The relation of embeddability between uncountable torsion-free - - PowerPoint PPT Presentation

The relation of embeddability between uncountable torsion-free abelian groups

Filippo Calderoni

University of Turin

March 14, 2017

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Borel reducibility

In the framework of (classical) Borel reducibility, relations are defined over Polish or standard Borel spaces.

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Borel reducibility

In the framework of (classical) Borel reducibility, relations are defined over Polish or standard Borel spaces.

Definition

Let P and Q be quasi-orders X and Y , respectively. We say that P Borel reduces to Q (or P ≤B Q) if and only if there is a Borel f : X → Y function such that x1 P x2 ⇔ f (x1) Q f (x2).

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Borel reducibility

In the framework of (classical) Borel reducibility, relations are defined over Polish or standard Borel spaces.

Definition

Let P and Q be quasi-orders X and Y , respectively. We say that P Borel reduces to Q (or P ≤B Q) if and only if there is a Borel f : X → Y function such that x1 P x2 ⇔ f (x1) Q f (x2). If E, F are equivalence relation and E ≤B F we say roughly that E is not more complicated than F.

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Analytic equivalence relations

Σ1

1 equivalence relation

idR

• E0
• E∞
• Ectble
• id2<ω1
• ES∞
• EG∞
• E1
• Eℓ∞
• EΣ1

1

• countable Borel

classifiable by N-structures Polish group actions

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Analytic equivalence relations

Σ1

1 equivalence relation

idR

• E0
• E∞
• Ectble
• id2<ω1
• ES∞
• EG∞
• E1
• Eℓ∞
• EΣ1

1

• EΣ1

1 is defined as the

≤B-maximum and called the complete Σ1

1 equiv-

alence relation.

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Complete Σ1

1 quasi-orders

Definition

Q is a complete Σ1

1 quasi-order if and only if Q is Σ1 1 and

P ≤B Q, for every Σ1

1 quasi-order P.

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Complete Σ1

1 quasi-orders

Definition

Q is a complete Σ1

1 quasi-order if and only if Q is Σ1 1 and

P ≤B Q, for every Σ1

1 quasi-order P.

If Q is a complete Σ1

1 quasi-order, then EQ := Q ∩ Q−1 is a

complete Σ1

1 equivalence relation.

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Complete Σ1

1 quasi-orders

Definition

Q is a complete Σ1

1 quasi-order if and only if Q is Σ1 1 and

P ≤B Q, for every Σ1

1 quasi-order P.

If Q is a complete Σ1

1 quasi-order, then EQ := Q ∩ Q−1 is a

complete Σ1

1 equivalence relation.

Theorem (Louveau-Rosendal 2005)

The relation of embeddability between countable graphs ⊑GRAPHS is a complete Σ1

1 quasi-order. Thus the bi-embeddability ≡GRAPHS

is a complete Σ1

1 equivalence relation.

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Embeddability between coutable structures

Definition

Denote by XGRAPHS the space of countable (undirected) graphs with domain N. It is a closed subset of 2N2, thus a Polish space.

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Embeddability between coutable structures

Definition

Denote by XGRAPHS the space of countable (undirected) graphs with domain N. It is a closed subset of 2N2, thus a Polish space. T ⊑GRAPHS V

def

⇐ ⇒ ∃h : N 1−1 − − → N h is an isomorphism from T to V ↾ Im(h).

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Embeddability between coutable structures

Definition

Denote by XGRAPHS the space of countable (undirected) graphs with domain N. It is a closed subset of 2N2, thus a Polish space. T ⊑GRAPHS V

def

⇐ ⇒ ∃h : N 1−1 − − → N h is an isomorphism from T to V ↾ Im(h). ⊑GRAPHS is a Σ1

1 quasi-order.

we can consider the relation of embeddability on any space of countable structures (graphs, groups, etc...);

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Embeddability of countable groups

Theorem (Williams 2014)

The embeddability between countable groups ⊑GROUPS is a complete Σ1

1 quasi-order.

