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

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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.

Sujets

• Mathematics

Informations

• Publié par: westfalische_wilhelms-universitat_munster
• Publié le: 01 janvier 2005
• Nombre de lectures: 12
• Langue: English

Extrait

Mathematik

The Behavior of Nil-Groups under Localization

Inaugural-Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften im Fachbereich Mathematik und Informatik derMathematisch-NaturwissenschaftlichenFakulta¨t derWestf¨alischenWilhelms-Universita¨tMu¨nster

vorgelegt von Joachim Grunewald aus Bielefeld – 2005 –

Dekan: Erster Gutachter: Zweiter Gutachter: Tagdermu¨ndlichenPr¨ufung: Tag der Promotion:

Prof. Dr. Klaus Hinrichs Prof.Dr.WolfgangLu¨ck PD Dr. Holger Reich 18.11.05 8.2.06

Abstract

This thesis centers around the study of the behavior of Nil-groups under localization and its effect on the Farrell-Jones conjecture. We prove that under mild assumptions we can always write the Nil-groups of the localized ring as a certain colimit over the Nil-groups of the ring, generalizing a result of Vorst. For applications it is fruitful to improve this result. For this purpose, we define Frobenius and Verschiebung operations on certain Nil-groups. These operations provide the tool to prove that Nil-groups are modules over the ring of Witt-vectors and are either trivial or not finitely generated as abelian groups. We use the improved localization results to show that Nil-groups of polycyclic-by-finite groups are torsion. As an important corollary, we obtain the result that the Nil-groups appearing in the decomposition of theK-groups of virtually cyclic groups are torsion groups. This implies that the Farrell-Jones conjecture predicts that rationally the building blocks of theK-groups of groups are theK-groups of finite groups.

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Farrell Nil-Groups . . . . Waldhausen Nil-Groups of Waldhausen Nil-Groups of

1.2.2 1.2.3 1.2.4

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and Nil-Groups End-Groups and END-categories 1.1.1 End(R)-Groups . . . . . . 1.1.2 Waldhausen End-Groups Nil-Groups and NIL-categories . 1.2.1 Bass Nil-Groups . . . . .

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End- 1.1

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3.4

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Applications 4.1 Improved Localization Results . . . . . 4.2 Transfer and Induction on Nil-Groups 4.3 Regular Group Rings . . . . . . . . . .

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53 53 55 55 57 60 60 63 66

ell and Waldhausen Nil-Groups as Modules over the Ring of Witt ors Nil-Groups as Modules over End 0 . . . . . . . . . . . . . . . . .. . . . Operations on Farrell and Waldhausen Nil-Groups . . . . . . . . . . . 3.2.1 Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Verschiebung . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Non Finiteness Results . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The Relation V n (y∗F n x) = (V n y)∗x .on Farrell Nil-Groups . 3.3.3 The Relation V n (y∗F n x) = (V n y)∗xon Waldhausen Nil-Groups Farrell and Waldhausen Nil-Groups as Modules over the Ring of Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Farr vect 3.1

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Behavior of Nil-Groups under Localization Homological Facts about NIL-categories . . . . . . . . . . . . . . . Colimits in the Category of Exact Categories . . . . . . . . . . . . The Long Exact Localization Sequence of NIL-Categories . . . . . The Behavior of Nil-Groups under Localization . . . . . . . . . . .

The 2.1 2.2 2.3

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Introduction

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Torsion Results . . . . The Relative Assembly

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Introduction

In general, Nil-groups measure the difference between theK-groups of a ground ring and theK-groups of a certain ring built up out of this ring. We start by briefly recalling the importance of the various kinds of Nil-groups. The most basic kind of Nil-group is given by Bass’s definition of the abelian groups Nil i (R) [Bas68, Section 12.6]. The so calledFundamental Theorem of algebraicK- theory[BHS64] gives a description of theK-groups of the Laurent polynomial ring of a ringRin terms of theK-groups ofRand the groups Nil i (R) for alli∈Z: K i (R[t t −1 ]) =K i (R)⊕K i−1 (R)⊕Nil i−1 (R)⊕Nil i−1 (R).

