THE GENERALIZED BURNSIDE RING AND THE

K–THEORY OF A RING WITH ROOTS OF UNITY

W. G. Dwyer^{∗}, E. M. Friedlander^{∗∗} and S. A. Mitchell

University of Notre Dame Northwestern University University of Washington

§1. Introduction

Determining the algebraic K-theory of rings of integers in number fields has been the goal of much research. In [10] D. Quillen showed that the Hurewicz map h : Q_{0}(S^{0}) → B GL(ℤ)^{+} (see 1.1 for the notation) induces an interesting map on homotopy groups from the stable homotopy groups of spheres to the algebraic K-theory of the ring ℤ of rational integers. Quillen observed that if ℓ is an odd prime and if p ≠ ℓ is another prime generating the ℓ-adic units, then the composite map

Q_{0}(S^{0}) ⟶^{h} B GL(ℤ)^{+} ⟶^{ρ} B GL(𝔽_{p})^{+}

induces a surjection

π_{∗}^{s}(S^{0}, ℤ/ℓ) → K_{∗}(ℤ, ℤ/ℓ) → K_{∗}(𝔽_{p}, ℤ/ℓ)

on mod ℓ homotopy groups. In geometric terms, one can write Q_{0}(S^{0}) ≃ Im J × Coker J where Im J is the factor of Q_{0}(S^{0}) associated to the J-homomorphism J : O → Q_{0}(S^{0}). The space Im J is then equivalent after localization at ℓ to B GL(𝔽_{p})^{+} in such a way that the composite

Im J × Coker J ≃ Q_{0}(S^{0}) ⟶^{h} B GL(ℤ)^{+} ⟶^{ρ} B GL(𝔽_{p})^{+}

can be identified after localization at ℓ with the projection r : Im J × Coker J ⟶ Im J.

In [6] the third author complemented Quillen’s result by proving that the Hurewicz map h does not in fact detect anything in stable homotopy theory beyond what Quillen

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∗ Partially supported by the N.S.F. ^{∗∗} Partially supported by the N.S.F. and NSA Grant # MDA904-90-H-4006 .

found. He did this by verifying that the localization of h at ℓ factors through Im J, or, more precisely, that the maps

h, h · s · r : Q_{0}(S^{0}) → B GL(ℤ)^{+}

are homotopic after localization at ℓ, where s : Im J → Q_{0}(S^{0}) is a suitable right inverse after localization at ℓ for the factor projection r : Q_{0}(S^{0}) ⟶ Im J. It follows that Quillen’s map h_{∗} : π_{∗}^{s}(S^{0}, ℤ/ℓ) → K_{∗}(ℤ, ℤ/ℓ) annihilates the homotopy of Coker J, and that for any ring R the structure of K_{∗}(R, ℤ/ℓ) as a module over π_{∗}^{s}(S^{0}, ℤ/ℓ) extends (uniquely) to a structure of K_{∗}(R, ℤ/ℓ) as a module over π_{∗}(Im J, ℤ/ℓ) = K_{∗}(𝔽_{p}, ℤ/ℓ).

Let ζ_{a} denote a primitive ℓ^{a}-th root of unity and μ the group of ℓ-primary roots of unity of ℤ[ζ_{a}]. The inclusion μ ⊂ GL_{1}(ℤ[ζ_{a}]) leads to an enriched analogue

h : Q_{0}(Bμ_{+}) → B GL(ℤ[ζ_{a}])^{+}

of Quillen’s Hurewicz map. Let ℓ and p be primes as before (ℓ is odd and p generates the ℓ-adic units) and let 𝔽_{q} denote 𝔽_{p}[ζ_{a}]. In ℤ[ζ_{a}] there is a unique prime above p with residue class field 𝔽_{q}. The main result of B. Harris and G. Segal [3] implies that the composition

Q_{0}(Bμ_{+}) ⟶^{h} B GL(ℤ[ζ_{a}])^{+} ⟶^{ρ} B GL(𝔽_{q})^{+}

induces a surjection of mod ℓ homotopy groups. In fact, for a suitable space X there is after localization at ℓ a product decomposition Q_{0}(Bμ_{+}) ≃ B GL(𝔽_{q})^{+} × X in such a way that the (localized) composition

B GL(𝔽_{q})^{+} × X ≃ Q_{0}(Bμ_{+}) → B GL(ℤ[ζ_{a}])^{+} → B GL(𝔽_{q})^{+}

can be identified with the projection r : B GL(𝔽_{q})^{+} × X ⟶ B GL(𝔽_{q})^{+}.

The purpose of this paper is to extend the results of the third author to this wider context. Theorem 4.1 asserts that after localization at ℓ the maps

h, h · s · r : Q_{0}(Bμ_{+}) → B GL(ℤ[ζ_{a}])^{+}

are homotopic, where s : B GL(𝔽_{q})^{+} ⟶ Q_{0}(Bμ_{+}) is a suitable right inverse for the factor projection r. It follows that for any algebra R over ℤ[ζ_{a}] the structure of K_{∗}(R, ℤ/ℓ) as a module over π_{∗}^{s}(Bμ_{+}, ℤ/ℓ) extends (uniquely) to a structure of K_{∗}(R, ℤ/ℓ) as a module over K_{∗}(𝔽_{q}, ℤ/ℓ). We also make some further remarks (4.10) about the relationship of our work to [6], and briefly consider the situation of infinite cyclotomic extensions (4.12).

Let G be a finite ℓ-group. To prove Theorem 4.1 we show that h, h·s·r have homotopic compositions with any map BG → Q_{0}(Bμ_{+}). Our technique is to use the generalized Burnside ring A(G, μ) to study maps from BG to Q_{0}(Bμ_{+}) and to use the representation ring R_{R}G of G over R to study maps from BG to B GL(R)^{+}. The key algebraic result is Theorem 3.5, which amounts to an algebraic analogue of (4.1) and asserts that the map from A(G, μ) to R_{ℤ[ζ_{a}]}G naturally factors through a surjection A(G, μ) ⟶ R_{𝔽_{q}}G.

We conjecture that, after localization at ℓ, the composition

h · s : B GL(𝔽_{q})^{+} → B GL(ℤ[ζ_{a}])^{+}

extends to an infinite loop map. One can view Proposition 4.9 as evidence that such an extension exists. The reader is referred to [7] for a discussion of this conjecture and related matters.