Williams defines a reduction from ⊑GRAPHS to ⊑GROUPS.

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Embeddability of countable groups

Theorem (Williams 2014)

The embeddability between countable groups ⊑GROUPS is a complete Σ1

1 quasi-order.

Williams defines a reduction from ⊑GRAPHS to ⊑GROUPS. The groups Williams builds to reduce ⊑GRAPHS to ⊑GROUPS are nonabelian.

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Embeddability on abelian groups

Question

How about the embeddability between countable abelian groups?

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Embeddability on abelian groups

Question

How about the embeddability between countable abelian groups? And the more challenging...

Question

How about the embeddability between countable torsion-free abelian groups? Is it a complete Σ1

1 quasi-order?

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What I’m not talking about today

A few weeks ago...

Theorem (C.-Thomas)

The relation of embeddability ⊑TFA between countable torsion-free abelian groups is a complete Σ1

1 quasi-order.

An old problem

Is the isomorphism between countable torsion-free abelian groups S∞-complete (Borel complete)?

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The “generalized” result

A few more weeks ago...

Theorem (C.)

If κ is uncountable such that κ<κ = κ, then the relation ⊑κ

TFA of

embeddability between TFA groups of size κ is a complete Σ1

1

quasi-order.

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The “generalized” result

A few more weeks ago...

Theorem (C.)

If κ is uncountable such that κ<κ = κ, then the relation ⊑κ

TFA of

embeddability between TFA groups of size κ is a complete Σ1

1

quasi-order. Generalized Descriptive Set Theory is not the mere generalization of Classical Descriptive Set Theory.

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The “generalized” result

A few more weeks ago...

Theorem (C.)

If κ is uncountable such that κ<κ = κ, then the relation ⊑κ

TFA of

embeddability between TFA groups of size κ is a complete Σ1

1

quasi-order. Generalized Descriptive Set Theory is not the mere generalization of Classical Descriptive Set Theory. The proof relies on the existence of an almost-full embedding G : Graphs → Ab (Prze´ zdziecki 2014).

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Generalied Descriptive Set Theory

Let κ be an uncountable cardinal.

Definition

The generalized Baire space on κ is κκ := {x | x : κ → κ} endowed with the (bounded) topology, i.e. the one generated by the sets of the form Ns := {x ∈ κκ | x ⊇ s}, where s ∈ <κκ.

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Generalied Descriptive Set Theory

Let κ be an uncountable cardinal.

Definition

The generalized Baire space on κ is κκ := {x | x : κ → κ} endowed with the (bounded) topology, i.e. the one generated by the sets of the form Ns := {x ∈ κκ | x ⊇ s}, where s ∈ <κκ. The generalized Cantor space is the closed subspace κ2 of κκ consisting of the binary sequences.

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Generalied Descriptive Set Theory

Let κ be an uncountable cardinal.

Definition

The generalized Baire space on κ is κκ := {x | x : κ → κ} endowed with the (bounded) topology, i.e. the one generated by the sets of the form Ns := {x ∈ κκ | x ⊇ s}, where s ∈ <κκ. The generalized Cantor space is the closed subspace κ2 of κκ consisting of the binary sequences. They are not metrizable (unless cof (κ) = ω), and their density character is κ<κ and 2<κ.

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The most common approach

We assume κ<κ = κ

κκ is a κ-space (i.e., it has a base of size κ).

κ+-Borel sets are the closure under complements and ≤ κ-unions of the basic open sets. They form a proper subset

• f P(κκ) that can be stratified in a hierarchy with exactly

κ+-many levels (the Σ0

α’s and Π0 α’s).

A κ-space is standard Borel if it is κ+-Borel isomorphic to a κ+-Borel subset of κκ. Let X be standard Borel. Then a set A ⊆ X is (κ-)analytic or Σ1

1 if it is a projection of a closed subset of κκ × X.

