INDUCTION AND COMPUTATION OF BASS NIL GROUPS FOR FINITE GROUPS

IAN HAMBLETON AND WOLFGANG LÜCK

Abstract. 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.

1. Introduction

In this note we study the Bass Nil-groups (see [3, Chap. XII])

NK_{n}(RG) = ker(K_{n}(RG[t]) → K_{n}(RG)),

where R is an associative ring with unit, G is a finite group, n ∈ Z, and the augmentation map sends t ↦ 0. Alternately, the isomorphism

NK_{n}(RG) ≅ K̃_{n−1}(NIL(RG))

identifies the Bass Nil-groups with the K-theory of the category NIL(RG) of nilpotent endomorphisms (Q, f) on finitely-generated projective RG-modules [3, Chap. XII], [12, Theorem 2]. Farrell [8] proved that NK_{1}(RG) is not finitelygenerated as an abelian group whenever it is non-zero, and the corresponding result holds for NK_{n}(RG), n ∈ Z (see [26, 4.1]), so some organizing principle is needed to better understand the structure of the Nil-groups. Our approach is via induction theory. The functors NK_{n} are Mackey functors on the subgroups of G, and we ask to what extent they can be computed from the Nil-groups of proper subgroups of G.

The Bass-Heller-Swan formula [3, Chap. XII, §7], [11, p. 236] relates the Bass Nil-groups with the K-theory of the infinite group G × Z. There are two (split) surjective maps

N_{±}: K_{n}(R[G × Z]) → NK_{n}(RG)

Date: June 27, 2007.

Research partially supported by NSERC Discovery Grant A4000 and by the Max Planck Forschungspreis of the second author. The first author would like to thank the SFB 478, Universität Münster, for its hospitality and support.

which form part of the Bass-Heller-Swan direct sum decomposition

K_{n}(R[G × Z]) = K_{n}(RG) ⊕ K_{n−1}(RG) ⊕ NK_{n}(RG) ⊕ NK_{n}(RG).

Notice that both K_{n}(R[(−) × Z]) and NK_{n}(R[−]) are Mackey functors on the subgroups of G (see Farrell-Hsiang [9, §2] for this observation about infinite groups). We observe that the maps N_{±} are actually natural transformations of Mackey functors (see Section 6). It follows from Dress induction [7] that the functors NK_{n}(RG) and the maps N_{±} are computable from the hyperelementary family. We will show how the results of Farrell [8] and techniques of Farrell-Hsiang [10] lead to a better induction statement for the Bass Nil-groups.

We need some notation to state the main result. For each prime p, we denote by 𝔓_{p}(G) the set of finite p-subgroups of G, and by 𝔈_{p}(G) the set of p-elementary subgroups of G. Recall that a p-elementary group has the form E = C × P, where P is a finite p-group, and C is a finite cyclic group of order prime to p. For each element g ∈ C, we let

I(g) = {k ∈ ℕ | a prime q divides k ⇒ q divides |g|}

where |g| denotes the order of g. For each P ∈ 𝔓_{p}(G), let

C_{G}^{⊥}(P) = {g ∈ G | gx = xg, ∀x ∈ P, and p ∤ |g|}

and for each g ∈ C_{G}^{⊥}(P) we define a functor

ϕ(P, g): NIL(RP) → NIL(RG)

by sending a nilpotent RP -endomorphism f: Q → Q of a finitely-generated projective RP -module Q to the nilpotent RG-endomorphism

RG ⊗_{RP} Q → RG ⊗_{RP} Q, x ⊗ q ↦ xg ⊗ f(q) .

Note that this RG-endomorphism is well-defined since g ∈ C_{G}^{⊥}(P). The functor ϕ(P, g) induces a homomorphism

ϕ(P, g): NK_{n}(RP) → NK_{n}(RG)

for each n ∈ Z. For each p-subgroup P in G, define a homomorphism

Φ_{P} : NK_{n}(RP) → NK_{n}(RG)

by the formula

Φ_{P} = ∑_{g∈C_{G}^{⊥}(P), k∈I(g)}V_{k} ∘ ϕ(P, g),

where

V_{k}: NK_{n}(RG) → NK_{n}(RG)

denotes the Verschiebung homomorphism, k ≥ 1, recalled in more detail in Section 2.

Theorem A. Let R be an associative ring with unit, and G be a finite group. For each prime p, the map

Φ = (Φ_{P}): ⨁_{P ∈𝔓_{p}(G)}NK_{n}(RP)_{(p)} → NK_{n}(RG)_{(p)}

is surjective for all n ∈ Z, after localizing at p.

For every g ∈ C_{G}^{⊥}(P), the homomorphism ϕ(P, g)_{n} factorizes as

NK_{n}(RP) ⟶^{ϕ(P,g)} NK_{n}(R[C × P]) ⟶^{i_{∗}} NK_{n}(RG)

where C = ⟨g⟩ and i: C ×P → G is the inclusion map. Since the Verschiebung homomorphisms are natural with respect to the maps induced by group homomorphisms, we obtain:

Corollary B. The sum of the induction maps

⨁_{E∈𝔈_{p}(G)}NK_{n}(RE)_{(p)} → NK_{n}(RG)_{(p)}

is surjective, for all n ∈ Z.

