K-THEORY HOMOLOGY OF SPACES

ERIK KJÆR PEDERSEN AND CHARLES A. WEIBEL

Abstract. Let KR be a nonconnective spectrum whose homotopy groups give the algebraic K-theory of the ring R. We give a description of the associated homology theory KR_{∗}(X) associated to KR. We also show that the various constructions of KR in the literature are homotopy equivalent, and so give the same homology theory

0. Introduction

There is a generalized homology theory E_{∗} associated to every spectrum E, namely

E_{n}(X) = lim_{i→∞} π_{n+i}(E_{i} ∧ X)

In particular this is true if E is the nonconnective K-theory spectrum KR of a ring R. In this paper, we give a geometric interpretation of KR_{n}(X) for n ≤ 0 ( and a new interpretation for n > 0)

Let X be a subcomplex of S^{n}, and form the open cone O(X) on X inside R^{n+1} (which is the open cone on S^{n}). There is a category C = C_{O(X)}(R), whose objects are based free R-modules parameterized in a locally finite way by O(X), and whose morphisms are linear maps moving the bases a bounded amount. (Compare with Quinn’s geometric R-modules in [13]). The group K_{1}(C) is generated by the automorphisms in C, with well-known relations [1]; our main theorem yields the formula:

KR_{0}(X) ≅ K_{1}(C)

When R = ℤπ, these groups appear as obstruction groups of bounded (or thin) h-cobordisms parameterized by O(X) with constant fundamental group π. (See [8] for a relatively elementary proof of this. This, of course, is in accordance with the basic results of Chapman [2, 3] and Quinn [11, 12]).

The negative KR-homology groups of X can be obtained from the formula KR_{−n}(X) = KR_{0}(S^{n}X). The positive KR-homology groups are Quillen’s higher K-groups of C:

KR_{n}(X) ≅ K_{n+1}(C) for n ≥ 0.

To make the proofs easier, it turns out to be better to generalize the above discussion, replacing the category F_{R} of finitely generated based free R-modules by any additive category A. We impose the \underline{semisimple} exact structure of [1], i. e., declaring every short exact sequence split, in order to compute the K-theory of A; this makes Bass’ groups K_{1}(A) the same as Quillen’s by [17].

The first author wants to thank the Sonderforschungsbereich für Geometrie und Analysis in Göttingen for a most agreeable year.

Partially supported by NSF grant MCS-8301686.

This being said, we can generalize the spectrum KR (for A = F_{R}) to the nonconnective spectrum KA constructed in [9] and ask about KA-homology. There is a category C_{O(X)}(A), generalizing C_{O(X)}(R) and described in §1 below. In §2 we make this construction functorial in X. Our Main Theorem is carefully stated in §3 and proved in §4:

Main Theorem. The KA-homology of X is naturally isomorphic to the algebraic K-theory of the idempotent completion C^{∧} of C = C_{O(X)}(A), with a degree shift:

KA_{∗−1}(X) ≅ K_{∗}(C_{O(X)}(A)^{∧}).

Note that K_{∗}(C^{∧}) = K_{∗}(C) for ∗ ≥ 1 but that K_{0}(C^{∧}) ≠ K_{0}(C) in general. For example if C = F_{R}, then K_{0}(C) is ℤ (or a quotient of ℤ) but K_{0}(C^{∧}) = K_{0}(R). If ∗ < 0 and C ≠ C^{∧}, the groups K_{∗}(C) are not even defined. As in [9], the key technical step is an application of Thomason’s double mapping cylinder construction from [15].

The knowledgeable reader will wonder about the relationship between the spectrum KA and other nonconnective spectra in the literature. We pin down this loose end in §6. When A = F_{R} we show that our spectrum KF_{R} is homotopy equivalent to the Gersten-Wagoner spectrum of [4, 16]. For general A, we prove that our spectrum KA is homotopy equivalent to Karoubi’s spectrum [6]. Actually, the discussion in [6] only mentions K_{0} and K_{1}, and not spectra, since it was written before higher K-theory emerged. We devote §5 to showing that Karoubi’s prescription in [6] actually gives an infinite loop spectrum. The authors want to thank W. Vogell for pointing out an error in an earlier draft of this paper.

1. The functor C

In this section we generalize the functor C_{i}(A) considered in [9] to a functor in two variables. We have already described how C_{i}(−) is an endofunctor of the category of filtered additive categories [9]. We remind the reader that a \underline{filtered} additive category is an additive category A, that comes with a filtration of the homsets Hom(A, B)

0 ⊆ F_{0} Hom(A, B) ⊆ F_{1} Hom(A, B) ⊆ … ⊂ Hom(A, B)

such that a) F_{i} Hom(A, B) is a subgroup and Hom(A, B) = ∪F_{i} Hom(A, B).

b) F_{0} Hom(A, A) contains 0_{A} and 1_{A} for each A, all coherence isomorphisms of A, all projections A ⊕ B → A and all inclusions A → A ⊕ B.

c) if f ∈ F_{i} Hom(A, B) and g ∈ F_{j} Hom(B, C) then g ∘ f ∈ F_{i+j} Hom(A, C). Note that any additive category may be endowed with “discrete filtration”, in which

F_{0} Hom(A, B) = Hom(A, B)

for every A, B.

Thinking of the lower index i as the metric space ℤ^{i} or ℝ^{i}, we shall now turn C into a functor of that variable. (See [9, Remark 1,2,3]).

Definition 1.1. Let X be a metric space and A a filtered additive category. We then define the filtered category C_{X}(A) as follows:

1) An object A of C_{X}(A) is a collection of objects A(x) of A, one for each x ∈ X, satisfying the condition that for each ball B ⊂ X, A(x) ≠ 0 for only finitely many x ∈ B.

2) A morphism ϕ : A → B is a collection of morphisms ϕ_{y}^{x} : A(x) → B(y) in A such that there exists r depending only on ϕ so that

(a) ϕ_{y}^{x} = 0 for d(x, y) > r

(b) all ϕ_{y}^{x} are in F_{r} Hom(A(x), B(y)) (We then say that ϕ has filtration degree ≤ r.

Composition of ϕ : A → B with ψ : B → C is given by (ψϕ)_{z}^{x} = ∑_{y∈X} ψ_{z}^{y}ϕ_{y}^{x}. Notice that the sum makes sense because the category is additive and because the sum will always be finite. The category C_{ℤ^{i}}(A) is the category C_{i}(A) of [9].

We now introduce the category M of metric spaces and proper, eventually Lipschitz maps.

Definition 1.2. The category M has objects metric spaces X. A morphism f : X → Y must be both proper and eventually Lipschitz. We remind the reader that a map f : X → Y is \underline{Lipschitz} if there exists a number k ∈ ℝ_{+} such that

d(f(x), f(y)) ≤ kd(x, y)

We say that f is \underline{eventually Lipschitz} if there exist r and k, only depending on f, so that

∀x, y ∈ X, ∀s ∈ ℝ_{+} : if s > r and d(x, y) < s then d(f(x), f(y)) < k · s.

Finally, we call f \underline{proper} if the inverse image of a bounded set is bounded.

Example 1.3. One should note that maps in M are not necessarily continuous, but any jumps allowed must be universally bounded. For example the map ℝ → ℤ sending a real number x to the greatest integer smaller than x is a map in M.

Given a proper eventually Lipschitz map f : X → Y we obtain a functor f_{∗} : C_{X}(A) → C_{Y} (A) by defining (f_{∗}(A))_{y} = ⊕_{z∈f^{−1}(y)}A(z) for objects A in C_{X}(A). Since the inverse of a bounded set is bounded, there are only finitely many nonzero modules in a ball in Y, and f_{∗}(A)) is well-defined. On morphisms f_{∗} is induced by the identity. The eventually Lipschitz conditions on f ensures that we indeed do get morphisms in the category C_{Y} (A). Hence C_{−}(A) is a functor from M to (semisimple filtered) additive categories.

Lemma 1.4. Let X and Y be metric spaces and give X × Y the max metric,

d_{X×Y} ((x_{1}, y_{1}), (x_{2}, y_{2})) = max(d_{X}(x_{1}, x_{2}), d_{Y} (y_{1}, y_{2})),

then C_{X×Y} (A) = C_{X}(C_{Y} (A)).

Proof. This is where we use that C takes values in \underline{filtered} categories. The internal filtration degree will control distances in the Y component, while the external filtration degree will control distances in the X component and thus the max distance will be controlled. □

Remark. Note that the isomorphism class in M is not affected if we change the metric to a proper Lipschitz equivalent metric, i. e. a metric so that the identity map is a proper Lipschitz equivalence both ways. Therefore Lemma 1.4 remains true up to natural equivalence if the max metric is replaced by the usual product metric

d_{X×Y} ((x_{1}, x_{2}), (y_{1}, y_{2})) = \sqrt{d_{X}(x_{1}, x_{2})^{2} + d_{Y} (y_{1}, y_{2})^{2}}

The concept \underline{homotopy} is introduced in the category M in a standard fashion using the inclusions X = X × {0} → X × I, X = X × {1} → X × I and the projections X × I → X Since X × I has the max metric, these are maps in the category M. Note that the inclusion ℤ → ℝ is a homotopy equivalence in M, with homotopy inverse the greatest integer function of example 1.3.

Lemma 1.5. A compact metric space X is homotopy equivalent to a point in M and hence C_{X}(A) is equivalent to the category A.

Proof. Since maps are not required to be continuous and a compact metric space is globally bounded, any map X × I → X which is constant on the top and the identity on the bottom will be a contracting homotopy. The second assertion follows from the evident fact that C_{X}(A) = A when X is a point. □

Proposition 1.6. The functor C_{−}(A) is homotopy invariant. That is, for each metric space X, the inclusion X ⊂ X × I as X × 0 and the projection X × I → X induce an equivalence of categories

C_{X}(A) → C_{X×I}(A).

Proof. By 1.4 and 1.5, C_{X×I}(A) = C_{I}(C_{X}(A)) is homotopic to C_{X}(A). □

2. Open Cones

In this section we construct an open cone functor O(X) from finite PL complexes to M, so that C_{O(X)}(A) depends functorially on X in a homotopy invariant way. To fix notation let S^{0} = {−1, 1} ⊂ ℝ. Then the n-sphere is the join S^{n} = S^{0} ∗ S^{0} ∗ ⋯ ∗ S^{0} as a sub PL-complex of ℝ^{n+1}. We shall be considering the category of finite sub PL-complexes of S^{∞} (PL complexes that are subcomplexes of some S^{n}) and PL morphisms. We denote this category PL. This is essentially the category of finite PL-complexes, since any such may be embedded in some S^{n}, but we need the way the complex sits in S^{n} as part of the structure. We think of ℝ^{n+1} as a metric space using the max metric

d(\underline{x}, \underline{y}) = max |x_{i} − y_{i}|.

This induces a metric on the n-sphere and hence on any subcomplex.

We now construct a functor O from PL to M.

Definition 2.1. O sends an object X ⊂ S^{n} to O(X) = {t · x ∈ ℝ^{n+1}|t ∈ [0, ∞), x ∈ X} with metric induced from ℝ^{n+1}. For example O(S^{n}) = ℝ^{n+1}. On morphisms f : X → Y we extend to O(f) : O(X) → O(Y) by linearity: f(t · x) = t · f(x). One checks easily that O(f) is proper and Lipschitz (and therefore eventually Lipschitz) using the following well-known

Lemma 2.2. A PL-map f : X → Y between finite complexes is Lipschitz.

Proof. Triangulate so that f is linear on each simplex. Since there are only finitely many simplices, f will be Lipschitz. □

Remark 2.3. For a PL complex X, O(X) does not really depend on the embedding X ⊂ S^{n}, since a PL homeomorphism X_{1} ⊂ S^{n} to X_{2} ⊂ S^{m} will induce a proper Lipschitz homeomorphism from O(X_{1}) → O(X_{2}).

Lemma 2.4. Let X and Y be two PL complexes and let ∗ denote the join. Then

O(X ∗ Y) ≅ O(X) × O(Y).

In particular, O(ΣX) = O(X ∗ S^{0}) ≅ O(X) × ℝ and O(CX) ≅ O(X) × [0, ∞).

Proof. Embed X ⊂ S^{n} and Y ⊂ S^{m} so X ∗ Y ⊂ S^{n} ∗ S^{m} = S^{n+m+1}. A point in O(X ∗ Y) is s · (t · x + (1 − t) · y) = stx + s(1 − t)y and stx lies in O(X) ⊂ ℝ^{n+1} × 0, whereas s(1 − t)y lies in O(Y) ⊂ 0 × ℝ^{m+1}. The last sentence follows from the identities ΣX = X ∗ S^{0}, CX = X ∗ (point), O(S^{0}) = ℝ and O(point) = [0, ∞). □

Remark 2.5. The functor O is \underline{not} homotopy invariant. We only obtain homotopy invariance after passing to K-theory.

3. The Main Theorem

So far we have constructed functors

O : PL → M and

C_{−}(A) : M → (semisimple) additive categories

For our main theorem, we need a functor K_{∗} from additive categories to graded abelian groups. For n ≥ 1, there is no problem: given an additive category A we take K_{n}A = π_{n}(ΩBQA) as in [10], using the semisimple exact structure (in which all exact sequences split). However, unless A is idempotent complete, the groups K_{0}A may be wrong for our purposes, and A’s negative K-groups will not even be defined.

For example consider the category F_{R} of finitely generated based free R-modules. when R is a group ring, we know that the geometrically interesting group is not K_{0}(F_{R}) = ℤ, but rather K_{0}(R), which measures projective modules.

To handle this problem, we pass to the idempotent completion A^{∧} of A. This provides the correct group K_{0}(A^{∧}) and does not change the higher groups, since K_{n}(A) = K_{n}(A^{∧}) for n ≥ 1 For example F_{R}^{∧} is equivalent to the category of finitely generated projective R-modules.

Scholium. A^{∧} inherits the structure of a filtered additive category from A. The objects of A^{∧} are pairs (A, p), where p : A → A is idempotent. A morphism ϕ from (A_{1}, ϕ_{1}) to (A_{2}.ϕ_{2}) is an A-morphism ϕ : A_{1} → A_{2} with ϕ = p_{2}ϕp_{1} . The filtration degree of ϕ is the smallest d such that ϕ = p_{2}fp_{1} for some f ∈ F_{d} Hom(A_{1}, A_{2}) satisfying fp_{1} = p_{2}f. This filtration should have been stated explicitly in [9, 1.4].

If A is idempotent complete, the negative K-groups of A were defined by Karoubi in [6], and agree with the definition in [9] (as we shall see in §6 below). If A = F_{R}, they agree with Bass’ negative K-groups K_{n}(R).

The construction in [9] actually gives us slightly more information. If A is idempotent complete, it yields a nonconnective infinite loop spectrum KA, and the homotopy groups of KA are the groups K_{∗}A above.

Now associated to any spectrum such as KA is a reduced homology theory KA_{∗}. It is defined by

KA_{n}(X) = lim π_{n+i}((KA)_{n} ∧ X).

The coefficients of this homology theory are the groups

KA_{n}(S^{0}) = π_{n}(KA) = K_{n}(A).

We can now state our main theorem.

Theorem 3.1. If A is an idempotent complete additive category, the functor from PL to graded abelian groups sending X to K_{∗}(C_{O(X)}(A)^{∧}) is the KA_{∗}-homology theory of X, with a degree shift:

K_{∗}(C_{O(X)}(A)^{∧}) ≅ KA_{∗}(X).

Remark 3.2. This theorem had previously been known for spheres. For X = S^{i}, O(S^{i}) = ℝ^{i+1}, which is homotopy equivalent to ℤ^{i+1} in M, so therefore by 1.4

K_{∗}(C_{ℝ^{i}}(A)^{∧}) = K_{∗}(C_{ℤ^{i}}(A)^{∧}),

which was shown in [9] to equal K_{∗−i−1}(A^{∧}). The category C_{ℤ^{i+1}}(A) was first studied in [8], where A was the category of finitely generated R-modules. There it was shown that

K_{1}(C_{ℤ^{i+1}}(R)) = K_{−i}(R)

which is equal to KR_{i}(S^{0}) = KR_{0}(S^{i}), thus agreeing with our main theorem.

4. Proof of main theorem

Define the functor f as the composite

PL ⟶^{O} M ⟶^{C_{−}(A)} filtered add. categ. ⟶^{□} add. cat. ⟶^{∧}

idempotent complete add. cat. ⟶^{Ω^{∞}K} Top. spaces

Here □ is the forgetful functor, ^{∧} is idempotent completion and Ω^{∞}K is the zeroth space of the infinite loop spectrum K giving the K-theory of the category, either ΩBQ or the result of applying an infinite loop machine to the symmetric monoidal category of isomorphisms (see Thomason for a very functorial construction [15]). That is :

f(X) = Ω^{∞}K(C_{O(X)}(A)^{∧}).

Lemma 4.1. If X is a cone, X = CK, then f(X) is contractible.

Proof. C_{O(CK)}(A) = C_{[0,∞)×O(K)}(A) = C_{1}(C_{O(K)}(A)) in the notation of [9]. It was proven in [9, (3.1)] that Ω^{∞}KC_{+}(A) is contractible for an arbitrary category, A, and the argument given there applies verbatim to show that

f(X) = Ω^{∞}K((C_{+}(C_{O(K)}A)^{∧})

is also contractible.

□

Proposition 4.2. For each X, Ωf(ΣX) is homotopy equivalent to f(X). In particular f(X) is an infinite loop space.

Proof. By 1.6 and 2.4 we have

C_{O(ΣX)} = C_{O(X)×ℝ} = C_{ℝ}(C_{O(X)}(A)) = C_{1}(C_{O(X)}(A)).

By [9, Theorem (3.2)], applied to the filtered additive category C_{O(X)}(A)^{∧}, we know that the loop space of Ω^{∞}KC_{O(ΣX)}(A) is homotopy equivalent to f(X). But by the cofinality theorem, the spaces Ω^{∞}K(C) and Ω^{∞}K(C^{∧}) have homotopy equivalent connected components, and hence homotopy equivalent loop spaces, for any C. □

Theorem 4.3. The functor f sends cofibrations to fibrations.

Before proving this result , let us draw a quick consequence. It follows from Lemma 4.1 that f is homotopy invariant, since X → X×I → CX is a cofibration. The homotopy groups of the spectrum {f(X), f(ΣX), f(Σ^{2}X), …} coincide with the groups K_{∗}(C_{O(X)}(A)^{∧}). These groups are homotopy invariants of X which vanish when X is contractible by 4.1. From 4.3 we immediately obtain

Corollary 4.4. The functor from PL to graded abelian groups sending X to K_{∗}(C_{O(X)}(A)^{∧}) is a reduced homology theory.

Proof of Theorem 4.3. The special case X ⊂ CX ⊂ ΣX follows from 4.1 and 4.2. To do the general case, we proceed in a manner very much like the proof of Theorem 3.4 in [9]. Consider a cofibration A ⊂ X → X ∪ CA. Then we get

O(A) ⊂ O(X) ⊂ O(X) ∪_{O(A)} O(CA)

and a diagram

\xymatrix{C_{O(A)}(A) \ar[d] \ar[r] & C_{O(CA)}(A) \ar[d] \\ C_{O(X)}(A) \ar[r] & C_{O(X∪CA)}(A)}

By 1.6 and 2.4 we have O(CA) = O(A) × [o, ∞) and

C_{O(CA)}(A) = C_{[0,∞)}(C_{O(A)}(A)) = C_{+}(C_{O(A)}(A)).

To simplify notation, let us write C =X for the category of isomorphisms in C_{O(X)}(A)^{∧}. Using

the “double mapping cylinder” pushout construction P = of Thomason [15, (5.1)], we get

\xymatrix{C =A \ar[d] \ar[r] & C =CA \ar[d] \\ _{C =X} \ar[r] & P = \ar[dr]_{Σ} \\  &  & _{C =X ∪ CA}}

We wish to show that Σ induces a homotopy equivalence on certain components, i. e. that Σ induces a π_{0}-monomorphism and π_{i}-isomorphism. That π_{0} behaves correctly is then obtained by applying the argument to the suspension of this cofibration. It is therefore enough to show that for every object Y of C_{O(X∪CA)}(A), considered as an object of C =X ∪ CA,

that the category Y ↓ Σ is a contractible category. At this point we follow the proof of [9, Theorem (3.4)] very closely. We use the bound d to filter Y ↓ Σ as the increasing union of subcategories fil_{d} and show each of these has an initial object ∗_{d}.

Fil_{d} is the full subcategory of all iso’s α : Y → Σ(B^{A}, B^{X}, A^{+}), where B^{X} is an object of C =X, B^{A} and object of C =A and A^{+} an object of C =CA. We define Y_{d}^{A}, Y_{d}^{X} and Y_{d}^{+} in C =A,

C =X and C =CA as follows: Let

N_{d} = {x ∈ O(X ∪ CA)|∃y ∈ O(A) : d(x, y) ≤ d}.

Since A ⊆ is a cofibration it is easy to see that N_{d} is proper Lipschitz homotopy equivalent to O(A). Choose some proper Lipschitz homotopy equivalence h : N_{d} → O(A) and proceed as follows:

Y_{d}^{X}(x) = \begin{cases}Y (x) & for x ∈ O(X) − N_{d} \\ 0 & otherwise \end{cases}

Y_{d}^{+}(x) = \begin{cases}Y (x) & for x ∈ O(A) × (d, ∞) = O(CA) − N_{d} \\ 0 & otherwise \end{cases}

and

Y_{d}^{A} = h_{∗}(Y_{d}^{A}) where
Y_{d}^{A} = \begin{cases}Y (x) & for x ∈ N_{d} \\ 0 & otherwise \end{cases}

There is now an obvious isomorphism

σ : Y ≅ Y_{d}^{X} ⊕ Y_{d}^{A} ⊕ Y_{d}^{+} = Σ(Y_{d}^{A}, Y_{d}^{X}, Y_{d}^{+})

bounded by d (essentially the identity) and we may proceed to prove this is an initial object in Fil_{d} exactly as in the proof of [9, Theorem 3.4]. □

We finish off the proof of the main theorem as follows:

Proof of main theorem 3.1. We need to identify the homology theory of 4.4 as the homology theory associated to th spectrum K(A). It was proven by Thomas Gunnarson [5] that when a homotopy functor f sends cofibrations to fibrations and contractible spaces to contractible spaces, then the homology theory obtained by applying homotopy groups has as its representing spectrum {f(S^{0}), f(S^{1}), f(S^{2}), …}. This is a well-known fact when the homology theory is connective - see e. g. [18, theorem 1.14], [14] or [7] - but in the general case we need to use [5] Now we have by Lemma 2.4 that

f(S^{i}) = Ω^{∞}K(C_{i+1}(A)^{∧}).

The space f(S^{i}) is therefore the (i+1)^{st}-space of the spectrum K(A) constructed in Theorem B of [9]. The representing spectrum for the homology theory of 4.4 is therefore Ω^{−1}K(A), and the main theorem follows. □

We are finished, except we need to show that the delooping given in [9] agrees with the other deloopings in the literature. This is the subject of the final 2 sections.

5. Karoubi’s nonconnective spectra

In this section, we follow Karoubi’s ideas in [6] and construct nonconnective K-theory spectra, whose negative homotopy groups are the negative K-groups defined by Karoubi in op. cit. We then show that these agree with the spectra constructed in [9] (and in special cases, in [4], [16]). More explicitly, but also more technically, we show that C_{1}(A) → C_{+}(A)/A induces an isomorphism on Quillen K-theory. We are indebted to Karoubi, who suggested this possibility to us in 1983.

We shall use Karoubi’s delooping construction from [6], so we begin by recalling the construction. Let A be a full subcategory of an additive (hence semisimple exact) category U. We shall use the notation that letters A − F (resp U − Z) denote the objects of A (resp. U), and that U = E_{α} ⊕ U_{α} means an internal direct sum decomposition of U with E_{α} ∈ A. We say that U is A- \underline{filtered} if every object U has a family of decompositions \{U = E_{α} ⊕ U_{α}\} (called a \underline{filtration}) of U) satisfying the following axioms (cf. [6, pp. 114 ff.]):

(F1) For each U, the decompositions form a poset under the partial order that E_{α} ⊕ U_{α} ≤

E_{β} ⊕ U_{β} whenever U_{β} ⊆ U_{α} and E_{α} ⊆ E_{β}. (F2) Every map A → U factors A → E_{α} → E_{α} ⊕ U_{α} = U for some α. (F3) Every map U → A factors U = E_{α} ⊕ U_{α} → E_{α} → A for some α. (F4) For each U, V the filtration on U ⊕ V is equivalent to the sum of the filtrations

{U = E_{α} ⊕ U_{α}} and {V = F_{β} ⊕ V_{β}} i. e., to {U ⊕ V = E_{α} ⊕ F_{β}) ⊕ (U_{α} ⊕ V_{β})}. We shall assume each filtration is saturated in the sense that if U = E_{α} ⊕ U_{α} is in the

filtration and E_{α} = A ⊕ B in A, then U = A ⊕ (B ⊕ U_{α}) is also in the filtration. Finally, we say that U is \underline{flasque} if there is a functor ∞ : U → U and a natural transformation U^{∞} ≅ U ⊕ U^{∞} ([6, p. 147]). Our favorite selection of U is the following:

Example 5.1. The category C_{+}(A) of [9, (1.2.4)] is flasque and A-filtered. Objects of C_{+}(A) are sequences (A_{0}, A_{1}, …) of objects in A, and the morphisms are given by “bounded” matrices. A is the full subcategory of objects (A_{0}, 0, 0, …). The A-filtration on an object U = (A_{0}, A_{1}, …) contains the decompositions

U ≅ Fil_{n}(U) ⊕ (0, …, 0, A_{n+1}, …)
Fil_{n}(U) = A_{0} ⊕ … ⊕ A_{n} in A.

We proved that C_{+}(A) was flasque in [9, (1.3)], using the translation t(A_{0}, A_{1}, …) = (0, A_{0}, A_{1}, …) on C_{+}(A).

We now suppose given an A-filtered category U. Call a map U → V \underline{completely} \underline{continuous} (cc) if it factors through an object of A. Karoubi defined U/A to be the category with the same objects as U, but with Hom_{U/A}(U, V) = Hom_{U}(U, V)/{cc maps}.

Lemma 5.2. Suppose A is idempotent complete and that \overline{ϕ} : U → V is an isomorphism in U/A. Then there are decompositions U = E_{β} ⊕ U_{α} and V = F_{α} ⊕ V_{α} in the (saturated) filtrations and a U-isomorphism U_{α} ≅ V_{α} such that \overline{ϕ} is represented by U → U_{α} ≅ V_{α} → V.

Proof. Choose representatives ϕ, ψ for \overline{ϕ}, \overline{ϕ}^{−1}. Since 1_{U} − ψϕ is cc, there is a decomposition U = E_{α} ⊕ U_{β} with 1 = ψϕ on U_{β}. Replace U by U_{β} to assume 1_{U} = ψϕ. Similarly, write V = F ⊕ W with ϕψ = 1_{W}. Write ϕ = (ϵ, i) : U → F ⊕ W and ψ = (δ, j) : F ⊕ W → U.

Observe that ij = 1_{W} and that

(*) δϵ + ji = 1_{U}

Multiplying (*) by i, j and by (*) yields the equations iδϵ = 0, δϵj = 0 and (δϵ)^{2} = δϵ. Replacing ϵ by ϵδϵ makes (ϵδ)^{2} = ϵδ without affecting \overline{ϕ} or (*). Set F_{α} = ker(ϵδ) and V_{α} = (ϵδF) ⊕ W; F_{α} ∈ A because A is idempotent complete. The rest of the proof is straightforward, and left to the reader. □

Remark. The failure of this lemma when A is not idempotent complete is the Bass-Heller-Swan phenomenon. See [8, (1.16)].

Theorem 5.3. Let A be semisimple and idempotent complete. Then for every A-filtered category U the sequence

K^{Q}(A) → K^{Q}(U) → K^{Q}(U/A)

is a homotopy fibration, where K^{Q}(A) denotes the space whose homotopy groups give the algebraic K-theory of A.

Before proving this theorem, we draw our main conclusion. Suppose that in addition U is flasque; from additivity and the equation ∞ ≅ 1 + ∞ we conclude that K^{Q}(U) ≃ ∗. Since U/A shares the same objects as U, we conclude that K_{0}(U/A) = K_{0}(U) = 0. Finally, applying 5.3 to the diagram U → C_{+}(U) ← C_{+}(A) shows that K^{Q}(U/A) ≃ K^{Q}(C_{+}(A)/A). This proves:

Theorem/Definition 5.4. Let A be semisimple and idempotent complete. Choose a flasque, A-filtered category U and define SA to be U/A. Then K^{Q}(SA) is a connected space with ΩK^{Q}(SA) ≃ K^{Q}(A), and the homotopy type of K^{Q}(SA) is independent of the choice of U.

Our proof of Theorem 5.3 follows the proof of [9, (3.4)]. Let A =, U = and S = denote the

categories of isomorphisms of A, U and U/A. It is well-known that K^{Q}(A) is the group completion of BA =, and that K^{Q}(A), BA =^{−1}A = and Spt_{0}(A =) are homotopy equivalent. Let

Q = denote the double mapping cylinder of (0 ← A = → U =) given by Thomason in [15, (5.1)].

Thus objects of Q = are pairs (A, U), and a Q =-map from (A, U) to (B, V) is an equivalence

class of data (E, F, A ≅ E ⊕ B ⊕ F, F ⊕ U ≅ V). Thomason proves in [15, (5.2) and (5.5)] that Spt_{0}(A =) → Spt_{0}(U =) → Spt_{0}(Q =) is a homotopy fibration. Since U = → S = factors through

Q =, we see that 5.3 follows from [15, (2.3)] and the following result:

Proposition 5.5. The functor Σ : Q = → S = given by Σ(A, U) = A ⊕ U is a homotopy

equivalence when A is idempotent complete.

Proof. Fix an object S of S =; we will show that S ↓ Σ is a contractible category. The desired

result will the follow from Quillen’s Theorem A [10]. In order to do this, we need to thicken S ↓ Σ up a bit. Let S denote the category whose objects are tuples

α = (A, U, S ≅ D_{α} ⊕ S_{α}, U ≅ E_{α} ⊕ U_{α}, f_{α} : S_{α} ≅ U_{α})

where A ∈ A, U ∈ U, f_{α} is a U-isomorphism, and the direct sum decompositions belong to the A-filtrations of S and U. A map from α to

β = (B, V, S ≅ D_{β} ⊕ S_{β}, V ≅ F_{β} ⊕ V_{β}, f_{β} : S_{β} ≅ V_{β})

is just a map in Q = from (A, U) to (B, V), say given by

e = (E, F, A ≅ E ⊕ B ⊕ F, F ⊕ U ≅ V)

such that D_{α} ⊕ S_{α} ≥ D_{β} ⊕ S_{β}, (F ⊕ E_{α}) ⊕ U_{α} ≥ F_{β} ⊕ V_{β} in the filtrations of S and V, and such that there is a commutative square in U:

\xymatrix{S_{β} & ≅ & V_{β} \\ S_{α} \ar[u] & ≅ & U_{α} \ar[u]}

There is a natural functor pr : S → (S ↓ Σ) sending α to the object pr_{α} : S → S_{α} ≅ U_{α} → U → A ⊕ U = Σ(A, U) and e to itself. By Lemma 5.2, pr is onto. In fact pr is cofibered. We assert that pr is a homotopy equivalence, which follows from [10, p. 93] and the following: □

Sublemma 5.6. Given an isomorphism \overline{ϕ} : S → Σ(A, U) in U/A, the fiber category pr^{−1}(\overline{ϕ}) is a cofiltered poset, and hence contractible.

Proof. Set Φ = pr^{−1}(\overline{ϕ}); it is clear from the definition of S that Φ is a poset. We need only show that for each α, β in Φ there is a diagram α ← γ → β in Φ. To do this, we introduce some notation. Write α = (A, U, S ≅ D_{α} ⊕ S_{α}, U ≅ E_{α} ⊕ U_{α}, f_{α}) and β = (A, U, S ≅ D_{β} ⊕ S_{β}, U ≅ E_{β} ⊕ U_{β}, f_{β}). If D_{α} ⊕ S_{α} ≤ D_{γ} ⊕ S_{γ} we set D_{γα} = D_{γ} ∩ S_{α}, E_{γα} = f_{α}(D_{γα}) and α^{1} = (A, U, S ≅ D_{γ} ⊕ S_{γ}, U ≅ (E_{α} ⊕ E_{γα}) ⊕ f_{α}(S_{γ}), f_{α}|S_{γ} : S_{γ} ≅ f_{α}(S_{γ})). We shall refer to α^{1} as “α cut down to D_{γ} ⊕ S_{γ}”; note that there is a map α^{1} → α in Φ.

By axion (F1), there is a decomposition S ≅ D_{0} ⊕ S_{0} larger than both D_{α} ⊕ S_{α} and D_{β} ⊕ S_{β}. Cutting α and β down, we can assume that D_{α} = D_{β} = D_{0} and S_{α} = S_{β} = S_{0}. Now f_{α} − f_{β} : S_{0} → D is completely continuous, so after cutting α and β down further we can assume that f_{α} = f_{β}. Note that we still may have E_{α} ≠ E_{β}. Consider the maps

E_{α} → U → U_{β} ≅ S_{0} and E_{β} → U → U_{α} ≅ S_{0}.

By axiom (F2) there is a decomposition S ≅ D_{γ} ⊕ S_{γ} for which these two maps factor through D_{γ}. Let γ be α cut down to D_{γ} ⊕ S_{γ} ; evidently γ is also β cut down to D_{γ} ⊕ S_{γ}.

The resulting maps α ← γ → β in Φ were what we needed to prove sublemma 5.6, so we are done. □

Resuming the proof of 5.5 we let e : α → β be the map described at the proof’s outset. Let D_{αβ} = D_{α} ∩S_{β} and E_{αβ} = f_{β}(D_{αβ}), so that D_{αβ} ⊕D_{β} = D_{α} and D_{αβ} ⊕S_{α} = S_{β}. We first observe that from the definition of the map e there is a natural identification of subobjects of V:

(*) F_{β} ⊕ E_{αβ} = F ⊕ E_{α}.

Using this, there is a natural isomorphism in A:

s_{e} : D_{α} ⊕ A ⊕ E_{α} ≅ (D_{αβ} ⊕ D_{β}) ⊕ (E ⊕ B ⊕ F) ⊕ E_{α} ≅ (D_{αβ} ⊕ E) ⊕ (D_{β} ⊕ B ⊕ F_{β}) ⊕ E_{αβ}.

Now define q_{α} to be the object S → S_{α} → Σ(D_{α} ⊕ A ⊕ E_{α}, S_{α}) of S ↓ Σ defined naturally by α, and let q_{e} : q_{α} → q_{β} be the map

(D_{αβ} ⊕ E, E_{αβ}, s_{e}, E_{αβ} ⊕ S_{α} ≅ S_{β}).

It is easy to see that q is a functor from S to S ↓ Σ.

Now let z denote the object 1 : S → Σ(0, S) of S ↓ Σ. There is a map θ_{α} : q_{α} → z given by the data

(A ⊕ E_{α}, D_{α}, D_{α} ⊕ A ⊕ E_{α} ≅ (A ⊕ E_{α}) ⊕ 0 ⊕ D_{α}, D_{α} ⊕ S_{α} ≅ S)

and a map η_{α} : q_{α} → pr_{α} given by the data

(D_{α}, E_{α}, D_{α} ⊕ A ⊕ E_{α} = D_{α} ⊕ A ⊕ E_{α}, E_{α} ⊕ S_{α} ≅ E_{α} ⊕ U_{α} ≅ U).

Using (*), it is a straightforward matter to check that θ and η are natural transformations. This proves that the maps z, q and pr from BS to B(S ↓ Σ) are homotopic. Since pr is a homotopy equivalence, this shows that B(S ↓ Σ) is contractible. This finishes the proof of Proposition 5.5, and hence of Theorem 5.3.

Next we show how to remove the hypothesis that A is idempotent complete from Theorem 5.4 Let U be a flasque A-filtered category. Let A^{∧}, and let U^{∧} be the full subcategory of the idempotent completion of U on objects P ⊕ U, P in A^{∧}, and U in U. Then U^{∧} is A^{∧}-filtered but not flasque. However it is easy to see that K^{Q}(U^{∧}) is contractible, and Theorem 5.3 applies to show that ΩK^{Q}(U^{∧}/A^{∧}) ≅ K^{Q}(A^{∧}). On the other hand, U^{∧}/A^{∧} ≅ U/A, so we have proven:

Corollary 5.7. If A^{∧} denotes the idempotent completion of A, and U is a flasque A-filtered category, then

(i) K^{Q}(U/A) ≃ K^{Q}(U^{∧}/A^{∧})

(ii) ΩK^{Q}(U/A) ≃ K^{Q}(A^{∧})

(iii) K^{Q}(A) → K^{Q}(U) → K^{Q}(U/A) is a homotopy fibration if and only if K_{0}(A) ≅ K_{0}(A^{∧})

Definition 5.8. (Karoubi [6]) Given a semisimple exact category A, we define K_{−n}(A) to be K_{1}(S^{n+1}A). Note that K_{−0}(A) = K_{0}(A^{∧}) by 5.7 (ii). This is well-defined because by 5.4 the homotopy type of S^{n+1}A is independent of the choice of flasque category U used to construct SA = U/A. Note that K_{0}(S^{n}A) = 0 for n ≥ 1 because S^{n}A need not be idempotent complete. In fact K_{0}((S^{n}A)^{∧}) = K_{−n}(A). (Cf. [6, p.151]

Definition 5.9. By 5.4, 5.7 and 5.8, there is an Ω-spectrum

|K_{−n}(A) × K^{Q}(S^{n}A)| ≃ |K^{Q}((S^{n}A)^{∧})|

We shall call it Karoubi’s non-connective K-theory spectrum for A, since Karoubi gave the prescription for this spectrum in [6].

6. Agreement of spectra

Our task is now to show that Karoubi’s spectrum agrees with the other spectra in the literature. We first recall Wagoner’s construction in [16]. Given a ring R, let lR denote the ring of locally finite ℕ- indexed matrices over R, i. e. matrices (r_{ij}) with 1 ≤ i, j < ∞ such that each row and each column has only finitely many nonzero entries. The finite matrices form an ideal mR of lR, and we let μR = lR/mR. Wagoner’s spectrum is

{K_{0}(μ^{n}R) × BGL^{+}(μ^{n}R)}.

We shall show that this spectrum is the same as Karoubi’s spectrum for the category F_{R} of (based) finitely generated free R-modules, Note that F^{∧}(R) is equivalent to the category of finitely generated projective R-modules, so K(F^{∧}(R)) is the usual space K_{0}^{Q}(R)×BGL^{+}(R). The following argument was shown to us by H. J. Munkholm and A. A. Ranicki.

Proposition 6.1. Let U denote the category of countably (but not necessarily infinitely) generated based free R-modules and locally finite matrices over R. Then U/F(R) is equivalent to the category F(μR). Consequently, Wagoner’s spectrum for R is homotopy equivalent to Karoubi’s K-theory spectrum for F(R).

Proof. (Munkholm-Ranicki) Choose an infinitely generated based R-module R^{∞} in U, and observe that End_{U}(R^{∞}) = lR. Now U is F(R)-filtered, and the completely continuous endomorphisms of R^{∞} form the ideal mR. Thus End_{U/F}(R^{∞}) ≅ μR. The additive functor F(μR) → U/F(R) which sends 1_{μR} to 1_{R^{∞}} is therefore full and faithful. But every object of U/F(R) is either isomorphic to 0 or to R^{∞}, so this functor is also an equivalence. Done. □

Gersten has also constructed a nonconnective spectrum for the K-theory of a ring in [4]. Since Wagoner showed in [16] that it agreed with Wagoner’s spectrum, Gersten’s spectrum is also homotopy equivalent to Karoubi’s spectrum.

Finally we must compare Karoubi’s K-theory spectrum with the spectrum {f(S^{n})} of section 4 above, constructed in [9] using the categories C_{n}(A). We assume that A is filtered in the sense of [9, (1.1)], or §2 above, so that in the notation of \underline{op.} \underline{cit.} we have C_{n+1}(A) = C_{1}(C_{n}(A)). The choice of the filtration affects the morphisms allowed in C_{n}(A) and C_{+}(A), but not the homotopy type of K^{Q}(C_{n}(A)), as [9, (3.2)] shows. The real point of the filtration on A is to reduce the discussion to the case n = 1.

Recall the objects of C_{1}(A) = C_{ℤ}(A) are ℤ-graded sequences A = (…, A_{−1}, A_{0}, A_{1}, …) in A. A map ϕ : A → B is a matrix of maps ϕ_{ij} : A_{i} → B_{j} such that for some bound b = b(ϕ) we have ϕ_{ij} = 0 whenever |i − j| > b. Composition is given by matrix multiplication. Define trunc(A) to be the object (A_{0}, A_{1}, …) of C_{+}(A) and trunc(ϕ) to be the submatrix of ϕ_{ij} with i, j > 0. If ψ : B → C, then trunc(ψϕ) − trunc(ψ) trunc(ϕ) is completely continuous being bounded by b(ψ) + b(ϕ). Hence trunc defines a functor from C_{1}(A) → C_{+}(A)/A.

Theorem 6.2. The functor trunc induces a homotopy equivalence

K^{Q}(C_{1}(A)) ⟶^{∼} K^{Q}(C_{+}(A)/A) ≃ K^{Q}(SA)

Assuming this result, it follows directly from the above remarks that we have K^{Q}(C_{n+1}(A))) ⟶^{∼} K^{Q}(C_{1}(C_{n}(A)) ≃ K^{Q}(SC_{n}(A)) ≃ K^{Q}(S^{n+1}A). Hence the spaces B̂_{n} of [9] are K_{−n}(A) × K^{Q}(C_{n}(A)), and we have

Corollary 6.3. The nonconnective spectra of [9] agree with Karoubi’s. In fact, trunc induces a homotopy equivalence of spectra:

{K_{−n}(A) × K^{Q}(C_{n}A)} ⟶^{∼} {K_{−n}(A) × K^{Q}(S^{n}A)}

Proof of Theorem 6.2. We shall use the notation of [9] , only remarking that C =ϵ is the

category of isomorphisms of C_{ϵ}(A), and that Spt_{0}(C =ϵ ≃ K^{Q}(C_{ϵ}(A)). By 5.7 and [9, §3], we

can assume that A is idempotent complete. There is a map of squares

[figure: A =; C =+; →; A =; C =+; _{C =−}; _{C =1}; 0; S =]

By Theorem 5.3 and [9, §3], applying Spt_{0} yields homotopy cartesian squares with Spt_{0}(C =1) ≃

K^{Q}(C_{1}(A)) and Spt_{0}(S =) ≃ K^{Q}(SA) connected. Since Spt(C =−) and Spt(C =+) are con-

tractible, it follows that K^{Q}(C_{1}(A)) → K^{Q}(SA) is a weak homotopy equivalence, hence a homotopy equivalence. □

References

1. H. Bass, Algebraic K-theory, Benjamin, 1968.

2. T. A. Chapman, Controlled boundary and h-cobordism theorems, Trans. Amer. Math. Soc. 280 (1983), 73–95.

3. , Controlled Simple Homotopy Theory and Applications, Lecture Notes in Mathematics, vol. 1009, Springer, 1983.

4. S. Gersten, On the spectrum of algebraic K-theory, Bull. Amer. Math. Soc. (N.S.) 78 (1972), 216–219.

5. T. Gunnarson, Algebraic K-theory of spaces as K-theory of monads, Aarhus preprint series, 21, 1981/1982.

6. M. Karoubi, Foncteur dérivés et K-théorie, Lecture Notes in Mathematics, vol. 136, Springer, 1970.

7. P. May, E^{∞}-spaces, group completions and permutative categories, new developments in topology, Proc. Sympos. Algebraic Topology, (Oxford, 1972), London Math. Soc. Lecture Notes, vol. 11, Cambridge Univ. Press, Cambridge, 1974, pp. 61–94.

8. E. K. Pedersen, On the K_{−i} functors, J. Algebra 90 (1984), 461–475.

9. E. K. Pedersen and C. Weibel, A nonconnective delooping of algebraic K-theory, Algebraic and Geometric Topology, (Rutgers, 1983), Lecture Notes in Mathematics, vol. 1126, Springer, Berlin, 1985, pp. 166–181.

10. D. G. Quillen, Higher algebraic K-theory I, Algebraic K-theory, I: Higher K-theories, (Battelle Memorial Inst., Seattle, Washington, 1972), Lecture Notes in Mathematics, vol. 341, Springer, Berlin, 1973, pp. 85– 147.

11. F. Quinn, Ends of maps, I, Ann. of Math. (2) 110 (1979), 275–331.

12. , Ends of maps, II, Invent. Math. 68 (1982), 353–424.

13. , Geometric algebra, Algebraic and Geometric Topology, (Rutgers, 1983), Lecture Notes in Mathematics, vol. 1126, Springer, Berlin, 1985, pp. 182–198.

14. G.B. Segal, Categories and cohomology theories, Topology 13 (1974), 293–312.

15. R. W. Thomason, First quadrant spectral sequences in algebraic K-theory via homotopy colimits, Comm. Algebra 10 (1982), 1589–1668.

16. J. Wagoner, Delooping classifying spaces in algebraic K-theory, Topology 11 (1972), 349–370.

17. C. A. Weibel, K-theory of Azumaya Algebras, Proc. Amer. Math. Soc. 81 (1981), 1–7.

18. R. Woolfson, Hyper γ-spaces and hyperspectra, Quart. J. Math. Oxford Ser. (2) 30 (1979), 229–255.

Matematisk Institut, Odense Universitet, Odense Denmark

Department of Mathematics, Rutgers University