Cellular Covers of Cotorsion-free Modules R udiger G obel, Jos e - - PowerPoint PPT Presentation

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Cellular Covers of Cotorsion-free Modules

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Conference on Algebraic Topology CAT’09 6-11 July 2009 Warszawa, Poland

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Presentation

Summer 2008, with Lutz, R¨ udiger, and ...

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Presentation

... and Peter Loth!

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Cellular covers

Fix R a commutative ring with unit. Definition A homomorphism π : G → H of R-modules is a cellular cover

• ver H if:

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Cellular covers

Fix R a commutative ring with unit. Definition A homomorphism π : G → H of R-modules is a cellular cover

• ver H if:

G

π

− → H

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Cellular covers

Fix R a commutative ring with unit. Definition A homomorphism π : G → H of R-modules is a cellular cover

• ver H if:

G ց G

π

− → H

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Cellular covers

Fix R a commutative ring with unit. Definition A homomorphism π : G → H of R-modules is a cellular cover

• ver H if:

G ∃! ↓ ց G

π

− → H

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Cellular covers

Fix R a commutative ring with unit. Definition A homomorphism π : G → H of R-modules is a cellular cover

• ver H if:

G ∃! ↓ ց G

π

− → H π∗ : EndR(G) ∼ = HomR(G, H), ϕ → π ◦ ϕ.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Cellular covers

Fix R a commutative ring with unit. Definition A homomorphism π : G → H of R-modules is a cellular cover

• ver H if:

G ∃! ↓ ց G

π

− → H π∗ : EndR(G) ∼ = HomR(G, H), ϕ → π ◦ ϕ. A short exact sequence 0 → K → G

π

→ H → 0 with π a cellular cover is called a cellular exact sequence.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Cellular covers

Fix R a commutative ring with unit. Definition A homomorphism π : G → H of R-modules is a cellular cover

• ver H if:

G ∃! ↓ ց G

π

− → H π∗ : EndR(G) ∼ = HomR(G, H), ϕ → π ◦ ϕ. A short exact sequence 0 → K → G

π

→ H → 0 with π a cellular cover is called a cellular exact sequence.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Our results

We have constructed exact sequences of cotorsion-free R-modules 0 → K − → G

π

− → H → 0 with a prescribed K or H, such that the following “rigidity” properties hold:

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Our results

We have constructed exact sequences of cotorsion-free R-modules 0 → K − → G

π

− → H → 0 with a prescribed K or H, such that the following “rigidity” properties hold: End(G) = End(H) = R. and Hom(G, H) = πR

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Our results

We have constructed exact sequences of cotorsion-free R-modules 0 → K − → G

π

− → H → 0 with a prescribed K or H, such that the following “rigidity” properties hold: End(G) = End(H) = R. and Hom(G, H) = πR In particular G → H is a cellular cover.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Our results

We have constructed exact sequences of cotorsion-free R-modules 0 → K − → G

π

− → H → 0 with a prescribed K or H, such that the following “rigidity” properties hold: End(G) = End(H) = R. and Hom(G, H) = πR In particular G → H is a cellular cover.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Motivation

Some problems from homotopical localization theory needed to look at the algebraic counterpart. Bousfield’97, Casacuberta–R–Tai’98, Cascuberta–Guti´ errez’05. R-Scherer’01, 08 Algebraic problem: Determine all possible cellular exact sequences

• f groups:

0 → K → G → H → 1.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Motivation

Some problems from homotopical localization theory needed to look at the algebraic counterpart. Bousfield’97, Casacuberta–R–Tai’98, Cascuberta–Guti´ errez’05. R-Scherer’01, 08 Algebraic problem: Determine all possible cellular exact sequences

• f groups:

0 → K → G → H → 1.

1 Which properties of G are inherit from H? R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Motivation

Some problems from homotopical localization theory needed to look at the algebraic counterpart. Bousfield’97, Casacuberta–R–Tai’98, Cascuberta–Guti´ errez’05. R-Scherer’01, 08 Algebraic problem: Determine all possible cellular exact sequences

• f groups:

0 → K → G → H → 1.

