The groupsNrK0(ℤπ) are finitely generated ℤ[ℕr]-modules if π is a finite group

Connolly, Francis X.; Silva, Mário O. M. Da

doi:10.1007/bf00965456

Cited by 13 publications

(15 citation statements)

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“…We also get from Theorem A an improved estimate on the exponent of NK 0 (ZG), using a result of Connolly-da Silva [5]. If n is a positive integer, and n q = q k is its q-primary part, then let c q (n) = q l , where l ≥ log q (kn).…”

Section: Xii])mentioning

confidence: 99%

“…If n is a positive integer, and n q = q k is its q-primary part, then let c q (n) = q l , where l ≥ log q (kn). According to [5], the exponent of NK 0 (ZG) divides c(n) = q|n c q (n), where n = |G|, but according to Theorem A, the exponent of NK 0 (ZG) divides Acknowledgement: This paper is dedicated to Tom Farrell and Lowell Jones, whose work in geometric topology has been a constant inspiration. We are also indebted to Frank Quinn, who suggested that the Farrell-Hsiang induction techniques should be re-examined (see [20]).…”

Section: Xii])mentioning

confidence: 99%

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Induction and Computation of Bass Nil Groups for Finite Groups

Hambleton¹,

Lück²

2012

Pure and Applied Mathematics Quarterly

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Let G be a finite group. We show that the Bass Nil-groups NK n (RG), n ∈ Z, are generated from the p-subgroups of G by induction, certain twistings maps depending on elements in the centralizers of the psubgroups, and the Verschiebung homomorphisms. As a consequence, the groups NK n (RG) are generated by induction from elementary subgroups. For NK 0 (ZG) we get an improved estimate of the torsion exponent.

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Section: Xii])mentioning

confidence: 99%

Section: Xii])mentioning

confidence: 99%

Induction and Computation of Bass Nil Groups for Finite Groups

Hambleton¹,

Lück²

2012

Pure and Applied Mathematics Quarterly

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“…To derive this result we develop Frobenius and Verschiebung operations on certain kinds of Nil-groups. For Bass Nil-groups and End-groups Frobenius and Verschiebung operations are well understood [Blo78,CdS95,Sti82,Wei81].…”

Section: Operations On Nil-groupsmentioning

confidence: 99%

The behavior of Nil-groups under localization and the relative assembly map

Grunewald

2008

Topology

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We study the behavior of the Nil-subgroups of K-groups under localization. As a consequence of our results we obtain that the relative assembly map from the family of finite subgroups to the family of virtually cyclic subgroups is rationally an isomorphism. Combined with the equivariant Chern character we obtain a complete computation of the rationalized source of the K-theoretic assembly map that appears in the Farrell-Jones conjecture in terms of group homology and the K-groups of finite cyclic subgroups.Specifically we prove that under mild assumptions we can always write the Nil-groups and End-groups of the localized ring as a certain colimit over the Nil-groups and End-groups of the ring, generalizing a result of Vorst. 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. Combining the localization results with the Witt-vector module structure we obtain that Nil and localization at an appropriate multiplicatively closed set S commute, i.e. S −1 Nil = Nil S −1 . An important corollary is that the Nil-groups appearing in the decomposition of the K-groups of virtually cyclic groups are torsion groups. *

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“…is a monoid with respect to multiplication. In the algebraic K-theory of rings, the corresponding structure was studied in [2,10].…”

Section: Introductionmentioning

confidence: 99%

“…In the linear situation, there are partial results pointing in the direction that the homotopy groups of the nil-terms could be finitely-generated as Z[N × ]-modules; see [2,10]. However, Theorem 1.2 shows that this is not the case in the non-linear situation.…”

Section: Introductionmentioning

confidence: 99%

Operations on the A-theoretic nil-terms

Grunewald

Klein

Macko

2007

Journal of Topology

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For a space X, we define Frobenius and Verschiebung operations on the nil-terms NA fd ± (X) in the algebraic K-theory of spaces, in three different ways. Two applications are included. Firstly, we show that the homotopy groups of NA fd ± (X) are either trivial or not finitely generated as abelian groups. Secondly, the Verschiebung operation defines a Z[N×]-module structure on the homotopy groups of NA fd ± (X), with N× the multiplicative monoid. We also give a calculation of the homotopy type of the nil-terms NA fd ± ( * ) after p-completion for an odd prime p and their homotopy groups as Zp [N×]-modules up to dimension 4p−7. We obtain non-trivial groups only in dimension 2p − 2, where it is finitely generated as a Zp [N×]-module, and in dimension 2p − 1, where it is not finitely generated as a Zp [N×]-module.where A fd (X) denotes the finitely-dominated version of the algebraic K-theory of the space X, BA fd (X) denotes a certain canonical non-connective delooping of A fd (X), and NA fd + (X), and NA fd -(X) are two homeomorphic nil-terms. Thus, the study of A fd (X × S 1 ) splits naturally into studying A fd (X) and the nil-terms. Over time, there has been steady progress in understanding A fd (X) for some X, mostly for X = * (see [14,18,19]), but not much is known about the nil-terms. These are the subjects of the present paper.Recall that the splitting (1.1) is analogous to the splitting of the fundamental theorem of algebraic K-theory of rings [9] for any ring R( 1.2) where K(R) denotes the (−1)-connective K-theory space of the ring R and BK(R) is a certain canonical non-connective delooping of K(R). In fact, for any space X there is a linearization map l : A fd (X) → K(Z[π 1 X]) and the two splittings are natural with respect to this map. We pursue the analogy between the algebraic K-theory of spaces (the non-linear situation) and the algebraic K-theory of rings (the linear situation) further. We define the Frobenius and Verschiebung operations on the nil-terms NA fd ± (X), which are analogs of such operations defined in the linear case. As a consequence, we obtain the following result (compare with [5]).Theorem 1.1. The homotopy groups π * NA fd ± (X) and all of their p-primary subgroups are either trivial or not finitely generated as abelian groups.

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The groupsNrK0(ℤπ) are finitely generated ℤ[ℕr]-modules if π is a finite group

Cited by 13 publications

References 9 publications

Induction and Computation of Bass Nil Groups for Finite Groups

Induction and Computation of Bass Nil Groups for Finite Groups

The behavior of Nil-groups under localization and the relative assembly map

Operations on the A-theoretic nil-terms

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