Note that Theorem A does not show that NK_{n}(RG) is generated by induction from p-groups, because the maps ϕ(P, g) for g ≠ 1 are not the usual maps induced by inclusion P ⊂ G.

Bass Nil-groups are non-finitely generated torsion groups (if non-zero) so remain difficult to calculate explicitly, but we have some new qualitative results. For example, Theorem A shows that the order of every element of NK_{n}(RG) is some power of m = |G|, whenever NK_{n}(R) = 0 (since its q-localization is zero for all q ∤ m). If R is a regular (noetherian) ring (e.g. R = Z), then NK_{n}(R) = 0 for all n ∈ Z. For R = Z and some related rings, this is a result of Weibel [26, (6.5), p. 490]. In particular, we know that every element of NK_{n}(ZP) has p-primary order, for every finite p-group P.

Note that an exponent that holds uniformly for all elements in NK_{n}(RP), over all p-groups of G, will be an exponent for NK_{n}(RG). As a special case, we have NK_{n}(Z[Z/p]) = 0 for n ≤ 1, for p a prime (see Bass-Murthy [4]), so Theorem A implies:

Corollary 1.1. Let G be a finite group and let p be a prime. Suppose that p^{2} does not divide the order of G. Then

NK_{n}(ZG)_{(p)} = 0

for n ≤ 1.

As an application, we get a new proof of the fact that NK_{n}(ZG) = 0, for n ≤ 1, if the order of G is square-free (see Harmon [16]).

We also get from Theorem A an improved estimate on the exponent of NK_{0}(ZG), using another 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). 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

d(n) = ∏_{q|n} c(n_{q}) .

For example, c(60) = 1296000, but d(60) = 120.

Acknowledgement: This paper is dedicated to Tom Farrell, 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 [19]).

2. Bass Nil-groups

Standard constructions in algebraic K-theory for exact categories or Waldhausen categories yield only K-groups in degrees n ≥ 0 (see Quillen[18], Waldhausen [25]). One approach on the categorial level to negative K-groups has been developed by Pedersen-Weibel (see [17], [15, §2.1]). Another ring theoretic approach is given as follows (see Bartels-Lück [1, Section 9], Wagoner [24]). The cone ring ΛZ of Z is the ring of column and row finite ℕ × ℕmatrices over Z, i.e., matrices such that every column and every row contains only finitely many non-zero entries. The suspension ring ΣZ is the quotient of ΛZ by the ideal of finite matrices. For an associative (but not necessarily commutative) ring A with unit, we define ΛA = ΛZ ⊗_{Z} A and ΣA = ΣZ ⊗_{Z} A. Obviously Λ and Σ are functors from the category of rings to itself. There are identifications, natural in A,

(2.1)  K_{n−1}(A) = K_{n}(ΣA)
(2.2)  NK_{n−1}(A) = NK_{n}(ΣA)

for all n ∈ Z. In our applications, usually A = RG where R is a ring with unit and G is a finite group.

Using these identifications it is clear how to extend the definitions of certain maps between K-groups given by exact functors to negative degrees. Moreover, we will explain constructions and proofs of the commutativity of certain diagrams only for n ≥ 1, and will not explicitly mention that these carry over to all n ∈ Z, because of the identifications (2.1) and (2.2) and the obvious identification Σ(RG) = (ΣR)G or because of Pedersen-Weibel [17]).

We have a direct sum decomposition

(2.3)  K_{n}(A[t]) = K_{n}(A) ⊕ NK_{n}(A)

which is natural in A, using the inclusion A → A[t] and the ring map A[t] → A defined by t ↦ 0.

Let P(A) be the exact category of finitely generated projective A-modules, and let NIL(A) be the exact category of nilpotent endomorphism of finitely generated projective A-modules. The functor P(A) → NIL(A) sending P ↦ (P, 0) and the functor NIL(A) → P(A) sending (P, f) ↦ P are exact functors. They yield a split injection on the K-groups K_{n}(A) := K_{n}(P(A)) → K_{n}(NIL(A)) for n ∈ Z. Denote by K̃_{n}(NIL(A)) the cokernel for n ∈ Z. There is an identification (see Grayson [12, Theorem 2])

(2.4)  K̃_{n−1}(NIL(A)) = NK_{n}(A),

for n ∈ Z, essentially given by the passage from a nilpotent A-endomorphism (Q, f) to the AZ-automorphism

AZ ⊗_{A} Q → AZ ⊗_{A} Q, u ⊗ q ↦ u ⊗ q − ut ⊗ f(q),

for t ∈ Z a fixed generator.

The Bass Nil-groups appear in the Bass-Heller-Swan decomposition for n ∈ Z (see [3, Chapter XII], [2], [11, p. 236], [18, p. 38], and [23, Theorem 10.1] for the original sources, or the expositions in [20, Theorems 3.3.3 and 5.3.30], [21, Theorem 9.8] ).

