Forcing and liberable groups Filippo Calderoni Master of Pure and - - PowerPoint PPT Presentation

▶
forcing and liberable groups
SMART_READER_LITE
LIVE PREVIEW

Forcing and liberable groups Filippo Calderoni Master of Pure and - - PowerPoint PPT Presentation

Dec 05, 2022 420 likes •793 views

Forcing and liberable groups Filippo Calderoni Master of Pure and Applied Logic Universitat de Barcelona calderonifilippo@gmail.com www.calderonifilippo.com 16th April 2014 Filippo Calderoni (UB) XXV incontro AILA 1/17 Outline Almost

slide-1
SLIDE 1

Forcing and liberable groups

Filippo Calderoni

Master of Pure and Applied Logic Universitat de Barcelona calderonifilippo@gmail.com www.calderonifilippo.com

16th April 2014

Filippo Calderoni (UB) XXV incontro AILA 1/17

slide-2
SLIDE 2

Outline

Almost Freeness Filtrations and Γ-invariant Liberating groups

Filippo Calderoni (UB) XXV incontro AILA 2/17

slide-3
SLIDE 3

Some History

“The modern era of set-theoretic methods in algebra can be said to have begun on July 11, 1973 when Saharon Shelah borrowed L´ aszl´

  • Fuchs’ Infinite Abelian Groups

from the Hebrew University Library.” (P. Eklof, A. Mekler)1 In 1974, could prove that the Whitehead’s Problem is independent from ZFC.

  • 1P. Eklof, A. Mekler. Almost Free Modules. North Holland, 2002.

Filippo Calderoni (UB) XXV incontro AILA 3/17

slide-4
SLIDE 4

Some History

“The modern era of set-theoretic methods in algebra can be said to have begun on July 11, 1973 when Saharon Shelah borrowed L´ aszl´

  • Fuchs’ Infinite Abelian Groups

from the Hebrew University Library.” (P. Eklof, A. Mekler)1 In 1974, could prove that the Whitehead’s Problem is independent from ZFC.

  • 1P. Eklof, A. Mekler. Almost Free Modules. North Holland, 2002.

Filippo Calderoni (UB) XXV incontro AILA 3/17

slide-5
SLIDE 5

Almost Free Groups

Definition

An (abelian) group G is free if and only if it has a basis. Ex.

α<κ Z

Definition

A group G is torsion-free if and only if every finitely generated subgroup of G is free.

  • Ex. (Q, +)

Definition (Fuchs 1958)

A group G is ℵ1-free if and only if every countably generated subgroup of G is free.

Filippo Calderoni (UB) XXV incontro AILA 4/17

slide-6
SLIDE 6

Almost Free Groups

Definition

An (abelian) group G is free if and only if it has a basis. Ex.

α<κ Z

Definition

A group G is torsion-free if and only if every finitely generated subgroup of G is free.

  • Ex. (Q, +)

Definition (Fuchs 1958)

A group G is ℵ1-free if and only if every countably generated subgroup of G is free.

Filippo Calderoni (UB) XXV incontro AILA 4/17

slide-7
SLIDE 7

Almost Free Groups

Definition

An (abelian) group G is free if and only if it has a basis. Ex.

α<κ Z

Definition

A group G is torsion-free if and only if every finitely generated subgroup of G is free.

  • Ex. (Q, +)

Definition (Fuchs 1958)

A group G is ℵ1-free if and only if every countably generated subgroup of G is free.

Filippo Calderoni (UB) XXV incontro AILA 4/17

slide-8
SLIDE 8

Examples and basic facts

Since every subgroup of a free subgroup is free, it is clear that G free ⇒ G ℵ1-free. However, G free ⇐ G ℵ1-free

Baer-Specker Group

The direct product of ω copies of Z, Zω :=

  • i<ω

Z is ℵ1-free (Specker 1950) but it is not free (Baer 1937).

Filippo Calderoni (UB) XXV incontro AILA 5/17

slide-9
SLIDE 9

Examples and basic facts

Since every subgroup of a free subgroup is free, it is clear that G free ⇒ G ℵ1-free. However, G free ⇐ G ℵ1-free

Baer-Specker Group

The direct product of ω copies of Z, Zω :=

  • i<ω

Z is ℵ1-free (Specker 1950) but it is not free (Baer 1937).

Filippo Calderoni (UB) XXV incontro AILA 5/17

slide-10
SLIDE 10

We shall focus on almost free groups of cardinality ℵ1.

Question

Can we force any almost free group of cardinality ℵ1 to be free? This is not possible in general. In fact, ZFC “Zω is not free”.

True Question

When is it possible to force an almost free group of cardinality ℵ1 to be free?

