n-amalgamation Mgeq n-simplicity Fields in simple theories

Geometric Simplicity Theory

Byunghan Kim

Dept. Math. Yonsei University

ALC 10September 1-6, 2008

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

Outline

1 n-amalgamation

2 Mgeq

3 n-simplicity

4 Fields in simple theories

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

Geometric Simplicity Theory

Byunghan Kim

Dept. Math. Yonsei University

ALC 10September 1-6, 2008

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

Let CT be a category of the algebraically closed substructures ofM. Recall that any poset is a category.For n ∈ ω, write P(n)− := P(n) − {n}.

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

Let CT be a category of the algebraically closed substructures ofM. Recall that any poset is a category.For n ∈ ω, write P(n)− := P(n) − {n}.

Definition

A functor a : W (⊆ P(n)) → CT is said to be independencepreserving (i.p.) if

1 for any w0, w1 ⊆ w ∈ W , aw0^| aw0∩w1

aw1holds within aw ;

2 for w ∈ W , aw = acl(⋃{a{i}| i ∈ w}).

We say T has n-amalgamation if any i.p. functora : P(n)− → CT can be extended to i.p. a : P(n) → CT .

3-amalgamation is type-amalgamation (the IndependenceTheorem). Hence any simple T has 3-amalgamation.

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

3-amalgamation

3-amalgamation

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

4-amalgamation

4-amalgamation

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

4-amalgamation over acl base set

4-amalgamation

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

T stable.Then n-amalgamation always holds over any model. Butn-amalgamation need not hold if the base set is not a model!

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

T stable.Then n-amalgamation always holds over any model. Butn-amalgamation need not hold if the base set is not a model!

Example

Consider [A]2 = {{a, b}|a 6= b ∈ A} where A infinite. LetB = [A]2 × {0, 1} where {0, 1} = Z2.Also let E ⊆ A × [A]2 be a membership relation, and let P be asubset of B3 such that ((w1, δ1)(w2, δ2)(w3, δ3)) ∈ P iff there aredistinct a1, a2, a3 ∈ A such that for {i , j , k} = {1, 2, 3},wi = {aj , ak}, and δ1 + δ2 + δ3 = 0.Let M = (A, [A]2,B ; E ,P ; Pr1 : B → [A]2). Then M is stable.

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

The stable example does not have 4-amalgamation over ∅ = acl(∅).

Why

Note first that dcl(∅) = acl(∅), and for a ∈ A, dcl(a) = acl(a).Now choose distinct a1, a2, a3, a4 ∈ A. For {i , j , k} ⊆ {1, 2, 3, 4},fix an enumeration aiaj = (bij , ...) of acl(aiaj) wherebij = ({ai , aj}, δ) ∈ B = [A]2 × {0, 1}. Let rij(xij) = tp(aiaj), andlet x1

ij be the variable for bij . Note that bij = ({ai , aj}, δ) andb′ij = ({ai , aj}, δ + 1) have the same type over aiaj . Hence there is

(aiaj)′ = (b′

ij , ...) also realizing rij(xij). Therefore we have completetypes rijk(xijk), r ′ijk(x ′

ijk) both extending rij(xij) ∪ rik(xik) ∪ rjk(xjk)realized by some enumerations of acl(aiajak) such thatP(x1

ijx1ikx1

jk) ∈ rijk whereas ¬P(x1ijx

1ikx1

jk) ∈ r ′ijk . Then it is easy tosee that r123 ∪ r124 ∪ r134 ∪ r ′234 is inconsistent.

n-amalgamation Mgeq n-simplicity Fields in simple theories

Resolution

In his recent preprint [http://arxiv.org/abs/math/0603413v1],Hrushovski showed that if M |= T is stable, then there isCM∗ |= T ∗ in L∗(⊇ L) such that M is stably embedded into M∗,and M∗ has n-amalgamation over any acl bases.We may write M∗ as Mgeq.In short, like Meq, wlog, we can assume M = Mgeq when T isstable.

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

Resolution

In his recent preprint [http://arxiv.org/abs/math/0603413v1],Hrushovski showed that if M |= T is stable, then there isCM∗ |= T ∗ in L∗(⊇ L) such that M is stably embedded into M∗,and M∗ has n-amalgamation over any acl bases.We may write M∗ as Mgeq.In short, like Meq, wlog, we can assume M = Mgeq when T isstable.

Open Problem

Can we construct such M∗ for simple T? If yes, then possibly wecan remove the assumption of 4-amalgamation in the groupconfiguration theorem.

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

In the following notions of n-simplicity and K (n)-simplicity, forconvenience, we use imprecise definitions good enough howeverrepresenting the essence of notions.Also for convenience, we describe notions only over ∅ = acl(∅).

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

In the following notions of n-simplicity and K (n)-simplicity, forconvenience, we use imprecise definitions good enough howeverrepresenting the essence of notions.Also for convenience, we describe notions only over ∅ = acl(∅).

Fact

Let I = 〈ai | i ∈ ω〉 be Morley, and let b | a0. Then there isb′ ≡a0

b such that I is Morley over b′ and b′ | I .

Above fact is a particular case of 3-amalgamation (IT), and usedcrucially in showing IT for simple T .

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

Definition

T is K (n)-simple if for k ≤ n and any Morley I = 〈ai | i ∈ ω〉,whenever Ik = 〈ai |i < k〉 is Morley over b with b | Ik , there isb′ ≡Ik b such that I is Morley over b′ and b′ | I .