B: K_{n}(A) ⊕ K_{n−1}(A) ⊕ NK_{n}(A) ⊕ NK_{n}(A) ⟶^{≅} K_{n}(AZ). (2.5)

The isomorphism B is natural in A and comes from the localization sequence

(2.6) 0 → K_{n}(A) ⟶^{K_{n}(i)⊕−K_{n}(i)} K_{n}(A[t]) ⊕ K_{n}(A[t]) ⟶^{K_{n}(j_{+})+K_{n}(j_{−})} K_{n}(AZ)

⟶^{∂_{n}} K_{n−1}(A) → 0

where i: A → A[t] is the obvious inclusion and the inclusion j_{±}: A[t] → AZ sends t to t^{±1} if we write AZ = A[t, t^{−1}], the splitting of ∂_{n}

s_{n}: K_{n−1}(A) → K_{n}(AZ) (2.7)

which is given by the natural pairing

K_{n−1}(A) ⊗ K_{1}(Z[t, t^{−1}]) → K_{n}(A ⊗_{Z} Z[t, t^{−1}]) = K_{n}(AZ)

evaluated at the class of unit t ∈ Z[t, t^{−1}] in K_{1}(Z[t, t^{−1}]), and the canonical splitting (2.3). Let B be the direct sum of K_{n}(j_{+} ∘ i), s_{n}, and the restrictions of the maps K_{n}(j_{+}) and K_{n}(j_{−})) to NK_{n}(A).

In particular we get two homomorphisms, both natural in A, from the Bass-Heller-Swan decomposition (2.5)

i_{n}: NK_{n}(A) → K_{n}(AZ)
r_{n}: K_{n}(AZ) → NK_{n}(A),

by focusing on the first copy of NK_{n}(A), such that r_{n} ∘ i_{n} is the identity on NK_{n}(A).

Let σ_{k}: Z → Z be the injection given by t ↦ t^{k}. We may consider the ring A[t] as an A[t] − A[t] bimodule with standard left action, and right action a(t) · b(t) = a(t)b(t^{k}) induced by σ_{k}. This map induces an induction functor

Ind_{k}: P(A[t) → P(A[t])

defined by P ↦ A[t] ⊗_{σ_{k}} P. There is also a restriction functor

Res_{k}: P(A[t]) → P(A[t])

defined by equipping P with the new A[t]-module structure a(t) · p = a(t^{k})p, for all a(t) ∈ A[t] and all p ∈ P. The induction and restriction functors yield two homomorphisms

Ind_{k}: K_{n}(AZ) → K_{n}(AZ)
Res_{k}: K_{n}(AZ) → K_{n}(AZ) .

See [22] or [18, p. 27] for more details.

There are also Verschiebung and Frobenius homomorphisms

(2.8)  V_{k}, F_{k}: NK_{n}(A) → NK_{n}(A)

induced on the Nil-groups (and related to Ind_{k} and Res_{k} respectively). The Frobenius homomorphism is induced by the functor NIL(A) → NIL(A) sending

(f: Q → Q) ↦ (f^{k}: P → P),

while the Verschiebung homomorphism is induced by the functor

⨁_{i=1}^{k} P → ⨁_{i=1}^{k} P, (q_{1}, q_{2}, … q_{k}) ↦ (f(q_{k}), q_{1}, q_{2}, …, q_{k−1}) .

The next result is proven by Stienstra [22, Theorem 4.7]). (Notice that Stienstra considers only commutative rings A, but his argument goes through in our case since the set of polynomials T we consider is {t^{n} | n ∈ Z, n ≥ 0} and each polynomial in T is central in A[t] with respect to the multiplicative structure.)

Lemma 2.9. The following diagrams commute for all n ∈ Z and k ∈ Z, k ≥ 1

\xymatrix{NK_{n}(A) \ar[r]^{i_{n}} \ar[d]^{F_{k}} & K_{n}(AZ) \ar[d]^{_{Res_{k}}} \\ NK_{n}(A) \ar[r]^{i_{n}} & K_{n}(AZ)}

and

\xymatrix{NK_{n}(A) \ar[r]^{i_{n}} \ar[d]^{V_{k}} & K_{n}(AZ) \ar[d]^{_{Ind_{k}}} \\ NK_{n}(A) \ar[r]^{i_{n}} & K_{n}(AZ)}

The next result is well-known for n = 1 (see [8, Lemma 3]).

Lemma 2.10. For every n ∈ Z and each x ∈ NK_{n}(A), there exists a positive integer M(x) such that Res_{m} ∘ i_{n}(x) = 0 for m ≥ M(x).

Proof. The Frobenius homomorphism F_{m}: NK_{n}(A) → NK_{n}(A) is induced by the functor sending (P, f) in NIL(A) to (P, f^{m}). For a given object (P, f) in NIL(A) there exists a positive integer M(f) with (P, f^{m}) = (P, 0) for m ≥ M(f). This implies by a filtration argument, and the fact that K-theory commutes with directed limits, that for x ∈ NK_{n}(A) there exists a positive integer M(x) with F_{m}(x) = 0 for m ≥ M(x). Now the claim follows from Lemma 2.9. □

3. Subgroups of G × Z

A finite group G is called p-hyperelementary if is isomorphic to an extension

1 → C → G → P → 1

where P is a p-group, and C is a cyclic group of order prime to p. Such an extension is a semi-direct product, and hence determined by the action map α: P → Aut(C) defined by conjugation. The group G is p-elementary precisely when α is the trivial map, or in other words, when there exists a retraction G → C. Notice that for a cyclic group C = Z/q^{k}, where q ≠ p is a prime, Aut(C) = Z/q^{k−1}(q − 1), if q odd, or Aut(C) = Z/2^{k−2} × Z/2, k ≥ 2, if q = 2. In either case, Aut(C)_{(p)} ≅ Aut(Q)_{(p)} by projection to any non-trivial quotient group C → Q.