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Generalized Borel reducibility

We denote by X κ

GRAPHS, X κ TFA the standard Borel κ-spaces of the

respective structures over κ. We denote by ⊑κ

GRAPHS, ⊑κ TFA the Σ1 1 quasi-orders of

embeddability on the respective spaces.

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Generalized Borel reducibility

We denote by X κ

GRAPHS, X κ TFA the standard Borel κ-spaces of the

respective structures over κ. We denote by ⊑κ

GRAPHS, ⊑κ TFA the Σ1 1 quasi-orders of

embeddability on the respective spaces.

Definition

Let X and Y be standard Borel κ-space, and P, Q be binary relations over X and Y , respectively. We say that P Borel reduces to Q (or P ≤B Q) if and only if there is a κ+-Borel f : X → Y such that x1 P x2 ⇔ f (x1) Q f (x2). The definition of Σ1

1-completeness is translated verbatim.

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Generalized Louveau-Rosendal Theorem

Theorem (Motto Ros 2013)

If κ is weakly compact, then ⊑κ

GRAPHS is a complete Σ1 1

quasi-order.

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Generalized Louveau-Rosendal Theorem

Theorem (Motto Ros 2013)

If κ is weakly compact, then ⊑κ

GRAPHS is a complete Σ1 1

quasi-order.

Theorem (Mildenberger-Motto Ros)

If κ is uncountable such that κ<κ = κ, then ⊑κ

GRAPHS is a

complete Σ1

1 quasi-order.

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Theorem (C.)

If κ is uncountable such that κ<κ = κ, then the relation ⊑κ

TFA of

embeddability between TFA groups of size κ is a complete Σ1

1

quasi-order.

Proof (sketch)

It suffices to show that ⊑κ

GRAPHS≤B⊑κ TFA.

Let Γ be a skeleton of the category Graphctble Suppose that every graph in Γ has some subset of N as vertex set. Let Wκ be a universal graph of size κ. I.e. every graph

• f size κ embeds into it.

If T is a graph, then let [T]<ω1 be the poset of countable induced subgraphs of T ordered by inclusion.

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Theorem (C.)

If κ is uncountable such that κ<κ = κ, then the relation ⊑κ

TFA of

embeddability between TFA groups of size κ is a complete Σ1

1

quasi-order.

Proof (sketch)

It suffices to show that ⊑κ

GRAPHS≤B⊑κ TFA.

Let Γ be a skeleton of the category Graphctble Suppose that every graph in Γ has some subset of N as vertex set. Let Wκ be a universal graph of size κ. I.e. every graph

• f size κ embeds into it.

If T is a graph, then let [T]<ω1 be the poset of countable induced subgraphs of T ordered by inclusion.

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Theorem (C.)

If κ is uncountable such that κ<κ = κ, then the relation ⊑κ

TFA of

embeddability between TFA groups of size κ is a complete Σ1

1

quasi-order.

Proof (sketch)

It suffices to show that ⊑κ

GRAPHS≤B⊑κ TFA.

Let Γ be a skeleton of the category Graphctble Suppose that every graph in Γ has some subset of N as vertex set. Let Wκ be a universal graph of size κ. I.e. every graph

• f size κ embeds into it.

If T is a graph, then let [T]<ω1 be the poset of countable induced subgraphs of T ordered by inclusion.

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Theorem (C.)

If κ is uncountable such that κ<κ = κ, then the relation ⊑κ

TFA of

embeddability between TFA groups of size κ is a complete Σ1

1

quasi-order.

Proof (sketch)

It suffices to show that ⊑κ

GRAPHS≤B⊑κ TFA.

Let Γ be a skeleton of the category Graphctble Suppose that every graph in Γ has some subset of N as vertex set. Let Wκ be a universal graph of size κ. I.e. every graph

• f size κ embeds into it.

If T is a graph, then let [T]<ω1 be the poset of countable induced subgraphs of T ordered by inclusion.

