Georgian Mathematical Journal Volume 9 (2002), Number 4, 691–702

ON THE EQUIVALENCE OF QUILLEN’S AND SWAN’S K-THEORIES

PAUL EBELING AND FRANS KEUNE

Abstract. The K-theory of rings can be defined in terms of nonabelian derived functors as described in [9]; see also the books [7] and [8] of Inassaridze for a similar approach. In fact both Swan’s theory and Quillen’s theory can be described this way. The equivalence of both K-theories is proved by Gersten [5]. In this paper we give a proof using these descriptions that involve nonabelian derived functors.

2000 Mathematics Subject Classification: 19D50.

Key words and phrases: Higher algebraic K-theory, simplicial resolutions.

Introduction

Quillen’s higher algebraic K-groups of a unital ring R are defined as the homotopy groups of the space that is obtained from the classifying space of the elementary group of the ring R:

K_{n}(R) = π_{n}(BE(R)^{+}) (for n ⩾ 2).

In Section 4 we replace this space by a simplicial set ℤ_{∞}W̄ E(R) also depending functorially on R, having a geometric realization which is homotopy equivalent to BE(R)^{+}. This description depends on the notion of integral completion in the sense of Bousfield and Kan [3].

Swan’s K-groups of a (nonunital) ring R are defined by means of a free simplicial resolution of R. In Section 8 we consider a simplicial group H(R) depending functorially on R, having Swan’s K-groups of R as homotopy groups. Writing K′_{n}(R) for these groups the formula becomes

K′_{n}(R) = π_{n−2}(H(R)).

The main result is the theorem in Section 8, which says that Swan’s K-groups coincide with Quillen’s when extended to the category of nonunital rings in the standard way: K′_{n}(R) ≅ K_{n}(R^{+}, R). Here R^{+} stands for the ring obtained from R by formally adjoining a unity element. The proof uses Gersten’s result [5]: free associative nonunital rings have trivial K-theory which historically was the missing part of the earliest proof of the equivalence of these K-theories by Anderson [1].

It should be noted that the proof in Section 8 is a corrected and improved version of the proof in the unpublished paper [11].

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

1. The Plus Construction

The so-called plus construction is used in Quillen’s definition of the higher K-groups of a unital ring. Its defining properties are described in the theorem below; see also Property P1. Good references for the plus construction are Loday’s thesis [13] and the book by Berrick [2].

Theorem and definition ([13], Théorème 1.1.1). Let X be a connected CW-complex with basepoint ∗, and let N be a perfect normal subgroup of π (= π_{1}(X, ∗)). Then there exists a connected CW-complex X^{+} and a map j: X → X^{+} such that

(i) π_{1}(j): π_{1}(X, ∗) → π_{1}(X^{+}, j(∗)) identifies with the canonical projection π → π/N.

(ii) The map j induces an isomorphism on integral homology.

In the sequel the following properties will be used.

Property P1 ([13], Proposition 1.1.2). The pair (X^{+}, j) is universal (in the homotopy category) among pairs (Y, f), where Y is a connected space and f: X → Y a continuous map satisfying π_{1}(f)(N) = 0.

A consequence is that the plus construction is functorial up to homotopy ([13], Corollaire 1.1.3).

Property P2 ([13], Proposition 1.1.7). Let G be a group with a perfect commutator subgroup [G, G]. Then B[G, G]^{+} is up to homotopy the universal covering of BG^{+}, where in both cases the plus construction is relative to the subgroup [G, G]. (As usual BG is the classifying space of the group G.)

The plus construction is used in the definition of higher K-groups as follows. Let R be a unital ring. The general linear group GL(R) of R has the elementary subgroup E(R) as the commutator subgroup (the ‘Whitehead Lemma’). Apply the plus construction to the classifying space BGL(R) of GL(R) relative to the subgroup E(R) (which is perfect), and finally take homotopy groups.

Definition. K_{n}(R) = π_{n}(BGL(R)^{+}) for n ⩾ 1.

From Property P2 one deduces

K_{n}(R) = π_{n}(BE(R)^{+}) for n ⩾ 2

(and of course π_{1}(BE(R)^{+}) = 0). In this paper this identity is taken as definition of K_{n} for n ⩾ 2. An important property of the space BE(R)^{+} is the following.

