Morley rank sets in differentially closed fields, seems to be an interesting question. I would like to thank Ehud Hrushovski, Martin Ziegler, and Boris Zilber, for questions and discussions around the topics of this paper. Thanks are also due to Andreas Baudisch and the Humboldt University of Berlin for their hospitality in Autumn 2001 when work on this paper was begun.
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Iterative linear differential equations and Picard-Vessiot extensions
In this section I will define the algebraic objects. We will work with fields and rings of characteristic p > 0 equipped with a single Hasse derivation.
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Everything we say extends to the case of fields equipped with a finite number of commuting Hasse derivations. We will use the expressions “Hasse derivation” and “iterative derivation” interchangeably. Definition 2.1 (a) A Hasse derivation ∂ on a ring R is a sequence (∂0 , ∂1 , ...) of additive endomorphisms of R such that (i) ∂0 = identity. P (ii) ∂n (xy) = i+j ∂i (x)∂j (y). ! n+m (iii) ∂n (∂m (x)) = ∂n+m (x). n (b) Let (R, ∂) be an iterative differential ring. An iterative differential module for (R, ∂) is an R-module V together with a sequence D = (D0 , D1 , ...) of additive maps from V to V , such that (i) D0 is the identity. P (ii) for r ∈ R and v ∈ V , Dn! (rv) = i+j=n ∂i rDj (v), n+m (iii) Dn (Dm (v)) = Dn+m (v) for v ∈ V . n The ring of constants CR of an iterative differential ring (R, ∂) is by definition {r ∈ R : ∂n (r) = 0 for all n}. Lemma 2.2 Let (V, D) be an s-dimensional dimensional ∂-module over the iterative differential field (K, ∂). Then V ∂ = {v ∈ V : Dn (v) = 0 for all n} is a vector space over CK of dimension at most s. Proof. Clearly V ∂ is a vector space over CK . We will show by induction on r that if v1 , .., vr ∈ V ∂ are linearly independent over CK then they are linearly independent over K. Suppose v1 , .., vr+1 ∈ V ∂ are linearly independent over CK . By induction hypothesis v1 , .., vr are linearly independent over K. Suppose by way of contradiction that vr+1 = λ1 v1 + ... + λr vr where the λi are P from K and are not all zero. For each n, 0 = Dn (vr+1 ) = Dn ( i=1,..r λi vi ) = P P P i=1,..,r ∂n (λi )vi . Hence ∂n (λi ) = 0 for all i=1,..,n ( k+l=n ∂k (λi )Dl (vi )) = n ≥ 1 and i = 1, .., r. It follows that the λi are all constants, a contradiction. Suppose now that (V, D) is a finite-dimensional ∂-module over the iterative differential field (K, ∂). After fixing a basis (e1 , .., es ) of V over K, the sequence of equations (Dn (v) = 0 : n = 0, 1, ..) on V translates into a sequence of equations (∂n (y) = Bn y : n = 0, 1, ..) on K n . (Here y is a 1 × s column vector of unknowns and Bn is an s × s matrix over K.) 3
Definition 2.3 (i) By an iterative linear differential equation over the iterative differential field (K, ∂) we mean a sequence (*) (∂n (y) = Bn y : n = 0, 1, ...) associated to some finite dimensional ∂-module (V, D) over (K, ∂). (ii) By a fundamental matrix of solutions of (*) we mean an invertible square matrix U (s × s if y is 1 × s) over some iterative field extension (F, ∂) of (K, ∂), such that ∂n (U ) = Bn U for all n. Definition 2.4 Let (K, D) be an iterative differential field such that CK is algebraically closed. Let (*) (∂n (y) = Bn y : n < ω) be an iterative linear differential equation over K. By a Picard-Vessiot extension of (K, ∂) for the system (*) we mean an iterative differential field extension (F, ∂) of (K, ∂) such that (i) there is a fundamental matrix U of solutions of (*), with entries from F , (ii) F is generated (as an iterative differential field, and so also as a field) by K together with the entries of U , (iii) CF = CK . The above definition is the natural one, in analogy with the characteristic zero case. In [7] and [8], the definition of a Picard-Vessiot extension is on the face of it slightly stronger: it is also required that the iterative ring over K generated by the entries of U together with K be a simple iterative differential ring, namely has no nontrivial differential ideal. Existence and uniqueness of the Picard-Vessiot extension are proved rather quickly under this definition. Only after developing elements of the Galois theory is it shown that their definition coincides with our 2.4. (See Proposition 4.8 of [7]).