1.1 Notation and conventions. The symbol “+” as a subscript denotes a disjoint basepoint; as a superscript it denotes Quillen’s plus construction. We write Σ^{∞}X_{+} for the suspension spectrum of the pointed space X_{+} and QX_{+} (= Q(X_{+})) for the corresponding infinite loop space, where as usual Q = Ω^{∞}Σ^{∞}. Similarly Q_{0}X_{+} ≡ Q_{0}(X_{+}) denotes the basepoint component of QX_{+}. We employ the notation [–, –] to denote pointed homotopy classes of maps between pointed spaces or spectra.

§2. Preliminaries on representation rings and the generalized Burnside ring.

Let G be a finite group. We will let AG denote the Burnside ring of G, i.e., the Grothendieck ring of finite G-sets, with sum and product induced respectively by disjoint union and cartesian product of G-sets. Suppose that H is another finite group. The Burnside module A(G, H) is defined as follows (cf. [5]). A (G, H)-set is a set P with a left G-action and a free right H-action such that the two actions commute; equivalently, P is an H-free (G × H)-set, but it is convenient to put the two actions on opposite sides. Then A(G, H) is the free abelian group on the isomorphism classes [P] of finite (G, H)-sets, subject to the relations [P ⨿ Q] = [P] + [Q]. We now list some properties of the group A(G, H) and describe some ways in which it is related to representation rings.

(2.1). A (G, H)-set is indecomposable if and only if it has the form P = P_{K,θ} where K ⊂ G, θ : K → H is a homomorphism and P_{K,θ} = G ×_{K} H. There is an obvious conjugation action of G × H on the set of all pairs (K, θ), and P_{K,θ} is isomorphic to P_{K′,θ′} if and only if (K, θ) and (K′, θ′) lie in the same orbit of this action. As an abelian group, A(G, H) is freely generated by the elements [P_{K,θ}] as (K, θ) runs through a set of orbit representatives.

(2.2). The group A(G, H) is a module over AG, with multiplication [P][Q] = [P × Q],

[P] ∈ AG, [Q] ∈ A(G, H). Here G acts diagonally on P × Q and H acts only on the Qfactor. There is a natural map of AG-modules σ : A(G, H) → AG given by [P] ↦ [P/H]; in addition, σ has a canonical right inverse η, given by η([S]) = [S×H] (here H has a trivial G-action). Composing σ with the usual augmentation ϵ : AG→ℤ, we get an augmentation ϵ′ : A(G, H)→ℤ. The augmentation ϵ is split by the ring homomorphism ℤ → A(G) which sends an integer n ≥ 0 to the trivial G-set [n] with n elements.

(2.3). Let IG = Ker ϵ and I(G, H) = Ker ϵ′. Then if G is an ℓ-group, the IG-adic and ℓ-adic filtrations on A(G, H) are topologically equivalent on I(G, H).

(2.4). Suppose that H is abelian. Then A(G, H) is a commutative ring, with product [P][Q] = [P ×_{H} Q]. The maps η and σ of (2.2) make A(G, H) into an augmented algebra over AG.

Given a (G, H)-set P, define a map of spectra α_{P} : Σ^{∞}BG_{+} → Σ^{∞}BH_{+} as the composite

Σ^{∞}BG_{+} →^{τ} Σ^{∞}(EG ×_{G} (P/H)_{+}) →^{Σ^{∞}h} Σ^{∞}BH_{+}

where τ is the transfer and h classifies the principal H-bundle EG×_{G}P over EG×_{G}(P/H). In the special case P = P_{K,θ}, α_{P} is the familiar composition of transfer τ : Σ^{∞}BG_{+} → Σ^{∞}BK_{+} and Bθ : Σ^{∞}BK_{+} → BH_{+}.

(2.5). The correspondence P ↦ α_{P} extends to a homomorphism

α : A(G, H) → [Σ^{∞}BG_{+}, Σ^{∞}BH_{+}]

of AG-modules which is a ring homomorphism if H is abelian. Here [Σ^{∞}BG_{+}, Σ^{∞}BH_{+}] is a module over the stable cohomotopy ring π_{s}^{0}(BG_{+}), and so receives an AG-module structure from the ring homomorphism AG = A(G, ∗) → π_{s}^{0}BG_{+}. Note that Σ^{∞}BH_{+} is a commutative augmented ring spectrum if H is abelian.

(2.6). Recall that QBH_{+} can be identified with the group completion of the monoid ∐_{n}(Σ_{n} ≀ H), and Q_{0}BH_{+} can be identified with B(Σ_{∞} ≀ H)^{+}. Now a (G, H)-set P corresponds to a conjugacy class of homomorphisms G → Σ_{n} ≀H, n = |P/H|, and so determines a unique homotopy class BG → B(Σ_{n} ≀ H). The map α can then be reinterpreted in an obvious in terms of group completion or in terms of the plus construction.

(2.7). For any commutative ring R, let R_{R}G denote the representation ring of G over R, i.e., the Grothendieck group of finitely generated R-projective RG-modules. It is clear that R_{R}G is a ring under tensor product. Let R′_{R}G denote the Grothendieck group of all finitely generated RG-modules. If R is a Dedekind domain, then the natural map R_{R}G → R′_{R}G is an isomorphism. If G is an ℓ-group and E is a field of characteristic ℓ, then the augmentation ideal of EG is nilpotent and every finitely generated module over EG has a finite filtration with each filtration quotient isomorphic to the trivial G-module E; consequently R′_{E}G ≅ ℤ, generated by the trivial G-module E.

(2.8). For a commutative ring R there is an obvious ring homomorphism AG → R_{R}G induced by mapping a G-set P to the permutation module RP. More generally, suppose given a homomorphism j : H → R^{∗}. Then there is a natural homomorphism

φ(j) : A(G, H) → R_{R}G

of AG-modules, given by P ↦ RP ⊗_{RH} R. Here R is an RH-module via j. If H is abelian, φ(j) is a ring homomorphism. We remark that the image of φ(j) lies in the subroup of R_{R}G generated by the R-free RG-modules. Observe that if P = P_{K,θ}, then φ(j)(P_{K,θ}) is the induced representation RG⊗_{RK} R, where the action of K on R is given by j · θ. Equivalently, φ(j)(P_{K,θ}) is a “monomial representation”, that is, the corresponding homomorphism G → GL_{n} R maps into the monomial matrices Σ_{n} ≀ R^{∗} (up to conjugacy), where n = |G/K|. If μ is a subgroup of R^{∗} and j : μ → R^{∗} is the inclusion we will write φ_{R} instead of φ(j).