Property P3. BE(R)^{+} is an H-space. ([13], §1.3.4.)

2. The Integral Completion of a Group

In Section 4 the space BE(R)^{+} will be replaced by the integral completion (in the sense of Bousfield and Kan [3]) of BE(R). The definition and properties of this completion are given in the next section. We will need the notion of integral completion of a group.

ON THE EQUIVALENCE OF QUILLEN’S AND SWAN’S K-THEORIES 693

Let G be a group. The subgroups Γ_{i}G (for i ⩾ 1) are defined inductively by

Γ_{1}G = G,
Γ_{i+1}G = [Γ_{i}G, G].

As usual [H_{1}, H_{2}] denotes the subgroup generated by the commutators [h_{1}, h_{2}] = h_{1}h_{2}h_{1}^{−1}h_{2}^{−1} with h_{1} ∈ H_{1} and h_{2} ∈ H_{2}. Clearly, the subgroups Γ_{i}G are normal subgroups of G. The series of subgroups

G ⊵ Γ_{2}G ⊵ Γ_{3}G ⋯

is known as the lower central series of the group G. It induces a tower of groups

⋯ → G/Γ_{3}G → G/Γ_{2}G → 1

in which every group homomorphism G/Γ_{i+1}G → G/Γ_{i}G is a central extension.

The integral completion CG (= G_{ℤ}^{∧} in the terminology of [3]) of G is defined as the inverse limit of the tower of groups:

CG = \underleftarrow{lim}_{i} G/Γ_{i}G.

It was pointed out to the authors that this construction also appears in the literature under the name “pronilpotent completion”.

In an obvious way C is a functor Gr → Gr, where Gr denotes the category of groups. This functor can be viewed as the inverse limit of the functor which assigns to every homomorphism G → N, with N a nilpotent group, the group N, and to every commutative triangle

\xymatrix{ & N \ar[dd] \\ G \ar@{-->}[ur] \ar[dr] \\  & N′}

with N and N′ nilpotent, the map N → N′. The existence of this inverse limit follows from the existence of small cofinal diagrams, e.g. given by the tower of groups above.

3. The Integral Completion of a Space

We will use here the simplicial terminology. For a category C the category of simplicial C-objects is denoted by sC. The category of reduced simplicial sets is denoted by rsSet. It is the full subcategory of sSet, the category of simplicial sets, consisting of those X ∈ sSet which have only one vertex, i.e. X_{0} is a one-element set.

The functor G: rsSet → sGr assigns to a reduced simplicial set its loop group GX, which is a simplicial group satisfying π_{i}(GX) ≅ π_{i+1}(X) for all i ⩾ 0. This functor G has a right adjoint W̄: sGr → rsSet, which is the simplicial analogue of the classifying space functor.

The reduced simplicial set W̄ H is called the classifying complex of the simplicial group H. The adjunction of G and W̄ induces a natural simplicial map X → W̄ GX, which induces isomorphisms on the homotopy groups when X is a Kan complex. A good reference for this is May [14].

A functor T: Gr → Gr determines a functor T̃: rsSet → rsSet in the following way: let X be a reduced simplicial set; first form its loop group GX, next apply T dimension-wise to obtain a simplicial group T GX, and finally take the classifying complex. In a formula: T̃ = W̄ T G, where T stands for T applied dimension-wise. In particular the integral completion C: Gr → Gr as defined in Section 2 determines a functor

ℤ_{∞} = C̃: rsSet → rsSet.

This functor assigns to a reduced simplicial set its so-called integral completion. This integral completion functor was introduced by Bousfield and Kan [3]. The definition given here is in fact one of various possible definitions: it is the definition they give in Chapter III of [3]. Some of the main properties of the integral completion functor are:

Property I1 ([3], Ch. I, Lemma 5.5, p. 25). Let f: X → Y be a map in rsSet. Then ℤ_{∞}f: ℤ_{∞}X → ℤ_{∞}Y is a homotopy equivalence if and only if f induces an isomorphism on integral homology.

Property I2 ([3], Ch. V, Proposition 3.4, p. 134). For X ∈ rsSet there is a natural map i: X → ℤ_{∞}X which is a weak homotopy equivalence if X is nilpotent.

