CHAPTER IV DEFINITIONS OF HIGHER K-THEORY

The higher algebraic K-groups of a ring R are defined to be the homotopy groups K_{n}(R) = π_{n}K(R) of a certain topological space K(R), which we shall construct in this chapter. Of course, the space K(R) is rigged so that if n = 0, 1, 2 then π_{n}K(R) agrees with the groups K_{n}(R) constructed in chapters II and III.

We shall also define the higher K-theory of a category A in each of the three settings where K_{0}(A) was defined in chapter II: when A is a symmetric monoidal category (§4), an exact category (§6) and a Waldhausen category (§8). In each case we build a “K-theory space” KA and define the group K_{n}A to be its homotopy groups: K_{n}A = π_{n}KA. Of course the group π_{0}KA will agree with the corresponding group K_{0}A defined in chapter II.

We will show these definitions of K_{n}A coincide whenever they coincide for K_{0}. For example, the group K_{0}(R) of a ring R was defined in §II.2 as K_{0} of the category P(R) of finitely generated projective R-modules, but to define K_{0}P(R) we could also regard the category P(R) as being either a symmetric monoidal category (II.5.2), an exact category (II.7.1) or a Waldhausen category (II.9.1.3). We will show that the various constructions give homotopy equivalent spaces KP(R), and hence the same homotopy groups. Thus the groups K_{n}(R) = π_{n}KP(R) will be independent of the construction used.

Many readers will not be interested in the topological details, so we have designed this chapter to allow “surfing.” Since the most non-technical way to construct K(R) is to use the “+”-construction, we will do this in §1 below. The short section 2 defines K-theory with finite coefficients, as the homotopy groups of K(R) with finite coefficients. These have proved to be remarkably useful in describing the structure of the groups K_{n}(R), especially as related to étale cohomology. This is illustrated in chapter VI.

In §3, we summarize the basic facts about the geometric realization BC of a category C, and the basic connection between category theory and homotopy theory needed for the rest of the constructions. Indeed, the K-theory space KA is constructed in each setting using the geometric realization BC of some category C, concocted out of A. For this, we assume only that the reader has a slight familiarity with cell complexes, or CW complexes, which are spaces obtained by successive attachment of cells, with the weak topology.

Sections 4–9 give the construction of the K-theory spaces. Thus in §4 we have group completion constructions for a symmetric monoidal category S, such as the S^{−1}S construction and the connection with the +-construction). It is used in §5 to construct λ-operations on K(R). Quillen’s Q-construction for abelian and

Typeset by AMS-TEX

2 IV. DEFINITIONS OF HIGHER K-THEORY

exact categories is given in §6; in §7 we prove the “+ = Q” theorem, that the Q-construction and group completion constructions agree for split exact categories (II.7.1.2). The wS_{·} construction for Waldhausen categories is in §8, along with its connection to the Q-construction. In §9 we give an alternative construction for exact categories, due to Gillet and Grayson.

Section 10 gives a construction of the non-connective spectrum for algebraic Ktheory of a ring, whose negative homotopy groups are the negative K-groups of Bass developed in chapter III.4. Sections 11 and 12 are devoted to Karoubi-Villamayor K-theory and the homotopy-invariant version KH of K-theory. We will return to this topic in chapter V.

§1. The BGL^{+} definition for Rings

Let R be an associative ring with unit. Recall from chapter III that the infinite general linear group GL(R) is the union of the groups GL_{n}(R), and that its commutator subgroup is the perfect group E(R) generated by the elementary matrices e_{ij}(r). Moreover the group K_{1}(R) is defined to be the quotient GL(R)/E(R).

In 1969, Quillen proposed defining the higher K-theory of a ring R to be the homotopy groups of a certain topological space, which he called “BGL(R)^{+}.” Before describing the elementary properties of Quillen’s construction, and the related subject of acyclic maps, we present Quillen’s description of BGL(R)^{+} and define the groups K_{n}(R) for n ≥ 1.

For any group G, we can naturally construct a connected topological space BG whose fundamental group is G, but whose higher homotopy groups are zero. Details of this construction are in §3 below (see 3.1.3). Moreover, the homology of the topological space BG (with coefficients in a G-module M) coincides with the algebraic homology of the group G (with coefficients in M); the homology of a space X with coefficients in a π_{1}(X)-module is defined in [Wh, VI.1–4]. For G = GL(R) we obtain the space BGL(R), which is central to the following definition.

Definition 1.1. The notation BGL(R)^{+} will denote any CW complex X which has a distinguished map BGL(R) → BGL(R)^{+} such that

(1) π_{1}BGL(R)^{+} ≅ K_{1}(R), and the natural map from GL(R) = π_{1}BGL(R) to π_{1}BGL(R)^{+} is onto with kernel E(R);

(2) H_{∗}(BGL(R); M) ⟶^{≅} H_{∗}(BGL(R)^{+}; M) for every K_{1}(R)-module M. We will sometimes say that X is a model for BGL(R)^{+}.

For n ≥ 1, K_{n}(R) is defined to be the homotopy group π_{n}BGL(R)^{+}.

By Theorem 1.5 below, any two models are homotopy equivalent, i.e., the space BGL(R)^{+} is uniquely defined up to homotopy. Hence the homotopy groups K_{n}(R) of BGL(R)^{+} are well-defined up to a canonical isomorphism.

By construction, K_{1}(R) agrees with the group K_{1}(R) = GL(R)/E(R) defined in chapter III. We will see in 1.7.1 below that K_{2}(R) = π_{2}BGL^{+}(R) agrees with the group K_{2}(R) defined in chapter III.

Several distinct models for BGL(R)^{+} are described in 1.9 below. We will construct even more models for BGL(R)^{+} in the rest of this chapter: the space P^{−1}P(R) of §3, the space ΩBQP(R) of §5 and the space Ω(iso S_{·}S) arising from the Waldhausen construction in §8.

Definition 1.1.1. Write K(R) for the product K_{0}(R) × BGL(R)^{+}. That is, K(R) is the disjoint union of copies of the connected space BGL(R)^{+}, one for each element of K_{0}(R). By construction, K_{0}(R) = π_{0}K(R). Moreover, it is clear that π_{n}K(R) = π_{n}BGL(R)^{+} = K_{n}(R) for n ≥ 1.

Functoriality 1.1.2. Each K_{n} is a functor from rings to abelian groups, while the topological spaces BGL(R)^{+} and K(R) are functors from rings to the homotopy category of topological spaces. However, without more information about the models used, the topological maps BGL(R)^{+} → BGL(R′)^{+} are only well-defined up to homotopy.

To see this, note that any ring map R → R′ induces a natural group map GL(R) → GL(R′), and hence a natural map BGL(R) → BGL(R′). This induces a map BGL(R)^{+} → BGL(R′)^{+}, unique up to homotopy, by Theorem 1.5 below. Thus the group maps K_{n}(R) → K_{n}(R′) are well defined. Since the identity of R induces the identity on BGL(R)^{+}, only composition remains to be considered. Given a second map R′ → R′′, the composition BGL(R) → BGL(R′) → BGL(R′′) is induced by R → R′′ because BGL is natural. By uniqueness in Theorem 1.5, the composition BGL(R)^{+} → BGL(R′)^{+} → BGL(R′′)^{+} must be homotopy equivalent to any a priori map BGL(R)^{+} → BGL(R′′)^{+}.

It is possible to modify the components of K(R) = K_{0}(R) × BGL(R)^{+} up to homotopy equivalence in order to form a homotopy-commutative H-space in a functorial way, using other constructions (see 4.11.1). Because the map K_{1}(R/I)→ K_{0}(R, I) is nontrivial (see III.2.3), K(R) is not the product of the H-space BGL(R)^{+} and the discrete group K_{0}(R) in a natural way.

Transfer maps 1.1.3. If R → S is a ring map such that S ≅ R^{d} as an Rmodule, the isomorphisms S^{m} ≅ R^{md} induce a group map GL(S) → GL(R) and hence a map BGL(S)^{+} → BGL(R)^{+}, again unique up to homotopy. On homotopy groups, the maps K_{n}(S) → K_{n}(R) are called transfer maps. We will see another construction of these maps in 6.3.2 below.

We shall be interested in the homotopy fiber of the map BGL(R) → BGL(R)^{+}.

Homotopy Fiber 1.2. The maps π_{∗}E → π_{∗}B induced by a continuous map E ⟶^{f} B can always be made to fit into a long exact sequence, in a natural way. The homotopy fiber F (f) of a f, relative to a basepoint ∗_{B} of B, is the space of pairs (e, γ), where e ∈ E and γ: [0, 1] → B is a path in B starting at the basepoint γ(0) = ∗_{B}, and ending at γ(1) = f(e). A sequence of based spaces F → E ⟶^{f} B with F → B constant is called a homotopy fibration sequence if the evident map F → F (f) (using γ(t) = ∗_{B}) is a homotopy equivalence.

The key property of the homotopy fiber is that (given a basepoint ∗_{E} with f(∗_{E}) = ∗_{B}) there is a long exact sequence of homotopy groups/pointed sets

⋯ π_{n+1}B ⟶^{∂}π_{n}F (f) → π_{n}E → π_{n}B ⟶^{∂} π_{n−1}F (f) → ⋯
⋯ ⟶^{∂}π_{1}F (f) → π_{1}E → π_{1}B ⟶^{∂} π_{0}F (f) → π_{0}E → π_{0}B.

When E → B is an H-map of H-spaces, F (f) is also an H-space, and the maps ending the sequence are product-preserving.

Acyclic Spaces and Acyclic Maps

The definition of BGL(R)^{+} fits into the general framework of acyclic maps, which we now discuss. Our discussion of acyclicity is taken from [HH] and [Berrick].

Definition 1.3 (Acyclic spaces). We call a topological space F acyclic if it has the homology of a point, that is, if H̃_{∗}(F; ℤ) = 0.

Lemma 1.3.1. Let F be an acyclic space. Then F is connected, its fundamental group G = π_{1}(F) is a perfect group, and H_{2}(G; ℤ) = 0 as well.

Proof. The acyclic space F must be connected, as H_{0}(F) = ℤ. Because G/[G, G] = H_{1}(F; ℤ) = 0, we have G = [G, G], i.e., G is a perfect group. To calculate H_{2}(G), observe that the universal covering space F̃ has H_{1}(F̃; ℤ) = 0. Moreover, the homotopy fiber (1.2) of the canonical map F → BG is homotopy equivalent to F̃; to see this, consider the long exact sequence of homotopy groups 1.2. The Serre Spectral Sequence for this homotopy fibration is E_{pq}^{2} = H_{p}(G; H_{q}(F̃; ℤ)) ⇒ H_{p+q}(F; ℤ) and the conclusion that H_{2}(G; ℤ) = 0 follows from the associated exact sequence of low degree terms:

H_{2}(F; ℤ) → H_{2}(G; ℤ) ⟶^{d^{2}} H_{1}(F̃; ℤ)^{G} → H_{1}(F; ℤ) → H_{1}(G; ℤ). □

Example 1.3.2 (Volodin Spaces). The Volodin space X(R) is an acyclic subspace of BGL(R), constructed as follows. For each n, let T_{n}(R) denote the subgroup of GL_{n}(R) consisting of upper triangular matrices with 1’s on the diagonal. As n varies, the union of these groups forms a subgroup T (R) of GL(R). Similarly we may regard the permutation groups Σ_{n} as subgroups of GL_{n}(R) by their representation as permutation matrices, and their union (the infinite permutation group Σ_{∞}) is a subgroup of GL(R). For each σ ∈ Σ_{n}, let T_{n}^{σ}(R) denote the subgroup of GL_{n}(R) obtained by conjugating T_{n}(R) by σ. For example, if σ = (n … 1) then T_{n}^{σ}(R) is the subgroup of lower triangular matrices.

Since the classifying spaces BT_{n}(R) and BT_{n}(R)^{σ} are subspaces of BGL_{n}(R), and hence of BGL(R), we may form their union over all n and σ: X(R) = ⋃_{n,σ} BT_{n}(R)^{σ}. The space X(R) is acyclic (see [Su81]). Since X(R) was first described by Volodin in 1971, it is usually called the Volodin space of R.

The image of the map π_{1}X(R) → π_{1}BGL(R) = GL(R) is the group E(R). To see this, note that π_{1}(X) is generated by the images of the π_{1}BT_{n}(R)^{σ}, the image of the composition π_{1}BT_{n}^{σ}(R) → π_{1}(X) → π_{1}BGL(R) = GL(R) is the subgroup T_{n}^{σ}(R) of E(R), and every generator e_{ij}(r) of E(R) is contained in some T_{n}^{σ}(R).

Definition 1.4 (Acyclic maps). Let X and Y be based connected CW complexes. A map f: X → Y is called acyclic if the homotopy fiber F (f) of f is acyclic (has the homology of a point). This implies that F (f) is connected and π_{1}F (f) is a perfect group.

From the exact sequence π_{1}F (f) → π_{1}(X) → π_{1}(Y) → π_{0}F (f) of homotopy groups/pointed sets, we see that if X → Y is acyclic then the map π_{1}(X) → π_{1}(Y) is onto, and its kernel P is a perfect normal subgroup of π_{1}(X).

Definition 1.4.1. Let P be a perfect normal subgroup of π_{1}(X), where X is a based connected CW complex. An acyclic map f: X → Y is called a +–construction on X (relative to P) if P is the kernel of π_{1}(X) → π_{1}(Y).

Example 1.4.2. If X is acyclic, the map X → point is acyclic. By Ex. 1.2, it is a +-construction.

When Quillen introduced the notion of acyclic maps in 1969, he observed that both Y and the map f are determined up to homotopy by the subgroup P. This is the content of the following theorem; its proof uses topological obstruction theory. Part (1) is proven in Ex. 1.4; an explicit proof may be found in §5 of [Berrick].

Theorem 1.5 (Quillen). Let P be a perfect normal subgroup of π_{1}(X). Then (1) There is a +–construction f: X → Y relative to P

(2) Let f: X → Y be a +–construction relative to P, and g: X → Z a map such that P vanishes in π_{1}(Z). Then there is a map h: Y → Z, unique up to homotopy, such that g = hf.

(3) In particular, if g is another +–construction relative to P, then the map h in (2) is a homotopy equivalence: h: Y ⟶^{∼} Z.

Remark 1.5.1. Every group G has a unique largest perfect subgroup P, called the perfect radical of G, and it is a normal subgroup of G; see Ex. 1.5. If no mention is made to the contrary, the notation X^{+} will always denote the +–construction relative to the perfect radical of π_{1}(X).

The first construction along these lines was announced by Quillen in 1969, so we have adopted Quillen’s term “+–construction” as well as his notation. A good description of his approach may be found in [HH] or [Berrick].

Lemma 1.6. Let X and Y be connected CW complexes. A map f: X → Y is acyclic if and only if H_{∗}(X, M) ≅ H_{∗}(Y, M) for every π_{1}(Y)-module M.

Proof. Suppose first that f is acyclic, with homotopy fiber F (f). Since the map π_{1}F (f) → π_{1}Y is trivial, π_{1}F (f) acts trivially upon M. By the Universal Coefficient Theorem, H_{q}(F (f); M) = 0 for q ≠ 0 and H_{0}(F (f); M) = M. Therefore E_{pq}^{2} = 0 for q ≠ 0 in the Serre Spectral Sequence for f:

E_{pq}^{2} = H_{p}(Y; H_{q}(F (f); M)) ⇒ H_{p+q}(X; M).

Hence the spectral sequence collapses to yield H_{p}(X; M) ⟶^{≅} H_{p}(Y; M) for all p.

Conversely, we suppose first that π_{1}Y = 0 and H_{∗}(X; ℤ) ≅ H_{∗}(Y; ℤ). By the Comparison Theorem for the Serre Spectral Sequences for F (f) → X ⟶^{f} Y and ∗ → Y ⟶^{=} Y, we have H̃_{∗}(F (f); ℤ) = 0. Hence F (f) and f are acyclic.

The general case reduces to this by the following trick. Let Ỹ denote the universal covering space of Y, and X̃ = X ×_{Y} Ỹ the corresponding covering space of X. Then there are natural isomorphisms H_{∗}(Ỹ; ℤ) ≅ H_{∗}(Y; M) and H_{∗}(X̃; ℤ) ≅ H_{∗}(X; M), where M = ℤ[π_{1}(Y)]. The assumption that H_{∗}(X; M) ≅ H_{∗}(Y; M) implies that the map f̃: X̃ → Ỹ induces isomorphisms on integral homology. But π_{1}(Ỹ) = 0, so by the special case above the homotopy fiber F (f̃) of f̃ is an acyclic space. But by path lifting we have F (f̃) ≅ F (f), so F (f) is acyclic. Thus f is an acyclic map. □

Recall from III.5.4 that every perfect group P has a universal central extension E → P, and that the kernel of this extension is the abelian group H_{2}(P; ℤ).

Proposition 1.7. Let P be a perfect normal subgroup of a group G, with corresponding +–construction f: BG → BG^{+}. If F (f) is the homotopy fiber of f then π_{1}F (f) is the universal central extension of P, and π_{2}(BG^{+}) ≅ H_{2}(P; ℤ).

Proof. We have an exact sequence π_{2}(BG) → π_{2}(BG^{+}) → π_{1}F (f) → G → G/P → 1. But π_{2}(BG) = 0, and π_{2}(BG^{+}) is in the center of π_{1}F (f) by [Wh, IV.3.5]. Thus π_{1}F (f) is a central extension of P with kernel π_{2}(BG^{+}). But F (f) is acyclic, so π_{1}F (f) is perfect and H_{2}(F; ℤ) = 0 by 1.3.1. By the Recognition Theorem III.5.4, π_{1}F (f) is the universal central extension of P. □

Recall from Theorem III.5.5 that the Steinberg group St(R) is the universal central extension of the perfect group E(R). Thus we have:

Corollary 1.7.1. The group K_{2}(R) = π_{2}BGL(R)^{+} is isomorphic to the group K_{2}(R) ≅ H_{2}(E(R); ℤ) of chapter III.

In fact, we will see in Ex. 1.8 and 1.9 that K_{n}(R) ≅ π_{n}(BE(R)^{+}) for all n ≥ 2, and K_{n}(R) ≅ π_{n}(BSt(R)^{+}) for all n ≥ 3, with K_{3}(R) ≅ H_{3}(St(R); ℤ).

Corollary 1.7.2. The fundamental group π_{1}X(R) of the Volodin space (1.3.2) is the Steinberg group St(R).

Construction Techniques

One problem with the +construction approach is the fact that BGL(R)^{+} is not a uniquely defined space. It is not hard to see that BGL(R)^{+} is an H-space (see Ex. 1.11). Quillen proved that that it is also an infinite loop space, and extends to an Ω-spectrum K(R). We omit the proof here, because it will follow from the + = Q theorem in section 7.

Here is one of the most useful recognition criteria, due to Quillen. The proof is is an application of obstruction theory, which we omit (but see [Ger72, 1.5].)

Theorem 1.8. The map i: BGL(R) → BGL(R)^{+} is universal for maps into H-spaces. That is, for each map f: BGL(R) → H, where H is an H-space, there is a map g: BGL(R)^{+} → H so that f = gi, and such that the induced map π_{i}(BGL(R)^{+}) → π_{i}(H) is independent of g.

Remark 1.8.1. If f_{∗} : H_{∗}(BGL(R), ℤ) ≅ H_{∗}(H, ℤ) is an isomorphism, then f is acyclic and g is a homotopy equivalence: BGL(R)^{+} ≃ H. This gives another characterization of BGL(R)^{+}. The proof is indicated in Exercise 1.3.

Constructions 1.9. Here are some ways that BGL(R)^{+} may be constructed:

(i) Using point-set topology, e.g., by attaching 2-cells and 3-cells to BGL(R). If we perform this construction over ℤ and let BGL(R)^{+} be the pushout of BGL(ℤ)^{+} and BGL(R) along BGL(ℤ), this gives a construction which is functorial in R. This method is described Ex. 1.4, and in the books [Berrick] and [Rosenberg].

(ii) By the Bousfield-Kan integral completion functor ℤ_{∞}: we set BGL(R)^{+} = ℤ_{∞}BGL(R). This approach, which is also functorial in R, is used in [Dror] and [Ger72].

(iii) “Group completing” the H-space ∐_{n=0}^{∞} BGL_{n}(R) yields an infinite loop space whose basepoint component is BGL(R)^{+}. This method will be discussed more in section 3, and is due to G. Segal [Segal].

(iv) By taking BGL of a free simplicial ring F_{∗} with an augmentation F_{0} → R such that F_{∗} → R is a homotopy equivalence, as in [Swan]. Swan showed that the simplicial group GL(F_{∗}) and the simplicial space BGL(F_{∗}) are independent (up to simplicial homotopy) of the choice of resolution F_{∗} → R, and that π_{1}BGL(F_{∗}) = π_{0}GL(F_{∗}) = E(R). The Swan K-theory space ΩK^{Sw}(R) is defined to be the homotopy fiber of BGL(F_{∗}) → BGL(R), and we set K_{i}^{Sw}(R) = π_{i−1}ΩK^{Sw}(R) for i ≥ 1 so that K_{1}^{Sw}(R) = K_{1}(R) by construction. The space ΩK^{Sw}(R) is a model for the loop space ΩBGL(R)^{+}.

As an application, if F is a free ring (without unit), we may take F_{∗} to be the constant simplicial ring, so ΩK^{Sw}(F) is contractible, and K_{i}^{Sw}(F) = 0 for all i. Gersten proved in [Ger74] (see V.6.5) that BGL(F)^{+} is contractible; this was used by Don Anderson [And72] to prove that the canonical map from GL(R) = ΩBGL(R) to ΩK^{Sw}(R) induces a homotopy equivalence ΩBGL(R)^{+} → ΩK^{Sw}(R).

(v) Volodin’s construction. Let X(R) denote the acyclic Volodin space of Example 1.3.2. By Ex. 1.6, the quotient group BGL(R)/X(R) is a model for BGL(R)^{+}.

An excellent survey of these constructions may be found in [Ger72], except for details on Volodin’s construction, which are in [Su81].

Products

If A and B are rings, any natural isomorphism φ_{pq} : A^{p} ⊗ B^{q} ≅ (A ⊗ B)^{pq} of A ⊗ B-modules allows us to define a “tensor product” homomorphism GL_{p}(A) × GL_{q}(B) → GL_{pq}(A⊗B). This in turn induces continuous maps φ_{p,q} : BGL_{p}(A)^{+}× BGL_{q}(B)^{+} → BGL_{pq}(A ⊗ B)^{+} → BGL(A ⊗ B)^{+}. A different choice of φ yields a tensor product homomorphism conjugate to the original, and a new map φ_{p,q} freely homotopic to the original. It follows that φ_{p,q} is compatible up to homotopy with stabilization in p and q.

Since the target is an H-space (Ex. 1.11), we can define new maps γ_{p,q}(a, b) = φ_{p,q}(a, b) − φ_{p,q}(a, ∗) − φ_{p,q}(∗, b), where ∗ denotes the basepoint. Since γ_{p,q}(a, ∗) = γ_{p,q}(∗, b) = ∗, and γ_{p,q} is compatible with stabilization in p, q, it induces a map, well defined up to weak homotopy equivalence

γ : BGL(A)^{+} ∧ BGL(B)^{+} → BGL(A ⊗ B)^{+}.

Combining γ with the reduced join π_{p}(X) ⊗ π_{q}(Y) → π_{p+q}(X ∧ Y) [Wh, p. 480] allows us to define a product map :

K_{p}(A) ⊗ K_{q}(B) → K_{p+q}(A ⊗ B).

Loday proved the following result in [Lo76].

Theorem 1.10. (Loday) The product map is natural in A and B, bilinear and associative, and if A is commutative, the induced product

K_{p}(A) ⊗ K_{q}(A) → K_{p+q}(A ⊗ A) → K_{p+q}(A)

is graded-commutative. Moreover, the special case K_{1}(A) ⊗ K_{1}(B) → K_{2}(A ⊗ B) coincides with the product defined in III.5.12.

Example 1.10.1. (Steinberg symbols) If r_{1}, ..., r_{n} are units of a commutative ring R, the product of the r_{i} ∈ K_{1}(R) is an element {r_{1}, ..., r_{n}} of K_{n}(R). These elements are called Steinberg symbols, since the products {r_{1}, r_{2}} ∈ K_{2}(R) agree with the Steinberg symbols of III.5.10. If F is a field, the universal property (III.7.1) of Milnor K-theory implies that there is a ring homomorphism K_{∗}^{M}(F) → K_{∗}(F). We will see in Ex. 1.12 that it need not be an injection.

Example 1.10.2. Associated to the unit x of Λ = ℤ[x, x^{−1}] we choose a map S^{1} → BGL(Λ)^{+}, representing [x] ∈ π_{1}BGL(Λ)^{+}. The pairing γ induces a map BGL(R)^{+}∧S^{1} to BGL(R[x, x^{−1}])^{+}. By adjunction, this yields a map BGL(R)^{+} → ΩBGL(R[x, x^{−1}])^{+}. A spectrum version of this map is given in Ex. 4.14.

Example 1.10.3. (The K-theory Assembly Map) If G is any group, the inclusion G ⊂ ℤ[G]^{×} = GL(ℤ[G]) induces a map BG → BGL(ℤ[G])^{+}. If R is any ring, the product map BGL(R)^{+} ∧ BGL(ℤ[G])^{+} → BGL(R[G])^{+} induces a map from BGL(R)^{+} ∧ (BG_{+}) to BGL(R[G])^{+}, where BG_{+} denotes the disjoint union of BG and a basepoint. By Ex. 1.14, there is also a map from K(R) ∧ (BG_{+}) to K(R[G]).

Now for any infinite loop space (or spectrum) E, and any pointed space X, the homotopy groups of the space E ∧ X give the generalized homology of X with coefficients in E, H_{n}(BG; E). For E = K(R), H_{n}(BG; K(R)) is the generalized homology of BG with coefficients in K(R).

The map H_{n}(BG; K(R)) = π_{n}(BGL(R)^{+} ∧BG_{+}) → K(R_{n}[G]) which we have

just constructed is called the K-theory Assembly Map, and it plays a critical role in the K-theory of group rings. It is due to Quinn and Loday [Lo76], who observed that for n = 0 it is just the map K_{0}(R) → K_{0}(R[G]), while for n = 1 it is the map K_{1}(R) ⊕ G/[G, G] → K_{1}(R[G]). The higher Whitehead Group W h_{n}(G) is defined to be π_{n−1} of the homotopy fiber of the map K(ℤ) ∧ (BG_{+}) → K(ℤ[G]). The above calculations show that W h_{0}(G) is Wall’s finiteness obstruction (II.2.4), and the classical Whitehead group W h_{1}(G) = K_{1}(ℤ[G])/{±G} of III.1.9.

If G is a torsionfree group, the Isomorphism Conjecture for G states that the assembly map H_{n}(BG; K(R)) → K_{n}(R[G]) should be an isomorphism for any regular ring R. There is a more general Isomorphism Conjecture for infinite groups with torsion, due to Farrell-Jones [FJ]; it replaces H_{n}(BG; K(R)) by the equivariant homology of E_{vc}G, an equivariant version of the universal covering space EG of BG relative to the class of virtually cyclic subgroups of G.

Relative K-groups

Relative groups 1.11.1. Given a ring homomorphism f: R → R′, let K(f) be the homotopy fiber of K(R) → K(R′), and set K_{n}(f) = π_{n}K(f). This construction is designed so that these relative groups fit into a long exact sequence:

⋯ K_{n+1}(R′) ⟶^{∂}K_{n}(f) → K_{n}(R) → K_{n}(R′) ⟶^{∂} ⋯
⋯ K_{1}(f) → K_{1}(R) → K_{1}(R′) ⟶^{∂}K_{0}(f) → K_{0}(R) → K_{0}(R′).

Using the functorial homotopy-commutative H-space structure on K(R) (see 1.1.2), it follows that each K_{n}(f), including K_{0}(f), is an abelian group.

When R′ = R/I for some ideal I, we write K(R, I) for K(R → R/I). It is easy to see (Ex. 1.15) that K_{0}(R, I) and K_{1}(R, I) agree with the relative groups defined in Ex. II.2.3 and III.2.2, and that the ending of this sequence is the exact sequence of III.2.3 and III.5.7. Keune and Loday have shown that K_{2}(R, I) agrees with the relative group defined in III.5.7.

Absolute Excision 1.11.2. A non-unital ring I is said to satisfy absolute excision for K_{n} if K_{n}(ℤ ⊕ I, I) ⟶^{≅} K_{n}(R, I) is an isomorphism for every unital ring R containing I as an ideal; ℤ ⊕ I is the canonical augmented ring (see Ex. I.1.10.) By II, Ex. 2.3, every I satisfies absolute excision for K_{0}. By III, Remark 2.2.1, I satisfies absolute excision for K_{1} if and only if I = I^{2}.

Suslin proved in [Su95] that I satisfies absolute excision for K_{n} if and only if the groups Tor_{i}^{ℤ⊕I}(ℤ, ℤ) vanish for i = 1, …, n. (Since Tor_{1}(ℤ, ℤ) = I/I^{2}, this recovers the result for K_{1}.) In homological algebra, a non-unital ring I is called H-unital if every Tor_{i}(ℤ, ℤ) vanishes; Suslin’s result says that I satisfies absolute excision for all K_{n} if and only if I is H-unital.

Together with a result of Suslin and Wodzicki [SuW], this implies that I satisfies absolute excision for K_{n} ⊗ ℚ if and only if I ⊗ ℚ satisfies absolute excision for K_{n}.

Suppose now that I = I^{2}. In this case the commutator subgroup of GL(I) is perfect (III, Ex. 2.10). By Theorem 1.5 there is a +-construction BGL(I)^{+} and a map from BGL(I)^{+} to the basepoint component of K(R, I). When I is H-unital, this is a homotopy equivalence; π_{n}BGL(I)^{+} ≅ K_{n}(R, I) for all n ≥ 1. This concrete version of absolute excision was proven by Suslin and Wodzicki in [SuW, 1.7].

Suspension Rings 1.11.3. Let C(R) be the cone ring of row-and-column finite matrices over a fixed ring R (Ex. I.1.8); by II.2.1.3, C(R) is flasque, so K(C(R)) is contractible by Ex. 1.17. The suspension ring S(R) of III, Ex. 1.15 is C(R)/M(R), where M(R) is the ideal of finite matrices over R. Since M(R) ≅ M(M(R)) and GL(R) = GL_{1}(M(R)), we have GL(R) ≅ GL(M(R)) and hence BGL(R)^{+} ≅ BGL(M(R))^{+}. Since M(R) is H-unital (Ex. 1.20), it satisfies absolute excision and we have a fibration sequence

K_{0}(R) × BGL(R)^{+} → BGL(C(R))^{+} → BGL(S(R))^{+}.

Since the middle term is contractible, this proves that K_{0}(R) × BGL(R)^{+} ≃ ΩBGL(S(R))^{+} so that K_{n+1}S(R) ≅ K_{n}(R) for all n ≥ 1. (K_{0}S(R) ≅ K_{−1}(R) by III, Ex. 4.10.) This result was first proven by Gersten and Wagoner.

K-theory of finite fields

Next, we describe Quillen’s construction for the K-theory of finite fields, arising from his work on the Adams Conjecture [Q70]. Adams had shown that the Adams operations ψ^{k} on topological K-theory (II.4.4) are represented by maps ψ^{k}: BU → BU in the sense that for each X the Adams operations on K̃U(X) are the maps:

K̃U(X) = [X, BU] ⟶^{[X,ψ^{k}]} [X, BU] = K̃U(X).

Fix a finite field 𝔽_{q} with q = p^{ν} elements. For each n, the Brauer lifting of the trivial and standard n-dimensional representations of GL_{n}(𝔽_{q}) are n-dimensional complex representations, given by homomorphisms 1_{n}, id_{n}: GL_{n}(𝔽_{q}) → U. Since BU is an H-space, we can form the difference ρ_{n} = B(id_{n}) − B(1_{n}) as a map BGL_{n}(𝔽_{q}) → BU. Quillen observed that ρ_{n} and ρ_{n+1} are compatible up to homotopy with the inclusion of BGL_{n}(𝔽_{q}) in BGL_{n+1}(𝔽_{q}). (See 5.3.1 below.) Hence there is a map ρ: BGL(𝔽_{q}) → BU, well defined up to homotopy. By Theorem 1.8, ρ induces a map from BGL(𝔽_{q})^{+} to BU, and hence maps ρ_{∗} : K_{n}(𝔽_{q}) → π_{n}(BU) = K̃U(S^{n}).

We will define operations λ^{k} and ψ^{k} on K_{∗}(𝔽_{q}) in 5.3.1 and Ex. 5.2 below, and show (5.5.2) that ψ^{p} is induced by the Frobenius on 𝔽_{q}, so that ψ^{q} is the identity map on K_{n}(𝔽_{q}). We will also see in 5.7 and 5.8 below that ρ_{∗} commutes with the operations λ^{k} and ψ^{k} on K_{n}(𝔽_{q}) and K̃U(S^{n}).

Theorem 1.12. (Quillen) The map BGL(𝔽_{q})^{+} → BU identifies BGL(𝔽_{q})^{+} with the homotopy fiber of ψ^{q} − 1. That is, the following is a homotopy fibration.

BGL(𝔽_{q})^{+} ⟶^{ρ} BU ⟶^{ψ^{q}−1} BU

On homotopy groups, II.4.4.1 shows that ψ^{q} is multiplication by q^{i} on π_{2i}BU = K̃U(S^{2i}). Using the homotopy sequence 1.2 and 1.12, we immediately deduce:

Corollary 1.13. For every finite field 𝔽_{q}, and n ≥ 1, we have

K_{n}(𝔽_{q}) = π_{n}BGL(𝔽_{q})^{+} ≅ \{ \begin{matrix}ℤ/(q^{i} − 1) & n = 2i − 1, \\ 0 & n even. \end{matrix}

Moreover, if 𝔽_{q} ⊂ 𝔽_{q′} then K_{n}(𝔽_{q}) → K_{n}(𝔽_{q′}) is an injection, identifying K_{n}(𝔽_{q}) with K_{n}(𝔽_{q′})^{G}, where G = Gal(𝔽_{q′}/𝔽_{q}); the transfer map K_{n}(𝔽_{q′}) → K_{n}(𝔽_{q}) is onto (see 1.1.3).

Remark 1.13.1. Clearly all products in the ring K_{∗}(𝔽_{q}) are trivial. We will see in section 2 that it is also possible to put a ring structure on the homotopy groups with mod-ℓ coefficients, K_{n}(𝔽_{q}; ℤ/ℓ) = π_{n}(BGL(𝔽_{q}); ℤ/ℓ).

If ℓ | (q − 1), the long exact sequence for homotopy with mod-ℓ coefficients (2.2) shows that K_{n}(𝔽_{q}; ℤ/ℓ) ≅ ℤ/ℓ for all n ≥ 0. The choice of a primitive unit ζ ∈ 𝔽_{q}^{×} and a primitive ℓth root of unity ω gives generators ζ for K_{1}(𝔽_{q}; ℤ/ℓ) and the Bott element β for K_{2}(𝔽_{q}; ℤ/ℓ), respectively. (The Bockstein sends β to ω ∈ K_{1}(𝔽_{q}).) Browder has shown [Br] that K_{∗}(𝔽_{q}; ℤ/ℓ) ≅ ℤ/ℓ[β, ζ]/(ζ^{2}) as a graded ring, and that the natural isomorphism from the even part ⊕_{n}K_{2n}(𝔽_{q}; ℤ/ℓ) ≅ ℤ/ℓ[β] to ⊕π_{2n}(BU; ℤ/ℓ) is a ring isomorphism.

If p ≠ ℓ, the algebraic closure 𝔽̄_{p} is the union of the 𝔽_{q} where q = p^{ν} and ℓ | (q − 1). Hence the ring K_{∗}(𝔽̄_{p}; ℤ/ℓ) is the direct limit of the K_{∗}(𝔽_{q}; ℤ/ℓ). As each ζ vanishes and the Bott elements map to each other, we have:

K_{∗}(𝔽̄_{p}; ℤ/ℓ) ≅ ℤ/ℓ[β] ≅ π_{∗}(BU; ℤ/ℓ).

Remark 1.13.2. Browder has also shown in [Br, 2.4] that the Bott element β in K_{2}(ℂ; ℤ/m) maps to a generator of π_{2}(BU; ℤ/m) = ℤ/m under the change-of-topology map. Hence the map K_{∗}(ℂ; ℤ/m) → π_{∗}(BU; ℤ/m) is also onto. We will see in VI.1.4 that it is an isomorphism.

Homological Stability

Homological stability, the study of how the homology of a group like GL_{n}(R) depends upon n, plays an important role in algebraic K-theory. The following theorem was proven by Suslin in [Su82], using Volodin’s construction of K(R). Recall from Ex. I.1.5 that the stable range of R, sr(R), is defined in terms of unimodular rows; if R is commutative and noetherian it is at most dim(R) + 1.

Theorem 1.14. Let R be a ring with stable range sr(R). For r ≥ max{2n + 1, n + sr(R)} the maps π_{n}BGL_{r}(R)^{+} → π_{n}BGL_{r+1}(R)^{+} are isomorphisms.

Now assume that r > sr(R) + 1, so that E_{r}(R) is a perfect normal subgroup of GL_{r}(R) by Ex. III.1.3. The universal covering space of BGL_{r}(R)^{+} is then homotopy equivalent to BE_{r}(R)^{+} for (by Ex. 1.8). Applying the Hurewicz theorem (and the Comparison Theorem) to these spaces implies:

Corollary 1.14.1. In the range r ≥ max{2n + 1, n + sr(R)}, the following maps are isomorphisms:

H_{n}(BGL_{r}(R)) → H_{n}(BGL_{r+1}(R)) → H_{n}(BGL(R)^{+})
H_{n}(BE_{r}(R)) → H_{n}(BE_{r+1}(R)) → H_{n}(BE(R)^{+})

For example, suppose that R is an Artinian ring, so that sr(R) = 1 by Ex. I.1.5. Then π_{n}BGL_{r}(R)^{+} ≅ K_{n}(R) and H_{n}(BGL_{r}(R)) ≅ H_{n}(BGL(R)^{+}) for all r > 2n. The following result, due to Suslin [Su-KM], improves this bound for fields.

Proposition 1.15. (Suslin) If F is an infinite field, H_{n}(GL_{r}(F))→H_{n}(GL(F)) is an isomorphism for all r ≥ n. In addition, there is a canonical isomorphism H_{n}(GL_{n}(F))/im H_{n}(GL_{n−1}(F)) ≅ K_{n}^{M}(F).

Proposition 1.16. (Kuku) If R is a finite ring, then K_{n}(R) is a finite abelian group for all n > 0.

Proof. The case n = 1 follows from III.1.2.5 (or Ex. III.1.2): K_{1}(R) is a quotient of R^{×}. Since K_{n}(R) = π_{n}BE(R)^{+} for n > 1 by Ex. 1.8, it suffices to show that the homology groups H_{n}(E(R); ℤ) are finite for n > 0. But each E_{r}(R) is a finite group, so the groups H_{n}(BE_{r}(R); ℤ) are indeed finite for n > 0. □

Rank of K_{n} over number fields

It is a well known theorem of Cartan and Serre that the “rational” homotopy groups π_{n}(X) ⊗ ℚ of an H-space X inject into the rational homology groups H_{n}(X; ℚ), and that π_{∗}(X)⊗ℚ forms the primitive elements in the coalgebra structure on H_{∗}(X; ℚ). (See [MM, p. 163].) For X = BGL(R)^{+}, which is an H-space by Ex. 1.11, this means that the groups K_{n}(R) ⊗ ℚ = π_{n}(BGL(R)^{+}) ⊗ ℚ inject into the groups H_{∗}(GL(R); ℚ) = H_{∗}(BGL(R); ℚ) = H_{∗}(BGL(R)^{+}; ℚ) as the primitive elements. For X = BSL(R)^{+}, this means that the groups K_{n}(R) ⊗ ℚ inject into H_{∗}(SL(R); ℚ) as the primitive elements for n ≥ 2.

Now suppose that A is a finite dimensional semisimple algebra over ℚ, such as a number field, and that R is a subring of A which is finitely generated over ℤ and has R ⊗ℚ = A (R is an order). In this case, Borel determined the ring H^{∗}(SL_{m}(R); ℚ) and hence the dual coalgebra H_{∗}(SL_{m}(R); ℚ) and hence its primitive part, K_{∗}(R)⊗ ℚ. (See the review MR0387496 of Borel’s paper [Bor] by Garland.) The answer only depends upon the semisimple ℝ-algebra A ⊗_{ℚ} ℝ.

More concretely, let 𝔤 and 𝔨 be the Lie algebras (over ℚ) of SL_{m}(A ⊗ ℝ) and one of its maximal compact subgroups K. Borel first proved in [Bor, Thm. 1] that

H^{q}(SL_{m}(R); ℝ) ≅ H^{q}(SL_{m}(A); ℝ) ≅ H^{q}(𝔤, 𝔨; ℝ) for m ≫ q.

