CHAPTER V THE FUNDAMENTAL THEOREMS OF HIGHER K-THEORY

We now restrict our attention to exact categories and Waldhausen categories, where the extra structure enables us to use the following types of comparison theorems: Additivity (1.2), Cofinality (2.3), Approximation (2.4), Resolution (3.1), Devissage (4.1), and Localization (2.1, 2.5, 5.1 and 7.3). These are the extensions to higher K-theory of the corresponding theorems of chapter II. The highlight of this chapter is the so-called “Fundamental Theorem” of K-theory (6.3 and 8.2), comparing K(R) to K(R[t]) and K(R[t, t^{−1}]), and its analogue (6.13.2 and 8.3) for schemes.

§1. The Additivity theorem

If F′ → F → F′′ is a sequence of exact functors F′, F, F′′: B → C between two exact categories (or Waldhausen categories), the Additivity Theorem tells us when the induced maps K(B) → K(C) satisfy F_{∗} = F′_{∗} + F′′_{∗}. To state it, we need to introduce the notion of a short exact sequence of functors, which was mentioned briefly in II(9.1.8).

Definition 1.1. (a) If B and C are exact categories, we say that a sequence F′ → F → F′′ of exact functors and natural transformations from B to C is a short exact sequence of exact functors, and write F′ ↣ F ↠ F′′, if

0 → F′(B) → F (B) → F′′(B) → 0

is an exact sequence in C for every B ∈ B.

(b) If B and C are Waldhausen categories, we say that F′ ↣ F ↠ F′′ is a short exact sequence, or a cofibration sequence of exact functors if each F′(B) ↣ F (B) ↠ F′′(B) is a cofibration sequence and if for every cofibration A ↣ B in B, the evident map F (A) ∪_{F′(A)} F′(B) ↣ F (B) is a cofibration in C.

When exact categories are regarded as Waldhausen categories, these two notions of “short exact sequence” of exact functors between exact categories are easily seen to be the same.

Universal Example 1.1.1. Recall from chapter II, 9.3, that the extension category E = E(C) is the category of all exact sequences E: A ↣ B ↠ B/A in C. If C is an exact category, or a Waldhausen category, so is E. The source s(E) = A, target t(E) = B, and quotient q(E) = C of such a sequence are exact functors from E to C, and s ↣ t ↠ q is a short exact sequence of functors. This example is universal in the sense that giving an exact sequence of exact functors from B to C is the same thing as giving an exact functor B → E(C).

Typeset by AMS-TEX

2 V. THE FUNDAMENTAL THEOREMS OF HIGHER K-THEORY

Additivity Theorem 1.2. Let F′ ↣ F ↠ F′′ be a short exact sequence of exact functors from B to C, either between exact categories or between Waldhausen categories. Then F_{∗} ≃ F′_{∗} + F′′_{∗} as H-space maps K(B) → K(C), and therefore on the homotopy groups we have F_{∗} = F′_{∗} + F′′_{∗} : K_{i}(B) → K_{i}(C).

Proof of the Additivity Theorem. By universality of E, we may assume that B is E and prove that t_{∗} = s_{∗} + q_{∗}. The map s_{∗} + q_{∗} is induced by the exact functor s ∐ q: E → C which sends A ↣ B ↠ C to A ⨿ C, because the H-space structure on K(C) is induced from ⨿ (see IV, 6.4 and 8.5.1). The compositions of t and s ∐ q with the coproduct functor ⨿ : C × C → E agree:

C × C \underrightarrow{∐} E ⇉^{t}_{s∨q} C,

and hence give the same map on K-theory. The Extension Theorem 1.3 below proves that K(⨿) is a homotopy equivalence, from which we conclude that t ≃ s ∐ q, as desired. □

Given an exact category (or a Waldhausen category) A, we say that a sequence 0 → A_{n} → ⋯ → A_{0} → 0 is admissibly exact if each map decomposes as A_{i+1} ↠ B_{i} ↣ A_{i}, and each B_{i} ↣ A_{i} ↠ B_{i−1} is an exact sequence.

Corollary 1.2.1. (Additivity for characteristic exact sequences). If

0 → F^{0} → F^{1} → ⋯ → F^{n} → 0

is an admissibly exact sequence of exact functors B → C, then ∑(−1)^{p}F_{∗}^{p} = 0 as maps from K_{i}(B) to K_{i}(C).

Proof. This follows from the Additivity Theorem 1.2 by induction on n. □

Remark 1.2.2. Suppose that F′ and F are the same functor in the Additivity Theorem 1.2. Using the H-space structure we have F′′_{∗} ≃ F_{∗}−F_{∗} ≃ 0. It follows that

the homotopy fiber of K(B) ⟶^{F′′} K(C) is homotopy equivalent to K(B) ∨ ΩK(C).

Example 1.2.3. Let C be a Waldhausen category with a cylinder functor (IV.8.8). Then the definition of cone and suspension imply that 1 ↣ cone ↠ Σ is an exact sequence of functors: each A ↣ cone(A) ↠ ΣA is exact. If C satisfies the Cylinder Axiom (IV.8.8), the cone is null-homotopic. The Additivity Theorem implies that Σ_{∗} + 1 = cone_{∗} = 0. It follows that Σ: K(C) → K(C) is a homotopy inverse with respect to the H-space structure on K(C).

The following calculation of K(E), used in the proof of the Additivity Theorem 1.2, is due to Quillen [Q341] for exact categories and to Waldhausen [W1126] for Waldhausen categories. (The K_{0} version of this Theorem was presented in II.9.3.1.)

Extension Theorem 1.3. The exact functor (s, q) : E = E(C) → C × C induces homotopy equivalences wS.E ≃ (wS.C)^{2} and K(E) ⟶^{∼} K(C) × K(C). The coproduct functor ⨿, sending (A, B) to the sequence A ↣ A⨿B ↠ B, is a homotopy inverse.

Proof for Waldhausen categories. Let C_{m}^{w} denote the category of sequences A ⟶^{≃} B ⟶^{≃} ⋯ of m weak equivalences; this is a category with cofibrations (defined termwise). The set s_{n}C_{m}^{w} of sequences of n cofibrations in C_{m}^{w} (IV.8.5.2)

A_{0} ↣ A_{1} ↣ ⋯ ↣ A_{n}
↓∼ ↓∼ ↓∼
B_{0} ↣ B_{1} ↣ ⋯ ↣ B_{n}
↓∼  ↓∼  ↓∼
⋯  ⋯  ⋯

is naturally isomorphic to the m-simplices in the nerve of the category wS_{n}C. That is, the bisimplicial sets wS_{.}C and s_{.}C_{.}^{w} are isomorphic. By Ex. IV.8.10 applied to C_{m}^{w}, each of the maps s_{.}E(C_{m}^{w}) → s_{.}C_{m}^{w} × s_{.}C_{m}^{w} is a homotopy equivalence. As m varies, we get a bisimplicial map s_{.}E(C_{.}^{w}) → s_{.}C_{.}^{w} × s_{.}C_{.}^{w}, which must then be a homotopy equivalence. But we have just seen that this is isomorphic to the bisimplicial map wS_{.}E(C) → wS_{.}C × wS_{.}C of the Extension Theorem. □

We include a proof of the Extension Theorem 1.3 for exact categories, because it is short and uses a different technique.

Proof of 1.3 for Exact Categories. By Quillen’s Theorem A (IV.3.7), it suffices to show that, for every pair (A, C) of objects in C, the comma category T = (s, q)/(A, C) is contractible. A typical object in this category is a triple T = (u, E, v), where E is an extension A_{0} ↣ B_{0} ↠ C_{0} and both u : A_{0} → A and v : C_{0} → C are morphisms in QC. We will compare T to its subcategories T_{A} and T_{C}, consisting of those triples T such that: u is an admissible epi, respectively, v is an admissible monic. The contraction of BT is illustrated in the following diagram.

T :  A ⟵^{u} A_{0} ↣ B_{0} ↠ C_{0} ⟶^{v} C
η_{T} ↓  ||  i∨ ↓  ∨ ↓  ||  ||
p(T) :  A ↠^{j} A_{1} ↣ B ↠ C_{0} ⟶^{v} C
π_{p(T)} ↑ || || ↑ ↑ ↑ ↑ ||
qp(T) :  A ↠ A_{1} ↣ B_{1} ↠ C_{1} ↣ C
↑ || ↓ ↓ ↑ ↑ ∧ ||
0  A ↠ 0 ↣ 0 ↠ 0 ↣ C

Given a triple T, choose a factorization of u as A_{0}↣^{i}A_{1}↞^{j}A, and let B be the pushout of A_{1} and B_{0} along A_{0}; B is in C and p(E) : A_{1} ↣ B ↠ C_{0} is an exact sequence by II, Ex. 7.8(2). Thus p(T) = (j, p(E), v) is in the subcategory T_{A}. The construction shows that p is a functor from T to T_{A}, and that there is a natural transformation η_{T} : T → p(T). This provides a homotopy Bη between the identity of BT and the map Bp : BT → BT_{A} ⊂ BT .

By duality, if we choose a factorization of v as C_{0} ↞ C_{1} ↣ C and let B_{1} be the pullback of B_{0} and C_{1} along C_{0}, then we obtain a functor q : T → T_{C}, and a natural transformation π_{T} : q(T) → T. This provides a homotopy Bπ between Bq : BT → BT_{C} ⊂ BT and the identity of BT .

The composition of Bη and the inverse of Bπ is a homotopy BT × I → BT between the identity and Bqp : BT → B(T_{A} ∩ T_{C}) ⊂ BT . Finally, the category (T_{A} ∩ T_{C}) is contractible because it has an initial object: (A ↠ 0, 0, 0 ↣ C). Thus Bqp (and hence the identity of BT ) is a contractible map, providing the contraction BT ≃ 0 of the comma category. □

It is useful to have a variant of the Extension Theorem involving two Waldhausen subcategories A, C of a Waldhausen category B. Recall from II.9.3 that the extension category E = E(A, B, C) of C by A is the Waldhausen subcategory of E(B) consisting of cofibration sequences A ↣ B ↠ C with A in A and C in C.

Corollary 1.3.1. Let A and C be Waldhausen subcategories of a Waldhausen category B, and E = E(A, B, C) the extension category. Then (s, q) : E → A × C induces a homotopy equivalence K(E) → K(A) × K(C).

Proof. Since (s, q) is a left inverse to ⨿ : A×C → E, it suffices to show that the identity of K(E) is homotopic to ⨿(s, q)_{∗} = ⨿(s, 0)_{∗} + ⨿(0, t)_{∗}. This follows from Additivity applied to the short exact sequence of functors ⨿(s, 0) ↣ id_{E} ↠ ⨿(0, t) displayed in the proof of the corresponding Extension Theorem II.9.3.1 for K_{0}. □

The rest of this section is devoted to applications of the Additivity Theorem.

Exact Sequences 1.4. Let A_{exact}^{[0,n]} denote the category of admissibly exact sequences of length n. If n = 1 it is equivalent to the category A; for n = 2 it is the category E of cofibration sequences of 1.1.1. In fact, A_{exact}^{[0,n]} is a Waldhausen category in a way which extends the structure in 1.1.1: A_{∗} → A′_{∗} is a weak equivalence if each A_{i} → A′_{i} is, and is a cofibration if the B_{i} ↣ B′_{i} and B′_{i} ∪_{B_{i}} A_{i} ↣ A_{1}^{i} are all cofibrations. Since A_{exact}^{[0,n]} is the extension category E of A ≅ A_{exact}^{[0,0]} by A_{exact}^{[1,n]} (1.3.1), the Extension Theorem 1.3 implies that the functors A_{∗} ↦ B_{i} (i = 0, ..., n−1) from A_{exact}^{[0,n]} to A induce a homotopy equivalence KA_{exact}^{[0,n]} ≅ ∏_{i=1}^{n} K(A).

Projective bundles

Let E be a vector bundle of rank r + 1 over a quasi-projective scheme X, and consider the projective space bundle ℙ = ℙ(E) ⟶^{π} X. We saw in II.8.5 that K_{0}(ℙ) is a free K_{0}(X)-module with basis {[O(−i)] : i = 0 …, r}. The summands of this decomposition arise from the exact functors

u_{i} : VB(X) → VB(ℙ), u_{i}(N) = π^{∗}(N)(−i).

Projective Bundle Theorem 1.5. Let ℙ(E) be a projective bundle over a quasi-projective scheme X. Then the u_{i} induce an equivalence K(X)^{r+1} ≃ K(ℙ(E)). Thus K_{∗}(X) ⊗_{K_{0}(X)} K_{0}(ℙ(E)) → K_{∗}(ℙ(E)) is a ring isomorphism.

When E is a trivial bundle, so ℙ(E) = ℙ_{X}^{n}, we have the following special case.

Corollary 1.5.1. As a ring, K_{∗}(ℙ_{X}^{n}) ≅ K_{∗}(X)⊗_{ℤ}K_{0}(ℙ_{ℤ}^{n}) ≅ K_{∗}(X)[z]/(z^{r+1}).

To prove Theorem 1.5, recall from II.8.7.1 that we call a vector bundle F Mumford-regular if R^{q}π_{∗}F(−q) = 0 for all q > 0. We write MR for the exact category of all Mumford-regular vector bundles. The direct image π_{∗} : MR → VB(X) is an exact functor, so the following lemma allows us to define the transfer map π_{∗} : K(ℙ) → K(X).

Lemma 1.5.2. MR ⊂ VB(ℙ) induces an equivalence KMR ≃ K(ℙ).

Proof. Write MR(n) for the category of all F for which F(−n) is Mumfordregular. Then MR = MR(0) ⊆ MR(−1) ⊆ ⋯, and VB(ℙ(E)) is the union of the MR(n) as n → −∞. Thus K(ℙ) = \underrightarrow{lim} KMR(n). Hence it suffices to show that each inclusion ι_{n} : MR(n) ⊂ MR(n − 1) induces a homotopy equivalence on Ktheory. We saw in the proof of II.8.7.10 that the exact functors λ_{i} : MR(n − 1) → MR(n), λ_{i}(F) = F(i) ⊗ π^{∗}(∧^{i}E), fit into a Koszul resolution of F:

0 → F → F(1) ⊗ π^{∗}E → ⋯ → F(r + 1) ⊗ π^{∗} ∧^{r+1} E → 0.

By Additivity (1.2.1), ι_{n} has ∑_{i=1}^{r}(−1)^{i}λ_{i} as a homotopy inverse. □

Proof of Theorem 1.5. By II.8.7.9 we have exact functors T_{i} : MR → VB(X), with T_{0} = π_{∗}, which assemble to form a map t : MR → ∐ VB(X). By Quillen’s canonical resolution of a Mumford-regular bundle (II.8.7.8) is an exact sequence of exact functors:

0 → π^{∗}(T_{r})(−r) ⟶^{ε(−r)} ⋯ → π^{∗}(T_{i})(−i) ⟶^{ε(−i)} ⋯ ⟶^{ε(−1)} π^{∗}(T_{0}) ⟶^{ε} F → 0.

Again by Additivity (1.2.1), ∑(−1)^{i}u_{i}T_{i} is homotopic to the identity on KMR, so u_{∗} is split up to homotopy.