Property I3 ([3], Ch. II, Lemma 5.4, p. 63). Let p: E → B (in rsSet) be a fibration with connected fibre F such that the Serre action of π_{1}(B) on H_{i}(F; ℤ) is nilpotent for all i ⩾ 0. Then ℤ_{∞}(p): ℤ_{∞}E → ℤ_{∞}B is a fibration and the inclusion ℤ_{∞}F → ℤ_{∞}(p)^{−1}(∗) is a homotopy equivalence ([3],Ch. II, Lemma 5.1, p. 62). The action of π_{1}(B) on H_{∗}(F; ℤ) is in particular nilpotent if π_{1}(E) acts nilpotently on π_{i}(F) for all i ⩾ 1.

4. The Integral Completion of W̄ E(R)

For any group H, there is a simplicial group which is H in every dimension, having the identity as degeneracy and boundary maps. This object is a constant simplicial group and we denote it by H again.

Proposition 1. Let R be a unital ring. Then the geometric realization |ℤ_{∞}W̄ E(R)| of ℤ_{∞}W̄ E(R) is homotopy equivalent to BE(R)^{+}, the equivalence being functorial in R.

Proof. The plus construction has its simplicial analogue in rsSet, the category of reduced simplicial sets. There exists a map j: W̄ E(R) → W̄ E(R)^{+} in rsSet such that its geometric realization |j|: BE(R) → |W̄ E(R)^{+}| is the map j: BE(R) → BE(R)^{+} of Section 1. Consider the commutative square (which is functorial in R)

\xymatrix{W̄ E(R) \ar[r]^{j} \ar[d]_{i} & W̄ E(R)^{+} \ar[d]_{i} \\ ℤ_{∞}W̄ E(R) \ar[r]^{ℤ_{∞}(j)} & ℤ_{∞}(W̄ E(R)^{+})}

It suffices to prove that i: W̄ E(R)^{+} → ℤ_{∞}(W̄ E(R)^{+}) and ℤ_{∞}(j) are homotopy equivalences. The first map is a homotopy equivalence because of Property I2 and Property P3: the space BE(R)^{+} is an H-space, so it is nilpotent. The map ℤ_{∞}(j) is a homotopy equivalence because of Property I1. □

The proof above can also be found in [5], where it is attributed to E. Dror. In exactly the same way one proves the homotopy equivalence of the spaces |ℤ_{∞}W̄ GL(R)| and BGL(R)^{+}.

Corollary. For n ⩾ 2 we have K_{n} ≅ π_{n}ℤ_{∞}W̄ E.

5. Derived Functors

Let Gr be the category of groups. In this section we will review the theory of (left) derived functors of a given functor T: Gr → Gr as introduced in [9]. The situation is analogous to the Abelian case where projective resolutions are used.

Let G be a simplicial group. For each n ⩾ 1 we define

Z_{n}(G) = { (x_{0}, …, x_{n+1}) ∈ G_{n}^{n+2} | d_{i}x_{j} = d_{j−1}x_{i} for all 0 ⩽ i < j ⩽ n + 1 },

a subgroup of G_{n}^{n+2} (= G_{n} × ⋯ × G_{n}, n + 2 times). The elements of Z_{n}(G) are those n + 2 -tuples of elements of G_{n} that fit together in exactly the same way as the n + 2 faces of an n + 1 -simplex do. The group Z_{n}(G) will be called the group of n-spheres in the simplicial group G. There is an obvious homomorphism d: G_{n+1} → Z_{n}(G) which assigns to an n + 1 -simplex x the n-sphere (d_{0}x, …, d_{n+1}x) of its faces. A simplicial group is called aspherical if for each n ⩾ 1 the map d: G_{n+1} → Z_{n}(G) is surjective, that is if every n-sphere is the boundary of an n + 1 -simplex.

For n > 0 there is an isomorphism

ᾱ: π_{n}(G) → Z_{n}(G)/dG_{n+1},

which is induced by the homomorphism

α: G̃_{n} → Z_{n}(G), g ↦ (1, …, 1, g),

where G̃_{n} = ⋂_{i} Ker d_{i}.

The isomorphism ᾱ has the useful property that it respects the natural action of G_{0} on G, given by conjugating G dimension-wise by the images of elements of G_{0} under the degeneracy maps. The action obviously induces actions on G̃_{n} and Z_{n}(G) by restricting the actions on G_{n} and the n + 2 -fold product G_{n} × ⋯ × G_{n} respectively.