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Separably closed fields equipped with a Hasse derivation
The theory of separably closed fields equipped with Hasse derivation(s) was developed by Messmer and Wood [9] and subseqently by Ziegler [16]. In the case of Ersov invariant 1 the approaches coincide. We will be working in that context. We follow Ziegler’s presentation. Again everything generalizes to the case of Ersov invariant e with e commuting Hasse derivations. So 4
the language we work with is the language of rings together with a sequence (∂i : i < ω) of unary function symbols. The theory SCHp,1 (or just SCHp ) says that K is a separably closed field of Ersov invariant 1 (namely dimK p K = p), that (∂n : n < ω) is an (iterative) Hasse derivation on K, and that K p = {x ∈ K : ∂1 (x) = 0}. It follows that the field of constants of K is ∞ n K p = ∩n K p . In From [9] and [16] we have: Fact 3.1 (i) SCHp is complete, stable, has quantifier-elimination, and is the model companion of the class of iterative differential fields of characteristic p. (ii) SCHp has elimination of imaginaries. (iii) Let (K, ∂) |= SCHp and let F be a ∂-subfield. Then dcl(F ) in the sense n of the structure (K, ∂) is precisely {x ∈ K : xp ∈ K for some n ≥ 0}. It follows from (iii) above that: Remark 3.2 Let (K, ∂) be a model of SCHp and F a ∂-subfield, such that CF is an algebraically closed field. Then dcl(F ) ∩ CK = CF . The prime model of SCHp can be obtained as follows: Equip Fp (t) with an iterative derivation by defining ∂ by defining ∂1 (t) = 1 and ∂j (t) = 0 for j > 1. ∂ extends uniquely to the separable closure of Fp (t) and that is our model. Let now (U, ∂) be a saturated model of SCHp of “large” cardinality κ say, and let C denote its field of constants. Every iterative differential field of cardinality ≤ κ embeds in U. Moreover if (K, ∂) is an iterative differential subfield of U of cardinality < κ and L is an iterative differential extension of K of cardinality ≤ κ then there is an embedding of L into U over F . “Small” will mean of cardinality < κ. Let us fix a ∂-subfield K of U, such that the field CK of constants of K is algebraically closed, and also fix an iterative linear differential equation (*) ∂n (y) = Bn y, n = 0, .. where y is a 1 × s vector of indeterminates and the Bn are s × s matrices over K. We assume that this system is associated to a ∂-module (V, DV ) over K (after choosing a basis for V ). We will prove, using the model theory of SCHp : Proposition 3.3 There is a Picard-Vessiot extension F of K for (*). Moreover if F1 , F2 are Picard-Vessiot extensios of K for (*), then F1 and F2 are isomorphic over F . 5
We proceed via a few lemmas. Lemma 3.4 The system (*) has a fundamental matrix of solutions with coefficients from U (so by 2.2, the set of solutions of (*) in U is an s-dimensional vector space over C). Proof. Let R = U[xi,j : 1 ≤ i, j ≤ s] be the polynomial ring over U in s2 -indeterminates. As the system (*) comes from a ∂-module, one can show that, defining ∂n (x) = Bn x for all n (where x is the s × s matrix with entries the xi,j ), makes R into an iterative ring extension of (U, ∂). As (U, ∂) is a saturated existentially closed iterative differential field, the quantifier-free type of x over the coefficients of the Bn ’s, can be realized in U, giving us our fundamental matrix of solutions in U. Let X denote the set of solutions of (*) in U. So, by 3.4, X is an s-dimensional vector space over C. Note that both C and X are type-definable (over K). Let us introduce an auxiliary structure which we call M . M will be the 2-sorted structure whose sorts are C and X and whose relations are those induced from K-definable sets in U. That is, we add a predicate for a set Y ⊆ C n × X m if Y is the intersection of some K-definable set in U with C n × X m . Note that in the structure M , both the field structure on C and the C-vector space structure on X, are ∅-definable. Lemma 3.5 (i) Suppose Z ⊆ U n is definable in U n with parameters a. Let b = dcl(a) ∩ C. Then Z ∩ C n is definable in (C, +, ·) with parameters b. (ii) M is saturated, and T h(M ) has quantifier-elimination and finite Morley rank. Proof. (i) is of course well-known: First, by stability and quantifier-elimination we may assume that Z is quantifier-free definable in U with parameters b from C. But all the ∂i are zero on C, so Z ∩ C n is (quantifier-free) definable in (C, +, ·) as required. (ii) Let b be a basis for X over C. Then clearly the structure (M, b) is biinterpretable (without parameters) with the structure N which has universe C and relations induced from K ∪ {b}-definable sets in U. By (i) the latter structure is saturated. Thus (M, b) is too. In particular the structure M is saturated. By (i), T h(N ) has finite Morley rank. So T h(M, b) does too, and
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hence also T h(M ). Finally, note that the structure M is quantifier-free homogeneous: if finite tuples a1 , a1 from M have the same quantifier-free type in M then they have the same type over K in U so there is an automorphism of U over K taking a1 to a2 and thus an automorphism of M taking a1 to a2 . As M is also saturated, T h(M ) has quantifier-elimination.