Define v_{i} : MR → VB(X) by v_{i}(F) = π_{∗}(F(i)). Then by II.8.7.2:

v_{i}u_{j}(N) = \{ \begin{matrix}0, & i < j; \\ N , & i = j; \\ Sym_{i−j}E ⊗ π^{∗}(N), & i > j. \end{matrix}

It follows that v_{∗} ∘ u_{∗} : K(X)^{r+1} → K(X)^{r+1} is given by a triangular matrix whose diagonal entries are homotopic to the identity. Thus v_{∗} ∘ u_{∗} is a homotopy equivalence, as desired. □

Variant 1.5.3. Theorem 1.5 remains valid if X is noetherian instead of quasiprojective, but the proof is more intricate because in this case K(X) is defined to be KCh_{perf}(X) (see 2.7.3). This generalization was proven by Thomason in [TT, 4.1] by (a) replacing VB(X) by the category of perfect complexes of coherent sheaves; (b) replacing MR by the category of all Mumford-regular coherent sheaves (II.8.7.1) and passing to the Waldhausen category of bounded perfect complexes of Mumford-regular sheaves (Lemma 1.5.2 remains valid); and (c) observing that Quillen’s canonical resolution (II.8.7.8) makes sense for Mumford-regular sheaves. With these modifications, the proof we have given for Theorem 1.5 goes through. By Ex. 1.10, Theorem 1.5 even remains valid for all quasi-compact and quasi-separated schemes X.

The projective line over a ring

Let R be any associative ring. We define mod- ℙ_{R}^{1} to be the abelian category of triples F = (M_{+}, M_{−}, α), where M_{±} is in mod-R[t^{±1}] and α is is an isomorphism M_{+}⊗_{R[t]}R[t, 1/t] ⟶^{≃} M_{−}⊗_{R[1/t]}R[t, 1/t]. It has a full (exact) subcategory VB(ℙ_{R}^{1}) consisting of triples where M_{±} are finitely generated projective modules, and we write K(ℙ_{R}^{1}) for KVB(ℙ_{R}^{1}).

If R is commutative, it is well known that mod- ℙ_{R}^{1} is equivalent to the category of quasi-coherent sheaves on ℙ_{R}^{1}, and VB(ℙ_{R}^{1}) is equivalent to the usual category of vector bundles on the line ℙ_{R}^{1}; thus K(ℙ_{R}^{1}) agrees with the definition in IV.6.3.4, and both π_{∗} and R^{1}π_{∗} have their usual meanings.

We define the functors π_{∗}, R^{1}π_{∗} : mod- ℙ_{R}^{1} → mod-R via the exact sequence

0 → π_{∗}(F) → M_{+} × M_{−} ⟶^{d} M_{−} ⊗_{R[1/t]} R[t, 1/t] → R^{1}π_{∗}(F) → 0.

where d(x, y) = α(x)−y. If R is commutative, these are the usual functors π_{∗}, R^{1}π_{∗}.

There are exact functors u_{i} : P(R) → VB(ℙ_{R}^{1}), sending P to (P [t], P [1/t], t^{i}); for commutative R the u_{i} are the functors u_{i}(P) = π^{∗}(P) ⊗ O(−i) of Theorem 1.5.

Theorem 1.5.4. The functors u_{0}, u_{1} induce an equivalence K(R) ⊕ K(R) ≃ K(ℙ_{R}^{1}). In addition, (u_{i+1})_{∗} + (u_{i+1})_{∗} ≃ (u_{i})_{∗} + (u_{i+2})_{∗} for all i.

Proof. If F = (M_{+}, M_{−}, α), we define F(n) to be (M_{+}, M_{−}, t^{−n}α), and let X_{0}, X_{1} : F(n − 1) → F(n) be the maps (1, 1/t) and (t, 1), respectively. Then we have an exact sequence (the Koszul resolution of F).

0 → F(−2) ⟶^{(X_{1},−X_{0})} F(−1)^{2} ⟶^{(X_{0},X_{1})} F → 0.

Applying this to u_{i}(P) and using u_{i}(P)(n) = u_{i−n}(P) yields the exact sequence u_{i+2} ↣ u_{i+1}^{2} ↠ u_{i} of functors, and the relations follow from Additivity 1.2. The proof of Theorem 1.5 now goes through to prove Theorem 1.5.4 (see Ex. 1.3). □

Severi-Brauer schemes

Let A be a central simple algebra over a field k, and ℓ a maximal subfield of A. Then A ⊗_{k} ℓ ≅ M_{r}(ℓ) for some r, and the set of minimal left ideals of M_{r}(ℓ) correspond to the ℓ-points of the projective space ℙ_{ℓ}^{r−1}; if I is a minimal left ideal corresponding to a line L of ℓ^{r} then the rows of matrices in I all lie on L. The Galois group Gal(ℓ/k) acts on this set, and it is well known that there is a variety X, defined over k, such that X_{ℓ} = X ×_{k} ℓ is ℙ_{ℓ}^{r−1} with this Galois action. The variety X is called the Severi-Brauer variety of A. For example, the Severi-Brauer variety associated to A = M_{r}(k) is just ℙ_{k}^{r−1}.

Historically, these varieties arose in the 1890’s (over ℝ) as forms of a complex variety, together with a real structure given by an involution with no fixed points.

Example 1.6.1. Let X be a non-singular projective curve over k defined by an irreducible quadratic aX^{2} + bY^{2} = Z^{2} (a, b ∈ k). Then X ≅ ℙ_{k}^{1} if and only if X has a k-point, which holds if and only if the quadratic form q(x, y) = ax^{2} + by^{2} has a solution to q(x, y) = 1 in k. The associated algebra is the quaternion algebra A(a, b) of III.6.9. For example, if X is the plane curve X^{2} + Y^{2} + Z^{2} = 0 over ℝ then A is the usual quaternions ℍ.

Here are some standard facts about Severi-Brauer varieties. By faithfully flat descent, the vector bundle O^{r}(−1) on ℙ_{ℓ}^{r−1} descends to a vector bundle J on X of rank r, and A ≅ H^{0}(X, End_{X}(J)) because End_{ℙ}_{ℓ}^{r−1}(O^{r}(−1)) is a sheaf of matrix algebras having M_{r}(ℓ) as its global sections. Moreover, if π : X → S = Spec(k) is the structure map then π^{∗}(A) ≅ End_{X}(J), as can be checked by pulling back to ℓ.

There is a canonical surjection O_{ℙ}^{r}(−1) → O_{ℙ} (Ex. II.6.14); by descent it defines a surjection J ↠ O_{X}. Hence there is a Koszul resolution:

0 → ∧^{r}J → ⋯ → ∧^{2}J → J → O_{X} → 0.

The n-fold tensor product A^{⊗n} of A over k is also a central simple algebra, isomorphic to End_{X}(J^{⊗n}). Moreover, since J^{⊗n} is a right module over A^{⊗n} = End_{X}(J^{⊗n}) there is an exact functor J^{⊗n}⊗ :P(A^{⊗n}) → VB(X) sending P to J^{⊗n} ⊗_{A^{⊗n}} P.

Theorem 1.6.2. (Quillen) If X is the Severi-Brauer variety of A, the functors J^{⊗n}⊗ define an equivalence ∏_{n=0}^{r−1} K(A^{⊗n}) ⟶^{∼} K(X), and an isomorphism

⨁_{n=0}^{r−1} K_{∗}(A^{⊗n}) ⟶^{≃} K_{∗}(X).

Example 1.6.3. If X is the nonsingular curve aX^{2} + bY^{2} + Z^{2} associated to the quaternion algebra A = A(a, b), then K_{∗}(X) ≅ K_{∗}(k) ⊕ K_{∗}(A).

The proof of Theorem 1.6.2 is a simple modification of the proof of Theorem 1.5. First, we define a vector bundle F to be regular if F ⊗_{k} ℓ is Mumford-regular on X_{ℓ} = ℙ_{ℓ}^{r−1}. The regular bundles form an exact subcategory of O_{X}-mod, and F ↦ π_{∗}F = H^{0}(X, F) is an exact functor from regular bundles to k-modules, as one checks by passing to ℓ and applying II.8.7.4. To get the analogue of the Quillen Resolution Theorem II.8.7.8, we modify Definition II.8.7.6 using J.

Definition 1.6.4 (T_{n}). Given a regular O_{X}-module F, we define a natural sequence of k-modules T_{n} = T_{n}(F) and O_{X}-modules Z_{n} = Z_{n}(F), starting with T_{0}(F) = π_{∗}F and Z_{−1} = F. Let Z_{0} be the kernel of the natural map J⊗_{A}π_{∗}F → F. Inductively, we define T_{n}(F) = π_{∗} Hom_{X}(J^{⊗n}, Z_{n−1}) and define Z_{n} to be the kernel of J^{⊗n} ⊗_{A^{⊗n}} T_{n}(F) → Z_{n−1}(F).

These fit together to give a natural sequence of O_{X}-modules

(1.6.5)  0 → J^{⊗r−1} ⊗_{A^{⊗r−1}} T_{r−1}(F) → ⋯ → O_{X} ⊗_{k} T_{0}(F) → F → 0.

When lifted to X_{ℓ}, it is easy to see that these are exactly the functors T_{n} and Z_{n} of II.8.7.6 for F ⊗_{k} ℓ. By faithfully flat descent, (1.6.5) is exact, i.e., a resolution of the regular bundle F.

Thus all the tools used in the Projective Bundle Theorem 1.5 are available for Severi-Brauer varieties. The rest of the proof is routine (Exercise 1.4).

Theorem 1.6.2 may be generalized over any base scheme S. Here are the key points. An Azumaya algebra over S is a sheaf of rings A which is locally isomorphic to M_{r}(O_{S}) for the étale topology. That is, there is a faithfully flat map T → S so that A ⊗_{S} O_{T} is the sheaf of rings M_{r}(O_{T}). Let VB(A) denote the exact category of vector bundles on S which are left modules for A, and define

K(A) = KVB(A); K_{n}(A) = K_{n}VB(A).

By a Severi-Brauer scheme over S we mean a scheme X which is locally isomorphic to projective space for the étale topology, i.e., such that X ×_{S} T ≅ ℙ_{T}^{r−1} for some faithful étale map T → S. In this situation, we may define a vector bundle J on X by faithfully flat descent so that J ⊗_{S} T = O_{T}^{r} (−1), as above, and then A = π_{∗} End_{X}(J) will be an Azumaya algebra over S. Again, each J^{⊗n} is a right module over A^{⊗n} and we have exact functors J^{⊗n}⊗_{A^{⊗n}} : VB(A^{⊗n}) → VB(X). Replacing H^{0}(X, −) with π_{∗}, Definition 1.6.4 makes sense and (1.6.5) is admissibly exact. Therefore the proof still works in this generality, and we have:

Theorem 1.6.6. (Quillen) If X is a Severi-Brauer variety over S, with associated Azumaya algebra A, the functors J^{⊗n}⊗_{A^{⊗n}} define an isomorphism

∐_{n=0}^{r−1} K_{∗}(A^{⊗n}) ⟶^{≃} K_{∗}(X).

Our next application of Additivity was used in IV, 8.5.3–5 to show that K(C) is an infinite loop space. To do this, we defined the relative K-theory space to be K(f) = Ω^{2}|wS.(S.f)|, and invoked the following result.

Proposition 1.7. If f : B → C is an exact functor, the following sequence is a homotopy fibration:

Ω|wS.(S.B)| → |wS.C| → |wS.(S.f)| → |wS.(S.B)|.

Proof. Each category S_{n}f of IV.8.5.3 is equivalent to the extension category E(B, S_{n}f, S_{n}C) of B by S_{n}C (see II.9.3). By 1.3.1, the map

(s, q) : wS.(S_{n}f.) ⟶^{≃} wS.B × wS.(S_{n}C)

is a homotopy equivalence. That is, |wS.B| → |wS.(S_{n}f.)| → |wS.(S_{n}C)| is a (split) fibration of connected spaces for each n. But if X. → Y. → Z. is any sequence of simplicial spaces, and each X_{n} → Y_{n} → Z_{n} is a fibration with Z_{n} connected, then Ω|Z.| → |X.| → |Y.| → |Z.| is a homotopy fibration sequence; see [Wa78, 5.2]. This applies to our situation by realizing in the wS. direction first (so that X_{n} = |wS.B| for all n), and the result follows. □

Remark 1.7.1. As observed in IV.8.5.4, wS.S.f is contractible when f is the identity of B. It follows that |wS.B| ≃ Ω|wS.(S.B)|, yielding the formulation of 1.7 given in IV.8.5.3.

Let F : A → B be an exact functor between exact categories. An admissible filtration of F is a sequence 0 ↣ F_{1} ↣ F_{2} ↣ ⋯ ↣ F_{n} = F of functors and admissible monomorphisms, sending an object A in A to the sequence

0 = F_{0}(A) ↣ F_{1}(A) ↣ ⋯ ↣ F_{n}(A) = F (A).

It follows that the quotient functors F_{p}/F_{p−1} exist, but they may not be exact.

Proposition 1.8. (Admissible Filtrations) If 0 ↣ F_{1} ↣ F_{2} ↣ ⋯ ↣ F_{n} = F is an admissible filtration of F, and the quotient functors F_{i}/F_{i−1} are exact, then F : KA → KB is homotopic to ∑ F_{i}/F_{i−1}. In particular,

F_{∗} = ∑(F_{i}/F_{i−1})_{∗} : K_{∗}(A) → K_{∗}(B).

Proof. Apply the Additivity Theorem 1.2 to F_{i−1} ↣ F_{i} ↠ F_{i}/F_{i−1}, and use induction on n. □

Here is a simple application of Proposition 1.8. Let S = R ⊕ S_{1} ⊕ S_{2} ⊕ ⋯ be a graded ring, and consider the category P_{gr}(S) of finitely generated graded projective S-modules. Its K-groups are naturally modules over ℤ[σ, σ^{−1}], where σ acts by the shift automorphism σ(P) = P [−1] of graded modules. If S = R, K_{∗}(R)[σ, σ^{−1}] ≅ K_{∗}(P_{gr}(R)) by Ex. IV.6.11. Thus the base change map ⊗_{R}S : P_{gr}(R) → P_{gr}(S) induces a morphism K_{∗}(R)[σ, σ^{−1}] → K_{∗}(P_{gr}(S)).

Corollary 1.8.1. If S = R ⊕ S_{1} ⊕ S_{2} ⊕ ⋯ is graded then the base change map induces an isomorphism K_{∗}(R)[σ, σ^{−1}] ≅ K_{∗}(P_{gr}(S)).

Proof. For each a ≤ b, let P_{[a,b]}(S) denote the (exact) subcategory of P_{gr}(S) consisting of graded modules P generated by the P_{i} with i ≤ b, and with P_{i} = 0 for i < a. By Ex. 1.9, the identity functor on this category has an admissible filtration: 0 = F_{a} ↣ F_{a+1} ↣ ⋯ ↣ F_{b} = id, where F_{n}P denotes the submodule of P generated by the P_{i} with i ≤ n. Moreover, there is a natural isomorphism between F_{n}/F_{n−1} and the degree n part of the exact functor ⊗_{S}R : P_{gr}(S) → P_{gr}(R). By Proposition 1.8, the homomorphism ⊕_{n=a}^{b}K_{∗}(R)⊗σ^{n} ≅ K_{∗}P_{[a,b]}(R) → K_{∗}P_{[a,b]}(S) is an isomorphism with inverse ⊗_{S}R. Since P_{gr}(S) is the filtered colimit of the P_{[a,b]}(S), the result follows from IV.1.4. □

Flasque Categories 1.9. Call an exact (or Waldhausen) category A flasque if there is an exact functor ∞: A → A and a natural isomorphism ∞(A) ≅ A⨿∞(A), i.e., ∞ ≅ 1 ⨿ ∞, 1 being the identity functor. By additivity, ∞_{∗} = 1_{∗} + ∞_{∗}, and hence the identity map 1_{∗} : K(A) → K(A) is null-homotopic. Therefore K(A) is contractible, and K_{i}(A) = 0 for all i.