For any set X let F X be the free group on the elements of X. A simplicial group G is called free if there is a subset X_{n} of G_{n} for each n ⩾ 0 such that G ≅ F X_{n} and moreover the degeneracy maps s_{i}: G_{n} → G_{n+1} (i = 0, …, n) map X_{n} into X_{n+1} for each n ⩾ 0. An example of a free simplicial group is the loop group GX of a reduced simplicial set X.

Let H be a group. A free resolution (G, ε), or simply G, of H consists of:

(1) a free aspherical simplicial group G;

(2) a group homomorphism ε: G_{0} → H, which induces an isomorphism π_{0}(G) → H.

Free resolutions do exist. What is more, there are functorial free resolutions G(H). One example is the cotriple resolution G_{n}(H) = F^{n+1}(H). Another example is GW̄ (H), the loop group on the classifying complex of H.

Let G be a free resolution of a group H and G′ a free resolution of a group H′. In [9] it is proved that a homomorphism h: H → H′ can be covered by a simplicial homomorphism g: G → G′, i.e. π_{0}(g): π_{0}(G) → π_{0}(G′) induces f via the isomorphisms π_{0}(G) → H and π_{0}(G′) → H′. Moreover, two such simplicial homomorphisms are Gr-homotopic. As a consequence one can define derived functors of a functor T: Gr → Gr. On objects the n-th derived functor L_{n}T: Gr → Gr is defined as follows. Let H ∈ Gr; take a free resolution G of H; then put

L_{n}T = π_{n}(T G),

where T G means: T applied dimension-wise to G. On morphisms L_{n}T is defined by

(L_{n}T)(h) = π_{n}(T g),

where g: G → G′ covers h: H → H′, G and G′ being free resolutions of H and H′ respectively.

Example. Let T: Gr → Gr be the Abelianization funtor. What are its derived functors? GW̄ H is a free resolution of H ∈ Gr. Therefore

(L_{n}T)(H) = π_{n}(T GW̄ H) = π_{n}(GW̄ H/[GW̄ H, GW̄ H])
= H_{n+1}(W̄ H; ℤ) = H_{n+1}(H; ℤ)

(cf. [14], p.121). So L_{n}T is the homology functor H_{n+1}(−; ℤ).

One can also consider functors T: Rg → Gr on the category Rg of nonunital rings instead of Gr. The role of the free group is then taken over by the free ring: for X a set F X is the ring of polynomials without constant term in the non-commuting variables x ∈ X, with coefficients in ℤ. Analogously one then considers: free simplicial rings, rings of n-spheres in a simplicial ring, etc. Then too one has a theory of derived functors for functors from Rg to Gr. More generally, the procedure is applicable to functors T: A → Set_{∗}, where A is a category of triple algebras and Set_{∗} the category of pointed sets. In fact this is how the theory is presented in [9].

6. Derived Functors of the Integral Completion

We will consider the derived functor L_{n}C of C: Gr → Gr, the integral completion functor as described in Section 2. Let H be a group. Note that for any simplicial set X the simplicial group GX is free. It follows that the simplicial group GW̄ H is a free resolution of H. The groups (L_{n}C)(H) are therefore the homotopy groups of CGW̄ H, a simplicial group which has as classifying complex the integral completion of W̄ H (cf. Section 3):

ℤ_{∞}W̄ H = W̄ CGW̄ H.

Hence

(L_{n}C)(H) = π_{n}(CGW̄ H) ≅ π_{n+1}(W̄ CGW̄ H) = π_{n+1}(ℤ_{∞}W̄ H).

In the special case H = GL(R) with R a unital ring, we obtain Lemma. For each n ⩾ 0 there is a canonical isomorphism

(L_{n}C)(GL(R)) ≅ π_{n+1}(ℤ_{∞}W̄ GL(R)) ≅ K_{n+1}(R).

7. Derived Functors of GL

In [9] higher K-functors were defined as derived functors of GL: Rg → Gr by the formula

K′_{n}(R) = L_{n−2}GL (n ⩾ 3)

and K′_{1} and K′_{2} are then defined by the exactness of

1 → K′_{2} → L_{0}GL → GL → K′_{1} → 1.