By the above remarks, this proves:

Theorem 1.17 (Borel). Let A be a finite dimensional semisimple ℚ-algebra. Then for every order R in A we have K_{n}(R) ⊗ ℚ ≅ K_{n}(A) ⊗ ℚ for all n ≥ 2.

Borel also calculated the ranks of these groups. Since A is a finite product of simple algebras A_{i}, and K_{n}(A) is the product of the K_{n}(A_{i}) by Ex. 1.7, we may assume that A is simple, i.e., a matrix algebra over a division algebra. The center of A is then a number field F. It is traditional to write r_{1} and r_{2} for the number of factors of ℝ and ℂ in the ℝ-algebra F ⊗_{ℚ} ℝ, so that F ⊗_{ℚ} ℝ ≅ (ℝ)^{r_{1}} × (ℂ)^{r_{2}}. Borel proved in [Bor, thm. 2] [Bor1, 12.2] that H^{∗}(SL(A), ℝ) is a tensor product of r_{1} exterior algebras having generators x_{i} in degrees 4i + 1 (i ≥ 1) and r_{2} exterior algebras having generators x_{j} in degrees 2j + 1 (j ≥ 1). Taking primitive parts, this proves the following result:

Theorem 1.18 (Borel). Let F be a number field, and A a central simple F - algebra. Then for n ≥ 2 we have K_{n}(A) ⊗ ℚ ≅ K_{n}(F) and

rank K_{n}(A) ⊗ ℚ = \{ \begin{matrix}r_{2}, & n ≡ 3 (mod 4) \\ r_{1} + r_{2}, & n ≡ 1 (mod 4) \\ 0, & else. \end{matrix}

By Theorem 1.17, this also gives the rank of K_{n}(R) for every order R. In particular, these groups are torsion for every even n ≥ 2.

Regulator Maps 1.18.1. Borel’s construction provides a specific map from K_{n}(R) to the real vector space P_{n} of primitives in H_{n}(SL(R), ℝ); Borel observed that the image is a lattice Λ. There is another canonical lattice Λ′ in P_{n}: the image of π_{n}(X) for the symmetric space X contained in K\GL_{m}(ℂ). The higher regulator of R is defined to be the positive real number R_{n} such that the volume of P_{n}/Λ is R_{n} times the volume of P_{n}/Λ′. Borel also proved that R_{2i+1} was a positive rational number times \sqrt{D} π^{−m(i+1)}ζ_{F} (i + 1), where D is the discriminant of F/ℚ and ζ_{F} is the zeta function of F.

Group Rings 1.18.2. The group ring ℤ[G] of a finite group G is an order in the semisimple algebra ℚ[G]. Therefore Theorem 1.18 gives the rank of the groups K_{n}(ℤ[G]) for n ≥ 2. The rank of K_{1}(ℤ[G]) was given in III.1.8, and does not follow this pattern. For example, if C_{p} is a cyclic group of prime order p ≥ 3 then r_{1} + r_{2} = (p + 1)/2 yet K_{1}(ℤ[C_{p}]) has rank (p − 3)/2.

K_{3}(R) and H_{3}(E(R), ℤ)

The following material is due to Suslin [Su91]. Given an element α of π_{n}(X) and an element β of π_{m}(S^{n}), the composition product α ∘ β is the element of π_{m}(X) represented by S^{m} ⟶^{β} S^{n} ⟶^{α} X. We will apply this to the Hopf element η ∈ π_{3}(S^{2}), using the following observation.

If Y_{n} is the wedge of n copies of S^{2}, the Hilton-Milnor Theorem [Wh, XI(8.1)] says that

ΩΣY_{n} ≃ ∏_{i=1}^{n} ΩS^{3} × ∏_{i≠j} ΩS^{5} × Y′_{n},

where Y′_{n} is 5-connected and Σ is suspension. Note that π_{3}(ΩΣS^{2}) = π_{4}(S^{3}) = ℤ/2, on the image of η ∈ π_{3}(S^{2}). Hence π_{3}(ΩΣY_{n}) ≅ (ℤ/2)^{n}. If Y_{I} is a wedge of copies of S^{2} indexed by an infinite set I then (taking the filtered colimit over finite subsets of I) it follows that π_{3}(ΩΣY_{I}) ≅ ⊕_{I} ℤ/2, generated by the factors S^{2} → ΩS^{2} → Y_{I}.

Lemma 1.19. If X is a simply connected loop space, the composition product with η and the Hurewicz map h : π_{3}(X) → H_{3}(X, ℤ) fit into an exact sequence

π_{2}(X) ⟶^{∘η} π_{3}(X) ⟶^{h} H_{3}(X, ℤ) → 0.

Proof. Let I be a set of generators of π_{2}(X); the maps f(i) : S^{2} → X induce

a map f : Y → X, where Y = ∨_{I} S^{2}. The map f factors as Y → ΩΣY ⟶^{Ωf^{∗}} X, where X = ΩX′ and f^{∗} : ΣY → X′ is the adjoint of f. Since π_{2}(Y) → π_{2}(X) is onto, the sequence π_{3}(ΩΣY) → π_{3}(X) → H_{3}(X) → 0 is exact by Exercise 1.25.

As above, π_{3}(ΩΣY) ≅ ⊕_{I} π_{3}(ΩΣS^{2}), and the i^{th} factor is the image of π_{3}(S^{2}), generated by η. The map π_{3}(ΩΣY) → π_{3}(X) sends the generator of the i^{th} factor to the composition product f(i) ∘ η : S^{3} ⟶^{η} S^{2} → ΩΣS^{2} ⟶ X. Since π_{2}(X) is generated by the f(i), the result follows. □

Remark 1.19.1. (Suslin) Lemma 1.19 holds for any simply connected H-space X. To see this, note that the Hilton-Milnor Theorem for Y_{n} = ∨S^{2} states that the space ΩY_{n} is homotopy equivalent to ∏_{i} ΩS^{2} × ∏_{i≠j} ΩS^{3} × Y′′ where Y′′ is 3-connected. Thus π_{3}(Y_{n}) = π_{2}(ΩY_{n}) is the sum of ℤ^{n} = ⊕π_{3}(S^{2}) and ⊕_{i≠j}π_{3}(S^{3}), where the second summand is generated by the Whitehead products [ι_{i}, ι_{j}] of the generators of π_{2}(Y). These Whitehead products map to [f(i), f(j)], which vanish in π_{3}(X) when X is any H-space by [Wh, X(7.8)]. With this modification, the proof of Lemma 1.19 goes through.

Corollary 1.20. For any ring R the product with [−1] ∈ K_{1}(ℤ) fits into an exact sequence

K_{2}(R) ⟶^{[−1]} K_{3}(R) ⟶^{h} H_{3}(E(R), ℤ) → 0.

Proof. By Ex. 1.12(a), the map π_{3}(S^{2}) → K_{1}(ℤ) sends η to [−1]. Since X = BGL(R^{+}) is an H-space, the composition product π_{2}(X) ⟶^{η} π_{3}(X) is multiplication by the image of η in π_{1}(X) = K_{1}(R), namely [−1]; see Ex. 1.12(e). The result follows from Lemma 1.19 and the observation in Exercise 1.8 that π_{n}BE(R)^{+} → K_{n}(R) is an isomorphism for n ≥ 2. □

EXERCISES

1.1 (Kervaire) Let X be a homology n-sphere, i.e., a space with H_{∗}(X) = H_{∗}(S^{n}). Show that there is a homotopy equivalence S^{n} → X^{+}. Hint: Show that π_{1}(X) is perfect if n ≠ 1, so X^{+} is simply connected, and use the Hurewicz theorem.

The binary icosohedral group Γ = SL_{2}(𝔽_{5}) embeds in O_{3}(ℝ) as the symmetry group of both the dodecahedron and icosahedron. Show that the quotient X = S^{3}/Γ is a homology 3-sphere, and conclude that the canonical map S^{3} → X^{+} is a homotopy equivalence. (The fact that it is a homology sphere was discovered by Poincaré in 1904, and X is sometimes called the Poincaré sphere.) 1.2 a) If F is an acyclic space, show that F^{+} is contractible.

b) If X ⟶^{f} Y is acyclic and f_{∗} : π_{1}(X) ≅ π_{1}(Y), show that f is a homotopy equivalence. 1.3 Prove the assertions in Remark 1.8.1 using the following standard result: Let X and Y be H-spaces having the homotopy type of a CW complex. If f : X → Y is a map which induces an isomorphism H_{∗}(X, ℤ) ≅ H_{∗}(Y, ℤ), show that f is a homotopy equivalence. Hint: Since π_{1}(Y) acts trivially on the homotopy fiber F by [Wh, IV.3.6], the relative Hurewicz theorem [Wh, IV.7.2] shows that π_{∗}(F) = 0. 1.4 Here is a point-set construction of X^{+} relative to a perfect normal subgroup P. Form Y by attaching one 2-cell e_{p} for each element of P, so that π_{1}(Y) = π_{1}(X)/P. Show that H_{2}(Y; ℤ) is the direct sum of H_{2}(X; ℤ) and the free abelian group on the set {[e_{p}] : p ∈ P }. Next, prove that each homology class [e_{p}] is represented by a map h_{p}: S^{2} → Y, and form Z by attaching 3-cells to Y (one for each p ∈ P) using the h_{p}. Finally, prove that Z is a model for X^{+}. 1.5 Perfect Radicals. Show that the subgroup generated by the union of perfect subgroups of any group G is itself a perfect subgroup. Conclude that G has a largest perfect subgroup P, called the perfect radical of G, and that it is a normal subgroup of G.

1.6 Let cone(i) denote the mapping cone of a map F ⟶^{i} X. If F is an acyclic space, show that the map X → cone(i) is acyclic. If F is a subcomplex of X then cone(i) is homotopy equivalent to the quotient space X/F, so X → X/F is also acyclic. Conclude that if X(R) is the Volodin space of Example 1.3.2 then BGL(R)/X(R) is a model for BGL(R)^{+}. Hint: Consider long exact sequences in homology. 1.7 Show that BGL(R_{1} × R_{2})^{+} ≃ BGL(R_{1})^{+} × BGL(R_{2})^{+} and hence K_{n}(R_{1} × R_{2}) ≅ K_{n}(R_{1}) × K_{n}(R_{2}) for every pair of rings R_{1}, R_{2} and every n. Hint: Use 3.1(6) below to see that BGL(R_{1} × R_{2}) ≅ BGL(R_{1}) × BGL(R_{2}). 1.8 Let P be a perfect normal subgroup of G, and let BG → BG^{+} be a +- construction relative to P. Show that BP^{+} is homotopy equivalent to the universal covering space of BG^{+}. Hence π_{n}(BP^{+}) ≅ π_{n}(BG^{+}) for all n ≥ 2. Hint: BP is homotopy equivalent to a covering space of BG.

For G = GL(R) and P = E(R), this shows that BE(R)^{+} is homotopy equivalent to the universal covering space of BGL(R)^{+}. Thus K_{n}(R) ≅ π_{n}BE(R)^{+} for n ≥ 2.

(a) If R is a commutative ring, show that SL(R) ↪ GL(R) induces isomorphisms π_{n}BSL(R)^{+} ≅ K_{n}(R) for n ≥ 2, and π_{1}BSL(R)^{+} ≅ SK_{1}(R). Conclude that the map BSL(R)^{+} × B(R^{×}) → BGL(R)^{+} is a homotopy equivalence.

(b) If A is a finite semisimple algebra over a field, the subgroups SL_{n}(A) of GL_{n}(A) were defined in III.1.2.4. Show that SL(A) ↪ GL(A) induces isomorphisms π_{n}BSL(A)^{+} ≅ K_{n}(A) for n ≥ 2, and π_{1}BSL(A)^{+} ≅ SK_{1}(A).

1.9 Suppose that A → S → P is a universal central extension (III.5.3). In particular, S and P are perfect groups. Show that there is a homotopy fibration BA → BS^{+} → BP^{+}. Conclude that π_{n}(BS^{+}) = 0 for n ≤ 2, and that π_{n}(BS^{+}) ≅ π_{n}(BP^{+}) ≅ π_{n}(BG^{+}) for all n ≥ 3. In particular, π_{3}(BP^{+}) ≅ H_{3}(S; ℤ).

Since the Steinberg group St(R) is the universal central extension of E(R), this shows that K_{n}(R) ≅ π_{n}St(R)^{+} for all n ≥ 3, and that K_{3}(R) ≅ H_{3}(St(R); ℤ).

1.10 For n ≥ 3, let P_{n} denote the normal closure of the perfect group E_{n}(R) in GL_{n}(R), and let BGL_{n}(R)^{+} denote the +–construction on BGL_{n}(R) relative to P_{n}. Corresponding to the inclusions GL_{n} ⊂ GL_{n+1} we can choose a sequence of maps BGL_{n}(R)^{+} → BGL_{n+1}(R)^{+}. Show that \underrightarrow{lim} BGL_{n}(R)^{+} is BGL(R)^{+}.

1.11 For each m and n, the group map □: GL_{m}(R) × GL_{n}(R) → GL_{m+n}(R) ⊂ GL(R) induces a map BGL_{m}(R) × BGL_{n}(R) → BGL(R) → BGL(R)^{+}. Show that these maps induce an H-space structure on BGL(R)^{+}.

1.12 In this exercise, we develop some properties of BΣ_{∞}^{+}, where Σ_{∞} denotes the union of the symmetric groups Σ_{n}. We will see in 4.9.3 that π_{n}(BΣ_{∞}^{+}) is the stable homotopy group π_{n}^{s}. The permutation representations Σ_{n} → GL_{n}(ℤ) (1.3.2) induce a map BΣ_{∞}^{+} → BGL(ℤ)^{+} and hence homomorphisms π_{n}^{s} → K_{n}(ℤ).

(a) Show that η ∈ π_{1}^{s} ≅ ℤ/2 maps to [−1] ∈ K_{1}(ℤ).

(b) Show that the subgroups Σ_{m} × Σ_{n} of Σ_{m+n} induce an H-space structure on BΣ_{∞}^{+} such that BΣ_{∞}^{+} → BGL(ℤ)^{+} is an H-map. (See Ex. 1.11.)

(c) Modify the construction of Loday’s product (1.10) to show that product representations Σ_{m} × Σ_{n} → Σ_{mn} induce a map BΣ_{∞}^{+} ∧ BΣ_{∞}^{+} → BΣ_{∞}^{+} compatible with the corresponding map for BGL(ℤ)^{+}. The resulting product π_{m}^{s} ⊗ π_{n}^{s} → π_{m+n}^{s} makes the stable homotopy groups into a gradedcommutative ring, and makes π_{∗}^{s} → K_{∗}(ℤ) into a ring homomorphism.

(d) Show that the Steinberg symbol {−1, −1, −1, −1} of 1.10.1 vanishes in K_{4}(ℤ) and K_{4}(ℚ). Since this symbol is nonzero in K_{4}^{M}(ℚ) by III.7.2(c,d), this shows that the Milnor K-groups of a field need not inject into its Quillen K-groups. Hint: η^{3} ≠ 0 in π_{3}^{s} but η^{4} = 0 in π_{4}^{s}.

(e) If β ∈ π_{n+t}(S^{n}) and α ∈ K_{n}(R), show that the composition product α ∘ β in K_{n+t}(R) agrees with the product of α with [β] ∈ π_{t}^{s}.

1.13 Let A_{∞} denote the union of the alternating groups A_{n}; it is a subgroup of Σ_{∞} of index 2. A_{∞} is a perfect group, since the A_{n} are perfect for n ≥ 5.

(a) Show that BΣ_{∞}^{+} ≃ BA_{∞}^{+} × B(ℤ/2), so π_{n}BA_{∞}^{+} ≅ π_{n}^{s} for all n ≥ 2.

(b) Use Lemma 1.19 and π_{3}^{s} ≅ ℤ/24 to conclude that H_{3}(A_{∞}, ℤ) ≅ ℤ/12.

(c) Use the Künneth formula and (a) to show that H_{3}(Σ_{∞}, ℤ) ≅ H_{3}(A_{∞}, ℤ) ⊕ (ℤ/2)^{2}. This calculation was first done by Nakaoka [Nak].

1.14 Extend the product map γ of Theorem 1.10 to a map K(A) ∧ K(B) → K(A ⊗ B), so that the induced maps K_{0}(A) × K_{n}(B) → K_{n}(A ⊗ B) agree with the products defined in III.1.6.1 and Ex. III.5.4.

1.15 Let I be an ideal in R. Show that the group π_{0}K(R, I) of 1.11.1 is isomorphic to the group K_{0}(I) of Ex. II.2.3, and that the maps K_{1}(R/I) → K_{0}(I) → K_{0}(R) in loc. cit. agree with the maps of 1.11.1. Hint: π_{0}K(R ⊕ I, 0 ⊕ I) must be K_{0}(I).

Use Ex. III.2.7 to show that π_{1}K(R → R/I) is isomorphic to the group K_{1}(R, I) of III.2.2, and that the maps K_{2}(R/I) → K_{1}(R, I) → K_{1}(R) in III.5.7.1 agree with those of 1.11.1.

1.16 If f : R → S is a ring homomorphism, show that the relative group K_{0}(f) of 1.11.1 agrees with the relative group K_{0}(f) of II.2.10.

1.17 (Wagoner) We say GL(R) is flabby if there is a homomorphism τ : GL(R) → GL(R) so that for each n the restriction τ_{n} : GL_{n}(R) → GL(R) of τ is conjugate to the map (1, τ_{n}) : g ↦ ({g \atop 0} {0 \atop τ_{n}(g)}). In particular, τ_{n} and (1, τ_{n}) induce the same map H_{∗}(BGL_{n}(R)) → H_{∗}(BGL(R)) by [WHomo, 6.7.8].

(a) Assuming that GL(R) is flabby, show that BGL(R) is acyclic. By Ex. 1.2, this implies that BGL(R)^{+} is contractible, i.e., that K_{n}(R) = 0 for n > 0. Hint: The H-space structure (Ex. 1.11) makes H_{∗}(BGL(R)) into a ring.

(b) Show that GL(R) is flabby for every flasque ring R (see II.2.1.3). This shows that flasque rings have K_{n}(R) = 0 for all n. Hint: Modify Ex. II.2.15(a).

1.18 Suppose that I is a nilpotent ideal and that p^{ν}I = 0 for some ν. Show that H_{∗}(GL(R); M) ≅ H_{∗}(GL(R/I); M) for every uniquely p-divisible module M. Conclude that the relative groups K_{∗}(R, I) are p-groups.

1.19 Suppose that I is a nilpotent ideal in a ring R, and that I is uniquely divisible as an abelian group. Show that H_{∗}(GL(R); M) ≅ H_{∗}(GL(R/I); M) for every torsion module M. Conclude that the relative groups K_{∗}(R, I) are uniquely divisible abelian groups.

1.20 Show that every ring with unit is H-unital (see 1.11.2). Then show that a non-unital ring I is H-unital if every finite subset of I is contained in a unital subring. (This shows that the ring M(R) of finite matrices over R is H-unital.)

Finally, show that I is H-unital if for every finite subset {a_{j}} of I there is an e ∈ I such that ea_{j} = a_{j}. (An example of such an I is the non-unital ring of functions with compact support on ℂ^{n}.)

1.21 (Morita invariance) For each n > 0, we saw in III.1.1.4 that GL(R) ≅ GL(M_{n}(R)) via isomorphisms M_{m}(R) ≅ M_{m}(M_{n}(R)). Deduce that there is a homotopy equivalence BGL(R)^{+} ≃ BGL(M_{n}(R))^{+} and hence isomorphisms K_{∗}(R) ≅ K_{∗}(M_{n}(R)). (The cases ∗ = 0, 1, 2 were given in II.2.7, III.1.6.4 and III.5.6.1.) We will give a more categorical proof in 6.3.5 below.

Compare this to the approach of 1.11.2, using M(R).

1.22 (Loday symbols) Let a_{1}, ..., a_{n} be elements of A so that a_{n}a_{1} = 0 and each a_{i}a_{i+1} = 0. Show that the elementary matrices e_{n,1}(a_{n}) and e_{i,i+1}(a_{i}) commute and define a ring homomorphism B = ℤ[x_{1}, 1/x_{1}, …, x_{n}, 1/x_{n}] → M_{n}(A). Using Ex. 1.21, we define the Loday symbol ⟨⟨a_{1}, ..., a_{n}⟩⟩ in K_{n}(A) to be the image of {x_{1}, ..., x_{n}} under K_{n}(B) → K_{n}(M_{n}(A)) ≅ K_{n}(A).

1.23 Let F → E ⟶^{p} B and F′ → E′ ⟶^{p′} B′ be homotopy fibrations (1.2), and suppose given pairings e : E ∧ X → E′, b : B ∧ X → B′ so that p′e = b(p ∧ 1).

ΩB ∧ X ⟶^{∂∧1} F ∧ X ⟶ E ∧ X ⟶^{p∧1} B ∧ X
↓^{Ωb}  ↓^{f}  ↓^{e}  ↓^{b}
ΩB′  ⟶^{∂′} F′ ⟶ E′  ⟶^{p′} B′.

Show that there is a pairing F ∧ X ⟶^{f} F′ compatible with e and such that for β ∈ π_{∗}(B) and γ ∈ π_{∗}(X) the reduced join [Wh, p. 480] satisfies ∂(β ∧_{b} γ) = ∂(β) ∧_{f} γ. 1.24 Let f : A → B be a ring homomorphism and let K(f) (resp., K(f_{C}) be the relative groups (1.11.1), i.e., the homotopy fiber of K(A) → K(B) (resp., K(A ⊗ C) → K(B ⊗ C)). Use Ex. 1.23 to show that there is an induced pairing K_{∗}(f) ⊗ K_{∗}(C) → K_{∗}(f_{C}) such that for β ∈ K_{∗}(B) and γ ∈ K_{∗}(C) we have ∂(β ∧ γ) = ∂(β) ∧ γ in K_{∗}(f_{C}).

When f is an R-algebra homomorphism, show that K_{∗}(f) is a right K_{∗}(R)- module and that the maps in the relative sequence K_{n+1}(B) → K_{n}(f) → K_{n}(A) → K_{n}(B) of (1.11.1) are K_{∗}(R)-module homomorphisms. 1.25 Suppose given a homotopy fibration sequence F → Y → X with X, Y and F simply connected. Compare the long exact homotopy sequence (see 1.2) with the exact sequence of low degree terms in the Leray-Serre Spectral sequence (see [WHomo, 5.3.3]) to show that there is an exact sequence π_{3}(Y) → π_{3}(X) → H_{3}(X) → 0. 1.26 The Galois group G = Gal(𝔽_{q^{i}}/𝔽_{q}) acts on the group μ of units of 𝔽_{q^{i}}, and also on the i-fold tensor product μ^{⊗i} = μ ⊗ ⋯ ⊗ μ. By functoriality 1.1.2, G acts on K_{∗}(𝔽_{q}). Show that K_{2i−1}(𝔽_{q}) is isomorphic to μ^{⊗i} as a G-module. 1.27 Monomial matrices Let F be a field and consider the subgroup M of GL(F) consisting of matrices with only one nonzero entry in each row and column.

(a) Show that M is the wreath product F^{×} ≀ Σ_{∞}, and contains F^{×} ≀ A_{∞}.

(b) Show that [M, M] is the kernel of det : F^{×}≀A_{∞} → F^{×}, so H_{1}(M) ≅ F^{×}×Σ_{2}.

(c) Show that [M, M] is perfect, and BM^{+} ≃ B[M, M]^{+} × B(F^{×}) × BΣ_{2}.

§2. K-theory with finite Coefficients

In addition to the usual K-groups K_{i}(R), or the K-groups K_{i}(C) of a category C, it is often useful to study its K-groups with coefficients “mod ℓ” K_{i}(R; ℤ/ℓ) (or K_{i}(C; ℤ/ℓ),) where ℓ is a positive integer. In this section we quickly recount the basic construction from mod ℓ homotopy theory. Basic properties of mod ℓ homotopy theory may be found in [N].

Recall [N] that if m ≥ 2 the mod ℓ Moore space P^{m}(ℤ/ℓ) is the space formed from the sphere S^{m−1} by attaching an m-cell via a degree ℓ map. It is characterized as having only one nonzero reduced integral homology group, namely H̃^{m}(P) = ℤ/ℓ. The suspension of P^{m}(ℤ/ℓ) is the Moore space P^{m+1}(ℤ/ℓ), and as m varies these fit together to form a suspension spectrum P^{∞}(ℤ/ℓ), called the Moore spectrum.

Definition 2.1. If m ≥ 2, the mod ℓ homotopy “group” π_{m}(X; ℤ/ℓ) of a based topological space X is defined to be the pointed set [P^{m}(ℤ/ℓ), X] of based homotopy classes of maps from the Moore space P^{m}(ℤ/ℓ) to X.

For a general space X, π_{2}(X; ℤ/ℓ) isn’t even a group, but the π_{m}(X; ℤ/ℓ) are always groups for m ≥ 3 and abelian groups for m ≥ 4 [N]. If X is an H-space, such as a loop space, then these bounds improve by one. If X = ΩY then we can define π_{1}(X; ℤ/ℓ) as π_{2}(Y; ℤ/ℓ); this is independent of the choice of Y by Ex. 2.1. More generally, if X = Ω^{k}Y_{k} for k >> 0 and P^{m} = P^{m}(ℤ/ℓ) then the formula

π_{m}(X; ℤ/ℓ) = [P^{m}, X] = [P^{m}, Ω^{k}Y_{k}] ≅ [P^{m+k}, Y_{k}] = π_{m+k}(Y_{k}; ℤ/ℓ)

shows that we can ignore these restrictions on m, and that π_{m}(X; ℤ/ℓ) is an abelian group for all m ≥ 0 (or even negative m, as long as k > 2 + |m|).

In particular, if X is an infinite loop space then abelian groups π_{m}(X; ℤ/ℓ) are defined for all m ∈ ℤ, using the explicit sequence of deloopings of X provided by the given structure on X.

2.1.1. If F → E → B is a Serre fibration there is a long exact sequence of groups/pointed sets (which is natural in the fibration):

⋯ → π_{m+1}(B; ℤ/ℓ) →π_{m}(F; ℤ/ℓ) → π_{m}(E; ℤ/ℓ) →
π_{m}(B; ℤ/ℓ) →π_{m−1}(F; ℤ/ℓ) → ⋯ → π_{2}(B; ℤ/ℓ).

This is just a special case of the fact that ⋯ → [P, F] → [P, E] → [P, B] is exact for any CW complex P; see [Wh, III.6.18*].

If m ≥ 2, the cofibration sequence S^{m−1} ⟶^{ℓ} S^{m−1} ⟶ P^{m}(ℤ/ℓ) defining P^{m}(ℤ/ℓ) induces an exact sequence of homotopy groups/pointed sets

π_{m}(X) ⟶^{ℓ} π_{m}(X) → π_{m}(X; ℤ/ℓ) ⟶^{∂} π_{m−1}(X) ⟶^{ℓ} π_{m−1}(X).

If ℓ is odd, or divisible by 4, F. Peterson showed that there is even a non-canonical splitting π_{m}(X; ℤ/ℓ) → π_{m}(X)/ℓ (see [Br, 1.8]).

It is convenient to adopt the notation that if A is an abelian group then _{ℓ}A denotes the subgroup of all elements a of A such that ℓ · a = 0. This allows us to restate the above exact sequence in a concise fashion.

Universal Coefficient Sequence 2.2. For all m ≥ 3 there is a natural short exact sequence

0 → (π_{m}X) ⊗ ℤ/ℓ → π_{m}(X; ℤ/ℓ) ⟶^{∂} _{ℓ}(π_{m−1}X) → 0.

This sequence is split exact (but not naturally) when ℓ ≢ 2 mod 4.

For π_{2}, the sequence 2.2 of pointed sets is also exact in a suitable sense; see [N, p. 3]. However this point is irrelevant for loop spaces, so we ignore it.

Example 2.2.1. When ℓ = 2, the sequence need not split. For example, it is known that π_{m+2}(S^{m}; ℤ/2) = ℤ/4 for m ≥ 3, and that π_{2}(BO; ℤ/2) = π_{3}(O; ℤ/2) = ℤ/4; see [AT65].

Here is another way to define mod ℓ homotopy groups, and hence K_{∗}(R; ℤ/ℓ).

Proposition 2.3. Suppose that X is a loop space, and let F denote the homotopy fiber of the map X → X which is multiplication by ℓ. Then π_{m}(X; ℤ/ℓ) ≅ π_{m−1}(F) for all m ≥ 2.

Proof. (Neisendorfer) Let Maps(A, X) be the space of pointed maps. If S = S^{k} is the k-sphere then the homotopy groups of Maps(S^{k}, X) are the homotopy groups of X (reindexed by k), while if P = P^{k}(ℤ/ℓ) is a mod ℓ Moore space, the homotopy groups of Maps(P, X) are the mod ℓ homotopy groups of X (reindexed by k).

Now applying Maps(−, X) to a cofibration sequence yields a fibration sequence, and applying Maps(A, −) to a fibration sequence yields a fibration sequence; this may be formally deduced from the axioms (SM0) and (SM7) for any model structure, which hold for spaces. Applying Maps(−, X) to S^{k} → S^{k} → P^{k+1}(ℤ/ℓ) shows that Maps(P, X) is the homotopy fiber of Maps(S^{k}, X) → Maps(S^{k}, X). Applying Maps(S^{k}, −) to F → X → X shows that Maps(S^{k}, F) is also the homotopy fiber, and is therefore homotopy equivalent to Maps(P, X). Taking the homotopy groups yields the result. □

Spectra 2.3.1. For fixed ℓ, the Moore spectrum P^{∞}(ℤ/ℓ) is equivalent to the (spectrum) cofiber of multiplication by ℓ on the sphere spectrum. If E is a spectrum, then (by S-duality) the homotopy groups π_{∗}(E; ℤ/ℓ) = \underrightarrow{lim} π_{∗+r}(E_{r}; ℤ/ℓ) are the same as the homotopy groups of the spectrum E ∧ P^{∞}(ℤ/ℓ).

Now suppose that C is either a symmetric monoidal category, or an exact category, or a Waldhausen category. We will construct a K-theory space K(C) below (in 4.3, 6.3 and 8.5); in each case K(C) is an infinite loop space.

Definition 2.4. The mod ℓ K-groups of R are defined to be the abelian group:

K_{m}(R; ℤ/ℓ) = π_{m}(K(R); ℤ/ℓ), m ∈ ℤ.

Similarly, if the K-theory space K(C) of a category C is defined then the mod ℓ K-groups of C are defined to be K_{m}(C; ℤ/ℓ) = π_{m}(K(C); ℤ/ℓ).

By 2.1.1, if C_{1} → C_{2} → C_{3} is a sequence such that K(C_{1}) → K(C_{2}) → K(C_{3}) is a fibration, then there is a long exact sequence of abelian groups

⋯ → K_{n+1}(C_{3}; ℤ/ℓ) → K_{n}(C_{1}; ℤ/ℓ) → K_{n}(C_{2}; ℤ/ℓ) → K_{n}(C_{3}; ℤ/ℓ) ⋯

If m ≥ 2 this definition states that K_{m}(R; ℤ/ℓ) = [P^{m}(ℤ/ℓ), K(R)]. Because K(R) ≃ ΩY, we can define K_{1}(R; ℤ/ℓ) in a way that is independent of the choice of Y (Ex. 2.1); it agrees with the definition in III.1.7.4 (see Ex. 2.2). However, the groups K_{0}(R; ℤ/ℓ) and K_{m}(R; ℤ/ℓ) for m < 0 depend not only upon the loop space K(R), but also upon the choice of the deloopings of K(R) in the underlying Ktheory spectrum K(R). In fact, the literature is not consistent about K_{m}(R; ℤ/ℓ) when m < 2, even for K_{1}(R; ℤ/ℓ). Similar remarks apply to the definition of K_{m}(C; ℤ/ℓ).

By Universal Coefficients 2.2, the mod ℓ K-groups are related to the usual Kgroups:

Universal Coefficient Theorem 2.5. There is a short exact sequence

0 → K_{m}(R) ⊗ ℤ/ℓ → K_{m}(R; ℤ/ℓ) → _{ℓ}K_{m−1}(R) → 0

for every m ∈ ℤ, C, and ℓ. It is split exact unless ℓ ≡ 2 mod 4. Ex. 2.3 shows that the splitting is not natural in R.

Similarly, if the K-theory of a category C is defined then we have an exact sequence

0 → K_{m}(C) ⊗ ℤ/ℓ → K_{m}(C; ℤ/ℓ) → _{ℓ}K_{m−1}(C) → 0

Example 2.5.1. (ℓ = 2) Since the isomorphism Ω^{∞}Σ^{∞} → ℤ × BO factors through K(ℤ) and K(ℝ), the universal coefficient theorem and 2.2.1 show that

K_{2}(ℤ; ℤ/2) ≅ K_{2}(ℝ; ℤ/2) ≅ π_{2}(BO; ℤ/2) = ℤ/4.

It turns out [AT65] that for ℓ = 2 the sequence for K_{m}(R; ℤ/2) is split whenever multiplication by [−1] ∈ K_{1}(ℤ) is the zero map from K_{m−1}(R) to K_{m}(R). For example, this is the case for the finite fields 𝔽_{q}, an observation made in [Br].

Example 2.5.2 (Bott elements). Suppose that R contains a primitive ℓ^{th} root of unity ζ. The Universal Coefficient Theorem 2.5 provides an element β ∈ K_{2}(R; ℤ/ℓ), mapping to ζ ∈ _{ℓ}K_{1}(R). This element is called the Bott element, and it plays an important role in the product structure of the ring K_{∗}(R; ℤ/ℓ). For finite fields, this role was mentioned briefly in Remark 1.13.1.

Remark 2.5.3. A priori, β depends not only upon ζ but also upon the choice of the splitting in 2.5. One way to choose β is to observe that the inclusion of μ_{ℓ} in GL_{1}(R) induces a map Bμ_{ℓ} → BGL(R) → BGL(R)^{+} and therefore a set function μ_{ℓ} → K_{2}(R; ℤ/ℓ). A posteriori, it turns out that this is a group homomorphism unless ℓ ≡ 2 (mod 4).

Example 2.6. Let k be the algebraic closure of the field 𝔽_{p}. Quillen’s computation of K_{∗}(𝔽_{q}) in 1.13 shows that K_{n}(k) = 0 for m even (m ≥ 2), and that K_{m}(k) = ℚ/ℤ[\frac{1}{p}] for m odd (m ≥ 1). It follows that if ℓ is prime to p then:

K_{m}(k; ℤ/ℓ) = \{ \begin{matrix}ℤ/ℓ & if m is even, m ≥ 0 \\ 0 & otherwise. \end{matrix}

In fact, K_{∗}(k; ℤ/ℓ) is the polynomial ring ℤ/ℓ[β] on the Bott element β ∈K_{2}(k; ℤ/ℓ), under the K-theory product of 2.8 below. See 1.13.1 (and Chapter VI, 1.3.1) for more details.

The next result shows that we may always assume that ℓ is a power of a prime.

Proposition 2.7. If ℓ = q_{1}q_{2} with q_{1} and q_{2} relatively prime, then π_{m}(X; ℤ/ℓ) is naturally isomorphic to π_{m}(X; ℤ/q_{1}) × π_{m}(X; ℤ/q_{1}).

Proof. Set P_{1} = P^{m}(ℤ/q_{1}), P_{2} = P^{m}(ℤ/q_{2}) and P = P_{1}∨P_{2}. Since P has only one nonzero reduced integral homology group, namely H̃^{m}(P) = ℤ/q_{1} × ℤ/q_{2} ≅ ℤ/ℓ, the natural map P → P^{m}(ℤ/ℓ) must be a homotopy equivalence. But then π_{m}(X; ℤ/ℓ) is naturally isomorphic to

[P^{m}(ℤ/q_{1}) ∨ P^{m}(ℤ/q_{2}), X] ≅ [P^{m}(ℤ/q_{1}), X] × [P^{m}(ℤ/q_{2}), X],

which is the required group π_{m}(X; ℤ/q_{1}) × π_{m}(X; ℤ/q_{1}). □

Products

If ℓ ≥ 3 is prime, there is a homotopy equivalence P^{m}(ℤ/ℓ^{ν}) ∧ P^{n}(ℤ/ℓ^{ν}) ≃ P^{m+n}(ℤ/ℓ^{ν})∨P^{m+n−1}(ℤ/ℓ^{ν}). The projections onto the first factor give a spectrum “product” map P^{∞}(ℤ/ℓ^{ν})∧P^{∞}(ℤ/ℓ^{ν}) → P^{∞}(ℤ/ℓ^{ν}) which is homotopy associative and commutative unless ℓ^{ν} = 3. (The same thing is true when ℓ = 2, except the product map does not exist if 2^{ν} = 2, it is not homotopy associative if 2^{ν} = 4 and it is not homotopy commutative when 2^{ν} = 4, 8.) These facts are due to Araki and Toda, and follow by S-duality from [N, 8.5–6]. So from now on, we shall exclude the pathological cases ℓ^{ν} = 2, 3, 4, 8.

If E is a homotopy associative and commutative ring spectrum, then so is the spectrum E ∧ P^{∞}(ℤ/ℓ^{ν}), unless ℓ^{ν} = 2, 3, 4, 8. Applying this to E = K(R) yields the following result.

Theorem 2.8. Let R be a commutative ring, and suppose ℓ^{ν} ≠ 2, 3, 4, 8. Then K(R) ∧ P^{∞}(ℤ/ℓ^{ν}) is a homotopy associative and commutative ring spectrum. In particular, K_{∗}(R; ℤ/ℓ^{ν}) is a graded-commutative ring.

Scholium 2.8.1. Browder [Br] has observed that if π_{m}(E) = 0 for all even m > 0 (and m < 0) then E∧P^{∞}(ℤ/ℓ^{ν}) is a homotopy associative and commutative ring spectrum even for ℓ^{ν} = 2, 3, 4, 8. This applies in particular to E = K(𝔽_{q}), as remarked in 1.13.1 and 2.6 above.

Corollary 2.8.2. If ℓ ≥ 3 and R contains a primitive ℓ^{th} root of unity ζ, and β ∈ K_{2}(R; ℤ/ℓ) is the Bott element (2.5.2), there is a graded ring homomorphism ℤ/ℓ[ζ, β] → K_{∗}(R; ℤ/ℓ).

If ζ ∉ R, there are elements β′ ∈ K_{2ℓ}(R; ℤ/ℓ) and ζ′ ∈ K_{2ℓ−1}(R; ℤ/ℓ) whose images in K_{2ℓ}(R[ζ]; ℤ/ℓ) and K_{2ℓ−1}(R[ζ]; ℤ/ℓ) are β^{ℓ−1} and β^{ℓ−2}ζ, respectively.

Proof. The first assertion is immediate from 2.8 and 2.5.2. For the second assertion we may assume that R = ℤ. Then the Galois group G of ℤ[ζ] over ℤ is cyclic of order ℓ − 1, and we define β′ to be the image of −β^{ℓ−1} ∈ K_{2ℓ}(ℤ[ζ]; ℤ/ℓ) under the transfer map i_{∗} (1.1.3). Since i^{∗}i_{∗} is ∑_{g∈G} g^{∗} by Ex. 6.13,

i^{∗}β′ = − ∑ g^{∗}β^{ℓ−1} = −(ℓ − 1)β^{ℓ−1}.

Similarly, ζ′ = i_{∗}(−ζβ^{ℓ−2}) has i^{∗}ζ′ = ζβ^{ℓ−2}. □

2.9 The ℓ-adic completion. Fix a prime ℓ. The ℓ-adic completion of a spectrum E, Ê_{ℓ}, is the homotopy limit (over ν) of the spectra E ∧ P^{∞}(ℤ/ℓ^{ν}). We let π_{n}(E; ℤ_{ℓ}) denote the homotopy groups of this spectrum; if E = K(R) we write K_{n}(R; ℤ_{ℓ}) for the homotopy groups π_{n}(K(R); ℤ_{ℓ}) of the ℓ-adic completion K̂(R)_{ℓ}. There is an extension

0 → \underleftarrow{lim}^{1}π_{n+1}(E; ℤ/ℓ^{ν}) → π_{n}(E; ℤ_{ℓ}) → \underleftarrow{lim} π_{n}(E; ℤ/ℓ^{ν}) → 0.

If the homotopy groups π_{n+1}(E; ℤ/ℓ^{ν}) are finite, the \underleftarrow{lim}^{1} term vanishes and, by Universal Coefficients 2.5, π_{n}(E; ℤ_{ℓ}) is an extension of the Tate module of π_{n−1}(E) by the ℓ-adic completion of π_{n}(E). (The (ℓ-primary) Tate module of an abelian group A is the inverse limit of the groups Hom(ℤ/ℓ^{ν}, A).) For example, the Tate module of K_{1}(ℂ) = ℂ^{×} is ℤ_{ℓ}, so K_{1}(ℂ; ℤ_{ℓ}) = π_{1}(K(ℂ); ℤ_{ℓ}) is ℤ_{ℓ}.

If E is a homotopy associative and commutative ring spectrum then so is the homotopy limit Ê_{p}. Thus π_{∗}(E; ℤ_{ℓ}), and in particular K_{∗}(R; ℤ_{ℓ}), is also a gradedcommutative ring.

