QACFA
Alice Medvedev
University of California, Berkeley
Algebra and Model Theory Satellite
Pucon, Chile
December 13, 2010
Exercises in pure model theory.
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Signatures L1 ⊂ L2 ⊂ . . . and L := ∪n Ln
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Ln -theories Sn ⊂ Sn+1 and Tn ⊂ Tn+1
Suppose both S := ∪n Sn and T := ∪n Tn are consistent.
Exercises in pure model theory.
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Signatures L1 ⊂ L2 ⊂ . . . and L := ∪n Ln
If L0 ⊂ L is finite, L0 ⊂ Ln for some n.
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Ln -theories Sn ⊂ Sn+1 and Tn ⊂ Tn+1
Suppose both S := ∪n Sn and T := ∪n Tn are consistent.
Exercises in pure model theory.
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Signatures L1 ⊂ L2 ⊂ . . . and L := ∪n Ln
If L0 ⊂ L is finite, L0 ⊂ Ln for some n.
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Ln -theories Sn ⊂ Sn+1 and Tn ⊂ Tn+1
Suppose both S := ∪n Sn and T := ∪n Tn are consistent.
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∀n Tn is model-complete ⇒ T is model-complete.
Exercises in pure model theory.
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Signatures L1 ⊂ L2 ⊂ . . . and L := ∪n Ln
If L0 ⊂ L is finite, L0 ⊂ Ln for some n.
Ln -theories Sn ⊂ Sn+1 and Tn ⊂ Tn+1
Suppose both S := ∪n Sn and T := ∪n Tn are consistent.
∀n Tn is model-complete ⇒ T is model-complete.
Poof: M ≤ N both models of T ⇒ ∀n M|Ln ≤ N|Ln both
models of Tn ⇒ ∀n M|Ln N|Ln ⇒ M N.
Exercises in pure model theory.
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Signatures L1 ⊂ L2 ⊂ . . . and L := ∪n Ln
If L0 ⊂ L is finite, L0 ⊂ Ln for some n.
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Ln -theories Sn ⊂ Sn+1 and Tn ⊂ Tn+1
Suppose both S := ∪n Sn and T := ∪n Tn are consistent.
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∀n Tn is model-complete ⇒ T is model-complete.
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∀n Tn is the model-companion of Sn ⇒
T is the model-companion of S.
Exercises in pure model theory.
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Signatures L1 ⊂ L2 ⊂ . . . and L := ∪n Ln
If L0 ⊂ L is finite, L0 ⊂ Ln for some n.
Ln -theories Sn ⊂ Sn+1 and Tn ⊂ Tn+1
Suppose both S := ∪n Sn and T := ∪n Tn are consistent.
∀n Tn is model-complete ⇒ T is model-complete.
∀n Tn is the model-companion of Sn ⇒
T is the model-companion of S.
Poof: M |= S ⇒ ∀n M|Ln |= Sn ⇒
∀n Diag (M|Ln )∪Tn is consistent ⇒ Diag (M)∪T is consistent
Exercises in pure model theory.
I
Signatures L1 ⊂ L2 ⊂ . . . and L := ∪n Ln
If L0 ⊂ L is finite, L0 ⊂ Ln for some n.
I
Ln -theories Sn ⊂ Sn+1 and Tn ⊂ Tn+1
Suppose both S := ∪n Sn and T := ∪n Tn are consistent.
I
∀n Tn is model-complete ⇒ T is model-complete.
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∀n Tn is the model-companion of Sn ⇒
T is the model-companion of S.
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Quantifier elimination results, characterization of algebraic
closure, etc pass from Tn to T .
Exercises in pure model theory.
I
Signatures L1 ⊂ L2 ⊂ . . . and L := ∪n Ln
If L0 ⊂ L is finite, L0 ⊂ Ln for some n.
I
Ln -theories Sn ⊂ Sn+1 and Tn ⊂ Tn+1
Suppose both S := ∪n Sn and T := ∪n Tn are consistent.
I
∀n Tn is model-complete ⇒ T is model-complete.
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∀n Tn is the model-companion of Sn ⇒
T is the model-companion of S.
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Quantifier elimination results, characterization of algebraic
closure, etc pass from Tn to T .
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An Ln -formula that is stable in Tn is stable in T .
Exercises in pure model theory.
