Cotorsion pairs for Bext and a generalization of Whithead's problem [Elektronische Ressource] / von Nicole Hülsmann

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COTORSION PAIRS FOR BEXT AND AGENERALIZATION OF WHITEHEAD’S PROBLEMDissertationzur Erlangung des akademischen Grades einesDoktors der Naturwissenschaften(Dr. rer. nat.)vorgelegt beim Fachbereich Mathematikder Universit¨at Duisburg–EssenCampus EssenvonNicole Hu¨lsmannaus EssenVorlage der Dissertation: 04.10.2006Tag der mund¨ lichen Pruf¨ ung: 17.11.2006Prufungs¨ ausschuss:Vorsitzender: Prof. Dr. M. KunzeGutachter: Prof. Dr. R. G¨obelProf. Dr. L. SalceNicht weil es unerreichbar ist, wagen wir es nicht,sondern weil wir es nicht wagen, ist es unerreichbar.(Seneca)AcknowledgementI would like to thank Prof. Dr. Rud¨ iger G¨obel and PD Dr. Lutz Strun¨ gmann for theirinvaluable advice. Without their daily support and concern this work could never havebeen completed.I am also grateful to Dr. Simone Wallutis for her instructive advice and support and to mycolleagues Dr. Daniel Herden, Christian Mu¨ller and Dr. Sebastian Pokutta for a pleasentworking atmosphere and many productive discussions.Furthermore, I would like to thank all my friends for giving me an enjoyable time apartfrom mathematics.Last but not least, I owe more than I can say to the support and encouragement of mypartner, Dr. Georg Hennecke.I would like to thank the German–Israeli Foundation for Scientific Research and Develop-ment for granting me a scholarship.Dedicated to my father.ContentsIntroduction 3List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Sujets

• Mathematics

Informations

• Publié par: universitat_duisburg-essen
• Publié le: 01 janvier 2006
• Nombre de lectures: 35
• Langue: English

Extrait

COTORSION

PAIRS

GENERALIZATION

OF

FOR

BEXT

AND

WHITEHEAD’S

Dissertation

zur Erlangung des akademischen Grades eines

Doktors der Naturwissenschaften

(Dr. rer. nat.)

vorgelegt beim Fachbereich Mathematik

derUniversita¨tDuisburg–Essen

Campus Essen

von

NicoleHu¨lsmann

aus Essen

A

PROBLEM

Vorlage der Dissertation:

Tagdermu¨ndlichenPr¨ufung:

Pr¨ufungsausschuss:

Vorsitzender:

Gutachter:

04.10.2006

17.11.2006

Prof. Dr. M. Kunze

Prof.Dr.R.Go¨bel

Prof. Dr. L. Salce

Nicht weil es unerreichbar ist, wagen wir es nicht, sondern weil wir es nicht wagen, ist es unerreichbar. (Seneca)

Acknowledgement

IwouldliketothankProf.Dr.R¨udigerGo¨belandPDDr.LutzStr¨ungmannfortheir invaluable advice. Without their daily support and concern this work could never have been completed. I am also grateful to Dr. Simone Wallutis for her instructive advice and support and to my colleaguesDr.DanielHerden,ChristianM¨ullerandDr.SebastianPokuttaforapleasent working atmosphere and many productive discussions. Furthermore, I would like to thank all my friends for giving me an enjoyable time apart from mathematics.

Last but not least, I owe more than I can say to the support and encouragement of my partner, Dr. Georg Hennecke.

I would like to thank the German–Israeli Foundation for Scientific Research and Develop- ment for granting me a scholarship.

Dedicated to my father.

4.4

Uniformization and the existence of non–R 0

References

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R–Whitehead groups assuming V=L . . . . . . . . . . . . . . . . . .

R–Whitehead groups

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R–Whitehead

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B–cotorsion pairs

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Balanced subgroups

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Basics of set theory .

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The functor Bext

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Preliminaries

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List of symbols

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Introduction

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Torsion–free groups of rank 1 . . . . . . . . . . . .

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Algebraically compact groups and their relatives .

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Introduction

In 1979 Salce [Sa] introduced the notion of cotorsion pairs. A pair (G,H) of classes of abelian groups is called a cotorsion pair ifGandHare maximal with respect to the property that Ext(G, H) = 0 for allG∈ GandH∈ H cotorsion pair (. AG,H) is called generated by a classAof abelian groups if ( ⊥ (A ⊥ ),A ⊥ ) = (G,H), whereA ⊥ ={X|Ext(A, X) = 0 for allA∈ A}and ⊥ A={Y|Ext(Y, A) = 0 for allA∈ A}.Likewise, a cotorsion pair (G,H) is called cogenerated by a classAof abelian groups if ( ⊥ A,( ⊥ A) ⊥ ) = (G,H). A partial ordering of the cotorsion pairs is defined by (G,H)≤(G 0 ,H 0 ) iffG ⊆ G 0 for two cotorsion pairs (G,H) and (G 0 ,H 0 ). Salce defined the ordering conversely ((G,H)≤(G 0 ,H 0 ) iffG 0 ⊆ G) but, of course, his results hold mutatis mutandis for this ordering. He showed that the cotorsion pairs form a complete lattice and he proved that every cotorsion pair has enough projectives if and only if it has enough injectives. Moreover, he showed that there is a bijection from the set of all cotorsion pairs between the classical cotorsion pair (the pair generated byQ) and the maximal one (the pair generated by the class of all abelian groups) to the power set of the set of all primes. Salce started a characterization of the groupsAsuch that Ext(R, A) = 0 for a rational groupR⊆Q. With the help of these resultsGo¨bel,ShelahandWallutis[GSW]showedthatthesublatticeofallcotorsionpairs singly generated by a rational group is isomorphic to the lattice of all types in the sense of Baer [B], i.e. isomorphism classes of rank 1 groups. Furthermore, they proved that there is an embedding from any power set into the lattice of all cotorsion pairs. Hence there are ascending and descending chains as well as anti–chains of arbitrary length in the lattice of allcotorsionpairs.FortheproofofthisembeddingGo¨bel,ShelahandWallutisusedan important result due to Eklof and Trlifaj [ET]. For every moduleBover any ring Eklof and Trlifaj constructed a related moduleAsuch that Ext(B, A This construction) = 0. can also be used to obtain splitters, i.e. modulesAsuch that Ext(A, A the With) = 0. help of these results Bican, El Bashir and Enochs [BEE] proved the flat cover conjecture, namely that every module has a flat cover. This question had been open for a long time. In Chapter 3 we will transfer these results to the functor Bext. The functor Bext is defined as a subfunctor of Ext, where the group Bext(C, A) is the subgroup of Ext(C, A) that consists of all balanced–exact sequences

0→A→B→C→0.

Balanced–exact sequences of arbitrary abelian groups were defined by Hunter [Hu] in 1976. Hunter characterized the balanced–injective groups and the balanced–projective groups. A

groupG The torsion–free balanced–is balanced–injective if and only if it is pure–injective.

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