Since St(F X) ≅ GL(F X) for free rings F X, we have L_{0}GL = L_{0} St, where St denotes the Steinberg group. It is easily seen that L_{0} St = St, i.e. St is a right exact functor, see [10] for details. Hence the exact sequence above becomes

1 → K′_{2} → St → GL → K′_{1} → 1,

which shows that the functors K′_{1} and K′_{2} coincide with the classical ones. It will be proven in Section 8 that the functors as defined above are isomorphic to the functors K_{n} as defined in Section 1.

Remark. The groups K′_{n}(R) defined in this section coincide with the groups K_{n}(R) as defined by Gersten [4], since Gersten uses the cotriple resolution of R for their definition, which is simply one of possible resolutions of R. In [17] Swan proved that his functor K_{n}, which he defined in [16], coincides with Gersten’s.

8. Comparison of Both K-theories

In this section we prove the main theorem.

Theorem. Let R ∈ Rg. Then for all n ⩾ 2,

K′_{n}(R) ≅ K_{n}(R^{+}, R).

Let R be a nonunital ring. Form the simplicial ring F R by applying the free resolution functor. Adjoining a unit in every dimension of F R we get a split homomorphism of simplicial unital rings,

[figure: (F R)^{+}; ℤ ,]

where the right-hand side is interpreted as a constant simplicial ring.

The adjunction of G and W̄ induces a cotriple on sGr. By first applying this cotriple resolution to the diagram E((F R)^{+}) → E(ℤ), and then the integral completion functor C, we define the two following bisimplicial groups together with a split homomorphism.

Q_{pq}^{+} = (C(GW̄)^{q+1}E((F R)^{+}))_{p} ⟶ Z_{pq} = (C(GW̄)^{q+1}E(ℤ))_{p}

Let Q denote the fibre (or kernel) of this map. Taking homotopy in each row of these bisimplicial groups, we can consider the usual long exact sequence of a fibration. Since the homomorphism splits, this sequence of simplicial groups degenerates into split short exact sequences

[figure: 1; π_{q}^{h}Q; π_{q}^{h}Q^{+}; π_{q}^{h}Z; 1.]

Again computing homotopy, each of these fibrations induces a long exact sequence, which too is split. Hence for every p and q we have a split short exact sequence

[figure: 1; π_{p}^{v}π_{q}^{h}Q; π_{p}^{v}π_{q}^{h}Q^{+}; π_{p}^{v}π_{q}^{h}Z; 1.]

Repeating this process, but now taking vertical homotopy groups first, we have

[figure: 1; π_{q}^{h}π_{p}^{v}Q; π_{q}^{h}π_{p}^{v}Q^{+}; π_{q}^{h}π_{p}^{v}Z; 1.]

Note that for each p, the simplicial group Q_{p∗}^{+} is the result of an application of the functor C to a free Gr-resolution of E((F_{p}R)^{+}). Hence,

π_{q}^{h}π_{p}^{v}Q^{+} = (L_{p}C)(E((F_{q}R)^{+})) = π_{p+1}ℤ_{∞}W̄ E((F_{q}R)^{+}).

Using the description of the Quillen K-groups from Section 6, it follows that for all p and q we have a short exact sequence

0 ⟶ π_{q}^{h}π_{p}^{v}Q ⟶ K_{p+1}((F_{q}R)^{+}) ⟶ K_{p+1}(ℤ) ⟶ 0.

Using Gersten’s theorem on the K-theory of free rings [5], we find that for all p and q

π_{q}^{h}π_{p}^{v}Q = 0,

and hence, e.g. by Quillen’s spectral sequence [15] we also have

π_{p}^{v}π_{q}^{h}Q = 0.

By letting p = 0 and applying this formula to the relevant split exact sequence above, we obtain an isomorphism

π_{0}^{v}π_{q}^{h}Q^{+} →^{∼} π_{0}^{v}π_{q}^{h}Z.

We will now determine these homotopy groups. Note that the functorial homomorphisms GW̄ (H) → H are homotopy equivalences for any simplicial group H. From this it follows that all the maps d_{i}^{v}: Q_{p,q+1}^{+} → Q_{p,q}^{+} are homotopy equivalences too, since the functor C is applied dimension-wise. Hence,

π_{p}^{v}π_{q}^{h}Q^{+} = 0 for p > 0

and

π_{0}^{v}π_{q}^{h}Q^{+} = π_{q}CGW̄ E((F R)^{+}) = π_{q+1}ℤ_{∞}W̄ E((F R)^{+}).