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For every S ∈ [Wκ]<ω1 fix a graph isomorphism θS : S → σ(S) where σ(S) is the unique graph in Γ isomorphic to S. Define A := Z[Arw(Γ) ∪ {1} ∪ Pfin(ω)] i.e., the free abelian group on Arw(Γ) ∪ {1} ∪ Pfin(ω). We give A a ring structure as follows. For every a, b ∈ Arw(Γ) ∪ Pfin(ω), ab =            a ◦ b if a and b are composable a′′b if

• b ⊆ dom(a)

a ↾ b is a graph isomorphism

• therwise

with the additional requirement that 1 is the multiplicative unit.

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For every S ∈ [Wκ]<ω1 fix a graph isomorphism θS : S → σ(S) where σ(S) is the unique graph in Γ isomorphic to S. Define A := Z[Arw(Γ) ∪ {1} ∪ Pfin(ω)

• not in [Prz14]

] i.e., the free abelian group on Arw(Γ) ∪ {1} ∪ Pfin(ω). We give A a ring structure as follows. For every a, b ∈ Arw(Γ) ∪ Pfin(ω), ab =            a ◦ b if a and b are composable a′′b if

• b ⊆ dom(a)

a ↾ b is a graph isomorphism

• therwise

with the additional requirement that 1 is the multiplicative unit.

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For every S ∈ [Wκ]<ω1 fix a graph isomorphism θS : S → σ(S) where σ(S) is the unique graph in Γ isomorphic to S. Define A := Z[Arw(Γ) ∪ {1} ∪ Pfin(ω)] i.e., the free abelian group on Arw(Γ) ∪ {1} ∪ Pfin(ω). We give A a ring structure as follows. For every a, b ∈ Arw(Γ) ∪ Pfin(ω), ab =            a ◦ b if a and b are composable a′′b if

• b ⊆ dom(a)

a ↾ b is a graph isomorphism

• therwise

with the additional requirement that 1 is the multiplicative unit.

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Notice that A is a ring of cardinality 2ℵ0 whose additive group is free.

Theorem (Corner 1963; Prze´ zdziecki 2014)

If A is a ring of cardinality at most the continuum whose additive group is free, then there exists a torsion-free abelian group M ⊆ A such that |A| = |M| End(M) ∼ = A A ⊆ M as A-algebras.

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Notice that A is a ring of cardinality 2ℵ0 whose additive group is free.

Theorem (Corner 1963; Prze´ zdziecki 2014)

If A is a ring of cardinality at most the continuum whose additive group is free, then there exists a torsion-free abelian group M ⊆ A such that |A| = |M| End(M) ∼ = A A ⊆ M as A-algebras. All of the elements of M are represented by Cauchy nets in A. So the A-module structure of M is given by coordinate-wise product a · [(an)] = [(aan)].

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For every T ∈ [Wκ]κ we define a direct system of torsion-free abelian group indexed by [T]<ω1.

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For every T ∈ [Wκ]κ we define a direct system of torsion-free abelian group indexed by [T]<ω1. First for every C ∈ Γ, set GC := idC · M.

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For every T ∈ [Wκ]κ we define a direct system of torsion-free abelian group indexed by [T]<ω1. First for every C ∈ Γ, set GC := idC · M. For every S, S′ ∈ [T]<ω1 such that S ⊆ S′, the inclusion S ֒ → S′ induces an injective map γS

S′ : σ(S) → σ(S′)

between the isomorphic copies in Γ of S, S′ .

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For every T ∈ [Wκ]κ we define a direct system of torsion-free abelian group indexed by [T]<ω1. First for every C ∈ Γ, set GC := idC · M. For every S, S′ ∈ [T]<ω1 such that S ⊆ S′, the inclusion S ֒ → S′ induces an injective map γS

S′ : σ(S) → σ(S′)

between the isomorphic copies in Γ of S, S′ . The γS

S′ induces

in turn an injective map τ S

S′ : Gσ(S) → Gσ(S′),

g → γS

S′ · g

by left multiplication.