Proof of Proposition 3.3. Note that U is a fundamental matrix of solutions for (*) (with coefficients from U just if the columns of U form a basis of X over C. By Lemma 3.5 (ii) T h(M ) is totally transcendental, so has a prime model M0 say, which is an elementary substructure of M . Then in M0 we may find a basis (b1 , .., bs ) of X over C. Claim 1. M0 ∩ C = CK . Proof. Let φ(x) be a formula over ∅ in the language of M which implies x ∈ C. Let Z be the subset of C defined by φ(x). By 3.2 and 3.5 (i), Z is definable over (CK in the structure C, +, ·), and thus meets CK , as the latter is an elementary substructure. As the type of any element of M0 ∩ C is isolated in T h(M ), this proves Claim 1. Now let F be the field generated by K together with the elements of b1 , .., bs . So clearly F is a ∂-subfield of U. Claim 2. CF = CK . Proof. Let a ∈ F ∩ C. So a = f (b1 , .., bs ) for some K-definable function f (−) in the sense of U. The graph of f intersect X ×..×X ×C is then a ∅-definable function in T h(M ), and thus a ∈ M0 ∩ C. By Claim 1, a ∈ CK . As (b1 , .., bs ) is a fundamental matrix of solutions of (*), we see, from Claim 2, that F is a Picard-Vessiot extension of K for (*). Now for uniqueness. First, as (b1 , .., bs ) is a tuple from M0 , its type in T h(M ) is isolated, by a formula ψ(x1 , .., xs ) say. Let (F 0 , ∂) be another Picard Vessiot extension of K for (*). We may assume that F 0 is a ∂-subfield of U containing K. Let (b01 , .., b0s ) be the fundamental matrix of solutions to (*) which generates F 0 over K. Considering b01 , ., b0s as elements of M let M1 ⊂ M be a prime model over b01 , .., b0s . Claim 3. M1 ∩ C = CK . Proof. Let Z be a (b01 , ., b0s )-definable subset of C in the sense of M . As T h(M ) has quantifier-elimination, and using 3.5 (i), Z is definable over some
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c ∈ C in the structure (C, +, ·), where c is in dcl(K, b01 , .., b0s ) in the sense of U. That is c ∈ dcl(F 0 ) ∩ C which by 3.2 is precisely CK . As in the proof of Claim 1, Z must meet CK . The claim is proved. Claim 4. (F 0 , ∂) is isomorphic to (F, ∂) over K. Proof. As M1 is a model of T h(M ) let (b001 , .., b00s ) realize ψ(x1 , .., xs ) in M1 (where recall ψ isolates the type of (b1 , .., bs ) in M ). So both (b01 , .., b0s ) and (b001 , .., b00s ) are bases for X ∩ M1 over C ∩ M1 . Using Claim 3, (b01 , .., b0s ) and (b001 , .., b00s ) can be exchanged by nonsingular matrices over CK . It follows that F 0 = K(b001 , .., b00s ). Now (b1 , .., bs ) and (b001 , .., b00s ) have the same type in M and so by definition of the structure on M , they have the same type over K in U. Thus F and F 0 are isomorphic (as ∂-fields) over K. The following may be of interest to model theorists. Remark 3.6 Let F be a Picard-Vessiot extension of K for (*), where F is a ∂-subfield of U. Then there is an elementary substructure K1 of U which contains F and satisfies CK1 = CK . Moreover X ∩ K1s = X ∩ F s . Proof. As SCHp is a countable theory, there are “locally atomic” models over all sets. (See [15].) In particular there is a locally atomic model K1 over F . What this means is that for every finite tuple a from K1 and formula φ(x, y) of the language of SCHp there is a formula ψ(x) ∈ tp(a/F ) such that ψ(x) implies φ(x, c) whenever φ(x, c) ∈ tp(a/F ). We claim that CK1 = CF (= CK ). Suppose for the sake of contradiction that there is a new contant a ∈ K1 . By the proof of Proposition 3.3, we have F = K(b1 , .., bs ) where b1 , .., bn are in M0 . So taking φ(x, y) to be x 6= y there is a formula ψ(x, b1 , .., bs ) = ψ(x, b) over K ∪ {b} such that |= ψ(a, b) and ψ(x, b) → x 6= c for each c ∈ CK . Let Z be the subset of U defined by ψ(x, b). Then Z ∩ C 6= ∅ and by 3.5(i), it is defined by a formula over CK in the structure (C, +, ·), and thus meets CK , a contradiction. As there are no new constants in K1 it follows that all solutions of (*) with coefficients from K1 are already in F . Let me finally note that there is also a model-theoretic account of the Galois theory, which amusingly makes heavy use of our auxiliary structure M . So I will sketch a proof of:
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Proposition 3.7 Let (F, ∂) be a Picard-Vessiot extension of (K, ∂) for the equation (*). Then Aut∂ (F/K) can be identified with an algebraic subgroup G of GLs (CK ). Moreover there is a Galois correspondence between algebraic subgroups of G and ∂-fields lying between K and F . Proof. We will make use of the following Fact 3.8 Any Picard-Vessiot extension of a ∂-field is a separable extension. By virtue of this fact we see that if E is a ∂-field lying between K and F then F is a separable extension of E, so by 3.1(iii), E is definably closed in F in the structure U. We work in the structure M . Let b = (b1 , .., bs ) be our basis for X over C which lies in the prime model M0 and generates F over K. By total transcendentality of T h(M ), M0 and also M0eq are homogeneous structures in the sense of model theory. Remember that M0 is a two-sorted structure with sorts X ∩ M0 and C ∩ M0 , and where moreover C ∩ M0 is precisely CK . We will be using this as well as other observations made during the proof of 3.2. Let us first set up an identification of Aut∂ (F/K) with Aut(M0 ) the group of automorphisms of the structure M0 . Let σ ∈ Aut∂ (F/K). Then σ(b) = b0 say. So b0 is another fundamental matrix of solutions for (*) hence b0 = b.A for some matrix A ∈ GLs (C). As A = b−1 b0 , A has coefficients from C ∩ F = CK . Thus b0 is in X ∩ M0 . By quantifier-elimination in U, b and b0 have the same type over K in U hence the same type in M0 . By homogeneity of M0 there is an automorphism σ 0 of M0 taking b to b0 . As b and b0 are both bases for X ∩ M0 over C ∩ M0 and σ 0 fixes C ∩ M0 = CK pointwise, σ 0 is determined by the fact that it takes b to b0 , so σ 0 is determined by σ. It is clear by this discussion that this establishes an isomorphism between Aut∂ (F/K) and Aut(M0 ). We will simply identify the two groups, and note we have seen that σ is determined by σ(b) (and b). Next we have an isomorphism ρ between Aut∂ (F/K) = Aut(M0 ) and a subgroup G of GLs (CK ) given by (**). σ(b) = b.ρ(σ). Note that G is type-definable in M0 , and thus is an algebraic subgroup of GLn (CK ). G is an algebraic subgroup. Note that M0 and hence also M0eq is contained in the definable closure of b. So, clearly if e ∈ M0eq and e = f (b)
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for some definable ∅-definable function in M0 , and σ ∈ Aut(M0 ) then σ(e) = f (σ(b)) = f (b.ρ(σ)). Next we set up a Galois correspondence between algebraic subgroups of G and definably closed subsets of M0eq . This is of course very well-known but we give some details for completeness. For H a subgroup of G let Ψ1 (H) = {e ∈ M0eq : (ρ−1 (g))(e) = e for all g ∈ H}. For C a subset of M0eq let Ψ2 (C) = {g ∈ G : ρ−1 (g)(e) = e for all e ∈ C}. Lemma 3.9 (i) For any subgroup H of G, Ψ1 (H) is definably closed in M0eq . (ii) For any subset C of M0eq , Ψ2 (C) is an algebraic subgroup of G. (iii) Ψ1 induces a (inclusion reversing) bijection between algebraic subgroups of G and definably closed subsets of M0eq , whose inverse is Ψ2 . Proof. (i) Note that Ψ1 (H) is the set of elements fixed by every automorphism of M0 which is in ρ−1 (H). So Ψ1 (H) is definably closed in M0eq . (ii) Clearly Ψ2 (C) is a subgroup H of G which is type-definable in M0 , hence an algebraic subgroup. (iii) Let H be an algebraic subgroup of G. Then the orbit of b under ρ−1 (H) is definable in M0 (by (**)), and so is an element e ∈ M0eq . Thus ρ−1 (H) is precisely {σ ∈ Aut(M0 ) : σ(e) = e}, so clearly H = Ψ2 (Ψ1 (H)). On the other hand if C is a definably closed subset of M0eq , then by ωstability C = dcleq (c) for some finite tuple from C. If d ∈ M0eq \ C then as M0eq is a prime, atomic, homogeneous, model there is an automorphism σ of M0 which fixes c and moves d. It follows that Ψ1 (Ψ2 (C)) is precisely C. This completes the proof of the Lemma. To complete the proof of Proposition 3.7 we describe an essentially tautological bijection between the set of definably closed subsets of M0eq and the set of ∂-fields lying between K and F . First any e ∈ M0eq is, by quantifier elimination, of the form b/E where E is a quantifier-free ∅-definable equivalence relation in M . E is the restriction to X s of some K-definable set E 0 in U. By stability we may assume E 0 is an equivalence relation. That is, we may assume e = b/E where E is a Kdefinable equivalence relation in U. By elimination of imaginaries in U, b/E is interdefinable over K in U with some tuple c from U which is in dcl(K, b). By 3.1(iii), F rn (c) lies in F for some n. Thus b/E is interdefinable over K in U with a tuple from F . 10
Conversely, if c ∈ F then c = f (b) for some K-rational function f . Let E(x, y) be f (x) = f (y). So the restriction of E to X s is ∅-definable in M and b/E ∈ M0eq . Aain b/E is “interdefinable” over K with c. So we have established a natural bijection between definably closed subsets of F containing K and definably closed subsets of M0eq . By the sentence following 3.8, the definably closed subsets of F containing K are precisely the ∂-fields between K and F . The identification of Aut∂ (L/F ) with Aut(M0 ) and then with G clearly yields a Galois correspondence between algebraic subgroups H of G and ∂-fields L between K and F , where H maps to L = F ix(H) < F . This completes the proof of 3.7. The interesting thing from the model-theoretic point of view is that we have more or less “interpreted” the ∂-field F in a many-sorted manner in M0eq .
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Finite-dimensional sets definable in differentially closed fields.