Lemma 3.1. Let p be a prime, and let G be a finite p-hyperelementary group. Suppose that for every prime q ≠ p which divides the order of G, there exists an epimorphism f_{q}: G → Q_{q} onto a non-trivial cyclic group Q_{q} of q-power order. Then G is p-elementary.

Proof. Let Q be the product of the groups Q_{q} over all primes q ≠ p which divide the order of G. Let f: G → Q be the product of the given epimorphisms. Since every subgroup in G of order prime to p is characteristic, we have a diagram

[figure: 1; C; G; P; 1; 1; Q; Ḡ; P; 1]

But the epimorphism f: G → Q induces a retraction Ḡ → Q of the lower sequence, hence its action map ᾱ: P → Aut(Q) is trivial. As remarked above, this implies that α is also trivial and hence G is p-elementary. □

We now combine this result with the techniques of [10]. Given positive integers m, n and a prime p, we choose an integer N = N(m, n, p) satisfying the following conditions:

(i) p ∤ N, but q | N if and only if q | n for all primes q ≠ p. (ii) k ≥ log_{q}(mn) (i.e. q^{k} ≥ mn) for each full prime power q^{k}∥N.

The Farrell-Hsiang technique is to compute K-theory via p-hyperelementary subgroups H ⊂ G × Z/N, and their inverse images Γ_{H} = pr^{−1}(H) ⊂ G × Z via the second factor projection map pr: G × Z → G × Z/N.

Lemma 3.2. Let G be a finite group and let M be a positive integer. Let p be a prime dividing the order of |G|, and choose an integer N = N(M, |G|, p). For every p-hyperelementary subgroup H ⊂ G × Z/N, one of the following holds:

(i) the inverse image Γ_{H} ⊂ G × m · Z, for some m ≥ M, or (ii) the group H is p-elementary.

In the second case, we have the following additional properties:

(a) There exists a finite p-group P and isomorphism

α: P × Z ⟶^{≅} Γ_{H}.

(b) There exists a positive integer k, a positive integer ℓ with (ℓ, p) = 1, an element u ∈ Z/ℓ and an injective group homomorphism

j: P × Z/ℓ → G

such that the following diagram commutes

[figure: P × Z; α; id_{P} ×β; Γ_{H}; i; P × Z/ℓ × Z; j×k·id_{Z}; G × Z]

where i: Γ_{H} → G × Z is the inclusion and β: Z → Z/ℓ × Z sends n to (nu, n).

Proof. In the proof we will write elements in Z/N additively and elements in G multiplicatively. Let H ⊂ G × Z/N be a p-hyperelementary subgroup, and suppose that Γ_{H} is not contained in G × m · Z for any m ≥ M. We have a pull-back diagram

[figure: G′′; G′′; Z/N′′; H; G′; Z/N′′; Z/N′; Z/ℓ]

where G′ ⊂ G and Z/N′ are the images of H ⊂ G × Z under the first and second factor projection, respectively. Notice that Z/ℓ is the common quotient group of G′ and Z/N′. In terms of this data, H ⊆ G′ × Z/N′ and hence the pre-image Γ_{H} ⊆ G′ × m · Z ⊆ G × m · Z, where m = N/N′.

We now show that G′′ is a p-group. Suppose, if possible, that some other prime q ≠ p divides |G′′|. Since H is p-hyperelementary the Sylow q-subgroup of H is cyclic. However G′′ × Z/N′′ ⊆ H, so q ∤ N′′. But N′ = N′′ · ℓ, hence this implies that the q-primary part N′_{q} = ℓ_{q} ≤ |G′| ≤ |G|. Now

m = N/N′ ≥ q^{k}/N′_{q} ≥ q^{k}/|G| ≥ M

by definition of N = N(M, |G|, p). This would imply that Γ_{H} ⊂ G × m · Z for some m ≥ M, contrary to our assumption. Hence P := G′′ is a p-group, or more precisely the p-Sylow subgroup of G′ since p ∤ ℓ.

Alternative (ii) is an immediate consequence. If q ≠ p is a prime dividing |H|, then q | N′ since G′′ is a p-group. Hence H admits an epimorphism onto a non-trivial finite cyclic q-group. By Lemma 3.1, this implies that H is p-elementary. Note that there is an isomorphism

j′ = (id_{P} × s): P × Z/ℓ ⟶^{≅} G′

defined by the inclusion id_{P} : P ⊂ G′ and a splitting s: Z/ℓ → G′ of the projection G′ → Z/ℓ.

Next we consider assertion (a). A similar pull-back diagram exists for the subgroup Γ_{H} ⊂ G × Z. We obtain a pull-back diagram of exact sequences

[figure: 1; P; Γ_{H}; Z; 1; 1; P; G′; Z/ℓ; s; 1]

since P = Γ_{H} ∩ (G × 0), and pr_{Z}(Γ_{H}) = k · Z for some positive integer k. This exact sequence splits, since it is the pull-back of the lower split sequence: we can choose the element (g_{0}, k) ∈ Γ_{H} ⊆ G′ × Z which projects to a generator of pr_{Z}(Γ_{H}), by taking g_{0} = s(u) where u ∈ Z/ℓ is a generator. The isomorphism α: P × Z ⟶^{≈} Γ_{H} is defined by α(g, n) = (gg_{0}^{n}, n) for g ∈ P and n ∈ Z.