The Eilenberg Swindle 1.9.1. For example, the category of countably generated R-modules is flasque, so its K-theory is trivial. To see this, let ∞(M) = M^{∞} be the direct sum M ⊕ M ⊕ ⋯ of infinitely many copies of M. The isomorphism M^{∞} ≅ M ⊕ M^{∞} is the shift

(M ⊕ M ⊕ ⋯) ≅ M ⊕ (M ⊕ M ⊕ ⋯).

This infinite shifting trick is often called the “Eilenberg swindle” (see I.2.8, II.6.1.4 and II.9.1.4); it is why we restrict to finitely generated modules in defining K(R).

Example 1.9.2 (flasque rings). Here is another example of a flasque category, due to Karoubi. Recall from II.2.1.3 that a ring R is called flasque if there is an R-bimodule M_{∞}, finitely generated projective as a right module, and a bimodule isomorphism θ : R ⊕ M_{∞} ≅ M_{∞}. If R is flasque, then P(R) and M(R) are flasque categories in the sense of 1.9, with ∞(M) = M ⊗_{R} M_{∞}. The contractibility of K(R) = K_{0}(R) × BGL(R)^{+} for flasque rings, established in Ex. IV.1.17, may be viewed as an alternative proof that KP(R) = ΩBQP(R) is contractible, via the ‘+ = Q’ theorem IV.7.2.

As mentioned in IV.10.4.1, this contractibility was used by Karoubi, Gersten and Wagoner to define deloopings of K(R) in terms of the suspension ring S(R) of IV.1.11.3, forming a nonconnective spectrum K^{GW} (R) homotopy equivalent to the spectrum K^{B}(R) of IV.10.4.

EXERCISES

1.1 In the proof of the Extension Theorem 1.3, show that the functors p and q are left and right adjoint, respectively, to the inclusions of T_{A} and T_{C} in T . This proves that Bp and Bq are homotopy equivalences. 1.2 Let 0 → E′ → E → E′′ → 0 be a short exact sequence of vector bundles on a scheme (or a ringed space). Show that the map (− ⊗ E)_{∗} : K(X) → K(X) given by the exact functor F → F ⊗ E satisfies (− ⊗ E)_{∗} = (− ⊗ E′)_{∗} + (− ⊗ E′′)_{∗}.

1.3 Complete the proof of Theorem 1.5.4, by modifying the proof of 1.5 for ℙ_{R}^{1}. (See [Q341, 8.4.1].) 1.4 Complete the proof of Theorems 1.6.2 and 1.6.6, by modifying the proof of 1.5. (See [Q341, 8.4.1].) For extra credit, describe the ring structure on K_{∗}(X) using the pairings mod-A^{⊗i} × mod-A^{⊗j} → mod-A^{⊗i+j} (tensor product over k) and the Morita equivalence of A^{⊗i} and A^{⊗i+n}.

1.5 Given an exact category A and integers a ≤ b, let iso Ch^{[a,b]}(A) denote the category of chain complexes C_{b} → ⋯ → C_{a} in A. We may consider it as a Waldhausen category whose cofibrations are degreewise admissible monics, with isomorphisms as the weak equivalences (II.9.1.3). Use the Additivity Theorem 1.2 to show that the “forget differentials” functor iso Ch^{[a,b]}(A) → ∏_{i=a}^{b} A induces a homotopy equivalence on K-theory.

1.6 If A is an exact category, the category A_{exact}^{[0,n]} of admissibly exact sequences (Example 1.4) may be viewed as a subcategory of the category iso Ch^{[0,n]}(A) of the previous exercise. Use the Additivity Theorem to show that the “forget differentials” functor A_{exact}^{[0,n]} → ∏_{i=0}^{n} A and the functor A_{∗} ↦ (B_{1}, B_{1} ⊕ B_{2}, ..., B_{n−1} ⊕ B_{n}, B_{n}) induce homotopy equivalent maps on K-theory. 1.7 If f : B → C is exact, show that the composite wS.B → Ω|wS.S.B| → |wS.C| in Proposition 1.7 is the map induced by f. Hint: Use wS.S.id_{B} → wS.S.f. 1.8 Recall from IV.8.7 that A(∗) is the K-theory of the category R_{f}(∗) of finite based CW complexes. Let R_{f}^{(2)}(∗) be the subcategory of simply connected com-

plexes. Show that KR_{f}^{(2)}(∗) ⟶^{∼} KR_{f}(∗) = A(∗) is a homotopy equivalence, with Y ↦ Σ^{2}Y as inverse. Then formulate a version for A(X). 1.9 (Swan) If S = R ⊕ S_{1} ⊕ ⋯ is a graded ring and P is a graded projective S-module, show that the map (P ⊗_{S} R) ⊗_{R} S → P is an isomorphism. If F_{n}P is the submodule of P generated by the P_{i} with i ≤ n, show that F_{n}P and P/F_{n}P are graded projective modules, and that F_{n} and P ↦ P/F_{n}P are exact functors from P_{gr}(S) to itself. Conclude that ⋯ ↣ F_{n}P ↣ F_{n+1}P ↣ ⋯ is an admissible filtration of P. Is there a natural isomorphism P ≅ F_{n}P ⊕ P/F_{n}P? 1.10 Let X be a quasi-compact, quasi-separated scheme. Show that the variant 1.5.3 of the Projective Bundle Theorem 1.5 holds for X. Hint: X is the inverse limit of an inverse system of noetherian schemes X_{α} with affine bonding maps by [TT, C.9]. Show that any vector bundle E on X is the pullback of a vector bundle E_{α} over some X_{α}.

§2. Waldhausen localization and Approximation

Here are two fundamental results about Waldhausen K-theory that, although technical in nature, have played a major role in the development of K-theory.

Waldhausen Localization

The first fundamental result involves a change in the category of weak equivalences, with the same underlying category of cofibrations. The K_{0} version of this result, which needed fewer hypotheses, was presented in II.9.6.

Waldhausen Localization Theorem 2.1. Let A be a category with cofibrations, equipped with two categories of weak equivalences, v(A) ⊂ w(A), such that (A, v) and (A, w) are both Waldhausen categories. In addition, we suppose that (A, w) has a cylinder functor satisfying the Cylinder Axiom (IV.8.8), and that w(A) satisfies the Saturation and Extension Axioms (II.9.1.1 and IV.8.2.1). Then

K(A^{w}) → K(A, v) → K(A, w)

is a homotopy fibration, where A^{w} denotes the Waldhausen subcategory of (A, v) consisting of all A in A for which 0 → A is in w(A). In particular, there is a long exact sequence:

⋯ → K_{i+1}(A, w) → K_{i}(A^{w}) → K_{i}(A, v) → K_{i}(A, w) → ⋯,

ending in the exact sequence K_{0}(A^{w}) → K_{0}(A, v) → K_{0}(A, w) → 0 of II.9.6.

Proof. Consider the bicategory v.w.C (IV.3.10) whose bimorphisms are commutative squares in C

\begin{matrix}· & ⟶^{w′} & · \\ ^{v}↓ & & ↓_{v′} \\ · & ⟶^{w} & · \end{matrix}

in which the vertical maps are in vC and the horizontal maps are in wC. Considering wC as a bicategory which is vertically constant, we saw in IV, 3.10.2 and Ex. 3.13 that wC → v.w.C is a homotopy equivalence. Applying this construction to S_{n}C, we get equivalences wS_{n}C ≃ v.wS_{n}C and hence wS.C ≃ v.wS.C.

Let v. co w.C denote the sub-bicategory of those squares in v.w.C whose horizontal maps are also cofibrations. We claim that the inclusions v. co w.C ⊂ v.w.C are homotopy equivalences. To see this, we fix m and consider the column category v_{m}w.C to be the category of weak equivalences in the category C(m, v) of diagrams C_{0} ⟶^{∼} ⋯ ⟶^{∼} C_{m} in C whose maps are in vC. Because (C(m, v), w) inherits the saturation and cylinder axioms from (C, w), it follows from IV, Ex. 8.14 that the inclusion v_{m} co w.C ⊂ v_{m}w.C is a homotopy equivalence. Since this is true for each m, the claim follows.

Now each S_{n}C inherits a cylinder functor from C. Replacing C by the S_{n}C shows that the simplicial bicategory v.w.S.C contains a simplicial bicategory v. co w.S.C, and that the inclusion v. co w.S.C ⊂ v.w.S.C is a homotopy equivalence. This means that the right vertical map is a homotopy equivalence in the following diagram; the bottom horizontal map is a homotopy equivalence by the first paragraph of this proof.

vS.C^{w} ⟶ vS.C ⟶ v. co w.S.C
↓  ↓^{≃}
wS.C ⟶^{≃} v.w.S.C

Thus it suffices to show that the top row is a homotopy fibration. We will identify it with the homotopy fibration vS.C^{w} → vS.C → vS.(S.f) arising from the relative K-theory space construction (IV.8.5.3), applied to the inclusion f : (C^{w}, v) → (C, v).

By the extension axiom, a trivial cofibration in (C, w) is just a cofibration whose quotient lies in C^{w}. In particular, there is an equivalence S_{1}f → co wC. Forgetting the choices of the C_{i}/C_{j} yields an equivalence S_{n}f → co w_{n}C, where co w_{n}C is the category of all trivial cofibration sequences C_{0} ↣^{∼} C_{1} ↣^{∼} ⋯ ↣^{∼} C_{n}, and an equivalence between vS_{n}f and the vertical category v. co w_{n}C of the bicategory v. co w.C. Similarly, forgetting choices yields an equivalence between the categories vS_{m}(S_{n}f) and v. co w_{n}(S_{m}C), and thus a homotopy equivalence vS.S.f → v. co w.S.C.

Now vS_{m}C → v. co w_{n}(S_{m}C) factors through vS_{m}(S_{n}f), so vS.C → v. co w.S.C factors through the homotopy equivalence vS.S.f → v. co w.S.C, as required. □

For us, the most important application of Waldhausen Localization is the following theorem, which allows us to replace the K-theory of any exact category A by the K-theory of the category Ch^{b}(A) of bounded chain complexes, which is a Waldhausen category with a cylinder functor. This result was first worked out by Waldhausen in special cases, and generalized by Gillet. Our presentation is taken from [TT, 1.11.7].

Let A be an exact category, and consider the category Ch^{b}(A) of bounded chain complexes in A. We saw in II.9.2 that Ch^{b}(A) is a Waldhausen category; the cofibrations are degreewise admissible monomorphisms, and the weak equivalences are quasi-isomorphisms (as computed in a specified ambient abelian category). The isomorphism K_{0}(A) ≅ K_{0}Ch^{b}(A) of Theorem II.9.2.2 generalizes as follows.

Theorem 2.2 (Gillet-Waldhausen). Let A be an exact category, closed under kernels of surjections in an abelian category (in the sense of II.7.0.1.) Then the exact inclusion A ⊂ Ch^{b}(A) induces a homotopy equivalence K(A) ⟶^{∼} KCh^{b}(A).

In particular, K_{n}(A) ≅ K_{n}Ch^{b}(A) for all n.

Proof. We will apply Waldhausen’s Localization Theorem 2.1 to the following situation. For a ≤ b, let Ch^{[a,b]} denote the full subcategory of all complexes A_{∗} in Ch(A) for which the A_{i} are zero unless a ≤ i ≤ b. This is a Waldhausen subcategory of Ch^{b}(A) with w the quasi-isomorphisms. We write iso Ch^{[a,b]} for the Waldhausen category with the same underlying category with cofibrations, but with isomorphisms as weak equivalences. Because A is closed under kernels of surjections, the subcategory of quasi-isomorphisms in iso Ch^{[a,b]} is just the Waldhausen category A_{exact}^{[a,b]} of Example 1.4 (see Ex. 2.4). We claim that there is a homotopy fibration

KA_{exact}^{[a,b]} → Kiso Ch^{[a,b]} ⟶^{χ} K(A).

By Example 1.4 and Ex. 1.5, the first two spaces are products of n = b−a and n+1 copies of K(A), respectively. By Ex. 1.6, the induced map ∏_{a+1}^{b} K(A) → ∏_{a}^{b} K(A) is equivalent to that induced by the exact functor

(B_{a+1}, ..., B_{b}) ↦ (B_{a+1}, B_{a+1} ⊕ B_{a+2}, ..., B_{b−1} ⊕ B_{b}, B_{b}).

The homotopy cofiber of this map is K(A), with the map ∏_{a}^{b} K(A) → K(A) being the alternating sum of the factors, i.e., the Euler characteristic χ. This shows that KCh^{[a,b]} ≃ K(A) for each a and b.

Taking the direct limit as a → −∞ and b → +∞ yields a homotopy fibration

KA_{exact}^{[−∞,∞]} → Kiso Ch^{[−∞,∞]} ⟶^{χ} K(A),

where χ is the Euler characteristic. But by Waldhausen Localization 2.1, the cofiber is KCh^{b}(A). □

Remark 2.2.1. When A is not closed under kernels in its ambient abelian category, K_{0}(A) may not equal K_{0}Ch^{b}(A); see Ex. II.9.11. However, the following trick shows that the extra assumption is harmless in Theorem 2.2, provided that we allow ourselves to change the ambient notion of quasi-isomorphism slightly in Ch^{b}(A). Consider the Yoneda embedding of A in the abelian category L of contravariant left exact functors (Ex. II.7.8). As pointed out in loc. cit., the idempotent completion Â of A (II.7.3) is closed under surjections in L.

Let A′ be the full subcategory of Â consisting of all B with [B] in the subgroup K_{0}(A) of K_{0}(Â). We saw in Ex. IV.8.13 that A′ is exact and closed under admissible epis in Â (and hence in L), so that Theorem 2.2 applies to A′. By K_{0}-cofinality (II, 7.2 and 9.4), K_{0}(A) = K_{0}(A′) = K_{0}Ch^{b}(A) = K_{0}Ch^{b}(A′). By Waldhausen Cofinality (IV.8.9), K(A) ≃ K(A′) and KCh^{b}(A) ≃ KCh^{b}(A′). Hence K(A) ≃ KCh^{b}(A).

Cofinality Theorem 2.3. Let (A, v) be a Waldhausen category with a cylinder functor satisfying the cylinder axiom (IV.8.8). Suppose that we are given a surjective homomorphism π: K_{0}(A) → G, and let B denote the full Waldhausen subcategory of all B in A with π[B] = 0 in G.

Then vs.B → vs.A → BG and its delooping K(B) → K(A) → G are homotopy fibrations. In particular, K_{n}(B) ≅ K_{n}(A) for all n > 0 and (as in II.9.6.2) there is a short exact sequence:

0 → K_{0}(B) → K_{0}(A) ⟶^{π} G → 0.

Proof. (Thomason) As in II.9.6.2, we can form the Waldhausen category (A, w), where w(A) is the set of maps A → A′ in A with π[A] = π[A′]. It is easy to check that w(A) is saturated (II.9.1.1), B = A^{w}, and that (A, w) satisfies the Extension Axiom IV.8.2.1. By the Waldhausen Localization Theorem 2.1, there is a homotopy fibration

K(B) → K(A) → K(A, w).

