The model companion of partial differential fields with an automorphism

Transcription

1 The model companion of partial differential fields with an automorphism University of Waterloo May 8, 2003

2 Goals Axiomatize the existentially closed models of the theory of partial differential fields of characteristic zero with an automorphism in terms of characteristic sets of prime differential ideals. Establish the basic model theoretic properties of this theory. Use generalized Jet spaces to show the canonical base property and Zilber Dichotomy for finite dimensional types.

3 Model companions Determining the existence of a model companion is generally an interesting and rather difficult question. For several geometrically well behaved theories one can adapt the geometric axiomatization of ACFA: Differential fields (ordinary and partial) Fields with commuting Hasse-Schmidt derivations Fields with free operators Theories having a geometric notion of genericity Remark The theory of ordinary differential fields with an automorphism has a model companion. However, the techniques do not seem to extend to the partial setting.

4 Model companions Determining the existence of a model companion is generally an interesting and rather difficult question. For several geometrically well behaved theories one can adapt the geometric axiomatization of ACFA: Differential fields (ordinary and partial) Fields with commuting Hasse-Schmidt derivations Fields with free operators Theories having a geometric notion of genericity Remark The theory of ordinary differential fields with an automorphism has a model companion. However, the techniques do not seem to extend to the partial setting.

5 Notation We work in the language L,σ = L fields {δ 1,..., δ m } {σ} K {x} denotes the -ring of -polynomials over K If (K, ) = DCF 0,m, by a -closed subset of K n we mean the solution set in K n of a system of -polynomials over K. Irreducible -closed sets are defined naturally. A differential-difference field is an L,σ -structure (K,, σ) such that K is a field of characteristic zero, is set of m commuting derivations and σ is a -automorphism.

6 Notation We work in the language L,σ = L fields {δ 1,..., δ m } {σ} K {x} denotes the -ring of -polynomials over K If (K, ) = DCF 0,m, by a -closed subset of K n we mean the solution set in K n of a system of -polynomials over K. Irreducible -closed sets are defined naturally. A differential-difference field is an L,σ -structure (K,, σ) such that K is a field of characteristic zero, is set of m commuting derivations and σ is a -automorphism.

7 Notation We work in the language L,σ = L fields {δ 1,..., δ m } {σ} K {x} denotes the -ring of -polynomials over K If (K, ) = DCF 0,m, by a -closed subset of K n we mean the solution set in K n of a system of -polynomials over K. Irreducible -closed sets are defined naturally. A differential-difference field is an L,σ -structure (K,, σ) such that K is a field of characteristic zero, is set of m commuting derivations and σ is a -automorphism.

8 Notation We work in the language L,σ = L fields {δ 1,..., δ m } {σ} K {x} denotes the -ring of -polynomials over K If (K, ) = DCF 0,m, by a -closed subset of K n we mean the solution set in K n of a system of -polynomials over K. Irreducible -closed sets are defined naturally. A differential-difference field is an L,σ -structure (K,, σ) such that K is a field of characteristic zero, is set of m commuting derivations and σ is a -automorphism.

9 Characterization of the existentially closed models Theorem (Guzy and Rivière, 2007) A differential-difference field (K,, σ) is existentially closed if and only if (i) (K, ) = DCF 0,m (ii) If V and W are irreducible -closed sets such that W V V σ and W projects -dominantly onto both V and V σ, then there is a V such that (a, σa) W. Problem It is not known if irreducibility of -closed sets and -dominance are definable properties.

10 Characterization of the existentially closed models Theorem (Guzy and Rivière, 2007) A differential-difference field (K,, σ) is existentially closed if and only if (i) (K, ) = DCF 0,m (ii) If V and W are irreducible -closed sets such that W V V σ and W projects -dominantly onto both V and V σ, then there is a V such that (a, σa) W. Problem It is not known if irreducibility of -closed sets and -dominance are definable properties.

11 Characteristic sets Let P K {x} be a prime differential ideal. There exists a finite set Λ P, called a characteristic set of P, such that Λ determines P in the following sense: P = [Λ] : HΛ := {f K {x} : Hl Λf [Λ] for some l < ω} (1) where H Λ K {x} is uniformly determined by Λ. Remark The actual definition of characteristic set if more technical, but for our purposes this presentation is sufficient.

12 Characteristic sets Let P K {x} be a prime differential ideal. There exists a finite set Λ P, called a characteristic set of P, such that Λ determines P in the following sense: P = [Λ] : HΛ := {f K {x} : Hl Λf [Λ] for some l < ω} (1) where H Λ K {x} is uniformly determined by Λ. Remark The actual definition of characteristic set if more technical, but for our purposes this presentation is sufficient.