We conclude with Gabber’s Rigidity Theorem [Gabber]. If I is an ideal in a commutative ring A, we say that (A, I) is a Hensel pair if for every finite commutative A-algebra C the map C → C/IC induces a bijection on idempotents. A Hensel local ring is a commutative local ring R such that (A, 𝔪) is a Hensel pair. These conditions imply that I is a radical ideal (Ex. I.2.1), and (A, I) is a Hensel pair whenever I is complete by Ex. I.2.2(i).

Rigidity Theorem 2.10. If (A, I) is a Hensel pair and 1/ℓ ∈ A, then for all

n ≥ 1 we have K_{n}(A; ℤ/ℓ) ⟶^{≅} K_{n}(A/I; ℤ/ℓ), and H̃_{∗}(GL(I), ℤ/ℓ) = 0

Gabber proves that H̃_{∗}(GL(I), ℤ/ℓ) = 0, and observes that this is equivalent to K_{n}(A; ℤ/ℓ) → K_{n}(A/I; ℤ/ℓ) being onto.

Example 2.10.1. If 1/ℓ ∈ R then K_{n}(R[[x]]; ℤ/ℓ) ≅ K_{n}(R; ℤ/ℓ) for all n ≥ 0.

Examples 2.10.2. A restriction like n ≥ 0 is necessary. Les Reid [Reid] has given an example of a 2-dimensional hensel local ℚ-algebra with K_{−2}(A) = ℤ, and Drinfeld [Drin] has shown that K_{−1}(I) = 0.

EXERCISES

2.1 Suppose that X is a loop space. Show that π_{1}(X; ℤ/ℓ) is independent of the choice of Y such that X ≃ ΩY. This shows that K_{1}(R; ℤ/ℓ) and even K_{1}(C; ℤ/ℓ) are well defined. 2.2 Show that the group K_{1}(R; ℤ/ℓ) defined in 2.4 is isomorphic to the group defined in III.1.7.4. Using the Fundamental Theorem III.3.7 (and III.4.1.2), show that K_{0}(R; ℤ/ℓ) and even the groups K_{n}(R; ℤ/ℓ) for n < 0 which are defined in 2.4 are isomorphic to the corresponding groups defined in Ex. III.4.6. 2.3 Let R be a Dedekind domain with fraction field F. Show that the kernel of the map K_{1}(R; ℤ/ℓ) → K_{1}(F; ℤ/ℓ) is SK_{1}(R)/ℓ. Hence it induces a natural map

ℓ Pic(R) ⟶^{ρ} F^{×}/F^{×ℓ}R^{×}.

Note that F^{×}/R^{×} is a free abelian group by I.3.6, so the target is a free ℤ/ℓ-module for every integer ℓ. Finally, use I.3.6 and I.3.8.1 to give an elementary description of ρ.

In particular, If R is the ring of integers in a number field F, the Bass-Milnor-Serre Theorem III.2.5 shows that the extension K_{1}(R; ℤ/ℓ) of _{ℓ}Pic(R) by R^{×}/R^{×ℓ} injects into F^{×}/F^{×ℓ}, and that _{ℓ}K_{0}(R) is not a natural summand of K_{1}(R; ℤ/ℓ). (If 1/ℓ ∈ R, the étale Chern class K_{1}(R; ℤ/ℓ) → H_{et}^{1}(Spec(R), μ_{ℓ}) of V.11.10 is an isomorphism.) 2.4 If n ≥ 2, there is a Hurewicz map π_{n}(X; ℤ/ℓ) → H_{n}(X; ℤ/ℓ) sending the class of a map f : P^{n} → X to f_{∗}[e], where [e] ∈ H_{n}(P^{n}; ℤ/ℓ) ≅ H_{n}(S^{n}; ℤ/ℓ) is the canonical generator. Its restriction to π_{n}(X)/ℓ is the reduction modulo ℓ of the usual Hurewicz homomorphism π_{n}(X) → H_{n}(X; ℤ).

(a) If n ≥ 3, show that the Hurewicz map is a homomorphism. (If n = 2 and ℓ is odd, it is also a homomorphism.) Hint: Since P^{n} is a suspension, there is a comultiplication map P^{n} → P^{n} ∨ P^{n}.

If n = 2 and ℓ is even, the Hurewicz map h may not be a homomorphism, even if X is an infinite loop space. The precise formula is: h(a+b) = h(a)+h(b)+(ℓ/2){∂a, ∂b}. (See [We93].)

(b) In example 2.5.1, show that the Hurewicz map from K_{2}(ℝ; ℤ/2) ≅ ℤ/4 to H_{2}(SL(ℝ); ℤ/2) ≅ ℤ/2 is nonzero on β and 2β = {−1, −1}, but zero on 3β.

(c) If n = 2, show that the Hurewicz map is compatible with the action of π_{2}(X) on π_{2}(X; ℤ/ℓ) and on H_{2}(X; ℤ/ℓ).

2.5 Show that K_{∗}(R; ℤ/ℓ^{ν}) is a graded module over K_{∗}(R), associated to the evident pairing K(R) ∧ K(R) ∧ P^{∞}(ℤ/ℓ^{ν}) → K(R) ∧ P^{∞}(ℤ/ℓ^{ν}). 2.6 Fix a prime ℓ and let ℤ/ℓ^{∞} denote the union of the groups ℤ/ℓ^{ν}, which is a divisible torsion group. Show that there is a space P^{m}(ℤ/ℓ^{∞}) = \underrightarrow{lim} P^{m}(ℤ/ℓ^{ν}) such that π_{m}(X; ℤ/ℓ^{∞}) = [P^{m}(ℤ/ℓ^{∞}), X] is the direct limit of the π_{m}(X; ℤ/ℓ^{ν}). Then show that there is a universal coefficient sequence for m ≥ 3:

0 → (π_{m}X) ⊗ ℤ/ℓ^{∞} → π_{m}(X; ℤ/ℓ^{∞}) ⟶^{∂} (π_{m−1}X)_{ℓ-tors} → 0.

§3. Geometric realization of a small category

Recall (II.6.1.3) that a “small” category is a category whose objects form a set. If C is a small category, its geometric realization BC is a CW complex constructed naturally out of C. By definition, BC is the geometric realization |NC| of the nerve NC of C; see 3.1.4 below. However, it is characterized in a simple way.

Characterization 3.1. The realization BC of a small category C is the CW complex uniquely characterized up to homeomorphism by the following properties. Let n denote the category with n objects {0, 1, ⋯, n − 1}, with exactly one morphism i → j for each i ≤ j; n is an ordered set, regarded as a category.

(1) (Naturality) A functor F: C → D induces a cellular map BF: BC → BD, BF ∘ BG = B(F G) and B(id_{C}) is the identity map on BC.

(2) Bn is the standard (n − 1)–simplex Δ^{n−1}. The functor ϕ : i → n induces the simplicial map Δ^{i−1} → Δ^{n−1} sending vertex j to vertex ϕ(j).

(3) BC is the colimit colim_{Φ} Bi, where Φ is the category whose objects are functors n → C, and whose morphisms are factorizations i → n → C. The corresponding map Bi → Bn is given by (2).

The following useful properties are consequences of this characterization: (4) If C is a subcategory of D, BC is a subcomplex of BD;

(5) If C is the coproduct of categories C_{α}, BC = ∐ BC_{α};

(6) B(C × D) is homeomorphic to (BC) × (BD), where the product is given the compactly generated topology;

Here are some useful special cases of (2) for small n:

B0 = ∅ is the empty set, because 0 is the empty category.

B1 = {0} is a one-point space, since 1 is the one object-one morphism category. B2 = [0, 1] is the unit interval, whose picture is: 0 · ⟶ · 1. B3 is the 2-simplex; the picture of this identification is:

1_{·}

f_{0} ↗ ↘^{f_{1}}

0 · ⟶_{f_{1}∘f_{0}} · 2

The small categories form the objects of a category CAT, whose morphisms are functors. By (1), we see that geometric realization is a functor from CAT to the category of CW complexes and cellular maps.

Recipe 3.1.1. The above characterization of the CW complex BC gives it the following explicit cellular decomposition. The 0-cells (vertices) are the objects of C. The 1-cells (edges) are the morphisms in C, excluding all identity morphisms, and they are attached to their source and target. For each pair (f, g) of composable maps in C, attach a 2-simplex, using the above picture of B3 as the model. (Ignore pairs (f, g) where either f or g is an identity.) Inductively, given an n-tuple of composable maps in C (none an identity map), c_{0} → c_{1} → ⋯ → c_{n}, attach an n-simplex, using B(n + 1) as the model. By (3), BC is the union of these spaces, equipped with the weak topology.

Notice that this recipe implies a canonical cellular homeomorphism between BC and the realization BC^{op} of the opposite category C^{op}. In effect, the recipe doesn’t notice which way the arrows run.

Example 3.1.2. Let C_{2} be the category with one object and two morphisms, 1 and σ, with σ^{2} = 1. The recipe tells us that BC_{2} has exactly one n-cell for each n, attached to the (n−1)-cell by a map of degree 2 (corresponding to the first and last faces of the n-simplex). Therefore the n-skeleton of BC_{2} is the projective n-space ℝℙ^{n}, and their union BC_{2} is the infinite projective space ℝℙ^{∞}.

Example 3.1.3. Any group G (or monoid) may be regarded as a category with one object. The realization BG of this category is the space studied in Section 1. The recipe 3.1.1 shows that BG has only one vertex, and one 1-cell for every nontrivial element of G.

Although the above recipe gives an explicit description of the cell decomposition of BC, it is a bit vague about the attaching maps. To be more precise, we shall assume that the reader has a slight familiarity with the basic notions in the theory of simplicial sets, as found for example in [WHomo] or [May]. A simplicial set X is a contravariant functor Δ → Sets, where Δ denotes the subcategory of ordered sets on the objects {0, 1, ..., n, ...}. Alternatively, it is a sequence of sets X_{0}, X_{1}, …, together with “face” maps ∂_{i}: X_{n} → X_{n−1} and “degeneracy maps” σ_{i}: X_{n} → X_{n+1} (0 ≤ i ≤ n), subject to certain identities for the compositions of these maps.

We may break down the recipe for BC into two steps: we first construct a simplicial set NC, called the nerve of the category C, and then set BC = |NC|.

Definition 3.1.4 (The nerve of C). The nerve NC of a small category C is the simplicial set defined by the following data. Its n-simplices are functors c: n + 1 → C, i.e., diagrams in C of the form

c_{0} → c_{1} → ⋯ → c_{n}.

The i^{th} face ∂_{i}(c) of this simplex is obtained by deleting c_{i} in the evident way; to get the i^{th} degeneracy σ_{i}(c), one replaces c_{i} by c_{i} ⟶^{=} c_{i}.

The geometric realization |X_{·}| of a simplicial set X_{·} is defined to be the CW complex obtained by following the recipe 3.1.1 above, attaching an n-cell for each nondegenerate n-simplex x, identifying the boundary faces of the simplex with the (n − 1)-simplices indexed by the ∂_{i}x. See [WHomo, 8.1.6] or [May, §14] for more details.

BC is defined as the geometric realization |NC| of the nerve of C. From this prescription, it is clear that BC is given by recipe 3.1.1 above.

By abuse of notation, we will say that a category is contractible, or connected, or has any other topological property if its geometric realization has that property. Similarly, we will say that a functor F: C → D is a homotopy equivalence if BF is a homotopy equivalence BC ≃ BD.

Homotopy-theoretic properties 3.2. A natural transformation η: F_{0} ⇒ F_{1} between two functors F_{i}: C → D gives a homotopy BC × [0, 1] → BD between the maps BF_{0} and BF_{1}. This follows from (4) and (6) of 3.1, because η may be viewed as a functor from C × 2 to D whose restriction to C × {i} is F_{i}.

As a consequence, any adjoint pair of functors L: C → D, R: D → C induces a homotopy equivalence between BC and BD, because there are natural transformations LR ⇒ id_{D} and id_{C} ⇒ RL.

Example 3.2.1 (Smallness). Any equivalence C_{0} ⟶^{F} C between small categories induces a homotopy equivalence BC_{0} ⟶^{∼} BC, because F has an adjoint.

In practice, we will often work with a category C, such as P(R) or M(R), which is not actually a small category, but which is skeletally small (II.6.1.3). This means that C is equivalent to a small category, say to C_{0}. In this case, we can use BC_{0} instead of the mythical BC, because any other choice for C_{0} will have a homotopy equivalent geometric realization. We shall usually overlook this fine set-theoretic point in practice, just as we did in defining K_{0} in chapter II.

Example 3.2.2 (Initial objects). Any category with an initial object is contractible, because then the natural functor C → 1 has a left adjoint. Similarly, any category with a terminal object is contractible.

For example, suppose given an object d of a category C. The comma category C/d of objects over d has as its objects the morphisms f: c → d in C with target d. A morphism in the comma category from f to f′: c′ → d is a morphism h: c → c′ so that f = f′h. The comma category C/d is contractible because it has a terminal object, namely the identity map id_{d}: d ⟶^{=} d. The dual comma category d\C with objects d → c is similar, and left to the reader.

Example 3.2.3 (Comma categories). Suppose given a functor F: C → D and an object d of D. The comma category F/d has as its objects all pairs (c, f) with c an object in C and f a morphism in D from F (c) to d. By abuse of notation, we shall write such objects as F (c) ⟶^{f} d. A morphism in F/d from this object to

F (c′) ⟶^{f′} d is a morphism h: c → c′ in C so that the following diagram commutes in D.

F (c) ⟶^{F (h)} F (c′)
f ↘  ↙_{f′}
d

There is a canonical forgetful functor j: F/d → C, j(c, f) = c, and there is a natural transformation η_{(c,f)} = f from the composite F ∘j: F/d → D to the constant functor with image d. So B(F ∘ j) is a contractible map. It follows that there is a natural continuous map from B(F/d) to the homotopy fiber of BC → BD.

There is a dual comma category d\F, whose objects are written as d ⟶ F (c), and morphisms are morphisms h: c → c′ in C. It also has a forgetful functor to C, and a map from B(d\F) to the homotopy fiber of BC → BD. In fact, d\F = (d/F^{op})^{op}.

In the same spirit, we can define comma categories F/D (resp., D\F); an object is just an object of F/d (resp., of d\F) for some d in D. A morphism in F/D from (c, F (c) → d) to (c′, F (c′) → d′) is a pair of morphisms c → c′, d → d′ so that the two maps F (c) → d′ agree; there is an evident forgetful functor F/D → C × D. A morphism in D\F from (c, d → F (c)) to (c′, d′ → F (c′)) is a pair of morphisms c → c′, d′ → d so that the two maps d′ → F (c′) agree; there is an evident forgetful functor D\F → D^{op} × C.

The set π_{0} of components of a category

The set π_{0}(X) of connected components of any CW complex X can be described as the set of vertices modulo the incidence relation of edges. For BC this takes the following form. Let obj(C) denote the set of objects of C, and write π_{0}(C) for π_{0}(BC).

Lemma 3.3. Let ∼ be the equivalence relation on obj(C) which is generated by the relation that c ∼ c′ if there is a morphism in C between c and c′. Then

π_{0}(C) = obj(C)/ ∼ .

Translation categories 3.3.1. Suppose that G is a group, or even a monoid, acting on a set X. The translation category G∫ X is defined as the category whose objects are the elements of X, with Hom(x, x′) = {g ∈ G|g·x = x′}. By Lemma 3.3, π_{0}(G∫ X) is the orbit space X/G. The components of G∫ X are described in Ex. 3.2.

Thinking of a G-set X as a functor G → CAT, the translation category becomes a special case of the following construction, due to Grothendieck.

Example 3.3.2. Let I be a small category. Given a functor X: I → Sets, let I∫ X denote the category of pairs (i, x) with i an object of I and x ∈ X(i), in which a morphism (i, x) → (i′, x′) is a morphism f: i → i′ in I with X(f)(x) = x′. By Lemma 3.3 we have π_{0}(I∫ X) = colim_{i∈I} X(i).

More generally, given a functor X: I → CAT, let I∫ X denote the category of pairs (i, x) with i an object of I and x an object of X(i), in which a morphism (f, ϕ): (i, x) → (i′, x′) is given by a morphism f: i → i′ in I and a morphism ϕ: X(f)(x) → x′ in X(i′). Using Lemma 3.3, it is not hard to show that π_{0}(I∫ X) = colim_{i∈I} π_{0}X(i).

For example, if F : C → D is a functor then d ↦ F/d is a functor on D, and D∫(F/−) is F/D, while d ↦ d\F is a functor on D^{op}, and D∫(−\F) is D^{op}\F.

The fundamental group π_{1} of a category

Suppose that T is a set of morphisms in a category C. The graph of T is the 1-dimensional subcomplex of BC consisting of the edges corresponding to T and their incident vertices. We say that T is a tree in C if its graph is contractible (i.e., a tree in the sense of graph theory). If C is connected then a tree T is maximal (a maximal tree) just in case every object of C is either the source or target of a morphism in T. By Zorn’s Lemma, maximal trees exist when C ≠ ∅.

Classically, the fundamental group π_{1}(Γ) of the 1-skeleton Γ of BC is a free group on symbols [f], one for every non-identity morphism f in C not in T. (The loop is the composite of f with the unique paths in the tree between the basepoint and the source and target of f.) The following well known formula for the fundamental group of BC is a straight-forward application of Van Kampen’s Theorem.

Lemma 3.4. Suppose that T is a maximal tree in a small connected category C. Then the group π_{1}(BC) has the following presentation: it is generated by symbols [f], one for every morphism in C, modulo the relations that

(1) [t] = 1 for every t ∈ T, and [id_{c}] = 1 for the identity morphism id_{c} of each object c.

(2) [f] · [g] = [f ∘ g] for every pair (f, g) of composable morphisms in C.

This presentation does not depend upon the choice of the object c_{0} of C chosen as the basepoint. Geometrically, the class of f: c_{1} → c_{2} is represented by the unique path in T from c_{0} to c_{1}, followed by the edge f, followed by the unique path in T from c_{2} back to c_{0}.

Application 3.4.1 (Groups). Let G be a group, considered as a category with one object. Since BG has only one vertex, BG is connected. By Lemma 3.4 (with T empty) we see that π_{1}(BG) = G. In fact, π_{i}(BG) = 0 for all i ≥ 2. (See Ex. 3.2.) BG is often called the classifying space of the group G, for reasons discussed in Examples 3.9.2 and 3.9.3 below.

Application 3.4.2 (Monoids). If M is a monoid then BM has only one vertex. This time, Lemma 3.4 shows that the group π = π_{1}(BM) is the group completion (Ex. II.1.1) of the monoid M.

For our purposes, one important thing about BG is that its homology is the same as the ordinary Eilenberg-Mac Lane homology of the group G (see [WHomo, 6.10.5 or 8.2.3]). In fact, if M is any G-module then we may consider M as a local coefficient system on BG (see 3.5.1). The cellular chain complex used to form the homology of BG with coefficients in M is the same as the canonical chain complex used to compute the homology of G, so we have H_{∗}(BG; M) = H_{∗}(G; M). As a special case, we have H_{1}(BG; ℤ) = H_{1}(G; ℤ) = G/[G, G], where [G, G] denotes the commutator subgroup of G, i.e., the subgroup of G generated by all commutators [g, h] = ghg^{−1}h^{−1} (g, h ∈ G).

(3.5) The homology of C and BC. The i^{th} homology of a CW complex X such as BC is given by the homology of the cellular chain complex C_{∗}(X). By definition, C_{n}(X) is the free abelian group on the n-cells of X. If e is an n + 1-cell and f is an n-cell, then the coefficient of [f] in the boundary of [e] is the degree of the map S^{n} ⟶^{ε} X^{(n)} ⟶^{f} S^{n}, where ε is the attaching map of e and the second map is the projection from X^{(n)} (the n-skeleton of X) onto S^{n} given by the n-cell f.

For example, H_{∗}(BC; ℤ) is the homology of the unreduced cellular chain complex C_{∗}(BC), which in degree n is the free abelian group on the set of all n-tuples (f_{1}, ..., f_{n}) of composable morphisms in C, composable in the order c_{0} ⟶^{f_{1}} c_{1} → ⋯ ⟶^{f_{n}} c_{n}. The boundary map in this complex sends the generator (f_{1}, ..., f_{n}) to the alternating sum obtained by succesively deleting the c_{i} in the evident way:

(f_{2}, ..., f_{n}) − (f_{2}f_{1}, f_{3}, ..., f_{n}) + ⋯ ± (..., f_{i+1}f_{i}, ...) ∓ ⋯ ± (..., f_{n}f_{n−1}) ∓ (..., f_{n−1}).

More generally, for each functor M: C → Ab we let H_{i}(C; M) denote the i^{th} homology of the chain complex

⋯ → ∐_{c_{0}→···→c_{n}} M(c_{0}) → ⋯ → ∐_{c_{0}→c_{1}} M(c_{0}) → ∐_{c_{0}} M(c_{0}).

The final boundary map sends the copy of M(c_{0}) indexed by c_{0} ⟶^{f} c_{1} to M(c_{0}) ⊕ M(c_{1}) by x ↦ (−x, fx). The cokernel of this map is the usual description for the colimit of the functor M, so H_{0}(C; M) = colim_{c∈C} M(c).

Local coefficients 3.5.1. A functor C → Sets is said to be morphisminverting if it carries all morphisms of C into isomorphisms. By Ex. 3.1, morphisminverting functors are in 1–1 correspondence with covering spaces of BC. Therefore the morphism-inverting functors M: C → Ab are in 1–1 correspondence with local coefficient systems on the topological space BC. In this case, the groups H_{i}(C; M) are canonically isomorphic to H_{i}(BC; M), the topologist’s homology groups of BC with local coefficients M. The isomorphism is given in [Wh, VI.4.8].

Bisimplicial Sets

A bisimplicial set X is a contravariant functor Δ × Δ → Sets, where Δ is the subcategory of ordered sets on the objects {0, 1, ..., n, ...}. Alternatively, it is a doubly indexed family X_{p,q} of sets, together with “horizontal” face and degeneracy maps (∂_{i}^{h} : X_{p,q} → X_{p−1,q} and σ_{i}^{h} : X_{p,q} → X_{p+1,q}) and “vertical” face and degeneracy maps (∂_{i}^{v} : X_{p,q} → X_{p,q−1} and σ_{i}^{v} : X_{p,q} → X_{p,q+1}), satisfying the horizontal and vertical simplicial identities and such that horizontal maps commute with vertical maps. In particular, each X_{p,·} and X_{·,q} is a simplicial set.

Definition 3.6. The geometric realization BX of a bisimplicial set X is obtained by taking one copy of the product Δ^{p} × Δ^{q} for each element of X_{p,q}, inductively identifying its horizontal and vertical faces with the appropriate Δ^{p−1}×Δ^{q} or Δ^{p} × Δ^{q−1}, and collapsing horizontal and vertical degeneracies. This construction is sometimes described as a coend: BX = ∫_{p,q} X_{p,q} × Δ^{p} × Δ^{q}.

There is a diagonalization functor diag from bisimplicial sets to simplicial sets (diag(X)_{p} = X_{p,p}), and it is well known (see [BF,B.1]) that BX is homeomorphic to B diag(X). The following theorem is also well known; see [Wa78, p. 164–5] or [Q341, p. 98] for example.

Theorem 3.6.1. Let f : X → Y be a map of bisimplicial sets.

(i) If each simplicial map X_{p,∗} → Y_{p,∗} is a homotopy equivalence, so is BX → BY. (ii) If Y is the nerve of a category I (constant in the second simplicial coordinate), and f^{−1}(i) → f^{−1}(j) is a homotopy equivalence for every i → j in I, then each B(f^{−1}(i)) → BX → B(I) is a homotopy fibration sequence.

Example 3.6.2. (Quillen) If F : C → D is a functor, the canonical functor D\F → C is a homotopy equivalence, where D\F is the comma category of Example 3.2.3. To see this, let X denote the bisimplicial set such that X_{p.q} is the set of all pairs of sequences

(d_{q} → ⋯ → d_{0} → F (c_{0}), c_{0} → ⋯ → c_{p});

the horizontal and vertical faces come from the nerves of C and D. Consider the projection of X onto the nerve of C. Since NC_{p} is the discrete set of all sequences c_{0} → ⋯ → c_{p}, the inverse image of this sequence is isomorphic to the nerve of D\F (c_{0}), and D\F (c_{0}) is contractible since it has a terminal object. Theorem 3.6.1 applies to yield BX ⟶^{∼} BC. The simplicial set diag(X) is the nerve of D\F, and the composition B(D\F) ⟶^{∼} BX ⟶^{∼} BC is the canonical map, whence the result.

Homotopy Fibers of Functors

If F: C → D is a functor, it is useful to study the realization map BF: BC → BD in terms of homotopy groups, and for this we want a category-theoretic interpretation of the homotopy fiber (1.2). The naïve approximations to the homotopy fiber are the realization of the comma categories F/d and its dual d\F. Indeed, we saw in 3.2.3 that there are continuous maps from both B(F/d) and B(d\F) to the homotopy fiber.

Here is the fundamental theorem used to prove that two categories are homotopy equivalent. It was proven by Quillen in [Q341]. Note that it has a dual formulation, replacing d\F by F/d, because BD ≃ BD^{op}.

3.7 Quillen’s Theorem A. Let F: C → D be a functor such that d\F is contractible for every d in D. Then BF: BC ⟶^{≃} BD is a homotopy equivalence.

Proof. Consider the comma category D\F of 3.2.3, which is equipped with functors C ⟵ D\F ⟶ D^{op} such that BC ⟵ B(D\F) is a homotopy equivalence (by 3.6.2). The functor D\F → D\D, sending (d → F (c), c) to (d → F (c), F (c)), fits into a commutative diagram of categories

C ⟵^{≃} D\F ⟶ D^{op}
F ↓  ↓  ∥
D ⟵^{≃} D\D ⟶ D^{op}

Therefore it suffices to show that B(D\F) → BD^{op} is a homotopy equivalence. This map factors as B(D\F) ≃ BX ⟶^{π} BD^{op}, where X is the bisimplicial set of Example 3.6.2 and π is the projection. Consider the simplicial map π_{∗,q} from X_{∗,q} to the qth component of the nerve of D^{op}, which is the discrete set of all sequences d_{q} → ⋯ → d_{0} in D. For each such sequence, the inverse image in X_{∗,q} is the nerve of d_{0}\F, which is assumed to be contractible. By Theorem 3.6.1, B(D\F) ≃ BX → BD is a homotopy equivalence, as required. □

Example 3.7.1. If F: C → D has a left adjoint L, then d\F is isomorphic to the comma category L(d)\C, which is contractible by Example 3.2.2. In this case, Quillen’s Theorem A recovers the observation in 3.2 that C and D are homotopy equivalent.

Example 3.7.2. Consider the inclusion of monoids i: ℕ ↪ ℤ as a functor between categories with one object ∗. Then ∗\i is isomorphic to the translation category ℕ∫ ℤ, which is contractible (why?). Quillen’s Theorem A shows that Bℕ ≃ Bℤ ≃ S^{1}.

The inverse image F^{−1}(d) of an object d is the subcategory of C consisting of all objects c with F (c) = d, and all morphisms h in C mapping to the identity of d. It is isomorphic to the full subcategory of F/d consisting of pairs (c, F (c) ⟶^{=} d), and also to the full subcategory of pairs (d ⟶^{=} F (c), c) of d\F. It will usually not be homotopy equivalent to either F/d or d\F.

One way to ensure that F^{−1}(d) is homotopic to a comma category is to assume that F is either pre-fibered or pre-cofibered in the following sense.

Fibered and Cofibered functors 3.7.3. (Cf. [SGA 1, Exp. VI]) We say that a functor F: C → D is pre-fibered if for every d in D the inclusion F^{−1}(d) ↪ d\F has a right adjoint. This implies that BF^{−1}(d) ≃ B(d\F), and the base change functor f^{∗}: F^{−1}(d′) → F^{−1}(d) associated to a morphism f: d → d′ in D is defined as the composite F^{−1}(d′) ↪ (d\F) → F^{−1}(d). F is called fibered if it is pre-fibered and g^{∗}f^{∗} = (fg)^{∗} for every pair of composable maps f, g, so that F^{−1} gives a contravariant functor from D to CAT.

Dually, we say that F is pre-cofibered if for every d the inclusion F^{−1}(d) ↪ F/d has a left adjoint. In this case we have BF^{−1}(d) ≃ B(F/d). The cobase change functor f_{∗}: F^{−1}(d) → F^{−1}(d′) associated to a morphism f: d → d′ in D is defined as the composite F^{−1}(d) ↪ (F/d′) → F^{−1}(d′). F is called cofibered if it is precofibered and (fg)_{∗} = f_{∗}g_{∗} for every pair of composable maps f, g, so that F^{−1} gives a covariant functor from D to CAT.

These notions allow us to state a variation on Quillen’s Theorem A.

Corollary 3.7.4. Suppose that F : C → D is either pre-fibered or pre-cofibered, and that F^{−1}(d) is contractible for each d in D. Then BF is a homotopy equivalence BC ≃ BD.

Example 3.7.5. Cofibered functors over D are in 1–1 correspondence with functors D → CAT. We have already mentioned one direction: if F: C → D is cofibered, F^{−1} is a functor from D to CAT. Conversely, for each functor X: D → CAT, the category D∫ X of Example 3.3.2 is cofibered over D by the forgetful functor (d, x) ↦ d. It is easy to check that these are inverses: C is equivalent to D∫ F^{−1}.

Here is the fundamental theorem used to construct homotopy fibration sequences of categories. It was originally proven in[Q341]. Note that it has a dual formulation, in which d\F is replaced by F/d; see Ex. 3.6.

3.8 Quillen’s Theorem B. Let F: C → D be a functor such that for every morphism d → d′ in D the induced functor d′\F → d\F is a homotopy equivalence. Then for each d in D the geometric realization of the sequence

d\F ⟶^{j} C ⟶^{F} D

is a homotopy fibration sequence. Thus there is a long exact sequence

⋯ → π_{i+1}(BD) ⟶^{∂} π_{i}B(d\F) ⟶^{j} π_{i}(BC) ⟶^{F} π_{i}(BD) ⟶^{∂} · · · .

Proof. We consider the projection functor X ⟶^{p} ND^{op} of 3.6.2. Since p^{−1}(d) is the nerve of d\F, we may apply Theorem 3.6.1 to conclude that B(d\F) → BX → BD^{op} is a homotopy fibration sequence. Since B(d\F) → B diag(X) = B(D\F) ⟶^{≃} BC is induced from j : d\F → C, the theorem follows from the diagram

d\F ⟶ D\F ⟶ D^{op}
↓  ^{F} ↓  ∥
∗ ≃ d\D ⟶ D\D ⟶^{≃} D^{op}. □

Corollary 3.8.1. Suppose that F is pre-fibered, and for every f: d → d′ in D the base change f^{∗} is a homotopy equivalence. Then for each d in D the geometric realization of the sequence

F^{−1}(d) ⟶^{j} C ⟶^{F} D

is a homotopy fibration sequence. Thus there is a long exact sequence

⋯ → π_{i+1}(BD) ⟶^{∂} π_{i}BF^{−1}(d) ⟶^{j} π_{i}(BC) ⟶^{F} π_{i}(BD) ⟶^{∂} · · · .

Topological Categories 3.9. If C = C^{top} is a topological category (i.e., the object and morphism sets form topological spaces), then the nerve of C^{top} is a simplicial topological space. Using the appropriate geometric realization of simplicial spaces, we can form the topological space BC^{top} = |NC^{top}|. It has the same underlying set as our previous realization BC^{δ} (the δ standing for “discrete,” i.e., no topology), but the topology of BC^{top} is more intricate. Since the identity may be viewed as a continuous functor C^{δ} → C^{top} between topological categories, it induces a continuous map BC^{δ} → BC^{top}.

For example, any topological group G = G^{top} is a topological category, so we need to distinguish between the two connected spaces BG^{δ} and BG^{top}. It is traditional to write BG for BG^{top}, reserving the notation BG^{δ} for the less structured space. As noted above, BG^{δ} has only one nonzero homotopy group: π_{1}(BG^{δ}) = G^{δ}. In contrast, the loop space Ω(BG^{top}) is G^{top}, so π_{i}BG^{top} = π_{i−1}G^{top} for i > 0.

Example 3.9.1. Let G = ℝ be the topological group of real numbers under addition. Then Bℝ^{top} is contractible because ℝ^{top} is, but Bℝ^{δ} is not contractible because π_{1}(Bℝ^{δ}) = ℝ.

Example 3.9.2 (BU). The unitary groups U_{n} are topological groups, and we see from I.4.10.1 that BU_{n} is homotopy equivalent to the infinite complex Grassmannian manifold G_{n}, which classifies n-dimensional complex vector bundles by Theorem I.4.10. The unitary group U_{n} is a deformation retract of the complex general linear group GL_{n}(ℂ)^{top}. Thus BU_{n} and BGL_{n}(ℂ)^{top} are homotopy equivalent spaces. Taking the limit as n → ∞, we have a homotopy equivalence BU ≃ BGL(ℂ)^{top}.

By Theorem II.3.2, KU(X) ≅ [X, ℤ × BU] and K̃U(X) ≅ [X, BU] for every compact space X. By Ex. II.3.11 we also have KU^{−n}(X) ≅ [X, Ω^{n}(ℤ×BU)] for all n ≥ 0. In particular, for the one-point space ∗ the groups KU^{−n}(∗) = π_{n}(ℤ × BU) are periodic of order 2: ℤ if n is even, 0 if not. This follows from the observation in II.3.2 that the homotopy groups of BU are periodic — except for π_{0}(BU), which is zero as BU is connected.

A refinement of Bott periodicity states that ΩU ≃ ℤ × BU. Since Ω(BU) ≃ U, we have Ω^{2}(ℤ×BU) ≃ Ω^{2}BU ≃ ℤ×BU and Ω^{2}U ≃ U. This yields the periodicity formula: KU^{−n}(X) = KU^{−n−2}(X).

Example 3.9.3 (BO). The orthogonal group O_{n} is a deformation retract of the real general linear group GL_{n}(ℝ)^{top}. Thus the spaces BO_{n} and BGL_{n}(ℝ)^{top} are homotopy equivalent, and we see from I.4.10.1 that they are also homotopy equivalent to the infinite real Grassmannian manifold G_{n}. In particular, they classify n-dimensional real vector bundles by Theorem I.4.10. Taking the limit as n → ∞, we have a homotopy equivalence BO ≃ BGL(ℝ)^{top}.

Bott periodicity states that the homotopy groups of BO are periodic of order 8 — except for π_{0}(BO) = 0, and that the homotopy groups of ℤ × BO are actually periodic of order 8. These homotopy groups are tabulated in II.3.1.1. A refinement of Bott periodicity states that Ω^{7}O ≃ ℤ × BO. Since Ω(BO) ≃ O, we have Ω^{8}(ℤ × BO) ≃ Ω^{8}(BO) ≃ ℤ × BO and Ω^{8}O ≃ O.

By Definition II.3.5 and Ex. II.3.11, the (real) topological K-theory of a compact space X is given by the formula KO^{−n}(X) = [X, Ω^{n}(ℤ×BO)], n ≥ 0. This yields the periodicity formula: KO^{−n}(X) = KO^{−n−8}(X).

Bicategories

One construction that has proven useful in constructing spectra is the geometric realization of a bicategory. Just as we could have regarded a small category A as a special type of simplicial set, via its nerve 3.1.2 (A_{0} and A_{1} are the objects and morphisms, all other A_{n} are pullbacks and ∂_{1} : A_{2} = A_{1} ×_{A_{0}} A_{1} → A_{1} defines composition), we can do the same with small bicategories.

Definition 3.10. A small bicategory C is a bisimplicial set such that every row C_{·,q} and column C_{p,·} is the nerve of a category. We refer to elements of C_{0,0}, C_{1,0}, C_{0,1} and C_{1,1} as the objects, horizontal and vertical morphisms, and bimorphisms. A bifunctor between bicategories is a morphism of the underlying bisimplicial sets.

Example 3.10.1. If A and B are categories, we can form the product bicategory A ⊗ B. Its objects (resp., bimorphisms) are ordered pairs of objects (resp., morphisms) from A and B. Its (p, q)-morphisms are pairs of functors p + 1 → A, q + 1 → B.

It is easy to see that diag(A⊗B) is the product category A×B, and that B(A⊗B) is BA × BB. In particular B((p + 1) × (q + 1)) = Δ^{p} × Δ^{q}.

Bicategory terminology arose (in the 1960’s) from the following paradigm.

Example 3.10.2. For any category B, biB is the bicategory whose degree (p, q) part consists of commutative diagrams arising from functors p + 1×q + 1 → B. In particular, bimorphisms are commutative squares in B; the horizontal and vertical edges of such a square are its associated horizontal and vertical morphisms. If A is a subcategory, AB is the sub-bicategory of biB whose vertical maps are in A.

We may also regard the small category B as a bicategory which is constant in the vertical direction (B_{p,q} = NB_{p}); this does not affect the homotopy type BB since diag B recovers the category B. The natural inclusion into biB is a homotopy equivalence by Ex. 3.13. It follows that any bifunctor A ⊗ B → biC induces a continuous map

BA × BB → B biC ≃ BC.

EXERCISES

3.1 Covering spaces. If X: I → Sets is a morphism-inverting functor (3.5.1), use the recipe 3.1.1 to show that the forgetful functor I∫ X → I of Example 3.3.2 makes B(I∫ X) into a covering space of BI with fiber X(i) over each vertex i of BI.

Conversely, if E ⟶^{π} BI is a covering space, show that X(i) = π^{−1}(i) defines a morphism-inverting functor on I, where i is considered as a 0-cell of BI. Conclude that these constructions give a 1–1 correspondence between covering spaces of BI and morphism-inverting functors. (See [Q341, p. 90].) 3.2 Translation categories. Suppose that a group G acts on a set X, and form the translation category G∫ X. Show that B(G∫ X) is homotopy equivalent to the disjoint union of the classifying spaces BG_{x} of the stabilizer subgroups G_{x}, one space for each orbit in X. For example, if X is the coset space G/H then B(G∫ X) ≃ BH.

In particular, if X = G is given the G-set structure g · g′ = gg′, this shows that B(G∫ G) is contractible, i.e., the universal covering space of BG. Use this to calculate the homotopy groups of BG, as described in Example 3.4.1. 3.3 Let H be a subgroup of G, and ι: H ↪ G the inclusion as a subcategory.

(a) Show that ι/∗ is the category H∫ G of Ex. 3.1. Conclude that the homotopy fiber of BH → BG is the discrete set G/H, while Bι^{−1}(∗) is a point.

(b) Use Ex. 3.2 to give another proof of (a). 3.4 If C is a filtering category [WHomo, 2.6.13], show that BC is contractible. Hint: It suffices to show that all homotopy groups are trivial (see [Wh, V.3.5]). Any map from a sphere into a CW complex lands in a finite subcomplex, and every finite subcomplex of BC lands in the realization BD of a finite subcategory D of C; D lies in another subcategory D′ of C which has a terminal object. 3.5 Mapping telescopes. If ∪n denotes the union of the categories n of 3.1, then a functor ∪n ⟶^{C} CAT is just a sequence C_{0} → C_{1} → C_{2} → ⋯ of categories. Show that the geometric realization of the category L = (∪n)∫C of Example 3.3.2 is homotopy equivalent to BC, where C is the colimit of the C_{n}. In particular, this shows that BL ≃ lim_{n→∞} BC_{n}. Hint: C_{n} ≃ n∫ C. 3.6 Suppose that F: C → D is pre-cofibered (Definition 3.7.3).

(a) Show that F^{op}: C^{op} → D^{op} is pre-fibered. If F is cofibered, F^{op} is fibered.

(b) Derive the dual formulation of Quillen’s theorem B, using F/d and F^{op}.

(c) If each cobase change functor f_{∗} is a homotopy equivalence, show that the geometric realization of F^{−1}(d) ⟶ C ⟶^{F} D is a homotopy fibration sequence for each d in D, and there is a long exact sequence:

⋯ → π_{i+1}(BD) ⟶^{∂} π_{i}BF^{−1}(d) ⟶ π_{i}(BC) ⟶^{F} π_{i}(BD) ⟶^{∂} · · · .

3.7 Let F: C → D be a cofibered functor (3.7.3). Construct a first quadrant double complex E^{0} in which E_{pq}^{0} is the free abelian group on the pairs (d_{p} → ⋯ → d_{0} → F (c_{0}), c_{0} → ⋯ → c_{q}) of sequences of composable maps in C and D. By filtering the double complex by columns, show that the homology of the total complex Tot E^{0} is H_{q}(Tot E^{0}) ≅ H_{q}(C; ℤ). Then show that the row filtration yields a spectral sequence converging to H_{∗}(C; ℤ) with E_{pq}^{2} = H_{p}(D; H_{q}F^{−1}), the homology of D with coefficients in the functor d ↦ H_{q}(F^{−1}(d); ℤ) described in 3.5. 3.8 A lax functor M: I → CAT consists of functions assigning: (1) a category M(i) to each object i; (2) a functor f_{∗}: M(i) → M(j) to every map i ⟶^{f} j in I; (3) a natural transformation (id_{i})_{∗} ⇒ id_{M(i)} for each i; (4) a natural transformation (fg)_{∗} ⇒ f_{∗}g_{∗} for every pair of composable maps in I. This data is required to be “coherent” in the sense that the two transformations (fgh)_{∗} ⇒ f_{∗}g_{∗}h_{∗} agree, and so do the various transformations f_{∗} ⇒ f_{∗}. For example, a functor is a lax functor in which (3) and (4) are identities.