By IV.8.10, ws.(A, w) ≃ BG and hence K(A, w) ≃ Ω(BG) = G, as required. □

Combining this with the Waldhausen Cofinality Theorem IV.8.9.1, we obtain the following variation. Recall from Theorem II.9.4 that a Waldhausen subcategory B is said to be cofinal in A if for each A in A there is an A′ so that A ⨿ A′ is in B, and that this implies that K_{0}(B) → K_{0}(A) is an injection.

Corollary 2.3.1. Let B be a cofinal Waldhausen subcategory of A closed under extensions. Suppose that A has a cylinder functor satisfying the cylinder axiom (IV.8.8), and restricting to a cylinder functor on B.

Then for G = K_{0}(A)/K_{0}(B) there is a homotopy fibration sequence

K(B) → K(A) → G.

Proof. Clearly B is contained in the Waldhausen subcategory A^{w} associated to K_{0}(A) → G. By the Waldhausen Cofinality Theorem IV.8.9.1, K(B) ≃ K(A^{w}). The result now follows from Theorem 2.3. □

Waldhausen Approximation

The second fundamental result is the Approximation Theorem, whose K_{0} version was presented in II.9.7. Consider the following “approximate lifting property,” which is to be satisfied by an exact functor F: A → B: (App) Given any map b: F (A) → B in B, there is a map a: A → A′ in A and a

weak equivalence b′ : F (A′) ≃ B in B so that b = b′ ∘ F (a). Roughly speaking, this axiom says that every object and map in B lifts up to weak equivalence to A. Note that if we replace A′ by the mapping cylinder T (a) of IV.8.8, a by A ↣ T (a) and b′ by F (T (a)) ≃ F (A′) ≃ B, then we may assume that a is a cofibration. The following result is taken from [W1126, 1.6.7].

Waldhausen Approximation Theorem 2.4. Suppose that F : A → B is an exact functor between saturated Waldhausen categories, satisfying the conditions:

(a) A morphism f in A is a weak equivalence if and only if F (f) is a w.e. in B.

(b) A has a cylinder functor satisfying the cylinder axiom.

(c) The approximate lifting property (App) is satisfied. Then wS.A ⟶^{∼} wS.B and K(A) ⟶^{∼} K(B) are homotopy equivalences. In particular, the groups K_{∗}A and K_{∗}B are isomorphic.

Proof. (Waldhausen) Each of the exact functors S_{n}A → S_{n}B also satisfies (App); see Ex. 2.1. Applying Proposition 2.4.1 below to these functors, we see that each wS_{n}A → wS_{n}B is also a homotopy equivalence. It follows that the bisimplicial map wS_{.}A → wS_{.}B is also a homotopy equivalence, as required. □

Proposition 2.4.1. (Waldhausen) Suppose that F : A → B is an exact functor between Waldhausen categories, satisfying the three hypotheses of Theorem 2.4. Then wF : wA ⟶^{∼} wB is a homotopy equivalence.

Proof. By Quillen’s Theorem A (IV.3.7), it suffices to show that the comma categories wF/B are contractible. The condition (App) states that, given any object (A, b) of F/B there is a map a in F/B to an object (A′, b′) of wF/B. Applying it to (0, 0) shows that wF/B is nonempty. For any finite set of objects (A_{i}, b_{i}) in wF/B, the maps A_{i} → ⊕A_{i} yield maps in F/B to (⊕A_{i}, b), where b : F (⊕A_{i}) ≅ ⊕F (A_{i}) → B, and hence maps a_{i} from each (A_{i}, b_{i}) to an object (A′, b′) in wF/B; the F (a_{i}) are in wB by saturation, so the a_{i} are in wA and represent maps in wF/B. The same argument shows that if we are given any finite diagram D on these objects in wF/B, an object (A, b) in F/B and maps (A_{i}, b_{i}) → (A, b) forming a (larger) commutative diagram D_{+} in F/B, then by composing with (A, b) → (A′, b′), we embed D into a diagram D′_{+} in wF/B with a terminal object. This implies that |D| is contractible in |wF/B|.

The rest of the proof consists of finding such a diagram D_{+} for every “nonsingular” finite subcomplex of |wF/B|, using simplicial methods. We omit this part of the proof, which is lengthy (5 pages), and does not seem relevant to this book, and refer the reader to [W1126, 1.6.7]. □

Remark 2.4.2. The Approximation Theorem can fail in the absence of a cylinder functor. For example, if A is an exact category then A^{⊕} ⊂ A satisfies (App), yet K_{0}(A^{⊕}) and K_{0}(A) are often different; see II.7.1.

Combining Waldhausen Localization 2.1 and Approximation 2.4 yields the following useful result, applicable to exact functors F : A → B which are onto up to weak equivalence. Let A^{w} denote the Waldhausen subcategory of A consisting of all A such that F (A) is weak equivalent to 0 in B.

Proposition 2.5. Let F : A → B be an exact functor between two saturated Waldhausen categories having cylinder functors, with B extensional (IV.8.2.1). If every object B and every map F (A) → B in B lifts to A up to weak equivalence, then K(A^{w}) → K(A, v) → K(B) is a homotopy fibration sequence, and there is a long exact sequence:

⋯ ⟶^{F} K_{n+1}(B) → K_{n}(A^{w}) → K_{n}(A) ⟶^{F} K_{n}(B) → ⋯,

ending in the exact sequence K_{0}(A^{w}) → K_{0}(A) → K_{0}(B) → 0.

Proof. (Thomason) Let v(A) and w(B) denote the respective categories of weak equivalences in A and B, and set w(A) = F^{−1}(w(B)). Replacing v(A) with w(A) yields a new Waldhausen category, which we write as (A, w) for clarity. The Approximation Theorem 2.4 states that K(A, w) ≃ K(B). Since (A, w) inherits the extension axiom from B, Waldhausen Localization 2.1 applies to give the fibration K(A^{w}) → K(A, v) → K(B) and hence the displayed long exact sequence. □

Changing cofibrations 2.5.1. (Hinich-Shektman). Let A = (A, coA, wA) be a saturated Waldhausen category with a cylinder functor, satisfying the cylinder axiom. Suppose that coA ⊂ co_{1}A ⊂ A is such that A_{1} = (A, co_{1}A, wA) is also a Waldhausen category. Then K(A) ≃ K(A_{1}), by Waldhausen Approximation.

2.6. Combining Proposition 2.5 with the Gillet-Waldhausen Theorem 2.2 yields several useful localization sequences.

G-theory localization for rings 2.6.1. Localization at a central multiplicatively closed set S in a ring R induces an exact functor M(R) → M(S^{−1}R) satisfying (App). Passing to Ch^{b}M(R) → Ch^{b}M(S^{−1}R) does not change the Ktheory (by 2.2) but does add a cylinder functor, so (App) still holds (see II, Ex. 9.7).

Hence Proposition 2.5 applies, with A^{w} being the category Ch_{S}^{b}M(R) of complexes E such that S^{−1}E is exact.

We define G(R on S) to be KCh_{S}^{b}M(R), and G_{n}(R on S) = K_{n}Ch_{S}^{b}M(R), so that we get a homotopy fibration G(R on S) → G(R) → G(S^{−1}R), and a long exact sequence

⋯ → G_{n+1}(S^{−1}R) → G_{n}(R on S) → G_{n}(R) → G_{n}(S^{−1}R) → ⋯

ending in the surjection G_{0}(R) → G_{0}(S^{−1}R) of II.6.4.1. When R is noetherian, we will identify G(R on S) = KCh_{S}^{b}M(R) with KM_{S}(R) in 6.1 below.

G-theory localization for schemes 2.6.2. If Z is a closed subscheme of a noetherian scheme X, we define G(X on Z) to be KCh_{Z}^{b}M(X), where Ch_{Z}^{b}M(X) is the (Waldhausen) category of bounded complexes which are acyclic on X − Z.

Now G(X) = KM(X) by IV.6.3.4, and the localization M(X) → M(X − Z) satisfies (App). Since A^{w} = Ch_{Z}^{b}M(X), Proposition 2.5 and Theorem 2.2 yield a homotopy fibration G(X on Z) → G(X) → G(X − Z) and a long exact sequence

⋯ → G_{n+1}(X − Z) → G_{n}(X on Z) → G_{n}(X) → G_{n}(X − Z) → ⋯

ending in the surjection G_{0}(X) → G_{0}(X − Z) of II.6.4.2. Later on (in 3.10.2, 6.11 and Ex. 4.3), we will identify G(X on Z) with KM_{Z}(X) and G(Z).

Let S be a central multiplicatively closed set of central elements in a noetherian ring R. If S consists of nonzerodivisors, we will see in Theorem 7.1 that the analogue of KM_{S}(R) for projective modules is the K-theory of the category H_{S}(R) of Storsion perfect modules (generalizing II.7.7.4). Otherwise, this is not correct; see Exercises 2.9 and 7.3 below. Instead, as in II.9.8, we define K(R on S) to be the Ktheory of Ch_{S}^{b}P(R), the Waldhausen category of bounded complexes P of finitely generated projective modules such that S^{−1}P is exact (II.9.7.5).

Theorem 2.6.3. If S is a central multiplicatively closed set in a ring R, there is a homotopy fibration K(R on S) → K(R) → K(S^{−1}R), and hence a long exact sequence

⋯ K_{n+1}(S^{−1}R) → K_{n}(R on S) → K_{n}(R) → K_{n}(S^{−1}R) ⋯

ending in the exact sequence K_{0}(R on S) → K_{0}(R) → K_{0}(S^{−1}R) of II.9.8.

Proof. As in the proof of II.9.8, we consider the category P of S^{−1}R-modules of the form S^{−1}P for P in P(R). We saw in II.9.8.1 that (by clearing denominators in the maps), the localization from A = Ch^{b}(P(R)) to B = Ch^{b}(P) satisfies (App), so Proposition 2.5 applies with A^{w} = Ch_{S}^{b}P(R). Thus we have a homotopy fibration KCh_{S}^{b}P(R) → K(R) → K(P). By Cofinality (IV.6.4.1), K(P) → K(S^{−1}R) → G is a homotopy fibration, and the result follows. □

We conclude with a few useful models for K-theory, arising from the Waldhausen Approximation Theorem 2.4.

Homologically bounded complexes 2.7.1. If A is an abelian category, let Ch^{hb}(A) denote the Waldhausen category of homologically bounded chain complexes of objects in A, and Ch_{±}^{hb}(A) the subcategory of bounded below (resp., bounded above) complexes. We saw in II.9.7.4 that Ch^{b}(A) ⊂ Ch_{−}^{hb}(A) and Ch_{+}^{hb}(A)⊂Ch^{hb}(A) satisfy (App), by good truncation; dually, Ch^{b}(A) ⊂ Ch_{+}^{hb}(A) and Ch_{+}^{hb}(A) ⊂ Ch^{hb}(A) satisfy the dual of (App). By Waldhausen Approximation (and 2.2), this yields

K(A) ≃ KCh^{b}(A) ≃ KCh_{−}^{hb}(A) ≃ KCh_{+}^{hb}(A) ≃ KCh^{hb}(A).

The K_{0} version of the resulting isomorphism K_{n}(A) ≅ K_{n}Ch^{b}(A) ≅ K_{n}Ch^{hb}(A) was given in II.9.7.4. We will see another argument for this in 3.8.1 below.

Perfect complexes 2.7.2. A perfect complex of R-modules is a complex M which is quasi-isomorphic to a bounded complex of finitely generated projective Rmodules, i.e., to a complex in Ch^{b}(P(R)). We saw in II.9.7.5 that the perfect complexes of R-modules form a Waldhausen subcategory Ch_{perf}(R) of Ch(mod-R), and that (App) holds for the inclusions Ch^{b}(P(R)) ⊂ Ch_{perf}^{−}(R) ⊂ Ch_{perf}(R). Thus (invoking Theorems 2.2 and 2.4) we have that

K(R) ≃ KCh^{b}(P(R)) ≃ KCh_{perf}^{−}(R) ≃ KCh_{perf}(R).

If S is a central multiplicatively closed set in R, then K(R on S) = KCh_{S}^{b}P(R) is also the K-theory of the category Ch_{perf,S}(R) of perfect complexes P with S^{−1}P exact. This follows from Waldhausen Approximation; the Approximation Property for the inclusion Ch_{S}^{b}P(R) ⊂ Ch_{perf,S}(R) was established in II, Ex. 9.2.

K-theory of schemes 2.7.3. If X is any scheme, we define K(X) to be KCh_{perf}(X). Thus our K_{0}(X) is the group K_{0}^{der}(X) of II, Ex. 9.10.

When X is a quasi-projective scheme over a commutative ring, we defined K(X) = KVB(X) in IV.6.3.4. These definitions agree; in fact they also agree when X is a separated regular noetherian scheme (II.8.2), or more generally a (quasicompact, quasi-separated) scheme such that every coherent sheaf is a quotient of a vector bundle. Indeed, KVB(X) = KCh_{perf}(X), by Waldhausen Approximation applied to Ch^{b}(VB(X)) ⊂ Ch_{perf}(X). The condition (App) is given in [SGA6, II] or [TT, 2.3.1]: given a map P → C from a bounded complex of vector bundles to a perfect complex, there is a factorization P → Q ⟶^{∼} C in these settings.

G(R) and Pseudo-coherent complexes 2.7.4. If R is a noetherian ring, the discussion of 2.7.1 applies to the abelian category M(R) of finitely generated R-modules. Thus we have:

G(R) = KM(R) ≃ KCh^{b}M(R) ≃ KCh_{+}^{hb}M(R) ≃ KCh^{hb}M(R).

Instead of Ch_{+}^{hb}M(R), we could consider the (Waldhausen) category Ch_{+}^{hb}P(R) of bounded below, homologically bounded chain complexes of finitely generated projective modules, or even the category Ch_{pcoh}^{hb}(R) of homologically bounded pseudocoherent complexes (R-module complexes which are quasi-isomorphic to a bounded complex of finitely generated modules; see II.9.7.6). By II, Ex. 9.7, Waldhausen Approximation applies to the inclusions of M(R) and Ch_{+}^{hb}P(R) in Ch_{pcoh}^{hb}(R). Hence we also have G(R) ≃ KCh_{+}^{hb}P(R) ≃ KCh_{pcoh}^{hb}(R).

If R is not noetherian, we can consider the exact category M(R) of pseudocoherent modules (II.7.1.4), which we saw is closed under kernels of surjections, and the Waldhausen category Ch_{pcoh}^{hb}(R) of homologically bounded pseudo-coherent complexes (II.9.7.6). Since (App) holds by Ex. II.9.7, the same proof gives:

G(R) = KM(R) ≃ KCh^{b}M(R) ≃ KCh_{+}^{hb}P(R) ≃ KCh_{pcoh}^{hb}(R).

Now suppose that S is a multiplicatively closed set of central elements in R. Anticipating Theorem 5.1 below, we consider the category M_{S}(R) of S-torsion modules in M(R). If R is noetherian, this is an abelian category by II.6.2.8; if not, it is the exact category of pseudo-coherent S-torsion modules (II.7.1.4). By Theorem 2.2, KM_{S}(R) ≃ KCh^{b}M_{S}(R).