(2.9). There is a natural ring homomorphism

β : R_{R}G → [BG_{+}, B GL(R)^{+} × K_{0}R]

obtained essentially by associating to an R-projective G-module M the map determined by the G-structure map of M, G → Aut(M) and viewing B GL(R)^{+} × K_{0}R as the homotopytheoretic group completion of ∐_{M} B Aut(M). If H is abelian, then j : H → R^{∗} induces a functor P ↦ RP ⊗_{RH} R of bipermutative categories (free right H-sets) → (f.g. projective R-modules) and hence a map of ring spectra Σ^{∞}BH_{+} → KR. The adjoint map

Φ(j) : BH_{+} → BGLR^{+} × K_{0}R

restricted to BH is just the obvious composite BH →^{Bj} B GL_{1}(R) → B GL(R)^{+} × {1}. It induces a ring homomorphism

Φ(j)_{∗} : [BG_{+}, QBH_{+}] → [BG_{+}, B GL(R)^{+} × K_{0}R].

We will write Φ_{0}(j) for the restricted map Q_{0}BH_{+} → B GL(R)^{+}. If μ is a subgroup of R^{∗} and j : μ → R^{∗} is the inclusion, we will write Φ_{R} instead of Φ_{0}(j).

(2.10). Let Â(G, H) and R̂_{R}G denote the IG-adic completions of respectively A(G, H) and R_{R}G. Then there are maps α̂ and β̂ which fit into a commutative diagram

A(G, H) ⟶ Â(G, H) ⟶^{α̂}  [BG_{+}, QBH_{+}]
φ(j)↓  ^{φ̂(j)}↓  ^{Φ(j)_{∗}}↓
R_{R}G ⟶ R̂_{R}G ⟶^{β̂} [BG_{+}, B GL(R)^{+} × K_{0}R]

in which the composite of the two upper arrows is α (2.5) and the composite of the two lower arrows is β (2.9).

The first assertion of the following theorem is a result of G. Lewis, J.P. May, and J. McClure [4] derived from the affirmation of the Segal Conjecture; the second assertion is a result of D. Rector [8].

2.11 Theorem. ([4], [8]) Let G, H be finite groups. Then the map α̂ of (2.10) is an isomorphism. Moreover, if R is a finite field, then map β̂ of (2.10) is also an isomorphism.

§3. Monomial representations of finite ℓ-groups

Throughout this section G denotes a finite ℓ-group. If R is a ring, let μ(R) denote the ℓ-torsion subgroup of R^{∗}, that is, the group of all ℓ^{n}-th roots of unity, n ≥ 0, and let a(R) be defined by the equation |μ(R)| = ℓ^{a(R)}, 0 ≤ a(R) ≤ ∞. The condition |μ(R)| > 2 is equivalent to a(r) ≥ 1 if ℓ is odd or to a(R) ≥ 2 if ℓ = 2.

Our goal in this section is to investigate the map φ(j) : A(G, μ) → R_{R}G (2.8) in the special situation in which μ is a subgroup of μ(R) and j : μ → R^{∗} is the inclusion; recall (2.8) that in this case we denote φ(j) by φ_{R}.

We begin with the following (known) description of the structure of a simple G-module over a field of characteristic different from ℓ. Our discussion follows the analysis of P. Roquette [11].

3.1 Proposition. Let F be a field of characteristic different from ℓ with |μ(F)| > 2, and let V be a simple right F G module. Then End_{F G} V is a cyclotomic extension field F′ of F, and there exists a subgroup K ⊂ G and homomorphism θ : K → μ(F′) such that F′ = F [θ(K)] and V ≅ F′ ⊗_{F K} F G as an (F′, F G)-bimodule.

Proof. Denote by D the division algebra End_{F G} V. We must show that D is a cyclotomic field extension of F. Clearly, we may assume that G acts faithfully on V. If V admits a non-trivial decomposition V = W_{1} ⊕ ⋯ ⊕ W_{k} as D-modules such that G permutes the W_{i}, then V is obtained by induction from a proper subgroup of G. Arguing inductively on the order of G, we may assume that V admits no such decomposition. By [11], G must then be a group of “normal rank one” and hence either G is cyclic or ℓ = 2 and G is dihedral, semidihedral or quaternionic. If G is cyclic, then one readily verifies that D is a cyclotomic extension F′ of F and that the action of G on V is given by a homomorphism θ : G → Aut_{F G} V ≅ (F′)^{∗} such that F′ = F [θ(G)]. We proceed to show that other normal rank one groups do not occur.

Assume then that ℓ = 2, a(F) ≥ 2, and that G is normal rank one but not cyclic; in this case G fits into an extension ℤ/2^{n} → G → ℤ/2, n ≥ 2. Let y generate ℤ/2^{n} and let x ∈ G project to the generator of ℤ/2. Then either xyx^{−1} = y^{−1} (dihedral and quaternionic cases) or xyx^{−1} = y^{2^{n−1}−1} (semidihedral case). In all cases xT x^{−1} = T^{−1}, where T = y^{2^{n−2}} has order four. Consider V as a module over ⟨T ⟩ = ℤ/4. Since ⟨T ⟩ is normal and V is primitive, the restriction of V to ⟨T ⟩ is isotypical, and nontrivial since V is faithful. Since a(F) ≥ 2, we conclude that T ν = iν for all ν, where i ∈ μ(F) is a fixed element of order 4. But then T (xν) = xT^{−1}ν = −i · xν, which is a contradiction. □

Proposition 3.1 enables us to prove the following surjectivity result, which asserts that sometimes the representation ring of G is generated by monomial representations.

3.2 Proposition. Let F be a field of characteristic different from ℓ with |μ(F)| > 2. Let μ denote μ(F). Then

φ_{F} : A(G, μ) → R_{F} G

is surjective with right inverse σ : R_{F} G → A(G, μ).