Show that the definitions of objects and morphisms in Example 3.3.2 define a category I∫ M, where the map ϕ′′ in the composition (f′f, ϕ′′) of (f, ϕ) and (f′, ϕ′) is (f′f)_{∗}(x) → f′_{∗}f_{∗}(x) → f′_{∗}(x′) → x′′. Show that the projection functor π: I∫ M → I is pre-cofibered. 3.9 Subdivision. If C is a category, its Segal subdivision Sub(C) is the category whose objects are the morphisms in C; a morphism from i: A → B to i′: A′ → B′ is a pair of maps (A′ ⟶ A, B ⟶ B′) so that i′ is A′ ⟶ A ⟶^{i} B ⟶ B′.

(a) Draw the Segal subdivisions of the unit interval 2 and the 2-simplex 3.

(b) Show that the source and target functors C^{op} ⟵ Sub(C) ⟶ C are homotopy equivalences. Hint: Use Quillen’s Theorem A and 3.2.2.

3.10 Given a simplicial set X, its Segal subdivision Sub(X) is the sequence of sets X_{1}, X_{3}, X_{5}, …, made into a simplicial set by declaring the face maps ∂′_{i}: X_{2n+1} → X_{2n−1} to be ∂_{i}∂_{2n+1−i} and σ′_{i}: X_{2n+1} → X_{2n+3} to be σ_{i}σ_{2n+1−i} (0 ≤ i ≤ n).

If X is the nerve of a category C, show that Sub(X) is the nerve of the Segal subdivision category Sub(C) of Ex. 3.9. 3.11 (Waldhausen) Let f : X → Y be a map of simplicial sets. For y ∈ Y_{n}, define the simplicial set f/(n, y) to be the pullback of X and the n-simplex Δ^{n} along f : X → Y and the map y : Δ^{n} → Y. Thus an m-simplex consists of a map α : m → n in Δ and an x ∈ X_{m} such that f(x) = α^{∗}(y). Prove that: (A) If each f/(n, y) is contractible, then f is a homotopy equivalence; (B) If for every m ⟶^{α} n in Δ and every y ∈ Y_{n} the map f/(m, α^{∗}y) → f/(n, y) is a homotopy equivalence, then each |f/(n, y)| → X → Y is homotopy fibration sequence. Hint: Any simplicial set X determines a category Δ^{op} ∫ X cofibered over Δ^{op}, by 3.3.2 and 3.7.5. Now apply Theorems A and B. 3.12 If C is a category, its arrow category C/C has the morphisms of C as its objects, and a map (a, b): f → f′ in C/C is a commutative diagram in C:

A ⟶^{f} B
a↓  ↓^{b}
A′ ⟶^{f′} B′

If f: A → B, the source s(f) = A and target t(f) = B of f define functors C/C → C. Show that s is a fibered functor, and that t is a cofibered functor. Then show that both s and t are homotopy equivalences. 3.13 Swallowing Lemma. If A is a subcategory of B, show that the bicategory inclusion B ⊂ AB of 3.10.2 induces a homotopy equivalence BB ≃ B(AB). When A = B this proves that BB ≃ B(biB). Hint: Show that B ≃ N_{p}(A)B for all p. 3.14 Diagonal Category. (Waldhausen [Wa78]) Show that the functor from small categories to small bicategories sending B to biB (3.10.2) has a left adjoint, sending C to its diagonal category, and that the diagonal category of the bicategory A⊗B is the product category A × B. Hint: both the horizontal and vertical morphisms of a bicategory C yield morphisms, and every bimorphism yields an equivalence relation for the composition of horizontal and vertical morphisms.

§4. Symmetric Monoidal Categories

The geometric realization BS of a symmetric monoidal category is an H-space with a homotopy-commutative, homotopy-associative product. To see this, recall from definition II.5.1 that a symmetric monoidal category is a category S with a functor □: S × S → S which has a unit object “e” and is associative and is commutative, all up to coherent natural isomorphism. By 3.1(6) the geometric realization of □ is the “product” map (BS) × (BS) ≅ B(S × S) → BS. The natural isomorphisms s□e ≅ s ≅ e□s imply that the vertex e is an identity up to homotopy, i.e., that BS is an H-space. The other axioms imply that the product on BS is homotopy commutative and homotopy associative.

In many cases e is an initial object of S, and therefore the H-space BS is contractible by Example 3.2.2. For example, any additive category A is a symmetric monoidal category (with □ = ⊕), and e = 0 is an initial object, so BA is contractible. Similarly, the category Sets_{fin} of finite sets is symmetric monoidal (□ being disjoint union) by I.5.2, and e = ∅ is initial, so BSets_{fin} is contractible.

Here is an easy way to modify S in order to get an interesting H-space.

Definition 4.1. Let iso S denote the subcategory of isomorphisms in S. It has the same objects as S, but its morphisms are the isomorphisms in S. Because iso S is also symmetric monoidal, B(iso S) is an H-space.

By Lemma 3.3, the abelian monoid π_{0}(iso S) is just the set of isomorphism classes of objects in S — the monoid S^{iso} considered in II.5. In fact, iso S is equivalent to the disjoint union ∐ Aut_{S}(s) of the 1–object categories Aut_{S}(s), and B(iso S) is homotopy equivalent to the disjoint union of the classifying spaces B Aut(s), s ∈ S^{iso}.

Examples 4.1.1. B(iso S) is often an interesting H-space.

(a) In the category Sets_{fin} of finite (pointed) sets, the group of automorphisms of any n-element set is isomorphic to the permutation group Σ_{n}. Thus the subcategory iso Sets_{fin} is equivalent to ∐ Σ_{n}, the disjoint union of the one-object categories Σ_{n}. Thus the classifying space B(iso Sets_{fin}) is homotopy equivalent to the disjoint union of the classifying spaces BΣ_{n}, n ≥ 0.

(b) The additive category P(R) of finitely generated projective R-modules has 0 as an initial object, so BP(R) is a contractible space. However, its subcategory P = iso P(R) of isomorphisms is more interesting. The topological space BP is equivalent to the disjoint union of the classifying spaces B Aut(P) as P runs over the set of isomorphism classes of finitely generated projective R-modules.

(c) Fix a ring R, and let F(R) be the category ∐ GL_{n}(R) whose objects are the based free R-modules {0, R, R^{2}, ⋯, R^{n}, ⋯ } (these objects are distinct because the bases have different orders; see Section I.1). There are no maps in F(R) between R^{m} and R^{n} if m ≠ n, and the self-maps of R^{n} form the group GL_{n}(R). This is a symmetric monoidal category: R^{m}□R^{n} = R^{m+n} by concatenation of bases; if a and b are morphisms, a□b is the matrix ({A \atop 0} {0 \atop B} ). The space BF(R) is equivalent to the disjoint union of the classifying spaces BGL_{n}(R).

If R satisfies the Invariant Basis Property (I.1.1), then F(R) is a full subcategory of iso P(R). In this case, we saw in II.5.4.1 that F(R) is cofinal in iso P(R).

(d) Fix a commutative ring R, and let S = Pic(R) be the category of invertible R-modules and their isomorphisms. This is a symmetric monoidal category in which □ is tensor product and e is R; see II.5.2(5). In this case, S = iso S and S^{iso} is the Picard group Pic(R) discussed in section I.3. By Lemma I.3.3, Aut(L) = R^{×} for every L. Thus Pic(R) is equivalent to a disjoint union of copies of R^{×}, and B(Pic) is homotopy equivalent to the product Pic(R) × B(R^{×}).

(e) If F is a field, we saw in II.5.7 that the categories SBil(F) and Quad(F) = Quad^{+}(F) of symmetric inner product spaces and quadratic spaces are symmetric monoidal categories. More generally, let A be any ring with involution, and ϵ = ±1. Then the category Quad^{ϵ}(A) of nonsingular ϵ-quadratic A-modules is a symmetric monoidal category with □ = ⊕ and e = 0. See [B72, Bak] for more details.

(f) If G is a group, consider the category G-Sets_{fin} of free G-sets X having a finite number of orbits. This is symmetric monoidal under disjoint union (cf. II.5.2.2). If X has n orbits, then Aut(X) is the wreath product G ≀ Σ_{n}. As in (a), B(G-Sets_{fin}) is equivalent to the disjoint union of the classifying spaces B(G ≀ Σ_{n}).

There is a monoidal functor G-Sets_{fin} → P(ℤ[G]) which sends X to the free abelian group on the set X.

The S^{−1}S Construction

In [GQ], Quillen gave a construction of a category S^{−1}S such that K(S) = B(S^{−1}S) is a “group completion” of BS (see 4.4 below), provided that every map in S is an isomorphism and every translation s□: Aut_{S}(t) → Aut_{S}(s□t) is an injection. The motivation for this construction comes from the construction of the universal abelian group completion of an abelian monoid given in Chapter II, §1.

Definition 4.2 (S^{−1}S). The objects of S^{−1}S are pairs (m, n) of objects of S. A morphism in S^{−1}S is an equivalence class of composites

(m_{1}, m_{2}) ⟶^{s□} (s□m_{1}, s□m_{2}) ⟶^{(f,g)} (n_{1}, n_{2}).

This composite is equivalent to

(m_{1}, m_{2}) ⟶^{t□} (t□m_{1}, t□m_{2}) ⟶^{(f′,g′)} (n_{1}, n_{2})

exactly when there is an isomorphism α: s ⟶^{≃} t in S so that composition with α□m_{i} sends f′ and g′ to f and g.

A (strict) monoidal functor S → T induces a functor S^{−1}S → T^{−1}T.

Explanation 4.2.1. There are two basic types of morphisms in S^{−1}S. The first type is a pair of maps (f_{1}, f_{2}): (m_{1}, m_{2}) → (n_{1}, n_{2}) with f_{i}: m_{i} → n_{i} in S, arising from the inclusion of S × S in S^{−1}S. The second type is a formal map s□: (m, n) → (s□m, s□n).

We shall say that translations are faithful in S if every translation Aut(s) → Aut(s□t) in S is an injection. In this case every map in S^{−1}S determines s, f and g up to unique isomorphism.

Remark 4.2.2. S^{−1}S is a symmetric monoidal category, with (m, n)□(m′, n′) = (m□m′, n□n′), and the functor S → S^{−1}S sending m to (m, e) is monoidal. Hence the natural map BS → B(S^{−1}S) is an H-space map, and π_{0}(S) → π_{0}(S^{−1}S) is a map of abelian monoids.

In fact π_{0}(S^{−1}S) is an abelian group, the inverse of (m, n) being (n, m), because of the existence of a morphism η in S^{−1}S from (e, e) to (m, n)□(n, m) = (m□n, n□m). Warning: η is not a natural transformation! See Ex. 4.3.

Definition 4.3. Let S be a symmetric monoidal category in which every morphism is an isomorphism. Its K-groups are the homotopy groups of B(S^{−1}S):

K_{n}^{□}(S) = π_{n}(BS^{−1}S).

It is sometimes convenient to write K^{□}(S) for the geometric realization B(S^{−1}S), and call it the K-theory space of S, so that K_{n}^{□}(S) = π_{n}K^{□}(S). By 4.2, a (strict) monoidal functor S → T induces a map K^{□}(S) → K^{□}(T) and hence homomorphisms K_{n}^{□}(S) → K_{n}^{□}(T).

In order to connect this definition up with the definition of K_{0}^{□}(S) given in section II.5, we recall from 4.2.2 that the functor S → S^{−1}S induces a map of abelian monoids from π_{0}(S) = S^{iso} to π_{0}(S^{−1}S).

Lemma 4.3.1. The abelian group K_{0}^{□}(S) = π_{0}(S^{−1}S) is the group completion of the abelian monoid π_{0}(S) = S^{iso}. Thus definition 4.3 agrees with the definition of K_{0}^{□}(S) given in II.5.1.2.

Proof. Let A denote the group completion of π_{0}(S), and consider the function α(m, n) = [m] − [n] from the objects of S^{−1}S to A. If s ∈ S and f_{i}: m_{i} → n_{i} are morphisms in S then in A we have α(m, n) = α(s□m, s□n) and α(m_{1}, m_{2}) = [m_{1}]− [m_{2}] = [n_{1}]−[n_{2}] = α(n_{1}, n_{2}). By Lemma 3.3, α induces a set map π_{0}(S^{−1}S) → A. By construction, α is an inverse to the universal homomorphism A → π_{0}(S^{−1}S). □

Group Completions

Group completion constructions for K-theory were developed in the early 1970’s by topologists studying infinite loop spaces. These constructions all apply to symmetric monoidal categories.

Any discussion of group completions depends upon the following well-known facts (see [Wh, III.7]). Let X be a homotopy commutative, homotopy associative H-space. Its set of components π_{0}X is an abelian monoid, and H_{0}(X; ℤ) is the monoid ring ℤ[π_{0}(X)]. Moreover, the integral homology H_{∗}(X; ℤ) is an associative graded-commutative ring with unit.

We say that a homotopy associative H-space X is group-like if it has a homotopy inverse; see [Wh, III.4]. Of course this implies that π_{0}(X) is a group. When X is a CW complex, the converse holds: if the monoid π_{0}(X) is a group, then X is group-like. (If π_{0}(X) = 0 this is [Wh, X.2.2]; if π_{0}(X) is a group, the proof in loc. cit. still goes through as the shear map π_{0}(X)^{2} → π_{0}(X)^{2} is an isomorphism.)

For example, if S = iso S then π_{0}(BS) is the abelian monoid S^{iso} of isomorphism classes, and H_{0}(BS; ℤ) is the monoid ring ℤ[S^{iso}]. In this case, the above remarks show that BS is grouplike if and only S^{iso} is an abelian group under □.

Definition 4.4 (Group Completion). Let X be a homotopy commutative, homotopy associative H-space. A group completion of X is an H-space Y, together with an H-space map X → Y, such that π_{0}(Y) is the group completion of the abelian monoid π_{0}(X) (in the sense of section I.1), and the homology ring H_{∗}(Y; k) is isomorphic to the localization π_{0}(X)^{−1}H_{∗}(X; k) of H_{∗}(X; k) by the natural map, for all commutative rings k.

If X is a CW complex (such as X = BS), we shall assume that Y is also a CW complex. This hypothesis implies that the group completion Y is group-like.

Lemma 4.4.1. If X is a group-like H-space then X its own group completion, and any other group completion f: X → Y is a homotopy equivalence.

Proof. Since f is a homology isomorphism, it is an isomorphism on π_{0} and π_{1}. Therefore the map of basepoint components is a +–construction relative to the subgroup 1 of π_{1}(X), and Theorem 1.5 implies that X ≃ Y. □

Example 4.4.2 (Picard groups). Let R be a commutative ring, and consider the symmetric monoidal category S = Pic(R) of Example II.5.2(5). Because π_{0}(S) is already a group, S and S^{−1}S are homotopy equivalent (by lemma 4.4.1). Therefore we get

K_{0}Pic(R) = Pic(R), K_{1}Pic(R) = U(R) and K_{n}Pic(R) = 0 for n ≥ 2.

The determinant functor from P = iso P(R) to Pic(R) constructed in section I.3 gives a map from K(R) = K(P) to KPic(R). Upon taking homotopy groups, this yields the familiar maps det: K_{0}(R) → Pic(R) of II.2.6 and det: K_{1}(R) → R^{×} of III.1.1.1.

A phantom map ϕ : X → Y is a map such that, for every finite CW complex A, every composite A → X → Y is null homotopic, i.e., ϕ_{∗} : [A, X] → [A, Y] is the zero map. If f : X → Y is a group completion then so is f + ϕ : X → Y for every phantom map ϕ. Thus the group completion is not unique up to homotopy equivalence whenever phantom maps exist.

The following result, taken from [CCMT, 1.2], shows that phantom maps are essentially the only obstruction to uniqueness of group completions. We say that two maps X → Y are weakly homotopic if they induce the same map on homotopy classes [A, X] → [A, Y]; if Y is an H-space, this means that their difference is a phantom map.

Theorem 4.4.3. Let X be an H-space such that π_{0}(X) is either countable or contains a countable cofinal submonoid. If f′ : X → X′ and f′′ : X → X′′ are two group completions, then there is a homotopy equivalence g : X′ → X′′, unique up to weak homotopy, such that gf′ and f′′ are weak homotopy equivalent. (The map g is also a weak H-map.)

The fact that gf′ and f′′ are weak homotopy equivalent implies that g is a homology isomorphism, and hence is a homotopy equivalence by 4.4.1.

4.5. One can show directly that ℤ × BGL(R)^{+} is a group completion of BS when S = ∐ GL_{n}(R); see Ex. 4.9. We will see in theorem 4.8 below that the Ktheory space B(S^{−1}S) is another group completion of BS, and then give an explicit homotopy equivalence between B(S^{−1}S) and ℤ × BGL(R)^{+} in 4.9. Here are some other methods of group completion:

Segal’s ΩB Method 4.5.1.

If X is a topological monoid, such as ∐ BGL_{n}(R) or ∐ BΣ_{n}, then we can form BX, the geometric realization of the (one-object) topological category X (see 3.9). In this case, ΩBX is an infinite loop space and the natural map X → ΩBX is a group completion. For example, if X is the one-object monoid ℕ then Bℕ ≃ S^{1}, and ΩBℕ ≃ ΩS^{1} ≃ ℤ. That is, π_{0}(ΩBℕ) is ℤ, and every component of ΩBℕ is contractible. See [Adams] for more details.

Machine Methods 4.5.2. (See [Adams].) If X isn’t quite a monoid, but the homotopy associativity of its product is nice enough, then there are constructions called “infinite loop space machines” which can construct a group completion Y of X, and give Y the structure of an infinite loop space. All machines produce the same infinite loop space Y (up to homotopy); see [MT]. Some typical machines are described in [Segal], and [May74].

The realization X = BS of a symmetric monoidal category S is nice enough to be used by infinite loop space machines. These machines produce an infinite loop space K(S) and a map BS → K(S) which is a group completion. Most infinite loop machines will also produce explicit deloopings of K(S) in the form of an Ωspectrum K(S), the K-theory spectrum of S, which is connective in the sense that π_{n}K(S) = 0 for n < 0. The production of K(S) is natural enough that monoidal functors between symmetric monoidal categories induce maps of the corresponding spectra.

Pairings and Products

A pairing of symmetric monoidal categories is a functor ⊗ : S_{1} × S_{2} → S such that s ⊗ 0 = 0 ⊗ s = 0, and there is a coherent natural bi-distributivity law

(a + a′) ⊗ (b + b′) ≅ (a ⊗ b)□(a ⊗ b′)□(a′ ⊗ b)□(a′ ⊗ b′).

If S_{1} = S_{2} = S, we will just call this a pairing on S. Instead of making this technical notion precise, we refer the reader to [May80, §2] and content ourselves with two examples from 4.1.1: the product of finite sets is a pairing on Sets_{fin}, and the tensor product of based free modules is a pairing F(A) × F(B) → F(A ⊗ B). The free module functor from Sets_{fin} to F(A) preserves these pairings. The following theorem was proven by Peter May in [May80, 1.6 and 2.1].

Theorem 4.6. A pairing S_{1} × S_{2} → S of symmetric monoidal categories determines a natural pairing K(S_{1}) ∧ K(S_{2}) → K(S) of infinite loop spaces in 4.5.2, as well as a pairing of Ω-spectra K(S_{1})∧K(S_{2}) → K(S). This in turn induces bilinear products K_{p}(S_{1}) ⊗ K_{q}(S_{2}) → K_{p+q}(S). There is also a commutative diagram

BS_{1} × BS_{2} ⟶ BS_{1} ∧ BS_{2}  ⟶^{B⊗} BS
↓  ↓  ↓
K(S_{1}) × K(S_{2}) ⟶ K(S_{1}) ∧ K(S_{2}) ⟶^{B⊗} K(S).

From theorem 4.6 and the constructions in 1.10 and 4.9, repectively Ex. 1.12 and 4.9.3, we immediately deduce:

Corollary 4.6.1. When S is Sets_{fin} or F(R), the product defined by Loday (in 1.10) agrees with the product in Theorem 4.6.

Remark 4.6.2. If there is a pairing S×S → S which is associative up to natural isomorphism, then K(S) can be given the structure of a ring spectrum. This is the case when S is Sets_{fin} or F(R) for commutative R.

Actions on other categories

To show that B(S^{−1}S) is a group completion of BS, we need to fit the definition of S^{−1}S into a more general framework.

Definition 4.7. A monoidal category S is said to act upon a category X by a functor □: S × X → X if there are natural isomorphisms s□(t□x) ≅ (s□t)□x and e□x ≅ x for s, t ∈ S and x ∈ X, satisfying coherence conditions for the products s□t□u□x and s□e□x analogous to the coherence conditions defining S.

For example, S acts on itself by □. If X is a discrete category, then S acts on X exactly when the monoid π_{0}(S) acts on the underlying set of objects in X.

Here is the analogue of the translation category construction (3.3.1) associated to a monoid acting on a set.

Definition 4.7.1. If S acts upon X, the category ⟨S, X⟩ has the same objects as X. A morphism from x to y in ⟨S, X⟩ is an equivalence class of pairs (s, s□x ⟶^{ϕ} y), where s ∈ S and ϕ is a morphism in X. Two pairs (s, ϕ) and (s′, ϕ′) are equivalent in case there is an isomorphism s ≅ s′ identifying ϕ′ with s′□x ≅ s□x ⟶^{ϕ} y.

We shall write S^{−1}X for ⟨S, S × X⟩, where S acts on both factors of S × X. Note that when X = S this definition recovers the definition of S^{−1}S given in 4.2 above. If S is symmetric monoidal, then the formula s□(t□x) = (s□t, x) defines an action of S on S^{−1}X.

For example, if every arrow in S is an isomorphism, then e is an initial object of ⟨S, S⟩ and therefore the space S^{−1}1 = B⟨S, S⟩ is contractible.

We say that S acts invertibly upon X if each translation functor s□: X → X is a homotopy equivalence. For example, S acts invertibly on S^{−1}X (if S is symmetric) by the formula s□(t, x) = (s□t, x), the homotopy inverse of the translation (t, x) ↦ (s□t, x) being the translation (t, x) ↦ (t, s□x), because of the natural transformation (t, x) ↦ (s□t, s□x).

Now π_{0}S is a multiplicatively closed subset of the ring H_{0}(S) = ℤ[π_{0}S], so it acts on H_{∗}(X) and acts invertibly upon H_{∗}(S^{−1}X). Thus the functor X → S^{−1}X sending x to (0, x) induces a map

(4.7.2)  (π_{0}S)^{−1}H_{q}(X) → H_{q}(S^{−1}X).

Theorem 4.8 (Quillen). If every map in S is an isomorphism and translations are faithful in S, then the map (4.7.2) is an isomorphism for all X and q.

In particular, B(S^{−1}S) is a group completion of the H-space BS.

Proof. (See [GQ, p. 221].) By Ex. 4.5, the projection functor ρ: S^{−1}X → ⟨S, S⟩ is cofibered with fiber X. By Ex. 3.7 there is an associated spectral sequence E_{pq}^{2} = H_{p}(⟨S, S⟩; H_{q}(X)) ⇒ H_{p+q}(S^{−1}X). Localizing this at the multiplicatively closed subset π_{0}S of H_{0}(S) is exact, and π_{0}S already acts invertibly on H_{∗}(S^{−1}X) by Ex. 3.7, so there is also a spectral sequence E_{pq}^{2} = H_{p}(⟨S, S⟩; M_{q}) ⇒ H_{p+q}(S^{−1}X), where M_{q} = (π_{0}S)^{−1}H_{q}(X). But the functors M_{q} are morphism-inverting, so by Ex. 3.1 and the contractibility of ⟨S, S⟩, the group H_{p}(⟨S, S⟩; M_{q}) is zero for p ≠ 0, and equals M_{q} for p = 0. Thus the spectral sequence degenerates to the claimed isomorphism (4.7.2).

The final assertion is immediate from this and definition 4.4, given Remark 4.2.2 and Lemma 4.3.1. □

Bass gave a classical definition of K_{1}(S) and K_{2}(S) in [B72, p. 197]; we gave them implicitly in III.1.6.3 and III.5.6. We can now state these classical definitions, and show that they coincide with the K-groups defined in this section.

Corollary 4.8.1. If S = iso S and translations are faithful in S, then:

K_{1}(S) = \underrightarrow{lim}_{s∈S} H_{1}(Aut(s); ℤ),

K_{2}(S) = \underrightarrow{lim}_{s∈S} H_{2}([Aut(s), Aut(s)]; ℤ).

Proof. ([We-Az]) The localization of H_{q}(BS) = ⊕_{s∈S}H_{q}(Aut(s)) at π_{0}(X) = S^{iso} is the direct limit of the groups H_{q}(Aut(s)), taken over the translation category of all s ∈ S. Since π_{1}(X) = H_{1}(X; ℤ) for every H-space X, this gives the formula for K_{1}(S) = π_{1}B(S^{−1}S).

For K_{2} we observe that any monoidal category S is the filtered colimit of its monoidal subcategories having countably many objects. Since K_{2}(S) and Bass’ H_{2} definition commute with filtered colimits, we may assume that S has countably many objects. In this case the proof is relegated to exercise 4.10. □

Relation to the +–construction

Let S = F(R) = ∐ GL_{n}(R) be the monoidal category of based free R-modules, as in example 4.1.1(c). In this section, we shall establish the following result, identifying the +–construction on BGL(R) with the basepoint component of K(S) = B(S^{−1}S).

Theorem 4.9. When S is ∐ GL_{n}(R), K(S) = B(S^{−1}S) is the group completion of BS = ∐ BGL_{n}(R), and

B(S^{−1}S) ≃ ℤ × BGL(R)^{+}.

As theorems 4.8 and 1.8 suggest, we first need to find an acyclic map from BGL(R) to the connected basepoint component of B(S^{−1}S). This is done by the following “mapping telescope” construction (illustrated in Figure 4.9.1).

.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} BGL_{1} BGL_{2} BGL_{3} etc.  ⟶ BS^{−1}S

Figure 4.9.1. The mapping telescope of BGL(R) and B(S^{−1}S).

Any group map η from GL_{n}(R) to Aut_{S^{−1}S}(R^{n}, R^{n}) gives a map from BGL_{n}(R) to B(S^{−1}S). For the specific maps η = η_{n} defined by η_{n}(g) = (g, 1), the diagram

GL_{n}(R) ⟶^{η} Aut(R^{n}, R^{n})
□R↓  ^{□(R,R)}↓
GL_{n+1}(R) ⟶^{η} Aut(R^{n+1}, R^{n+1})

commutes, i.e., there is a natural transformation from η to η(□R). The resulting homotopy of maps η ≃ η(□R) : BGL_{n}(R) → B(S^{−1}S) gives the map from the “mapping telescope” construction of BGL(R) to B(S^{−1}S); see Ex. 3.5. In fact, this map lands in the connected component Y_{S} of the identity in B(S^{−1}S). Since B(S^{−1}S) is an H-space, so is the connected component Y_{S} of the identity.

Proof of Theorem 4.9. (Quillen) We shall show that the map BGL(R) → Y_{S} is an isomorphism on homology with coefficients ℤ. By the remark following Theorem 1.8, this will induce a homotopy equivalence BGL(R)^{+} → Y_{S}.

Let e ∈ π_{0}BS be the class of R. By theorem 4.8, H_{∗}B(S^{−1}S) is the localization of the ring H_{∗}(BS) at π_{0}(S) = {e^{n}}. But this localization is the colimit of the maps H_{∗}(BS) → H_{∗}(BS) coming from the translation ⊕R: S → S. Hence H_{∗}B(S^{−1}S) ≅ H_{∗}(Y_{S}) ⊗ ℤ[e, e^{−1}], where Y_{S} denotes the basepoint component of B(S^{−1}S), and H_{∗}(Y_{S}) ≅ colim H_{∗}(BGL_{n}(R)) = H_{∗}(BGL(R)). This means that the map BGL(R) → Y_{S} is a homology isomorphism, as required. □

Example 4.9.2. (Segal) Consider the symmetric monoidal category S = ∐ Σ_{n}, equivalent to the category Sets_{fin} of example 4.1.1(a). The infinite symmetric group Σ_{∞} is the union of the symmetric groups Σ_{n} along the inclusions □1 from Σ_{n} to Σ_{n+1}, and these inclusions assemble to give a map from the mapping telescope construction of BΣ_{∞} to B(S^{−1}S), just as they did for GL(R) (see Figure 4.9.1). Moreover the proof of theorem 4.9 formally goes through to prove that B(S^{−1}S) ≃ K(Sets_{fin}) is homotopy equivalent to ℤ × BΣ_{∞}^{+}. This is the equivalence of parts (a) and (b) in the following result. We refer the reader to [BP71] and [Adams, §4.2] for the equivalence of parts (b) and (c).

The Barratt-Priddy-Quillen-Segal Theorem 4.9.3. The following three infinite loop spaces are the same:

(a) the group completion K(Sets_{fin}) of BSets_{fin};

(b) ℤ × BΣ_{∞}^{+}, where Σ_{∞} is the union of the symmetric groups Σ_{n}; and

(c) The infinite loop space Ω^{∞}S^{∞} = lim_{n→∞} Ω^{n}S^{n}. Hence the groups K_{n}(Sets_{fin}) are the stable homotopy groups of spheres, π_{n}^{s}.

More generally, suppose that S has a countable sequence of objects s_{1}, … such that s_{n+1} = s_{n}□a_{n} for some a_{n} ∈ S, and satisfying the cofinality condition that for every s ∈ S there is an s′ and an n so that s□s′ ≅ s_{n}. In this case we can form the group Aut(S) = colim_{n→∞} Aut_{S}(s_{n}).

Theorem 4.10. Let S = iso S be a symmetric monoidal category whose translations are faithful, and suppose the above condition is satisfied, so that the group Aut(S) exists. Then the commutator subgroup E of Aut(S) is a perfect normal subgroup, K_{1}(S) = Aut(S)/E, and the +–construction on B Aut(S) is the connected component of the identity in the group completion K(S). Thus

K(S) ≃ K_{0}(S) × B Aut(S)^{+}.

Proof. ([We-Az]) The assertions about E are essentially on p. 355 of [Bass]. On the other hand, the mapping telescope construction mentioned above gives an acyclic map from B Aut(S) to the basepoint component of B(S^{−1}S), and such a map is by definition a +–construction. □

Example 4.10.1. Consider the subcategory ∐ G ≀ Σ_{n} of the category G-Sets_{fin} of free G sets introduced in 4.1.1(f). The group Aut(S) is the (small) infinite wreath product G ≀ Σ_{∞} = ∪G ≀ Σ_{n}, so we have K(G-Sets_{fin}) ≃ ℤ × B(G ≀ Σ_{∞})^{+}. On the other hand, the Barratt-Priddy theorem [BP71] identifies this with the infinite loop space Ω^{∞}S^{∞}(BG_{+}) associated to the disjoint union BG_{+} of BG and a point.

The monoidal functor G-Sets_{fin} → P(ℤ[G]) of 4.1.1(f) induces a group homomorphism G ≀ Σ_{∞} → GL(ℤ[G]) and hence maps B(G ≀ Σ_{∞})^{+} → BGL(ℤ[G])^{+} and Ω^{∞}S^{∞}(BG_{+}) ≃ K(G-Sets_{fin}) → K(ℤ[G]).

This map is a version of the “assembly map” in the following sense. If R is any ring, there is a product map K(R) ∧ K(ℤ[G]) → K(R[G]); see 1.10 and 4.6. This yields a map from K(R)∧Ω^{∞}S^{∞}(BG_{+}) to K(R[G]). Now the space K(R)∧BG is included as a direct factor in K(R) ∧ Ω^{∞}S^{∞}(BG_{+}) (split by the “Snaith splitting”). Since the homotopy groups of the first space give the generalized homology of BG with coefficients in K(R), H_{n}(BG; K(R)), we get homomorphisms H_{n}(BG; K(R)) → K(R[G]). It is not known if K(R) ∧ BG has a complementary factor which maps trivially.

Cofinality

A monoidal functor f: S → T is called cofinal if for every t in T there is a t′ and an s in S so that t□t′ ≅ f(s); cf. II.5.3. For example, the functor F(R) → P(R) of example 4.1.1(c) is cofinal, because every projective module is a summand of a free one. For Pic(R), the one-object subcategory R^{×} is cofinal.

Cofinality Theorem 4.11. Suppose that f: S → T is cofinal. Then

(a) If T acts on X then S acts on X via f, and S^{−1}X ≃ T^{−1}X.

(b) If Aut_{S}(s) ≅ Aut_{T} (fs) for all s ∈ S then the basepoint components of K(S) and K(T) are homotopy equivalent. Thus K_{n}(S) ≅ K_{n}(T) for all n ≥ 1.

Proof. By cofinality, S acts invertibly on X if and only if T acts invertibly on X. Hence Ex. 4.6 yields

S^{−1}X ⟶^{≃} T^{−1}(S^{−1}X) ≅ S^{−1}(T^{−1}X) ⟵^{≃} T^{−1}X.

An alternate proof of part (a) is sketched in Ex. 4.8.

For part (b), let Y_{S} and Y_{T} denote the connected components of B(S^{−1}S) and B(T^{−1}T). Writing the subscript s ∈ S to indicate a colimit over the translation category 3.3.1 of π_{0}(S), and similarly for the subscript t ∈ T, theorem 4.8 yields:

H_{∗}(Y_{S}) = colim_{s∈S} H_{∗}(B Aut(s)) = colim_{s∈S} H_{∗}(B Aut(fs))
≅ colim_{t∈T} H_{∗}(B Aut(t)) = H_{∗}(Y_{T}).

Hence the connected H-spaces Y_{S} and Y_{T} have the same homology, and this implies that they are homotopy equivalent. □

Note that K_{0}(F(R)) = ℤ is not the same as K_{0}(P(R)) = K_{0}(R) in general, although K_{n}(F) ≅ K_{n}(P) for n ≥ 1 by the Cofinality Theorem 4.11(b). By Theorem 4.9 this establishes the following important result.

Corollary 4.11.1. Let S = iso P(R) be the category of finitely generated projective R-modules and their isomorphisms. Then

B(S^{−1}S) ≃ K_{0}(R) × BGL(R)^{+}.

Remark 4.11.2. Consider the 0-connected cover K(R)⟨0⟩ of K(R), the spectrum constructed by an infinite loop space machine from iso P(R), as in 4.5.2. By 4.8 and 4.11.1, BGL(R)^{+} is the 0^{th} space of the spectrum K(R)⟨0⟩. In particular, it provides a canonical way to view BGL(R)^{+} as an infinite loop space.

4.12. Let’s conclude with a look back at the other motivating examples in 4.1.1. In each of these examples, every morphism is an isomorphism and the translations are faithful, so the classifying space of S^{−1}S is a group completion of BS.

Example 4.12.1 (Stable homotopy). The “free R-module” on a finite set determines a functor from Sets_{fin} to P(R), or from the subcategory ∐ Σ_{n} of Sets_{fin} to ∐ GL_{n}(R). This functor identifies the symmetric group Σ_{n} with the permutation matrices in GL_{n}(R). Applying group completions, theorem 4.9 and 4.9.3 show that this gives a map from Ω^{∞}S^{∞} to K(R), hence maps π_{n}^{s} → K_{n}(R).

Example 4.12.2 (L-theory). Let S = Quad^{ϵ}(A) denote the category of nonsingular ϵ-quadratic A-modules, where ϵ = ±1 and A is any ring with involution [B72, Bak]. The K-groups of this category are the L-groups _{ϵ}L_{n}(A) of Karoubi and others. For this category, the sequence of hyperbolic spaces H^{n} is cofinal (by Ex. II.5.11), and the automorphism group of H^{n} is the orthogonal group _{ϵ}O_{n}. The infinite orthogonal group _{ϵ}O = _{ϵ}O(A), which is the direct limit of the groups _{ϵ}O_{n}, is the group Aut(S) in this case. By theorem 4.10, we have

K(Quad^{ϵ}(A)) ≃ _{ϵ}L_{0}(A) × B_{ϵ}O^{+}.

When A = ℝ, the classical orthogonal group O is _{+1}O. When A = ℂ and the involution is complex conjugation, the classical unitary group U is _{+1}O(ℂ). For more bells and whistles, and classical details, we refer the reader to [Bak].

Example 4.12.3. When R is a topological ring (such as ℝ or ℂ), we can think of P(R) as a topological symmetric monoidal category. Infinite loop space machines (4.5.2) also accept topological symmetric monoidal categories, and we write K(R^{top}) for K(P(R)^{top}). The change-of-topology functor P(R) → P(R)^{top} induces natural infinite loop space maps from K(R) to K(R^{top}). The naturality of these maps allows us to utilize infinite loop space machinery. As an example of the usefulness, we remark that

K(ℝ^{top}) ≃ ℤ × BO  and  K(ℂ^{top}) ≃ ℤ × BU.

EXERCISES

4.1 Let ℕ be the additive monoid {0, 1, ...}, considered as a symmetric monoidal category with one object. Show that ⟨ℕ, ℕ⟩ is the union ∪n of the ordered categories n, and that ℕ^{−1}ℕ is a poset, each component being isomorphic to ∪n.

4.2 Show that a sequence X_{0} → X_{1} → ⋯ of categories determines an action of ℕ on the disjoint union X = ∐ X_{n}, and that ⟨ℕ, X⟩ is the mapping telescope category ∪n∫ X of Ex. 3.5.

4.3 (Thomason) Let S be symmetric monoidal, and let ι: S^{−1}S → S^{−1}S be the functor sending (m, n) to (n, m) and (f_{1}, f_{2}) to (f_{2}, f_{1}). Show that there is no natural transformation 0 ⇒ id□ι. Hint: The obvious candidate is given in 4.2.2.

Thomason has shown that Bι is the homotopy inverse for the H-space structure on B(S^{−1}S), but for subtle reasons.

4.4 If S is a symmetrical monoidal category, so is its opposite category S^{op}. Show that the group completions K(S) and K(S^{op}) are homotopy equivalent.

4.5 (Quillen) Suppose that S = isoS, and that translations in S are faithful (4.2.1). Show that the projection S^{−1}X ⟶^{ρ} ⟨S, S⟩ is cofibered, where ρ(s, x) = s.

4.6 Let S = iso S be a monoidal category whose translations are faithful (4.2.1). Suppose that S acts invertibly upon a category X. Show that the functors X → S^{−1}X (x ↦ (s, x)) are homotopy equivalences for every s in S. If S acts upon a category Y, then S always acts invertibly upon S^{−1}Y, so this shows that S^{−1}Y ≃ S^{−1}(S^{−1}Y). Hint: Use exercises 3.6 and 4.5, and the contractibility of ⟨S, S⟩.

4.7 Suppose that every map in X is monic, and that each translation Aut_{S}(s) ⟶^{□x} Aut_{X}(s□x) is an injection. Show that the sequence S^{−1}S ⟶^{□x} S^{−1}X ⟶^{π} ⟨S, X⟩ is a homotopy fibration for each x in X, where π is projection onto the second factor. In particular, if ⟨S, X⟩ is contractible, this proves that S^{−1}S ⟶^{□x} S^{−1}X is a homotopy equivalence. Hint: Show that π and S^{−1}π: S^{−1}(S^{−1}X) → ⟨S, X⟩ are cofibered, and use the previous exercise.

4.8 Use exercises 4.5 and 4.6 to give another proof of the Cofinality Theorem 4.11(b).

4.9 Fix a ring R and set S = ∐ GL_{n}(R). The maps BGL_{n}(R) → BGL(R) → {n}×BGL(R)^{+} assemble to give a map from BS to ℤ×BGL(R)^{+}. Use Ex. 1.11 to show that it is an H-space map. Then show directly that this makes ℤ×BGL(R)^{+} into a group completion of BS.

4.10 Let S be a symmetric monoidal category with countably many objects, so that the group Aut(S) exists and its commutator subgroup E is perfect, as in 4.10. Let F denote the homotopy fiber of the H-space map B Aut(S)^{+} → B(K_{1}S).

(a) Show that π_{1}(F) = 0 and H_{2}(F; ℤ) ≅ π_{2}(F) ≅ K_{2}(S).

(b) ([We-Az]) Show that the natural map BE → F induces H_{∗}(BE) ≅ H_{∗}(F), so that F = BE^{+}. Hint: Show that K_{1}S acts trivially upon the homology of BE and F, and apply the comparison theorem for spectral sequences.

(c) Conclude that K_{2}(S) ≅ H_{2}(E) ≅ \underrightarrow{lim}_{s∈S} H_{2}([Aut(s), Aut(s)]; ℤ).

4.11 If f: X → Y is a functor, we say that an action of S on X is fiberwise if S × X ⟶^{□} X ⟶^{f} Y equals the projection S × X → X followed by f.