Filippo Calderoni (UB) XXV incontro AILA 6/17

slide-11
SLIDE 11

We shall focus on almost free groups of cardinality ℵ1.

Question

Can we force any almost free group of cardinality ℵ1 to be free? This is not possible in general. In fact, ZFC “Zω is not free”.

True Question

When is it possible to force an almost free group of cardinality ℵ1 to be free?

Filippo Calderoni (UB) XXV incontro AILA 6/17

slide-12
SLIDE 12

We shall focus on almost free groups of cardinality ℵ1.

Question

Can we force any almost free group of cardinality ℵ1 to be free? This is not possible in general. In fact, ZFC “Zω is not free”.

True Question

When is it possible to force an almost free group of cardinality ℵ1 to be free?

Filippo Calderoni (UB) XXV incontro AILA 6/17

slide-13
SLIDE 13

We shall focus on almost free groups of cardinality ℵ1.

Question

Can we force any almost free group of cardinality ℵ1 to be free? This is not possible in general. In fact, ZFC “Zω is not free”.

True Question

When is it possible to force an almost free group of cardinality ℵ1 to be free?

Filippo Calderoni (UB) XXV incontro AILA 6/17

slide-14
SLIDE 14

Filtrations

Definition

Given a group G of cardinality ℵ1, a filtration of G is a sequence {Gα : α ∈ ℵ1} of subgroups of G whose union is G and such that for all α, β < κ: Gα is a countable subgroup of G; if α < β, then Gα ⊆ Gβ; if γ is a limit ordinal, then Gγ =

α<γ Gα.

Theorem

A group G of cardinality ℵ1 is ℵ1-free if and only if it has a filtration {Gα : α < ℵ1} consisting of free groups.

Filippo Calderoni (UB) XXV incontro AILA 7/17

slide-15
SLIDE 15

Filtrations

Definition

Given a group G of cardinality ℵ1, a filtration of G is a sequence {Gα : α ∈ ℵ1} of subgroups of G whose union is G and such that for all α, β < κ: Gα is a countable subgroup of G; if α < β, then Gα ⊆ Gβ; if γ is a limit ordinal, then Gγ =

α<γ Gα.

Theorem

A group G of cardinality ℵ1 is ℵ1-free if and only if it has a filtration {Gα : α < ℵ1} consisting of free groups.

Filippo Calderoni (UB) XXV incontro AILA 7/17

slide-16
SLIDE 16

Γ-invariant

Theorem (Eklof 1977)

Let G be an ℵ1-free group of cardinality ℵ1. Then, G is free if and

  • nly if E := {α < ℵ1 : G/Gα is not ℵ1-free} is not stationary, for

some filtration {Gα : α < ℵ1}. This criterion does NOT depend on the filtration! In fact, any two filtrations agree on a club.

Definition

[E] in P(ℵ1)/club is the so called Γ-invariant of G.

Filippo Calderoni (UB) XXV incontro AILA 8/17

slide-17
SLIDE 17

Γ-invariant

Theorem (Eklof 1977)

Let G be an ℵ1-free group of cardinality ℵ1. Then, G is free if and

  • nly if E := {α < ℵ1 : G/Gα is not ℵ1-free} is not stationary, for

some filtration {Gα : α < ℵ1}. This criterion does NOT depend on the filtration! In fact, any two filtrations agree on a club.

Definition

[E] in P(ℵ1)/club is the so called Γ-invariant of G.

Filippo Calderoni (UB) XXV incontro AILA 8/17

slide-18
SLIDE 18

Γ-invariant

Theorem (Eklof 1977)

Let G be an ℵ1-free group of cardinality ℵ1. Then, G is free if and

  • nly if E := {α < ℵ1 : G/Gα is not ℵ1-free} is not stationary, for

some filtration {Gα : α < ℵ1}. This criterion does NOT depend on the filtration! In fact, any two filtrations agree on a club.

Definition

[E] in P(ℵ1)/club is the so called Γ-invariant of G.

Filippo Calderoni (UB) XXV incontro AILA 8/17

slide-19
SLIDE 19

Proof.

⇒) Let {xα : α < ℵ1} be a basis for G. Define the filtration {Gα : α < ℵ1} such that Gα = xξ : ξ < α, then it turns out that E = ∅. ⇐) Let {Gα : α < ℵ1} be any filtration of G and assume that E is not stationary. That is, there exists a club C ⊆ ω1 such that G/Gα is ℵ1-free, for each α ∈ C. Moreover, {Gα : α ∈ C} is a filtration of G. Claim: given any function f : ℵ1 → C enumerating C, Gf (α+1)/Gf (α) is free, for every α < ℵ1. We conclude that G is free because G =

  • α∈ℵ1

Gf (α+1)/Gf (α).