Definition

T is K (n)-simple if for k ≤ n and any Morley I = 〈ai | i ∈ ω〉,whenever Ik = 〈ai |i < k〉 is Morley over b with b | Ik , there isb′ ≡Ik b such that I is Morley over b′ and b′ | I .

T is n-simple if for k ≤ n and any Morley I = 〈ai | i ≤ k〉,whenever Ik = 〈ai |i < k〉 is Morley over b with b | Ik , there isb′ ≡Ik b such that I is Morley over b′ and b′ | I .

Definition

T is K (n)-simple if for k ≤ n and any Morley I = 〈ai | i ∈ ω〉,whenever Ik = 〈ai |i < k〉 is Morley over b with b | Ik , there isb′ ≡Ik b such that I is Morley over b′ and b′ | I .

T is n-simple if for k ≤ n and any Morley I = 〈ai | i ≤ k〉,whenever Ik = 〈ai |i < k〉 is Morley over b with b | Ik , there isb′ ≡Ik b such that I is Morley over b′ and b′ | I .

We say T has n-CA if T has k-amalgamation for all k ≤ n.

Definition

T is K (n)-simple if for k ≤ n and any Morley I = 〈ai | i ∈ ω〉,whenever Ik = 〈ai |i < k〉 is Morley over b with b | Ik , there isb′ ≡Ik b such that I is Morley over b′ and b′ | I .

T is n-simple if for k ≤ n and any Morley I = 〈ai | i ≤ k〉,whenever Ik = 〈ai |i < k〉 is Morley over b with b | Ik , there isb′ ≡Ik b such that I is Morley over b′ and b′ | I .

We say T has n-CA if T has k-amalgamation for all k ≤ n.

Both n-simplicity, K (n)-simplicity are particular cases of(n + 2)-CA.

Question

Are those 3 notions equivalent?

simple = 1-simple = K(1)-simple = 3-amalgamation = 3-CA

Yes

(Kolesnikov) 2-simple = K(2)-simple = 4-amalgamation = 4-CA

Yes

(Kolesnikov) 2-simple = K(2)-simple = 4-amalgamation = 4-CA

Yes and No

(K, Kolesnikov, Tsuboi) Yes:n-simple = (n + 2)-CA

Yes

(Kolesnikov) 2-simple = K(2)-simple = 4-amalgamation = 4-CA

Yes and No

(K, Kolesnikov, Tsuboi) Yes:n-simple = (n + 2)-CA

No: For each n > 3, there is an example of K (n)-simple but nothaving (n + 2)-CA.L = {R}, R is a n-ary relation.

K := {A| A is a finite R-structure;R is symmetric and irreflexive;for any A0 ⊆ A with |A0| = n + 1, the no. of

n-element subsets of A0 holding R is even }

The Fraısse limit of K is the simple ω-categorical example.

n-amalgamation Mgeq n-simplicity Fields in simple theories

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

Definition

F a field.

F is said to be PAC if any absolutely irreducible algebraic setV defined over F has F -rational point.

F is pseudo-finite if it is an infinite field of the theory of allfinite fields.

An extension field E of F is separable if it is algebraic over F ,and for each e ∈ E , irrF (e) has no multiple roots.

F perfect if every algebraic extension is separable; equivalentlyeither Char(F ) = 0 or x 7→ xp where p = Char(F ) > 0 is onto.

F is bounded for each n > 1, there are only finitely manyseparably algebraic extensions of degree n.

F is separably closed if it has no proper separable extension.

Fact

F PAC.

(Chatzidakis) F is simple iff F is bounded.

(Hrshovski, Pillay, Poizat) F is supersimple iff F is perfect andbounded.

(Duret, Wood) F is stable iff F is separably closed.

Fact

F PAC.

(Chatzidakis) F is simple iff F is bounded.

(Hrshovski, Pillay, Poizat) F is supersimple iff F is perfect andbounded.

(Duret, Wood) F is stable iff F is separably closed.

Historically the first simple unstable fields studied seriously arepseudo-finite fields by J. Ax in late 60s.

Fact

F PAC.

(Chatzidakis) F is simple iff F is bounded.

(Hrshovski, Pillay, Poizat) F is supersimple iff F is perfect andbounded.

(Duret, Wood) F is stable iff F is separably closed.

Historically the first simple unstable fields studied seriously arepseudo-finite fields by J. Ax in late 60s.

Fact

(Ax)

A field F is pseuo-finite iff 1. it is perfect2. it is PAC, and3. for each n it has exactlyone algebraic extension of degree n

Hence pseuo-finite fields are unstable supersimple (of SU-rank 1).

n-amalgamation Mgeq n-simplicity Fields in simple theories

Fact

(Macintyre; Cherlin, Shelah) Superstable division ring is analgebraically closed field.

(Pillay, Scanlon, Wagner) Supersimple division ring is a field.

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory

n-amalgamation Mgeq n-simplicity Fields in simple theories

Fact

(Macintyre; Cherlin, Shelah) Superstable division ring is analgebraically closed field.

(Pillay, Scanlon, Wagner) Supersimple division ring is a field.

Open Problem

Is any stable field separably closed?

Is any supersimple field PAC?

Byunghan Kim Dept. Math. Yonsei University

Geometric Simplicity Theory