1 Which properties of G are inherit from H? 2 Is there any bound in the cardinality of G when we fix either

K or H?

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Motivation

Some problems from homotopical localization theory needed to look at the algebraic counterpart. Bousfield’97, Casacuberta–R–Tai’98, Cascuberta–Guti´ errez’05. R-Scherer’01, 08 Algebraic problem: Determine all possible cellular exact sequences

• f groups:

0 → K → G → H → 1.

1 Which properties of G are inherit from H? 2 Is there any bound in the cardinality of G when we fix either

K or H? Many partial answers (also for localizations): http://jlrodri.wordpress.com/

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Motivation

Some problems from homotopical localization theory needed to look at the algebraic counterpart. Bousfield’97, Casacuberta–R–Tai’98, Cascuberta–Guti´ errez’05. R-Scherer’01, 08 Algebraic problem: Determine all possible cellular exact sequences

• f groups:

0 → K → G → H → 1.

1 Which properties of G are inherit from H? 2 Is there any bound in the cardinality of G when we fix either

K or H? Many partial answers (also for localizations): http://jlrodri.wordpress.com/

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Motivation

Some algebraic structures preserved or non preserved under group (co)-localization Localization Modules Rings Algebras Finite groups Nilpotent groups (???) Solvable groups Perfect groups Simple groups Classes of abelian groups (?) Others (??) Co-localization Modules Rings Algebras Finite groups Nilpotent groups Solvable groups (?) Perfect groups Simple groups Classes of abelian groups (?) Others (??)

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Notation from abelian group theory

Given S = {1, s2, s3, s4, ...} a distinguished countable multiplicatively closed subset of Z. We denote Z the completion of Z, with respect to its S-adic topology, having {sZ} as a basis of neighborhoods of 0. Any abelian group A has endowed an S-adic topology, ( {sA} basis of neighborhoods of 0); we denote A be the completion

• f A with respect to its S-adic topology.

The abelian group A is S-reduced (i.e.

s∈S sA = 0) if and

• nly if it is Hausdorff.

A is called S-cotorsion-free if it is S-torsion-free, S-reduced and Hom( Z, A) = 0. For example: if S = Z \ 0, then A is cotorsion-free ⇐ ⇒ A ⊇ Q, Zp, Z/pZ, for any prime p. If |A| < 2ℵ0, then S-cotorsion-free ⇔ S-torsion-free and S-reduced.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Notation from abelian group theory

A subgroup A ⊂ B is S-pure if s A ∩ B = sB, for all s ∈ S, i.e. A is a subspace of B w.r.t. the S-adic topology. A ⊂ A is pure and dense (therefore A/A is divisible). If A ⊂ B ⊂ B∗ ⊂ A, where B∗ is the purification in A, then B∗/B is torsion and A/B∗ is divisible. Every homomorphism A → B extends uniquely to a homomorphism A → B of Z-modules.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Some known results on cellular covers of abelian groups

Suppose 0 → K → G

π

→ H → 0 is a cellular exact sequence of abelian groups. THEN: K is reduced and torsion-free (Farjoun–G¨

• bel–Segev’07)

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Some known results on cellular covers of abelian groups

Suppose 0 → K → G

π

→ H → 0 is a cellular exact sequence of abelian groups. THEN: K is reduced and torsion-free (Farjoun–G¨

• bel–Segev’07)

If H is divisible, then G can be described explicitly (Chach´

• lski–Farjoun–G¨
• bel–Segev’07 / Fuchs–G¨
• bel’09)

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Some known results on cellular covers of abelian groups

Suppose 0 → K → G

π

→ H → 0 is a cellular exact sequence of abelian groups. THEN: K is reduced and torsion-free (Farjoun–G¨

• bel–Segev’07)

If H is divisible, then G can be described explicitly (Chach´

• lski–Farjoun–G¨
• bel–Segev’07 / Fuchs–G¨
• bel’09)