Filippo Calderoni (UB) XXV incontro AILA 9/17

slide-20
SLIDE 20

Proof.

⇒) Let {xα : α < ℵ1} be a basis for G. Define the filtration {Gα : α < ℵ1} such that Gα = xξ : ξ < α, then it turns out that E = ∅. ⇐) Let {Gα : α < ℵ1} be any filtration of G and assume that E is not stationary. That is, there exists a club C ⊆ ω1 such that G/Gα is ℵ1-free, for each α ∈ C. Moreover, {Gα : α ∈ C} is a filtration of G. Claim: given any function f : ℵ1 → C enumerating C, Gf (α+1)/Gf (α) is free, for every α < ℵ1. We conclude that G is free because G =

  • α∈ℵ1

Gf (α+1)/Gf (α).

Filippo Calderoni (UB) XXV incontro AILA 9/17

slide-21
SLIDE 21

Proof.

⇒) Let {xα : α < ℵ1} be a basis for G. Define the filtration {Gα : α < ℵ1} such that Gα = xξ : ξ < α, then it turns out that E = ∅. ⇐) Let {Gα : α < ℵ1} be any filtration of G and assume that E is not stationary. That is, there exists a club C ⊆ ω1 such that G/Gα is ℵ1-free, for each α ∈ C. Moreover, {Gα : α ∈ C} is a filtration of G. Claim: given any function f : ℵ1 → C enumerating C, Gf (α+1)/Gf (α) is free, for every α < ℵ1. We conclude that G is free because G =

  • α∈ℵ1

Gf (α+1)/Gf (α).

Filippo Calderoni (UB) XXV incontro AILA 9/17

slide-22
SLIDE 22

Proof.

⇒) Let {xα : α < ℵ1} be a basis for G. Define the filtration {Gα : α < ℵ1} such that Gα = xξ : ξ < α, then it turns out that E = ∅. ⇐) Let {Gα : α < ℵ1} be any filtration of G and assume that E is not stationary. That is, there exists a club C ⊆ ω1 such that G/Gα is ℵ1-free, for each α ∈ C. Moreover, {Gα : α ∈ C} is a filtration of G. Claim: given any function f : ℵ1 → C enumerating C, Gf (α+1)/Gf (α) is free, for every α < ℵ1. We conclude that G is free because G =

  • α∈ℵ1

Gf (α+1)/Gf (α).

Filippo Calderoni (UB) XXV incontro AILA 9/17

slide-23
SLIDE 23

Proof.

⇒) Let {xα : α < ℵ1} be a basis for G. Define the filtration {Gα : α < ℵ1} such that Gα = xξ : ξ < α, then it turns out that E = ∅. ⇐) Let {Gα : α < ℵ1} be any filtration of G and assume that E is not stationary. That is, there exists a club C ⊆ ω1 such that G/Gα is ℵ1-free, for each α ∈ C. Moreover, {Gα : α ∈ C} is a filtration of G. Claim: given any function f : ℵ1 → C enumerating C, Gf (α+1)/Gf (α) is free, for every α < ℵ1. We conclude that G is free because G =

  • α∈ℵ1

Gf (α+1)/Gf (α).

Filippo Calderoni (UB) XXV incontro AILA 9/17

slide-24
SLIDE 24

The hint

Question

Can we force any almost free group of cardinality ℵ1 to be free? Well, if we could destroy stationary sets...

Filippo Calderoni (UB) XXV incontro AILA 10/17

slide-25
SLIDE 25

Shooting a club

Forcing notion (Baumgartner 1976)

Given a stationary S ⊆ ω1, let PS be the sets of bounded closed subsets of S. Given any p, q ∈ PS p ≤ q iff q = p ∩ α, for some α < κ. If H is a PS-generic filter, then H ⊆ S and is a club of ω1.

Filippo Calderoni (UB) XXV incontro AILA 11/17

slide-26
SLIDE 26

Shooting a club

Forcing notion (Baumgartner 1976)

Given a stationary S ⊆ ω1, let PS be the sets of bounded closed subsets of S. Given any p, q ∈ PS p ≤ q iff q = p ∩ α, for some α < κ. If H is a PS-generic filter, then H ⊆ S and is a club of ω1.

Filippo Calderoni (UB) XXV incontro AILA 11/17

slide-27
SLIDE 27

Theorem

Let G be an almost free group. Then, G is free in some forcing extension that preserves ℵ1 iff E = {α < ℵ1 : G/Gα is not ℵ1-free} is stationary and co-stationary in ℵ1 (i.e., Γ(G) = [ℵ1]).

Proof.