The notion of a Zariski structure appears in work of Boris Zilber in the early 90’s [17],[18]. The expression “Zariski-type structure” was used, and completeness (the analogue of compactness) was built into the definition. In [5], Hrushovski and Zilber introduced the notion of a Zariski geometry, which is, roughly speaking a possiby incomplete strongly minimal Zariski structure. They proved that a non locally modular Zariski geometry interprets an algebraically closed field. As a matter of fact the strong classification results in [5] show that possibly incomplete Zariski geometries can be “compactified”. In [18], Zilber noted that any compact complex manifold is a (complete) Zariski structure in a natural way. Various analogies between compact complex manifolds and finite-dimensional differential algebraic varieties have been observed in the meantime. Zilber asked us whether these finitedimensional differential algebraic varieties (essentially just finite Morley rank sets definable in a differentially closed field) are (possibly incomplete) Zariski structures. We will give an affirmative answer here. The result should be well-known to several people. It is implicit in Hrushovski’s “geometric” interpretation of differential equations in [2], as well as in [12]. For strongly minimal sets in differentially closed fields, the answer already appears in [4]. 11
In any case we consider our treatment here to be essentially expository. Let us start with the definition of a (possibly incomplete) Zariski structure. So by a Zariski structure we mean a set M together with a family D of subsets of M, M × M, ..., which we call the closed sets, and a map dim : D → N ∪ {−1}, satisfying axioms (i)-(iv) below: (i) For each n, the family of closed subsets of M n equips M n with a Noetherian topology. In particular, any closed S ⊆ M n is uniquely a finite irredundant union of closed irreducible subsets, the irreducible components of S. We also stipulate that each M n is irreducible. (ii) Any singleton in M n is closed. Any finite Cartesian product of closed sets is closed. The graph of equality is closed. The projections M n+1 → M n are continuous with respect to the topologies on M n+1 , M n , repectively. (iii) For closed sets S1 , S2 ⊆ M n , dim(S1 ∪ S2 ) = max{dim(S1 ), dim(S2 )}. The dimension of a singleton is zero. The dimension of the empty set is −1. If S is irreducible closed and S 0 is a proper closed subset of S then dim(S 0 ) < dim(S). (iv) Let π be the projection from M n+m to M n , and let S ⊆ M n+m be closed and irreducible. Let k = min{dim(π −1 (a) ∩ S : a ∈ π(S)}, and let π(S) be the closure of π(S). Then (a) dim(S) = dim(π(S)) + k, and (b) {a ∈ π(S) : dim(π −1 (a) ∩ S) = k} contains a nonempty open subset of π(S). The Zariski structure (M, D, dim) is said to be complete if it satisfies in addition (iv)’ Whenever π : M n+m → M n is a projection and S ⊆ M n+m is closed, then for every k ∈ ω, {a ∈ M n : dim(π −1 (a) ∩ S) ≥ k} is closed. (Note that (iv)’ implies directly that π(S) is closed.) The Zariski structure (M, D, dim) will be called pre-smooth if it also satisfies the dimension theorem: (v) Let S1 , S2 ⊆ M n be closed and irreducible. Then any irreducible component of S1 ∩ S2 has dimension ≥ dim(S1 ) + dim(S2 ) − dim(M n ). Note that an irreducible algebraic variety X (over an algebraically closed field) equipped with the Zariski topology on each of X, X × X,.., is a Zariski 12
structure, where dim(−) is the usual algebraic-geometric dimension. Moreover if X is nonsingular then it satisfies pre-smoothness. Let us start with an easy observation, showing that our set-up agrees with Zilber’s in [19]. We will call a subset X of M n constructible if it is a finite Boolean combination of closed sets (that is a finite union of locally closed sets). Lemma 4.1 Let (M, D, dim) be a Zariski structure. Let π be a projection from M n+m to M n and let S be an irreducible closed subset of M n+m . Then (i) π(S) is constructible. (ii) For each r ≥ 0, {a ∈ π(S) : dim(π −1 (a) ∩ S) = r} is constructible. Proof. Note that π(S) is irreducible. Also note that (ii) implies (i). We will prove (ii) by induction on s = dim(π(S)). If s = 0 it is clear. Suppose s > 0. Let k be as in Axiom (v) and let (by axiom (v)), U be an open nonempty subset of π(S) such that dim(π −1 (a)∩S) = k for a ∈ U . Let U 0 = π −1 (U )∩S, an open nonempty subset of S. Let S1 , .., St be the irreducible components of S \ U 0 . For each i, π(Si ) is a proper closed subset of π(S) so has dimension < s. So we can apply the induction hypothesis and this clearly gives the desired conclusion (using Axiom (iii)). A Zariski structure (M, D, dim) can (and will) be considered as a first order structure with universe M and basic relations the closed sets. We will just call this structure M , hopefully with no ambiguity. By Lemma 4.1 and Theorems 3.1 and 3.5 from [19] T h(M ) has quantifier-elimination (every definable set is constructible) and finite Morley rank. Moreover the Morley rank of a definable set is bounded by the dimension of its closure. M is not saturated because to all intents and purposes we have constants for all elements of M . If M ∗ is a saturated elementary extension of M , then M ∗ becomes a Zariski structure in its own right, by defining a subset X of (M ∗ )n to be closed if there is a closed subset S of M n+m for some m, and b ∈ (M ∗ )m such that X = {a ∈ (M ∗ )n : (a, b) ∈ S(M ∗ )}. We also have to define the dimension of such a set X. By Lemma 4.1 for any r, {b0 ∈ M m : dim(Sb0 ) = r} is definable in M by a formula φr (y). We put dim(X) = r if M ∗ |= φr (b). One cannot and should not expect dimension and Morley rank (or U -rank) to coincide. Moreover, although both in M and in the elementary extension M ∗ , dimension is definable (by (v)), there are examples where Morley rank is not definable in M ∗ (see [13]). 13
