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¨ obel, Jos´ e L. Rodr´ ıguez, Lutz Str¨ ungmann Conference on Algebraic Topology CAT’09 6-11 July 2009 Warszawa, Poland R¨ udiger G¨ obel, 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¨ obel, 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¨ obel, 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 over H if: R¨ udiger G¨ obel, 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 over H if: π G − → H R¨ udiger G¨ obel, 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 over H if: G ց π G − → H R¨ udiger G¨ obel, 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 over H if: G ∃ ! ↓ ց π G − → H R¨ udiger G¨ obel, 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 over H if: G ∃ ! ↓ ց π G − → H π ∗ : End R ( G ) ∼ = Hom R ( G , H ) , ϕ �→ π ◦ ϕ. R¨ udiger G¨ obel, 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 over H if: G ∃ ! ↓ ց π G − → H π ∗ : End R ( G ) ∼ = Hom R ( G , H ) , ϕ �→ π ◦ ϕ. π A short exact sequence 0 → K → G → H → 0 with π a cellular cover is called a cellular exact sequence . R¨ udiger G¨ obel, 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 over H if: G ∃ ! ↓ ց π G − → H π ∗ : End R ( G ) ∼ = Hom R ( G , H ) , ϕ �→ π ◦ ϕ. π A short exact sequence 0 → K → G → H → 0 with π a cellular cover is called a cellular exact sequence . R¨ udiger G¨ obel, 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 π 0 → K − → G − → H → 0 with a prescribed K or H , such that the following “rigidity” properties hold: R¨ udiger G¨ obel, 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 π 0 → K − → G − → H → 0 with a prescribed K or H , such that the following “rigidity” properties hold: End ( G ) = End ( H ) = R . and Hom ( G , H ) = π R R¨ udiger G¨ obel, 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 π 0 → K − → G − → H → 0 with a prescribed K or H , such that the following “rigidity” properties hold: End ( G ) = End ( H ) = R . and Hom ( G , H ) = π R In particular G → H is a cellular cover. R¨ udiger G¨ obel, 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 π 0 → K − → G − → H → 0 with a prescribed K or H , such that the following “rigidity” properties hold: End ( G ) = End ( H ) = R . and Hom ( G , H ) = π R In particular G → H is a cellular cover. R¨ udiger G¨ obel, 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 of groups: 0 → K → G → H → 1 . R¨ udiger G¨ obel, 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 of groups: 0 → K → G → H → 1 . 1 Which properties of G are inherit from H ? R¨ udiger G¨ obel, 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 of 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¨ obel, 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 of 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¨ obel, 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 of 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¨ obel, 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 Co-localization Modules Modules Rings Rings Algebras Algebras Finite groups Finite groups Nilpotent groups (???) Nilpotent groups Solvable groups Solvable groups (?) Perfect groups Perfect groups Simple groups Simple groups Classes of abelian groups (?) Classes of abelian groups (?) Others (??) Others (??) R¨ udiger G¨ obel, Jos´ e L. Rodr´ ıguez, Lutz Str¨ ungmann Cellular Covers of Cotorsion-free Modules