EXERCISES

2.1 If an exact functor F : A → B satisfies the approximate lifting property (App), show (by induction on n) that each S_{n}F : S_{n}A → S_{n}B also satisfies (App). 2.2 If A is a strictly cofinal exact subcategory of A′, show that Ch^{b}(A) ⊂ Ch^{b}(A′) satisfies (App), and that KCh^{b}(A) ≃ KCh^{b}(A′). 2.3 Let splitCh^{b}(A) denote the category Ch^{b}(A), made into a Waldhausen category by restricting the cofibrations to be the degreewise split monomorphisms whose quotients lie in A (a priori they lie in Â; see II.7.3). Generalize II.9.2.4 by showing that splitCh^{b}(A) → Ch^{b}(A) induces a homotopy equivalence on K-theory, so that K_{n}(splitCh^{b}(A)) ≅ K_{n}(A) for all n. 2.4 If A is an exact subcategory of an abelian category M, the Waldhausen category A_{exact}^{[0,n]} of admissibly exact complexes of length n (Example 1.4) is contained in the category Ch^{[0,n]}(A)^{qiso} of complexes in A which are acyclic as complexes in M. If A is closed under kernels of surjections in M, show that these categories are the same. 2.5 Consider the exact category F(R) of finite free R-modules (II.5.4.1). Analyze Remark 2.2.1 to show that KF(R) ≃ KCh^{b}(F(R)). If S is a central multiplicative set in R, compare KCh_{S}^{b}(F(R)) to K(R on S). Is Ch_{S}^{b}(F(R)) cofinal in Ch_{S}^{b}(P(R))? 2.6 Let S be a central multiplicative set in a ring R. Mimick the proof of 2.7.4 to show that KCh_{S}^{b}M(R) is equivalent to KCh_{pcoh,S}^{+}(R) and KCh_{pcoh,S}^{hb}(R). A fancier proof of this equivalence will be given in 3.10.1 below. 2.7 Let X be a noetherian scheme. Show that G(X) ≃ KCh_{pcoh}^{hb}(X), generalizing II, Ex. 9.8. Hint: Mimick the proof of II, Ex. 9.7. A fancier proof of this equivalence will be given in 3.10.2 below. 2.8 Let X be a noetherian scheme, and F the Waldhausen category of bounded above perfect cochain complexes of flat O_{X}-modules. Show that VB(X) ⊂ F induces an equivalence K(X) ≃ K(F). 2.9 Let R = k[s, t]/(st) and S = {s^{n}}, where k is a field, so that S^{−1}R = k[s, 1/s]. Show that every R-module M with s^{n}M = 0 for some n has infinite projective dimension, so that the category H_{S}(R) consists only of 0. Then use the Mayer-Vietoris sequence III.2.6 to show that K_{0}(R on S) = ℤ. Conclude that KH_{S}(R) is not the homotopy fiber of K(R) → K(S^{−1}R).

§3. The Resolution Theorems and transfer maps

In this section we establish the Resolution Theorems for exact categories (3.1) and Waldhausen categories of chain complexes (3.9). We first give the version for exact categories, and some of its important applications. The second Resolution Theorem 3.9 requires the properties of derived categories which are listed in 3.8.

The Fundamental Theorem 3.3–4 that K_{∗} ≅ G_{∗} for regular rings and schemes (proven for K_{0} in II, 7.8 and 8.2), and the existence of transfer maps f_{∗} (3.3.2, 3.5 and 3.7), are immediate consequences of the first Resolution Theorem, as applied to P(R) ⊆ H(R) and VB(X) ⊆ H(X).

Recall from II.7.0.1 that P is said to be closed under kernels of admissible surjections in an exact category H if whenever A ↣ B ↠ C in H is an exact sequence with B, C in P then A is also in P. (A prototype is P = H_{n}(R), H ⊆ mod-R.)

Resolution Theorem 3.1. Let P be a full exact subcategory of an exact category H, such that P is closed under extensions and under kernels of admissible surjections in H. Suppose in addition that every object M of H has a finite Presolution:

0 → P_{n} → ⋯ → P_{1} → P_{0} → M → 0.

Then K(P) ≃ K(H), and thus K_{i}(P) ≅ K_{i}(H) for all i.

The proof will reduce the theorem to the special case in which objects of H have a P-resolution of length one, which will be handled in Proposition 3.1.1.

Proof. The category H is the union of the subcategories H_{n} of objects with resolutions of length at most n, and H_{0} = P. Since the kernel of any admissible P ↠ P′ is also in P, H_{n−1} ⊆ H_{n} is closed under admissible subobjects and extensions (see Ex. 3.1). Applying 3.1.1, we see that each K(P) → K(H_{n−1}) → K(H_{n}) is a homotopy equivalence. Taking the colimit over n yields the result. □

Proposition 3.1.1. Let P ⊂ H be as in Theorem 3.1, and suppose that every M in H fits into an exact sequence 0 → P_{1} → P_{0} → M → 0 with the P_{i} in P. Then K(P) → K(H) is a homotopy equivalence, and K_{∗}(P) ≅ K_{∗}(H).

Proof. (Quillen) The inclusion QP ⊂ QH is not full, so we consider the full subcategory Q on the objects of QP, and write i for the inclusion QP ⊂ Q. For each P in QP, the objects of the comma category i/P are pairs (P_{2}, u) with u of the form P_{2} ↞ P_{1} ↣ P and P/P_{1} in H. Set z(P_{2}, u) = (P_{1}, P_{1} ↣ P) and note that P_{2} ↞ P_{1} and 0 ↣ P_{1} are morphisms of QP. They define natural transformations (P_{2}, u) → z(P_{2}, u) ← (0, 0 ↣ P) in i/P. This shows that i/P is contractible, and hence by Theorem A (IV.3.6) that i : QP → Q is a homotopy equivalence.

It now suffices to show that the inclusion j : Q → QH is a homotopy equivalence. We shall resort to the dual of Theorem A, so we need to show that for each M in H, the comma category M\j is contractible. The objects of M\j are pairs (P, u : M ↞ P_{1} ↣ P) with P in P; it is nonempty by the assumption that some P_{0} ↠ M exists. Let C denote the full subcategory on the pairs (P, M ↞ P). The inclusion C ⊂ (M\j) is a homotopy equivalence because it has a right adjoint, namely r(P, u) = (P_{1}, M ↞ P_{1}). And the category C is contractible because if we fix any (P_{0}, u_{0} : M ↞ P_{0}), then p(P, u) = (P ×_{M} P_{0}, M ↞ P ×_{M} P_{0}) is in C (because P is subobject-closed) and there are natural transformations (P, u) ← p(P, u) → (P_{0}, u_{0}). □

Remark 3.1.2. It is not known how to generalize the Resolution Theorem to Waldhausen categories. Other proofs of the Resolution Theorem for exact categories, using Waldhausen K-theory, have been given in [Gra87] and [Staf].

Here is the main application of the Resolution Theorem. It is just the special case in which P = P(R) and H = H(R).

Theorem 3.2. For every ring R, the inclusion of P(R) in H(R) induces an equivalence K(R) = KP(R) ≃ KH(R), so K_{∗}(R) = K_{∗}P(R) ≅ K_{∗}H(R).

If S is a multiplicatively closed set of central nonzero-divisors of R, we introduced the categories H_{S}(R) and H_{1,S}(R) in II.7.7.3. The proof there using Resolution applies verbatim to yield:

Corollary 3.2.1. KH_{1,S}(R) ≃ KH_{S}(R), and K_{∗}H_{1,S}(R) ≅ K_{∗}H_{S}(R).

By definition, a ring R is regular if every R-module has a finite projective resolution, i.e., finite projective dimension (see I.3.8.). We say R is coherent if the category M(R) of pseudo-coherent R-modules (II.7.1.4) is abelian.

Theorem 3.3. If R is a noetherian (or coherent) regular ring, K(R) ≃ G(R). Thus for every n we have K_{n}(R) ≅ G_{n}(R).

Proof. In either case, H(R) is the category M(R). The Resolution Theorem 3.2 gives the identification. □

Corollary 3.3.1. If f : R → S is a homomorphism, R is regular noetherian and S is finite as an R-module, then there is a transfer map f_{∗} : K_{∗}(S) → K_{∗}(R), defined by the G-theory transfer map (IV.6.3.3):

K(S) → G(S) = KM(S) ⟶^{f_{∗}} KM(R) = G(R) ≃ K(R).

Transfer Maps for K_{∗}(R) 3.3.2. Let f: R → S be a ring homomorphism such that S has a finite R-module resolution by finitely generated projective Rmodules. Then the restriction of scalars defines a functor P(S) → H(R). By 3.2, we obtain a transfer map f_{∗}: K(S) → KH(R) ≃ K(R), and hence maps f_{∗} : K_{n}(S) → K_{n}(R). If S is projective as an R-module, f_{∗} is the transfer map of IV.6.3.2.

The projection formula states that f_{∗} is a K_{∗}(R)-module homomorphism when R is commutative. That is, if x ∈ K_{∗}(S) and y ∈ K_{∗}(R) then f_{∗}(x · f^{∗}(y)) = f_{∗}(x) · y in K_{∗}(R). To see this, we note that the biexact functor H(S) × P(R) → H(R), (M, P) ↦ M ⊗_{R} P, produces a pairing KH(S) ∧ K(R) → KH(R) representing the right side via Theorem 3.2. Since M ⊗_{R} P ≅ M ⊗_{S} (S ⊗_{R} P), it is naturally homotopic to the pairing representing the left side.

We have already seen special cases of the transfer map f_{∗}. It was defined for K_{0} in II.7.9, and for K_{1} in a special case in III, Ex. 1.11. If S is projective as an R-module then f_{∗} was also constructed for K_{2} in III.5.6.3, and for all K_{n} in IV.1.1.3.

Recall from II.8.2 that a separated noetherian scheme X is regular if every coherent O_{X}-module F has a finite resolution by vector bundles; see [SGA6, II, 2.2.3 and 2.2.7.1] or II.8.2–3. The Resolution Theorem applies to VB(X) ⊂ M(X), and we have:

Theorem 3.4. If X is a separated regular noetherian scheme, then K(X) = KVB(X) satisfies:

K(X) ≃ G(X) and K_{∗}(X) ≅ G_{∗}(X).

Variant 3.4.1. If X is quasi-projective (over a commutative ring), we defined K(X) to be KVB(X) in IV.6.3.3. We saw in II.8.3.1 that the Resolution Theorem applies to VB(X) ⊂ H(X) so we have K(X) ≃ KH(X).

Remark 3.4.2. Theorem 3.4 does not hold for non-separated regular noetherian schemes. This is illustrated when X is the affine line with a double origin over a field, since (as we saw in II, 8.2.4 and Ex. 9.10) G_{0}(X) = ℤ ⊕ ℤ but K_{0}VB(X) = ℤ. The analogue of Theorem 3.4 for quasi-compact regular schemes is given in Exercise 3.9.

Base change maps for G_{∗}(R) 3.5. Let f : R → S be a homomorphism of noetherian rings such that S has finite flat dimension fd_{R} S as a right R-module. Let F ⊂ M(R) be the full subcategory of all R-modules M which are Tor-independent of S in the sense that

Tor_{n}^{R}(S, M) = 0 for n ≠ 0.

As observed in II.7.9, the usual properties of Tor show that every finitely generated R-module M has a finite resolution by objects of F, that F is an exact subcategory closed under kernels, and that M ↦ M ⊗_{R} S is an exact functor from F to M(S). By the Resolution Theorem 3.1, there is a natural map

G(R) ⟵^{≃} K(F) → G(S),

giving maps f^{∗}: G_{∗}(R) → G_{∗}(S). If g : S → T is another map, and T has finite flat dimension over S, then the natural isomorphism (M ⊗_{R} S) ⊗_{S} T ≅ M ⊗_{R} T shows that g^{∗}f^{∗} ≃ (gf)^{∗}.

Note that if the ring S is finite over R then the forgetful functor M(S) → M(R) is exact and induces a contravariant “finite transfer” map f_{∗} : G(S) → G(R) (see II.6.2 and IV.6.3.3). The seemingly strange notation (f^{∗} and f_{∗}) is chosen with an eye towards schemes: if Y = Spec(R) and X = Spec(S) then f maps X to Y, so f^{∗} is contravariant and f_{∗} is covariant as functors on schemes.

Example 3.5.1. Let i : R → R[s] be the inclusion and f : R[s] → R the map f(s) = 0. Since i is flat, we have the flat base change i^{∗}: G(R) → G(R[s]). Now f has finite flat dimension, since Tor_{n}^{R[s]}(R, −) = 0 for n ≥ 2, so we also have a base change map f^{∗}: G(R[s]) → G(R) by 3.5. Since fi is the identity on R, the composite f^{∗}i^{∗} : G(R) → G(R[s]) → G(R) is homotopic to the identity map. The Fundamental Theorem for G(R) (6.2 below) will show that these are inverse homotopy equivalences.

In contrast, the transfer maps K(R) ⟶^{f_{∗}} K(R[s]) and G(R) ⟶^{f_{∗}} G(R[s]) are zero. This follows from the Additivity Theorem applied to the sequence of functors i^{∗} ↣ i^{∗} ↠ f_{∗} sending an R-module M to

0 → M[s] ⟶^{s} M[s] ⟶ M → 0.

Example 3.5.2. Suppose that S = R⊕S_{1}⊕S_{2}⊕⋯ is a graded noetherian ring, and let M_{gr}(S) be the abelian category of finitely generated graded S-modules. Its K-groups are naturally modules over ℤ[σ, σ^{−1}], where σ acts by the shift automorphism σ(M) = M(−1) of graded modules. (See Exercises II.6.12 and II.7.14.)

Now assume that S is flat over R, so that tensoring with S gives a functor from M(R) to M_{gr}(S), and hence a ℤ[σ, σ^{−1}]-module map

β: G_{i}(R) ⊗ ℤ[σ, σ^{−1}] → K_{i}M_{gr}(S).

In the special case S = R, M_{gr}(R) is just a coproduct of copies of M(R), and the map β is an isomorphism: G_{∗}(R)[σ, σ^{−1}] ≅ K_{∗}M_{gr}(R). If R has finite flat dimension over S (via S → R) then the Resolution Theorem 3.1, applied to the category P_{gr} of graded S-modules Tor-independent of R, induces a map

K_{i}M_{gr}(S) → K_{i}M_{gr}(R) ≅ G_{i}(R) ⊗ ℤ[σ, σ^{−1}]

which is a left inverse to β, because ⊗_{R} sends M_{gr}(R) to P_{gr} and there is a natural isomorphism (M ⊗_{R} S) ⊗_{S} R ≅ M. In fact, β is an isomorphism (see Ex. 3.3).

Similarly, if M_{gr,≥0}(S) is the subcategory of positively graded S-modules, there is a natural map β : G_{i}(R) ⊗ ℤ[σ] → K_{i}M_{gr,≥0}(S). If R has finite flat dimension over S, it is an isomorphism (see Ex. 3.3).

Example 3.5.3. (Projection Formula) Let f : R → S be a homomorphism of commutative noetherian rings such that S is a finitely generated right R-module of finite projective dimension. Then G_{∗}(S) and G_{∗}(R) are K_{∗}(R)-modules by IV.6.6.5. The projection formula states that f_{∗}(x · f^{∗}y) = f_{∗}(x) · y in G_{∗}(R), provided that either (i) x ∈ G_{∗}(S) and y ∈ K_{∗}(R) or (ii) x ∈ K_{∗}(S) and y ∈ G_{∗}(R).

For (i), observe that the functor ⊗_{R} : M(S)×P(R) → M(S) → M(R) is biexact, so it induces a pairing G(S) ∧ K(R) → G(S) → G(R) representing the right side f_{∗}(x) · y. Since M ⊗_{R} P ≅ M ⊗_{S} (S ⊗_{R} P), it also represents the left side.