Now we pass to finite Morley rank sets definable in a differentially closed field. Let (K, +, ·, ∂) be a differentially closed field of characteristic zero. We will prove: Proposition 4.2 Let X ⊆ K n be a definable set of finite Morley rank and Morley degree 1. Then after possibly removing from X a set of smaller Morley rank than X, X can be equipped with a pre-smooth Zariski structure D, dim, in such a way that the subsets of X n definable in K are precisely those definable in (X, D). Remark 4.3 (i) The closed sets will be essentially differential algebraic sets and the dimension will correspond to the “order” of the defining equations. (ii) The Zariski structure on X will be closely related to the fact that X corresponds to an “algebraic D-variety” in the sense of Buium[1]. Before getting into the proof of 4.2, let us recall the construction of the “prolongation” of an algebraic variety defined over a differential field. In general we refer to [6] for background on the model theory of differential fields, although we will try to be as self-contained as possible. Fix a differentially closed field (K, +, ·, ∂) of characteristic zero. Let X ⊆ K n be an (affine) algebraic variety, possibly reducible. Let I(X) ⊆ K[x1 , .., xn ] be the ideal consisting of those polynomials vanishing on X. We define τ (X) to be the subvariety of K 2n defined by equations P (¯ x) = 0 and P ∂ ∂ x)ui + P (¯ x) = 0, for P ∈ I(X). Here P denotes the i=1,..,n (∂P/∂xi )(¯ polynomial obtained from P by hitting the coefficients with ∂. It is enough to require that the polynomials range over a generating set for I(X). There is a natural projection π : τ (X) → X. The pair (τ (X), π) is the first prolongation of X. If X is defined over the field CK of constants of K, then (τ (X), π) identifies with the tangent bundle of X. In general τ (X) is clearly a “torsor” for the tangent bundle T (X). That is there is a fibrewise action of T (X) on τ (X), such that each fibre of τ (X) is a principal homogeneous space for the corresponding fibre of T (X). Clearly if Y is a subvariety of X then τ (Y ) is a subvariety of τ (X) over the inclusion Y ⊆ X. Moreover, as for tangent bundles: Fact 4.4 Suppose that Y is an irreducible component of X. Then over some Zariski open subset U of Y , τ (X) = τ (Y ). It follows from [11] that: 14
Fact 4.5 Suppose that s : X → τ (X) is a polynomial map (over K) which is a section of π : τ (X) → X. Then for any irreducible component Y of X and Zariski open subset U of Y there is a ¯ ∈ U such that ∂(¯ a) = s(¯ a). Finally we note that τ (−) is a functor, so τ (X × Y ) = τ (X) × τ (Y ). Proof of Proposition 4.2. Let m be the Morley rank of X. Note that we are free to iterate either of the following operations: (a) replace X by something in definable bijection with X, and (b) remove from X a definable set of Morley rank < m. In particular we may assume (using quantifier-elimination) that X is defined by a finite system of ∂-equations. Let (U, +, ·, ∂) be a saturated elementary extension of (K, +, ·, ∂). The proof will involve three steps. First, by iterating (a) and (b) replace X by a more amenable object. Second, define what we want the closed sets to be. Third, show that this family of closed sets works. Step 1. Let p(x) be the unique type over K which contains X and has Morley rank m. Let the finite tuple a from U realize p. As tp(a/K) has finite Morley rank, tr.deg(K < a > /K is finite. So after replacing a by some (a, ∂(a), .., ∂ r (a)), we have that ∂(a) ∈ K(a). After adjoining to a some elements of K(a) we have that ∂(a) ∈ K[a], say ∂(a) = s(a) where s(x) ∈ K[x]. a is a generic point of an affine algebraic irreducible variety V defined over K, which we may assume to be nonsingular. Identify V with its set of U-points. s : V → τ (V ) is a regular section of τ (V ) → V . Let V ] = {x ∈ V : s(x) = ∂(x)}. V ] (K) is then something obtained from X by iterating the operations (a) and (b) above, so without loss of generality we will call it X. Step 2. For any n define Z ⊆ X to be closed if Z is defined by a finite system of ∂-polynomial equations over K. (A ∂-polynomial equation over K in indeterminates x¯ is something of the form P (¯ x, ∂(¯ x), .., ∂ r (¯ x)) = 0 where P is a polynomial over K.) D will be the set of closed subsets of X n as n varies. For Z ∈ D define dim(Z) to be the algebraic-geometric dimension (which we denote adim) of the Zariski closure of Z in V n . Before establishing the axioms, let us make the connection with “algebraic Dsubvarieties” of V . Firstly, the pair (V, s) is an algebraic D-variety (defined 15