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For every T ∈ [Wκ]κ we define a direct system of torsion-free abelian group indexed by [T]<ω1. First for every C ∈ Γ, set GC := idC · M. For every S, S′ ∈ [T]<ω1 such that S ⊆ S′, the inclusion S ֒ → S′ induces an injective map γS

S′ : σ(S) → σ(S′)

between the isomorphic copies in Γ of S, S′ . The γS

S′ induces

in turn an injective map τ S

S′ : Gσ(S) → Gσ(S′),

g → γS

S′ · g

by left multiplication. We claim that ({Gσ(S)}; {τ S

S′})S,S′∈[T]<ω1 is a direct system

• f torsion-free abelian group.

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Now we can define the reduction. For every T ∈ [Wκ]κ let GT := lim

− → S∈[T]<ω1

Gσ(S).

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Now we can define the reduction. For every T ∈ [Wκ]κ let GT := lim

− → S∈[T]<ω1

Gσ(S).

Lemma

One can “code” every GT as a group GT with underlying set κ such that the map [Wκ]κ → X κ

TFA,

T → GT is κ+-Borel.

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The first part of the reduction is proved straightforwardly as in [Prz14].

Lemma

If ϕ : T → V is an embedding then Gϕ : GT → GV is

• ne-to-one. Therefore

T ⊑κ

GRAPHS V =

⇒ GT ⊑κ

TFA GV .

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The first part of the reduction is proved straightforwardly as in [Prz14].

Lemma

If ϕ : T → V is an embedding then Gϕ : GT → GV is

• ne-to-one. Therefore

T ⊑κ

GRAPHS V =

⇒ GT ⊑κ

TFA GV .

We are left to prove that the converse also holds.

Lemma

GT ⊑κ

TFA GV =

⇒ T ⊑κ

GRAPHS V .

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Proof of Lemma. Next theorem is essetially by [Prz14].

Theorem (Almost-fullness)

There is a natural group isomorphism Ψ: Z[Hom(T, V )] → Hom(GT, GV ).

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Proof of Lemma. Next theorem is essetially by [Prz14].

Theorem (Almost-fullness)

There is a natural group isomorphism Ψ: Z[Hom(T, V )] → Hom(GT, GV ). Let h: GT → GV be an embedding. By almost-fullness, there are finitely many ϕi ∈ Hom(T, V ) so that h = Ψ(

• i∈I

kiϕi).

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Proof of Lemma. Next theorem is essetially by [Prz14].

Theorem (Almost-fullness)

There is a natural group isomorphism Ψ: Z[Hom(T, V )] → Hom(GT, GV ). Let h: GT → GV be an embedding. By almost-fullness, there are finitely many ϕi ∈ Hom(T, V ) so that h = Ψ(

• i∈I

kiϕi). We claim that there must be some i ∈ I such that ϕi is an embedding from T into V . Suppose not, then...

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Proof of Lemma. Next theorem is essetially by [Prz14].

Theorem (Almost-fullness)

There is a natural group isomorphism Ψ: Z[Hom(T, V )] → Hom(GT, GV ). Let h: GT → GV be an embedding. By almost-fullness, there are finitely many ϕi ∈ Hom(T, V ) so that h = Ψ(

• i∈I

kiϕi). We claim that there must be some i ∈ I such that ϕi is an embedding from T into V . Suppose not, then... h is not injective, a contradiction. [Find a detector in Pfin(ω)].

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A generalization to R-modules

Theorem (C.)

Let κ be an uncountable cardinal such that κ<κ = κ. If R is a commutative S-cotorsion-free ring of cardinality less than the continuum, then the relation ⊑κ

R-MOD of embeddability on the

space of R-modules of size κ is a complete Σ1

1 quasi-order.

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Calderoni F., On the complexity of the relation of embeddability between R-modules of uncountable cardinality, in preparation.

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Calderoni F., On the complexity of the relation of embeddability between R-modules of uncountable cardinality, in preparation. Thank you!

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