⇒) If E and ℵ1 E are stationary, consider the poset P := Pℵ1E let H be Pℵ1E-generic. Then, V [H] “G is free”. Moreover, ℵ1 does not collapse because PS is ω-distributivity. ⇐) If ℵ1 E is not stationary, then E contains a club. So, E cannot be destroyed.

Filippo Calderoni (UB) XXV incontro AILA 12/17

slide-28
SLIDE 28

Theorem

Let G be an almost free group. Then, G is free in some forcing extension that preserves ℵ1 iff E = {α < ℵ1 : G/Gα is not ℵ1-free} is stationary and co-stationary in ℵ1 (i.e., Γ(G) = [ℵ1]).

Proof.

⇒) If E and ℵ1 E are stationary, consider the poset P := Pℵ1E let H be Pℵ1E-generic. Then, V [H] “G is free”. Moreover, ℵ1 does not collapse because PS is ω-distributivity. ⇐) If ℵ1 E is not stationary, then E contains a club. So, E cannot be destroyed.

Filippo Calderoni (UB) XXV incontro AILA 12/17

slide-29
SLIDE 29

Theorem

Let G be an almost free group. Then, G is free in some forcing extension that preserves ℵ1 iff E = {α < ℵ1 : G/Gα is not ℵ1-free} is stationary and co-stationary in ℵ1 (i.e., Γ(G) = [ℵ1]).

Proof.

⇒) If E and ℵ1 E are stationary, consider the poset P := Pℵ1E let H be Pℵ1E-generic. Then, V [H] “G is free”. Moreover, ℵ1 does not collapse because PS is ω-distributivity. ⇐) If ℵ1 E is not stationary, then E contains a club. So, E cannot be destroyed.

Filippo Calderoni (UB) XXV incontro AILA 12/17

slide-30
SLIDE 30

There are liberable groups

Example

For every stationary E ⊆ ℵ1 there is a group G such that Γ(G) = [E]. Assume E ⊆ lim(ω1). Fix a ladder system on E {ηδ : δ ∈ E}. Consider the Q-vector space with basis {yδ,n : δ ∈ E, n ∈ ω} ∪ {xα : α ∈ ω1}. For each δ ∈ E and n ∈ ω, zδ,n := yδ,0 − n

i=0 2ixηδ(i)

2n+1 . Then, G := {xα : α ∈ ω1} ∪ {zδ,n : δ ∈ E, n ∈ ω}Z. G is ℵ1-free and Γ(G) = [E]. Thus, G is not free.

Filippo Calderoni (UB) XXV incontro AILA 13/17

slide-31
SLIDE 31

Further possible developments

To liberate almost free group of cardinality κ. We cannot reproduce the same result because shooting a club collapses κ in general! To focus on almost ℵ1-free groups. To enrich the English dictionary because“liberable” is NOT an English word.

Filippo Calderoni (UB) XXV incontro AILA 14/17

slide-32
SLIDE 32

Further possible developments

To liberate almost free group of cardinality κ. We cannot reproduce the same result because shooting a club collapses κ in general! To focus on almost ℵ1-free groups. To enrich the English dictionary because“liberable” is NOT an English word.

Filippo Calderoni (UB) XXV incontro AILA 14/17

slide-33
SLIDE 33

Further possible developments

To liberate almost free group of cardinality κ. We cannot reproduce the same result because shooting a club collapses κ in general! To focus on almost ℵ1-free groups. To enrich the English dictionary because“liberable” is NOT an English word.

Filippo Calderoni (UB) XXV incontro AILA 14/17

slide-34
SLIDE 34

References I

Fuchs, L. Abelian Groups. Publishing House of the Hungarian Academy of Science, Budapest, 1958. Eklof, P.C. - Mekler, A. Almost Free Modules: Set-Theoretic Methods. Revised Edition. North-Holland Mathematical Library, Amsterdam, 2002. Jech, T. Set Theory. 3d edition. Springer, New York, 2003. Baer R. Abelian groups without elements of finite order Duke Math. J. 368-122, 1937.

Filippo Calderoni (UB) XXV incontro AILA 15/17

slide-35
SLIDE 35

References II

Specker E. Additive Gruppen yon Folgen ganzer Zahlen Portugaliae Math. 9, 131-140, 1950. Baumgartner J. E., Harrington L. A., Kleinberg E.M. Adding a closed and unbounded set

  • J. Symbolic Logic 41, no. 2, 481-482, 1976.

Eklof, P. C. Methods of logic in abelian group theory, in Abelian Group Theory, Lecture Notes in Mathematics No. 616, Springer- Verlag, 251-269, 1977.

Filippo Calderoni (UB) XXV incontro AILA 16/17

slide-36
SLIDE 36

Thanks for your attention!

Filippo Calderoni (UB) XXV incontro AILA 17/17