If H is reduced, then K is cotorsion-free (Farjoun–G¨

• bel–Segev–Shelah’07 / Buckner–Dugas’06)

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Some known results on cellular covers of abelian groups

Suppose 0 → K → G

π

→ H → 0 is a cellular exact sequence of abelian groups. THEN: K is reduced and torsion-free (Farjoun–G¨

• bel–Segev’07)

If H is divisible, then G can be described explicitly (Chach´

• lski–Farjoun–G¨
• bel–Segev’07 / Fuchs–G¨
• bel’09)

If H is reduced, then K is cotorsion-free (Farjoun–G¨

• bel–Segev–Shelah’07 / Buckner–Dugas’06)

If H is torsion and reduced, then K = 1 (Fuchs–G¨

• bel’09)

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Some known results on cellular covers of abelian groups

Suppose 0 → K → G

π

→ H → 0 is a cellular exact sequence of abelian groups. THEN: K is reduced and torsion-free (Farjoun–G¨

• bel–Segev’07)

If H is divisible, then G can be described explicitly (Chach´

• lski–Farjoun–G¨
• bel–Segev’07 / Fuchs–G¨
• bel’09)

If H is reduced, then K is cotorsion-free (Farjoun–G¨

• bel–Segev–Shelah’07 / Buckner–Dugas’06)

If H is torsion and reduced, then K = 1 (Fuchs–G¨

• bel’09)

If K is cotorsion-free, then G can be arbitrarily large. (Buckner–Dugas’06)

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Some known results on cellular covers of abelian groups

Suppose 0 → K → G

π

→ H → 0 is a cellular exact sequence of abelian groups. THEN: K is reduced and torsion-free (Farjoun–G¨

• bel–Segev’07)

If H is divisible, then G can be described explicitly (Chach´

• lski–Farjoun–G¨
• bel–Segev’07 / Fuchs–G¨
• bel’09)

If H is reduced, then K is cotorsion-free (Farjoun–G¨

• bel–Segev–Shelah’07 / Buckner–Dugas’06)

If H is torsion and reduced, then K = 1 (Fuchs–G¨

• bel’09)

If K is cotorsion-free, then G can be arbitrarily large. (Buckner–Dugas’06) Our interest: What happens if H is torsion-free and reduced (or cotorsion-free)?

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Some known results on cellular covers of abelian groups

Suppose 0 → K → G

π

→ H → 0 is a cellular exact sequence of abelian groups. THEN: K is reduced and torsion-free (Farjoun–G¨

• bel–Segev’07)

If H is divisible, then G can be described explicitly (Chach´

• lski–Farjoun–G¨
• bel–Segev’07 / Fuchs–G¨
• bel’09)

If H is reduced, then K is cotorsion-free (Farjoun–G¨

• bel–Segev–Shelah’07 / Buckner–Dugas’06)

If H is torsion and reduced, then K = 1 (Fuchs–G¨

• bel’09)

If K is cotorsion-free, then G can be arbitrarily large. (Buckner–Dugas’06) Our interest: What happens if H is torsion-free and reduced (or cotorsion-free)?

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Prescribing endomorphism rings

Why are important the cotorsion-free abelian groups? Fuchs’s Problem 44: When R = End(G) for some G?

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Prescribing endomorphism rings

Why are important the cotorsion-free abelian groups? Fuchs’s Problem 44: When R = End(G) for some G? Corner (1963): Every countable reduced torsion-free ring is an endomorphism ring.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Prescribing endomorphism rings

Why are important the cotorsion-free abelian groups? Fuchs’s Problem 44: When R = End(G) for some G? Corner (1963): For rings R of size < 2ℵ0. Every countable reduced torsion-free ring is an endomorphism ring. Key idea: C ⊆ G ⊂∗ C, for some cotorsion-free R-module C. There exists a countable family of elements in R which are algebraically independent over C.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Prescribing endomorphism rings