I
Signatures L1 ⊂ L2 ⊂ . . . and L := ∪n Ln
If L0 ⊂ L is finite, L0 ⊂ Ln for some n.
I
Ln -theories Sn ⊂ Sn+1 and Tn ⊂ Tn+1
Suppose both S := ∪n Sn and T := ∪n Tn are consistent.
I
∀n Tn is model-complete ⇒ T is model-complete.
I
∀n Tn is the model-companion of Sn ⇒
T is the model-companion of S.
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Quantifier elimination results, characterization of algebraic
closure, etc pass from Tn to T .
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An Ln -formula that is stable in Tn is stable in T .
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If A ^
| nC B is a decent notion of independence in Tn , then
A^
| C B :⇔ ∀n A ^
| nC B is a decent notion of independence
for T .
Exercises in pure model theory.
I
Signatures L1 ⊂ L2 ⊂ . . . and L := ∪n Ln
If L0 ⊂ L is finite, L0 ⊂ Ln for some n.
I
Ln -theories Sn ⊂ Sn+1 and Tn ⊂ Tn+1
Suppose both S := ∪n Sn and T := ∪n Tn are consistent.
I
∀n Tn is model-complete ⇒ T is model-complete.
I
∀n Tn is the model-companion of Sn ⇒
T is the model-companion of S.
I
Quantifier elimination results, characterization of algebraic
closure, etc pass from Tn to T .
I
An Ln -formula that is stable in Tn is stable in T .
I
If A ^
| nC B is a decent notion of independence in Tn , then
A^
| C B :⇔ ∀n A ^
| nC B is a decent notion of independence
for T .
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Ranks may grow with n, and explode in the limit:
supersimplicity or ω-stability might not be preserved.
Please meet QACFA.
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L is the signature LR of rings with new unary function
symbols {σq | q ∈ Q}
S is the theory of fields with a (Q, +) action.
(Each σq is a field automorphism and σq ◦ σr = σq+r .)
Please meet QACFA.
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L := LR ∪ {σq | q ∈ Q}, S := Th(fields with (Q, +)-action).
(Each σq is a field automorphism and σq ◦ σr = σq+r .)
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Ln := LR ∪ {σ n!m | m ∈ Z} and Sn := S|Ln . Note Sn ⊂ Sn+1 .
Please meet QACFA.
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L := LR ∪ {σq | q ∈ Q}, S := Th(fields with (Q, +)-action).
(Each σq is a field automorphism and σq ◦ σr = σq+r .)
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Ln := LR ∪ {σ n!m | m ∈ Z} and Sn := S|Ln . Note Sn ⊂ Sn+1 .
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Any Ln -structure M |= Sn is a definitional expansion of
M|LR ∪{σ 1 }
n!
Please meet QACFA.
I
L := LR ∪ {σq | q ∈ Q}, S := Th(fields with (Q, +)-action).
(Each σq is a field automorphism and σq ◦ σr = σq+r .)
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Ln := LR ∪ {σ n!m | m ∈ Z} and Sn := S|Ln . Note Sn ⊂ Sn+1 .
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Any Ln -structure M |= Sn is a definitional expansion of
M|LR ∪{σ 1 } , so each Sn is just the theory of difference fields!
n!
Please meet QACFA.
I
L := LR ∪ {σq | q ∈ Q}, S := Th(fields with (Q, +)-action).
(Each σq is a field automorphism and σq ◦ σr = σq+r .)
I
Ln := LR ∪ {σ n!m | m ∈ Z} and Sn := S|Ln . Note Sn ⊂ Sn+1 .
I
Any Ln -structure M |= Sn is a definitional expansion of
M|LR ∪{σ 1 } , so each Sn is just the theory of difference fields!
n!
which has a model-companion ACFA (see next slide).
Please meet QACFA.
I
L := LR ∪ {σq | q ∈ Q}, S := Th(fields with (Q, +)-action).
(Each σq is a field automorphism and σq ◦ σr = σq+r .)
I
Ln := LR ∪ {σ n!m | m ∈ Z} and Sn := S|Ln . Note Sn ⊂ Sn+1 .
I
Any Ln -structure M |= Sn is a definitional expansion of
M|LR ∪{σ 1 } , so each Sn is just the theory of difference fields!
n!
which has a model-companion ACFA (see next slide).
I
Let Tn := ((K , σ 1 ) |= ACFA).
n!
Please meet QACFA.