over K) in the sense of Buium [1], as is (V n , s¯) for any n, where s¯ = (s, .., s). By definition an (algebraic) D-subvariety of V n defined over K will be a subvariety W of V n defined over K such that s¯|W is a map from W to τ (W ). Given such W , define W ] to be {a ∈ W : ∂(a) = s¯(a)}, and W ] (K) to be the points of W ] with coordinates in K. So W ] (K) = W ∩ X n . We will make a series of claims, some of which can be seen as a translation of facts from differential algebra into the language of D-varieties. Claim I. If Z ⊆ X n and W ⊆ V n is the Zariski closure of Z, then W is a D-subvariety of V n defined over K. If moreover Z ∈ D then Z = W ] (K). Proof. If a ∈ Z then a ∈ W . Applying ∂ to the equations defining W we see that (a, ∂(a)) ∈ τ (W ). But as a ∈ X n , ∂(a) = s¯(a). So for a Zariski-dense subset (namely Z) of W , (a, s¯(a)) ∈ τ (W ). It follows that this is true for all a ∈ W and so W is a D-subvariety of V n , clearly defined over K. For the second part: suppose that Z is defined by Pi (¯ x, ∂(¯ x), .., ∂ r (¯ x)) = 0 for i = 1, .., k. Let Qi (¯ x) be the polynomial obtained from Pi by substituting s¯(¯ x) for ∂(¯ x). So Z is defined (in the differential field K) by {Qi (¯ x) = 0 : i = 1, .., k} together with s¯(¯ x) = ∂(¯ x). Let W 0 be the variety defined by the {Qi (¯ x) = 0 : i = 1, .., k}. So Z = W 0 ∩ X n , and Z ⊆ W ⊆ W 0 , hence Z = W ∩ X n too, as required. Claim II. (i) Let W1 , W2 be D-subvarieties of V n , defined over K. Then W1 is properly contained in W2 if and only if W1] (K) is properly contained in W2] (K). (ii) Any irreducible component of a D-subvariety of V n is also a D-subvariety. Proof. (i) Right to left is obvious. For left to right: if W1 is properly contained in W2 , then W2 \ W1 contains a Zariski open subset U of an irreducible component of W2 , defined over K. By Fact 4.5, there is a ∈ U (K) such that ∂(a) = s¯(a). But then a ∈ W2] (K) \ W1 . (ii) This follows from 4.4. Claim III. Let Z ⊆ V n+m be an irreducible D-subvariety of V n+m , defined over K. Let π be the projection from V n+m to V n . Let W ⊆ V n be the Zariski closure of π(Z). Then (i) W is a D-subvariety of V n , defined over K. Moreover W is the Zariski closure of π(Z ] (K)). (ii) If a ∈ π(Z) ∩ X n then Za (=π −1 (a) ∩ Z) is a D-subvariety of V n+m , defined over K. Proof. (i) Clearly W is irreducible and defined over K. By Claim I, it 16
is enough to prove the second clause. Let U be a nonempty Zariski open subset of W , defined over K. Let U 0 = π −1 (U ) ∩ Z. So by 4.5, there is (a, b) ∈ U 0 ∩ X n+m . So a ∈ U ∩ π(Z ] (K)). (ii) The ideal of Za is generated by polynomials in the ideal of Z together with x1 − a1 , .., xn − an (where a = (a1 , ., an )). So (x, y, u, v) ∈ τ (Za ) iff (x, y, u, v) ∈ τ (Z) and x = a and u = ∂(a). As s¯(a) = ∂(a), we see that s¯|Za : Za → τ (Za ) whereby Za is a D-subvariety, defined over K. Claim IV. Let W1 , W2 be D-subvarieties of V n , defined over K. Then so is W1 ∩ W2 . Proof. Let I1 , I2 be the ideals of W1 , W2 respectively. Then the ideal I of W1 ∩ W2 is the radical of I1 + I2 . We have to show that for P ∈ I, P ∂P/∂xi (¯ x)¯ s(¯ x) + P ∂ (¯ x) = 0. As this is the case for P ∈ I1 and for P ∈ I2 it is also true for P ∈ I1 + I2 . The usual proof (see [6]) that the radical of a differential ideal is also a differential ideal, shows that it is true for P ∈ I too. Step 3. We will verify that (X, D, dim) satisfies axioms (i) to (iv) as well as (v). By Claim I, the sets in D are precisely those of the form W ] (K) = W ∩ X n for W a D-subvariety of V n defined over K. So by Claim II, we see that axiom (i) holds. Note also that if Z = W ] (K) for W a D-variety, Z is irreducible (in the sense of D) iff W is irreducible as a Zariski closed subset of V n . Axiom (ii) is immediate from the original definition of D. Axiom (iii) follows from the definitions as well as Claim II. Axiom (iv): Let Y ⊆ X n+m be D-closed and irreducible, and π : X n+m → X n the projection. Let Z ⊆ V n+m be the Zariski closure of Y . Allowing π to also denote the corresponding projection from V n+m to V n , let W be the Zariski closure of π(Z). So Z and W are irreducible D-subvarieties (over K) of V n+m and V n respectively. By definition dim(Y ) = adim(Z). By Claim III, W ] (K) is precisely the D-closure π(Y ) of π(Y ) ⊆ X n , so dim(π(Y )) = adim(W ). Let k = adim(Z) − adim(W ). By the validity of axiom (iv) for algebraic varieties, there is a nonempty Zariski open subset U of W , defined over K such that for a ∈ U , adim(Za ) = k (where again Za = π −1 (a) ∩ Z). Moreover for all a0 ∈ π(Z), adim(Za0 ) ≥ k. Let U 0 = U ∩ X n . Then U 0 is a nonempty open subset of π(Y ). Let a ∈ U 0 . So adim(Za ) = k, and by Claim III, Za is a D-subvariety of V n+m . Note that Za] (K) is precisely π −1 (a) ∩ Y . 17