Why are important the cotorsion-free abelian groups? Fuchs’s Problem 44: When R = End(G) for some G? Corner (1963): For rings R of size < 2ℵ0. Every countable reduced torsion-free ring is an endomorphism ring. Key idea: C ⊆ G ⊂∗ C, for some cotorsion-free R-module C. There exists a countable family of elements in R which are algebraically independent over C.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Prescribing endomorphism rings

Shelah’s Black Box (1984): For rings R of size ≥ 2ℵ0. A combinatorial theorem and endomorphism rings of abelian groups, Israel J. Math. vol. 49 (1984). Introduces his famous prediction principle valid in ordinary set theory (ZFC). Imitates Jensen’s diamond principle valid in G¨

• del’s constructible

universe.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Prescribing endomorphism rings

Shelah’s Black Box (1984): For rings R of size ≥ 2ℵ0. A combinatorial theorem and endomorphism rings of abelian groups, Israel J. Math. vol. 49 (1984). Introduces his famous prediction principle valid in ordinary set theory (ZFC). Imitates Jensen’s diamond principle valid in G¨

• del’s constructible

universe. It allows to realize almost every ring as the endomorphism ring of some torsion-free abelian group.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Prescribing endomorphism rings

Shelah’s Black Box (1984): For rings R of size ≥ 2ℵ0. A combinatorial theorem and endomorphism rings of abelian groups, Israel J. Math. vol. 49 (1984). Introduces his famous prediction principle valid in ordinary set theory (ZFC). Imitates Jensen’s diamond principle valid in G¨

• del’s constructible

universe. It allows to realize almost every ring as the endomorphism ring of some torsion-free abelian group. The idea is that every endomorphism C → C can be approximate by a family of “small” canonical homomorphisms {ϕβ}, β < λ, in a very well organized way.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Prescribing endomorphism rings

Shelah’s Black Box (1984): For rings R of size ≥ 2ℵ0. A combinatorial theorem and endomorphism rings of abelian groups, Israel J. Math. vol. 49 (1984). Introduces his famous prediction principle valid in ordinary set theory (ZFC). Imitates Jensen’s diamond principle valid in G¨

• del’s constructible

universe. It allows to realize almost every ring as the endomorphism ring of some torsion-free abelian group. The idea is that every endomorphism C → C can be approximate by a family of “small” canonical homomorphisms {ϕβ}, β < λ, in a very well organized way.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Prescribing endomorphisms rings

G¨

• bel–Shelah’85: ‘ ‘Modules over arbitrary domains”.

Generalizes Corner’s result to arbitrarily large cardinals.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Prescribing endomorphisms rings

G¨

• bel–Shelah’85: ‘ ‘Modules over arbitrary domains”.

Generalizes Corner’s result to arbitrarily large cardinals. Corner–G¨

• bel’85: “Prescribing endomorphism algebras - A

unified treatement”.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Prescribing endomorphisms rings

G¨

• bel–Shelah’85: ‘ ‘Modules over arbitrary domains”.

Generalizes Corner’s result to arbitrarily large cardinals. Corner–G¨

• bel’85: “Prescribing endomorphism algebras - A

unified treatement”. Dugas–Mader–Vinsonhaler’87: “Large E-rings exist”. Uses a strong version of Shelah’s Black-Box.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Prescribing endomorphisms rings

G¨

• bel–Shelah’85: ‘ ‘Modules over arbitrary domains”.

Generalizes Corner’s result to arbitrarily large cardinals. Corner–G¨

• bel’85: “Prescribing endomorphism algebras - A

unified treatement”. Dugas–Mader–Vinsonhaler’87: “Large E-rings exist”. Uses a strong version of Shelah’s Black-Box. G¨

• bel–May’90: “Four submodules suffice for realizing algebras
• ver commutative rings”.

A new purely algebraic method to construct rigid modules.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Prescribing endomorphisms rings

G¨

• bel–Shelah’85: ‘ ‘Modules over arbitrary domains”.