Proof. Since F G is semi-simple, R_{F} G is freely generated as an abelian group by the set of isomorphism classes of simple F G-modules. Thus, the surjectivity of φ_{F} as well as the existence of the left inverse σ follows immediately once we verify that every irreducible F G-module lies in the image of φ_{F}. As discussed in (2.8), any representation of G obtained by induction from a one-dimensional representation θ : K → μ of some subgroup K of G is of the form φ_{F} (P_{K,θ}). Thus, it suffices to prove that every irreducible representation of F G is induced from some one-dimensional representation of a subgroup K of G.

By (3.1), each irreducible F G-module is of the form V = L ⊗_{F H} F G, where H ⊂ G is a subgroup acting on the cyclotomic field extension L/F via a homomorphism θ : H → μ(L) with L = F [θ(H)]. Let K denote θ^{−1}(μ(F)) ⊂ H. Since F is preserved by the action of θ(K), we conclude the existence of a natural homomorphism of F G-modules

F ⊗_{F K} F G → L ⊗_{F H} F G.

This map is a surjective homomorphism of F -vector spaces of the same dimension, and thus an isomorphism. (The equality of these dimensions comes down to the observation that since |μ(F)| > 2 and L = F [θ(H)] is obtained by adjoining to F additional ℓ-primary roots of unity, the degree of L over F is equal to the order of the quotient group μ(L)/μ(F), which in turn is equal to the index of K in H.) □

Remarks. (a) Proposition 3.2 is obviously false if a(F) = 0, although if ℓ is odd it is possible to recover the conclusion by making an additional assumption about F [6, 1.15].

(b) If ℓ = 2 and F = 𝔽_{q} with q = 3 mod 4, Proposition 3.2 is false for G = ℤ/8. (c) The assertion of Proposition 3.2 for fields of characteristic 0 is a special case of a theorem of J. Tornehave [12] (which also treats the case a(F) = 1 for ℓ = 2 when char(F) = 0).

3.3 Representation ring calculations. We will now use Morita theory to carry out certain standard representation ring calculations in a very explicit way. As a matter of terminology, if F is a number field with ring of integers R, we will call R[1/ℓ] the ring of ℓ-integers of F. The situation we are interested in is as follows. We let F be a number field with |μ(F) > 2|, and V_{1}, …, V_{N} a complete nonredundant list of isomorphism classes of simple right F G-modules. For 1 ≤ i ≤ N we denote by F_{i} the endomorphism field (3.1) of V_{i} and by θ_{i} : K_{i} → μ(F_{i}) a homomorphism provided by 3.1, so that V_{i} ≅ F_{i} ⊗_{F K_{i}} F G as an (F_{i}, F G)-bimodule. We let S be the ring of ℓ-integers of F, S_{i} the ring of ℓ-integers of F_{i}, and V_{i}^{S} ⊂ V_{i} the (S_{i}, SG)-bimodule S_{i} ⊗_{SK_{i}} SG. (Observe that the tensor product defining V_{i}^{S} makes sense because θ_{i}(K_{i}) ⊂ μ(S_{i}) = μ(F_{i})). Finally, E is a finite residue class field of S such that μ(S) ⟶ μ(E) is an isomorphism, E_{i} is the field E ⊗_{S} S_{i}, and V_{i}^{E} the (E_{i}, EG)-bimodule E_{i} ⊗_{S_{i}} V_{i}^{S}. (The fact that E_{i} is a field follows from the hypothesis on a(F), the assumption μ(S) ≅ μ(E) and the fact (3.1) that F_{i} is obtained from F by adjoining ℓ-primary roots of unity: together these imply by elementary number theory that the prime of S determining E is inert in the extension F_{i}/F.)

There are maps f_{i}^{F} : R_{F} G → K_{0}F_{i} given by f_{i}^{F} [M] = [V_{i} ⊗_{F G} M], maps f_{i}^{S} : R_{S}G → K_{0}S_{i} given by f_{i}^{S}[M] = [V_{i}^{S} ⊗_{SG} M], and maps f_{i}^{E} : R_{E}G → K_{0}E_{i} given by f_{i}^{E}[M] = [V_{i}^{E} ⊗_{EG} M]. (Here K_{0}(–) is defined to be the Grothendieck group of finitely generated projective left modules.)

3.4 Proposition. In the above situation, there is a commutative diagram

R_{E}G  ⟵ R_{S}G ⟶  R_{F} G
×f_{i}^{E} ↓  ^{×f_{i}^{S}} ↓  ↓^{×f_{i}^{F}}
ℤ^{N} ≅ ∏_{i=1}^{N} K_{0}E_{i} ⟵ ∏_{i=1}^{N} K_{0}S_{i} ⟶ ∏_{i=1}^{N} K_{0}F_{i} ≅ ℤ^{N}

in which the horizontal arrows are induced by the evident ring homomorphisms. The vertical arrows in this diagram are isomorphisms. Moreover, if both of the groups ∏_{i=1}^{N} K_{0}E_{i} and ∏_{i=1}^{N} K_{0}F_{i} are identified with ℤ^{N} by using the standard bases provided by rank one free modules, then the two maps R_{S}G ⟶ ℤ^{N} given by this diagram are the same.

Proof. The action of G on the various modules involved gives a commutative diagram of rings and homomorphisms

EG  ⟵  SG  ⟶  F G
↓  ↓  ↓
∏_{i=1}^{N} End_{E_{i}} V_{i}^{E} ⟵ ∏_{i=1}^{N} End_{S_{i}} V_{i}^{S} ⟶ ∏_{i=1}^{N} End_{F_{i}} V_{i}

The Artin-Wedderburn theorem implies that in this diagram the right vertical arrow is an isomorphism. Let d_{i} = dim_{F_{i}} V_{i}. By construction V_{i}^{S} is a free module of rank d_{i} over S_{i}. Since |G| is invertible in S the ring SG is a maximal S-order in F G and for general reasons ∏_{i=1}^{N} End_{S_{i}} V_{i}^{S} ≅ ∏_{i=1}^{N} M_{d_{i}}(S_{i}) is a maximal S-order in ∏_{i=1}^{N} End_{F_{i}} V_{i} ≅ ∏_{i=1}^{N} M_{d_{i}}(F_{i}) (see [6, 1.7] and [9]); it follows that in the above diagram of rings the middle vertical arrow is an isomorphism. The left vertical arrow is then also an isomorphism, since it is the tensor product of the middle arrow over S with E.