Assertion (b) follows by composing the splitting (id_{P} ×s): P ×Z/ℓ ≅ G′ with the inclusion G′ ⊆ G to obtain an injection j: P ×Z/ℓ → G. By the definition of g_{0}, the composite (j × k · id_{Z}) ∘ (id_{P} × β) = i ∘ α, where i: Γ_{H} → G × Z is the inclusion. □

4. The Proof of Theorem A

We will need some standard results from induction theory for Mackey functors over finite groups, due to Dress (see [6], [7]), as well as as a refinement called the Burnside quotient Green ring associated to a Mackey functor (see [14, §1] for a description of this construction, and [13] for the detailed account).

For any homomorphism pr: Γ → G from an infinite discrete group to a finite group G, the functor

M(H) := K_{n}(RΓ_{H}),

where Γ_{H} = pr^{−1}(H), is a Mackey functor defined on subgroups H ⊆ G. The required restriction maps exist because the index [Γ_{H} : Γ_{K}] is finite for any pair of subgroups K ⊂ H in G. This point of view is due to Farrell and Hsiang [9, §2]. The Swan ring SW (G, Z) acts as a Green ring on M, and it is a fundamental fact of Dress induction theory that the Swan ring is computable from the family H of hyperelementary subgroups of G. More precisely, the localized Green ring SW (G, Z)_{(p)} is computable from the family H_{p} of p-hyperelementary subgroups of G, for every prime p. If follows from Dress induction the Mackey functor M(G)_{(p)} is also p-hyperelementary computable. We need a refinement of this result.

Theorem 4.1 ([14, Theorem 1.8]). Suppose that G is a Green ring which acts on a Mackey functor M. If G ⊗ Z_{(p)} is H-computable, then every x ∈ M(G) ⊗ Z_{(p)} can be written as

x = ∑_{H∈H_{p}} a_{H} Ind_{H}^{G}(Res_{G}^{H}(x))

for some coefficients a_{H} ∈ Z_{(p)}.

We fix a prime p. For each element x ∈ NK_{n}(RG), let M = M(x) as in Lemma 2.10 applied to the ring A = RG. Then

Res_{m}: K_{n}(R[G × Z]) → K_{n}(R[G × Z])

sends i_{n}(x) to zero for m ≥ M(x). Now let N = N(M, |G|, p), as defined in Section 3, and consider M(H) = K_{n}(RΓ_{H}) as a Mackey functor on the subgroups H ⊆ G × Z/N, via the projection pr: G × Z → G × Z/N.

Let H_{p}(x) denote the set of p-hyperelementary subgroups H ⊆ G × Z/N, such that Γ_{H} is not contained in G×m·Z, for any m ≥ M(x). By the formula of Theorem 4.1, applied to y = i_{n}(x), we see that x lies in the image of the composite map

(4.2) ⨁_{H∈H_{p}(x)} K_{n}(RΓ_{H})_{(p)} ⟶^{i_{∗}} K_{n}(R[G × Z])_{(p)} ⟶^{r_{n}} NK_{n}(RG)_{(p)}.

We conclude from Lemma 3.2 (b) (using that notation) that the composite

(4.3) K_{n}(R[P × Z]) ⟶^{α_{∗}} K_{n}(RΓ_{H}) ⟶^{i_{∗}} K_{n}(R[G × Z]) ⟶^{r_{n}} NK_{n}(RG)

agrees with the composite

(4.4) K_{n}(R[P × Z]) ⟶^{(id_{P} ×β)_{∗}} K_{n}(R[P × Z/ℓ × Z]) ⟶^{(j×id_{Z})_{∗}} K_{n}(R[G × Z])

⟶^{(id_{G} ×k·id_{Z})_{∗}} K_{n}(R[G × Z]) ⟶^{r_{n}} NK_{n}(RG).

Recall that β: Z → Z/ℓ × Z sends n to (nu, n) for some generator u ∈ Z/ℓ. Let NIL(RP) → NIL(R[P × Z/ℓ]) be the functor which sends a nilpotent RG-endomorphism f: Q → Q of a finitely generated RP -module Q to the nilpotent R[G × Z/ℓ])-endomorphism

R[P × Z/l] ⊗_{RP} Q ↦ R[P × Z/ℓ] ⊗_{RP} Q, x ⊗ q ↦ xu ⊗ f(q).

Let ϕ: NK_{n}(RP) → NK_{n}(R[P × Z/ℓ]) denote the induced homomorphism.

Lemma 4.5.

(a) The following diagram commutes

\xymatrix{K_{n}(R[P × Z]) \ar[r]^{(id_{P} ×β)_{∗}} \ar[d]^{r_{n}} & K_{n}(R[P × Z/ℓ × Z]) \ar[d]^{r_{n}} \\ NK_{n}(RP) \ar[r]^{ϕ} & NK_{n}(R[P × Z/ℓ])} .