Generalizes Corner’s result to arbitrarily large cardinals. Corner–G¨

• bel’85: “Prescribing endomorphism algebras - A

unified treatement”. Dugas–Mader–Vinsonhaler’87: “Large E-rings exist”. Uses a strong version of Shelah’s Black-Box. G¨

• bel–May’90: “Four submodules suffice for realizing algebras
• ver commutative rings”.

A new purely algebraic method to construct rigid modules. G¨

• bel–Trlifaj (2006)

“Approximations and endomorphism algebras of modules”, de Gruyter Expositions in Mathematics,

• 41. Walter de Gruyter GmbH & Co. KG, Berlin,
• 2006. xxiv+640 pp.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Prescribing endomorphisms rings

G¨

• bel–Shelah’85: ‘ ‘Modules over arbitrary domains”.

Generalizes Corner’s result to arbitrarily large cardinals. Corner–G¨

• bel’85: “Prescribing endomorphism algebras - A

unified treatement”. Dugas–Mader–Vinsonhaler’87: “Large E-rings exist”. Uses a strong version of Shelah’s Black-Box. G¨

• bel–May’90: “Four submodules suffice for realizing algebras
• ver commutative rings”.

A new purely algebraic method to construct rigid modules. G¨

• bel–Trlifaj (2006)

“Approximations and endomorphism algebras of modules”, de Gruyter Expositions in Mathematics,

• 41. Walter de Gruyter GmbH & Co. KG, Berlin,
• 2006. xxiv+640 pp.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Our results

We have constructed cellular exact sequences of cotorsion-free R-modules 0 → K − → G

π

− → H → 0 with a prescribed K or H. |G| < 2ℵ0 |G| ≥ 2ℵ0 Fixing K Theorem 2 Theorem 1 Fixing H Theorem 3 Theorem 4

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Our results

We have constructed cellular exact sequences of cotorsion-free R-modules 0 → K − → G

π

− → H → 0 with a prescribed K or H. |G| < 2ℵ0 |G| ≥ 2ℵ0 Fixing K Theorem 2 Theorem 1 Fixing H Theorem 3 Theorem 4 Use: 2, 3: Classical Corner’s methods.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Our results

We have constructed cellular exact sequences of cotorsion-free R-modules 0 → K − → G

π

− → H → 0 with a prescribed K or H. |G| < 2ℵ0 |G| ≥ 2ℵ0 Fixing K Theorem 2 Theorem 1 Fixing H Theorem 3 Theorem 4 Use: 2, 3: Classical Corner’s methods. 1, 4: General Shelah’s Black Box.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Our results

We have constructed cellular exact sequences of cotorsion-free R-modules 0 → K − → G

π

− → H → 0 with a prescribed K or H. |G| < 2ℵ0 |G| ≥ 2ℵ0 Fixing K Theorem 2 Theorem 1 Fixing H Theorem 3 Theorem 4 Use: 2, 3: Classical Corner’s methods. 1, 4: General Shelah’s Black Box.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Some assumptions

• R is a commutative ring with 1.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Some assumptions

• R is a commutative ring with 1.
• S a distinguished countable multiplicatively closed subset such

that R is S-reduced and S-torsion-free.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Some assumptions

• R is a commutative ring with 1.
• S a distinguished countable multiplicatively closed subset such

that R is S-reduced and S-torsion-free.

• Therefore, the induced S-adic topology on R is Hausdorff. We

let R be the S-adic completion of R.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Some assumptions

• R is a commutative ring with 1.
• S a distinguished countable multiplicatively closed subset such

that R is S-reduced and S-torsion-free.

• Therefore, the induced S-adic topology on R is Hausdorff. We

let R be the S-adic completion of R.

• Assume that R is cotorsion-free with respect to S, i.e.

Hom( R, R) = 0.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Some assumptions

• R is a commutative ring with 1.
• S a distinguished countable multiplicatively closed subset such

that R is S-reduced and S-torsion-free.

• Therefore, the induced S-adic topology on R is Hausdorff. We

let R be the S-adic completion of R.