For (ii), let F ⊂ M(R) be as in 3.5 and observe that the functor P(S) × F → M(S) → M(R), (P, M) ↦ P ⊗_{R} M is biexact. Hence it produces a pairing K(S) ∧ K(F) → G(S) → G(R), representing the left side f_{∗}(x · f^{∗}y) of the projection formula. But this pairing also factors through P(S) → H(R) followed by the tensor product pairing H(R) × F → M(R) representing the right side.

Base change maps for G(X) 3.6. If f : X → Y is a morphism of noetherian schemes such that O_{X} has finite flat dimension over f^{−1}O_{Y}, there is also a contravariant map f^{∗} from G(Y) to G(X). This is because every coherent O_{Y} -module has a finite resolution by coherent modules which are Tor-independent of f_{∗}O_{X}, locally on X, and f^{∗} is an exact functor on the category L(f) of these modules. If g : W → X is another map of finite flat dimension, then g^{∗}f^{∗} ≃ (fg)^{∗} by the natural isomorphism g^{∗}(f^{∗}F) ≅ (fg)^{∗}F.

Example 3.6.1. If X is a noetherian scheme we can consider the flat structure map p : X[s] → X and the zero-section f : X → X[s], where X[s] = X ×Spec(ℤ[s]) as in II.6.5.1. As in 3.5.1, f has finite flat dimension and pf is the identity on X, so f^{∗} : G(X[s]) → G(X) is defined and the composition f^{∗}p^{∗} is homotopic to the identity on G(X). The Fundamental Theorem 6.13 will show that p^{∗} : G(X) ≃ G(X[s]) is a homotopy equivalence.

In contrast the finite transfer map f_{∗} : G(X) → G(X[s]) is zero. This follows from the Additivity Theorem applied to the sequence of functors p^{∗} ↣ p^{∗} ↠ f_{∗} from M(X) to M(X[s]), analogous to the one in 3.5.1.

Proposition 3.7. If f: X → Y is a proper morphism of noetherian schemes, there is a “proper transfer” map f_{∗}: G(X) → G(Y). This induces homomorphisms f_{∗} : G_{n}(X) → G_{n}(Y) for each n. The transfer map makes G_{n}(X) functorial for proper maps.

The G_{0} version of Proposition 3.7, f_{∗}([F]) = ∑(−1)^{i}[R^{i}f_{∗}(F)], is given in II.6.2.6.

Proof. By Serre’s “Theorem B” (see II.6.2.6), the higher direct images R^{i}f_{∗}(F) of a coherent module are coherent, and vanish for i large. They are obtained by replacing F by a flasque resolution, applying f_{∗} and taking cohomology. The map f_{∗} : KM(X) → KM(Y) exists by Ex. 3.2. □

Proposition 3.7.1. If f: X → Y is a proper morphism of finite flat dimension. Then there is a “transfer” map f_{∗}: K(X) → K(Y). This induces homomorphisms f_{∗} : K_{n}(X) → K_{n}(Y) for each n. The transfer map makes K_{n}(X) functorial for projective maps.

Proof. As in II.8.4, let P(f) be the category of vector bundles E on X such that R^{i}f_{∗}(E) = 0 for i > 0. We saw in loc. cit. that f_{∗} : P(f) → H(Y) is an exact functor, i.e., that the O_{Y} -module f_{∗}(E) is perfect (II, Ex. 9.10). By Ex. 3.6(c) and left exactness of f_{∗}, the hypotheses of the Resolution Theorem 3.1 are satisfied, so we have K(X) ≃ KP(f). Thus we can define the transfer map to be the composite

K(X) ≃ KP(f) ⟶^{f_{∗}} KH(Y) ≃ KVB(Y) = K(Y).

Given a second map g : Y → Z of finite flat dimension, we can replace P(f) by P = P(f × gf), so that E in P satisfy R^{i}g_{∗}(f_{∗}(E)) = R^{i}(gf)_{∗}(E) = 0 for i > 0. Thus f_{∗}(P) lies in the subcategory H(g) of perfect g_{∗}-acyclic modules, on which g_{∗} is exact, and g_{∗}f_{∗} ≃ (gf)_{∗} because of the natural isomorphism (gf)_{∗}(E) ≅ g_{∗}f_{∗}(E) for E in P. Functoriality is now straightforward (Ex. 3.5). □

Base change Theorem 3.7.2. Let f : X → Y be a proper morphism of quasi-projective schemes and g : Y′ → Y a morphism of finite flat dimension, Tor-independent of X, and set X′ = X ×_{Y} Y′ so there is a cartesian square

\begin{matrix}X′ ⟶^{g′} X \\ f′ ↓ ↓ f \\ Y′ ⟶^{g} Y. \end{matrix}

Then g^{∗}f_{∗} ≃ f′_{∗}g′^{∗} as maps G(X) → G(Y′).

If in addition f has finite flat dimension, so that f_{∗} : K(X) → K(Y) is defined then g^{∗}f_{∗} ≃ f′_{∗}g′^{∗} as maps K(X) → K(Y′).

The idea of the proof is to use the following base change formula of [SGA6, IV.3.1.0]: if E is homologically bounded with quasi-coherent cohomology, then

Lg^{∗}(Rf_{∗}E) ⟶^{∼} Rf′_{∗}L(g′)^{∗}E.

Proof. Let A be the category of O_{X}-modules which are f_{∗}-acyclic and Torindependent of O_{X′}. For E in A, the base change formula implies that for all i ∈ ℤ:

Tor_{i}(f_{∗}E, O_{Y′}) = L_{i}g^{∗}(f_{∗}E) = R^{−i}f′_{∗}(g′^{∗}E).

These groups must vanish unless i = 0, because Tor_{i} and R^{i}f′_{∗} vanish for i < 0. That is, f_{∗}(E) is Tor-independent of O_{Y′}, and (g′)^{∗}E is f′_{∗}-acyclic, and we have g^{∗}f_{∗}(E) ≅ f′_{∗}(g′)^{∗}E. Therefore g^{∗}f_{∗} = f′_{∗}(g′)^{∗} as exact functors on A. It remains to apply the Resolution Theorem twice to show that A ⊂ L(f) ⊂ M(Y) induce equivalences on K-theory. The second was observed in 3.6, and the first follows from Ex. 3.6(e).

The proof is easier for K(X) → K(Y′) when f has finite flat dimension, using the category P(f); f_{∗} : P(f) → H(Y) and (g′)^{∗} : P(f) → VB(X′) are exact, and we saw in the proof of 3.7.1 that KP(f) ≃ K(X). □

Corollary 3.7.3. (Projection formula) If f : X → Y is a projective map of finite flat dimension, then for x ∈ K_{0}(X) and y ∈ G_{n}(Y) we have f_{∗}(x · f^{∗}y) = f_{∗}(x) · y in G_{n}(Y).

The G_{0} version of this projection formula was given in Ex. II.8.3. (Cf. Ex. 3.10.) We will generalize the projection formula to higher K-theory in 3.12 below.

Proof. As in the proof of 3.7.1, let P(f) be the category of f_{∗}-acyclic vector bundles E on X. By Ex. 3.6(c) and left exactness of f_{∗}, the hypotheses of the Resolution Theorem 3.1 are satisfied and we have K_{∗}(X) ≅ K_{∗}P(f). Thus it suffices to show that the projection formula holds when x = [E] for E in P(f). Let L_{E} denote the full subcategory of M(Y) consisting of modules which are Torindependent of f_{∗}E and O_{X}. By the Resolution Theorem, K(L_{E}) ≃ G(Y). The functor L_{E} → M(Y) given by F ↦ f_{∗}E ⊗ F is exact, and induces y ↦ f_{∗}(x) · y.

Similarly, the exact functors L_{E} → M(X), sending F to f^{∗}(F) and E ⊗ f^{∗}(F), induce y ↦ f^{∗}(y) and y ↦ x·f^{∗}(y), respectively. The projection formula of [SGA 6, III.3.7] shows that R^{i}f_{∗}(E ⊗f^{∗}F) = 0 for i > 0 and that f_{∗}(E ⊗f^{∗}F) ≅ f_{∗}(E)⊗F. Hence F ↦ f_{∗}(E ⊗ f^{∗}F) is an exact functor L_{E} → M(Y), and the projection formula follows. □

Derived Approximation

The third fundamental result for Waldhausen categories is an Approximation Theorem for the K-theory of categories based upon chain complexes, and is proven using Waldhausen Approximation 2.4. Roughly speaking, it says that the K-theory of C only depends on the derived category of C, defined as localization w^{−1}C of C at the set w of quasi-isomorphisms in C.

In order for the statement of this result to make more sense, C will be a subcategory of Ch(M) for some abelian category M; recall that the derived category D(M) is the localization of the category Ch(M) at the family of quasiisomorphisms. For basic facts about derived categories and triangulated categories, we refer the reader to the Appendix to chapter II, and to the standard references [Verd], [H20] and [WHomo, 10].

Triangulated and Localizing Categories 3.8. Let C be a full additive subcategory of Ch(M) which is closed under all the shift operators C ↦ C[n] and mapping cones. The localization w^{−1}C of C is the category obtained from C by formally inverting the multiplicatively closed set w = w(C) of all quasi-isomorphisms in C. (The usual construction, detailed in (II.A.5), uses a calculus of fractions to compose maps.) It is a triangulated category by [WHomo, 10.2.5].

Here are two key observations that make it possible for us to better understand these triangulated categories, and to even see that they exist. One is that chain homotopic maps in C are identified in w^{−1}C; see II, Ex. A.5 or [WHomo, 10.1.2]. Another is that if w is saturated then C is isomorphic to zero in w^{−1}C if and only if 0 → C is in w; see II.A.3.2 or [WHomo, 10.3.10].

We say that C is a localizing subcategory of Ch(M) if the natural map w^{−1}C → w^{−1}Ch(M) = D(M) is an embedding. This will be the case whenever the following condition holds: given any quasi-isomorphism C → B with C in C, there is a quasiisomorphism B → C′ with C′ in C. (See II.A.3 or [WHomo, 10.3.13].)

For example, if B is a Serre subcategory of M, the category Ch_{B}(M) of complexes in Ch(M) with homology in B is localizing ([WHomo, 10.4.3]), so D_{B}(M) = w^{−1}Ch_{B}(M) is a subcategory of D(M). The functor D(B) → D_{B}(A) need not be an equivalence; see [WHomo, Ex. 10.4.3].

Example 3.8.1. If B ⊂ C ⊆ Ch(M), the derived categories w^{−1}B and w^{−1}C are equivalent if for every complex C in C there is a quasi-isomorphism B ⟶^{∼} C with B in B. (See [TT, 1.9.7].)

For example, if A is an abelian category, the inclusion Ch^{b}(A) → Ch^{hb}(A) induces an equivalence of derived categories, where Ch^{hb}(A) is the category of homologically bounded complexes (II.9.7.4), because (as we saw in 2.7.1) every homologically bounded complex is quasi-isomorphic to a bounded complex.

If R is any ring, Ch^{b}M(R) has the same derived category as Ch_{pcoh}^{hb}(R) the homologically bounded pseudo-coherent complexes; see [WHomo, Ex. 10.4.6]. Similarly, the inclusions Ch_{perf}^{b}(R) ⊂ Ch_{perf}^{+}(R) ⊂ Ch_{perf}(R) induce equivalence on derived categories.

Homotopy Commutative Diagrams 3.8.2. As pointed out in II.A.4, the chain homotopy category of C satisfies a calculus of fractions, allowing us to perform constructions in w^{−1}C. Here is a typical example: any homotopy commutative diagram in C of the form

A′ → B_{1} ← B′
↘ ↓ f ↙
B

can be made into a commutative diagram by replacing B_{1} by the homotopy pullback of B_{1} ⟶^{f} B ⟵^{=} B, which is defined as the shifted mapping cone of B → cone(f), and is quasi-isomorphic to B_{1}.

We are now ready for the Waldhausen version of the Resolution Theorem, which is due to Thomason and Trobaugh [TT]. Fix an ambient abelian category M and consider the Waldhausen category Ch(M) of all chain complexes over M.

Thomason-Trobaugh Resolution Theorem 3.9. Let A ⊂ B be saturated Waldhausen subcategories of Ch(M), closed under mapping cones and all shift maps A ↦ A[n]. If w^{−1}A ⟶^{≃} w^{−1}B (i.e., the derived categories of A and B are equivalent), then KA ⟶^{≃} KB is a homotopy equivalence.

Remark 3.9.1. A map in A is a weak equivalence in A if and only if it is a weak equivalence in B. This is because, by saturation, both conditions are equivalent to the mapping cone being zero in the (common) derived category.

Proof. (Thomason-Trobaugh) Let A^{+} be the comma category whose objects are weak equivalences w : A ⟶^{∼} B with A in A and B in B; morphisms in A^{+} are commutative diagrams in B. It is a Waldhausen category in a way that makes A → A^{+} → B into exact functors; a morphism (A ⟶^{∼} B) → (A′ ⟶^{∼} B′) is a cofibration (resp., weak equivalence) if both its component maps A → A′ and B → B′ are. The forgetful functor A^{+} → A is right adjoint to the inclusion A → A^{+}, and exact, so K(A) ≃ K(A^{+}).

We will show that the Approximation Theorem 2.4 (or, rather, its dual) applies to the exact functor A^{+} → B sending A ⟶^{∼} B to B. This will imply that K(A^{+}) ≃ K(B), proving the theorem.

Condition 2.4(a) is satisfied, because given a map (A ⟶^{∼} B) → (A′ ⟶^{∼} B′) in A^{+} and a weak equivalence B ⟶^{∼} B′, the map A → A′ is a weak equivalence in B by the saturation axiom, and hence is in wA by 3.9.1. Condition 2.4(b) holds, because the cylinder functor on B induces one on A^{+}. Thus it suffices to check that the dual (App)^{op} of the approximation property holds for A^{+} → B; this will be a consequence of the hypothesis that A and B have the same derived category.

Using the Gabriel-Zisman Theorem (II.A.3), the approximation property (App)^{op} states that given B′ ⟶^{b} B ⟵^{∼} A with A in A, there is a commutative diagram

(3.9.2)  \begin{aligned}A′′ & ⟶ & A \\ & & ^{∼}↓ & ^{∼}↓ \\ B′ & ⟶ & B′′ & ⟶ & B \end{aligned}

with A′′ in A, such that the bottom composite is b.

By assumption, B′ is quasi-isomorphic to an object A_{1} of A; by calculus of fractions this is represented by a chain A_{1} ⟶^{∼} B_{1} ⟵^{∼} B′. Composing with B′ → B ⟵^{∼} A yields a map from A_{1} to A in the derived category, which must be represented by a chain A_{1} → A_{2} ⟵^{∼} A with A_{2} in A. Composing B_{1} ⟵^{∼} A_{1} → A → B (and using 3.8.2) yields a commutative diagram

(3.9.3)  \begin{aligned}A_{1} & ⟶ & A_{2} & ⟵^{∼} & A \\ & & ^{∼}↓ & & ^{∼}↓ & ^{∼}↓ \\ B′ & ⟶ & B_{1} & ⟶ & B_{2} & ⟵^{∼} & B \end{aligned}

whose bottom composite is chain homotopic to b. Let A′′ and B′′ denote the shifted mapping cylinders of A⊕A_{1} → A_{2} and B ⊕B_{1} → B_{2}, respectively; by the 5-lemma (the extension axiom), the induced map A′′ → B′′ is a quasi-isomorphism. By the universal property of mapping cylinders [WHomo, 1.5.3], the chain homotopic maps B′ ⟶^{b} B → B_{2} and B → B_{1} → B_{2} lift to a map B → B′′. We have now constructed a diagram like (3.9.2), which commutes up to chain homotopy. By 3.8.2, this suffices to construct a commutative diagram of this type. □

Applications

The Thomason-Trobaugh Resolution Theorem 3.9 provides a more convenient criterion than the Waldhausen Approximation Theorem 2.4 in many cases, because of the simplicity of the criterion 3.8.1: every complex in the larger category must be quasi-isomorphic to a complex in the smaller category.