I
L := LR ∪ {σq | q ∈ Q}, S := Th(fields with (Q, +)-action).
(Each σq is a field automorphism and σq ◦ σr = σq+r .)
I
Ln := LR ∪ {σ n!m | m ∈ Z} and Sn := S|Ln . Note Sn ⊂ Sn+1 .
I
Any Ln -structure M |= Sn is a definitional expansion of
M|LR ∪{σ 1 } , so each Sn is just the theory of difference fields!
n!
which has a model-companion ACFA (see next slide).
I
Let Tn := ((K , σ 1 ) |= ACFA).
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(Chatzidakis-Hrushovski =: C.-H.)
(K , τ ) |= ACFA ⇒ (K , τ n ) |= ACFA,
so T := ∪n Tn is a consistent L-theory, named QACFA
n!
From ACF to ACFA to QACFA.
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If A, B, C ⊂ K |= ACF, hAi is the field generated by A
aclfields (A) is the algebraic closure of A in K
A^
| fields
B :⇔ A is algebraically independent from B over C .
C
From ACF to ACFA to QACFA.
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If A, B, C ⊂ K |= ACF, hAi is the field generated by A
aclfields (A) is the algebraic closure of A in K
A^
| fields
B :⇔ A is algebraically independent from B over C .
C
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If (K , τ ) |= ACFA,
let hAiτ := h∪n∈Z τ n (A)i and aclτ (A) = aclfields (hAiτ )
(C.-H.) Model-theoretic algebraic closure in ACFA is aclτ .
Let A ^
| τC B :⇔ aclτ (AC ) ^
| fields
aclτ (BC ).
aclτ (C )
(C.-H.) ACFA is supersimple and ^
| τ is non-forking
independence.
From ACF to ACFA to QACFA.
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If A, B, C ⊂ K |= ACF, hAi is the field generated by A
aclfields (A) is the algebraic closure of A in K
A^
| fields
B :⇔ A is algebraically independent from B over C .
C
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If (K , τ ) |= ACFA,
let hAiτ := h∪n∈Z τ n (A)i and aclτ (A) = aclfields (hAiτ )
(C.-H.) Model-theoretic algebraic closure in ACFA is aclτ .
Let A ^
| τC B :⇔ aclτ (AC ) ^
| fields
aclτ (BC ).
aclτ (C )
(C.-H.) ACFA is supersimple and ^
| τ is non-forking
independence.
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If N := (K , {σq }q∈Q ) |= QACFA,
let hAiσ := h∪q∈Q σq (A)i and aclσ (A) = aclfields (hAiσ ).
Model-theoretic algebraic closure in QACFA is aclσ .
Let A ^
| σC B :⇔ aclσ (AC ) ^
| fields
aclσ (BC ).
aclσ (C )
σ
QACFAis simple and ^
| is non-forking independence.
From ACF to ACFA to QACFA.
I
If A, B, C ⊂ K |= ACF, hAi is the field generated by A
aclfields (A) is the algebraic closure of A in K
A^
| fields
B :⇔ A is algebraically independent from B over C .
C
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If (K , τ ) |= ACFA,
let hAiτ := h∪n∈Z τ n (A)i and aclτ (A) = aclfields (hAiτ )
(C.-H.) Model-theoretic algebraic closure in ACFA is aclτ .
Let A ^
| τC B :⇔ aclτ (AC ) ^
| fields
aclτ (BC ).
aclτ (C )
(C.-H.) ACFA is supersimple and ^
| τ is non-forking
independence.
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If N := (K , {σq }q∈Q ) |= QACFA,
let hAiσ := h∪q∈Q σq (A)i and aclσ (A) = aclfields (hAiσ ).
Model-theoretic algebraic closure in QACFA is aclσ .
Let A ^
| σC B :⇔ aclσ (AC ) ^
| fields
aclσ (BC ).
aclσ (C )
σ
QACFAis simple and ^
| is non-forking independence.
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ACFA is quantifier-free-ω-stable, so QACFA is
quantifier-free-stable.
Fixed fields
Let Fq := {a | σq (a) = a}, the fixed field of σq .
Fixed fields
Let Fq := {a | σq (a) = a}, the fixed field of σq .
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(C.-H.) All Fq are pseudofinite, and Fq ≤ Fnq with
[Fnq : Fq ] = n for all n ∈ Z≥1 .