• Assume that R is cotorsion-free with respect to S, i.e.

Hom( R, R) = 0. Definition: An R-module M has rank κ ≤ | R |: In case | M | > | R |, define rk(M) := | M |. In case | M | ≤ | R |, define rk(M) := κ if there is a free submodule E =

α<κ Reα of M such that M/E is S-torsion.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Some assumptions

• R is a commutative ring with 1.
• S a distinguished countable multiplicatively closed subset such

that R is S-reduced and S-torsion-free.

• Therefore, the induced S-adic topology on R is Hausdorff. We

let R be the S-adic completion of R.

• Assume that R is cotorsion-free with respect to S, i.e.

Hom( R, R) = 0. Definition: An R-module M has rank κ ≤ | R |: In case | M | > | R |, define rk(M) := | M |. In case | M | ≤ | R |, define rk(M) := κ if there is a free submodule E =

α<κ Reα of M such that M/E is S-torsion.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

• 1. Fixing the kernel. Case ≥ 2ℵ0

Buckner and Dugas showed that every cotorsion-free module is the kernel of arbitrarily large cellular covers. Fix infinite cardinals κ, µ such that: |R| ≤ κ, µκ = µ. Let λ := µ+. Theorem (Buckner–Dugas’06) Let K be a cotorsion-free R-module of size |K| ≤ κ. Then there exists a cellular exact sequence of cotorsion-free R-modules 0 → K → G → G/K → 0 with |G| = λ.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

• 1. Fixing the kernel. Case ≥ 2ℵ0

Buckner and Dugas showed that every cotorsion-free module is the kernel of arbitrarily large cellular covers. Fix infinite cardinals κ, µ such that: |R| ≤ κ, µκ = µ. Let λ := µ+. Theorem (Buckner–Dugas’06) Let K be a cotorsion-free R-module of size |K| ≤ κ. Then there exists a cellular exact sequence of cotorsion-free R-modules 0 → K → G → G/K → 0 with |G| = λ.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

• 1. Fixing the kernel. Case ≥ 2ℵ0

Buckner and Dugas showed that every cotorsion-free module is the kernel of arbitrarily large cellular covers. Fix infinite cardinals κ, µ such that: |R| ≤ κ, µκ = µ. Let λ := µ+. Theorem (Buckner–Dugas’06) Let K be a cotorsion-free R-module of size |K| ≤ κ. Then there exists a cellular exact sequence of cotorsion-free R-modules 0 → K → G → G/K → 0 with |G| = λ.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

• 1. Fixing the kernel. Case ≥ 2ℵ0

We improve Buckner–Dugas’s result by replacing the strong Black box by the general Black Box. Fix an infinite cardinal κ such that |R| < κℵ0. Take λ = κℵ0

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

• 1. Fixing the kernel. Case ≥ 2ℵ0

We improve Buckner–Dugas’s result by replacing the strong Black box by the general Black Box. Fix an infinite cardinal κ such that |R| < κℵ0. Take λ = κℵ0 Theorem (1) If K is a cotorsion-free R-module of size |K| ≤ κ. Then there is a cellular exact sequence of cotorsion-free R-modules 0 → K → G → G/K → 0 with |G| = λ.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

• 1. Fixing the kernel. Case ≥ 2ℵ0

We improve Buckner–Dugas’s result by replacing the strong Black box by the general Black Box. Fix an infinite cardinal κ such that |R| < κℵ0. Take λ = κℵ0 Theorem (1) If K is a cotorsion-free R-module of size |K| ≤ κ. Then there is a cellular exact sequence of cotorsion-free R-modules 0 → K → G → G/K → 0 with |G| = λ. Corollary If κ = κℵ0 then all members of the cellular exact sequence have the same size κ.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

• 1. Fixing the kernel. Case ≥ 2ℵ0

We improve Buckner–Dugas’s result by replacing the strong Black box by the general Black Box. Fix an infinite cardinal κ such that |R| < κℵ0. Take λ = κℵ0 Theorem (1) If K is a cotorsion-free R-module of size |K| ≤ κ. Then there is a cellular exact sequence of cotorsion-free R-modules 0 → K → G → G/K → 0 with |G| = λ. Corollary If κ = κℵ0 then all members of the cellular exact sequence have the same size κ.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

• 2. Fixing the kernel. Case < 2ℵ0

Using Corner’s method we complement Buckner–Dugas’s result for cardinals < 2ℵ0. We also improve a result in Fuchs–G¨

• bel’09.