Example 3.10.1 (Homologically bounded complexes). If A is abelian, we saw in 3.8.1 that Ch^{b}(A) ⊂ Ch^{hb}(A) have the same derived categories, so Theorem 3.9 applies applies, and (using Theorem 2.2) we recover the computation of 2.7.1: K(A) ≃ KCh^{b}(A) ≃ KCh^{hb}(A), and K_{n}(A) ≅ K_{n}Ch^{hb}(A) for all n.

If R is a ring, we saw in 3.8.1 that Ch^{b}M(R) and Ch_{pcoh}^{hb}(R) have the same derived categories. Again by Theorem 3.9, we recover the computation of 2.7.4: G(R) ≃ KCh_{pcoh}^{hb}(R).

If S is a central multiplicative set in R, it is easy to see by truncating that Ch_{S}^{b}M(R) → Ch_{pcoh,S}^{hb}(R) and Ch_{S}^{b}P(R) → Ch_{perf,S}(R) induce equivalences on derived categories. By Theorem 3.9, we obtain the calculation of 2.7.4 that they induce homotopy equivalences on K-theory.

Example 3.10.2. If X is a noetherian scheme, the discussion of 3.10.1 applies to the abelian category M(X) of coherent O_{X}-modules. We saw in Ex. II.9.8 that if a complex E has only finitely many nonzero cohomology sheaves, and these are coherent, then E is pseudo-coherent (i.e., it is quasi-isomorphic to a bounded above complex of vector bundles); by truncating below, it is quasi-isomorphic to a bounded complex of coherent modules. By 3.8.1, this proves that Ch^{b}M(X) and Ch_{pcoh}^{hb}(X) have the same derived categories, and hence G(X) ≃ KCh^{b}M(X) ≃ KCh_{pcoh}^{hb}(X).

Let Z be a closed subscheme of X. We saw in 2.6.2 that the relative K-theory of G(X) → G(X − Z) is the K-theory of the category Ch_{Z}^{b}M(X) of complexes of coherent modules which are acyclic on X − Z. It is contained in the category Ch_{pcoh,Z}^{hb}(X) of homologically bounded pseudo-coherent complexes acyclic on X − Z. The truncation argument in the previous paragraph shows that these two categories have the same derived categories, and hence the same K-theory: KCh_{Z}^{b}M(X) ≃ KCh_{pcoh,Z}^{hb}(X).

This argument works if X is quasi-compact but not noetherian, provided that we understand M(X) to be pseudo-coherent modules (see [TT, 3.11]); this is the case when X is quasi-projective over a commutative ring). However, it does not work for general X; see [SGA6, Is]. The following definition generalizes the definition of G_{0}(X) given in II, Ex. 9.9 and [SGA6, IV(2.2)].

Definition 3.10.3. If a scheme X is not noetherian, then we define G(X) to be KCh_{pcoh}^{hb}(X). If Z is closed in X, we define G(X on Z) to be KCh_{pcoh,Z}^{hb}(X).

By Proposition 2.5, G(X on Z) → G(X) → G(X − Z) is a homotopy fibration, and we get a long exact sequence on homotopy groups, exactly as in 2.6.2.

Example 3.10.4. (Thomason) If X is any quasi-compact scheme, the inclusion Ch_{perf}^{b}(X) ⊂ Ch_{perf}(X) induces an equivalence on derived categories by [TT, 3.5]. Comparing with Definition 2.7.3, we see that K(X) ≃ KCh_{perf}^{b}(X).

If X also has an ample family of line bundles (for example if X is quasiprojective), the inclusion of Ch^{b}VB(X) in Ch_{perf}(X) induces an equivalence on derived categories, by [TT, 3.6 and 3.8]. In this case, we get a fancy proof that K(X) ≃ KVB(X), which was already observed in 2.7.3.

Proper transfer f_{∗} and the Projection Formula

Proper Transfer 3.11. Here is a homological construction of the transfer f_{∗} of 3.7, associated to a proper map f : X → Y of noetherian schemes. It is based upon the fact (see [SGA4, V.4.9]) that the direct image f_{∗} sends flasque sheaves to flasque sheaves. Let F_{X} denote the Waldhausen category of homologically bounded complexes of flasque O_{X}-modules whose stalks have cardinality at most κ for a suitably large κ; f_{∗} is an exact functor from flasque modules to flasque modules, and from F_{X} to F_{Y}. Moreover, K(F_{X}) is independent of κ by Approximation 2.4, and F_{X} ⊂ Ch_{pcoh}^{hb}(X) induces a homotopy equivalence on K-theory by Resolution 3.9 (see Ex. 3.8). Hence we obtain the transfer map f_{∗} as the composite

G(X) ≃ KCh_{pcoh}^{hb}(X) ≃ K(F_{X}) ⟶^{f_{∗}} K(F_{Y}) ≃ KCh_{pcoh}^{hb}(Y) ≃ G(Y).

Given another proper map g : Y → Z (with Z noetherian), the composition g_{∗}f_{∗} : F_{X} → F_{Z} equals (gf)_{∗}. Thus X ↦ K(F_{X}) and hence X ↦ G(X) is a functor on the category of noetherian schemes and proper maps.

Suppose in addition that X has finite flat dimension over Y, so that f_{∗} sends perfect complexes to perfect complexes. (We say that f is a perfect map.) Then we have an exact functor f_{∗} : Ch_{perf}(X) → Ch_{perf}(Y) and hence a proper transfer

K(X) ≃ KCh_{perf}(X) ⟶^{f_{∗}} KCh_{perf}(Y) ≃ K(Y). The same argument shows that X ↦ K(X) is a functor on the category of noetherian schemes and perfect proper maps.

Variant 3.11.1. We can alter f_{∗} using the “Godement resolution” functor T, from complexes of O_{X}-modules to complexes of flasque sheaves; one takes the direct sum total complex of the Godement resolutions of the individual sheaves, as in [SGA4, XVII.4.2]. Since the direct image f_{∗} is exact on flasque sheaves, the functor Rf_{∗} = f_{∗} ∘ T is exact on all complexes of O_{X}-modules. The resulting map Rf_{∗} : KCh_{pcoh}^{hb}(X) → K(F_{Y}) ≃ KCh_{pcoh}^{hb}(Y) is homotopic to the functorial construction of G(X) → G(Y) in 3.11, so f_{∗}(x) = Rf_{∗}(x) for all x ∈ G_{m}(X).

Projection Formula 3.12. Let f : X → Y be a proper morphism of noetherian schemes. Then f_{∗} : G_{∗}(X) → G_{∗}(Y) is a graded K_{∗}(Y)-module homomorphism: for all x ∈ G_{m}(X) and y ∈ K_{n}(Y):

f_{∗}(x · f^{∗}y) = f_{∗}(x) · y.

Suppose in addition that f has finite flat dimension. Then f_{∗} : K_{∗}(X) → K_{∗}(Y) exists and is a graded K_{∗}(Y)-module homomorphism: the same formula holds in K_{m+n}(Y) for all x ∈ K_{m}(X), where f_{∗}(x) ∈ K_{m}(Y).

Proof. We will express each side as the pairing on K-theory arising from the construction of IV.8.11 applied to a biexact pairing of Waldhausen categories M(X) × VB(Y) → A_{X} × F → A_{Y} (II.9.5.2). Let A_{X} denote the category Ch_{pcoh}^{hb}(X) of homologically bounded pseudo-coherent complexes of O_{X}-modules, and let F denote the category of bounded above perfect complexes of flat O_{Y} - modules. Note that M(X) ⊂ A_{X} induces G(X) ≃ KA_{X} by 2.7.4 and VB(X) ⊂ F induces K(Y) ≃ K(F) by Ex. 2.8. Then the functors (E, F) ↦ (Rf_{∗}E) ⊗_{Y} F and (E, F) ↦ Rf_{∗}(E ⊗_{X} f^{∗}F) are biexact, where Rf_{∗} is defined in 3.11.1. By IV.8.11, they induce maps K(A_{X}) ∧ K(F) → KA_{Y} which on homotopy groups are the pairings sending (x, y) to f_{∗}(x · f^{∗}(y)) and f_{∗}(x) · y, respectively. The canonical map

(Rf_{∗}E) ⊗_{Y} F → f_{∗}(T E ⊗_{Y} f^{∗}F) → Rf_{∗}(E ⊗_{X} f^{∗}F)

is a natural quasi-isomorphism of pseudo-coherent complexes; see [SGA6, III.3.7]. Hence it induces a homotopy between the two maps, as desired.

If in addition X has finite flat dimension, we merely replace A_{X} (resp., A_{Y}) by the category of perfect complexes on X (resp., on Y), and the same proof works. □

EXERCISES

3.1 Under the hypothesis of the Resolution Theorem 3.1, show that H_{n−1} is closed under admissible subobjects and extensions in H_{n}. (It suffices to consider n = 1.)

3.2 Let f : B → C be a left exact functor between two abelian categories such that the right derived functors R^{i}f exist, and suppose that every object of B embeds in an f-acyclic object (an object for which R^{i}f vanishes when i > 0). Let A be the full subcategory of f-acyclic objects, and B_{f} the full subcategory of objects B such that only finitely many R^{i}f(B) are nonzero.

(a) Show that A and B_{f} are exact subcategories of B, closed under cokernels of admissible monomorphisms in B, and that f is an exact functor on A.

(b) Show that K(A) ≃ K(B_{f}). In this way we can define f_{∗} : K_{n}(B_{f}) → K_{n}(C) by K(B_{f}) ≃ K(A) → K(C).

3.3 Show that β is an isomorphism in Example 3.5.2. To do this, let F_{m}(M) be the submodule of M generated by M_{−m} ⊕ ⋯ ⊕ M_{m}, and consider the subcategory M_{m}(S) of graded B-modules M with M = F_{m}(M). Show that K_{i}M_{m}(R) is a sum of copies of G_{i}(R) and use the admissible filtration

0 ⊂ F_{0}(M) ⊂ ⋯ ⊂ F_{m}(M) = M

of modules in M_{m}(S) (see 1.8) to show that K_{i}M_{m}(S) ≅ K_{i}M_{m}(R) for all m.

3.4 If S = R ⊕ S_{1} ⊕ ⋯ is a graded noetherian ring, and both S/R and R/S have finite flat dimension, modify the previous exercise to show that Example 3.5.2 still holds. Hint: Consider S-modules which are acyclic for both ⊗_{S}R and ⊗_{S}S.

3.5 Suppose that X ⟶^{f} Y ⟶^{g} Z are proper morphisms of finite flat dimension. Show that f^{∗}g^{∗} = (gf)^{∗} as maps G_{∗}(Z) → G_{∗}(X). 3.6 (Quillen) In this exercise, we give another construction of the transfer map f_{∗} : G(X) → G(Y) associated to a projective morphism f: X → Y of noetherian schemes. Let A ⊂ M(X) denote the (exact) subcategory consisting of all coherent O_{X}-modules F such that Rf^{i}(F) = 0 for i > 0. Because X is projective, for every coherent O_{X}-module F there is an integer n_{0} such that, for all n ≥ n_{0}, the modules F(n) are in A, and are generated by global sections; see [Hart, III.8.8].

(a) Show that any coherent F embeds in F(n)^{r} for large n and r. Hint: To prove it for F = O_{X}, apply Hom(O_{X}(n), −) to a surjection O_{X}^{r} → O_{X}(n); now apply ⊗F.

(b) Show that every coherent module F has a finite A-resolution, starting with (a).

(c) Show that every vector bundle has a finite resolution by vector bundles in A.

(d) Use Ex. 3.2 to define f_{∗} to be the composition f_{∗} : G(X) ≃ K(A) → G(Y). Then show that this definition of f_{∗} is homotopy equivalent to the map defined in 3.7, so that the maps f_{∗} : G_{n}(X) → G_{n}(Y) agree.

(e) Given g : Z → X, let B ⊂ M(X) be the subcategory of modules Torindependent of O_{Z}. Show that every F in B has a finite A ∩ B-resolution. Hint: If F is in B, so is F(n).

3.7 Fix a noetherian ring R, and let ChM^{hb,+}(R) be the category of all bounded below chain complexes of R-modules which are quasi-isomorphic to a (bounded) complex in Ch^{b}(M(R)). For example, injective resolutions of finitely generated R-modules belong to ChM^{hb,+}(R). Let I ⊂ ChM^{hb,+}(R) denote the subcategory of

complexes of injective R-modules. Show that the derived categories of I, ChM^{hb,+}(R) and Ch^{b}(M(R)) are isomorphic. Conclude that

K(I) ≃ KChM^{hb,+}(R) ≃ KCh^{b}(M(R) ≃ G(R).

3.8 Fix a noetherian scheme X, and let ChM^{hb,+}(X) be the category of all bounded below chain complexes of O_{X}-modules which are quasi-isomorphic to a (bounded) complex in Ch^{b}(M(X)).

(a) Show that the derived categories of ChM^{hb,+}(X) and Ch^{b}(M(X)) are isomorphic,

and conclude that ChM^{hb,+}(X) ≃ KCh^{b}(M(X) ≃ G(X).

(b) Let I ⊂ F ⊂ ChM^{hb,+}(X) denote the subcategories of complexes of injective and flasque O_{X}-modules, respectively. Show that K(I) ≃ K(F) ≃ G(X) as well. 3.9 If X is a quasi-compact regular scheme (I.5.14), show that every pseudocoherent module is perfect (II, Ex. 9.10). Then show that Ch_{perf}(X) is the same as Ch_{pcoh}^{hb}. Conclude that K(X) ≃ G(X) by definition (2.7.3 and 3.10.3). 3.10 (Projection Formula) Suppose that f: X → Y is a proper map between quasiprojective schemes, both of which have finite flat dimension. Use a modification of the proof of 3.7.3 to show that the proper transfer map f_{∗} : G_{m}(X) → G_{m}(Y) is a K_{0}(Y)-module map, i.e., that f_{∗}(x·f^{∗}y) = f_{∗}(x)·y for y ∈ K_{0}(X) and x ∈ G_{m}(X). (The conclusion is a special case of 3.12; see [SGA6, IV.2.11.1].) 3.11 (Thomason) Let f : X → Y be a proper morphism of noetherian schemes, and let g : Y′ → Y be a map Tor-independent of f. Show that the Base Change Theorem 3.7.2 remains valid in this context: if g has finite flat dimension then g^{∗}f_{∗} ≃ f′_{∗}g′^{∗} as maps G(X) → G(Y′), while if f has finite flat dimension then g^{∗}f_{∗} ≃ f′_{∗}g′^{∗} as maps K(X) → K(Y′)

(a) First consider the category C of bounded above pseudo-coherent complexes of flat modules on X. Then g′^{∗} is exact on C and takes values in C′. Show that the Godement resolution T of 3.11.1 sends C to itself, and that E → T E is a quasiisomorphism. Then show that the inclusion of C in Ch_{pcoh}^{hb}(X) is an equivalence of derived categories, so G(X) ≃ K(C).