Ifαis a ring automorphism ofRFarrell introduced the abelian groups Nil i (R;α) [Far72]. He generalized, together with Hsiang, the Fundamental Theorem of algebraic K-theory toK 1  proved thatof the twisted Laurent polynomial ring [FH70]. They the sequence

K 1 (R) 1−α ∗ // K 1 (R)

K 1 (R α [t t −1 ])Nil 0 (R α)⊕Nil 0 (R α −1 )

// K 0 (R) 1−α ∗ // K 0 (R) is exact. This decomposition was extended by Grayson to higher algebraicK-theory [Gra88]. For applications in topology, theK-groups of group rings are of special importance. If Γ =Go α Zis the semidirect product of a groupG, an automorphismαofGand the infinite cyclic group, the group ring of Γ can be seen as a twisted Laurent polynomial ring. In this context, the statement gives that the sequence

K i (RG) 1−α ∗

K i (RG)

K i (RΓ)Nil i−1 (RG α)⊕Nil i−1 (RG α −1 )

− // K i−1 (RG) 1α ∗ // K i−1 (RG) // is exact. An inclusionα:C→Aof rings is calledpureifA=α(C)⊕A 0 asC-bimodules. It is calledpureandfreeif in additionA 0 is free as a leftC-module. Letα:C→A andβ:C→Bboth be pure and free. Thegeneralized free productofAandBis the push-out ofαandβ introduced, for Waldhausenin the category of rings.R- bimodulesXandY, the abelian groups Nil i (R;X Y) [Wal78a, Wal78b]. Nil-groups of this kind relate theK-groups of the generalized free product to theK-groups of the ground rings.

7

Introduction

Theorem (Waldhausen [Wal78a, Wal78b]).Letα:C ,→Aandβ:C ,→Bboth be pure and free. WriteA=α(C)⊕A 0 andB=β(C)⊕B 0 . LetRbe the generalized free product ofAandB. The groupsNil i (C;A 0  B 0 )are direct summands ofK i+1 (R) and there is a long exact sequence of groups // // K i+1 (R)Nil i  C;A 0  B 0  // K i (C) // ∙ ∙ ∙ 

. . . // K i+1 (C)

wherei≥0.

K i+1 (A)⊕K i+1 (B)

Fori <eserth04]L0.raetslnaLdu¨kcB[ultisextendedbyB A special case of a generalized free product comes from the amalgamated product of groups. LetG 1 andG 2 be groups with a common subgroupH. The group ring ofG 1 ∗ H G 2 is the generalized free product of the group rings ofH,G 1 andG 2 . In the context of group rings, the previous theorem reads as follows:

Corollary.LetΓ =G 1 ∗ H G 2 be the amalgamated product of the groupsH,G 1 andG 2  groups. TheNil i  ZH;Z[G 1 −H]Z[G 2 −H]  are direct summands ofK i+1 (Γ) and there is a long exact sequence of groups

∙ ∙ ∙

K i+1 (ZH)

K i+1 (ZG 1 )⊕K i+1 (ZG 2 )

K i+1 (Γ)Nil i  ZH;Z[G 1 −H]Z[G 2 −H] 

wherei∈Z.

K i (ZH)

 ∙ ∙ ∙

If the Nil-groups are ignored, the sequence given above is the analog of the Mayer- Vietoris sequence forK-groups. It is a general belief that Nil-groups are the ob- struction forK-theory to be a homology theory. Letα β:C→A Thebe pure and free.generalized Laurent extensionwith respect toα,βis the universal ringR=A α,β {t ±1 }which containsAand an invertible elementtwhich satisfies

α(c)t=tβ(c all) forc∈C.

The existence of such a ring is explained in [Wal78a]. ForR-bimodulesX,Y,Zand WWaldhausen introduced the abelian groups Nil i (R;X Y Z W) [Wal78a, Wal78b], which are the most general kind of Nil-groups. Nil-groups of this kind relate the K-groups of a generalized Laurent extension to theK-groups of the ground rings.