Suppose |R| < 2ℵ0. Theorem (2) Let K be any torsion-free and reduced R-module of rank κ < 2ℵ0. Then, there is a torsion-free and reduced R-module G of rank 3κ + 1, with submodule K, such that 0 → K → G → G/K → 0 is a cellular exact sequence. If κ = 1 we get G of rank 3.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

• 2. Fixing the kernel. Case < 2ℵ0

Using Corner’s method we complement Buckner–Dugas’s result for cardinals < 2ℵ0. We also improve a result in Fuchs–G¨

• bel’09.

Suppose |R| < 2ℵ0. Theorem (2) Let K be any torsion-free and reduced R-module of rank κ < 2ℵ0. Then, there is a torsion-free and reduced R-module G of rank 3κ + 1, with submodule K, such that 0 → K → G → G/K → 0 is a cellular exact sequence. If κ = 1 we get G of rank 3.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

• 2. Fixing the kernel. Case < 2ℵ0

Corollary If R = Z = K, then there exists a cellular exact sequence of torsion-free and reduced abelian groups 0 → Z → G → H → 0 where G has rank 3, and H has rank 2. Corollary If the rank κ of K is infinite (< 2ℵ0) we get a cellular exact sequence of torsion-free and reduced R-modules 0 → K → G → H → 0 all of the same rank κ.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

• 2. Fixing the kernel. Case < 2ℵ0

Corollary If R = Z = K, then there exists a cellular exact sequence of torsion-free and reduced abelian groups 0 → Z → G → H → 0 where G has rank 3, and H has rank 2. Corollary If the rank κ of K is infinite (< 2ℵ0) we get a cellular exact sequence of torsion-free and reduced R-modules 0 → K → G → H → 0 all of the same rank κ.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Sketch Proof

We do the finite rank case. E =

α<κ Reα ⊆ K of rank κ, such that K/E is torsion.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Sketch Proof

We do the finite rank case. E =

α<κ Reα ⊆ K of rank κ, such that K/E is torsion.

F =

α<κ Rfα of the same rank (or bigger).

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Sketch Proof

We do the finite rank case. E =

α<κ Reα ⊆ K of rank κ, such that K/E is torsion.

F =

α<κ Rfα of the same rank (or bigger).

C = K ⊕ F; then we define G as an S-pure submodule of C: G = K, F, (eαw + fα)wα, f ′w′ | α < κ∗ ⊆ C

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Sketch Proof

We do the finite rank case. E =

α<κ Reα ⊆ K of rank κ, such that K/E is torsion.

F =

α<κ Rfα of the same rank (or bigger).

C = K ⊕ F; then we define G as an S-pure submodule of C: G = K, F, (eαw + fα)wα, f ′w′ | α < κ∗ ⊆ C where w, w′ and wα is a family of alg. independent elts. / C, and f ′ = f0 + ... + fκ−1. Note G has rank 3κ + 1.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Sketch Proof

We do the finite rank case. E =

α<κ Reα ⊆ K of rank κ, such that K/E is torsion.

F =

α<κ Rfα of the same rank (or bigger).