The commutativity of the diagram of (3.4) is evident, and the isomorphism statements of (3.4) now follow from Morita theory; more precisely, what follows in the case of the map (×f_{i}^{F}) for instance is the fact that for each i the functor given by tensoring over End_{F_{i}} V_{i} with V_{i} gives an equivalence from the category of left End_{F_{i}} V_{i} modules to the category of F_{i} modules. The fact that the two indicated maps R_{S}G → ℤ^{N} agree amounts to the observation that both maps assign to a class [M] then vector (r_{0}, …, r_{N}), where r_{i} = rank_{S_{i}}(V_{i}^{S} ⊗_{SG} M).

The following theorem is a discrete analogue of Theorem 4.1. Theorem 3.5 extends a result of the third author [6, 1.12] from the Burnside ring A(G) to the generalized Burnside ring A(G, μ). Recall that ζ_{a} denotes a primitive ℓ^{a}-th root of unity.

3.5 Theorem. Let R = ℤ[ζ_{a}], where a ≥ 1 if ℓ is odd and a ≥ 2 if ℓ = 2. Let E be a residue field of R such that the reduction map R → E induces an isomorphism from μ = μ(R) to μ(E), and let σ : R_{E}G → A(G, μ) be a right inverse to φ_{E} (cf. 3.2). Then

φ_{R} · σ · φ_{E} = φ_{R} .

In other words, there is a unique map h = φ_{R} · σ making the following triangle[224z commute

\begin{matrix}A(G, μ) & →^{φ_{R}} & R_{R}G \\ ^{φ_{E}} ↓ & ↗ _{h} \\ R_{E}G \end{matrix} .

3.6 Remark. The maps φ_{E} and φ_{R} in (3.5) are maps of modules over AG; since φ_{E} is a surjection (3.2) the map h, if it exists, is also a map of modules over AG.

Proof of 3.5. Since φ_{E} is surjective by Proposition 3.2, it suffices to prove that ker φ_{E} = ker φ_{R}. Let S = R[1/ℓ] and let F be the quotient field of R. By (3.4) there is an isomorphism R_{E}G ≅ R_{F} G which fits in the following commutative diagram:

\begin{aligned}A(G, μ) & → & R_{S}G \\ ↓ & ↙ & ↓ \\ R_{E}G & ≅ & R_{F} G \end{aligned}

The localization theorem yields an exact sequence

⨁_{P|ℓ} R′_{R/P}G ⟶^{τ} R′_{R}G(≅ R_{R}G) ⟶^{η} R′_{S}G(≅ R_{S}G) → 0 .

(see 2.7) where R_{R/P}G ≅ ℤ, generated by the trivial G-module [R/P] of dimension one. The transfer map τ sends [R/P] to [R] − [P], where R and P are again regarded as trivial G-modules. Now the map R_{R}G → K_{0}R obtained by forgetting the G-action is canonically split by the construction which takes a projective R-module and gives it the trivial G-action. Hence K_{0}R is canonically embedded as a direct summand in R_{R}G, and the preceeding remarks show that ker η is contained in the class group K̃_{0}R, and is in fact equal to the subgroup of K̃_{0}R generated by the elements [R] − [P] as P runs through the non-principal primes of R over ℓ. Since φ_{R} maps A(G, μ) into the R-free RG-modules, im φ_{R} ∩ ker η = 0. Consequently, in order to show that ker φ_{E} = ker φ_{R} it is enough to show that ker φ_{S} = ker φ_{F}.

We will use the notation and indexing of (3.3) and (3.4). To show that ker φ_{S} = ker φ_{F}, it is enough to show that for any (G, μ)-set P and integer i between 1 and N the element f_{i}^{S}(φ_{S}(P)) ∈ K_{0}S_{i} is the class of a free S_{i}-module. Observe (2.8, 3.3) that by definition

f_{i}^{S}(φ_{S}(P)) = V_{i}^{S} ⊗_{SG} φ_{S}(P)
= S_{i} ⊗_{SK_{i}} SG ⊗_{SG} φ_{S}(P)
= S_{i} ⊗_{SK_{i}} φ_{S}(P)
= S_{i} ⊗_{SK_{i}} SP ⊗_{Sμ} S

where K_{i} acts on S_{i} via θ_{i} : K_{i} → μ(S_{i}). Choose such a P, fix the integer i; the above formula shows that f_{i}^{S}(φ_{S}(P)) depends only on the structure of P as a (K_{i}, μ)-set. We may decompose P as a disjoint union of indecomposable (K_{i}, μ) sets (see 2.1), each of the form P_{H,θ} where H ⊂ K_{i} is a subgroup and θ : H → μ a homomorphism. The summand of f_{i}^{S}(φ_{S}(P)) corresponding to such an P_{H,θ} is the module

S_{i} ⊗_{SK_{i}} SK_{i} ⊗_{SH} S = S_{i} ⊗_{SH} S .

where in the right-hand tensor product H acts on S by θ and on S_{i} by the composite of θ_{i} with the inclusion H → K_{i}. It is clear that S_{i} ⊗_{SH} S is a quotient module of S_{i}. Since this module is also a projective module over S_{i}, it must be either zero or isomorphic to S_{i}; in particular, it must be free.

§4. The main theorem

Let L denote the functor [1] on the category of spaces which assigns to each space X its localization LX at ℓ. If X is a connected loop space or a simply connected space, then the homotopy group π_{i}LX is isomorphic to ℤ_{(ℓ)} ⊗ π_{i}X, where ℤ_{(ℓ)} is the ring obtained from ℤ by inverting all primes except ℓ. There is a natural map X → LX which in all of the cases of interest to us nduces an isomorphism on homology or cohomology with coefficients in ℤ_{(ℓ)}. We will sometimes use the same notation for a map f : X → Y and for the induced map Lf : LX → LY. The notation LB GL(R)^{+} stands for L(B GL(R)^{+}).

For the definition of the maps Φ_{E} and Φ_{R} which appear in the following statement, see (2.9).

4.1 Theorem. Suppose that R is the ring ℤ[ζ_{a}], where a ≥ 1 if ℓ is odd and a ≥ 2 if ℓ = 2. Let μ = μ(R) and let E denote a residue class field of R with μ ≅ μ(E). Then there is a map s : LB GL(E)^{+} → LQ_{0}(Bμ_{+}) such that in the diagram

LB GL(E) ⟶^{s} LQ_{0}(Bμ_{+}) ⟶^{Φ_{E}} LB GL(E)^{+} ⟶^{s} LQ_{0}(Bμ_{+}) ⟶^{Φ_{R}} LB GL(R)^{+}

the following two conditions hold (1) the composite Φ_{E} · s is homotopic to the identity map of LB GL(E)^{+}, and

(2) the composite Φ_{R} · s · Φ_{E} is homotopic to the map Φ_{R}.