Thus dim(π −1 (a) ∩ Y ) = k. On the other hand, for any other a0 ∈ π(Y ), adim(Za0 ) ≥ k as remarked above. Again by Claim III, π −1 (a0 )∩Y = Za]0 (K), whereby dim(π −1 (a) ∩ Y ) ≥ k too. This proves Axiom (iv). Axiom (v) for (X, D, dim) follows from its validity for the nonsingular algebraic variety V , together with Claim IV and Claim II(ii). Finally let us note that, by quantifier-elimination in differentially closed fields, as well as as separation of parameters (any subset of X n definable in (K, +, ·, ∂) with parameter in K is definable with parameters from X), the definable sets in (X, D) agree with those definable in the ambient structure (K, +, ·, ∂). So Proposition 4.2 is proved. Let us finish this paper with some additional questions. In the proof of 4.2, we considered affine algebraic D-varieties. One can also consider arbitrary algebraic D-varieties. That is, for an algebraic variety V defined over (K, ∂) (a differentially closed field, say), we can define π : τ (V ) → V by glueing affine pieces, and consider regular sections s : V → τ (V ) of π, defined over K. Such a pair (V, s) is, by definition, an algebraic D-variety. Again we define (V, s)] to be set of K-points of V such that s(x) = ∂(x). Let us introduce a language L(V,s) , whose relation symbols correspond to algebraic D-subvarieties of V n as n varies. So V becomes tautologically an LV,s -structure, and (V, s)] as a subset of V also acquires an LV,s -structure. The proof of Proposition 4.2 shows also that ((V, s)] , LV,s ) is a Zariski-structure. In particular, (V, s)] has quantifier-elimination as an LV,s -structure. Conjecture 4.6 With the above notation, (V, s)] is an L(V,s) -elementary substructure of V . The conjecture implies that any finite-dimensional differential algebraic set (equipped with all induced structure from the ambient differentially closed field) is interpretable in a “pure” algebraically closed field. Conjecture 4.6 is equivalent to V as above having quantifier-elimination in the language L(V,s) . In fact it would be enough to show that projection of a D-subvariety of V n+m on V n is a Boolean combination of D-subvarieties. In spite of appearances this is not accomplished in the proof of 4.2. On the other hand Conjecture 4.6 has been proved by the author and Piotr Kowalski in the case where V is an algebraic D-group. A final question is whether the Zariski structures obtained in the conclusion 18
of 4.2 can be “compactified”. One should of course give a precise definition of a compactification. The natural definition is as follows: by a compactification of a Zariski structure (X, D, dim) we mean a complete Zariski structure (X 0 , D0 , dim0 ) such that (i) X is a D0 -open subset of X 0 , (ii) for each n, the D-closed subsets of X n are precisely the intersections with X n of the D0 closed subsets of X 0n , (iii) for each D-closed Y ⊆ X n , dim(Y ) = dim0 (Y 0 ) where Y 0 is the D-closure of Y in X 0n .
References [1] A. Buium, Finite dimensional differential algebraic groups, Lecture Notes in Math., 1506, Springer, 1992. [2] E. Hrushovski, Geometric Model Theory, Proceedings of ICM 1998, vol. 1, Documenta Mathematica, 1998. [3] E. Hrushovski, Computing the Galois group of a linear differential equation, Differential Galois Theory, proceedings of Bedlewo meeting, Banach Centre Publications, vol. 58, IMPAN, 2002. [4] E. Hrushovski and Z. Sokolovic, Minimal subsets of differentially closed fields, preprint 1992. [5] E. Hrushovski and B. Zilber, Zariski Geometries, Journal of AMS 9 (1996), 1-56. [6] D. Marker, Model theory of differential fields, in Model Theory of Fields, by D. Marker, M. Messmer, A. Pillay, Lecture Notes in Logic 5, Springer 1996. [7] H. Matzat, Differential Galois Theory in positive characteristic, notes written by Julia Hartmann, preprint 2001. [8] H. Matzat and M. van der Put, Iterative differential equations and the Abhyankar conjecture, preprint 2001. [9] M. Messmer and C. Wood, Separably closed fields with higher derivations I, Journal of Symbolic Logic, 60 (1995), 898-910.
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[10] K. Okugawa, Basic properties of differential fields in arbitrary characteristic and the Picard-Vessiot theory, J. Math. Kyoto Univ. 2 (1963), 294-322. [11] D. Pierce and A. Pillay, A note on the axioms for differentially closed fields of characteristic zero, Journal of Algebra 204 (1998), 108-115. [12] A. Pillay and M. Ziegler, Jet spaces of varieties over differential and difference fields, to appear in Selecta Math. [13] A. Pillay and T. Scanlon, Compact complex manifolds with the DOP and other properties, Journal of Symbolic Logic 67 (2002), 737-743. [14] B. Poizat, A Course in Model Theory, Springer, 2000. [15] S. Shelah, Classification Theory, 2nd edition, North-Holland, 1990. [16] M. Ziegler, Separably closed fields with Hasse derivations, to appear in Journal of Symbolic Logic. [17] B. Zilber, Lecture notes on Zariski-type structures, preprint 1991. [18] B. Zilber, Model theory and algebraic geometry, Proceedings of 10th Easter Conference in Berlin, 1993, Humboldt Univ. of Berlin. [19] B. Zilber, Notes on Zariski www.maths.ox.ac.uk/ zilber)
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geometries,
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(see