(a) Show that a fiberwise action on X restricts to an action of S on each fiber category X_{y} = f^{−1}(y), and that f induces a functor S^{−1}X → Y whose fibers are the categories S^{−1}(X_{y}).

(b) If f is a fibered functor (3.7.3), we say that a fiberwise action is cartesian if the base change maps commute with the action of S on the fibers. Show that in this case S^{−1}X → Y is a fibered functor. 4.12 Let G be a group, and G-Sets_{fin} as in 4.1.1(f).

(a) Using 4.10.1, show that K_{1}(G-Sets_{fin}) ≅ G/[G, G] × {±1}.

(b) Using Exercise II.5.9, show that the groups K_{n}(ℤ[G]) are modules over the Burnside ring A(G) = K_{0}G-Sets_{fin}.

(c) If G is abelian, show that the product of G-sets defines a pairing in the sense of Theorem 4.6. Conclude that K_{∗}G-Sets_{fin} is a ring. Using the free module functor, show that K_{∗}G-Sets_{fin} → K_{∗}(ℤ[G]) is a ring homomorphism.

4.13 (a) Show that the idempotent completion Ŝ (II.7.3) of a symmetric monoidal category S is also symmetric monoidal, and that S → Ŝ is a cofinal monoidal functor. Conclude that the basepoint components of K(S) and K(Ŝ) are homotopy equivalent.

(b) Show that a pairing S_{1} × S_{2} → S induces a pairing Ŝ_{1} × Ŝ_{2} → Ŝ and hence (by 4.6) a pairing of spectra K(Ŝ_{1}) ∧ K(Ŝ_{2}) → K(Ŝ).

(c) By (b), for every pair of rings A, B there is a pairing K(A)∧K(B) → K(A⊗B). Using 4.6.1, deduce that the induced product agrees with the extension of Loday’s product 1.10 described in Ex. 1.14. 4.14 Construct a morphism of spectra S^{1} → K(ℤ[x, x^{−1}]) which, as in 1.10.2, represents [x] ∈ K_{1}(ℤ[x, x^{−1}]). Using the previous exercise, show that it induces a product map ∪x : K(R) → ΩK(R[x, x^{−1}]), natural in the ring R.

§5. λ-operations in higher K-theory

Let A be a commutative ring. In Chapter II.4 we introduced the operations λ^{k} : K_{0}(A) → K_{0}(A) and showed that they endow K_{0}(A) with the structure of a special λ-ring (II.4.3.1). The purpose of this section is to extend this structure to operations λ^{k} : K_{n}(A) → K_{n}(A) for all n. Although many constructions of λ-operations have been proposed in more exotic settings, we shall restrict our attention in this section to operations defined using the +-construction.

We shall begin with a general construction, which produces the operations ∧^{k} as a special case. Fix an arbitrary group G. If ρ : G → Aut(P) is any representation of G in a finitely generated projective A-module P, any isomorphism P ⊕ Q ≅ A^{N} gives a map q(ρ) : BG → B Aut(P) → BGL_{N}(A) → BGL(A)^{+}. A different embedding of P in A^{N} will give a map which is homotopic to the first, because the two maps only differ by conjugation and BGL(A)^{+} is an H-space. (The action of π_{1}(H) on [X, H] is trivial for any H-space H and any space X; see [Wh, III.4.18]). Hence the map q(ρ) is well-defined up to homotopy.

Example 5.1. Recall from I.3 that the k^{th} exterior power ∧^{k}(P) of a finitely generated projective A-module P is also a projective module, of rank \binom{rank P}{k}.

Because ∧^{k} is a functor, it determines a group map ∧_{P}^{k} : Aut(P) → Aut(∧^{k}P), i.e., a representation, for each P. We write Λ_{P}^{k} for q(∧^{k}). Note that Λ_{P}^{0} = ∗.

Now any connected H-space, such as BGL(A)^{+}, has a multiplicative inverse (up to homotopy). Given a map f : X → H, this allows us to construct maps −f, and to take formal ℤ-linear combinations of maps.

Definition 5.2. If P has rank n, we define λ_{P}^{k} : B Aut(P) → BGL(A)^{+} to be the map

λ_{P}^{k} = ∑_{i=0}^{k−1}(−1)^{i}\binom{n + i − 1}{i}Λ_{P}^{k−i}.

One can show directly that the maps λ_{P}^{k} are compatible with the inclusions of P in P ⊕Q, up to homotopy of course, giving the desired operations λ^{k} : BGL(A)^{+} → BGL(A)^{+} (see Ex. 5.1). However, it is more useful to encode this bookkeeping in the Representation Ring R_{A}(G), an approach which is due to Quillen.

Recall from II, Ex. 4.2, that the representation ring R_{A}(G) is the Grothendieck group of the representations of G in finitely generated projective A-modules. We saw in loc. cit. that R_{A}(G) is a special λ-ring.

Proposition 5.3. If 0 → (P′, ρ′) → (P, ρ) → (P′′, ρ′′) → 0 is a short exact sequence of representations of G, then q(ρ) = q(ρ′) + q(ρ′′) in [BG, BGL(A)^{+}].

Hence there is a natural map q : R_{A}(G) → [BG, BGL(A)^{+}].

Proof. It is clear from the H-space structure on BGL(A)^{+} that q(ρ ⊕ ρ′) = q(ρ) + q(ρ′). By the above remarks, we may suppose that P′ and P′′ are free modules, of ranks m and n respectively. By universality, it suffices to consider the case in which G = G_{m,n} is the automorphism group of the sequence, i.e., the upper

triangular group ( \begin{matrix}Aut(P′) & Hom(P′′, P′) \\ 0 & Aut(P′′) \end{matrix} ). Quillen proved in [Q76] that in the

limit, the inclusions i : Aut(P′)×Aut(P′′) ↪ G_{m,n} induce a homology isomorphism

\underrightarrow{lim} H_{∗}(G_{m,n}) ≅ H_{∗}(GL(A) × GL(A)).

It follows that for any connected H-space H we have [\underrightarrow{lim} BG_{m,n}, H] ≅ [BGL(A) × BGL(A), H]. Taking H = BGL(A)^{+} yields the result. □

Example 5.3.1. If ρ is a representation on a rank n module P, the elements [ρ] − n and λ^{k}([ρ] − n) of R_{A}(G) determine maps BG → BGL(A)^{+}. When G = Aut(P) and ρ = id_{P} is the tautological representation, it follows from the formula of Ex. II.4.2 that λ^{k}([id_{P}] − n) is the map Λ_{P}^{k} of 5.1.

We can now define the operations λ^{k} on [BGL(A)^{+}, BGL(A)^{+}]. As n varies, the representations id_{n} of GL_{n}(A) are related by the relation i_{n}^{∗}id_{n+1} = id_{n} ⊕ 1, where i_{n} : GL_{n}(A) ↪ GL_{n+1}(A) is the inclusion. Hence the virtual characters ρ_{n} = id_{n} − n · 1 satisfy ρ_{n} = i_{n}^{∗}ρ_{n+1}. Since i^{∗} : R_{A}(GL_{n}A) → R_{A}(GL_{n+1}A) is a homomorphism of λ-rings, we also have λ^{k}ρ_{n} = i^{∗}(λ^{k}ρ_{n+1}). Hence we get a compatible family of homotopy classes λ_{n}^{k} ∈ [BGL_{n}(A), BGL(A)^{+}].

Because each BGL_{n}(A) → BGL_{n+1}(A) is a closed cofibration, it is possible to inductively construct maps λ_{n}^{k} : BGL_{n}(A) → BGL(A)^{+} which are strictly compatible, so that by passing to the limit they determine a continuous map λ_{∞}^{k} : BGL(A) → BGL(A)^{+} and even

λ^{k} : BGL(A)^{+} → BGL(A)^{+}.

The construction in Example 5.3.1 clearly applies to any compatible family of elements in the rings R_{A}(GL_{n}(A)). Indeed, we have a map

(5.3.2) \underleftarrow{lim} R_{A}(GL_{n}(A)) → \underleftarrow{lim}[BGL_{n}(A), BGL(A)^{+}] = [BGL(A)^{+}, BGL(A)^{+}].

For example, the operations ψ^{k} and γ^{k} may be defined in this way; see Ex. 5.2.

Definition 5.4. If X is any based space, and f : X → BGL(A)^{+} any map, we define λ^{k}f : X → BGL(A)^{+} to be the composition of f and λ^{k}. This defines operations on [X, BGL(A)^{+}] which we also refer to as λ^{k}. When X = S^{n}, we get operations λ^{k} : K_{n}(A) → K_{n}(A).

Example 5.4.1. When n = 1 and a ∈ A^{×} is regarded as an element of K_{1}(A), the formulas λ^{k}(a) = a and ψ^{k}(a) = a^{k} are immediate from the formula 5.2 for λ_{A}^{k}.

The abelian group [X, BGL(A)^{+}] inherits an associative multiplication from the product on BGL(A)^{+} described in 1.10: one uses the composition X ⟶^{Δ} X ∧ X → BGL(A)^{+} ∧ BGL(A)^{+}. If X = S^{n} for n > 0 (or if X is any suspension), this is the zero product because then the map X → X ∧ X is homotopic to 0.

Now recall from II.4 that a λ-ring must satisfy λ^{0}(x) = 1, which requires an identity. In contrast, our λ^{0} is zero. To fix this, we extend the operations to K_{0}(A) × [X, BGL(A)^{+}] by

λ^{k}(a, x) = (λ^{k}(a), λ^{k}(x) + a · λ^{k−1}(x) + ⋯ + λ^{i}(a)λ^{k−i}(x) + ⋯ + λ^{k−1}(a)x.

Thus λ^{0}(a, x) = (λ^{0}(a), λ^{0}(x)) = (1, 0), as required.

Theorem 5.5. For any based space X, the λ^{k} make K_{0}(A) × [X, BGL(A)^{+}] into a special λ-ring

Proof. It suffices to consider the universal case X = BGL(A)^{+}. Since π_{1}(X) = K_{1}(A), we have a map R_{A}(K_{1}A) → [X, X]. Via the transformation q of 5.3, we are reduced to checking identities in the rings R_{A}(GL_{n}(A)) by (5.3.2). For example, the formula λ^{k}(x + y) = ∑ λ^{i}(x)λ^{k−i}(y) comes from the identity

λ^{k} ∘ ⊕ = ∑ λ^{i} ⊗ λ^{k−i}

in R_{A}(GL_{m}(A)×GL_{n}(A)). Similarly, the formal identities for λ^{k}(xy) and λ^{n}(λ^{k}x), listed in II.4.3.1 and which need to hold in special λ-rings, already hold in R_{A}(G) and so hold in our setting via the map q. □

Corollary 5.5.1. If n > 0 then λ^{k} : K_{n}(A) → K_{n}(A) is additive, and we have ψ^{k}(x) = (−1)^{k−1}kλ^{k}.

Proof. Since the products are zero, this is immediate from the formulas in II.4.1 and II.4.4 for λ^{k}(x + y) and ψ^{k}(x). □

If A is an algebra over a field of characteristic p, the Frobenius endomorphism Φ of A is defined by Φ(a) = a^{p}. We say that A is perfect if Φ is an automorphism, i.e., if A is reduced and for every a ∈ A there is a b ∈ A with a = b^{p}.

Corollary 5.5.2. If A is an algebra over a field of characteristic p, ψ^{p} is the Frobenius Φ^{∗} on K_{n}(A), n>0, and more generally on [X, BGL(A)^{+}] for any X.

Proof. This follows from the fact (Ex. II.4.2) that ψ^{p} = Φ^{∗} on the representation ring R_{A}(G), together with the observation that q(Φ^{∗}) : K_{n}(A) → K_{n}(A) is induced by Φ : A → A by naturality in A. □

Proposition 5.6. If A is a perfect algebra over a field of characteristic p, then K_{n}(A) is uniquely p-divisible for all n > 0.

Proof. Since n > 0, we see from 5.5.1 that ψ^{p}(x) = (−1)^{p−1}pλ^{p}(x) for x ∈ K_{n}(A). Since ψ^{p} = Φ^{∗} is an automorphism, so is multiplication by p. □

For any based space X there is a space F X homotopy equivalent to B(π_{1}X) and a natural map X → F X with π_{1}(X) ⟶^{≅} π_{1}(F X); if X is a simplicial space, F X is just the 2-coskeleton of X. Composing this map with the q of 5.3 gives a natural transformation R_{A}(π_{1}X) → [X, BGL(A)^{+}] of functors from based spaces to groups.

Proposition 5.7. The natural transformation R_{A}(π_{1}X) ⟶^{q} [X, BGL(A)^{+}] is universal for maps to representable functors. That is, for any connected H-space H and any natural transformation η_{X} : R_{A}(π_{1}X) → [X, H] there is a map f : BGL(A)^{+} → H, unique up to homotopy, such that η_{X} is the composite

R_{A}(π_{1}X) ⟶^{q} [X, BGL(A)^{+}] ⟶^{f} [X, H].

Like Theorem 1.8, this is proven by obstruction theory. Essentially, one considers the system of spaces X = BGL_{n}(A) and the maps BGL_{n}(A) → H defined by η_{X}(id_{n}). See [Hiller, 2.4] for details.

Example 5.7.1. The above construction of operations works in the topological setting, allowing us to construct λ-operations on [X, BU] extending the operations in II.4.1.3. It follows that [X, BGL(ℂ)^{+}] → [X, BU] commutes with the operations λ^{k} and ψ^{k} for every X.

Example 5.8 (Finite fields). Let 𝔽_{q} be a finite field, and 𝔽_{q}^{×} → ℂ^{×} a homomorphism. It induces a homomorphism R_{𝔽_{q}}(G) → R_{ℂ}(G) called the Brauer lifting. The composition of Brauer lifting with R_{ℂ}(π_{1}X) → [X, BGL(ℂ)^{+}] induces the map BGL(𝔽_{q})^{+} → BGL(ℂ)^{+} → BU discussed in 1.12 above. Now an elementary calculation with characters (which we omit) shows that the Brauer lifting is actually a homomorphism of λ-rings. It follows from 5.7 and 5.7.1 that [X, BGL(𝔽_{q})^{+}] → [X, BGL(ℂ)^{+}] → [X, BU] are homomorphisms of λ-rings. This was used in Theorem 1.12 to calculate K_{n}(𝔽_{q}).

Compatibility with products

Theorem 5.9. The Adams operations ψ^{k} are compatible with the product on Ktheory, in the sense that ψ^{k}(x · y) = ψ^{k}(x) · ψ^{k}(y) for x ∈ K_{m}(A) and y ∈ K_{n}(A).

Proof. It suffices to show that the following diagram commutes up to weak homotopy:

BGL(A)^{+} ∧ BGL(A)^{+} ⟶ BGL(A)^{+}
ψ^{k}∧ψ^{k}↓  ↓_{ψ^{k}}
BGL(A)^{+} ∧ BGL(A)^{+} ⟶ BGL(A)^{+}.

Via Proposition 5.7, this follows from the fact that the R_{A}(GL_{m}(A)×GL_{n}(A)) are λ-rings. □

Example 5.9.1. If F is a field then ψ^{k} = k^{2} on K_{2}(F). This is because K_{2}(F) is generated by Steinberg symbols {a, b} (III.6.1), and Example 5.4.1 implies that ψ^{k}{a, b} = {a^{k}, b^{k}} = k^{2}{a, b}. The same argument shows that ψ^{k} = k^{n} on the image of K_{n}^{M}(F) → K_{n}(F); see III.7.

Example 5.9.2. For finite fields, we have ψ^{k}(x) = k^{i} x for x ∈ K_{2i−1}(𝔽_{q}). This follows from Example 5.8 and the fact (II.4.4.1) that ψ^{k} = k^{i} on π_{2i}BU = K̃U(S^{2i}).

The γ-filtration

Consider the γ-filtration (II.4.7) on K_{0}(A) × K_{n}(A); If n > 0 then F_{γ}^{k}K_{n}(A) is generated by all γ^{k′}(x) and a · γ^{j}(x) with k′ ≥ k, a ∈ F_{γ}^{i}K_{0}(A), x ∈ K_{n}(A), i > 0 and i + j ≥ k. (There are other possible definitions, using the ring structure on K_{∗}(A), but they coincide up to torsion [Sou85].) For this reason, we shall ignore torsion and deal with the γ-filtration on K_{n}(A) ⊗ ℚ.

Because x = γ^{1}(x) we have K_{n}(A) = F_{γ}^{1}K_{n}(A) for n > 0. The next layer F_{γ}^{1}/F_{γ}^{2} of the filtration is also small.

Proposition 5.10 (Kratzer). For all commutative A, SK_{1}(A) = F_{γ}^{2}K_{1}(A), and F_{γ}^{1}K_{1}(A)/F_{γ}^{2}K_{1}(A) = A^{×}, and for n ≥ 2: K_{n}(A) = F_{γ}^{2}K_{n}(A).

Proof. It suffices to compute in π_{n}BSL(A)^{+}, which equals SK_{1}(A) for n = 1 and K_{n}(A) for n > 1 (see Ex. 1.8(a)). For G = SL_{N}(A) the identity det(id_{N}) = 1 in R(G) may be written in terms of ρ = id_{N} −N as γ^{1}(ρ)+γ^{2}(ρ)+⋯+γ^{N}(ρ) = 0. Because γ^{i}(ρ) = 0 for i > N (Exercise 5.4), this yields the identity ∑_{1}^{∞} γ^{i}(x) = 0 for x ∈ π_{n}BSL_{N}(A)^{+}. Since x = γ^{1}(x), this shows that x ∈ F_{γ}^{2}π_{n}BSL_{N}(A). □

Remark 5.10.1. Soulé has proven [Sou85, Thm. 1] that if A has stable range sr(A) < ∞ (I, Ex. 1.5) then γ^{k} vanishes on K_{n}(A) for all k ≥ n + sr(A). This is a useful bound because sr(R) ≤ dim(A) + 1 for noetherian A. If F is a field, ψ^{k} = k^{n} and γ^{n} = (−1)^{n−1}(n − 1)! on the image of K_{n}^{M}(F) → K_{n}(F), by 5.9.1, so the bound is best possible. The proof uses Volodin’s construction of K-theory.

Theorem 5.11. For n > 0, the eigenvalues of ψ^{k} on K_{n}(A) ⊗ ℚ are a subset of {1, k, k^{2}, … }, and the subspace K_{n}^{(i)}(A) of eigenvectors for ψ^{k} = k^{i} is independent of k. Finally, the ring K_{∗}(A) ⊗ ℚ is isomorphic to the bigraded ring ⊕_{n,i}K_{n}^{(i)}(A).

Proof. Since every element of K_{n}(A) comes from the K-theory of a finitely generated subring, we may assume that sr(A) < ∞. As in the proof of II.4.10, the linear operator ∏_{1}^{N}(ψ^{k} − k^{i}) is trivial on each F_{γ}^{i}/F_{γ}^{i+1} for large N, and this implies that K_{n}(A)⊗ℚ is the direct sum of the eigenspaces for ψ^{k} = k^{i}, 1 ≤ i ≤ N.

Since ψ^{k} and ψ^{ℓ} commute, it follows by downward induction on i that they have the same eigenspaces, i.e., K_{n}^{(i)}(A) is independent of k. Finally, the bigraded ring structure follows from 5.9. □

Example 5.11.1. (Geller-Weibel) Let A = ℂ[x_{1}, …, x_{n}]/(x_{i}x_{j} = 0, i ≠ j) be the coordinate ring of the coordinate axes in ℂ^{n}. Then the Loday symbol ⟨⟨x_{1}, ..., x_{n}⟩⟩ of Ex. 1.22 projects nontrivially into K_{n}^{(i)}(A) for all i in the range 2 ≤ i ≤ n. In particular, K_{n}^{(i)}(A) ≠ 0 for each of these i. As sr(A) = 2, these are the only values of i allowed by Soulé’s bound in 5.10.1.

The ring of Example 5.11.1 is not regular. In contrast, it is widely believed that the following conjecture is true for all regular rings; it may be considered to be the outstanding problem in algebraic K-theory.

Vanishing Conjecture 5.12 (Beilinson-Soulé). If i < n/2 and A is regular then K_{n}(A) = F_{γ}^{i}K_{n}(A). (See Ex. VI.3.6 for the connection to motivic cohomology.)

EXERCISES

5.1 Show that the composition of the cofibration B Aut(P) → B Aut(P ⊕ Q) with λ_{P ⊕Q}^{k} is homotopic to the map λ_{P}^{k}. By modifying λ_{P ⊕Q}^{k}, we can make the composition equal to λ_{P}^{k}. Using the free modules A^{n} and induction, conclude that we have maps λ^{k} : BGL(A) → BGL(A)^{+} and hence operations λ^{k} on BGL(A)^{+}, well defined up to homotopy.

One could use 4.1.1(c), 4.10 and 4.11.1 to consider the limit over Aut(P) for all projective modules P; the same construction will work except that there will be more bookkeeping. 5.2 Modify the construction of 5.3.2 to construct operations ψ^{k} and γ^{k} on the ring K_{0}(A) × [X, BGL(A)^{+}] for all X. (See II.4.4 and II.4.5.)

5.3 Show that the λ-operations are compatible with K_{1}(A[t, 1/t]) ⟶^{∂} K_{0}(A), the map in the Fundamental Theorem III.3.6, in the sense that for every x ∈ K_{0}(A), t · x ∈ K_{1}(A[t, 1/t]) satisfies ∂λ^{k}(t · x) = (−1)^{k−1}ψ^{k}(x). 5.4 (γ-dimension) Consider the γ-filtration (II.4.7) on K_{0}(A) × K_{n}(A), and show that every element of K_{n}(A) has finite γ-dimension (II.4.5). Hint: Because S^{n} is a finite complex, each x ∈ K_{n}(A) comes from some π_{n}BGL_{N}(A)^{+}. If i > N, show that γ^{i} kills the representation [id_{N}] − N and apply the map q. 5.5 For any commutative ring A, show that the ring structure on R_{A}(G) induces a ring structure on [X, K_{0}(A) × BGL(A)^{+}]. 5.6 Suppose that a commutative A-algebra B is finitely generated and projective as an A-module. Use 5.3 to show that the restriction of scalars map R_{B}(G) → R_{A}(G) induces a “transfer” map BGL(B)^{+} → BGL(A)^{+}. Show that it agrees on homotopy groups with the transfer maps for K_{1} and K_{2} in III.1.7.1 and III.5.6.3, respectively. We will encounter other constructions of the transfer in 6.3.2.

5.7 Use Ex. 5.3 to give an example of a regular ring A such that K_{3}^{(2)}(A) and K_{3}^{(3)}(A) are both nonzero.

§6. Quillen’s Q-construction for exact categories

The higher K-theory groups of a small exact category A are defined to be the homotopy groups K_{n}(A) = π_{n+1}(BQA) of the geometric realization of a certain auxiliary category QA, which we now define. This category has the same objects as A, but morphisms are harder to describe. Here is the formal definition; we refer the reader to exercise 6.1 for a more intuitive interpretation of morphisms in terms of subquotients.

Definition 6.1. Let A be an exact category. A morphism from A to B in QA is an equivalence class of diagrams

(6.1.1)  A ↞^{j} B_{2} ↣^{i} B

where j is an admissible epimorphism and i is an admissible monomorphism in A. Two such diagrams are equivalent if there is an isomorphism between them which is the identity on A and B. The composition of the above morphism with a morphism B ↞ C_{2} ↣ C is A ↞ C_{1} ↣ C, where C_{1} = B_{2} ×_{B} C_{2}.

C_{1} ↣ C_{2} ↣ C
↓ ↓  ↓ ↓
A ↞ B_{2} ↣ B

Two distinguished types of morphisms play a special role in QA: the admissible monics A ↣ B (take B_{2} = A) and the oppositely oriented admissible epis A ↞ B (take B_{2} = B). Both types are closed under composition, and the composition of A ↞ B_{2} with B_{2} ↣ B is the morphism (6.1.1). In fact, every morphism in QA factors as such a composition in a way that is unique up to isomorphism.

Subobjects 6.1.2. Recall from [Mac] that (in any category) a subobject of an object B is an equivalence class of monics B_{2} ↣ B, two monics being equivalent if they factor through each other. In an exact category A, we call a subobject admissible if any (hence every) representative B_{2} ↣ B is an admissible monic.

By definition, every morphism from A to B in QA determines a unique admissible subobject of B in A. If we fix a representative B_{2} ↣ B for each subobject in A, then a morphism in QA from A to B is a pair consisting of an admissible subobject B_{2} of B and an admissible epi B_{2} ↠ A.

In particular, this shows that morphisms from 0 to B in QA are in 1-1 correspondence with admissible subobjects of B.

Isomorphisms in QA are in 1–1 correspondence with isomorphisms in A. To see this, note that every isomorphism i: A ≅ B in A gives rise to an isomorphism in

QA, represented either by A ↣^{i} B or by A ↞^{i^{−1}} B. Conversely, since the subobject determined by an isomorphism in QA must be the maximal subobject B ↣^{=} B, every isomorphism in QA arises in this way.

Remark 6.1.3. Some set-theoretic restriction is necessary for QA to be a category in our universe. It suffices for A to be well-powered, i.e., for each object of A to have a set of subobjects. We shall tacitly assume this, since we will soon need the stronger assumption that A is a small category.

We now consider the geometric realization BQA as a based topological space, the basepoint being the vertex corresponding to the object 0. In fact, BQA is a connected CW complex, because the morphisms 0 ↣ A in QA give paths in BQA from the basepoint 0 to every vertex A. (See Lemma 3.3.) The morphisms 0 ↞ A also give paths from 0 to A in QA.

Proposition 6.2. The geometric realization BQA is a connected CW complex with π_{1}(BQA) ≅ K_{0}(A). The element of π_{1}(BQA) corresponding to [A] ∈ K_{0}(A) is represented by the based loop composed of the two edges from 0 to A:

(6.2.1)  0 ↣ A ↠ 0.

Proof. Let T denote the family of all morphisms 0 ↣ A in QA. Since each nonzero vertex occurs exactly once, T is a maximal tree. By Lemma 3.4, π_{1}(BQA) has the following presentation: it is generated by the morphisms in QA, modulo the relations that [0 ↣ A] = 1 and [f] · [g] = [f ∘ g] for every pair of composable arrows in QA. Moreover, the element of π_{1}(BQA) corresponding to a morphism from A to B is the based loop following the edges 0 ↣ A → B ↢ 0.

Since the composition 0 ↣ B_{2} ↣ B is in T, this shows that [B_{2} ↣ B] = 1 in π_{1}(BQA). Therefore [A ↞ B_{2} ↣ B] = [A ↞ B_{2}]. Similarly, the composition 0 ↞ A ↞ B yields the relation [A ↞ B][0 ↞ A] = [0 ↞ B]. Since every morphism (6.1.1) factors, this shows that π_{1}(BQA) is generated by the morphisms [0 ↞ A].

If A ↣ B ↠ C is an exact sequence in A, then the composition 0 ↣ C ↞ B in QA is 0 ↞ A ↣ B. This yields the additivity relation

(6.2.2)  [0 ↞ B] = [C ↞ B][0 ↞ C] = [0 ↞ A][0 ↞ C]

in π_{1}(BQA), represented by the following picture in BQA:

0 ↣ A ↣ B
↓ ↓ ↗ ↓ ↓ ↘ ↘
0 ↣ C ↠ 0

Since every relation [f] · [g] = [f ∘ g] may be rewritten in terms of the additivity relation, π_{1}(BQA) is generated by the [0 ↞ A] with (6.2.2) as the only relation. Therefore K_{0}(A) ≅ π_{1}(BQA). □

Example 6.2.3. The presentation for π_{1}(BQA) in the above proof yields a function from morphisms in QA to K_{0}(A). It sends [A ↞ B_{2} ↣ B] to [B_{1}], where B_{1} is the kernel of B_{2} ↠ A.

Definition 6.3. Let A be a small exact category. Then KA denotes the space ΩBQA, and we set

K_{n}(A) = π_{n}KA = π_{n+1}(BQA) for n ≥ 0.

Proposition 6.2 shows that this definition of K_{0}(A) agrees with the one given in chapter II. Note that any exact functor F: A → B induces a functor QA → QB, hence maps BQA → BQB and K_{n}(A) → K_{n}(B). Thus the space KA = Ω BQA and all the groups K_{n}(A) are functors from exact categories and exact functors to spaces and abelian groups, respectively. Moreover, isomorphic functors induce the same map on K-groups, because they induce isomorphic functors QA → QA′.

Remark 6.3.1. If an exact category A is not small but has a set of isomorphism classes of objects then we define K_{n}(A) to be K_{n}(A′), where A′ is a small subcategory equivalent to A. By Ex. 6.2 this is independent of the choice of A′. From now on, whenever we talk about the K-theory of a large exact category A we will use this device, assuming tacitly that we have replaced it by a small A′. For example, this is the case in the following definitions.

Definition 6.3.2. Let R be a ring with unit, and let P(R) denote the exact category of finitely generated projective R-modules. We set K(R) = KP(R) and define the K-groups of R by K_{n}(R) = K_{n}P(R). For n = 0, lemma 6.2 shows that this agrees with the definition of K_{0}(R) in chapter II. For n ≥ 1, agreement with the (nonfunctorial) +–construction definition 1.1.1 of K(R) will have to wait until section 7.

Let f : R → S be a ring homomorphism such that S is finitely generated and projective as an R-module. Then there is a forgetful functor P(S) → P(R) and hence a “transfer” functor f_{∗} : K_{∗}(S) → K_{∗}(R).

Definition 6.3.3. If R is noetherian, let M(R) denote the category of finitely generated R-modules. Otherwise, M(R) is the category of pseudo-coherent modules defined in II.7.1.4. We set G(R) = KM(R) and define the G-groups of R by G_{n}(R) = K_{n}M(R). For n = 0, this also agrees with the definition in chapter II.

Let f : R → S be a ring map. When S is finitely generated as an R-module (and S is in M(R)), there is a contravariant “transfer” map f_{∗} : G(S) → G(R), induced by the forgetful functor f_{∗} : M(S) → M(R), as in II.6.2.

On the other hand, if S is flat as an R-module, the exact base change functor ⊗_{R}S : M(R) → M(S) induces a covariant map f^{∗} : G(R) → G(S) hence maps f^{∗} : G_{n}(R) → G_{n}(S) for all n. This generalizes the base change map G_{0}(R) → G_{0}(S) of II.6.2. We will see in V.3.5 that the base change map is also defined when S has finite flat dimension over R.

Definition 6.3.4. Similarly, if X is a scheme which is quasi-projective (over a commutative ring), we define K(X) = KVB(X) and K_{n}(X) = K_{n}VB(X). If X is noetherian, we define G(X) = KM(X) and G_{n}(X) = K_{n}M(X). For n = 0, this agrees with the definition of K_{0}(X) and G_{0}(X) in chapter II.

Morita Invariance 6.3.5. Recall from II.2.7 that if two rings R and S are Morita equivalent then there are equivalences P(R) ≅ P(S) and M(R) ≅ M(S). It follows that K_{n}(R) ≅ K_{n}(S) and G_{n}(R) ≅ G_{n}(S) for all n.

Elementary Properties 6.4. Here are some elementary properties of the above definition.

If A^{op} denotes the opposite category of A, then Q(A^{op}) is isomorphic to QA by Ex 6.3, so we have K_{n}(A^{op}) = K_{n}(A). For example, if R is a ring then P(R^{op}) ≅ P(R)^{op} by P ↦ Hom_{R}(P, R), so we have K_{n}(R) ≅ K_{n}(R^{op}).

The product or direct sum A ⊕ A′ of two exact categories is exact by Example II.7.1.6, and Q(A ⊕ A′) = QA × QA′. Since the geometric realization preserves products by 3.1(4), we have BQ(A ⊕ A′) = BQA × BQA′ and hence K_{n}(A ⊕ A′) ≅ K_{n}(A) ⊕ K_{n}(A′). For example, if R_{1} and R_{2} are rings then P(R_{1} × R_{2}) ≅ P(R_{1}) ⊕ P(R_{2}) and we have K_{n}(R_{1} × R_{2}) ≅ K_{n}(R_{1}) ⊕ K_{n}(R_{2}). (Cf. Ex. 1.7.) Similarly, if a quasi-projective scheme X is the disjoint union of two components X_{i}, then VB(X) is the sum of the VB(X_{i}) and we have K_{n}(X) ≅ K_{n}(X_{1}) ⊕ K_{n}(X_{2}).

The direct sum ⊕ : A × A → A is an exact functor, and its restriction to either factor is an isomorphism. It follows that B⊕ : BQA×BQA → BQA endows BQA with the structure of a homotopy-commutative H-space. (It is actually an infinite loop space; see 6.5.1).

Finally, suppose that i ↦ A_{i} is a functor from some small filtering category I to exact categories and exact functors. Then the filtered colimit A = \underrightarrow{lim} A_{i} is an exact category (Ex. II.7.9), and QA = \underrightarrow{lim} QA_{i}. Since geometric realization preserves filtered colimits by 3.1(3), we have BQA = \underrightarrow{lim} BQA_{i} and hence K_{n}(A) = \underrightarrow{lim} K_{n}(A_{i}). The K_{0} version of this result was given in chapter II, as 6.2.7 and 7.1.7.

For example, if a ring R is the filtered union of subrings R_{i} we have K_{n}(R) ≅ \underrightarrow{lim} K_{n}(R_{i}). However, i ↦ P(R_{i}) is not a functor. One way to fix this is to replace the category P(R_{i}) by the equivalent category P′(R_{i}) whose objects are idempotent matrices over R_{i}; P(R) is equivalent to the category P′(R) = \underrightarrow{lim} P_{i}. Alternatively one could use the Kleisli rectification, which is described in Ex. 6.5.

Cofinality 6.4.1. Let B be an exact subcategory of A which is closed under extensions in A, and which is cofinal in the sense that for every A in A there is an A′ in A so that A ⊕ A′ is in B. Then BQB is homotopy equivalent to the covering space of BQA corresponding to the subgroup K_{0}(B) of K_{0}(A) = π_{1}(BQA). In particular, K_{n}(B) ≅ K_{n}(A) for all n > 0.

A special case of this is sketched in Exercise 6.6; the general case follows from this case using the version 8.9.1 of Waldhausen Cofinality below. Note that K_{0}(B) is a subgroup of K_{0}(A) by II.7.2.

Waldhausen constructed a delooping of BQA in [Wa78, p. 194], using the “QQ” construction. This in turn provides a context for products.

Definition 6.5. When A is a small exact category, QQA is the bicategory whose bimorphisms are equivalence classes of commutative diagrams in A of the form

· ↞ · ↣ ·
↑ ↑ ↑ ↑ ↑ ↑
· ↞ · ↣ ·
∨ ↓  ∨ ↓  ∨ ↓
· ↞ · ↣ ·

in which the four little squares can be embedded in a 3 × 3 diagram with short exact rows and columns. Two such diagrams are equivalent if they are isomorphic by an isomorphism which restricts to the identity on each corner object.

Waldhausen proved that the loop space ΩQQA is homotopy equivalent to BQA (see [Wa78, p. 196] and Ex. 6.8). Thus we have K_{n}(A) = π_{n+1}BQA ≅ π_{n+2}BQQA.

Remark 6.5.1. There are also n-fold categories Q^{n}A, defined exactly as in 6.5, with ΩBQ^{n+1}A ≃ BQ^{n}A. The sequence of the BQ^{n}A (using ΩBQA if n = 0) forms an Ω-spectrum K(A), making K(A) into an infinite loop space.

Products

Definition 6.6. If A, B and C are exact categories, a functor ⊗ : A × B → C is called biexact if (i) each partial functor A ⊗ – : B → C and – ⊗ B : A → C is exact, and (ii) A ⊗ 0 = 0 ⊗ B = 0 for the distinguished zero objects of A, B and C.

This is the same as the definition of biexact functor in §II.7. Note that condition (ii) can always be arranged by replacing A, B and C by equivalent exact categories.

Given such a biexact functor, the bicategory map QA ⊗ QB → bi(QC) of 3.10.2 factors through the forgetful functor QQC → bi(QC). The functor QA⊗QB → QQC sends a pair of morphisms A_{0} ↞ A_{1} ↣ A_{2}, B_{0} ↞ B_{1} ↣ B_{2} to the bimorphism

(6.6.1)  \begin{matrix}A_{0} ⊗ B_{0} & ↞ & A_{1} ⊗ B_{0} & ↣ & A_{2} ⊗ B_{0} \\ ↑ ↑ & & ↑ ↑ & & ↑ ↑ \\ A_{0} ⊗ B_{1} & ↞ & A_{1} ⊗ B_{1} & ↣ & A_{2} ⊗ B_{1} \\ ∨ ↓ & & ∨ ↓ & & ∨ ↓ \\ A_{0} ⊗ B_{2} & ↞ & A_{1} ⊗ B_{2} & ↣ & A_{2} ⊗ B_{2} \end{matrix}

Now the geometric realization 3.6 of the bifunctor ⊗ : QA ⊗ QB → QQC is a map BQA × BQB → BQQC by 3.10.1. Since ⊗ sends QA ⊗ 0 and 0 ⊗ QB to 0, by the technical condition (ii), B⊗ sends BQA × 0 and 0 × BQB to the basepoint, and hence factors through a map

(6.6.2)  BQA ∧ BQB → BQQC,

and in fact a pairing K(A) ∧ K(B) → K(C) of spectra; see [Gillet, 7.12]. The reduced join operation [Wh, p. 480] yields bilinear maps

(6.6.3) K_{i}(A) ⊗ K_{j}(B) = π_{i+1}(BQA) ⊗ π_{j+1}(BQB) → π_{i+j+2}(BQA ∧ BQB) → π_{i+j+2}(BQC) ≅ K_{i+j}(C).

Remarks 6.6.4. We say that A acts upon B if there is a biexact A × B → B. If there is an object A_{0} of A so that A_{0} ⊗ – is the identity on B, the map S^{1} = B(0 ⇉ 1) → BQA given by the diagram 0 ↣ A_{0} ↠ 0 of 6.2 induces a map S^{1} ∧ BQA → B(QA ⊗ QB) → BQQB. Its adjoint BQA → ΩBQQA is the natural map of 6.5 (see Ex. 6.8).

When there is an associative pairing A × A → A, K_{∗}(A) becomes a graded ring; it has a unit [A_{0}] ∈ K_{0}(A) if A_{0} ⊗ – = – ⊗ A_{0} = id_{A}, by the preceeding paragraph, and K(A) is a ring spectrum. When K(A) acts on K(B) and the two evident functors A×A×B → B agree up to natural isomorphism, the pairing makes K_{∗}(B) into a left K_{∗}(A)-module.

Example 6.6.5. These remarks apply in particular to the category A = P(R) over a commutative ring R, and VB(X) over a scheme X. Tensor product makes K_{∗}(R) = K_{∗}P(R) and K_{∗}(X) = K_{∗}VB(X) into graded-commutative rings with unit. For every R-algebra A, K_{∗}(A) and G_{∗}(A) are 2-sided graded K_{∗}(R)-modules, and G_{∗}(X) is a graded K_{∗}(X)-module.

If f : A → B is an R-algebra map, and B is finite over A, the finite transfer f_{∗} : G(B) → G(A) is a K_{∗}(R)-module homomorphism: f_{∗}(x · y) = f_{∗}(x) · y for x ∈ G_{∗}(B) and y ∈ K_{∗}(R). This fact is sometimes referred to as the projection formula, and holds because f_{∗}(x·y) and x·f_{∗}(y) arise from the isomorphic functors M ⊗_{B} (B ⊗_{R} P) ≅ M ⊗_{A} (A ⊗_{R} P) of functors M(B) × P(R) → M(A).

The W (R)-module NK_{∗}(A)

6.7. Let k be a commutative ring. We saw in II.7.4.3 that the exact endomorphism category End(k) of pairs (P, α) has an associative, symmetric biexact pairing with itself, given by ⊗_{k}. This makes K_{∗}End(k) into a graded-commutative ring. As in loc. cit., the functors P(k) → End(k) → P(k) decompose this ring as a product of K_{∗}(k) and another graded-commutative ring which we call End_{∗}(k).

If R is an k-algebra, End(k) acts associatively by ⊗_{k} on the exact category Nil(R) of nilpotent endomorphisms (II.7.4.4), and on its subcategories F_{m}Nil(R) (Ex. II.7.17). As Nil(R) is their union, we see that K_{∗}Nil(R) = colim K_{∗}F_{m}Nil(R) is a filtered K_{∗}End(k)-module.

Let Nil(R) denote the fiber of the forgetful functor KNil(R) → K(R); since this is split, we have KNil(R) ≃ K(R) × Nil(R) and K_{∗}Nil(R) ≅ K_{∗}(R) × Nil_{∗}(R), where Nil_{∗}(R) = π_{∗} Nil(R) is a graded End_{∗}(k)-module.