Remark. The existence of infinitely many quotient fields E of R satisfying the condition of (4.1) is guaranteed by the Tchebotarev density theorem.

Remark. Theorem 4.1 implies that up to homotopy the following diagrams commute:

LB GL(E)^{+}  →^{id} LB GL(E)^{+}
s ↓  ↗ _{Φ_{E}}
LQ_{0}(Bμ_{+})

LQ_{0}(Bμ_{+})  →^{Φ_{R}} LB GL(R)^{+}
Φ_{E} ↓  ↗ _{Φ_{R}·s}
LB GL(E)^{+}

The composite map Φ_{R} ·s is determined uniquely up to homotopy (4.8) by the requirement that the diagram on the right commute. The map s itself does not appear to have any uniqueness property.

Remark. Since LQ_{0}(Bμ_{+}) and LB GL(R)^{+} are infinite loop spaces, Theorem 4.1 implies that up to homotopy there are product decompositions

LQ_{0}(Bμ_{+}) ≅ LB GL(E)^{+} × X_{1}
LB GL(R)^{+} ≅ LB GL(E)^{+} × X_{2}

with respect to which Φ_{R} amounts to the projection LB GL(E)^{+} × X_{1} ⟶ LB GL(E)^{+} followed by factor inclusion LB GL(E)^{+} ⟶ LB GL(E)^{+} × X_{2}.

Remark. The results of [2] show that if E is a field of characteristic different from ℓ which is a union of finite fields and |a(E)| > 2, then the homotopy type of LB GL(E)^{+} depends only on the isomorphism type of the group μ(E).

For the convenience of the reader we state the following standard propositions. The first one is due originally in a more general form to D. Sullivan, and can be proved using his technique of compact representable functors.

4.2 Proposition. Let X be a CW-complex and X_{0} ⊂ X_{1} ⊂ X_{2} ⊂ ⋯ X a filtration of X by subcomplexes such that ∪_{i}X_{i} = X. Let Y and Z be loop spaces. Assume that the groups H̃^{j}(X_{i}, ℤ_{(ℓ)}) are finite for all i, j, and that the homotopy groups of Y and Z are finitely generated modules over ℤ_{(ℓ)}. Then, given maps f : X → Z and g : Y → Z with f_{n} = f|_{X_{n}}, there exists h : X → Y with g · h ∼ f iff for each n there exists h_{n} : X_{n} ⟶ Y with g · h_{n} ∼ f_{n}.

4.3 Remark. Proposition 4.2 implies that if X and Z are spaces of the indicated type, then two maps f, f′ : X ⟶ Z are homotopic iff for each n their restrictions to X_{n} are homotopic (this follows from the observation that f and f′ are homotopic iff (f, f′) : X → Z × Z lifts up to homotopy over the diagonal map Z → Z × Z). We will apply (4.2) in situations in which the spaces Y and Z are derived from LQ_{0}(Bμ_{+}) (μ a finite group), from LB GL(E)^{+} (E a finite field) or from LB GL(R)^{+} (R the ring of integers in an algebraic number field). In the first case the relevant homotopy groups are finite by stable homotopy theory, in the second case they are finite by the calculation of Quillen, and in the third they are finitely generated, again by a theorem of Quillen.

4.4 Proposition. Let G be a finite group, H an ℓ-Sylow subgroup of G and i : H → G the inclusion. Then the map LQ(Bi_{+}) : LQ(BH_{+}) → LQ(BG_{+}) has a right inverse.

4.5 Remark. The right inverse referred to in in 4.4 is derived from the transfer (2.5). Proposition 4.4 implies that if G is a finite group with ℓ-Sylow subgroup H, and g : Y → Z is a map of ℓ-local infinite loop spaces, then a map f : BG → Z lifts up to homotopy to a map BG → Y iff the restriction of f to BH lifts up to homotopy to a map BH → Y. Similarly, a map BG → Y is null homotopic iff its restriction to BH is.

4.6 Proposition. If Y is a space which is local at ℓ and f : X_{1} → X_{2} is a map which induces an isomorphism H_{∗}(X_{1}, ℤ_{(ℓ)}) → H_{∗}(X_{2}, ℤ_{(ℓ)}), then f induces a bijection [X_{2}, Y] ⟶ [X_{1}, Y].

4.7 Proposition. Suppose that G is a finite ℓ-group and that Y is a connected loop space with finite homotopy groups. Then the natural map [BG, Y] ⟶ [BG, LY] is a bijection.

Remark. This follows from the fact that Y can be expressed up to homotopy as a product Y′ × LY, where Y′ is the localization of Y with respect to the ring ℤ[1/ℓ]; since the homotopy groups of Y′ are uniquely ℓ-divisible, the pointed set [BG, Y′] is trivial. We will apply this proposition with Y = Q_{0}Bμ_{+} (μ a finite group) or Y = B GL(E)^{+} (E a finite field).

Proof of 4.1. To find a map s : LB GL(E)^{+} → LQ_{0}(Bμ_{+}) satisfying 4.1(1) it is enough to find

(1) (by 4.6) a map B GL(E) = ∪_{n}B GL_{n}(E) → LQ_{0}(Bμ_{+}) which is a lift of the map

B GL(E) ⟶ LB GL(E)^{+}, or

(2) (by 4.2) for each n ≥ 1 a map B GL_{n}(E) → LQ_{0}(Bμ_{+}) which is a lift of the map

BGL_{n}(E) ⟶ LB GL(E)^{+}, or

(3) (by 4.4) for each n ≥ 1 a map BG_{n} ⟶ Q_{0}(Bμ_{+}) which is a lift of the map BG_{n} ⟶ B GL(E)^{+}, where G_{n} denotes some chosen ℓ-Sylow subgroup of GL_{n}(E).