(b) The following diagram commutes

\xymatrix{K_{n}(R[P × Z/ℓ × Z]) \ar[r]^{(j×id_{Z})_{∗}} \ar[d]^{r_{n}} & K_{n}(R[G × Z]) \ar[d]^{r_{n}} \\ NK_{n}(R[P × Z/ℓ) \ar[r]^{j_{∗}} & NK_{n}(RG)}

(c) The following diagram commutes

\xymatrix{K_{n}(R[G × Z]) \ar[r]^{^{Ind^{k}}} \ar[d]^{r_{n}} & K_{n}(R[G × Z]) \ar[d]^{r_{n}} \\ NK_{n}(RG) \ar[r]^{V_{k}} & NK_{n}(RG)}

Proof. (a) The tensor product ⊗_{Z} induces a pairing

μ_{R,Γ} : K_{n−1}(R) ⊗_{Z} K_{1}(ZΓ) → K_{n}(RΓ) (4.6)

for every group Γ, which is natural in R and Γ. It suffices to prove that the following diagram is commutative for every ring R (since we can replace R by RP). Let A = R[Z/ℓ] for short.

[figure: K_{n}(R) ⊕ K_{n−1}(R) ⊕ NK_{n}(R) ⊕ NK_{n}(R); _{( \begin{matrix}i_{1,1} & 0 & 0 & 0 \\ i_{1,2} & i_{2,2} & 0 & 0 \\ 0 & 0 & ϕ & 0 \\ 0 & 0 & 0 & ϕ \end{matrix} )}; B; ≅; K_{n}(RZ); β_{∗}; K_{n}(A) ⊕ K_{n−1}(A) ⊕ NK_{n}(A) ⊕ NK_{n}(A); B; ≅; K_{n}(AZ])]

Here the vertical arrows are the isomorphisms given by the Bass-Heller-Swan decomposition (2.5), the homomorphisms i_{1,1} and i_{2,2} are induced by the inclusion R → R[Z/ℓ] and the homomorphism i_{1,2} comes from the pairing

μ_{R,Z/ℓ}: K_{n−1}(R) ⊗_{Z} K_{1}(Z[Z/ℓ]) → K_{n}(R[Z/ℓ]) (4.7)

evaluated at the class of the unit u ∈ Z[Z/ℓ]) in K_{1}(Z[Z/ℓ]) and the obvious change of rings homomorphisms K_{1}(R[Z/ℓ]) → K_{1}(R[Z/ℓ × Z])

In order to show commutativity it suffices to prove its commutativity after restricting to one of the four summands in the left upper corner.

This is obvious for K_{n}(RG) since induction with respect to group homomorphisms is functorial.

For K_{n−1}(R) this follows from the naturality of the pairing (4.6) in R and the group Z/ℓ and the equality

K_{1}(β)(t) = K_{1}(R[j_{Z}])(t) + K_{1}(R[j_{Z/ℓ}])(u)

where j_{Z} : Z → Z/ℓ × Z and j_{Z/ℓ}: Z/ℓ → Z/ℓ × Z are the obvious inclusions.

The commutativity when restricted to the two Bass-Nil-groups follows from a result of Stienstra [22, Theorem 4.12 on page 78]).

(b) This follows from the naturality in R of r_{n}.

(c) It suffices to show that the following diagram commutes (since we can replace R by RG)

[figure: K_{n}(R) ⊕ K_{n−1}(R) ⊕ NK_{n}(R) ⊕ NK_{n}(R); _{( \begin{matrix}id & 0 & 0 & 0 \\ 0 & k · id & 0 & 0 \\ 0 & 0 & V_{k} & 0 \\ 0 & 0 & 0 & V_{k} \end{matrix} )}; B; ≅; K_{n}(RZ); _{Ind_{k}}; K_{n}(R) ⊕ K_{n−1}(R) ⊕ NK_{n}(R) ⊕ NK_{n}(R); B; ≅; K_{n}(RZ)]

where the vertical arrows are the isomorphisms given by the Bass-Heller-Swan decomposition (2.5).

In order to show commutativity it suffices to prove its commutativity after restricting to one of the four summands in the left upper corner.

This is obvious for K_{n}(R) since induction with respect to group homomorphisms is functorial.

Next we inspect K_{n−1}(R). The following diagram commutes

\xymatrix{K_{n−1}(R) ⊗_{Z} K_{1}(Z[Z]) \ar[r] \ar[d]^{id ⊗ Ind_{k}} & K_{n}(RZ) \ar[d]^{_{Ind_{k}}} \\ K_{n−1}(R) ⊗_{Z} K_{1}(Z[Z]) \ar[r] & K_{n}(RZ)}

where the horizontal pairings are given by μ_{R,Z} from (4.6). Since in K_{1}(Z[Z]), k times the class [t] of the unit t is the class [t^{k}] = Ind_{k}([t]), the claim follows for K_{n−1}(R).

The commutativity when restricted to the copies of NK_{n}(R) follows from Lemma 2.9. This finishes the proof of Lemma 4.5. □

Lemma 4.5 implies that the composite (4.4) and hence the composite (4.3) agree with the composite

K_{n}(R[P × Z]) ⟶^{r_{n}} NK_{n}(RP) ⟶^{ϕ} NK_{n}(R[P × Z/ℓ])

⟶^{j_{∗}} NK_{n}(RG) ⟶^{V_{k}} NK_{n}(RG).

Since we have already shown that the element x ∈ NK_{n}(RG)_{(p)} lies in the image of (4.2), we conclude that x lies in the image of the map

Φ = (Φ_{P}): ⨁_{P ∈𝔓_{p}(G)}NK_{n}(RP)_{(p)} → NK_{n}(RG)_{(p)}

subject only to the restriction k ∈ I(g) in the definition of Φ_{P}.