Completions of ACFA are given by specifying the
characteristic and the action of τ on aclfields (∅).
Fixed fields
Let Fq := {a | σq (a) = a}, the fixed field of σq .
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(C.-H.) All Fq are pseudofinite, and Fq ≤ Fnq with
[Fnq : Fq ] = n for all n ∈ Z≥1 .
Completions of ACFA are given by specifying the
characteristic and the action of τ on aclfields (∅).
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Both ∩q∈Q Fq and ∪q∈Q Fq are algebraically closed.
Fixed fields
Let Fq := {a | σq (a) = a}, the fixed field of σq .
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(C.-H.) All Fq are pseudofinite, and Fq ≤ Fnq with
[Fnq : Fq ] = n for all n ∈ Z≥1 .
Completions of ACFA are given by specifying the
characteristic and the action of τ on aclfields (∅).
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Both ∩q∈Q Fq and ∪q∈Q Fq are algebraically closed.
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So the algebraic closure of the prime field is fixed by all σq ,
and QACFA is complete after specifying characteristic.
Fixed fields
Let Fq := {a | σq (a) = a}, the fixed field of σq .
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(C.-H.) All Fq are pseudofinite, and Fq ≤ Fnq with
[Fnq : Fq ] = n for all n ∈ Z≥1 .
Completions of ACFA are given by specifying the
characteristic and the action of τ on aclfields (∅).
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Both ∩q∈Q Fq and ∪q∈Q Fq are algebraically closed.
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So the algebraic closure of the prime field is fixed by all σq ,
and QACFA is complete after specifying characteristic.
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F1 F 1 F 1 F 1 . . . is an infinite descending chain of
2
6
24
definable fields: this breaks supersimplicity and
quantifier-free-superstability.
Fixed fields
Let Fq := {a | σq (a) = a}, the fixed field of σq .
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(C.-H.) All Fq are pseudofinite, and Fq ≤ Fnq with
[Fnq : Fq ] = n for all n ∈ Z≥1 .
Completions of ACFA are given by specifying the
characteristic and the action of τ on aclfields (∅).
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Both ∩q∈Q Fq and ∪q∈Q Fq are algebraically closed.
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So the algebraic closure of the prime field is fixed by all σq ,
and QACFA is complete after specifying characteristic.
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F1 F 1 F 1 F 1 . . . is an infinite descending chain of
2
6
24
definable fields: this breaks supersimplicity and
quantifier-free-superstability.
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What is the induced structure on these Fq ? On ∪q∈Q Fq ?
Smells a little like separably closed fields, or PAC fields...
Away from the fixed fields
In ACFA in characteristic 0, fixed fields are the only source of
instability.
In QACFA, are they the only source of exploding U-rank?
Away from the fixed fields
In ACFA in characteristic 0, fixed fields are the only source of
instability.
In QACFA, are they the only source of exploding U-rank?
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Quest: Suppose p is a complete type in QACFA orthogonal to
all fixed fields.
Does it follow that U(p) < ∞?
Away from the fixed fields
In ACFA in characteristic 0, fixed fields are the only source of
instability.
In QACFA, are they the only source of exploding U-rank?
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Quest: Suppose p is a complete type in QACFA orthogonal to
all fixed fields.
Does it follow that U(p) < ∞?
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Quest: Suppose q is a complete type of finite rank in
L1 = LR ∪ {σ1 } orthogonal to F1 , and let
Nn := sup{U(r ) | q ⊂ r and r is a complete Ln -type}
(Ln = LR ∪ {σ 1 })
Is supn Nn finite?
n!
Away from the fixed fields
In ACFA in characteristic 0, fixed fields are the only source of
instability.
In QACFA, are they the only source of exploding U-rank?
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Quest: Suppose p is a complete type in QACFA orthogonal to
all fixed fields.
Does it follow that U(p) < ∞?
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Quest: Suppose q is a complete type of finite rank in
L1 = LR ∪ {σ1 } orthogonal to F1 , and let
Nn := sup{U(r ) | q ⊂ r and r is a complete Ln -type}
(Ln = LR ∪ {σ 1 })
Is supn Nn finite?
n!
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Theorem: If φ is an L1 -formula which is weakly minimal (in
S1 = ACFA about σ1 ) and orthogonal to F1 (in S1 ), then all
complete L-types in φ have finite rank.