Theorem (Waldhausen [Wal78a, Wal78b]).LetRbe the generalized Laurent extension of pure and free mapsα β:C ,→A. WriteA=α(C)⊕A 0 andA= β(C)⊕A 00  by. Denote β A α theC-bimoduleAwithC-action from the left viaβ and from the right viaα. Let α A 0 α , β A 0 β 0 and α A β  Thebe defined similarly. groups Nil i  C; α A 0 α  β A 0 β 0  β A α  α A β  are direct summands ofK i+1 (R)and there is a long exact sequence of groups

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∙ ∙ ∙

K i+1 (C)

α−β

K i+1 (A)

wherei≥0.

K i+1 (R)Nil i  C; α A 0 β A 0 β 0  β A α  α A β  α 

K i (C)

Introduction

 ∙ ∙ ∙

Again, this is extended toi <etraByb0u¨LdnaslL04]ck[B. A special case of a generalized Laurent extension comes from the HNN-extension of groups. LetHbe a group which is embedded into another groupGin two different ways. The group ring of the HNN-extension ofHandGis the generalized Laurent extension ofRHandRG. Insetting of group rings, the previous theorem reads the as follows:

Corollary.LetΓbe the HNN-extension of the embeddingsα β:H ,→G. The groups Nil i  ZH;Z[G−α(H)]Z[G−β(H)] β ZG α  α ZG β  are direct summands ofK i+1 (ZΓ) and there is a long exact sequence of groups

∙ ∙ ∙

K i+1 (ZH) α−β // K i+1 (ZG)

K i+1 (ZΓ)Nil i  ZH;Z[G−α(H)]Z[G−β(H)] β ZG α  α ZG β 

wherei∈Z.

 ∙ ∙ ∙

To avoid confusion, Nil-groups of the form Nil i (R) are calledBass Nil-groups, Nil-groups of the form Nil i (R;α) are calledFarrell Nil-groups, Nil-groups of the form Nil i (R;X Y) are calledWaldhausen Nil-groups of generalized free products and Nil-groups of the form Nil i (R;X Y Z W) are calledWaldhausen Nil-groups of generalized Laurent extensions. In general, Nil-groups are subgroups of theK- groups of NIL-categories. Bass Nil-groups are subgroups of theK-groups of the NIL- category NIL(R), for Farrell Nil-groups the NIL-category is denoted by NIL(R;α), for Waldhausen Nil-groups of generalized free products the NIL-category is denoted by NIL(R;X Yfor Waldhausen Nil-groups of generalized Laurent extensions) and the NIL-category is denoted by NIL(R;X Y Z W). For a definition of the different kind of Nil-groups and NIL-categories see Chapter 1.

The Behavior of Nil-Groups under Localization

Nil-groups seem to be hard to compute. It is for example known that higher Bass Nil- groups are either trivial or not finitely generated as abelian groups [Far77, Wei81]. However, in the first chapter of this thesis, it is proven that all these kinds of Nil- groups behave nicely under localization.

Definition.LetRbe a ring.

1.

2.

LetT⊆Rbe a multiplicatively closed subset of central non zero divisors. The ringT −1 Ris denoted byR T .

Letsbe a central non zero divisor and letSbe the multiplicatively closed set generated bys. We use the short hand notationR s forR S .

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Introduction

3. LetXbe anR define-bimodule. WeX T to be theR T -bimoduleR T ⊗ R X⊗ R R T .

4. Ifsis a central non zero divisor, we use the short hand notationX s forR s ⊗ R X⊗ R R s .

The main theorem of the first chapter is the following theorem:

Theorem.LetRbe a ring and let X, Y, Z and W be left flatR-bimodules. Letsbe an element of the center ofRwhich is not a zero divisor and satisfiess∙x=x∙s for all elementsx∈Xand similar conditions forY,ZandW obtain an. We isomorphism

Z[t t −1 ]⊗ Z[t] Nil i (R;X Y Z W) = ∼ Nil i (R s ;X s  Y s  Z s  W s )

fori∈Z, andtacts onNil i (R;X Y Z W)via the map induced by the functor

F s : NIL(R;X Y Z W)→NIL(R;X Y Z W) (P Q p q)7→(P Q p∙s q∙s).