By Almkvist’s Theorem II.4.3, End_{0}(k) is isomorphic to the subgroup of W (k) = (1+tk[[t]])^{×} consisting of all quotients f(t)/g(t) of polynomials in 1+tR[t]. Stienstra observed in [St85] (cf. [St82]) that the End_{0}(k)-module structure extended to a W (k)-module structure by the following device. There are exact functors F_{m}, V_{m} : Nil(R) → Nil(R) defined by F_{m}(P, ν) = (P, ν^{m}) and V_{m}(P, ν) = (P [t]/(t^{m} − ν), t) (see Ex. II.7.16). Stienstra proved in [St82] that (V_{m}α) · ν = V_{m}(α · F_{m}(ν)) for α ∈ End_{0}(k) and ν ∈ Nil_{∗}(R). Since F_{m} is zero on F_{m}Nil(R), the elements V_{m}(α) act as zero on the image F_{m} Nil_{∗}(R) of K_{∗}F_{m}Nil(R) → K_{∗}Nil(R) → Nil_{∗}(R).

For example, the class of α = [(k, a)] − [(k, 0)]) in End_{0}(k) ⊂ W (R) is 1 − at, so V_{m}(α) = (1 − at^{m}) acts as zero. Stienstra also proves in [St85] that if g(t) = 1 + ... has degree < m and f(t) is any polynomial then the element 1+t^{m}(f/g) of End_{0}(k) acts as zero on F_{m} Nil_{∗}(R). Hence the ideal End_{0}(R) ∩ (1 + t^{m}R[[t]]) is zero on F_{m} Nil_{∗}(R). Writing an element of W (k) as a formal factorization f(t) = ∏_{i=1}^{∞}(1 − a_{m}t^{m}), the formula f · ν = ∑(1 − a_{m}t^{m}) · ν makes sense as a finite sum.

Proposition 6.7.1. If k = ℤ/pℤ, Nil_{∗}(R) is a graded p-group.

If k = S^{−1}ℤ, or if k is a ℚ-algebra, Nil_{∗}(R) is a graded k-module.

Proof. If k = S^{−1}ℤ, or if k is a ℚ-algebra, the map m ↦ (1−t)^{m} defines a ring homomorphism from k into W (k), so any W (k)-module is a k-module. If p = 0 (or even p^{ν} = 0) in k then for each n the formal factorization of (1 − t)^{p^{N}} involves only (1 − a_{m}t^{m}) for m ≥ n. It follows that p^{N} annihilates the image of K_{∗}F_{n}Nil(R) in Nil_{∗}(R). Since Nil_{∗}(R) is the union of these images, the result follows. □

We will see in chapter V.9 that there is an isomorphism NK_{n+1}(R) ≅ Nil_{n}(R), so what we have really seen is that NK_{∗}(R) is a graded End_{∗}(k)-module, with the properties given by 6.7.1:

Corollary 6.7.2. If k = ℤ/pℤ, each NK_{n}(R) is a p-group.

If k = S^{−1}ℤ, or if k is a ℚ-algebra, each NK_{n}(R) is a k-module.

Example 6.7.3. If R is an algebra over the complex numbers ℂ, then each NK_{n}(R) has the structure of a ℂ-vector space. As an abelian group, it is either zero or else uniquely divisible and uncountable.

The endofunctor V_{m}(P, α) = (P [t]/t^{m} − α, t) of End(R) (Ex. II.7.16) sends Nil(R) to itself, and F_{m}V_{m}(P, ν) = ⊕_{1}^{m}(P ν). Hence V_{m} induces an endomorphism V_{m} on each Nil_{n}(R), such that F_{m}V_{m} is multiplication by m.

Proposition 6.7.4 (Farrell). If any NK_{n}(R) is nonzero, it cannot be finitely generated as an abelian group.

Proof. Since NK_{n}(R) = colim K_{n−1}F_{n}Nil(R), every element is killed by all sufficiently large F_{m}. If NK_{n}(R) were finitely generated, there would be an integer M so that the entire group is killed by F_{m} for all m > M. Pick β ≠ 0 in NK_{n}(R) and choose m > M so that mβ ≠ 0. But F_{m}(V_{m}β) = mβ is nonzero, a contradiction. □

Finite generation

The following conjecture is due to Bass.

Bass’ Finiteness Conjecture 6.8. Let R be a commutative regular ring, finitely generated as a ℤ-algebra. Then the groups K_{n}(R) are finitely generated for all n.

Quillen used a filtration of the Q-construction to prove in [Q73] that the groups K_{n}(R) are finitely generated for any Dedekind domain R such that (1) Pic(R) is finite and (2) the homology groups H_{n}(Aut(P), st(P ⊗_{R} F)) are finitely generated. He then verified (2) in [Q73] (number field case) and [GQ82] (affine curves). In other words:

Theorem 6.9. (Quillen) Let R be either an integrally closed subring of a number field F, finite over ℤ, or else the coordinate ring of a smooth affine curve over a finite field. Then K_{n}(R) is a finitely generated group for all n.

EXERCISES

6.1 Admissible subquotients. Let B be an object in an exact category A. An admissible layer in B is a pair of subobjects represented by a sequence B_{1} ↣ B_{2} ↣ B of admissible monics, and we call the quotient B_{2}/B_{1} an admissible subquotient of B. Show that a morphism A → B in QA may be identified with an isomorphism j: B_{2}/B_{1} ≅ A of A with an admissible subquotient of B, and that composition in QA arises from the fact that a subquotient of a subquotient is a subquotient. 6.2 If two exact categories A and A′ are equivalent (and the equivalence respects exactness), show that QA and QA′ are equivalent. If both are small categories, conclude that K_{n}(A) ≅ K_{n}(A′) for all n. 6.3 If A is an exact category, so is its opposite category A^{op} (see Example II.7.1.5). Show that Q(A^{op}) is isomorphic to QA. 6.4 Let B be an object in an exact category A. Show that the comma category QA/B is equivalent to the poset of admissible layers of B in the sense of Ex. 6.1. If P is an exact subcategory of A and i denotes the inclusion QP ⊂ QA, show that i/B is equivalent to the poset of admissible layers of B with B_{2}/B_{1} ∈ P. 6.5 Kleisli rectification. Let I be a filtering category, and let I → CAT be a lax functor in the sense of Ex. 3.8. Although the family of exact categories QA(i) is not filtering, the family of homotopy groups K_{n}A(i) is filtering. The following trick allows us make K-theoretic sense out of the phantom category A = \underrightarrow{lim} A(i).

Let A_{i} be the category whose objects are pairs (A_{j}, j ⟶^{f} i) with A_{j} in A(j)

and f a morphism in I. A morphism from (A_{j}, j ⟶^{f} i) to (A_{k}, k ⟶^{g} i) is a pair (j ⟶^{h} k, θ_{j}) where f = gh in I and θ_{j} is an isomorphism h_{∗}(A_{j}) ≅ A_{k} in A(k). Clearly A_{i} is equivalent to A(i), and i ↦ A_{i} is a functor. Thus if A denotes \underrightarrow{lim} A_{i} we have K_{n}A = \underrightarrow{lim} K_{n}A(i). 6.6 (Gersten) Suppose given a surjective homomorphism ϕ: K_{0}(A) → G, and let B denote the full subcategory of all B in A with ϕ[B] = 0 in G. In this exercise we show that if B is cofinal in A then K_{n}(B) ≅ K_{n}(A) for n > 0, and K_{0}(B) ⊂ K_{0}(A).

(a) Show that there is a functor ψ: QA → G sending the morphism (6.1.1) of QA to ϕ[B_{1}], B_{1} = ker(j), where G is regarded as a category with one object ∗. Using 6.2, show that the map π_{1}(QA) → π_{1}(G) is just ϕ.

(b) Show that the hypotheses of Theorem B are satisfied by ψ, so that B(ψ/∗) is the homotopy fiber of BQA → BG.

(c) Use Theorem A to show that QB → ψ^{−1}(∗) is a homotopy equivalence.

(d) Suppose in addition that B is cofinal in A (II.5.3), so that K_{0}(B) is the subgroup ker(ϕ) of K_{0}(A) by II.7.2. Use Theorem A to show that ψ^{−1}(∗) ≃ ψ/∗. This proves that BQB → BQA → BG is a homotopy fibration. Conclude that K_{n}(B) ≅ K_{n}(A) for all n ≥ 1. 6.7 (Waldhausen) If A is an exact category, let qA denote the bicategory (3.10) with the same objects as A, admissible monomorphisms and epimorphisms as the horizontal and vertical morphisms, respectively; the bimorphisms in qA are those bicartesian squares in A whose horizontal edges are admissible monomorphisms, and whose vertical edges are admissible epimorphisms.

A_{11} ↣ A_{10}
↓ ↓  ↓ ↓
A_{01} ↣ A_{00}

Show that the diagonal category (Ex. 3.14) of qA is the category QA. 6.8 (Waldhausen) Since the realization of the two-object category 0 ⇉ 1 is S^{1}, the realization of the bicategory (0 ⇉ 1) ⊗ A is S^{1} × BA. Given a morphism A_{0} ↞ A_{1} ↣ A_{2} show that the pair of bimorphisms in QQA

\begin{matrix}A_{0} & ↞ & A_{1} & ↣ & A_{2} \\ ↓ ↓ & & ↓ ↓ & & ↓ ↓ \\ 0 & ↞ & 0 & ↣ & 0, \end{matrix}  \begin{matrix}A_{0} & ↞ & A_{1} & ↣ & A_{2} \\ ↑ ∧ & & ↑ ∧ & & ↑ ∧ \\ 0 & ↞ & 0 & ↣ & 0 \end{matrix}

describe a map S^{1} ∧ BQA → BQQA. Waldhausen observed in [Wa78, p. 197] that this map is adjoint to the homotopy equivalence BQA ≃ ΩBQQA. 6.9 For every biexact A × B → C, show that the pairing K_{0}(A) ⊗ K_{0}(B) → K_{0}(C) of (6.6.3) agrees with the product of II.7.4. 6.10 Show that the functor QA⊗QB → QQC of 6.6 is a map of symmetric monoidal categories (the operation on QQC is slotwise direct sum). Conclude that BQA × BQB → BQQC is an H-space map. (In fact, it is an infinite loop space map.) 6.11 Let A be the direct sum ⊕_{i∈I}A_{i} of exact categories. Show that K_{n}(A) ≅ ⊕_{i∈I}K_{n}(A_{i}). 6.12 If f : R → S is such that S is in P(R), show that the transfer map f_{∗} : K_{0}(S) → K_{0}(R) of 6.3.2 agrees with the transfer functor for K_{0} given in II.2.8.

6.13 If f : R → S and S is in P(R), show that f_{∗}f^{∗} is multiplication by [S] ∈ K_{0}(R), and that f^{∗}f_{∗} is multiplication by [S ⊗_{R} S] ∈ K_{0}(S).

If f : k → ℓ is a purely inseparable field extension, show that both f^{∗}f_{∗} and f_{∗}f^{∗} are multiplication by [ℓ : k] = p^{r}.

If f : R → S is a Galois extension with group G, show that f^{∗}f_{∗} = ∑_{g∈G} g. 6.14 Let C be a category with a distinguished zero object ‘0’ and a coproduct ∨. We say that a family E of sequences of the form

0 → B ⟶^{i} C ⟶^{j} D → 0  (†)

is admissible if the following conditions hold (cf. Ex. II.7.8): (i) Any sequence in C isomorphic to a sequence in E is in E; (ii) If (†) is a sequence in E then i is a kernel for j (resp. j is a cokernel for i) in C; (iii) the class E contains all of the sequences 0 → B → B ∨ D → D → 0; (iv) the class of admissible epimorphisms is closed under composition and pullback along admissible monics; (v) the class of admissible monics is closed under composition and pullback along admissible epimorphisms.

A quasi-exact category is a pair (C, E), where E is admissible in the above sense. If C is small, show that there is a category QC, defined exactly as in 6.1, and that π_{1}(BQC) is the group K_{0}(C) defined exactly as in II.7.1: the group generated by the objects of C subject to the relations [C] = [B] + [D] arising from the admissible exact sequences. (This formulation is due to Deitmar.) 6.15 (Waldhausen) Show that the category Sets_{fin} of finite pointed sets is quasiexact, where E is the collection of split sequences 0 → B → B ∨ D → D → 0, and that K_{0}(Sets_{fin}) = ℤ, exacty as in II.5.2.1. The opposite category Sets_{fin}^{op} is not quasi-exact, because 0 → B → B ∧ D → D → 0 is not in E^{op}. 6.16 A monoid M with identity 1 is pointed if it has an element 0 with 0 · m = m · 0 = 0 for all m ∈ M. A pointed M-set is a pointed set X on which M acts and 0·x = ∗ for all x ∈ X. Show that the category of finitely generated pointed M-sets, and its subcategory of free pointed M-sets, are quasi-exact. Here a sequence (†) is admissible if i is an injection and j identifies D with C/B.

§7. The “+ = Q” Theorem

Suppose that A is an additive category. One way to define the K-theory of A is to consider the symmetric monoidal category S = iso A (where □ = ⊕) and use the S^{−1}S construction: K_{n}^{⊕}A = π_{n}B(S^{−1}S) and K^{⊕}A = K(S) = B(S^{−1}S).

Another way is to suppose that A has the structure of an exact category and form the Q-construction on A with the S^{−1}S construction on S. Comparing the definitions of K_{0}^{⊕}A and K_{0}A in II.5.1.2 and II.7.1, we see that the K_{0} groups are not isomorphic in general, unless perhaps every exact sequence splits in A, i.e., unless A is a split exact category in the sense of II.7.1.2.

Here is the main theorem of this section.

Theorem 7.1 (Quillen). If A is a split exact category and S = iso A, then ΩBQA ≃ B(S^{−1}S). Hence K_{n}(A) ≅ K_{n}(S) for all n ≥ 0.

In fact, B(S^{−1}S) is the group completion of BS by theorem 4.8 and exercise 7.1. In some circumstances (see 4.9, 4.10 and 4.11.1), the S^{−1}S construction is a +– construction. In these cases, theorem 7.1 shows that the Q–construction is also a +–construction. For A = P(R), this yields the “+ = Q” theorem:

Corollary 7.2 (+ = Q). For every ring R,

ΩBQP(R) ≃ K_{0}(R) × BGL(R)^{+}.

Hence K_{n}(R) ≅ K_{n}P(R) for all n ≥ 0.

Definition 7.3. Given an exact category A, we define the category EA as follows. The objects of EA are admissible exact sequences in A. A morphism from E′ : (A′ ↣ B′ ↠ C′) to E : (A ↣ B ↠ C) is an equivalence class of diagrams of the following form, where the rows are exact sequences in A:

(7.3.1)  \begin{aligned}E′ : & A′ & ↣ & B′ & ↠ & C′ \\ & α↑ ∧ & & || & & ↑ ↑ \\ ↓ & A & ↣ & B′ & ↠ & C′′ \\ & || & & ∨ ↓β & & ∨ ↓ \\ E : & A & ↣ & B & ↠ & C. \end{aligned}

Two such diagrams are equivalent if there is an isomorphism between them which is the identity at all vertices except for the C′′ vertex.

Notice that the right column in (7.3.1) is just a morphism φ in QA from C′ to C, so the target C is a functor t : EA → QA: t(A ↣ B ↠ C) = C. In order to improve legibility, it is useful to write E_{C} for the fiber category t^{−1}(C).

Fiber categories 7.4. If we fix φ as the identity map of C = C′, we see that the fiber category E_{C} = t^{−1}(C) of exact sequences with target C has for its morphisms all pairs (α, β) of isomorphisms fitting into a commutative diagram:

A′ ↣ B′ ↠ C
α ↑≅ ≅↓ β ||
A ↣ B ↠ C.

In particular, every morphism in E_{C} is an isomorphism.

Example 7.4.1. The fiber category E_{0} = t^{−1}(0) is homotopy equivalent to S = iso A. To see this, consider the functor from iso A to E_{0} sending A to the

trivial sequence A↣^{id}A ↠ 0. This functor is a full embedding. Moreover, every object of E_{0} is naturally isomorphic to such a trivial sequence, whence the claim.

Lemma 7.5. For any C in A, E_{C} is a symmetric monoidal category, and there is a faithful monoidal functor η_{C} : S → E_{C} sending A to A ↣ A ⊕ C ↠ C.

Proof. Given E_{i} = (A_{i} ↣ B_{i} ↠ C) in E_{C}, set E_{1} ∗ E_{2} equal to

(7.5.1)  A_{1} ⊕ A_{2} ↣ (B_{1} ×_{C} B_{2}) ↠ C.

This defines a symmetric product on E_{C} with identity e: 0 ↣ C ↠ C. It is now routine to check that S → E_{C} is a monoidal functor, and that it is faithful. □

Remark 7.5.2. If A is split exact then every object of E_{C} is isomorphic to one coming from S. In particular, the category ⟨S, E_{C}⟩ of 4.7.1 is connected. This fails if A has a non-split exact sequence.

Proposition 7.6. If A is split exact, each S^{−1}S → S^{−1}E_{C} is a homotopy equivalence.

Proof. By Ex. 4.7 and Ex. 7.1, S^{−1}S → S^{−1}E_{C} → ⟨S, E_{C}⟩ is a fibration, so it suffices to prove that L = ⟨S, E_{C}⟩ is contractible. First, observe that the monoidal product on E_{C} induces a monoidal product on L, so BL is an H-space (as in 4.1). We remarked in 7.5.2 that L is connected. By [Wh, X.2.2], BL is group-like, i.e., has a homotopy inverse.

For every exact sequence E, there is a natural transformation δ_{E}: E → E ∗ E in L, where ∗ is defined by (7.5.1), given by the diagonal.

E :  A ↣  B  ↠ C
↓  ↓  ↓  ||
E ∗ E : A ⊕ A ↣ B ×_{C} B ↠ C

Now δ induces a homotopy between the identity on BL and muliplication by 2. Using the homotopy inverse to subtract the identity, this gives a homotopy between zero and the identity of BL. Hence BL is contractible. □

We also need a description of how E_{C} varies with C.

Lemma 7.7. For each morphism φ: C′ → C in QA, there is a canonical functor φ^{∗}: E_{C} → E_{C′} and a natural transformation η_{E}: φ^{∗}(E) → E from φ^{∗} to the inclusion of E_{C} in EA.

In fact, t : EA → QA is a fibered functor with base change φ^{∗} (Ex. 7.2). It follows (from 3.7.5) that C ↦ E_{C} is a contravariant functor from QA to CAT.

Proof. Choose a representative C′ ↞ C′′ ↣ C for φ and choose a pullback B′ of B and C′′ along C. This yields an exact sequence A ↣ B′ ↠ C′′ in A. (Why?) The composite B′ ↠ C′′ ↠ C′ is admissible; if A′ is its kernel then set

φ^{∗}(A ↣ B ↠ C) = (A′ ↣ B′ ↠ C′).

Since every morphism in E_{C} is an isomorphism, it is easy to see that φ^{∗} is a functor, independent (up to isomorphism) of the choices made. Moreover, the construction yields a diagram (7.3.1), natural in E; the map β is an admissible monic because A ↣ B′ ⟶^{β} B′ is. Hence (7.3.1) constitutes the natural map η_{E} : E → φ^{∗}(E). □

Now the direct sum of sequences defines an operation ⊕ on EA, and S acts on EA via the inclusion of S in EA given by 7.4.1. That is, A′□(A ↣ B ↠ C) is the sequence A′ ⊕ A ↣ A′ ⊕ B ↠ C. Since t(A′□E) = t(E) we have an induced map T = S^{−1}t: S^{−1}EA → QA. This is also a fibered functor (Ex. 7.2).

Theorem 7.8. If A is a split exact category and S = iso A, then the sequence S^{−1}S → S^{−1}EA ⟶^{T} QA is a homotopy fibration.

Proof. We have to show that Quillen’s Theorem B applies, i.e., that the base changes φ^{∗} of 7.7 are homotopy equivalences. It suffices to consider φ of the form 0 ↣ C and 0 ↞ C. If φ is 0 ↣ C, the composition of the equivalence S^{−1}S → S^{−1}E_{C} of 7.6 with φ^{∗} is the identity by Ex. 7.5, so φ^{∗} is a homotopy equivalence.

Now suppose that φ is 0 ↞ C. The composition of the equivalence S^{−1}S → S^{−1}E_{C} of 7.6 with φ^{∗} sends A to A ⊕ C by Ex. 7.5. Since there is a natural transformation A → A ⊕ C in S^{−1}S, this composition is a homotopy equivalence. Hence φ^{∗} is a homotopy equivalence. □

Proof of theorem 7.1. This will follow from theorem 7.8, once we show that S^{−1}EA is contractible. By Ex. 7.3, EA is contractible. Any action of S on a contractible category must be invertible (4.7.1). By Ex. 4.6 and Ex. 7.1, EA → S^{−1}EA is a homotopy equivalence, and therefore S^{−1}EA is contractible. □

Agreement of Product Structures

Any biexact pairing A_{1} × A_{2} ⟶^{⊗} A_{3} of split exact categories (6.6) induces a pairing S_{1} × S_{2} ⟶^{□} S of symmetric monoidal categories, where S_{i} = iso A_{i}. We now compare the resulting pairings K(A_{1}) ∧ K(A_{2}) → K(A_{3}) of 6.6 and K(S_{1}) ∧ K(S_{2}) → K(S_{3}) of 4.6. Waldhausen’s Lemma [Wa78, 9.2.6] implies the following result; the details of the implication are given in [We81, 4.3]:

Theorem 7.9. The homotopy equivalences B(S_{i}^{−1}S_{i}) → ΩBQA_{i} of theorem 7.1 fit into a diagram which commutes up to basepoint-preserving homotopy:

B(S_{1}^{−1}S_{1}) ∧ B(S_{2}^{−1}S_{2}) ⟶^{□} B(S_{3}^{−1}S_{3})
≃↓  ↓^{≃}
(ΩBQA_{1}) ∧ (ΩBQA_{2}) ⟶^{γ} (ΩBQA_{3})
≃↓  ↓^{≃}
Ω^{2}(BQA_{1} ∧ BQA_{2}) ⟶^{Ω^{2}⊗} Ω^{2}(BQQA_{3}).

Hence there are commutative diagrams:

K_{p}(S_{1}) ⊗ K_{q}(S_{2}) ⟶^{□} K_{p+q}(S_{3})
≅↓  ↓≅
K_{p}(A_{1}) ⊗ K_{q}(A_{2}) ⟶^{⊗} K_{p+q}(A_{3}).

The middle map γ is induced from the H-space map ⊗ : ΩBQA_{1} × ΩBQA_{2} → Ω^{2}BQA_{3} of Ex. 6.10, since it sends x ⊗ 0 and 0 ⊗ y to 0.

EXERCISES 7.1 If A is an additive category, S = iso A is equivalent to the disjoint union of one-object categories Aut(A), one for every isomorphism class in A. Show that the translations Aut(A) → Aut(A⊕B) are injections. Then conclude using theorem 4.8 that B(S^{−1}S) is the group completion of the H-space BS = ∐ Aut(A). 7.2 Show that the target functor t: EA → QA is a fibered functor in the sense of Definition 3.7.3, with base change φ^{∗} given by 7.7. Then show that the action of S on EA is cartesian (Ex. 4.11), so that the induced functor S^{−1}EA → QA is also fibered, with fiber S^{−1}S over 0.

7.3 Let iQA denote the subcategory of QA whose objects are those of A but whose morphisms are admissible monomorphisms. Show that the category EA of 7.3 is equivalent to the subdivision category Sub(iQA) of Ex. 3.9. Conclude that the category EA is contractible. 7.4 Show that Quillen’s Theorem B can not apply to EA → QA unless A ≅ 0. Hint: Compare π_{0}S to K_{0}A. 7.5 If φ is the map 0 ↣ C, resp. 0 ↞ C, show that φ^{∗}: E_{C} → E_{0} ≅ S sends A ↣ B ↠ C to A, resp. to B. 7.6 Describe E′A = (EA)^{op}, which is cofibered over (QA)^{op} by Ex. 3.6 and 7.2. Use E′A to prove the + = Q Theorem 7.1. Hint: There is a new action of S. Use pushout instead of pullback in (7.5.1) to prove the analogue of proposition 7.6. 7.7 Finite Sets. Let Sets_{fin} denote the category of finite pointed sets, and form the category QSets_{fin} by copying the Q-construction 6.1 as in Ex. 6.15.

(a) Show that there is an extension category E′Sets_{fin}, defined as in Ex. 7.6, which is cofibered over (QSets_{fin})^{op} with S = iso Sets_{fin} as the fiber over the basepoint.

(b) Modify the proof of the + = Q theorem to prove that ΩBQSets_{fin} ≃ S^{−1}S.

(c) If G is a group, let F be the category of finitely generated free pointed Gsets, and QF as in Ex. 6.16. Using 4.10.1, show that ΩBQF ≃ S^{−1}S ≃ ℤ × Ω^{∞}S^{∞}(BG_{+}). 7.8 (π_{1}BQA) Given an object A in A, lift the morphisms (6.2.1) in QA to morphisms in EA, 0 → η_{A}(0) ← η_{0}(A). Conclude that the isomorphism between K_{0}(A) = π_{1}BQA and K_{0}(S) = π_{0}(S^{−1}S) of theorem 7.1 is the canonical isomorphism of II.7.1.2, identifying [A] with [A]. 7.9 (π_{2}BQA) Given an automorphism α of an object A in A, consider the continuous map [0, 1]^{2} → BQA given by the commutative diagram:

0 ↣ A ↠ 0
∥ α ↓ ∥
0 ↣ A ↠ 0.

Identifying the top and bottom edges to each other, the fact that the left and right edges map to the basepoint (0) means that we have a continuous function S^{2} → BQA, i.e., an element [α] of K_{1}(A) = π_{2}(BQA).

(a) Show that [α] + [α′] = [αα′] for every pair of composable automorphisms. Conclude that α ↦ [α] is a homomorphism Aut(A) → K_{1}(A).

(b) If β ∈ Aut(B), show that the automorphism α⊕β of A⊕B maps to [α]+[β]. Using 4.8.1, this given a map from K_{1}(iso A) to K_{1}(A).

(c) Finally, lift this diagram to EA using Ex. 7.8, representing a map I^{2} → BEA, and conclude that the isomorphism between K_{1}(A) = π_{2}BQA and K_{1}(S) = π_{1}(S^{−1}S) of theorem 7.1 identifies [α] with the class of α given by III.1.6.3 and 4.8.1.

7.10 (Canonical involution) Let R be a commutative ring. The isomorphism P(R) → P(R)^{op} sending P to Hom_{R}(P, R) induces an involution on QP(R) and hence K_{∗}(R) by 6.4; it is called the canonical involution. Show that the involution is a ring automorphism.

On the other hand, the “transpose inverse” involution of GL(R) (g ↦ ^{t}g^{−1}) induces a homotopy involution on BGL(R)^{+} and an involution on K_{n}(R) for n > 0. Show that these two involutions agree via the ‘+ = Q’ theorem 7.2.

§8. Waldhausen’s wS. construction

Our last construction of K-theory applies to Waldhausen categories, i.e., “categories with cofibrations and weak equivalences.” Unfortunately, this will occur only after a lengthy list of definitions, and we ask the reader to be forgiving.

Recall from chapter II, section 9 that a category with cofibrations is a category C with a distinguished zero object ‘0’ and a subcategory co(C) of morphisms in C called “cofibrations” (indicated with feathered arrows ↣). Every isomorphism in C is to be a cofibration, and so are the unique arrows 0 ↣ A for every object A in C. In addition, the pushout C ↣ B ∪_{A} C of any cofibration A ↣ B is a cofibration. (See Definition II.9.1 for more precise statements.) These axioms imply that two constructions make sense: the coproduct B ⨿ C = B ∪_{0} C of any two objects, and every cofibration A ↣ B fits into a cofibration sequence A ↣ B ↠ B/A, where B/A is the cokernel of A ↣ B. The following is a restatement of Definition II.9.1.1:

Definition 8.1. A Waldhausen category C is a category with cofibrations, together with a family w(C) of morphisms in C called “weak equivalences” (indicated with decorated arrows ⟶^{∼}). Every isomorphism in C is to be a weak equivalence, and weak equivalences are to be closed under composition (so we may regard w(C) as a subcategory of C). In addition, the “Glueing axiom” (W3) must be satisfied, which says that the pushout of weak equivalences is a weak equivalence (see II.9.1).

A functor f : A → C between two Waldhausen categories is called an exact functor if it preserves all the relevant structure: zero, cofibrations, weak equivalences and the pushouts along a cofibration.

A Waldhausen subcategory A of a Waldhausen category C is a subcategory which is also a Waldhausen category in such a way that: (i) the inclusion A ⊆ C is an exact functor, (ii) the cofibrations in A are the maps in A which are cofibrations in C and whose cokernel lies in A, and (iii) the weak equivalences in A are the weak equivalences of C which lie in A.

In order to describe Waldhausen’s wS. construction for K-theory, we need a sequence of Waldhausen categories S_{n}C. S_{0}C is the zero category, and S_{1}C is the category C, but whose objects A are thought of as the cofibrations 0 ↣ A. The category S_{2}C is the extension category E of II.9.3. For convenience, we repeat its definition here.

Extension Categories 8.2. The objects of the extension category S_{2}C are the cofibration sequences A_{1} ↣ A_{2} ↠ A_{12} in C. A morphism E → E′ in S_{2}C is a commutative diagram:

E :  A_{1} ↣ A_{2} ↠ A_{12}
↓ ↓^{u_{1}} ↓^{u_{2}} ↓^{u_{3}}
E′ :  A′_{1} ↣ A′_{2} ↠ A′_{12}

We make S_{2}C into a Waldhausen category as follows. A morphism E → E′ in S_{2}C is a cofibration if A_{1} → A′_{1}, A_{12} → A′_{12} and A′_{1} ∪_{A_{1}} A_{2} → A′_{2} are cofibrations in C. A morphism in S_{2}C is a weak equivalence if its component maps u_{i} : A_{i} → A′_{i} (i = 1, 2, 12) are weak equivalences in C.

A Waldhausen category C is called extensional if it satisfies the following technically convenient axiom: weak equivalences are “closed under extensions.”

Extension axiom 8.2.1. Suppose that f: E → E′ is a map between cofibration sequences, as in 8.2. If the source and quotient maps of f (A → A′ and C → C′) are weak equivalences, so is the total map of f (B → B′).

Definition 8.3. (S.C) If C is a category with cofibrations, let S_{n}C be the category whose objects A. are sequences of n cofibrations in C:

A.: 0 = A_{0} ↣ A_{1} ↣ A_{2} ↣ ⋯ ↣ A_{n}

together with a choice of every subquotient A_{ij} = A_{j}/A_{i} (0 < i < j ≤ n). These choices are to be compatible in the sense that there is a commutative diagram:

(8.3.0)  A_{n−1,n} ; ↑ ↑ ; ⋯; A_{23} ↣ ⋯ ↣ A_{2n} ; ↑ ↑  ↑ ↑ ; A_{12} ↣ A_{13} ↣ ⋯ ↣ A_{1n} ; ↑ ↑  ↑ ↑  ↑ ↑ ; A_{1} ↣ A_{2} ↣ A_{3} ↣ ⋯ ↣ A_{n}

The conventions A_{0j} = A_{j} and A_{jj} = 0 will be convenient at times. A morphism A. → B. in S_{n}C is a natural transformation of sequences.

If we forget the choices of the subquotients A_{ij} we obtain the higher extension category E_{n}(C) constructed in II.9.3.2. Since we can always make such choices, it follows that the categories S_{n}C and E_{n}(C) are equivalent. By Ex. II.9.4, when C is a Waldhausen category, so is E_{n}(C) and hence S_{n}C. Here are the relevant definitions for S_{n}, translated from the definitions II.9.3.2 for E_{n}.

A weak equivalence in S_{n}C is a map A. → B. such that each A_{i} → B_{i} (hence, each A_{ij} → B_{ij}) is a weak equivalence in C. A map A. → B. is a cofibration when for every 0 ≤ i < j < k ≤ n the map of cofibration sequences

A_{ij} ↣ A_{ik} ↠ A_{jk}
↓ ↓ ↓
B_{ij} ↣ B_{ik} ↠ B_{jk}

is a cofibration in S_{2}C.

The reason for including choices in the definition of the categories S_{n}C is that we can form a simplicial Waldhausen category. The maps ∂_{0}, ∂_{1} from C = S_{1}C to 0 = S_{0}C are trivial; the maps ∂_{0}, ∂_{1}, ∂_{2} from S_{2}C to C are q_{∗}, t_{∗} and s_{∗}, respectively.

Definition 8.3.1. For each n ≥ 0, the exact functor ∂_{0} : S_{n}C → S_{n−1}C is defined by deletion of the bottom row of (8.3.0). That is, ∂_{0} is defined by the formula

∂_{0}(A.): 0 = A_{11} ↣ A_{12} ↣ A_{13} ↣ ⋯ ↣ A_{1n}

together with the choices ∂_{0}(A.)_{ij} = A_{i+1,j+1}. By Ex. 8.1, ∂_{0}(A.) is in S_{n−1}C.

For 0 < i ≤ n we define the exact functors ∂_{i}: S_{n}C → S_{n−1}C by omitting the row A_{i∗} and the column containing A_{i} in (8.3.0), and reindexing the A_{jk} as needed. Similarly, we define the exact functors s_{i}: S_{n}C → S_{n+1}C by duplicating A_{i}, and reindexing with the normalization A_{i,i+1} = 0. (Exactness is checked in Ex. 8.2.)

By Ex. 8.2, the S_{n}C fit together to form a simplicial Waldhausen category S.(C), and the subcategories wS_{n}C of weak equivalences fit together to form a simplicial category wS.C. Hence their geometric realizations B(wS_{n}C) fit together to form a simplicial topological space BwS.C, and we write |wS.C| for the realization of BwS.C. Since S_{0}C is trivial, |wS.C| is a connected space.

Remark 8.3.2. In the realization of BwS.C, the spaces B(wS_{n}C)×Δ^{n} are glued together along the face maps. In particular, the suspension ΣB(wC) is a subspace of |wS.C|; the adjoint map is B(wC) → Ω|wS.C|. In this way, each object of C yields an element of π_{1}|wS.C|, and each weak equivalence A ≃ A in C yields an element of π_{2}|wS.C|.

Recall from chapter II, 9.1.2, that K_{0}(C) is defined as the group generated by the set of weak equivalence classes [A] of objects of C with the relations that [B] = [A] + [B/A] for every cofibration sequence

A ↣ B ↠ B/A.

Proposition 8.4. If C is a Waldhausen category then π_{1}|wS.C| ≅ K_{0}(C).

Proof. If X. is any simplicial space with X_{0} a point, then |X.| is connected and π_{1}|X.| is the free group on π_{0}(X_{1}) modulo the relations ∂_{1}(x) = ∂_{2}(x)∂_{0}(x) for every x ∈ π_{0}(X_{2}). For X. = BwS.C, π_{0}(BwS_{1}C) is the set of weak equivalence classes of objects in C, π_{0}(BwS_{2}C) is the set of equivalence classes of cofibration sequences, and the maps ∂_{i}: S_{2}C → S_{1}C of 8.3.1 send A ↣ B ↠ B/A to B/A, B and A, respectively. □

Definition 8.5. If C is a small Waldhausen category, its algebraic K-theory space K(C) = K(C, w) is the loop space

K(C) = Ω|wS.C|.

The K-groups of C are defined to be its homotopy groups:

K_{i}(C) = π_{i}K(C) = π_{i+1}|wS.C| if i ≥ 0.

We saw in Remark 8.3.2, there is a canonical map B(wC) → K(C).

Remark 8.5.1. Since the subcategory wC is closed under coproducts in C by axiom (W3), the coproduct gives an H-space structure to |wS.C| via the map

|wS.C| × |wS.C| ≅ |wS.C × wS.C| ⟶^{⨿} |wS.C|.

Simplicial Model 8.5.2. Suppose that C is a small Waldhausen category in which the isomorphisms iC are the weak equivalences. Let s_{n}C denote the set of objects of S_{n}C; as n varies, we have a simplicial set s.C. Waldhausen proved in [W1126, 1.4] that the inclusion |s.C| → |iS.C| is a homotopy equivalence. Therefore Ω|s.C| is a simplicial model for the space K(C).

Relative K-theory spaces 8.5.3. If f : B → C is an exact functor, let S_{n}f denote the category S_{n}B ×_{S_{n}C} S_{n+1}C whose objects are pairs

(B_{∗}, C_{∗}) = (B_{1} ↣ ⋯ ↣ B_{n}, C_{0} ↣ ⋯ ↣ C_{n})

such that f(B_{∗}) is ∂_{0}C_{∗} : C_{1}/C_{0} ↣ ⋯ ↣ C_{n}/C_{0}. Each S_{n}f is a Waldhausen category in a natural way, containing C as the (Waldhausen) subcategory of all (0, C = ⋯ = C), and the projection S_{n}f → S_{n}B is exact. We can apply the S. (and wS.) construction degreewise to the sequence C → S.f → S.B of simplicial Waldhausen categories, obtaining a sequence of bisimplicial Waldhausen categories S.C → S.(S.f) → S.(S.B), and a sequence wS.C → wS.(S.f) → wS.(S.B) of bisimplicial categories. We will see in V.1.7 (using 8.5.4) that the realization of the bisimplicial category sequence

wS.B → wS.C → wS.(S.f) → wS.(S.B),

is a homotopy fibration sequence. Thus we may regard K(f) = Ω^{2}|wS.(S.f)| as a relative K-theory space; the groups K_{n}(f) = π_{n}K(f) fit into a long exact sequence involving f_{∗} : K_{n}(B) → K_{n}(C), ending K_{0}(B) → K_{0}(C) → K_{−1}(f) → 0 (Ex. 8.11).

Lemma 8.5.4. If f : C → C is the identity, wS.f is contractible.

Proof. In this case the simplicial category S.f is just the simplicial path space construction of S.C, and wS.S.f is the simplicial path space construction of wS.S.C (see [WHomo, 8.3.14]). These are contractible since S_{0}f = 0 and wS.S_{0}f are. □

Infinite Loop Structure 8.5.5. Lemma 8.5.4 implies that there are natural homotopy equivalences |wS.C| ≃ Ω|wS.S.C|, and of course K(C) ≃ Ω^{2}|wS.S.C|. In fact K(C) is an infinite loop space.

To see this we just iterate the construction, forming the multisimplicial Waldhausen categories S.^{n}C = S.S. · · · S.C and the multisimplicial categories wS.^{n}C of their weak equivalences. By 8.5.4, we see that |wS.^{n}C| is the loop space of |wS.^{n+1}C|, and that the sequence of spaces

Ω|wS.C|, |wS.C|, |wS.S.C|, …, Ω|wS.^{n}C|, …

forms a connective Ω-spectrum KC, called the K-theory spectrum of C. Many authors think of the K-theory of C in terms of this spectrum. This does not affect the K-groups, because:

π_{i}(KC) = π_{i}K(C) = K_{i}(C), i ≥ 0.

An exact functor f induces a map f_{∗}: K(B) → K(C) of spaces, and spectra, and of their homotopy groups K_{i}(B) → K_{i}(C).

Exact Categories 8.6. We saw in II.9.1.3 that any exact category A becomes a Waldhausen category in which the cofibration sequences are just the admissible exact sequences, and the weak equivalences are just the isomorphisms. We write i(A) for the family of isomorphisms, so that we can form the K-theory space K(A) = Ω|iS.A|. Waldhausen proved in [W1126, 1.9] that there is a homotopy equivalence between |iS.A| and BQA, so that this definition is consistent with the definition of K(A) in definition 6.3. His proof is given in exercises 8.5 and 8.6 below.

Another important example of a Waldhausen category is R_{f}(X), introduced in II.9.1 and Ex. II.9.1. The so-called K-theory of spaces refers to the corresponding K-theory spaces A(X), and their homotopy groups A_{n}(X) = π_{n}A(X).

Example 8.7 (A(∗)). Recall from II.9.1.4 that the category R_{f} = R_{f}(∗) of finite based CW complexes is a Waldhausen category in which the family hR_{f} of weak equivalences is the family of weak homotopy equivalences. This category is saturated (II.9.1.1) and satisfies the extension axiom 8.2.1. Following Waldhausen [W1126], we write A(∗) for the space K(R_{f}) = Ω|hS.R_{f}|. We have A_{0}(∗) = K_{0}R_{f} = ℤ by II.9.1.5.

Example 8.7.1 (A(X)). More generally, let X be a CW complex. The category R(X) of CW complexes Y obtained from X by attaching cells, and having X as a retract, is a Waldhausen category in which cofibrations are cellular inclusions (fixing X) and weak equivalences are homotopy equivalences (see Ex. II.9.1). Consider the Waldhausen subcategory R_{f}(X) of those Y obtained by attaching only finitely many cells. Following Waldhausen [W1126], we write A(X) for the space K(R_{f}(X)) = Ω|hS.R_{f}(X)|. Thus A_{0}(X) = K_{0}R_{f}(X) is ℤ by Ex. II.9.1.

Similarly, we can form the Waldhausen subcategory R_{fd}(X) of those Y which are finitely dominated. We write A^{fd}(X) for K(R_{fd}(X)) = Ω|hS.R_{fd}(X)|. Note that A_{0}^{fd}(X) = K_{0}R_{fd}(X) is ℤ[π_{1}(X)] by Ex. II.9.1.