Consider k ≥ 1, P ∈ 𝔓_{p} and g ∈ C_{G}^{⊥}P_{p}. We write k = k_{0}k_{1} for k_{1} ∈ I(g) and (k_{0}, |g|) = 1. We have V_{k} = V_{k_{1}} ∘ V_{k_{0}} (see Stienstra [22, Theorem 2.12]). Since (k_{0}, |g|) = 1, we can find an integer l_{0} such that (l_{0}, |g|) = 1 and (g^{l_{0}})^{k_{0}} = g. We conclude from Stienstra [22, page 67]

V_{k_{0}} ∘ ϕ(P, g) = V_{k_{0}} ∘ ϕ(P, (g^{l_{0}})^{k_{0}})) = ϕ(P, g^{l_{0}}) ∘ V_{k_{0}}.

Hence the image of V_{k} ∘ ϕ(P, g) is contained in the image of V_{k_{1}} ∘ ϕ(P, g^{l_{0}}) and g^{l_{0}} ∈ C_{G}^{⊥}P_{p}. This finishes the proof of Theorem A. □

5. Examples

We briefly discuss some examples. As usual, p is a prime and G is a finite group. The first example shows that Theorem A gives some information about p-elementary groups.

Example 5.1. Let P be a finite p-group and let ℓ ≥ 1 an integer with (ℓ, p) = 1. Then Theorem A says that NK_{n}(R[P × Z/ℓ])_{(p)} is generated by the images of the maps

V_{k} ∘ ϕ(P, g): NK_{n}(RP)_{(p)} → NK_{n}(R[P × Z/ℓ])_{(p)}

for all g ∈ Z/ℓ, and all k ∈ I(g). Since the composite F_{k} ∘ V_{k} = k · id for k ≥ 1 (see [22, Theorem 2.12]) and (k, p) = 1, the map

V_{k}: NK_{n}(RG)_{(p)} → NK_{n}(RG)_{(p)}

is injective for all k ∈ I(g). For g = 1, ϕ(P, 1) is the map induced by the first factor inclusion P → P × Z/ℓ, and this is a split injection. In addition, the composition of ϕ(P, g) with the functor induced by the projection P ×Z/ℓ → P is the identity on NIL(RP), for any g ∈ Z/ℓ. Therefore, all the maps V_{k}∘ϕ(P, g) are split injective. It would be interesting to understand better the images of these maps as k and g vary. For example, what is the image of Φ_{{1}} where we take P = {1} ? □

In some situations the Verschiebung homomorphisms and the homomorphisms ϕ(P, g) for g ≠ 1 do not occur.

Example 5.2. Suppose that R is a regular ring. We consider the special situation where the p-Sylow subgroup G_{p} is a normal subgroup of G, and furthermore where C_{G}(P) ⊆ P holds for every non-trivial subgroup P ⊆ G_{p}. For P ≠ {1}, we have C_{G}^{⊥}(P) = {1} and the homomorphism Φ_{P} = ϕ(P, 1), which is the ordinary induction map. We can ignore the map Φ_{{1}} since NK_{n}(R) = 0 by assumption. Therefore, the (surjective) image of Φ in Theorem A is just the image of the induction map

NK_{n}(RG_{p})_{(p)} → NK_{n}(RG)_{(p)} .

Note that NK_{n}(RG_{p}) is p-local, and we can divide out the conjugation action on NK_{n}(RG_{p}) because inner automorphisms act as the identity on NK_{n}(RG).

However, G/G_{p} is a finite group of order prime to p, so that

H_{0}(G/G_{p}; NK_{n}(RG_{p})) = H^{0}(G/G_{p}; NK_{n}(RG_{p})) = NK_{n}(RG_{p})^{G/G_{p}} .

Hence induction on this fixed submodule

λ_{n}: NK_{n}(RG_{p})^{G/G_{p}} → NK_{n}(RG)_{(p)}

is surjective. An easy application of the double coset formula shows that the composition of λ_{n} with the restriction map Res_{G}^{G_{p}}: NK_{n}(RG)_{(p)} → NK_{n}(RG_{p})_{(p)} is given by |G/G_{p}|-times the inclusion NK_{n}(RG_{p})^{G/G_{p}} → NK_{n}(RG_{p}). Since (|G/G_{p}|, p) = 1 this composition, and hence λ_{n}, are both injective. We conclude that λ_{n} is an isomorphism.

Concrete examples are provided by semi-direct products G = P ⋊C, where P is a cyclic p-group, C has order prime to p, and the action map α: C → Aut(P) is injective. If we assume, in addition, that the order of C is square-free, then

NK_{n}(ZP)^{C} ⟶^{≅} NK_{n}(Z[P ⋊ C])

for all n ≤ 1 (this dimension restriction, and setting R = Z, are only needed to apply Bass-Murthy [4] in order to eliminate possible torsion in the Nil-group of orders prime to p). □

6. NK_{n}(A) as a Mackey functor

Let G be a finite group. We want to show that the natural maps

i_{n}: NK_{n}(RG) → K_{n}(R[G × Z])

and

r_{n}: K_{n}(R[G × Z]) → NK_{n}(RG)

in the Bass-Heller-Swan isomorphism are maps of Mackey functors (defined on subgroups of G). Hence NK_{n}(RG) is a direct summand of K_{n}(R[G × Z]) as a Mackey functor. Since RG is a finitely generated free RH-module, for any subgroup H ⊂ G, it is enough to apply the following lemma to A = RH and B = RG.