Performing the same calculation for Z and substituting this into the isomorphism above, we find the following proposition, which can be seen as a generalization of Gersten’s theorem to include some types of free simplicial rings:

Proposition 2. For each q ⩾ 0 we have

π_{q}ℤ_{∞}W̄ E((F R)^{+}) ≅ π_{q}ℤ_{∞}W̄ E(ℤ).

Let F R once again be the cotriple resolution of R in Rg. Then F_{0}R → R induces a surjective homomorphism E(F R) → E(R). Its kernel is denoted by HR. From the long exact sequence of the fibration HR → E(F R) → E(R) it follows that π_{n}HR = K′_{n+2}(R) for all n > 0 and π_{0}HR = Ker(St(R) → E(R)). Hence we have

π_{n}HR = K′_{n+2}(R) for all n ⩾ 0.

Note that the simplicial group HR is also the kernel of the homomorphism E((F R)^{+}) → E(R^{+}). To see this, apply the snake lemma to the following diagram having split exact rows:

\xymatrix{ & HR \ar[d] \\ 1 \ar[r] & E(F R) \ar[d] \ar[r] & E(((F R)^{+})) \ar[d] \ar[r] & E(ℤ) \ar[d]_{id} \ar[r] & 1 \\ 1 \ar[r] & E(R) \ar[r] & E(R^{+}) \ar[r] & E(ℤ) \ar[r] & 1}

This gives the desired identification of the simplicial group HR with the kernel of E((F R)^{+}) → E(R^{+}).

The point of introducing this simplicial group HR is that the fibration

1 → HR → E((F R)^{+}) → E(R^{+}) → 1

has good behavior under application of the composite functor ℤ_{∞}W̄. We need to verify the requirements of Property I3. Taking classifying complexes we obtain the following fibration in sSet:

W̄ HR → W̄ E((F R)^{+}) → W̄ E(R^{+}).

Recall that for any simplicial group G the usual action of π_{1}(W̄ G) = π_{0}(G) on π_{i}(W̄ G) is the action induced by dimension-wise conjugation of G by degenerate elements originating from G_{0}.

Lemma. There is a natural isomorphism GL(Z_{i}(F R)) → Z_{i}(GL(F R)) which is induced by the projection maps Z_{i}(F R) → F_{i}R.

Proof. The simplicial kernel Z_{i}(F R) is an inverse limit of a suitable system of rings and the functor GL preserves inverse limits. By inspecting the pullback diagrams, it is clear that Z_{i}(F R) is completely determined by the projection maps p_{∗}: Z_{i}(F R) → F_{i}R. We have a homomorphism (GL(p_{1}), …, GL(p_{i+2})) : GL(Z_{i}(F R)) → (GL(F_{i}R))^{i+2}. Its image is precisely Z_{i}(GL(F R)). □

Proposition 3. The usual action of π_{1}(W̄ E((F R)^{+})) on π_{i}(W̄ E((F R)^{+})) is trivial for i > 1.

Proof. For i > 0 we have the isomorphism

π_{i}(E(F R)^{+}) → Z_{i}(E((F R)^{+}))/dE((F R)^{+})_{i+1}

which commutes with the action of E((F R)^{+}) (see Section 5.) Using this we have that

Z_{i}(E((F R)^{+})) ≅ Z_{i}(E(F R) ⋊ E(ℤ)) = Z_{i}(GL(F R) ⋊ E(ℤ))
≅ Z_{i}(GL(F R)) ⋊ E(ℤ) ≅ GL(Z_{i}(F R) ⋊ E(ℤ)).

The image of the group dE((F R)^{+})_{i+1} under this composition is the group E(Z_{i}(F R)) ⋊ E(ℤ). To see this, note that d: (F R)_{i+1}^{+} → Z_{i}((F R)^{+}) is onto and that the functor E preserves such maps.

The images of the elements of E((F_{0}R)^{+}) ⊆ Z_{i}(E((F R)^{+})) are contained in E(Z_{i}(F R)) ⋊ E(ℤ). Hence the action on Z_{i}(E((F R)^{+})) corresponds to an action which becomes trivial when passing to quotients. It follows that the action of E((F R)^{+}) on π_{i+1}(W̄ E((F R)^{+})) is trivial. □

Corollary. The action of π_{1}(W̄ E((F R)^{+})) on π_{i}(W̄ HR) is trivial for all i.

Proof. For i > 1 this is a consequence of the previous lemma. For i = 1 the action is also trivial because π_{1}(W̄ HR) maps isomorphically onto the kernel of π_{1}(W̄ E((F R)^{+})) → E(R^{+}), which identifies with the central extension St(R)⋊ E(ℤ) → E(R) ⋊ E(ℤ). □

Corollary. The action of π_{1}(W̄ HR) on π_{i}(W̄ HR) is trivial.

Proof. The homomorphism π_{1}(W̄ HR) → π_{1}(W̄ E((F R)^{+})) is injective by the long exact sequence of a fibration. Now use the previous corollary. □

Proposition 4. The induced map ℤ_{∞}W̄ E((F R)^{+}) → ℤ_{∞}W̄ E(R^{+}) is a fibration and its fibre is homotopy equivalent to W̄ HR.

Proof. From Property I3 and the above corollary it follows that the map ℤ_{∞}W̄ E((F R)^{+}) → ℤ_{∞}W̄ E(R^{+}) is a fibration and that the canonical map from ℤ_{∞}W̄ HR to the fibre is a homotopy equivalence. From Property I2 and this corollary it follows that the natural map i: W̄ HR → ℤ_{∞}W̄ HR is a weak homotopy equivalence. Since all simplicial sets involved are Kan complexes, this map is a fortiori a homotopy equivalence. □

Now we can finish the proof of the theorem.

Proof. Let X be the fibre of the map ℤ_{∞}W̄ E(R^{+}) → ℤ_{∞}W̄ E(ℤ). We have the following diagram which consists horizontally and vertically of long exact sequences of suitable fibrations:

\xymatrix{π_{p+1}(X) \ar[d] \ar[r]^{∼} & π_{p}(⋯) \ar[d]^{≀} \ar[r] & 0 \ar[d] \ar[r] & π_{p}(X) \ar[d] \\ π_{p+1}ℤ_{∞}W̄ E(R^{+}) \ar[d] \ar[r] & π_{p}W̄ HR \ar[d] \ar[r] & π_{p}ℤ_{∞}W̄ E((F R)^{+}) \ar[d]^{≀} \ar[r] & π_{p}ℤ_{∞}W̄ E(R^{+}) \ar[d] \\ π_{p+1}ℤ_{∞}W̄ E(ℤ) \ar[r] & 0 \ar[r] & π_{p}ℤ_{∞}W̄ E(ℤ) \ar[r]^{=} & π_{p}ℤ_{∞}W̄ E(ℤ)}

The map in the third column is an isomorphism by Proposition 2. The other relations are evident from the diagram. The group K_{p+1}(R^{+}, R) equals π_{p+1}(X) for p ⩾ 1, which is in turn isomorphic to π_{p}(W̄ H(R)) by the above diagram. The latter group equals K′_{p+1}(R) for p ⩾ 1. □

Final remark. An alternative way to prove the equivalence of both algebraic K-theories is by showing that the Quillen K-theory satisfies the axioms for multirelative K-theory given in [12]. To do so, the Quillen K-theory has to be extended to include multirelative groups. The main concern is then to extend long exact sequences in such a way that they include K_{0}-groups as well.

References

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10. F. Keune, The relativization of K_{2}. J. Algebra 54(1978), 159–177.

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13. J. -L. Loday, K-théorie algébrique et représentations de groupes. Ann. Sci. École Norm. Sup. (4) 9(1976), No. 3, 309–377.

14. J. P. May, Simplicial objects in algebraic topology. Van Nostrand Mathematical Studies, No. 11. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967.

15. D. Quillen, Spectral sequences of a double simplicial group. Topology 5(1966), 155–158.

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17. R. G. Swan, Some relations between higher K-functors. J. Algebra 21(1972), 113–136.

(Received 30.04.2002)

Authors’ address: Department of Mathematics Katholieke Universiteit Nijmegen Postbus 9010, 6500 GL Nijmegen The Netherlands E-mail: ebeling@sci.kun.nl

keune@sci.kun.nl