Cylinder Functors

When working with Waldhausen categories, it is often technically convenient to have mapping cylinders. Recall from Ex. 3.12 that the category C/C of arrows in C has the morphisms of C as its objects, and a map (a, b): f → f′ in C/C is a commutative diagram in C:

(8.8.0)  \begin{matrix}A & ⟶^{f} & B \\ ^{a}↓ & & ↓^{b} \\ A′ & ⟶^{f′} & B′ \end{matrix}

The source s(f) = A and target t(f) = B of f define functors s, t: C/C → C.

Definition 8.8 (Cylinders). Let C be a Waldhausen category. A (mapping) cylinder functor on C is a functor T from the category C/C of arrows in C to the category C, together with natural transformations j_{1} : s ⇒ T, j_{2} : t ⇒ T and p: T ⇒ t so that for every f: A → B the diagram

A ⟶^{j_{1}} T (f) ⟵^{j_{2}} B
f ↘ ↓ p ↙=
B

commutes in C. The following conditions must also hold:

(i) T (0 ↣ A) = A, with p and j_{2} the identity map, for all A ∈ C. (ii) j_{1} ⨿ j_{2} : A ⨿ B ↣ T (f) is a cofibration for all f: A → B. (iii) Given a map (a, b): f → f′ in C/C, i.e., a commutative square (8.8.0), if a

and b are weak equivalences in C then so is T (f) → T (f′). (iv) Given a map (a, b): f → f′ in C/C, if a and b are cofibrations in C, then so

is T (f) → T (f′), and the following map, induced by condition (ii), is also a cofibration in C.

A′ ⨿_{A} T (f) ⨿_{B} B′ → T (f′)

We often impose the following extra axiom on the weak equivalences of C.

Cylinder Axiom 8.8.1. All maps p : T (f) → B are weak equivalences in C.

Suppose C has a cylinder functor T. The cone of an object A is cone(A) = T (A ↠ 0), and the suspension of A is ΣA = cone(A)/A. The cylinder axiom implies that cone(A) ⟶^{∼} 0 is a weak equivalence. Since A ↣ cone(A) ↠ ΣA is a cofibration sequence it follows from the description of K_{0}(C) in II.9.1.2 that [ΣA] = −[A] in K_{0}(C). (Cf. Lemma II.9.2.1.) In fact, the Additivity Theorem (see V.1.2 below) implies that the map Σ: K(C) → K(C) is a homotopy inverse with respect to the H-space structure on K(C), because Σ_{∗} + 1 = cone_{∗} = 0.

The name ‘cylinder functor’ comes from the following two paradigms.

Example 8.8.2. The Waldhausen categories R_{f}(∗) and R_{f}(X) of examples 8.7 and 8.7.1 have a cylinder functor: T (f) is the usual (based) mapping cylinder of f. By construction, the mapping cylinder satisfies the cylinder axiom 8.8.1. Because of this paradigm, j_{1} and j_{2} are sometimes called the front and back inclusions.

Example 8.8.3. Let Ch be the Waldhausen category of chain complexes and quasi-isomorphisms constructed from an abelian (or exact) category C; see II.9.2. The mapping cylinder of f: A. → B. is the usual mapping cylinder chain complex [WHomo, 1.5.5], in which

T (f)_{n} = A_{n} ⊕ A_{n−1} ⊕ B_{n}.

The suspension functor Σ(A.) = A.[−1] here is the shift operator: Σ(A.)_{n} = A_{n−1}.

Example 8.8.4. Exact categories usually do not have cylinder functors. This is reflected by the fact that for some A ∈ A there may be no B such that [A ⊕ B] = 0 in K_{0}(A). However, the Waldhausen category Ch^{b}(A) of bounded chain complexes does have a cylinder functor, and we used it to prove that K_{0}(A) ≅ K_{0}Ch^{b}(A) in II.9.2.2. In fact, K(A) ≃ K(Ch^{b}(A)) by the Gillet-Waldhausen theorem presented in V.2.2. Thus many results requiring mapping cylinders in Waldhausen K-theory can be translated into results for Quillen K-theory.

Cofinality

A Waldhausen subcategory B of C is said to be cofinal if for all C in C there is a C′ in C so that C ⨿ C′ is in B. The K_{0} version of the following theorem was proven in II.9.4. We will prove a stronger cofinality result in chapter V.

Waldhausen Cofinality 8.9. If B is a cofinal Waldhausen subcategory of C, closed under extensions, and such that K_{0}(B) = K_{0}(C). Then wS.B → wS.C and K(B) → K(C) are homotopy equivalences. In particular, K_{n}(B) ≅ K_{n}(C) for all n.

Remark 8.9.1. By Grayson’s Trick (Ex. II.9.14), the assumption that K_{0}(B) = K_{0}(C) is equivalent to saying that B is strictly cofinal in C, meaning that for every C in C there is a B in B so that B ∐ C is in B.

Proof. By 8.5.3, it suffices to show that the “relative” bisimplicial category wS.(S.f) is contractible, where f : B → C is the inclusion. For this it suffices to show that each wS_{n}(S.f) is contractible. Switching simplicial directions, we can rewrite wS_{n}(S_{m}f) as wS_{m}(S_{n}f_{n}), where f_{n} : S_{n}B → S_{n}C and S_{n}f_{n} is defined in 8.5.3. Since S_{n}B is equivalent to E_{n}(C) (see 8.3), we see from Ex. II.9.4 that K_{0}(S_{n}B) ≅ K_{0}(S_{n}C). Hence the hypothesis also applies to f_{n}. Replacing f by f_{n}, we have a second reduction: it suffices to show that the simplicial category wS.f is contractible.

Let B(m, w) denote the category of diagrams B_{0} ⟶^{≃} ⋯ ⟶^{≃} B_{m} in B whose maps are weak equivalences, and f_{(m,w)} the inclusion of B(m, w) in C(m, w). Then the bidegree (m, n) part w_{m}S_{n}f of wS.f is the set s_{n}f_{(m,w)} of objects of S_{n}f_{(m,w)}. Working with the nerve degreewise, it suffices to show that each w_{m}S.f = s.f_{(m,w)} is contractible. Since B is strictly cofinal in C (by Grayson’s trick), this implies that f_{(m,w)} is also strictly cofinal by Ex. 8.12(b). The theorem now follows from Lemma 8.9.2 below. □

Lemma 8.9.2. If f : B → C is strictly cofinal then s.f is contractible, where the elements of s_{n}f are the objects of S_{n}f.

Proof. Strict cofinality implies that for each finite set X of objects (B_{∗}^{i}, C_{∗}^{i}) of S_{n_{i}}f, there is an object B′ of B such that each (B′ ⨿ B_{∗}^{i}, B′ ⨿ C_{∗}^{i}) is in S_{n_{i}}id_{B}, because each B′ ⨿ C_{j}^{i} is in B.

We saw in 8.5.4 that s.id_{B} is the simplicial path space construction of s.B, and is contractible because s_{0}B is a point. We will show that s.f is contractible by showing that it is homotopy equivalent to s.id_{B}. For this we need to show that for any finite subcomplex L of s.f there is a simplicial homotopy h (in the sense of [WHomo, 8.3.11]) from the inclusion L ⊂ s.f to a map L → s.id_{B} ⊂ s.f, such that each component of h sends L ∩ s.id_{B} into s.id_{B}.

If X is the set of nondegenerate elements of L, we saw above that there is a B′ so that B′ ⨿ X (and hence B′ ⨿ L) is in s.id_{B}. The desired simplicial homotopy is given by the restriction of the maps h_{i} : s_{n}f → s_{n+1}f, sending (B_{∗}, C_{∗}) to

(⋯ ↣ B_{j} ↣ B′⨿B_{j} ↣ ⋯ ↣ B′⨿B_{n}, ⋯ ↣ C_{j} ↣ B′⨿C_{j} ↣ ⋯ ↣ B′⨿C_{n}). □

Question 8.9.3. If B is a cofinal Waldhausen subcategory of C, but is not closed under extensions, is K(B) ≃ K(C)? Using Ex. 8.12(a), the above proof shows that this is true if B is strictly cofinal in C.

At the other extreme of cofinality, we have the following theorem of Thomason, which shows that by changing the weak equivalences in A we can force all the higher K-groups to vanish. Let (A, co) be any category with cofibrations; recall from II.9.1.3 that the group K_{0}(A) = K_{0}(iso A) is defined in this context.

Suppose we are given a surjective homomorphism π : K_{0}(A) → G. Let w(A) denote the family of morphisms A → A′ in A such that π[A] = π[A′] in G. As observed in II.9.6.2, (A, w) is a Waldhausen category with K_{0}(A, w) = G.

Theorem 8.10. There is a homotopy equivalence wS.(A, w) → BG. Hence K(A) is homotopic to the discrete set G, and K_{n}(A, w) = 0 for all n ≠ 0.

Proof. (Thomason) By construction of w, the category wA is the disjoint union of the full subcategories π^{−1}(g) on the objects A with π[A] = g. For each g, fix an object A_{g} with π[A_{g}] = g. For n > 1, consider the function π : ws_{n}A → G^{n} sending the object A_{1} ↣ A_{2} ↣ ⋯ ↣ A_{n} of wS_{n}A to (π[A_{1}], π[A_{12}], π[A_{23}], …, π[A_{n−1,n}]). By the construction of w, it induces a decomposition of wS_{n}A into the disjoint union (indexed by G^{n}) of the full subcategories π^{−1}(g_{1}, ..., g_{n}) of objects mapping to (g_{1}, ..., g_{n}). We will show that each of these components is contractible.

Given g = (g_{1}, ..., g_{n}), π^{−1}(g) is not empty because it contains the object

A_{g} : A_{g_{1}} ↣ (A_{g_{1}} ⨿ A_{g_{2}}) ↣ (A_{g_{1}} ⨿ A_{g_{2}} ⨿ A_{g_{3}}) ↣ ⋯ ↣ (⨿_{i=1}^{n}A_{g_{i}})

of wS_{n}A. The subcategory π^{−1}(0) is contractible because it has initial object 0. For other g, there is a natural transformation from the identity of π^{−1}(g) to the functor F (B) = A_{g} ⨿ A_{−g} ⨿ B, given by the coproduct with the weak equivalence 0 → A_{g} ⨿ A_{−g}. But F is null-homotopic because it factors as the composite of F′ : π^{−1}(g) → π^{−1}(0), F′(B) = A_{−g} ⨿ B, and F′′ : π^{−1}(0) → π^{−1}g, F′′(C) = A_{g} ⨿ C. It follows that π^{−1}(g) is contractible, as desired. □

Products

8.11 Our discussion in 6.6 about products in exact categories carries over to the Waldhausen setting. The following construction is taken from [Wa1126, just after 1.5.3]. Let A, B and C be Waldhausen categories; recall from II.9.5.2 that a functor F: A × B → C is biexact if each F (A, −) and F (−, B) is exact, and the following condition is satisfied:

For every pair of cofibrations (A ↣ A′ in A, B ↣ B′ in B) the following map is a cofibration in C:

F (A′, B) ∪_{F (A,B)} F (A, B′) ↣ F (A′, B′).

We saw in II.9.5.1 that a biexact functor induces a bilinear map K_{0}(A) ⊗ K_{0}(B) → K_{0}(C). It also induces a morphism of bisimplicial bicategories

wS.A × wS.B → wwS.S.C

which resembles (6.6.1). Upon passage to geometric realization, this factors

K(A) ∧ K(B) → K(C).

EXERCISES

8.1 Show that for every 0 ≤ i < j < k ≤ n the diagram A_{ij} ↣ A_{ik} ↠ A_{jk} is a cofibration sequence, and that this gives an exact functor from S_{n}C to S_{2}C. 8.2 Show that each functor ∂_{i}: S_{n}C → S_{n−1}C is exact in 8.3.1. Then show that S.C is a simplicial category. 8.3 Let f, f′: A → B be exact functors. A natural transformation η: f → f′ is called a weak equivalence if each f(A) ⟶^{∼} f′(A) is a weak equivalence in B. Show that a weak equivalence induces a homotopy between the two maps K(A) → K(B). Hint: Show that the maps wS_{n}A → wS_{n}B are homotopic in a compatible way. 8.4 We saw in 8.3.2 that there is a canonical map from Bw(C) to K(C), and hence maps π_{i}B(wC) → K_{i}(C). The map π_{0}B(wC) → K_{0}(C) is described in 8.4.

(a) Every weak self-equivalence α: A ⟶^{∼} A determines an element [α] of K_{1}(C), by 3.4. If β is a weak self-equivalence of B, show that [α] + [β] = [α ∨ β]. If A = B, show that [α] + [β] = [βα]

(b) If A is an exact category, considered as a Waldhausen category, show that the map B(iso A) → K(A) induces a map from the group K_{1}^{⊕}A of 4.8.1 to K_{1}(A).

(c) In the notation of 8.2, show that a weak equivalence in S_{2}C with A_{i} = A′_{i} determines a relation [u_{1}] − [u_{2}] + [u_{12}] = 0 in K_{1}(C).

(d) Show that every pair of cofibration sequences A ↣ B ↠ C (with the same objects) determines an element of K_{2}(C). 8.5 (Waldhausen) Let A be a small exact category. In this exercise we produce a map from |iS.A| ≃ |s.A| to BQA, where s.A is defined in 8.5.2.

(a) Show that an object A. of iS_{3}A determines a morphism in QA from A_{12} to A_{3}.

(b) Show that an object A. of iS_{5}A determines a sequence A_{23} → A_{14} → A_{5} of row morphisms in QA.

(c) Recall from Ex. 3.10 that the Segal subdivision Sub(s.A) is homotopy equivalent to s.A. Show that (a) and (b) determine a simplicial map Sub(s.A) → QA. Composing with |iS.A| ≃ Sub(s.A), this yields a map from |iS.A| to BQA. 8.6 We now show that the map |iS.A| → BQA constructed in the previous exercise is a homotopy equivalence. Let iQ_{n}A denote the category whose objects are the degree n elements of the nerve of QA, i.e., sequences A_{0} → ⋯ → A_{n} in QA, and whose morphisms are isomorphisms.

(a) Show that iQ.A is a simplicial category, and that the nerve of QA is the simplicial set of objects. Waldhausen proved in [W1126, 1.6.5] that BQA → |iQ.A| is a homotopy equivalence.

(b) Show that for each n there is an equivalence of categories Sub(iS_{n}A) ⟶^{∼} iQ_{n}A, where Sub(iS_{n}A) is the Segal subdivision category of Ex. 3.9. Then show that the equivalences form a map of simplicial categories Sub(iS.A) → iQ.A. This map must be a homotopy equivalence, because it is a homotopy equivalence in each degree. Hint: The typical case Sub(iS_{3}A) → iQ_{3}A is illustrated in [W1126, 1.9].

(c) Show that the map of the previous exercise fits into a diagram

|s.A| ⟵^{≃} |Sub(s.A)| ⟶ BQA
≃↓  ^{≃}↓  ^{≃}↓
|iS.A| ⟵^{≃} |Sub(iS.A)| ⟶^{≃} |iQ.A|.

Conclude that the map |iS.A| → BQA of Ex. 8.5 is a homotopy equivalence. 8.7 Recall from 8.5.2 that K_{1}(A) ≅ π_{2}|s.A| for any exact category A. Given an automorphism α of an object A in A, show that the two 2-cells in |s.A| corresponding to the extensions 0 ↣ A↠^{α}A and A↣^{α}A ↠ 0 fit together to define an element of π_{2}|s.A|. Then show that the map of Ex. 8.5 identifies it with the element [α] of π_{2}BQA described in Ex. 7.8. 8.8 Finite Sets. Show that the category Sets_{fin} of finite pointed sets is a Waldhausen category, where the cofibrations are the injections and the weak equivalences are the isomorphisms. Then mimick Exercises 8.5 and 8.6 to show that the space BQSets_{fin} of Ex. 6.15 is homotopy equivalent to the Waldhausen space iS.Sets_{fin}. Using Theorem 4.9.3 and Ex. 7.7, conclude that the Waldhausen K-theory space K(Sets_{fin}) is ℤ×(BΣ_{∞})^{+} ≃ Ω^{∞}S^{∞}. Thus K_{n}(Sets_{fin}) ≅ π_{n}^{s} for all n. 8.9 G-Sets. If G is a group, show that the category G−Sets_{+} of finitely generated pointed G-sets, and its subcategory F of free pointed G-sets, are Waldhausen categories. Then mimick Exercises 8.5 and 8.6 to show that the spaces BQ(G−Sets_{+}) and BQF of Ex. 6.16 are homotopy equivalent to the Waldhausen spaces iS.(G−Sets_{+}) and iS.F. Using Ex. 7.7, conclude that the Waldhausen K-theory space K(F) is homotopy equivalent to Ω^{∞}S^{∞}(BG_{+}). 8.10 Given a category with cofibrations C, let E = E(C) denote the category of extensions in C (see II.9.3), and s_{.}C the simplicial set of 8.5.2. In this exercise we show that the source and quotient functors s, q : E → C induce s_{.}E ≃ s_{.}C × s_{.}C.

(a) Recall from Ex. 3.11 that for A in s_{n}C the simplicial set s/(n, A) is the pullback of s_{.}E and Δ^{n} along s and A : Δ^{n} → s_{.}C. Show that s/(0, 0) is equivalent to s_{.}C.

(b) For every vertex α of Δ^{n} and every A in s_{n}C that the map s/(0, 0) → s/(n, A) of Ex. 3.11 is a homotopy equivalence.

(c) Use (b) to show that s : s_{.}E → s_{.}C satisfies the hypothesis of Ex. 3.11(B).

(d) Use Ex. 3.11(B) to show that there is a homotopy fibration s_{.}C → s_{.}E → s_{.}C Conclude that s × q : s_{.}E → s_{.}C × s_{.}C is a homotopy equivalence. 8.11 Given an exact functor f : B → C, mimick the proof of 8.4 to show that the group K_{−1}(f) = π_{1}(wS.S.f) of 8.5.3 is the cokernel of K_{0}(B) → K_{0}(C). 8.12 Suppose that B is a strictly cofinal Waldhausen subcategory of C.

(a) Show that S_{n}B is strictly cofinal in S_{n}(C).

(b) Show that B(w, m) is strictly cofinal in C(w, m).

8.13 Any exact category A is cofinal in its idempotent completion Â, by definition (see II.7.3). Let A′ be the subcategory of Â consisting of all B in Â such that [B] lies in the subgroup K_{0}(A) of K_{0}(Â). Show that A′ is an exact category, closed under admissible epimorphisms in Â, and that A is strictly cofinal in A′. Hence K(A) ≃ K(A′). 8.14 Let Ch(C) be the Waldhausen category of chain complexes in an exact category C, as in 8.8.3. Show that Ch(C) and Ch^{b}(C) are saturated and satisfy the Extension axiom 8.2.1, and the Cylinder Axiom 8.8.1. 8.15 If (C, co, w) is a saturated Waldhausen category with a cylinder functor, satisfying the cylinder axiom, show that the category co wC of “trivial cofibrations” (cofibrations which are weak equivalences) is homotopy equivalent to wC. Hint: Use the cylinder to show that each i/C is contractible, and apply Theorem A.

§9. The Gillet-Grayson construction

Let A be an exact category. Following Grayson and Gillet [GG], we define a simplicial set G. = G.A as follows.

Definition 9.1. If A is a small exact category, G. is the simplicial set defined as follows. The set G_{0} of vertices consists of all pairs of objects (A, B) in A. The set G_{1} of edges consists of all pairs of short exact sequences with the same cokernel:

(9.1.0)  A_{0} ↣ A_{1} ↠ A_{01}, B_{0} ↣ B_{1} ↠ A_{01}.

The degeneracy maps G_{1} → G_{0} send (9.1.0) to (A_{1}, B_{1}) and (A_{0}, B_{0}), respectively.

The set G_{n} consists of all pairs of triangular commutative diagrams in A of the form

(9.1.1) A_{n−1,n} ; ↑ ↑ ; ⋯; A_{12} ↣ ⋯ ↣ A_{1n} ; ↑ ↑  ↑ ↑ ; A_{01} ↣ A_{02} ↣ ⋯ ↣ A_{0n} ; ↑ ↑  ↑ ↑  ↑ ↑ ; A_{0} ↣ A_{1} ↣ A_{2} ↣ ⋯ ↣ A_{n} A_{n−1,n} ; ↑ ↑ ; ⋯; A_{12} ↣ ⋯ ↣ A_{1n} ; ↑ ↑  ↑ ↑ ; A_{01} ↣ A_{02} ↣ ⋯ ↣ A_{0n} ; ↑ ↑  ↑ ↑  ↑ ↑ ; B_{0} ↣ B_{1} ↣ B_{2} ↣ ⋯ ↣ B_{n}

so that each sequence A_{i} ↣ A_{j} ↠ A_{ij} and B_{i} ↣ B_{j} ↠ A_{ij} is exact. As in the definition of S.A (8.3.1), the face maps ∂_{i}: G_{n} → G_{n−1} are obtained by deleting the row A_{i∗} and the columns containing A_{i} and B_{i}, while the degeneracy maps σ_{i}: G_{n} → G_{n+1} are obtained by duplicating A_{i} and B_{i}, and reindexing.

Suppressing the choices A_{ij} for the cokernels, we can abbreviate (9.1.1) as:

(9.1.2)  A_{0} ↣ A_{1} ↣ A_{2} ↣ ⋯ ↣ A_{n} ; B_{0} ↣ B_{1} ↣ B_{2} ↣ ⋯ ↣ B_{n} .

Remark 9.1.3. |G.| is a homotopy commutative and associative H-space. Its product |G.| × |G.| → |G.| arises from the simplicial map G. × G. → G. whose components G_{n} × G_{n} → G_{n} are termwise ⊕.

Note that for each isomorphism A ≅ A′ in A there is an edge in G_{1} from (0, 0) to (A, A′), represented by (0 ↣ A ↠ A, 0 ↣ A′ ↠ A). Hence (A, A′) represents zero in the group π_{0}|G.|.

Lemma 9.2. There is a group isomorphism π_{0}|G.| ≅ K_{0}(A).

Proof. As in 3.3, π_{0}|G.| is presented as the set of elements (A, B) of G_{0}, modulo the equivalence relation that for each edge (9.1.0) we have

(A_{1}, B_{1}) = (A_{0}, B_{0}).

It is an abelian group by 9.1.3, with operation (A, B) ⊕ (A′, B′) = (A ⊕ A′, B ⊕ B′). Since (A⊕B, B ⊕A) represents zero in π_{0}|G.|, it follows that (B, A) is the inverse of (A, B). From this presentation, we see that there is a map K_{0}(A) → π_{0}|G.| sending [A] to (A, 0), and a map π_{0}|G.| → K_{0}(A), sending (A, B) to [A] − [B]. These maps are inverses to each other. □

(9.3) We now compare G. with the loop space of the simplicial set s.A of 8.5.2. If we forget the bottom row of either of the two triangular diagrams in (9.1.1), we get a triangular commutative diagram of the form (8.3.0), i.e., an element of s_{n}A. The resulting set maps G_{n} → s_{n}A fit together to form a simplicial map ∂_{0} : G. → s.A.

Path Spaces 9.3.1. Recall from [WHomo, 8.3.14] that the path space P X. of a simplicial set X. has P X_{n} = X_{n+1}, its ith face operator is the ∂_{i+1} of X., and its ith degeneracy operator is the σ_{i+1} of X.. The forgotten face maps ∂_{0} : X_{n+1} → X_{n} form a simplicial map P X. → X., and π_{0}(P X.) ≅ X_{0}. In fact, σ_{0} induces a canonical simplicial homotopy equivalence from P X. to the constant simplicial set X_{0}; see [WHomo, Ex. 9.3.7]. Thus P X. is contractible exactly when X_{0} is a point.

Now there are two maps G_{n} → s_{n+1}A, obtained by forgetting one of the two triangular diagrams (9.1.1) giving an element of G_{n}. The face and degeneracy maps of G. are defined so that these yield two simplicial maps from G. to the path space P. = P (s.A). Clearly, either composition with the canonical map P. → s.A yields the map ∂_{0} : G. → s.A. Thus we have a commutative diagram

(9.3.2)  \begin{matrix}G. & ⟶ & P. \\ ↓ & & ↓ \\ P. & ⟶ & s.A. \end{matrix}

Since s_{0}A is a point, the path space |P.| is canonically contractible. Therefore this diagram yields a canonical map |G.| → Ω|s.A|. On the other hand, we saw in 8.6 that |s.A| ≃ BQA, so Ω|s.A| ≃ ΩBQA = K(A).

We cite the following result from [GG, 3.1]. Its proof uses simplicial analogues of Quillen’s theorems A and B.

Theorem 9.4. (Gillet-Grayson) Let A be a small exact category. Then the map of (9.3) is a homotopy equivalence:

|G.| ≃ Ω|s.A| ≃ K(A).

Hence π_{i}|G.| = K_{i}(A) for all i ≥ 0.

Example 9.5. A double s.e.s. in A is a pair ℓ of short exact sequences in A on the same objects:

ℓ: A ↣^{f} B ↠^{g} C, A ↣^{f′} B ↠^{g′} C.

Thus ℓ is an edge (in G_{1}) from (A, A) to (B, B). To ℓ we attach the element [ℓ] of K_{1}(A) = π_{1}|G.| given by the following 3-edged loop.

(A, A) ⟶^{ℓ} (B, B)
e_{A} ↖ ↗ e_{B}
(0, 0)

where e_{A} denotes the canonical double s.e.s. (0 ↣ A ↠ A, 0 ↣ A ↠ A).

The following theorem was proven by A. Nenashev in [Nen].

Nenashev’s Theorem 9.6. K_{1}(A) may be described as follows.

(a) Every element of K_{1}(A) is represented by the loop [ℓ] of a double s.e.s.;

(b) K_{1}(A) is presented as the abelian group with generators the double s.e.s. in A, subject to two relations:

(i) If E is a short exact sequence, the loop of the double s.e.s. (E, E) is zero; (ii) for any diagram of six double s.e.s. (9.6.1) such that the “first” diagram commutes, and the “second” diagram commutes, then

[r_{0}] − [r_{1}] + [r_{2}] = [c_{0}] − [c_{1}] + [c_{2}],

where r_{i} is the i^{th} row and c_{i} is the i^{th} column of (9.6.1).

(9.6.1)  \begin{matrix}A′ & ⇉ & A & ⇉ & A′′ \\ ↓↓ & & ↓↓ & & ↓↓ \\ B′ & ⇉ & B & ⇉ & B′′ \\ ↓↓ & & ↓↓ & & ↓↓ \\ C′ & ⇉ & C & ⇉ & C′′. \end{matrix}

Example 9.6.2. If α is an automorphism of A, the class [α] ∈ K_{1}(A) is the class of the double s.e.s. (0 ↣ A ↠^{α} A, 0 ↣ A ↠^{=} A).

If β is another automorphism of A, the relation [αβ] = [α] + [β] comes from relation (ii) for

0 ⇉  0  ⇉  0
↓↓  ↓↓  ↓↓
0 ⇉ A ⇉^{α}_{1} A
↓↓  1 ↓↓ 1  β ↓↓ 1
0 ⇉ A ⇉^{αβ}_{1} A.

EXERCISES

9.1 Verify that condition 9.6(i) holds in π_{1}|G.|. 9.2 Show that omitting the choice of quotients A_{ij} from the definition of G.A yields a homotopy equivalent simplicial set G′. A. An element of G′_{n}A is a diagram (9.1.2) together with a compatible family of isomorphisms A_{j}/A_{i} ≅ B_{j}/B_{i}. 9.3 Consider the involution on G. which interchanges the two diagrams in (9.1.1). We saw in 9.2 that it induces multiplication by −1 on K_{0}(A). Show that this involution is an additive inverse map for the H-space structure 9.1.3 on |G.|. 9.4 If α: A ≅ A is an isomorphism, use relation (ii) in Nenashev’s presentation 9.6 to show that [α^{−1}] ∈ K_{1}(A) is represented by the loop of the double s.e.s.:

A →^{α} A ; A →_{=} A

9.5 If A is a split exact category, use Nenashev’s presentation 9.6 to show that K_{1}(A) is generated by automorphisms (9.6.2).

§10. Non-connective spectra for algebraic K-theory

In III.4 we introduced the negative K-groups of a ring using Bass’ Fundamental Theorem for K_{0}(R[t, t^{−1}]), III.3.7. For many applications, it is useful to have a spectrum-level version of this construction, viz., a non-connective “Bass K-theory spectrum” K^{B}(R) with π_{n}K^{B}(R) = K_{n}(R) for all n < 0. In this section we constuct such a non-connective spectrum starting from any one of the functorial models of a connective K-theory spectrum K(R). (See 1.9(iii), 4.5.2 and 8.5.5.)

Let E be a functor from rings to spectra. Since the inclusions of E(R) in E(R[x]) and E(R[x^{−1}]) split, the homotopy pushout E(R[x]) ∨_{E(R)} E(R[x^{−1}]) is the wedge of E(R) and these two complementary factors.

Definition 10.1. Write LE(R) for the spectrum homotopy cofiber of the map f_{0} from this homotopy pushout to E(R[x, x^{−1}]), and ΛE(R) for its desuspension ΩLE(R).

Since the mapping cone is natural, LE and ΛE are functors and there is a cofibration sequence, natural in E and R:

ΛE(R) → E(R[x]) ∨_{E(R)} E(R[x^{−1}]) ⟶^{f_{0}} E(R[x, x^{−1}]) → LE(R).

The algebraic version of the Fundamental Theorem of higher K-theory, established in V.6.2 and V.8.2, states that there is a split exact sequence

0 → K_{n}(R) → K_{n}(R[x]) ⊕ K_{n}(R[x^{−1}]) → K_{n}(R[x, x^{−1}]) ⟶^{←} K_{n−1}(R) → 0,

in which the splitting is multiplication by x ∈ K_{1}(ℤ[x, x^{−1}]). Applying π_{n} to the case E = K of Definition 10.1 shows that π_{n}LK(R) ≅ K_{n−1}(R) for all n > 0. The Fundamental Theorems for K_{1} and K_{0} (III, 3.6 and 3.7) imply that π_{0}ΛK(R) = K_{0}(R), π_{−1}ΛK(R) = K_{−1}(R) and that π_{n}ΛK(R) = 0 for n < −1.

We will need the following topological version of the Fundamental Theorem, also established in the next chapter (in V.8.4). Fix a map S^{1} → K(ℤ[x, x^{−1}]), represented by the element x ∈ K_{1}(ℤ[x, x^{−1}]). Recall from 1.10.2 and Ex. 4.14 that this map induces a product map K(R) ⟶^{∪x} ΩK(R[x, x^{−1}]), natural in the ring R. Composing with ΩK(R[x, x^{−1}]) → ΩLK(R) ⟵^{≃} ΛK(R) yields a map of spectra K(R) → ΛK(R).

Fundamental Theorem 10.2. For any ring R, the map K(R) → ΛK(R) induces a homotopy equivalence between K(R) and the (−1)-connective cover of the spectrum ΛK(R). In particular, K_{n}(R) ≅ π_{n}ΛK(R) for all n ≥ 0.

By induction on k, we have natural maps

Λ^{k−1}K(R) ⟶^{∪x} Λ^{k−1}ΩK(R[x, x^{−1}]) → Λ^{k−1}ΩLK(R) ⟵^{≃} Λ^{k}K(R).

Corollary 10.3. For k > 0 the map Λ^{k−1}K(R) → Λ^{k}K(R) induces a homotopy equivalence between Λ^{k−1}K(R) and the (−k)-connective cover of Λ^{k}K(R), with K_{n}(R) ≅ π_{n}Λ^{k−1}K(R) ≅ π_{n}Λ^{k}K(R) for n > −k, and K_{−k}(R) ≅ π_{−k}Λ^{k}K(R).

Proof. We proceed by induction on k, the case k = 1 being Theorem 10.2. Set E = Λ^{k−1}K; we have a natural isomorphism K_{n}(R) ≅ π_{n}E(R) for n > −k, such that ∪x : K_{n}(R) → K_{n+1}(R[x, x^{−1}]) agrees with π_{n} of E(R) → ΩE(R[x, x^{−1}]) up to isomorphism. The map π_{n}E(R[x]) ∨_{E(R)} E(R[x^{−1}]) → π_{n}E(R[x, x^{−1}]) in 10.1 is an injection for all n, being either the injection from K_{n}(R[x])⊕K_{n}(R[x^{−1}])/K_{n}(R) to K_{n}(R[x, x^{−1}]) of III.4.1.2 and 10.2 (for n > −k) or 0 → 0 (for n ≤ −k). It follows from III.4.1.2 that for n > −k the maps K_{n}(R) ≅ π_{n}Λ^{k−1}K(R) → π_{n}Λ^{k}K(R) are isomorphisms, and that the composite

K_{−k}(R) ⟶^{∪x} K_{1−k}(R[x, x^{−1}]) ≅ π_{1−k}Λ^{k−1}K(R[x, x^{−1}]) → π_{−k}Λ^{k}K(R)

is an isomorphism. Since Λ^{k−1}K(R) is (−k)-connected, it is the (−k)-connected cover of Λ^{k}K(R). It is also clear from 10.1 that π_{n}Λ^{k}K(R) = 0 for n < −k. □

Definition 10.4. We define K^{B}(R) to be the homotopy colimit of the diagram

K(R) →ΩLK(R) ⟵^{≃} ΛK(R) → ⋯ Λ^{k−1}K(R) →Λ^{k−1}ΩLK(R) ⟵^{≃}Λ^{k}K(R) → ⋯

(The homotopy colimit may be obtained by inductively replacing each portion · ⟵^{≃} · → · by a pushout and then taking the direct limit of the resulting sequence of spectra.)

By 10.3, the canonical map K(R) → K^{B}(R) induces isomorphisms K_{n}(R) ≅ π_{n}K^{B}(R) for n ≥ 0, and K_{n}(R) ≅ π_{n}K^{B}(R) for all n ≤ 0 as well.

Variant 10.4.1. The “suspension ring” S(R) of R provides an alternative way of constructing a non-connective spectrum for K-theory. Recall from III, Ex. 1.15, that S(R) is defined to be C(R)/M(R). In III, Ex. 4.10, we saw that there are isomorphisms K_{n}(R) ≅ K_{0}S^{|n|}(R) for n ≤ 0. In fact, Gersten and Wagoner proved that K_{0}(R) × BGL(R)^{+} ≃ ΩBGL(S(R))^{+} so that K_{n}(R) ≅ K_{n+1}S(R) for all n ≥ 0. It follows that the sequence of spaces K^{GW} (R)_{i} = K_{0}(S^{i}(R)) × BGL(S^{i}(R))^{+} form a nonconnective spectrum with π_{n}K^{GW} (R) ≅ K_{n}(R) for all n. We leave it as an exercise to show that a homotopy equivalence between the 0th space of K(R) and K_{0}(R)×BGL(R)^{+} induces a homotopy equivalence of spectra K^{B}(R) ≃ K^{GW} (R).

We now introduce a delooping of Quillen’s space K(A) = ΩBQA (or spectrum) associated to an exact category A, as K(SA) for a different exact category SA. Iterating this yields a non-connective spectrum with connective cover K(A), which agrees with the construction of Definition 10.1 when A = P(R).

Big vector bundles 10.5. Many constructions require that K(X) be strictly functorial in X. For this we introduce the notion of big vector bundles, which I learned from Thomason; see Ex. 10.3. Let V be a small category of schemes over a fixed scheme X. By a big vector bundle over X we will mean the choice of a vector bundle E_{Y} on Y for each morphism Y → X in V, equipped with an isomorphism f^{∗}E_{Y} → E_{Z} for every f : Z → Y over X such that: (i) to the identity on Y we associate the identity on E_{Y}, and (ii) for each composition W ⟶^{g} Z ⟶^{f} Y, the map (fg)^{∗} is the compoition g^{∗}f^{∗}E_{Y} → g^{∗}E_{Z} → E_{W}. Let VB_{V}(X) denote the category of big vector bundles over X. The obvious forgetful functor VB_{V}(X) → VB(X) is an equivalence of categories, and X ↦ VB_{V}(X) is clearly a contravariant functor from V to exact categories. Since K-theory is a functor on exact categories, X ↦ KVB_{V}(X) is a presheaf of spectra on V.

If V is a small category of noetherian schemes and flat maps, a big coherent module over X for V is the choice of a coherent O_{Y} -module F on Y for each Y → X, equipped with a natural isomorphism f^{∗}F_{Y} → F_{Z} for every (flat) f : Z → Y over X, subject to the usual conditions on identity maps and compositions. Let M_{V}(X) denote the category of big coherent modules over X. The obvious forgetful functor M_{V}(X) → M(X) is an equivalence of categories, and X ↦ M_{V}(X) is clearly a contravariant functor from V to exact categories. Since K-theory is a functor on exact categories, X ↦ KM_{V}(X) is a presheaf of spectra on V.

Non-connective K-theory of schemes 10.6. Let V be a small category of quasi-projective schemes such that whenever X is in V then so are X × 𝔸^{1} and X × Spec(ℤ[x, x^{−1}]). Using big vector bundles on V, we may arrange that X ↦ K(X) is a functor from V to spectra. In this way, Construction 10.1 may be made functorial in X.

There is also a Fundamental Theorem like 10.2 for the algebraic K-theory of a quasi-projective scheme X (and even for quasi-compact, quasi-separated schemes), due to Thomason and Trobaugh [TT, 6.1]. Using this and functoriality of Λ^{k}K(X), the proof of 10.3 goes through, and we define K^{B}(X) to be the homotopy colimit of the Λ^{k}K(X). If X = Spec(R) then K(X) is homotopy equivalent to K(R) and hence K^{B}(X) is homotopy equivalent to K^{B}(R). As for rings, the canonical map K(X) → K^{B}(X) induces isomorphisms K_{n}(X) ≅ π_{n}K^{B}(X) for n ≥ 0, and K_{n}(X) ≅ π_{n}K^{B}(X) for all n ≤ 0 as well.

EXERCISES

10.1 Let I be an ideal in a ring R, and write K^{B}(R, I) for the homotopy fiber of K^{B}(R) → K^{B}(R/I). Let K^{≤0}(R, I) denote the homotopy cofiber of the 0connected cover K^{B}(R, I)⟨0⟩ → K^{B}(R, I), as in 4.11.2. Thus π_{n}K^{≤0}(R, I) = 0 for n > 0, and π_{0}K^{≤0}(R, I) ≅ K_{0}(I) by Ex. 1.15. Use III.2.3 to show that π_{n}K^{≤0}(R, I) ≅ K_{n}(I) for all n < 0. 10.2 Let A be the category VB(X). Use the method of 10.4.1 to produce a nonconnective spectrum with connective cover K(X). 10.3 Let V be a small category of schemes, so that X ↦ VB(X) is a contravariant lax functor on V. Recall the Kleisli rectification of VB_{X} in Exercise 6.5, whose objects are pairs (Y → X, E_{Y}), and whose morphisms are pairs (Z ⟶^{h} Y, h^{∗}(E_{Y}) ≅ E_{Z}). Given a morphism f : T → X in V, use the natural isomorphism h^{∗}f^{∗} ≅ f^{∗}h^{∗} to construct an exact functor f^{∗} : VB_{X} → VB_{T}. Compare this with the construction of big vector bundles in 10.5.

§11. Karoubi-Villamayor K-theory

Following Gersten, we say that a functor F from rings (or rings without unit) to sets is homotopy invariant if F (R) ≅ F (R[t]) for every R. Similarly, a functor F from rings to CW complexes (spaces) is called homotopy invariant if for every ring R the natural map R → R[t] induces a homotopy equivalence F (R) ≃ F (R[t]). Note that each homotopy group π_{n}F (R) also forms a homotopy invariant functor.

Of course, this notion may be restricted to functors defined on any subcategory of rings which is closed under polynomial extensions and contains the evaluations as well as the inclusion R ⊂ R[t]. For example, we saw in II, 6.5 and 7.9.3 that G_{0}(R) is a homotopy invariant functor defined on noetherian rings (and schemes) and maps of finite flat dimension.

Conversely, recall from III.3.4 that R is called F -regular if F (R) ≅ F (R[t_{1}, ..., t_{n}]) for all n. Clearly, any functor F from rings to sets becomes homotopy invariant when restricted to the subcategory of F -regular rings. For example, we see from II.7.8 that K_{0} becomes homotopy invariant when restricted to regular rings. The Fundamental Theorem in chapter V, 6.3 implies that the functors K_{n} are also homotopy invariant when restricted to regular rings.

There is a canonical way to make F into a homotopy invariant functor.

Strict homotopization 11.1. Let F be a functor from rings to sets. Its strict homotopization [F] is defined as the coequalizer of the evaluations at t = 0, 1: F (R[t]) ⇉ F (R). In fact, [F] is a homotopy invariant functor and there is a universal transformation F (R) → [F](R); see Ex. 11.1. Moreover, if F takes values in groups then so does [F]; see Ex. 11.3.

Example 11.1.1. Recall that a matrix is called unipotent if it has the form 1 + ν for some nilpotent matrix ν. Let Unip(R) denote the subgroup of GL(R) generated by the unipotent matrices. This is a normal subgroup of GL(R), because the unipotent matrices are closed under conjugation. Since every elementary matrix e_{ij}(r) is unipotent, this contains the commutator subgroup E(R) of GL(R).