Lemma 6.1. Let i: A → B be an inclusion of rings. Then the following diagram commutes

\xymatrix{K_{n}(A) ⊕ K_{n−1}(A) ⊕ NK_{n}(A) ⊕ NK_{n}(A) \ar[r]^{≅} \ar[d]^{i_{∗}⊕i_{∗}⊕i_{∗}⊕i_{∗}} & K_{n}(AZ) \ar[d]^{i[Z]_{∗}} \\ K_{n}(B) ⊕ K_{n−1}(B) ⊕ NK_{n}(B) ⊕ NK_{n}(B) \ar[r]^{≅} & K_{n}(BZ)}

where the vertical maps are given by induction, and the horizontal maps are the Bass-Heller-Swan isomorphisms. If B is finitely generated and projective, considered as an A-module, then

\xymatrix{K_{n}(B) ⊕ K_{n−1}(B) ⊕ NK_{n}(B) ⊕ NK_{n}(B) \ar[r]^{≅} \ar[d]^{i^{∗}⊕i^{∗}⊕i^{∗}⊕i^{∗}} & K_{n}(BZ) \ar[d]^{i[Z]^{∗}} \\ K_{n}(A) ⊕ K_{n−1}(A) ⊕ NK_{n}(A) ⊕ NK_{n}(A) \ar[r]^{≅} & K_{n}(AZ)}

where the vertical maps are given by restriction, and the horizontal maps are the Bass-Heller-Swan isomorphisms.

Proof. One has to show the commutativity of the diagram when restricted to each of the four summands in the left upper corner. One has to show that two composites of maps on the K-groups commutes. In each case these maps are induced by functors, and one shows that the two corresponding composites of functors are naturally equivalent. Hence the two composites induce the same map on K-theory. As an illustration we do this in two cases.

Consider the third summand NK_{n}(A) in the first diagram. The Bass-Heller-Swan isomorphism restricted to it is given by the the restriction of the map (j_{+})_{∗}: K_{n}(A[t]) → K_{n}(A[t, t^{−1}]) = K_{n}(AZ) induced by the obvious inclusion j_{+} : A[t] → A[t, t^{−1}] restricted to NK_{n}(A) = ker (ϵ_{∗}: K_{n}(A[t]) → K_{n}(A)), where ϵ: A[t] → A is given by t = 0. Since all these maps come from induction with ring homomorphisms, the following two diagram commute

\xymatrix{K_{n}(A[t]) \ar[r]^{ϵ_{∗}} \ar[d]^{i[t]_{∗}} & K_{n}(A) \ar[d]^{i_{∗}} \\ K_{n}(B[t]) \ar[r]^{ϵ_{∗}} & K_{n}(B)}

and

\xymatrix{K_{n}(A[t]) \ar[r]^{(j_{+})_{∗}} \ar[d]^{i[t]_{∗}} & K_{n}(A[t, t^{−1}]) \ar[d]^{i[t,t^{−1}]_{∗}} \\ K_{n}(B[t]) \ar[r]^{(j_{+})_{∗}} & K_{n}(B[t, t^{−1}])}

and the claim follows.

Consider the second summand K_{n−1}(B) in the second diagram. The restriction of the Bass-Heller-Swan isomorphism to K_{n−1}(B) is given by evaluating the pairing (4.6) for Γ = Z for the unit t ∈ Z[Z]. Hence it suffices to show that the following diagram commutes, where the horizontal maps are given by the pairing (4.6) for Γ = Z and the vertical maps come from restriction

\xymatrix{K_{n−1}(B) ⊗ K_{1}(Z[Z]) \ar[r] \ar[d]_{i^{∗}⊗id} & K_{n}(BZ) \ar[d]_{i[Z]^{∗}} \\ K_{n−1}(A) ⊗ K_{1}(Z[Z]) \ar[r] & K_{n}(AZ)}

This follows from the fact that for a finitely generated projective A-module P and a finitely generated projective Z[Z]-module Q there is a natural isomorphism of BZ-modules

(Res_{A} P) ⊗_{Z} Q ⟶^{≅} Res_{AZ}(P ⊗_{Z} Q), p ⊗ q ↦ p ⊗ q. □

Corollary 6.2. Let G be a finite group, and R be a ring. Then, for any subgroup H ⊂ G, the induction maps Ind_{H}^{G}: NK_{n}(RH) → NK_{n}(RG) and the restriction maps Res_{G}^{H}: NK_{n}(RG) → NK_{n}(RH) commute with the Verschiebung and Frobenius homomorphisms V_{k}, F_{k}, for k ≥ 1.

Proof. We combine the results of Lemma 6.1 with Stienstra’s Lemma 2.9 (note that these two diagrams also commute with i_{n} replaced by r_{n}). □

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Department of Mathematics & Statistics McMaster University Hamilton, ON L8S 4K1, Canada

E-mail address: ian@math.mcmaster.ca

Mathematisches Institut Universität Münster D-48149 Münster, Germany

E-mail address: lueck@math.uni-muenster.de