The behavior of Nil-groups under localization [Elektronische Ressource] / vorgelegt von Joachim Grunewald

MathematikThe Behavior of Nil-Groups underLocalizationInaugural-Dissertationzur Erlangung des Doktorgradesder Naturwissenschaften im FachbereichMathematik und Informatikder Mathematisch-Naturwissenschaftlichen Fakult¨atder Westf¨alischen Wilhelms-Universit¨at Munster¨vorgelegt vonJoachim Grunewaldaus Bielefeld– 2005 –Dekan: Prof. Dr. Klaus HinrichsErster Gutachter: Prof. Dr. Wolfgang Luc¨ kZweiter Gutachter: PD Dr. Holger ReichTag der mundlic¨ hen Prufung:¨ 18.11.05Tag der Promotion: 8.2.06AbstractThis thesis centers around the study of the behavior of Nil-groups under localizationanditseffectontheFarrell-Jonesconjecture. Weprovethatundermildassumptionswe can always write the Nil-groups of the localized ring as a certain colimit over theNil-groups of the ring, generalizing a result of Vorst. For applications it is fruitfulto improve this result. For this purpose, we define Frobenius and Verschiebungoperations on certain Nil-groups. These operations provide the tool to prove thatNil-groups are modules over the ring of Witt-vectors and are either trivial or notfinitely generated as abelian groups. We use the improved localization results toshow that Nil-groups of polycyclic-by-finite groups are torsion. As an importantcorollary, we obtain the result that the Nil-groups appearing in the decompositionof the K-groups of virtually cyclic groups are torsion groups.