To accomplish this last observe that the composite

BG_{n} ⟶ B GL_{n}(E) ⟶ B GL(E)^{+}

corresponds under the map β of (2.10) to (i_{n} − [n]), where i_{n} ∈ R_{E}G_{n} is given by the inclusion G_{n} ⊂ GL_{n}(E) and [n] is the dimension n trivial representation. By (3.2) the element i_{n} − [n] lifts to some element j_{n} of A(G_{n}, μ). The map α(j_{n}) then by (2.10) gives the required map BG_{n} ⟶ Q_{0}(Bμ_{+}).

A chain of reasoning parallel to the above (see 4.3) shows that 4.1(2) can be proved by verifying that for each n ≥ 1 the two maps

Φ_{R} , Φ_{R} · s · Φ_{E} : LQ_{0}(Bμ_{+}) ⟶ LB GL(R)^{+}

become homotopic after precomposition with the map

BK_{n} → B(Σ_{n} ≀ μ) → Q_{0}(Bμ_{+}) → LQ_{0}(Bμ_{+})

where K_{n} ⊂ Σ_{n} ≀ μ is an ℓ-Sylow subgroup. We will denote this map BK_{n} ⟶ LQ_{0}(Bμ_{+}) by f_{n}. Consider the map g_{n} = s · Φ_{E} · f_{n} : BK_{n} → LQ_{0}(Bμ_{+}), and let f̃_{n}, g̃_{n} : BK_{n} ⟶ Q_{0}(Bμ_{+}) be the lifts of f_{n} and g_{n} provided by (4.7). By (2.11) there are elements a_{n}, b_{n} ∈ Â(K_{n}, μ) such that f̃_{n} = α̂(a_{n}) and g̃_{n} = β̂(b_{n}). Since (LΦ_{E}) · f_{n} ∼ (LΦ_{E}) · g_{n}, it follows from (4.7) that Φ_{E} · f̃_{n} ∼ Φ_{E} · g̃_{n} and hence (2.11) that φ̂_{E}(a_{n}) = φ̂_{E}(b_{n}). The IG-adic completion of (3.5) (see 3.6) now implies that φ̂_{R}(a_{n}) = φ̂_{R}(b_{n}), which gives by (2.10) that Φ_{R} · f̃_{n} ∼ Φ_{R} · g̃_{n} and by naturality of localization that (LΦ_{R}) · f_{n} ∼ (LΦ_{R}) · g_{n}. □

4.8 Dyer-Lashof operations. Let R = ℤ[ζ_{a}]. In [3], the map B GL(R)^{+} → B GL(E)^{+} induced by the quotient map R → E is shown to have a right inverse after localization at ℓ, provided that μ(R) ≅ μ(E) and a ≥ 2 if ℓ = 2. Theorem 4.1 gives us a canonical homotopy class of such a right inverse if a ≥ 1, since (in view of the fact that the localization at ℓ of Φ_{E} has a right inverse up to homotopy) the map λ = Φ_{R} · s is clearly determined (up to homotopy) by the condition that λ · Φ_{E} = Φ_{R}. This observation leads to a simple proof of the following.

4.9 Proposition. Let R = ℤ[ζ_{a}], where a ≥ 1 and a ≥ 2 if ℓ = 2. Let E be a residue field of R with μ(R) ≅ μ(E), and let λ = Φ_{R} · s : LB GL(E)^{+} ⟶ LB GL(R)^{+} be the map provided by (4.1). Then λ commutes up to homotopy with the Dyer-Lashof action maps of LB GL(E)^{+} and LB GL(R)^{+} derived from the infinite loop structures of these spaces.

Proof. We indicate the proof briefly. Let X = LQ_{o}(Bμ(R)_{+}), Y = LB GL(E)^{+}, and Z = LB GL(R)^{+}. For any space W let D_{n}W = EΣ_{n} ×_{Σ_{n}} W^{n} (n ≥ 1). Construct the diagram

D_{n}X ⟶^{D_{n}Φ_{E}} D_{n}Y ⟶^{D_{n}λ} D_{n}Z
↓  ↓  ↓
X ⟶^{Φ_{E}} Y ⟶^{λ} Z

in which the vertical maps are Dyer-Lashof action maps arising from the appropriate infinite loop space structures. The left hand square commutes up to homotopy because the map Φ_{E} is an infinite loop map. The large rectangle obtained by omitting the central column commutes up to homotopy because Φ_{R} = λ · Φ_{E} and the map Φ_{R} is an infinite loop map. Commutativity of the right hand square now follows from the fact that D_{n}Φ_{E} has a right inverse up to homotopy, namely, D_{n}s. □

4.10 Fewer roots of unity. Theorem 4.1 easily leads to a proof at odd primes of the original [6] factorization and splitting results of the third author, a proof which in its totality is slightly different in detail from (but identical in concept to) the proof in [6]. Rather than formulate the most general statement we will just give one example.

4.11 Theorem. [6] Assume that ℓ is odd and that p is a prime which generates the ℓ-adic units. Then there is a map λ : LB GL(𝔽_{p})^{+} ⟶ LB GL(ℤ)^{+} such that the composite

LQ_{0}(B{1}_{+}) ≅ LQ_{0}S^{0} ⟶^{Φ_{𝔽p}} LB GL(𝔽_{p})^{+} ⟶^{λ} LB GL(ℤ)^{+}

is homotopic to the map Φ_{ℤ} : LQ_{0}S^{0} ⟶ LB GL(ℤ)^{+}.

Proof. Let R = ℤ[ζ_{1}] and let μ = μ(R). The hypothesis on p implies that there is a unique prime of R above p, and we will let E be the corresponding residue class field of R, so that μ(E) ≅ μ. Since ℤ[ζ_{1}] is a free module of rank ℓ − 1 over ℤ and ℓ − 1 is relatively prime to ℓ, the transfer construction in algebraic K-theory can be used to produce a map t : LB GL(R)^{+} ⟶ LB GL(ℤ)^{+} which is a left inverse up to homotopy to the map i_{ℤ} : LB GL(ℤ)^{+} ⟶ LB GL(R)^{+} induced by the ring map ℤ → R. Let i_{𝔽_{p}} : LB GL(𝔽_{p})^{+} ⟶ LB GL(E)^{+} be the map induced by 𝔽_{p} ⟶ E, and let λ_{ℤ} : LB GL(𝔽_{p})^{+} ⟶ LB GL(ℤ)^{+} be the composite

LB GL(𝔽_{p})^{+} ⟶^{i_{𝔽p}} LB GL(E)^{+} ⟶^{λ_{R}} LB GL(R)^{+} ⟶^{t} LB GL(ℤ)^{+}

where λ_{R} is the map Φ_{R} · s provided by (4.1). A simple direct calculation which combines (4.1) and naturality shows that λ_{ℤ} = λ has the desired property. □

4.12 Even more roots of unity. It is natural to ask for an analogue of (4.1) in the situation in which all ℓ-primary roots of unity are present. We can offer the following somewhat unsatsfactory result along these lines.

4.13 Theorem. Let R be the ring Z[ζ_{i}|i ≥ 1], let p be a prime which generates the ℓ-adic units, and let E be the quotient field R/p ≅ 𝔽_{p}[ζ_{i}|i ≥ 1]. Then there exists a map λ_{R} : LB GL(E)^{+} ⟶ LB GL(R)^{+}, such that, for any finite subgroup μ of μ(R) the two maps

Φ_{R} , λ_{R} · Φ_{E} : LQ_{0}(Bμ_{+}) ⟶ LB GL(R)^{+}

are homotopic.

Proof. Let R_{n} = ℤ[ζ_{n}], E_{n} = R_{n}/p, μ^{n} = μ(R_{n}), and let λ_{n} : LB GL(E_{n})^{+} ⟶ LB GL(R_{n})^{+} denote the map provided up to homotopy by (4.1). There are diagrams

LQ_{0}(Bμ_{+}^{n}) ⟶^{Φ_{En}} LB GL(E_{n})^{+} ⟶^{λ_{n}} LB GL(R_{n})^{+}
↓  ↓  ↓
LQ_{0}(Bμ_{+}^{n+1}) ⟶^{Φ_{E_{n+1}}} LB GL(E_{n+1})^{+} ⟶^{λ_{n+1}} LB GL(R_{n+1})^{+}

in which the left hand square evidently commutes up to homotopy, as does the large rectangle obtained by removing the center column. It thus follows from remarks above (4.8) that the right hand square commutes up to homotopy. After chosing if necessary models for the spaces LB GL(E_{n})^{+} so that the natural maps LB GL(E_{n})^{+} ⟶ LB GL(E_{m})^{+} (m > n) are represented by cofibrations, it is now easy to construct by induction on n a commutative ladder (of genuine maps)

LB GL(E_{1})^{+} ⟶ LB GL(E_{2})^{+} ⟶ ⋯ ⟶ LB GL(E_{n})^{+} ⟶ ⋯
Λ_{1}↓  ^{Λ_{2}}↓  ^{Λ_{n}}↓
LB GL(R_{1})^{+} ⟶ LB GL(R_{2})^{+} ⟶ ⋯ ⟶ LB GL(R_{n})^{+} ⟶ ⋯

such that Λ_{n} is homotopic to λ_{n}. The mapping telescope of the upper rail of this diagram is equivalent to LB GL(E)^{+} and the mapping telescope of the lower rail to LB GL(R)^{+}. It is easy to see that the induced map between these telescopes is the desired map λ_{R}. □

The reader may wonder about the restriction to finite subgroups μ ⊂ μ(R) in the above theorem. To remove this restriction it would be natural to try to add a third (upper) rail to the above ladder, a rail of the form

LQ_{0}(Bμ_{+}^{1}) ⟶ LQ_{0}(Bμ_{+}^{1}) ⟶ ⋯ LQ_{0}(Bμ_{+}^{n}) ⟶ · · · .

This new rail would be connected to the now center one by the maps Φ_{E_{n}} and to the lower one by the maps Φ_{R_{n}}. We are unable to construct such an extended ladder for the following reason. Let A_{n}, B_{n} and C_{n} denote the n’th spaces in respectively the upper, middle, and lower rails. The most direct approach to making the desired construction gives inductively, at the n’th stage, a map g_{n} : A_{n+1} ∐_{A_{n}} B_{n} ⟶ C_{n+1} and it is required to extend g_{n} over a map A_{n+1} ∐_{A_{n}} B_{n} ⟶ B_{n+1} to a map B_{n+1} ⟶ C_{n+1}. The homotopy class of maps B_{n+1} ⟶ C_{n+1} provided by (4.1) extends the restriction of g_{n} to A_{n+1} and also extends the restriction of g_{n} to B_{n}, but this is not enough. The difficulty is that the restriction map [A_{n+1} ∐_{A_{n}} B_{n}, C_{n+1}] ⟶ [A_{n+1}, C_{n+1}] × [B_{n}, C_{n+1}] is not necessarily injective.

References

[1] A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions and Localizations, Lecture Notes in Math. 304, Springer, Berlin, 1972.

[2] E. Friedlander and G. Mislin, Galois descent and cohomology for algebraic groups, Math. Zeitschrift 205 (1990), 177–190.

[3] B. Harris and G.Segal, K_{i} groups of rings of algebraic integers, Annals of Math. 101 (1975), 20–33.

[4] G. Lewis, P. May and J. McClure, Classifying G-spaces and the Segal conjecture, Current Trends in Algebraic Topology, CMS Conference Proc. no. 2, 1982, pp. 165–179.

[5] J.P. May, Stable maps between classifying spaces, Contemporary Math. vol. 37, American Math. Soc., Providence, 1985, pp. 121–129.

[6] S.A. Mitchell, The Morava K-theory of algebraic K-theory spectra, K-Theory 3 (1990), 607–626.

[7] S.A. Mitchell, On the Lichtenbaum-Quillen conjectures from a stable homotopy-theoretic viewpoint, preprint (l990).

[8] D. Rector, Modular characters and K-theory with coefficients in a finite field, J. Pure and Applied Algebrra 4 (1974), 135–158.

[9] I. Reiner, Maximal Orders, Academic Press, London, 1975.

[10] D. G. Quillen, Letter to J. Milnor, Lecture Notes in Math. 551, Springer, New York, 1976, pp. 182–188.

[11] P. Roquette, Realiserung von Darstellungen endlicher nilpotente Gruppen, Archiv der Math. 9 (1958), 224-250.

GENERALIZED BURNSIDE RING 15

[12] J. Tornehave, Relations among permutation representations of finite p-groups, preprint (1984). University of Notre Dame, Notre Dame, Indiana 46556 Northwestern University, Evanston, Illinois 60208 University of Washington, Seattle, Washington 98195

Processed March 17, 1992