C = K ⊕ F; then we define G as an S-pure submodule of C: G = K, F, (eαw + fα)wα, f ′w′ | α < κ∗ ⊆ C where w, w′ and wα is a family of alg. independent elts. / C, and f ′ = f0 + ... + fκ−1. Note G has rank 3κ + 1.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Fixing the cardinality of the cokernel

Known: the cardinality of the kernels of cellular covers is unbounded. For R = Z. Theorem (Farjoun–G¨

• bel–Segev–Shelah’07)

Let κ be any infinite cardinal. Then there exists a torsion-free and reduced abelian group H of cardinal κ, such that for any cardinal λ ≥ κ there is cellular exact sequence 0 → K → G → H → 0 with |G| = λ.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Fixing the cardinality of the cokernel

Known: the cardinality of the kernels of cellular covers is unbounded. For R = Z. Theorem (Farjoun–G¨

• bel–Segev–Shelah’07)

Let κ be any infinite cardinal. Then there exists a torsion-free and reduced abelian group H of cardinal κ, such that for any cardinal λ ≥ κ there is cellular exact sequence 0 → K → G → H → 0 with |G| = λ. No black box used, it is purely algebraic (based on G¨

• bel–May’90).

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

Fixing the cardinality of the cokernel

Known: the cardinality of the kernels of cellular covers is unbounded. For R = Z. Theorem (Farjoun–G¨

• bel–Segev–Shelah’07)

Let κ be any infinite cardinal. Then there exists a torsion-free and reduced abelian group H of cardinal κ, such that for any cardinal λ ≥ κ there is cellular exact sequence 0 → K → G → H → 0 with |G| = λ. No black box used, it is purely algebraic (based on G¨

• bel–May’90).

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

• 3. Fixing the cokernel: Case < 2ℵ0

Let κ be an infinite cardinal such that |R| ≤ κ ≤ λ < 2ℵ0. Theorem (3) Let H be any cotorsion-free R-module of size ≤ κ. Assume End(H) = R and Hom(H, M) = 0 for all ℵ0-free modules M. Then there exists a cellular exact sequence 0 → K → G → H → 0 where G a torsion-free and reduced R-module of size λ.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

• 3. Fixing the cokernel: Case < 2ℵ0

Let κ be an infinite cardinal such that |R| ≤ κ ≤ λ < 2ℵ0. Theorem (3) Let H be any cotorsion-free R-module of size ≤ κ. Assume End(H) = R and Hom(H, M) = 0 for all ℵ0-free modules M. Then there exists a cellular exact sequence 0 → K → G → H → 0 where G a torsion-free and reduced R-module of size λ. The assumptions hold for example if R = Z and H is a rank 1 torsion-free abelian group of type (1, 1, 1, ...).

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

• 3. Fixing the cokernel: Case < 2ℵ0

Let κ be an infinite cardinal such that |R| ≤ κ ≤ λ < 2ℵ0. Theorem (3) Let H be any cotorsion-free R-module of size ≤ κ. Assume End(H) = R and Hom(H, M) = 0 for all ℵ0-free modules M. Then there exists a cellular exact sequence 0 → K → G → H → 0 where G a torsion-free and reduced R-module of size λ. The assumptions hold for example if R = Z and H is a rank 1 torsion-free abelian group of type (1, 1, 1, ...).

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

• 4. Fixing the cokernel: Case ≥ 2ℵ0

Fix an infinite cardinal κ such that |R| < κℵ0. Take λ = κℵ0. Theorem (4) Let H be any cotorsion-free R-module of size ≤ κ. Assume End(H) = R and Hom(H, M) = 0 for all ℵ0-free modules M. Then there exists a cellular exact sequence of cotorsion-free R-modules 0 → K → G → H → 0 with |G| = λ.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

• 4. Fixing the cokernel: Case ≥ 2ℵ0

Fix an infinite cardinal κ such that |R| < κℵ0. Take λ = κℵ0. Theorem (4) Let H be any cotorsion-free R-module of size ≤ κ. Assume End(H) = R and Hom(H, M) = 0 for all ℵ0-free modules M. Then there exists a cellular exact sequence of cotorsion-free R-modules 0 → K → G → H → 0 with |G| = λ.

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules

Introduction Our results Motivation Known results Prescribing endomorphism rings Theorems

THANK YOU! www.ual.es/personal/jlrodri

R¨ udiger G¨

• bel, Jos´

e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules