A NONCONNECTIVE DELOOPING OF ALGEBRAIC K-THEORY

ERIK KJÆR PEDERSEN AND CHARLES A. WEIBEL

Abstract. Given a ring R, it is known that the topological space BGl(R)^{+} is an infinite loop space. One way to construct an infinite loop structure is to consider the category F = of free R-modules, or rather its classifying space BF =, as food for suitable infinite loop

space machines. These machines produce connective spectra whose zeroth space is (BF =)^{+} =

ℤ × BGl(R)^{+}. In this paper we consider categories C =_{1}(F =) = F =, C =_{1}(F =), … of parameterized

free modules and bounded homomorphisms and show that the spaces (BC =_{0})^{+} = (BF =)^{+},

(BC =_{1})^{+}, … are the connected components of a nonconnective Ω-spectrum BC =(F) with

K_{i}BC =(F) = K_{i}(R) even for negative i.

0. Introduction

Given a ring R, let F = be the category of finitely generated R-modules and isomorphisms.

Form the “group completion” category F =^{−1}F = of F = (see [5]); it is known that its classifying

space BF =^{−1}F = is the algebraic K-theory space BGl(R)^{+} × ℤ. The purpose of this paper is to

produce a nonconnective delooping of BGl(R)^{+} × K_{0}(R) by using a parameterized version C =_{0}(F =) = F =, C =_{1}(F =), … of F = given in [11]. Our main result is this:

Theorem A. Write B_{i} for the classifying space of the category C =^{−1}C =, except that B_{0} =

BGl(R)^{+}. Then the spaces B_{i} are connected, and for i ≥ 0 we have

ΩB_{i+1} = B_{i} × K_{−i}(R).

Thus the sequence of spaces B̂_{i} = B_{i} × K_{−i}(R) forms a nonconnective Ω-spectrum ˆ B = with

homotopy groups

π_{i}(ˆB =) = K_{i}(R), i any integer.

In particular, the negative homotopy groups of ˆ B = are the negative K-groups of Bass [2].

The second author was partially supported by an NSF grant .

Actually, we work in the generality of a small additive category A, rather than just the additive category F of finitely generated free R-modules. For example, one could take P, the category of finitely generated projective R-modules. The category P is the idempotent completion of F, and we recover the same spectrum ˆ B = if we replace F by P. Note that

BP =^{−1}P = is BGl(R)^{+} × K_{0}(R), where P = is the category of isomorphisms in P.

Given A, we consider the additive category C_{i}(A) of ℤ^{i}-graded objects and bounded homomorphisms (see section 1 for details). If A = F this definition specializes to the categories C_{i} of [11]. Let Ĉ_{i} be the idempotent completion of C_{i}(A), and let A =, C =_{i}, ˆ C = be the sub-categories

of isomorphisms in A, C_{i} and Ĉ_{i}, respectively. Our second result is this

Theorem B. Write B̂_{i} for the classifying space of the category ˆ C =_{i}^{−1} ˆC =_{i} and B_{i} for the classifying

space of C =_{i}^{−1}C =_{i}. Then

ΩB̂_{i+1} = B̂_{i}
Ω^{i}B̂_{i} = B̂_{0} = “group completion” (BA =)^{+} of BA =.

The connected component of B̂_{i} is B_{i} (except for i = 0), and the sequence of spaces B̂_{0}, B̂_{1}, … is a nonconnective Ω-spectrum. In particular, B̂_{i} is an i-fold delooping of (BA =)^{+}.

The outline of this paper is as follows. In section 1 we give the definitions of the ℤ^{i}-graded category C_{i}(A). In section 2, we recall the passage from categories to spectra, and review the main points of Thomason’s paper [13] that we need. In section 3, we prove Theorems A and B.

The authors would like to thank Bob Thomason for his lucid exposition in [13], which clarified a number of technical points.

The second author would also like to thank the Danish Natural Science Research Council and Odense University for its hospitality during the writing stage.

1. The categories C_{i}

In this section we give the definition of the categories C_{i}(A) associated to a small additive category A. We also review the notions of filtered additive categories and of the idempotent completion of A for the convenience of the reader.

1.1. Definition. An additive category is said to be \underline{filtered} if there is an increasing filtration

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

on hom(A, B) for every pair of objects A, B of A. Each F_{n}(A, B) is to be a subgroup of hom(A, B) and we must have ∪F_{n}(A, B) = hom(A, B). We require 0_{A} and 1_{A} to be in F_{0}(A, B), and assume that the composition of morphisms in F_{m}(A, B) and F_{n}(A, B) belongs to F_{m+n}(A, C). We also assume that the projections A ⊕ B → A, and inclusions A → A ⊕ B and coherence isomorphisms all belong to F_{0}. If ϕ is in F_{d}(A, B) we say that ϕ has \underline{filtration degree} d.

The reason for concerning ourselves with filtered categories is that the categories C_{i} come with a natural filtration. Of course every additive category has a trivial filtration, obtained by setting F_{0}(A, B) = hom(A, B).

1.1.1. Example. Given a ℤ-graded ring A such as R[t, t^{−1}], let A be the category of graded Amodules. We can filter A by legislating that homogeneous maps of degree ±d have filtration degree d.

We now give our definition of the filtered category C_{i}. Let the distance between points J = (j_{1}, …, j_{i}) and K = (k_{1}, …, k_{i}) in ℤ^{i} be given by

||J − K|| = max_{s} |j_{s} − k_{s}|.

1.2. Definition. Let A be a (filtered) additive category. We define C_{i}(A) to be the category of ℤ^{i}-graded objects and bounded homomorphisms. This means that an object A of C_{i} is a collection of objects A(J) in A, one for each J in ℤ^{i}. A morphism ϕ : A → B in C_{i} of filtration degree d is a collection

ϕ(J, K) : A(J) → B(K)

of A-morphisms, where we require ϕ(J, K) = 0 unless ||J − K|| ≤ d. If A is filtered, we also require each ϕ(J, K) to have filtration ≤ d. Composition of ϕ : A → B with ψ : B → C is defined by

(ψ ∘ ϕ)(J, L) = ∑_{K} ψ(K, L) ∘ ϕ(J, K).

Note that composition is well-defined because only finitely many elements in this sum are different from 0. It is easily seen that C_{0}(A) = A.

1.2.1. Example. If F is the category of finitely generated free R-modules (with trivial filtration), the category C_{i}(F) is the same as the category C_{i}(R) constructed in [11]. In that paper it was proven that

K_{1}(C_{i+1}(R)) = K_{−i}(R), i ≥ 0.

This indicated that C_{i+1} might be a delooping of K-theory, and was the original motivation for this paper. That it cannot be exactly the case follows from (1.3.1) below.

1.2.2. Example. Since C_{i}(A) is filtered, we can iterate the construction. It is easy to see that

C_{i}(C_{j}(A)) = C_{i+j}(A).

However, if we forget the filtrations on C_{j}(A) this is no longer the case.

1.2.3. Remark. If V is any metric space, we can define a category C_{V} (A) in a way generalizing the case V = ℤ^{i}. An object A of C_{V} is a collection of objects A(v), one for each v in V, subject to the following constraint: for every d > 0 and v, A(w) ≠ 0 for only finitely many w of distance less than d from v. Morphisms are defined as for C_{i}. It is easy to see that if V = ℝ^{i} then C_{V} is naturally equivalent to its subcategory C_{i}. This shows that the difference between C_{i} and C_{i+1} is the rate of growth of the number n(d, J) of points K within a distance of d from J.

1.2.4. Example. If we take V = (0, 1, 2, …) then we will let C_{+}(A) denote C_{V} (A). This is the full subcategory of C_{1}(A) whose objects satisfy A(j) = 0 for j < 0. Similarly, if we take V = (0, −1, −2, …), we will write C_{−}(A) for C_{V} (A). We can identify C_{+}(A) ∩ C_{−}(A) with A in the obvious way.

There is a shift functor T : C_{1}(A) → C_{1}(A) sending A to T A with T A(j) = A(j − 1), and T restricts to an endofunctor of C_{+}(A). There is an obvious natural isomorphism t from A to T A in both C_{1} and C_{+}. We include the following result here for expositional purposes, and will generalize it in section 3 below.

1.3. Lemma. Every object in C_{+}(A) is stably isomorphic to 0. In particular, the Grothendieck group K_{0}(C_{+}) is zero.

Proof. Given A in C_{+}, let B = ∑ T^{n}A. That is, B(j) = A(j) ⊕ A(j − 1) ⊕ … ⊕ A(0). It is clear that A ⊕ T A = B. The result follows from the observation that t : B ≅ T B is an isomorphisms in C_{+}(A). □

1.3.1. Corollary. If i ≠ 0 then every object of C_{i}(A) is stably isomorphic to 0. In particular, K_{0}(C_{i}) = 0.

Proof. By (1.2.2) we can assume that i = 1. But every object of C_{1} can be written A_{+} ⊕ A_{−} with A_{+} in C_{+} and A_{−} in C_{−}. Hence K_{0}(C_{1}) is a quotient of K_{0}(C_{+}) ⊕ K_{0}(C_{−}) = 0. □

1.4. Definition. (see, e. g., [3, p. 61]). Let A be an additive category. The \underline{idempotent} \underline{completion} Â of A has as objects all morphisms p : A → A from A satisfying p^{2} = p. An Â-morphism from p_{1} to p_{2} is an A-morphism ϕ from the domain A_{1} of p_{1} to the domain A_{2} of p_{2} satisfying that ϕ = p_{2}ϕp_{1}. It is easily seen that Â is an additive category and that hom(p_{1}, p_{2}) is a subgroup of hom(A_{1}, A_{2}). Hence Â inherits any filtered structure that A might have. There is a full embedding A in Â sending A to 1_{A}; if this is an equivalence of categories, we say that A is \underline{idempotent complete}.

1.4.1. Example. The idempotent completion of the category F of free R-modules is equivalent to the category P of projective R-modules.

1.4.2. Lemma. The categories A and C_{i}(A) are cofinal in their idempotent completions Â and Ĉ_{i}(A). Moreover, C_{i}(A) is cofinal in C_{i}(Â).

Proof. This is an easy computation. For example, if p is an object of C_{i}(Â), define q by q(J) = 1 − p(J). Then p ⊕ q belongs to C_{i}(A). □

To compute the K-theory of A, we need to know which sequences are “exact”: a different embedding of A in an ambient Abelian category will result in a different family of short exact sequences (see [12]). In particular, we cannot talk about K_{1}(C_{i}(A)) unless we know which sequences in C_{i} are “exact”. It is not clear what the notion of “exact” should be, unless either (a) all exact sequences in A split (we insist the same is true of C_{i}), or (b) A is embedded in an Abelian category Ã closed under countably infinite direct sum ( for then C_{i} is embeddable in Ã. In either case, it follows from (1.4.2) and Theorem 1.1 of [6] that

K_{n}(C_{i}(A)) = K_{n}C_{i}(Â) = K_{n}(Ĉ_{i}(A)), n ≥ 1.

Note that our proofs of theorem A and B only to situation (a).

1.5. Example. Let p_{−} be the idempotent natural transformation in C_{1}(A) given by

(p_{−})_{A} : A → A, p_{−}(j, k) = \begin{cases}1 & if j = k ≤ 0 \\ 0 & otherwise \end{cases}

Given an object A of A, let A_{−} denote the image of p_{−} on the constant object A(j) = A of C_{1}(A). Thus A_{−}(j) = 0 if j > 0 and A_{−}(j) = A if j ≤ 0. The map t is an endomorphism of the constant object A ≅ T A; write s for the restriction of p_{−}t to A_{−}. Then 1 − s : A_{−} → A_{−} is both a monomorphism and an epimorphism in C_{1}(A), but not an isomorphism. This is because the “inverse” ∑ s^{n} is not bounded. In particular, C_{1}(A) can never be an Abelian category, even if A is.

We conclude this section with the following result, which provides motivation for our Theorem B. It is also a consequence of Theorem B. Since we will not use this result, we merely sketch the proof.

1.6. Proposition. If all short exact sequences in A split, then K_{1}(C_{i+1}(A)) = K_{0}(Ĉ_{i}(A)). In particular, K_{1}C_{1}(A) = K_{0}(Â).

Sketch of proof. This is proven in section 1 of [11], modulo terminology.

First of all we can assume that A is idempotent complete and that i = 0 by (1.4.2) and (1.2.2). The map from K_{0}(A) to K_{1}C_{1}(A) sends the object A of A to the shift automorphism of the constant object A(j) = A of C_{1}(A). The map ϕ : K_{1}(C_{1}) → K_{0}(A) is defined by sending the class of α ∈ Aut(A) to the difference (for d ≫ 0) in K_{0}(A):

ϕ(α) = [(αp_{−}α^{−1})( ⨁_{j=−2d}^{2d} A(j))] − [p_{−}( ⨁_{j=−2d}^{2d} A(j))].

If α has filtration degree less than d, one shows as in [11, 1.11] that this map ϕ is well-defined and independent of d. Clearly the composition is the identity on K_{0}(A). The proof of [11, (1.20)] applies to show that ϕ is monic, which proves the proposition. □

1.6.1. Example. Again, let F be the category of finitely generated free R-modules. Then for i ≥ 1 we have K_{0}C_{i}(R) = 0 but K_{0}Ĉ_{i}(R) = K_{1}C_{i+1}(R) = K_{−i}(R).

Note. Example (1.6.1) follows from [11], not from (1.6).

2. The passage to topology

In this section we recall various results on the passage from the categories A, C_{i} etc. to infinite loop spaces and spectra. We also recall Thomason’s simplified double mapping cylinder from section 5 of [13]. We urge the reader to consult [13] for more details.

A \underline{symmetric monoidal category} S = is a category together with a functor ⊕ : S = × S = → S =

and natural isomorphisms

α : (A ⊕ B) ⊕ C ≅ A ⊕ (B ⊕ C)
γ : A ⊕ B ≅ B ⊕ A.

These natural isomorphism are subject to coherence conditions that certain diagrams commute. We refer he reader to [10] for a more detailed definition, contending ourselves with:

2.1. Example. If A is an additive category then A is a symmetric monoidal category under ⊕ = direct sum. The subcategory A = of the isomorphisms in A is also symmetric monoidal

under ⊕ = direct sum. It follows that C_{i}(A) and its category C =_{i}(A) of isomorphisms are

also symmetric monoidal.

There is a functor Spt from the category of small symmetric monoidal categories to the category of connective Ω-spectra ( i. e. sequences of spaces X_{n} with X_{n} being (n − 1)- connected and with X_{n} = ΩX_{n+1}). This functor satisfies

(a) A functor A = → B = preserving ⊕ up to coherent natural transformation, a “lax”

functor, induces a map Spt(A =) → Spt(B =) of infinite loop spectra.

(b) The zeroth space Spt_{0}(A =) is the “group completion” of BA =, the classifying space of

the category A =.

The construction of Spt is basically due to May and Segal, and Spt is unique up to homotopy equivalence. See [1]. One description of Spt may be found in the Appendix of

A NONCONNECTIVE DELOOPING OF ALGEBRAIC K-THEORY 7

2.2. Lemma. Suppose that A = → B = is a lax functor of small symmetric monoidal categories, and that BA = → BB = is a homotopy equivalence of topological spaces. Then Spt_{0}(A =) → Spt_{0}(B =) is a homotopy equivalence.

Proof. See (2.3) of [13]. □ 2.3. Lemma. Suppose that A = is a full, cofinal subcategory of the small symmetric monoidal category B =. Then the connected components of Spt_{0}(A =) and Spt_{0}(B =) are homotopy equivalent. Proof. This is well-known. The point is that

H_{∗}[Spt_{0}(A =)_{0}] = Colim_{A∈=^{A}} H_{∗}B Aut(A)
= Colim_{B∈=^{B}} H_{∗}B Aut(B)
= H_{∗}[Spt_{0}(B =)_{0}].
□

2.4. Lemma. (Quillen). Let S = be a small symmetric monoidal category in which all morphisms are isomorphisms, and assume that all translation S⊕ : S = → S = are faithful. Then there is a category S =S =^{−1} whose objects are pairs (S_{1}, S_{2}) of objects in S =, such that BS =S =^{−1} is homotopy equivalent to Spt_{0}(S =).

Proof. See [5, p. 221] or p. 1657 of [13]. □ 2.4.1. Corollary. If A is a small additive category, let A = denote the category of isomorphisms in A. Then BA =A =^{−1} is homotopy equivalent to Spt_{0}(A =).

2.4.2. Example. Let R be a ring for which R^{m} ≅ R^{n} implies that m = n, and let F = be the category of finitely generated free R-modules and isomorphisms. The basepoint component of F =^{−1}F = has objects R^{m} = (R^{m}, R^{m}) and

hom(R^{m}, R^{m+n}) = Gl_{m+n} ×_{Gl_{n}(R)} Gl_{m+n}(R).

In particular, hom(0, R^{m}) is Gl_{m}(R). The family of the hom(0, R^{m}) gives a map from BGl(R) to the basepoint component BGl^{+}(R) of BF =^{−1}F =

The main ingredient in the proof of Theorem B is the simplified mapping cylinder construction of R. W. Thomason, described in (5.1) of [13]. Let A = be a symmetric monoidal category

with all morphisms isomorphisms and u : A = → B =, v : A = → C = strong functors of symmetric

monoidal categories (i. e. functors preserving direct sum up to natural isomorphism). Define P = = (P =(A =, B =, C =, u, v) to be the category with objects triples (B, A, C) with A an object of A =,

B of B =, and C of C =. A morphism (B, A, C) → (B′, A′, C′) is a 5-tuple (ψ, ψ_{1}, ψ_{2}, U.V) where

U, V are objects of A, ψ : A ≅ U ⊕A′⊕V, ψ_{1} : B ⊕uU → B′ and ψ_{2} : C ⊕vV → C′. U and V may be varied up to isomorphism. Composition of (ψ, ψ_{1}, ψ_{2}, U, V) : (B, A, C) → (B′, A′, C′) with (\overline{ψ}, \overline{ψ}_{1}, \overline{ψ}_{2}, \overline{U}.\overline{V}) : (B′, A′, C′) → (B′′, A′′, C′′) is given by

A ≅ U ⊕ A′ ⊕ V ≅ (U ⊕ \overline{U}) ⊕ A′′ ⊕ (\overline{V} ⊕ V)
B ⊕ u(U ⊕ \overline{U}) ≅ (B ⊕ uU) ⊕ u\overline{U}) → B′ ⊕ u\overline{U} → B′′
v(\overline{V} ⊕ V) ⊕ C ≅ v\overline{V} ⊕ vV ⊕ C → v\overline{V} ⊕ C′ → C′′

and direct sum in P = is induced by direct sum in A =, B = and C =. We then have

2.5. Theorem. R. W. Thomason [13, (5.2)]. Up to homotopy the diagram

\xymatrix{Spt^{0} A = \ar[d] \ar[r] & Spt^{0} B = \ar[d] \\ _{Spt_{0} C =} \ar[r] & _{Spt P =_{0}}}

is a pullback diagram

3. The proof of Theorem A and B

In this section we prove Theorems A and B. We make the standing assumption that A is a small filtered additive category and that A = is the (symmetric monoidal) category of

isomorphisms of A. Similarly we write C =_{i}, C =_{+}, and C =_{−} for the categories of isomorphisms

of C_{i}(A), C_{+}(A) and C_{−}(A). The idea is to show that the diagram

\xymatrix{A = \ar[d] \ar[r] & C =_{+} \ar[d] \\ _{C =_{−}} \ar[r] & _{C =_{1}}}

induces a pullback diagram of spectra, and use the following result:

A NONCONNECTIVE DELOOPING OF ALGEBRAIC K-THEORY 9

3.1. Proposition. Spt_{0}(C =_{+}) and Spt_{0}(C =_{−}) are contractible. Proof. By symmetry it is enough to consider C =_{+}. Recall from the discussion before (1.3) that there is a shift functor T : C =_{+} → C =_{+} and a natural transformation t from A to T A.

The category C =_{+} has an endofunctor ∑_{n=0}^{∞} T^{n} with

(∑_{n=0}^{∞} T^{n})A(j) = ⨁_{n=0}^{j} A(j − n).

(Recall that A(j) = 0 for j < 0.) We can define ∑_{n=1}^{∞} T^{n} similarly. The natural isomorphism

t induces a natural isomorphism t from ∑_{n=0}^{∞} T^{n}A to ∑_{n=1}^{∞} T^{n}A. But as endofunctors of C =_{+} we

have 1 ⊕ ∑_{n=1}^{∞} T^{n} ≅ ∑_{n=0}^{∞} T^{n}. Hence as self maps of the H-space BC =_{+} we have

1 ∼ (∑_{n=0}^{∞} T^{n}) − (∑_{n=1}^{∞} T^{n}) ∼^{t} 0.

This shows that B is contractible. But then Spt(C =_{+}) is contractible by Lemma (2.2). □

Proof that Theorem B implies Theorem A. Write B̂_{i} for Spt_{0}(ˆC =_{i}). Since we have π_{0}(B̂_{i}) = K_{−i}(R) by (1.6.1) and since translations are faithful in ˆ C =_{i}, it follows that B̂_{i} is homotopy equivalent to B_{i} × K_{−i}(R). Since ΩB_{i} = ΩB̂_{i}, the result is now immediate. □

We now begin the proof of theorem B by making a series of reductions. Since

π_{0}(B_{i}) = π_{0} Spt_{0}(A =_{i}) = K_{0}(A =_{i}),

connectedness of the B_{i} for i ≠ 0 follows from (1.3.1). Noq C =_{i} is full and cofinal in C =_{i} by (1.4.2), so by (2.3) the connected space B_{i} = Spt(C =_{i}). By construction (or by (2.4.1)), B̂_{0} = Spt_{0}(ˆA =) is the group completion of B ˆA =. Thus the proof of Theorem B is reduced to showing that ΩB̂_{i+1} = B̂_{i} for i ≥ 0.

Next , observe that Ĉ_{i+1}(A =) = Ĉ_{1}Ĉ_{i}(A =), so that B̂_{i+1} = Spt_{0}(ˆC =_{1}(Ĉ_{i}(A)) and B̂_{i} = Spt_{o}(Ĉ_{i}(A)).

10 ERIK KJÆR PEDERSEN AND CHARLES A. WEIBEL

Since we can replace A by Ĉ_{i}(A), it is enough to prove that ΩB̂_{i+1} = B̂_{0} = Spt(ˆA =). There is also no loss in generality in assuming that A = is idempotent complete, since

ΩB̂_{1} = Ω Spt_{0}(Ĉ_{1}(A)) = Ω Spt_{0}(ˆC =_{1}(Â))

by (2.3). In fact by (2.3) we also have

Ω Spt_{0}(ˆC =_{1}) = Ω Spt_{0}(C =_{1}).

Therefore, Theorem B will follow from: 3.2. Theorem. Let A be a small, filtered additive category which is idempotent complete. Then Ω Spt(C =_{1}) is homotopy equivalent to Spt_{0}(A =). 3.3. Lemma. Let A be a small filtered additive category. Recall that C =_{+} and C =_{−} are subcategories of C =_{+} whose intersection is A =. Let P = be the simplified double mapping cylinder construction applied to A = → C =_{−} and A = → C =_{+}. Then Ω Spt_{0}(P =) is homotopy equivalent to Spt_{0}(A =). Proof. This is immediate from Thomason’s Theorem (2.5), since by (3.1) the spaces Spt_{0}(C =_{+}) and Spt_{0}(C =_{−}) are contractible. □

By the universal mapping property of P = (see p. 1648 of [13]), there is a strong symmetric monoidal functor Σ : P = → C =_{1}. This functor is defined on objects by

Σ(A^{−}, A, A^{+}) = A^{−} ⊕ A ⊕ A^{+}

where A^{−}, A, A^{+} are objects of C =_{+}, A = and C =_{−}, respectively. A morphism (ψ^{−}, ψ, ψ^{+}, U^{−}, U^{+}) is P = from (A^{−}, A, A^{+}) to (B^{−}, B, B^{+}) is sent by Σ to the composite

A^{−} ⊕ A ⊕ A^{+} ⟶^{1⊕ψ⊕1} A^{−} ⊕ U^{−} ⊕ A ⊕ U^{+} ⊕ A^{+} ⟶^{ψ^{−}⊕1⊕ψ^{+}} B^{−} ⊕ B ⊕ B^{+}.

3.4. Theorem. Let A be idempotent complete, and let P = be the double mapping cylinder of Lemma (3.3). Then the functor Σ : P = → C =_{1} induces a homotopy equivalence between the classifying spaces BP = and BC =_{1}.

Note that Theorem (3.4) immediately implies Theorem (3.2) by (3.3) and (2.2). Thus we have reduced the proof of Theorem B to the proof of Theorem (3.4).

Proof. We will show that this functor satisfies the conditions of Quillen’s Theorem A from [12]. Fix an object Y of C =_{1}; we need to show that Y ↓ Σ is a contractible category. To do

this, we use the bound d for C_{1}(A =) to filter Y ↓ Σ as the increasing union of sub-categories

Fil_{d}, and show that each Fil_{d} has an initial object ∗_{d}. Therefore Fil_{d} is contractible; their union Y ↓ Σ must also be contractible by standard topology.

The category Fil_{d} is the full subcategory of all α : Y → Σ(A^{−}, A, A^{+}) where both α and α^{−1} are bounded by d. Define Y_{d}, Y_{d}^{−} and Y_{d}^{+} in A =, C =_{−} and C =_{+} respectively by setting

Y_{d} = Y (−d) ⊕ … ⊕ Y (d) in A =
Y_{d}^{−} = Y (j) if j < −d, and = 0 otherwise
Y_{d}^{+} = Y (j) if j > −d, and = 0 otherwise.

The obvious isomorphism σ : Y ≅ Y_{d}^{−} ⊕ Y_{d} ⊕ Y_{d}^{+} in C =_{1} is bounded by d, and forms the

object ∗_{d} : Y → Σ(Y_{d}^{−}, Y_{d}, Y_{d}^{+}) of Fil_{d}. We will show that ∗_{d} is an initial object of Fil_{d}. Given an object α : Y → ∑(A^{−}, A, A^{+}), we have to show that there is a unique morphism

η = (ψ, ψ^{−}, ψ^{+}, e_{−}(Y_{d}), e_{+}(Y_{d})) : (Y_{d}^{−}, Y_{d}, Y_{d}^{+}) → (A^{−}, A, A^{+})

in P = so that Σ(η) = ασ^{−1} in C =_{1}. Let pr_{−}, pr, pr_{+} be the projections of Σ(A^{−}, A, A^{+}) onto

A^{−}, A and A^{+}, respectively. Since α^{−1} is bounded by d, α^{−1}(A) is contained in Y_{d}, or rather in the image σ^{−1}(Y_{d}) of Y_{d} in Y. Hence it makes sense to let e be σα^{−1}(pr)ασ^{−1} restricted to Y_{d}, and it is clear that e is an idempotent of Y_{d}. Similarly σα^{−1}(A^{−}) is contained in Y_{d}^{−} ⊕ Y_{d}, and α^{−1}(A^{+}) is contained in Y_{d} ⊕ Y_{d}^{+}. Let e_{−} and e_{+} be σα^{−1}(pr_{−})ασ^{−1} and σα^{−1}(pr_{+})ασ^{−1} restricted to Y_{d}. These maps are also idempotents of Y_{d}, and it is easy to see that e_{−} + e + e_{+} = 1. Since A is idempotent complete, the composition

Y_{d} ≅ e_{−}(Y_{d}) ⊕ e(Y_{d}) ⊕ e_{+}(Y_{d})

makes sense in A. Define ψ to be the composite

Y_{d} ≅ e_{−}(Y_{d}) ⊕ e(Y_{d}) ⊕ e_{+}(Y_{d}) ⟶^{1⊕α⊕1} e_{−}(Y_{d}) ⊕ A ⊕ e_{+}(Y_{d})

Similarly, define maps

ψ^{−} : Y_{d}^{−} ⊕ e_{−}(Y_{d}) ⟶^{ασ^{−1}} A^{−} in C =_{−}
ψ^{+} : e_{+}(Y_{d}) ⊕ Y_{d}^{+} ⟶^{ασ^{−1}} A^{+} in C =_{+}.

This completes the definition of the map η : (Y_{d}^{−}, Y_{d}, Y_{D}^{+}) → (A^{−}, A, A^{+}) in P =. By definition

of Σ we have Σ(η) = ασ^{−1}. Because all maps in A =, C =_{−} and C =_{+} are isomorphisms, it is an easy task to verify that η is the unique map with Σ(η) = ασ^{−1}. It follows that ∗_{d} is an initial object of Fil_{d} □

4. An overview

To place our construction in perspective, it is appropriate to review a little history. The definition of the functors K_{−i}(R) was given by Bass [2] in 1966 during an attempt to formalize his decomposition of K_{1}(R[t_{1}, t_{1}^{−1}, ….t_{n}, t_{n}^{−1}]). In 1967, Karoubi [8] gave another definition of K_{−i}(R) by defining K_{−i}(A) for any Abelian category. A third and fourth definition of K_{−i}(R) were given independently by Karoubi Villamayor [9] using the ring S(R) and by Wagoner [14] using the subring μ(R) of S(R). Happily all these definitions were shown to agree by Karoubi’s axiomatic treatment in [7].

In 1971, Gersten [4] constructed a nonconnective delooping of K_{0}(R) × BGl^{+}(R) using the fact that ΩBGl^{+}(S(R)) = K_{0}(R) × BGl^{+}(R). Wagoner [15] then constructed the Ωspectrum K_{0}(μ^{i}(R)) × BGl^{+}(μ^{i}(R)) and showed that the inclusions μ(R) → S(R) induced an equivalence of spectra. To our knowledge, nonconnective deloopings of K-theory of other additive categories besides F has not been studied until now.

The construction in [11] is very much in the spirit of the early definitions of K_{−i}(R), but works for any additive category. Needless to say, an open question in our work is whether or not the ΩBQC_{n}(A =)^{∧} yield a nonconnective delooping of any (idempotent complete) ad-

ditive category with exact sequences. A major difference between the categories C_{i}(A) and Karoubi’s categories S^{i}A is that SA is defined as a quotient of the flasque category CA (see [7]) while C_{1}(A) may be viewed as an enlargement of the flasque category C_{+}(A). It would be interesting to see if the natural inclusion of CA in C_{+}(A) could be made to induce an isomorphism between K-groups.

References

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

3. P. Freyd, Abelian Categories, Harper and Row, New York, 1966.

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

5. D. Grayson, Higher algebraic K-theory: II (after D. Quillen), Lecture Notes in Mathematics, vol. 551, Springer, 1976.

6. , Localization for flat modules in Algebraic K-theory, J. Algebra 90 (1984), 461–475.

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

8. , La périodicité de Bott en K-théorie générale, Ann. Sci. École Norm. Sup. (4) 4 (1971), 63–95.

9. M. Karoubi and O. Villamayor, K-théorie algébrique et K-théorie topologique, C. R. Acad. Sci. Paris Sér. I Math. 269 (1988), 416–419.

10. S. MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5, Springer, 1971.

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

12. 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.

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

14. J. Wagoner, On K_{2} of the Laurent polynomial ring, Amer. J. Math. 93 (1971), 123–138.

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

Matematisk Institut, Odense Universitet, Odense Denmark

Department of Mathematics, Rutgers University