We claim that [E]R = [Unip]R = 1 for every R. Indeed, if 1 + ν is unipotent, (1 + tν) is a matrix in Unip(R[t]) with ∂_{0}(1 + tν) = 1 and ∂_{1}(1 + tν) = (1 + ν). Since Unip(R) is generated by these elements, [Unip]R must be trivial. The same argument applies to the elementary group E(R).

We now consider GL(R) and its quotient K_{1}(R). A priori, [GL]R → [K_{1}]R is a surjection. In fact, it is an isomorphism.

Lemma 11.2. Both [GL]R and [K_{1}]R are isomorphic to GL(R)/Unip(R).

Definition 11.2.1. For each ring R, we define KV_{1}(R) to be GL(R)/Unip(R). Thus KV_{1}(R) is the strict homotopization of K_{1}(R) = GL(R)/E(R).

Proof. The composite Unip(R) → GL(R) → [GL]R is trivial, as it factors through [Unip]R = 1. Hence [GL]R (and [K_{1}]R) are quotients of GL(R)/Unip(R). By Higman’s trick III.3.5.1, if g ∈ GL(R[t]) is in the kernel of ∂_{0} then g ∈ Unip(R[t]) and hence ∂_{1}(g) ∈ Unip(R). Hence ∂_{1}(NGL(R)) = Unip(R). Hence GL(R)/Unip(R) is a strictly homotopy invariant functor; universality implies that the induced maps [GL]R → [K_{1}]R → GL(R)/Unip(R) must be isomorphisms. □

To define the higher Karoubi-Villamayor groups, we introduce the simplicial ring R[Δ^{·}], and use it to define the notion of homotopization. The simplicial ring R[Δ^{·}] also plays a critical role in the construction of higher Chow groups and motivic cohomology, which is used in Chapter VI.

Definition 11.3. For each ring R the coordinate rings of the standard simplices form a simplicial ring R[Δ^{·}]. It may described by the diagram

R ⇇ R[t_{1}] ← ; ← ; ← R[t_{1}, t_{2}] ⇇ ⇇ ⋯ R[t_{1}, …, t_{n}] ⋯

with R[Δ^{n}] = R[t_{0}, t_{1}, ⋯, t_{n}]/ (∑ t_{i} = 1) ≅ R[t_{1}, ⋯, t_{n}]. The face maps ∂_{i} are given by: ∂_{i}(t_{i}) = 0; ∂_{i}(t_{j}) is t_{j} for j < i and t_{j−1} for j > i. Degeneracies σ_{i} are given by: σ_{i}(t_{i}) = t_{i} + t_{i+1}; σ_{i}(t_{j}) is t_{j} for j < i and t_{j+1} for j > i.

Definition 11.4. Applying the functor GL to R[Δ^{·}] gives us a simplicial group GL. = GL(R[Δ^{·}]). For n ≥ 1, we define the Karoubi-Villamayor groups to be KV_{n}(R) = π_{n−1}(GL.) = π_{n}(BGL.).

Since π_{0}(GL.) is the coequalizer of GL(R[t]) ⇉ GL(R), we see from 11.2 that definitions 11.2.1 and 11.4 of KV_{1}(R) agree: KV_{1}(R) = GL(R)/Unip(R) ≅ π_{0}(GL.).

The proof in Ex. 1.11 that BGL(R)^{+} is an H-space also applies to BGL(R[Δ^{·}]) (exercise 11.9). From the universal property in theorem 1.8 we deduce the following elementary result.

Lemma 11.4.1. The map BGL(R) → BGL(R[Δ^{·}]) factors through an H-map BGL(R)^{+} → BGL(R[Δ^{·}]). Thus there are canonical maps K_{n}(R) → KV_{n}(R), n ≥ 1.

Remark 11.4.2. In fact, BGL(R[Δ^{·}])^{+} is an infinite loop space; it is the 0^{th} space of the geometric realization KV(R) of the simplicial spectrum K(R[Δ^{·}])⟨0⟩ of 4.11.2. (For any (−1)-connected simplicial spectrum E^{·}, the 0^{th} space of |E^{·}| is the realization of the 0^{th} simplicial space.) Since R[Δ^{0}] = R, there is a canonical morphism of spectra K(R) → K(R). This shows that the map BGL(R)^{+} → BGL(R[Δ^{·}]) of 11.4.1 is in fact an infinite loop space map.

It is useful to put the definition of KV_{∗} into a more general context:

Definition 11.5 (Homotopization). Let F be a functor from rings to CW complexes. Its homotopization F^{h}(R) is the geometric realization of the simplicial space F (R[Δ^{·}]). Thus F^{h} is also a functor from rings to CW complexes, and there is a canonical map F (R) → F^{h}(R).

Lemma 11.5.1. Let F be a functor from rings to CW complexes. Then:

(1) F^{h} is a homotopy invariant functor;

(2) π_{0}(F^{h}) is the strict homotopization [F_{0}] of the functor F_{0}(R) = π_{0}F (R);

(3) If F is homotopy invariant then F (R) ≃ F^{h}(R) for all R.

Corollary 11.5.2. The abelian groups KV_{n}(R) are homotopy invariant, i.e.,

KV_{n}(R) ≅ KV_{n}(R[x]) for every n ≥ 1.

Proof of 11.5.1. We claim that the inclusion R[Δ^{·}] ⊂ R[x][Δ^{·}] is a simplicial homotopy equivalence, split by evaluation at x = 0. For this, we define ring maps h_{i}: R[x][Δ^{n}] → R[x][Δ^{n+1}] by: h_{i}(f) = σ_{i}(f) if f ∈ R[Δ^{n}] and h_{i}(x) = x(t_{i+1} + ⋯ + t_{n+1}). These maps define a simplicial homotopy (see [WHomo]) between the identity map of R[x][Δ^{·}] and the composite

R[x][Δ^{·}] ⟶^{x=0} R[Δ^{·}] ⊂ R[x][Δ^{·}].

Applying F gives a simplicial homotopy equivalence between F^{h}(R[Δ^{·}]) and F^{h}(R[x][Δ^{·}]). Geometric realization converts this into a topological homotopy equivalence between F^{h}(R) and F^{h}(R[x]).

Part (2) follows from the fact that, for any simplicial space X., the group π_{0}(|X.|) is the coequalizer of ∂_{0}, ∂_{1} : π_{0}(X_{1}) ⇉ π_{0}(X_{0}). In this case π_{0}(X_{0}) = π_{0}F (R) and π_{0}(X_{1}) = π_{0}F (R[t]).

Finally, if F is homotopy invariant then the map from the constant simplicial space F (R) to F (R[Δ^{·}]) is a homotopy equivalence in each degree. It follows (see [Wa78]) that their realizations F (R) and F^{h}(R) are homotopy equivalent. □

It is easy to see that F → F^{h} is universal (up to homotopy equivalence) for natural transformations from F to homotopy invariant functors. A proof of this fact is left to Ex. 11.2.

Example 11.6. Suppose that G(R) is a group-valued functor. Then G^{h}(R) is the realization of the simplicial group G(R[Δ^{·}]). This shows that G^{h} may have higher homotopy groups even if G does not.

In fact, the groups π_{n}(G.) of any simplicial group G. may be calculated using the formula π_{p}(G.) = H_{n}(N^{∗}G.), where N^{∗}G. is the Moore complex; see [WHomo, 11.3.6] [May, 17.3]. By definition, the Moore complex of a simplicial group G. is the chain complex of groups with N^{0}G. = G_{0}, N^{1}G. = ker(∂_{0} : G_{1} → G_{0}) and N^{n}G. = ∩_{i=0}^{n−1} ker(∂_{i}) for n ≥ 1, with differential (−1)^{n}∂_{n}. See Ex. 11.4.

In the case that G.(R) = G(R[Δ^{·}]), N^{1}G.(R) is the group NG(R) of III.3.3, and N^{n}G.(R) ⊂ G(R[t_{1}, ..., t_{n}]) is the n^{th} iterate of this functor.

A related situation arises when F (R) = BG(R). Then |G(R[Δ^{·}])| is the loop space of F^{h}(R), which is a connected space with π_{n+1}F^{h}(R) = H_{n}(N^{∗}G(R[Δ^{·}])).

Example 11.6.1. Suppose that F (R) = |G.(R)| for some functor G. from rings to simplicial groups. Then F^{h}(R) is the geometric realization of a bisimplicial group G_{pq} = G_{q}(R[Δ^{p}]). We can calculate the homotopy groups of any bisimplicial space G.. using the standard spectral sequence [Q66]

E_{pq}^{1} = π_{q}(G_{p·}) ⇒ π_{p+q}|G..|.

As a special case, if F (R) ≃ F (R[t_{1}, ..., t_{n}]) for all n then G_{p•} ≃ G_{•}(R) for all p, so the spectral sequence degenerates to yield F (R) ≃ F^{h}(R).

Theorem 11.7. If F (R) is any functorial model of BGL(R)^{+} then we also have KV_{n}(R) = π_{n}F^{h}(R) for all n ≥ 1. Moreover, there is a first quadrant spectral sequence (for p ≥ 0, q ≥ 1):

(11.7.1)  E_{pq}^{1} = K_{q}(R[Δ^{p}]) ⇒ KV_{p+q}(R).

Proof. (Anderson) We may assume (by Ex. 11.2) that F (R) = |G.(R)| for a functor G. from rings to simplicial groups which is equipped with a natural transformation BGL → G. such that BGL(R) → |G.(R)| identifies |G.(R)| with BGL(R)^{+}. Such functors exist; see 1.9. The spectral sequence of 11.6.1 becomes (11.7.1) once we show that KV_{n}(R) = π_{n}G.(R[Δ^{·}]). Thus it suffices to prove that BGL^{h}(R) ≃ |G.(R)|^{h}. Since BGL^{h}(R) is an H-space (Ex. 11.9), the proof is standard, and relegated to Exercise 11.10. □

Theorem 11.8. If R is regular, then K_{p}(R) ≅ KV_{p}(R) for all p ≥ 1.

Proof. If R is regular, then each simplicial group K_{p}(R[Δ^{·}]) is constant (by the Fundamental Theorem in chapter V, 6.3). Thus the spectral sequence (11.7.1) degenerates at E^{2} to yield the result. □

We now quickly develop the key points in KV -theory.

Definition 11.9. We say that a ring map f: R → S is a GL-fibration if

GL(R[t_{1}, ..., t_{n}]) × GL(S) → GL(S[t_{1}, ..., t_{n}])

is onto for every n. Note that we do not require R and S to have a unit.

Example 11.9.1. If I is a nilpotent ideal in R, then R → R/I is a GL-fibration. This follows from Ex. I.1.12(iv), because each I[t_{1}, ..., t_{n}] is also nilpotent.

Remark 11.9.2. Any GL-fibration must be onto. That is, S ≅ R/I for some ideal I of R. To see this, consider the (1, 2) entry α_{12} of a preimage of the elementary matrix e_{12}(st). Since f(α_{12}) = st, evaluation at t = 1 gives an element of R mapping to s ∈ S. However, not every surjection is a GL-fibration; see Ex. 11.6(d).

Example 11.9.3. Any ring map R → S is homotopic to a GL-fibration. Indeed, the inclusion of R into the graded ring R′ = R⊕xS[x] = R×_{S} S[x] induces a homotopy equivalence GL(R[Δ^{·}]) ≃ GL(R′[Δ^{·}]) by Ex. 11.5, so that KV_{∗}(R) ≅ KV_{∗}(R′). Moreover, the map R′ → S sending x to 1 is a GL-fibration by Ex. 11.6(a,c).

The definition of KV_{n}(I) makes sense if I is a ring without unit using the group GL(I) of III.2: KV_{n}(I) = π_{n}BGL(I[Δ^{·}]). Since GL(R ⊕ I) is the semidirect product of GL(R) and GL(I), we clearly have KV_{n}(R ⊕ I) ≅ KV_{n}(R) ⊕ KV_{n}(I). This generalizes as follows.

Theorem 11.10 (Excision). If R → R/I is a GL-fibration, there is a long exact sequence

KV_{n+1}(R/I) ⟶ KV_{n}(I) → KV_{n}(R) → KV_{n}(R/I) ⟶ ⋯
→ KV_{1}(I) → KV_{1}(R) → KV_{1}(R/I) → K_{0}(I) → K_{0}(R) → K_{0}(R/I).

Theorem 11.10 is called an “Excision Theorem” because it says that (whenever R → R/I is a GL-fibration) KV_{n}(R, I) ≅ KV_{n}(I) for all n ≥ 1.

Proof. Let G_{n} ⊂ GL(R/I[Δ^{n}]) denote the image of GL(R[Δ^{n}]). Then there is an exact sequence of simplicial groups

(11.10.1)  1 → GL(I[Δ^{·}]) → GL(R[Δ^{·}]) → G. → 1.

Now any short exact sequence of simplicial groups is a fibration sequence, meaning there is a long exact sequence of homotopy groups. Moreover, the quotient GL(R/I[Δ^{·}])/G. is a constant simplicial group, by Ex. 11.7. It is now a simple matter to splice the long exact sequences together to get the result. The splicing details are left to Ex. 11.7. □

Corollary 11.10.2. For any ring map ϕ : R → S, set I′ = R ×_{S} xS[x] = {(r, x f(x)) ∈ R × xS[x] : ϕ(r) + f(1) = 0}. Then there is a long exact sequence

⋯ → KV_{n+1}(S) → KV_{n}(I′) → KV_{n}(R) → KV_{n}(S) ⟶ ⋯

ending in KV_{1}(I′) → KV_{1}(R) → KV_{1}(S) → K_{0}(I′).

Proof. Set R′ = R ⊕ xS[x] and note that R′ → S is a GL-fibration with kernel I′ by 11.9.3. Since R′ is graded, KV_{n}(R) ≅ KV_{n}(R′) for all n ≥ 1. The desired long exact sequence comes from Theorem 11.10. □

Remark 11.10.3. When R → R/I is a GL-fibration, then KV_{∗}(I) ≅ KV_{∗}(I′), and the long exact sequences of 11.10 and 11.10.2 coincide (with S = R/I). This follows from the 5-lemma, since the map ϕ factors through R′ → R/I yielding a morphism of long exact sequences.

Theorem 11.10 fails if R → R/I is not a GL-fibration. Not only does the extension of Theorem 11.10 to K_{0} fail (as the examples ℤ → ℤ/8 and ℤ → ℤ/25 show), but we need not even have KV_{∗}(I) ≅ KV_{∗}(I′), as exercise 11.14 shows.

Corollary 11.11. If I is a nilpotent ideal in a ring R, then KV_{n}(I) = 0 and KV_{n}(R) ≅ KV_{n}(R/I) for all n ≥ 1.

Proof. By 11.9.1, 11.10 and Lemma II.2.2, it suffices to show that KV_{n}(I) = π_{n}GL(I[Δ^{·}]) = 0. (A stronger result, that GL(I[Δ^{·}]) is simplicially contractible, is relegated to Ex. 11.11.) By exercise I.1.12(iii), GL_{m}(I[Δ^{n}]) consists of the matrices 1 + x in M_{m}(I[Δ^{n}]), so if T = (t_{0}t_{1} ⋯ t_{n−1}) then the degree n + 1 part of the Moore complex (11.6) consists of matrices 1 + xT, and ∂_{n}(1 + xT) = 1 exactly when x = t_{n}y for some matrix y. Regarding y as a matrix over I[t_{0}, ..., t_{n−1}], the element 1 + yT t_{n} in GL(I[Δ^{n+1}]) belongs to the Moore complex and ∂_{n+1} maps 1 + yT t_{n} to 1 + xT. □

Theorem 11.12 (Mayer-Vietoris). Let φ : R → S be a map of rings, sending an ideal I of R isomorphically onto an ideal of S. If S → S/I is a GL-fibration, then R → R/I is also a GL-fibration, and there is a long exact Mayer-Vietoris sequence

⋯ → KV_{n+1}(S/I) →KV_{n}(R) → KV_{n}(R/I) ⊕ KV_{n}(S) → KV_{n}(S/I) → ⋯
→ KV_{1}(R/I) ⊕ KV_{1}(S) →KV_{1}(S/I) → K_{0}(R) → K_{0}(R/I) ⊕ K_{0}(S).

It is compatible with the Mayer-Vietoris sequence for K_{1} and K_{0} in III.2.6.

Proof. To see that R → R/I is a GL-fibration, fix ḡ ∈ GL(R/I[t_{1}, ..., t_{n}]) with ḡ(0) = I. Since S → S/I is a GL-fibration, there is a g′ ∈ GL(S[t_{1}, ..., t_{n}]) which is φ(ḡ) modulo I. Since R is the pullback of S and R/I, there is a g in GL(R[t_{1}, ..., t_{n}]) mapping to g′ and ḡ. Hence R → R/I is a GL-fibration.

As in the proof of theorem III.5.8, there is a morphism from the (exact) chain complex of 11.10 for (R, I) to the corresponding chain complex for (S, I). Since every third term of this morphism is an isomorphism, the required Mayer-Vietoris sequence follows from a diagram chase. □

Here is an application of this result. Since R[x] → R has a section, it is a GLfibration. By homotopy invariance, it follows that KV_{n}(xR[x]) = 0 for all n ≥ 1. (Another proof is given in Ex. 11.5.)

Definition 11.13. For any ring R (with or without unit), define ΩR to be the ideal (x^{2} − x)R[x] of R[x]. Iterating yields Ω^{2}R = (x_{1}^{2} − x_{1})(x_{2}^{2} − x_{2})R[x_{1}, x_{2}], etc.

The following corollary of 11.10 shows that, for n ≥ 2, we can also define KV_{n}(R) as KV_{1}(Ω^{n−1}R), and hence in terms of K_{0} of the rings Ω^{n}R and Ω^{n}R[x].

Corollary 11.13.1. For all R, KV_{1}(R) is isomorphic to the kernel of the map K_{0}(ΩR) → K_{0}(xR[x]), and KV_{n}(R) ≅ KV_{n−1}(ΩR) for all n ≥ 2.

Proof. The map xR[x] ⟶^{x=1} R is a GL-fibration by Ex. 11.6(c), with kernel ΩR. The result now follows from Theorem 11.10. □

We conclude with an axiomatic treatment, due to Karoubi and Villamayor.

Definition 11.14. A positive homotopy K-theory (for rings) consists of a sequence of functors K_{n}^{h}, n ≥ 1, on the category of rings without unit, together with natural connecting maps δ_{n}: K_{n+1}^{h}(R/I) → K_{n}^{h}(I) and δ_{0} : K_{1}^{h}(R/I) → K_{0}(I), defined for every GL-fibration R → R/I, satisfying the following axioms:

(1) The functors K_{n}^{h} are homotopy invariant;

(2) For every GL-fibration R → R/I the resulting sequence is exact:

K_{n+1}^{h}(R/I) ⟶^{δ} K_{n}^{h}(I) → K_{n}^{h}(R) → K_{n}^{h}(R/I) ⟶^{δ} K_{n−1}^{h}(I) →
K_{1}^{h}(R) → K_{1}^{h}(R/I) ⟶^{δ} K_{0}(I) → K_{0}(R) → K_{0}(R/I).

Theorem 11.14.1. Up to isomorphism, there is a unique positive homotopy K-theory, namely K_{n}^{h} = KV_{n}.

Proof. The fact that KV_{n} form a positive homotopy K-theory is given by 11.4, 11.5.2 and 11.10. The axioms imply that any other positive homotopy K-theory must satisfy the conclusion of 11.13.1, and so must be isomorphic to KV -theory. □

EXERCISES

11.1 Let F be a functor from rings to sets. Show that [F] is a homotopy invariant functor, and that every natural transformation F (R) → H(R) to a homotopy invariant functor H factors uniquely through F (R) → [F](R). 11.2 Let F and H be functors from rings to CW complexes, and assume that H is homotopy invariant. Show that any natural transformation F (R) → H(R) factors through maps F^{h}(R) → H(R) such that for each ring map R → S the map F^{h}(R) → F^{h}(S) → H(S) is homotopy equivalent to F^{h}(R) → F^{h}(S) → H(S). 11.3 If G is a functor from rings to groups, let NG(R) denote the kernel of the map t = 0: G(R[t]) → G(R). Show that the image G_{0}(R) of the induced map t = 1: NG(R) → G(R) is a normal subgroup of G(R), and that [G]R = G(R)/G_{0}(R). Thus [G]R is a group.

11.4 (Moore) If G. is a simplicial group and N^{n}G. = ∩_{i=0}^{n−1} ker(∂_{i}) as in 11.6, show that ∂_{n+1}(N^{n+1}G.) is a normal subgroup of G_{n}. Conclude that π_{n}(G.) is also a group. Hint: conjugate elements of N^{n+1}G by elements of s_{n}G_{n}.

11.5 Let R = R_{0} ⊕ R_{1} ⊕ ⋯ be a graded ring. Show that for every homotopy invariant functor F on rings we have F (R_{0}) ≃ F (R). In particular, if F is defined on rings without unit then F (xR[x]) ≃ F (0) for every R. Hint: Copy the proof of III.3.4.1. 11.6 GL-fibrations. Let f: R → S be a GL-fibration with kernel I. Show that:

(a) If f factors as R → R′ → S, then R′ → S is a GL-fibration.

(b) Both xR[x] → xS[x] and ΩR → ΩS are GL-fibrations.

(c) The map xR[x] → R, f(x) ↦ f(1), is a GL-fibration with kernel ΩR.

(d) ℤ → ℤ/4 is not a GL-fibration, but GL(ℤ) → GL(ℤ/4) is onto.

(e) If S is a regular ring (with unit), then every surjection R → S is a GL-fibration. Hint: K_{1}(S) ≅ K_{1}(S[x]) by III.3.8. 11.7 Let f: R → S be a GL-fibration with kernel I, and define G_{.} as in the proof of Theorem 11.10. Show that GL(S[Δ^{·}])/G. is a constant simplicial group. Use this to show that π_{i}(G_{.}) = KV_{i+1}(S) for all i > 0, but that the cokernel of π_{0}(G_{.}) → π_{0}GL(S[Δ^{·}]) is the image of K_{1}(S) in K_{0}(I) under the map of III.2.3. Combining this with the long exact sequence of homotopy groups for (11.10.1), finish the proof of 11.10. 11.8 Consider the unit functor U on rings. The identity U(M_{m}(R)) = GL_{m}(R) implies that U^{h}(M(R)) = KV (R). If R is commutative, use I.3.12 to show that U^{h}(R) ≃ U(R_{red}). 11.9 Show that BGL(R[Δ^{·}]) is an H-space. This is used to prove Lemma 11.4.1. Hint: See Ex. 1.11; the permutation matrices lie in E(R), and E(R[Δ^{·}]) is path connected. 11.10 (Anderson) Use exercise 11.9 to complete the proof of theorem 11.7. 11.11 If I is nilpotent, show that the simplicial sets GL_{m}(I[Δ^{·}]) and GL(I[Δ^{·}]) have a simplicial contraction [WHomo, 8.4.6]. Hint: multiply by t_{n}. 11.12 If R is K_{i}-regular for all i ≤ n, show that K_{i}(R) ≅ KV_{i}(R) for all i ≤ n, and that KV_{n+1}(R) ≅ [K_{n}]R. 11.13 (Strooker) Consider the ring R = ℤ[x]/(x^{2} − 4). In this exercise, we use two different methods to show that the map K_{2}(R) → KV_{2}(R) is not onto, and that its cokernel is ℤ/2. Note that KV_{1}(R) = K_{1}(R) = R^{×} = {±1} by Ex. III.5.13.

(a) Compare the Mayer-Vietoris sequences III.5.8 and 11.12 to show that the natural map K_{2}(R) → K_{2}(ℤ)^{2} has cokernel ℤ/2, yet KV_{2}(R) ≅ K_{2}(ℤ)^{2} = (ℤ/2)^{2}.

(b) Use III.5.8 and Ex. III.5.15 to compute K_{1}(R[t]) and K_{1}(R[t_{1}, t_{2}]). Then show that the sequence N^{2}K_{1}(R) → NK_{1}(R) → K_{1}(R) is not exact. Use the spectral sequence (11.7.1) to conclude that the map K_{2}(R) → KV_{2}(R) is not onto, and that its cokernel is ℤ/2. 11.14 Let k be a field of characteristic 0, and set S = k[x, (x + 1)^{−1}]

(a) Show that K_{1}(ℤ ⊕ I, I) ≅ K_{1}(S, I). Hint: Use the obstruction described in III, Ex. 2.6, showing that 3ψ(a dx ⊗ x^{2}) = 0.

(b) Use III, Ex. 5.14(c) to show that K_{1}(S, I) is the cokernel of d ln : k^{×} → Ω_{k}, d ln(a) = da/a.

(c) Show that KV_{1}(I) = 0, and conclude that the sequence KV_{1}(I) → KV_{1}(S) → KV_{1}(S/I) is not exact.

§12. Homotopy K-theory

In order to define a truly homotopy invariant version of algebraic K-theory, we need to include K_{0} and even the negative K-groups. This is most elegantly done at the level of spectra, and that approach begins by constructing the non-connective “Bass K-theory spectrum” K^{B}(R) out of any one of the functorial models of a connective K-theory spectrum K(R). (See 1.9(iii), 4.5.2 and 8.5.5.)

Let R be an associative ring with unit. By 11.3 there is a simplicial ring R[Δ^{·}] and hence a simplicial spectrum K^{B}(R[Δ^{·}]).

Definition 12.1. Let KH(R) denote the (fibrant) geometric realization of the simplicial spectrum K^{B}(R[Δ^{·}]). For n ∈ ℤ, we write KH_{n}(R) for π_{n}KH(R).

It is clear from the definition that KH(R) commutes with filtered colimits of rings, and that there are natural transformations K_{n}(R) → KH_{n}(R) which factor through KV_{n}(R) when n ≥ 1. Indeed, the spectrum map K(R)⟨0⟩ → K^{B}(R) → KH(R) factors through the spectrum KV(R) = K(R[Δ^{·}])⟨0⟩ of 11.4.2

Theorem 12.2. Let R be a ring. Then:

(1) KH(R) ≃ KH(R[x]), i.e., KH_{n}(R) ≅ KH_{n}(R[x]) for all n.

(2) KH(R[x, x^{−1}]) ≃ KH(R) × Ω^{−1}KH(R), i.e.,

KH_{n}(R[x, x^{−1}]) ≅ KH_{n}(R) ⊕ KH_{n−1}(R) for all n.

(3) If R = R_{0} ⊕ R_{1} ⊕ ⋯ is a graded ring then KH(R) ≃ KH(R_{0}).

Proof. Part (1) is a special case of 11.5.1. Part (2) follows from the Fundamental Theorem 10.2 and (1). Part (3) follows from (1) and Ex. 11.5. □

The homotopy groups of a simplicial spectrum are often calculated by means of a standard right half-plane spectral sequence. In the case at hand, i.e., for KH(R), the edge maps are the canonical maps K_{q}(R) → KH_{q}(R), induced by K^{B}(R) → KH(R), and the spectral sequence specializes to yield:

Theorem 12.3. For each ring R there is an exhaustive convergent right halfplane spectral sequence:

E_{p,q}^{1} = N^{p}K_{q}(R) ⇒ KH_{p+q}(R).

The edge map from E_{0,q}^{1} = K_{q}(R) to KH_{q}(R) identifies E_{p,0}^{2} with the strict homotopization [K_{p}](R) of K_{p}(R), defined in 11.1.

The phrase “exhaustive convergent” in 12.3 means that for each n there is a filtration 0 ⊆ F_{0}KH_{n}(R) ⊆ ⋯ ⊆ F_{p−1}KH_{n}(R) ⊆ F_{p}KH_{n}(R) ⊆ ⋯ with union KH_{n}(R), zero for p < 0, and isomorphisms E_{p,q}^{∞} ≅ F_{p}KH_{n}(R)/F_{p−1}KH_{n}(R) for q = n − p. (A discussion of Convergence may be found in [WHomo, 5.2.11].)

As pointed out in 11.8, we will see in chapter V, 6.3 that regular rings are K_{q}- regular for all q, i.e., that N^{p}K_{q}(R) = 0 for every q and every p > 0. For such rings, the spectral sequence 12.3 degenerates at E^{1}, showing that the edge maps are isomorphisms. We record this as follows:

Corollary 12.3.1. If R is regular noetherian, then K(R) ≃ KH(R). In particular, K_{n}(R) ≃ KH_{n}(R) for all n.

For the next application, we use the fact that if R is K_{i}-regular for some i, then it is K_{q}-regular for all q ≤ i. If i ≤ 0, this was proven in III, 4.2.3. For i = 1 it was shown in III, Ex. 3.9. In the remaining case i > 1, the result will be proven in chapter V.

Corollary 12.3.2. Suppose that the ring R is K_{i}-regular for some fixed i. Then KH_{n}(R) ≅ K_{n}(R) for all n ≤ i, and KH_{i+1}(R) = [K_{i+1}]R.

If R is K_{0}-regular then KV_{n}(R) ≅ KH_{n}(R) for all n ≥ 1, and KH_{n}(R) ≅ K_{n}(R) for all n ≤ 0. In this case the spectral sequences of (11.7.1) and 12.3 coincide.

Proof. In this case, the spectral sequence degenerates below the line q = i, yielding the first assertion. If R is K_{0}-regular, the morphism KV(R) → KH(R) induces a morphism of spectral sequences, from (11.7.1) to 12.3, which is an isomorphism on E_{p,q}^{1} (except when p = 0 and q ≤ 0). The comparison theorem yields the desired isomorphism KV(R) → KH(R)⟨0⟩. □

Theorem 12.3.3. If 1/ℓ ∈ R then KH_{n}(R; ℤ/ℓ) ≅ K_{n}(R; ℤ/ℓ) for all n.

Proof. The proof of 12.3 goes through with finite coefficients to yield a spectral sequence with E_{p,q}^{1} = N^{p}K_{q}(R; ℤ/ℓ) ⇒ KH_{p+q}(R; ℤ/ℓ). When 1/ℓ ∈ R and p > 0, the groups N^{p}K_{q}(R) are ℤ[1/ℓ]-modules (uniquely ℓ-divisible groups) by 6.7.2. By the Universal Coefficient Sequence 2.2 we have N^{p}K_{q}(R; ℤ/ℓ) = 0, so the spectral sequence degenerates to yield the result. □

If I is a non-unital ring, we define KH(I) to be KH(ℤ ⊕ I)/KH(ℤ) and set KH_{n}(I) = π_{n}KH(I). If I is an ideal in a ring R, recall (from 1.11 or Ex. 10.1) that K^{B}(R, I) denotes the homotopy fiber of K^{B}(R) → K^{B}(R/I); it depends upon R. The following result, which shows that the KH-analogue does not depend upon R, is one of the most important properties of KH-theory.

Theorem 12.4 (Excision). Let I be an ideal in a ring R. Then KH(I) → KH(R) → KH(R/I) is a homotopy fibration. Thus there is a long exact sequence

⋯ → KH_{n+1}(R/I) ⟶ KH_{n}(I) → KH_{n}(R) → KH_{n}(R/I) → ⋯

Proof. Let KH(R, I) denote the homotopy fiber of KH(R) → KH(R/I). By standard simplicial homotopy theory, KH(R, I) is homotopy equivalent to |K^{B}(R[Δ^{·}], I[Δ^{·}])|. It suffices to prove that KH(I) → KH(R, I) is a homotopy equivalence.

We first claim that KH_{n}(I) → KH_{n}(R, I) is an isomorphism for n ≤ 0. For each p ≥ 0, let K^{B}(R[Δ^{p}], I[Δ^{p}])⟨0⟩ be the 0-connected cover of K^{B}(R[Δ^{p}], I[Δ^{p}]), and define K^{≤0}(R[Δ^{p}], I[Δ^{p}]) by the termwise “Postnikov” homotopy fibration:

K^{B}(R[Δ^{p}], I[Δ^{p}])⟨0⟩ → K^{B}(R[Δ^{p}], I[Δ^{p}]) → K^{≤0}(R[Δ^{p}], I[Δ^{p}]).

Let C_{R} denote the geometric realization of K^{≤0}(R[Δ^{·}], I[Δ^{·}]). Comparing the standard spectral sequence for C_{R} and the spectral sequence of Theorem 12.3, we see that KH_{n}(R, I) ≅ π_{n}(C_{R}) for all n ≤ 0. By Exercise 10.1, the ring map A = ℤ ⊕ I → R induces π_{n}K^{≤0}(A[Δ^{p}], I[Δ^{p}]) ≅ π_{n}K^{≤0}(R[Δ^{p}], I[Δ^{p}]) for all n and p. Hence we have homotopy equivalences for each p and hence a homotopy equivalence on realizations, C_{A} ≃ C_{R}. The claim follows.

For n > 0, we consider the homotopy fiber sequence KV(R, I) → KH(R, I) → C_{R}, where KV(R, I) is the geometric realization of K^{B}(R[Δ^{·}], I[Δ^{·}])⟨0⟩. Comparing with the spectrum KV(R) defined in 11.4.2, we see that KV(R, I) is the 0-connected cover of the homotopy fiber of KV(R) → KV(R/I).

The theorem now follows when R → R/I is a GL-fibration, since in this case KV_{n}(I) ≅ KV_{n}(R, I) for all n ≥ 1 by 11.10. Combining this with the above paragraph, the 5-lemma shows that in this case KH(I) ≃ KH(R, I), as required.

An important GL-fibration is given by the non-unital map xR[x] → R (or the unital ℤ ⊕ xR[x] → ℤ ⊕ R) with kernel ΩR; see Ex. 11.6(c). In the following diagram, the bottom two rows are homotopy fibration sequences by the previous paragraph, and the terms in the top row are defined so that the columns are homotopy fibrations:

KH(ℤ ⊕ ΩR, ΩI) ⟶ KH(ℤ ⊕ xR[x], xI[x]) ⟶ KH(R, I)
↓  ↓  ↓
KH(ΩR)  ⟶  KH(xR[x])  ⟶ KH(R)
↓  ↓  ↓
KH(ΩR/I) ⟶  KH(xR/I[x])  ⟶ KH(R/I).

Since KH(xR[x]) is contractible (by 12.2), the top middle term is contractible, and we have a natural homotopy equivalence ΩKH(R) ≃ KH(ΩR). Since the top row must also be a homotopy fibration, we also obtain a natural homotopy equivalence ΩK(R, I) → KH(ℤ ⊕ ΩR, ΩI). Applying π_{n} yields isomorphisms KH_{n+1}(R, I) ≅ KH_{n}(ℤ ⊕ ΩR, ΩI) for all n.

Now suppose by induction on n ≥ 0 that, for all rings R′ and ideals I′, the canonical map I′ → R′ induces an isomorphism KH_{n}(I′) ≅ KH_{n}(R′, I′). In particular, ΩI → ΩR induces KH_{n}(ΩI) ≅ KH_{n}(ℤ ⊕ ΩR, ΩI). It follows that the map from A = ℤ ⊕ I to R induces a commutative diagram of isomorphisms:

KH_{n+1}(I) = KH_{n+1}(A, I) ⟶^{≃} KH_{n}(ℤ ⊕ ΩA, ΩI)
↓  ↓^{≃}
KH_{n+1}(R, I) ⟶^{≃} KH_{n}(ℤ ⊕ ΩR, ΩI).

This establishes the inductive step. We have proven that for all R and I, KH_{n}(I) ≅ KH_{n}(R, I) for all n, and hence KH(I) ≃ KH(R, I), as required. □

Corollary 12.5. If I is a nilpotent ideal in a ring R, then the spectrum KH(I) is contractible and KH(R) ≃ KH(R/I). In particular, KH_{n}(I) = 0 and KH_{n}(R) ≅ KH_{n}(R/I) for all integers n.

Proof. By Ex. II.2.5, I is K_{0}-regular, and K_{n}(I) = 0 for n ≤ 0 (see III, Ex. 4.3). By 12.3.2 and 11.11, we have KH_{n}(I) ≅ KV_{n}(I) = 0 for n > 0, and KV_{n}(I) = 0 for n ≤ 0. Since KH_{n}(I) = 0 for all n, KH(I) is contractible. The remaining assertions now follow from Excision 12.4. □

Example 12.5.1. Let R be a commutative Artinian ring, with associated reduced ring R_{red} = R/nilradical(R). As R_{red} is regular, we see from 12.5 and 12.3.1 that KH_{n}(R) ≅ K_{n}(R_{red}) for all n. In particular, KH_{0}(R) = K_{0}(R) = H^{0}(R) and KH_{n}(R) = 0 for n < 0.

Example 12.5.2. Let R be a 1-dimensional commutative noetherian ring. Then KH_{0}(R) ≅ H^{0}(R) ⊕ [Pic]R, KH_{−1}(R) is torsionfree, and KH_{n}(R) = 0 for all n ≤ −2. This follows from 12.3.2 and Ex. III.4.4, which states that R is K_{−1}- regular and computes K_{n}(R) for n ≤ 0. An example in which KV_{1}(R) → KH_{1}(R) is not onto is given in Ex. 12.2.

If in addition R is seminormal, then R is Pic-regular by Traverso’s theorem I.3.12. In this case we also have KH_{0}(R) = K_{0}(R) = H^{0}(R) ⊕ Pic(R) and KV_{1}(R) ≅ KH_{1}(R).

Corollary 12.6 (Closed Mayer-Vietoris). Let R → S be a map of commutative rings, sending an ideal I of R isomorphically onto an ideal of S. Then there is a long exact Mayer-Vietoris sequence (for all integers n):

⋯ → KH_{n+1}(S/I) → KH_{n}(R) → KH_{n}(R/I) ⊕ KH_{n}(S) → KH_{n}(S/I) → ⋯

Recall (10.4) that K^{B}(R) is the homotopy colimit of a diagram of spectra L^{q}K(R). Since geometric realization commutes with homotopy colimits, at least up to weak equivalence, we have KH(R) = colim_{q} |L^{q}K(R[Δ^{·}])|.

Definition 12.7. Let X be a scheme. Using the functorial nonconnective spectrum K^{B} of 10.6, let KH(X) denote the (fibrant) geometric realization of the simplicial spectrum K^{B}(X × Δ^{·}), where Δ^{·} = Spec(R[Δ^{·}]) as in 11.3. For n ∈ ℤ, we write KH_{n}(X) for π_{n}KH(X).

Lemma 12.8. For any quasi-projective scheme X we have:

1. KH(X) ≃ KH(X × 𝔸^{1}).

2. KH(X × Spec(ℤ[x, x^{−1}]) ≃ KH(X) × Ω^{−1}KH(R), i.e.,

KH_{n}(X[x, x^{−1}]) ≅ KH_{n}(X) ⊕ KH_{n−1}(X) for all n.

3. If X is regular noetherian, then K(X) ≃ KH(X). In particular, K_{n}(X) ≃ KH_{n}(X) for all n.

Proof. The proof of 11.5.1 goes through to show (1). From the Fundamental Theorem (see 10.6), we get (2). We will see in V.6.13.2 that if X is a regular noetherian scheme then K(X) ≃ K(X × 𝔸^{1}) and hence K^{B}(X) ≃ K^{B}(X × 𝔸^{1}). It follows that KH(X) = K^{B}(X × Δ^{·}) is homotopy equivalent to the constant simplicial spectrum K^{B}(X). □

EXERCISES

12.1 Dimension shifting. Fix a ring R, and let Δ^{d}(R) denote the coordinate ring R[t_{0}, …, t_{d}]/(f), f = t_{0} ⋯ t_{d}(1 − ∑ t_{i}) of the d-dimensional tetrahehron over R. Show that for all n, KH_{n}(Δ^{d}(R)) ≅ KH_{n}((R) ⊕ KH_{d+n}(R). If R is regular, conclude that KH_{n}(Δ^{d}(R)) ≅ K_{n}(R) ⊕ K_{d+n}(R), and that K_{0}(Δ^{d}(R)) ≅ K_{0}(R) ⊕ K_{d}(R). Hint: Use the Mayer-Vietoris squares of III.4.3.1, where we saw that K_{j}(Δ^{n}(R)) ≅ K_{j+1}(Δ^{n−1}(R)) for j < 0. In III, Ex. 4.8 we saw that each Δ^{n}(R) is K_{0}-regular if R is. 12.2 (KV_{1} need not map onto KH_{1}.) Let k be a field of characteristic 0, I the ideal of S = k[x, (x + 1)^{−1}] generated by x^{2}, and R = k ⊕ I. Show that KH_{n}(R) ≅ KH_{n}(S) for all n, but that there is an exact sequence 0 → KV_{1}(R) → KH_{1}(R) → ℤ → 0. Hint: Use the Mayer-Vietoris sequence for R → S and apply I.3.12 to show that K_{0}(R) = ℤ⊕k/ℤ, NK_{0}(R) ≅ tk[t] and N^{2}K_{0}(R) ≅ t_{1}t_{2}k[t_{1}, t_{2}]. Alternatively, note that KV_{1}(I) = 0 by Ex. 11.14. 12.3 The seminormalization R^{+} of a reduced commutative ring R was defined in I, Ex. 3.15. Show that KH_{n}(R) ≅ KH_{n}(R^{+}) for all n. Hint: show that KH is invariant under subintegral extensions.