The condition that the bimodulesX,Y,ZandWare left flat does not seem to be overly restrictive since in all the cases considered by WaldhausenX,Y,ZandW are left free by the purity and freeness condition. The conditions∙x=x∙stranslates in Waldhausen’s setting of a generalized Laurent extension to the assumption that sis mapped, under the mapsαandβ, to central elements. In Remark 1.2.17, it is explained that Waldhausen Nil-groups of generalized Lau- rent extensions are a generalization of the other kind of Nil-groups. Thus we get the following corollaries:

Corollary.LetRbe a ring and let X and Y be left flatR-bimodules. Letsbe an element of the center ofRwhich is not a zero divisor and satisfiess∙x=x∙sfor all elementsx∈Xand a similar condition forY. We obtain an isomorphism

Z[t t −1 ]⊗ Z[t] Nil i (R;X Y) = ∼ Nil i (R s ;X s  Y s )

fori∈Z, andtacts onNil i (R;X Y)via the map induced by the functor

F s : NIL(R;X Y)→NIL(R;X Y) (P Q p q)7→(P Q p∙s q∙s).

Again, the assumption thatXandYare left flat modules is not overly restrictive since in the case of a generalized free productXandYare left free by the purity and freeness condition. The conditions∙x=x∙stranslates in the setting of a generalized free product to the condition thatsis mapped to central elements.

Corollary.LetRbe a ring and letα:R→Rbe an automorphism. Letsbe an element of the center ofRwhich is not a zero divisor and is fixed underα. We obtain an isomorphism

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Z[t t −1 ]⊗ Z[t] Nil i (R;α) = ∼ Nil i (R s ;α)

fori∈Z, andtacts onNil i (R α)via the map induced by the functor

F s : NIL(R;α)→NIL(R;α) (P ν)7→(P ν∙s).

Introduction

Corollary.LetRbe a ring. Letsbe an element of the center ofRwhich is not a zero divisor. We obtain an isomorphism

Z[t t −1 ]⊗ Z[t] Nil i (R) ∼ = Nil i (R s )

for alli∈Z, andtacts onNil i (R)via the map induced by the the functor

F s :

NIL(R)→NIL(R) (P ν)7→(P ν∙s).

For Nil i (R) the result was already known [Vor79].

Nil-Groups as Modules over the Ring of Witt vectors

For applications, it is fruitful to improve these results to an isomorphism between R s ⊗ R Nil(R) and Nil(R s ). To get this isomorphism, we develop in the third chapter a Witt vector module structure on certain Nil-groups. In this chapter, the ring is always assumed to be a either a group ring or more generally an algebra over a commutative ringR us briefly recall the definition of the ring of Witt vectors.. Let For an introduction to the ring of Witt vectors see [Blo77]. Thering of (big) Witt vectors +is the ring 1tRJtKof power series with constant term 1. The underlying additive group of the ring of Witt vectors is the multiplicative group of 1 +tRJtK. The multiplication is the unique continuous functorial operation∗for which

(1−at)∗(1−bt) = (1−abt)

holds for alla,b∈R. In the sequel, the ring of Witt vectors is denoted byW(R). We define idealsI N  +:= (1t N RJtK) for allN∈N. The resulting topology on the ring of Witt vectors is called thet-adic topology. There is a different approach to the ring of Witt vectors which emphasizes the re- lation toK-theory. Let END(R) be the exactcategory of endomorphisms of finitely generated projective rightR-modules. The objects of this category are pairs (B ϕ) whereBis a finitely generated projectiveR-module andϕis an endomorphism ofB. Maps from (B ϕ) to (B 0  ϕ 0 ) are module homomorphismsf:B→B 0 such thatϕ 0 ◦f=f◦ϕ. A sequence is calledexactif the underlying sequence of mod- ule homomorphisms is exact. SinceRis assumed to be commutative, the tensor product induces a ring structure onK 0  END(R)   by End. Denote 0 (R) the quotient ofK 0  END(R)  ideal generated by elements of the form [(by the B the0)]. Since characteristic polynomial χ  (B ϕ)  := det(id B −t∙ϕ)

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