13 Characteristic sets Suppose (K, ) = DCF 0,m. A -closed set is irreducible if and only if is a prime differential ideal. I(V ) = {f K {x} : f (V ) = 0} Hence to each irreducible -closed set we can associate a characteristic set. Conversely, to any characteristic set Λ we can associate the irreducible V([Λ] : H Λ ) = {a K n : f (a) = 0, for all f [Λ] : H Λ }

14 Characteristic sets Suppose (K, ) = DCF 0,m. A -closed set is irreducible if and only if is a prime differential ideal. I(V ) = {f K {x} : f (V ) = 0} Hence to each irreducible -closed set we can associate a characteristic set. Conversely, to any characteristic set Λ we can associate the irreducible V([Λ] : H Λ ) = {a K n : f (a) = 0, for all f [Λ] : H Λ }

15 Characteristic sets Suppose Λ is a characteristic of the prime differential ideal P. In general V(P) V(Λ) and strict containment may occur. However, Property (1) shows that V(Λ) \ V(H Λ ) = V(P) \ V(H Λ ). Hence, at least in the subset V (Λ) := V(Λ) \ V(H Λ ), we can control the behaviour of V(P) in terms of Λ.

16 The model companion Theorem (L.S., 2012) (K,, σ) is existentially closed if and only if 1 (K, ) = DCF 0,m 2 Suppose Λ and Γ are characteristic sets of prime differential ideals of K {x} and K {x, y}, respectively, such that V (Γ) V(Λ) V(Λ σ ). Suppose O and Q are nonempty -open subsets of V (Λ) and V (Λ σ ), respectively, such that O π x (V (Γ)) and Q π y (V (Γ)). Then there is a V (Λ) such that (a, σa) V (Γ).

17 Model companion Is condition (2) first-order? Fact (Tressl, 2005) The condition that Λ = {f 1,..., f s } is a characteristic set of a prime differential ideal of K {x} " is a definable property (in the language L ) of the coefficients of f 1,..., f s. Corollary The theory of partial differential fields of characteristic zero with an automorphism has a model companion DCF 0,m A.

18 Model companion Is condition (2) first-order? Fact (Tressl, 2005) The condition that Λ = {f 1,..., f s } is a characteristic set of a prime differential ideal of K {x} " is a definable property (in the language L ) of the coefficients of f 1,..., f s. Corollary The theory of partial differential fields of characteristic zero with an automorphism has a model companion DCF 0,m A.

19 Basic model theory Let (K,, σ), (L,, σ ) = DCF 0,m A and suppose A K. We denote by A the differential-difference field generated by A. The following are specializations of the work of Chatzidakis and Pillay on generic automorphisms: acl(a) = A alg. If K and L have a common algebraically closed differential-difference subfield F, then (K,, σ) F (L,, σ ). In particular, the completions of DCF 0,m A are determined by the difference field structure on Q alg.

20 Basic model theory Let (K,, σ), (L,, σ ) = DCF 0,m A and suppose A K. We denote by A the differential-difference field generated by A. The following are specializations of the work of Chatzidakis and Pillay on generic automorphisms: acl(a) = A alg. If K and L have a common algebraically closed differential-difference subfield F, then (K,, σ) F (L,, σ ). In particular, the completions of DCF 0,m A are determined by the difference field structure on Q alg.

21 Basic model theory Every completion of DCF 0,m A is supersimple. Moreover, if (K,, σ) is sufficiently saturated and A, B, C are small subsets of K, then A B if and only if A C is C algebraically disjoint from B C over C. One can also show that The field of constansts C K = {a K : σa = a, δa = 0 for all δ } is pseudofinite. Every completion of DCF 0,m A eliminates imaginaries.

22 Basic model theory Every completion of DCF 0,m A is supersimple. Moreover, if (K,, σ) is sufficiently saturated and A, B, C are small subsets of K, then A B if and only if A C is C algebraically disjoint from B C over C. One can also show that The field of constansts C K = {a K : σa = a, δa = 0 for all δ } is pseudofinite. Every completion of DCF 0,m A eliminates imaginaries.

23 Towards Zilber Dichotomy We make use of the results of Moosa and Scanlon on abstract prolongations and Jet spaces for generalized Hasse-Schmidt varieties. One can show that differential-difference fields are precisely the iterative Hasse-Schmidt fields with respect to an appropriate iterative Hasse-Schmidt system D. Hence (, σ)-closed sets can be viewed as generalized D-varieties and they are determined by their Jet spaces.

24 Towards Zilber Dichotomy We make use of the results of Moosa and Scanlon on abstract prolongations and Jet spaces for generalized Hasse-Schmidt varieties. One can show that differential-difference fields are precisely the iterative Hasse-Schmidt fields with respect to an appropriate iterative Hasse-Schmidt system D. Hence (, σ)-closed sets can be viewed as generalized D-varieties and they are determined by their Jet spaces.

25 Towards Zilber Dichotomy We make use of the results of Moosa and Scanlon on abstract prolongations and Jet spaces for generalized Hasse-Schmidt varieties. One can show that differential-difference fields are precisely the iterative Hasse-Schmidt fields with respect to an appropriate iterative Hasse-Schmidt system D. Hence (, σ)-closed sets can be viewed as generalized D-varieties and they are determined by their Jet spaces.

26 Towards Zilber Dichotomy We fix a sufficiently saturated (U,, σ) = DCF 0,m A. Definition Let K be an algebraically closed differential-difference subfield if U and a a tuple from U. Then tp(a/k ) is finite dimensional if trdeg(k a /K ) is finite. Lemma Suppose tp(a/k ) is finite dimensional. Then, for all m, the U-points of the m th -Jet space of the D-locus of a over K is a finite dimensional vector space over the constants C U.

27 Towards Zilber Dichotomy We fix a sufficiently saturated (U,, σ) = DCF 0,m A. Definition Let K be an algebraically closed differential-difference subfield if U and a a tuple from U. Then tp(a/k ) is finite dimensional if trdeg(k a /K ) is finite. Lemma Suppose tp(a/k ) is finite dimensional. Then, for all m, the U-points of the m th -Jet space of the D-locus of a over K is a finite dimensional vector space over the constants C U.

28 CBP and Zilber Dichotomy Canonical base property Suppose tp(a/k ) is finite dimensional and b is a tuple from U such that Cb(a/K b alg ) is interalgebraic with b over K. Then tp(b/k a ) is almost internal to C U. Zilber Dichotomy If p is a finite dimensional minimal type, then p is either locally modular or nonorthoganal to C U. Remark One could also introduce (, σ)-modules and extend the arguments of Pillay-Ziegler. However, this approach seems more direct.

29 CBP and Zilber Dichotomy Canonical base property Suppose tp(a/k ) is finite dimensional and b is a tuple from U such that Cb(a/K b alg ) is interalgebraic with b over K. Then tp(b/k a ) is almost internal to C U. Zilber Dichotomy If p is a finite dimensional minimal type, then p is either locally modular or nonorthoganal to C U. Remark One could also introduce (, σ)-modules and extend the arguments of Pillay-Ziegler. However, this approach seems more direct.

30 CBP and Zilber Dichotomy Canonical base property Suppose tp(a/k ) is finite dimensional and b is a tuple from U such that Cb(a/K b alg ) is interalgebraic with b over K. Then tp(b/k a ) is almost internal to C U. Zilber Dichotomy If p is a finite dimensional minimal type, then p is either locally modular or nonorthoganal to C U. Remark One could also introduce (, σ)-modules and extend the arguments of Pillay-Ziegler. However, this approach seems more direct.

31 Extensions (current work with R. Moosa) We can consider partial differential fields with free operators (in the sense of Moosa and Scanlon). Data D = (D(Q), {ɛ 0,..., ɛ l }) a finite dimensional Q-algebra with a fixed Q-basis. Definition A differential D-field is a triple (K,, ) where = ( 1,..., l ) are additive operators such that e : K D(K ) := D(Q) K given by e(x) = xɛ (x)ɛ l (x)ɛ l is a -homomorphism. Where the -structure on D(K ) is induced by δɛ i = 0 for all δ. Eg. if D(Q) = Q Q then differential D-fields are precisely differential-difference fields.

32 Extensions (current work with R. Moosa) We can consider partial differential fields with free operators (in the sense of Moosa and Scanlon). Data D = (D(Q), {ɛ 0,..., ɛ l }) a finite dimensional Q-algebra with a fixed Q-basis. Definition A differential D-field is a triple (K,, ) where = ( 1,..., l ) are additive operators such that e : K D(K ) := D(Q) K given by e(x) = xɛ (x)ɛ l (x)ɛ l is a -homomorphism. Where the -structure on D(K ) is induced by δɛ i = 0 for all δ. Eg. if D(Q) = Q Q then differential D-fields are precisely differential-difference fields.

33 Extensions (current work with R. Moosa) We have shown that the theory of differential D-fields has a model companion. This uses characteristic sets and nontrivial extension theorems. The completions are determined by the difference structure on Q alg. Each completion is simple and eliminates imaginaries. We expect to have a Zilber Dichotomy for finite dimensional types, but new machinery needs to be developed.