(b) Consider the category A whose objects consist of an E in C, a bounded above complex F of flat modules on Y with a quasi-isomorphism F → f_{∗}E, a bounded below complex G of flasque modules on X′ with a quasi-isomorphism (g′)^{∗}E → G. Show that A has the same derived category as C, and conclude that G(X) ≃ K(A).

(c) Show that g^{∗}f_{∗} is represented by the exact functor on A sending (E, F, G) to g^{∗}F, and that f′_{∗}(g′)^{∗} is represented by the exact functor sending (E, F, G) to f′_{∗}G.

(d) The canonical base change of [SGA4, XVII.4.2.12] is the natural isomorphism

g^{∗}F → g^{∗}f_{∗}(E) → g^{∗}f_{∗}g′_{∗}g′^{∗}E = g^{∗}g_{∗}f′_{∗}g′^{∗}E → f′_{∗}g′^{∗}E → f′_{∗}G.

Show that this canonical base change induces the desired homotopy g^{∗}f_{∗} ≃ f′_{∗}g′^{∗}. 3.12 Suppose that F : M^{1} → M^{2} is an additive functor between abelian categories, and that A^{i} ⊂ Ch(M^{i}) (i = 1, 2) are saturated Waldhausen subcategories, closed under mapping cones and shifts. If F sends A^{1} to A^{2} and induces an equivalence of derived categories, modify the proof of Theorem 3.9 to show that K(A^{1}) ≃ K(A^{2}). 3.13 Let ℙ_{R}^{1} denote the projective line over an associative ring R, as in 1.5.4, and let H_{n} denote the subcategory of mod- ℙ_{R}^{1} of modules having a resolution of length n by vector bundles. Show that K(ℙ_{R}^{1}) ≃ KH_{n} for all n. 3.14 Let S be a set of central nonzerodivisors in a ring R.

(a) Let Ch_{perf}(M_{S}) denote the category of perfect chain complexes of S-torsion R-modules. Show that the inclusion of Ch_{perf}(M_{S}) into Ch_{perf,S}(R) induces an equivalence KCh_{perf}(M_{S}) ≃ K(R on S). (See 2.7.2.) Hint: Consider the functor \underrightarrow{lim} Hom(R/sR, −) from Ch_{S}^{b}P(R) to Ch_{perf}(M_{S}).

(b) Let H be the additive subcategory of H_{S}(R) generated by the projective R/sRmodules. Show that KCh^{b}(H) ≃ KH_{S}(R).

(c) Show that the inclusion of Ch^{b}(H) in Ch_{perf}(M_{S}) satisfies property (App), and conclude that K(R on S) ≃ KH_{S}(R). By Theorem 2.6.3, this yields a long exact sequence

⋯ K_{n+1}(S^{−1}R) ⟶^{∂} K_{n}H_{S}(R) → K_{n}(R) → K_{n}(S^{−1}R) ⟶^{∂} · · · .

3.15 Let s be a nonzerodivisor in a commutative noetherian ring R, and assume that R is flat over a subring R_{0}, with R_{0} isomorphic to R/sR. Use the projection formula 3.5.3 to show that the transfer map G_{∗}(R/sR) → G_{∗}(R) is zero. 3.16 (Thomason [TT, 5.7]) Let X be a quasi-projective scheme, and let Z be a subscheme defined by an invertible ideal I of O_{X}. Let Ch_{perf,Z}(X) denote the category of perfect complexes on X which are acyclic on X − Z. For reasons that will become clear in §7, we write K(X on Z) for KCh_{perf,Z}(X).

(a) Let M_{Z}(X) denote the category of O_{X}-modules supported on Z, and let Ch_{perf}(M_{Z}) denote the category of perfect complexes of modules in M_{Z}(X). Show that the inclusion into Ch_{perf,Z}(X) induces an equivalence KCh_{perf}(M_{Z}) ≃ K(X on Z).

(b) Let H_{Z}(X) denote the subcategory of modules in H(X) supported on Z. If H is the additive subcategory of H_{Z}(X) generated by the O_{X}/I^{n}O_{X}(n), show that K(H) ≅ KH_{Z}(X).

(c) Show that the inclusion of Ch^{b}(H) in Ch_{perf}(M_{Z}) satisfies (App), so that Waldhausen Approximation 2.4 (and 2.2) imply that K(X on Z) ≃ KH_{Z}(X).

§4. Devissage

This is a result that allows us to perform calculations like G_{∗}(ℤ/p^{r}) ≅ K_{∗}(𝔽_{p}), which arise from the inclusion of the abelian category M(𝔽_{p}) of all finite elementary abelian p-groups into the abelian category M(ℤ/p^{r}) of all finite abelian p-groups of exponent p^{r}. It is due to Quillen and taken from [Q341].

Devissage Theorem 4.1. Let i : A ⊂ B be an inclusion of abelian categories such that A is an exact abelian subcategory of B (II.6.1.5) and A is closed in B under subobjects and quotients. Suppose that every object B of B has a finite filtration

0 = B_{r} ⊂ ⋯ ⊂ B_{1} ⊂ B_{0} = B

by objects in B such that every subquotient B_{i}/B_{i−1} lies in A. Then

K(A) ≃ K(B) and K_{∗}(A) ≅ K_{∗}(B).

Proof. By Quillen’s Theorem A (IV.3.7), it suffices to show that the comma categories Qi/B are contractible for every B in B. If B is in A, then Qi/B ≃ ∗ because B is a terminal object. Since B has a finite filtration, it suffices to show that the inclusion i/B′ → i/B is a homotopy equivalence for each B′ ↣ B in B with B/B′ in A.

By Ex. IV.6.1, we may identify the objects of Qi/B, which are pairs (A, A → B), with admissible layers u : B_{1} ↣ B_{2} ↣ B such that B_{2}/B_{1} is in A. Let J denote the subcategory of Qi/B consisting of all admissible layers of B with B_{1} ⊆ B′. We have functors s : Qi/B → J, s(u) = (B_{1} ∩ B′ ↣ B_{2} ↣ B) and r : J → Qi/B′, r(u) = (B_{1} ↣ B_{2} ∩ B′ ↣ B), because A is closed under subobjects. The natural transformations u → s(u) ← rs(u) defined by

(B_{1} ↣ B_{2} ↣ B) → (B_{1} ∩ B′ ↣ B_{2} ↣ B) ← (B_{1} ∩ B′ ↣ B_{2} ∩ B′ ↣ B)

show that s is left adjoint to the inclusion J ⊂ Qi/B and that r is right adjoint to the inclusion Qi/B′ ⊂ J. It follows (IV.3.2) that Qi/B′ ⊂ J ⊂ Qi/B are homotopy equivalences, as desired. □

Open Problem 4.1.1. Generalize the Devissage Theorem 4.1 to Waldhausen categories. Such a result should yield the above Devissage Theorem when applied to Ch^{b}(A).

Corollary 4.2. If I is a nilpotent ideal in a noetherian ring R, then G(R/I) ≃ G(R) and hence G_{∗}(R/I) ≅ G_{∗}(R).

Proof. As in II.6.3.1, Devissage applies to M(R/I) ⊂ M(R), because every finitely generated R-module M has a finite filtration by submodules MI^{n}. □

Example 4.2.1. Let R be an artinian local ring with maximal ideal 𝔪 (𝔪^{r} = 0) and quotient field k = R/𝔪. (E.g., R = ℤ/p^{r} and k = 𝔽_{p}). Then G_{∗}(R) ≅ K_{∗}(k).

It is instructive to deconstruct the argument slightly. In this case, M(k) is an exact subcategory of M(R) and every R-module M has the natural filtration

0 = 𝔪^{r}M ⊂ 𝔪^{r−1}M ⊂ ⋯ ⊂ 𝔪M ⊂ M.

Note that the Admissible Filtrations Proposition 1.8 does not apply because for example F (M) = M/𝔪M is not an exact functor.

Open Problem 4.2.2. Compute the K-groups K_{∗}(R) of an artinian local ring R, assuming K_{∗}(k) is known. If char(k) = 0, this can be done using cyclic homology (Goodwillie’s Theorem): the relative groups K_{n}(R, 𝔪) and HC_{n−1}(R, 𝔪) are isomorphic. If char(k) ≠ 0, this can be done using topological cyclic homology (McCarthy’s Theorem): K_{n}(R, 𝔪) and T C_{n}(R, 𝔪) are isomorphic. In terms of generators and relations, though, we only know K_{0}, K_{1}, K_{2} and sometimes K_{3} at present. (See II.2.2, III.2.4 and III.5.11.1.)

Application 4.3. Let A be an abelian category such that every object has finite length. By Devissage, K(A) is equivalent to K(A_{ss}), where A_{ss} is the subcategory of semisimple objects. By Schur’s Lemma, A_{ss} ≅ ⊕M(D_{i}), where the D_{i} are division rings. (The D_{i} are the endomorphism rings of non-isomorphic simple objects A_{i}.) It follows from Ex. IV.6.11 that

K_{∗}(A) ≃ ⊕_{i}K_{∗}(D_{i}).

This applies to finitely generated torsion modules over Dedekind domains and curves, and more generally to finitely generated modules of finite support over any commutative ring or scheme.

Application 4.4. (R-modules with support) Given a central element s in a noetherian ring R, let M_{s}(R) denote the abelian subcategory of M(R) consisting of all M such that Ms^{n} = 0 for some n. We saw in II.6.3.3 that these modules have finite filtrations with subquotients in M(R/sR). By Devissage, KM_{s}(R) ≃ KM(R/sR), so we have K_{∗}M_{s}(R) ≅ G_{∗}(R/sR). More generally, given any ideal I we can form the exact category M_{I}(R) of all M such that MI^{n} = 0 for some n. Again by Devissage, KM_{I}(R) ≃ KM(R/I) and we have K_{∗}M_{I}(R) ≅ G_{∗}(R/I). The case I = 𝔭 (K_{∗}M_{𝔭}(R) ≅ G_{∗}(R/𝔭)) will be useful in section 6 below.

If S is a central multiplicatively closed set in R, the exact category M_{S}(R) is the filtered colimit over s ∈ S of the M_{s}(R). By IV.6.4, KM_{S}(R) = ΩBQM_{S}(R) is \underrightarrow{lim} KM_{s}(R) and hence

K_{n}M_{S}(R) = \underrightarrow{lim} K_{n}M_{s}(R) = \underrightarrow{lim} G_{n}(R/sR).

EXERCISES

4.1 (Jordan-Hölder) Given a ring R, describe the K-theory of the category of Rmodules of finite length. (K_{0} is given by Ex. II.6.1.) 4.2 If X is a noetherian scheme, show that G(X_{red}) ≃ G(X) and hence that G_{∗}(X_{red}) ≅ G_{∗}(X). This generalizes the result II.6.3.2 for G_{0}. 4.3 (Quillen) Let Z be a closed subscheme of a noetherian scheme X, and let M_{Z}(X) be the exact category of all coherent X-modules supported on Z. Generalize 4.4 and II.6.3.4 by showing that

G(Z) = KM(Z) ≃ KM_{Z}(X).

4.4 Let S = R ⊕ S_{1} ⊕ ⋯ be a graded noetherian ring, and let M_{gr}^{b}(S) denote the category of graded modules M with M_{n} = 0 for all but finitely many n. (Ex. II.6.14) Via S → R, any R-module in M_{gr}^{b}(R) = M_{gr}(R) is a graded S-module in M_{gr}^{b}(S).

(a) Use Devissage to show that M_{gr}(R) ⊂ M_{gr}^{b}(S) induces an equivalence in Ktheory, so that (in the notation of Example 3.5.2) K_{∗}M_{gr}^{b}(S) ≃ G_{∗}(R)[σ, σ^{−1}].

(b) If t ∈ S_{1}, write M_{gr,t}(S) for the category of t-torsion modules in M_{gr}(S). Show that M_{gr}(S/tS) ⊂ M_{gr,t}(S) induces KM_{gr}(S/tS) ≃ KM_{gr,t}(S).

§5. The Localization Theorem for abelian categories

The K_{0} Localization Theorems for abelian categories (II.6.4) and certain exact categories (II.7.7.4) generalize to higher K-theory, in a way we shall now describe. It is also due to Quillen.

Recall from the discussion before II.6.4 that a Serre subcategory of an abelian category A is an abelian subcategory B which is closed under subobjects, quotients and extensions. The quotient abelian category A/B exists; Gabriel’s construction of A/B using the Calculus of Fractions is also described just before II.6.4.

Abelian Localization Theorem 5.1. Let B be a Serre subcategory of a (small) abelian category A. Then

K(B) → K(A) ⟶^{loc} K(A/B)

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

(5.1.1) ⋯ → K_{n+1}(A/B) ⟶^{∂} K_{n}(B) → K_{n}(A) ⟶^{loc} K_{n}(A/B) ⟶^{∂} K_{n−1}(B) → ⋯

ending in K_{0}(B) → K_{0}(A) → K_{0}(A/B) → 0, the exact sequence of II.6.4.

Proof. (Quillen) For any A in A, let us write Ā for the object loc(A) of A/B. Recall from IV.3.2.3 that for any object L in A/B the comma category L\Qloc consists of pairs (A, u) with A in A and u : L → Ā a morphism in QA/B. We will deduce the result from Theorem B (IV.3.7), so we need to show that 0\Qloc is homotopy equivalent to QB, and that for every L → L′ the map L′\Qloc → L\Qloc is a homotopy equivalence.

To do this, we introduce the full subcategory F_{L} of L\Qloc consisting of pairs (A, u) in which u is an isomorphism in A/B. (By IV.6.1.2, isomorphisms in QA/B are in 1–1 correspondence with isomorphisms in A/B, and are described in II.A.1.2.) In particular, the category F_{0} (L = 0) is equivalent to the subcategory QB because, by construction (II.6.4), the objects of B are exactly the objects of A isomorphic to 0 in A/B. Thus the functor QB → QA factors as QB ≅ F_{0} ↪ 0\Qloc → QA. Claim 5.1.2: The inclusion i : F_{0} → 0\Qloc is a homotopy equivalence.

This will follow from Quillen’s Theorem A (IV.3.7) once we observe that for every A in A the comma category i/(A, u : 0 → Ā) is contractible. By IV.6.1.2, the morphism u in QA/B is the same as a subobject L of Ā in A/B, and this is represented by a subobject of A. By Ex. IV.6.1, we can represent each object of i/(A, u) by an admissible layer A_{1} ↣ A_{2} ↣ A in A such that Ā_{1} = Ā_{2} = L, and it is easy to see that i/(A, u) is equivalent to the poset of such layers. But this poset is directed by the very construction of A/B: A_{1} ↣ A_{2} ↣ A and A′_{1} ↣ A′_{2} ↣ A both map to (A_{1} ∩ A′_{1}) ↣ (A_{2} + A′_{2}) ↣ A. This verifies our claim, showing that QB → 0\Qloc is a homotopy equivalence.

Claim 5.1.3: The inclusion F_{L} ⟶^{i} L\Qloc is a homotopy equivalence for every L.

This is proven with the same argument used to prove Claim 5.1.2. The only difference is that now the category i/(A, u) is equivalent to the poset of admissible layers A_{1} ↣ A_{2} ↣ Ā with \overline{A_{2}/A_{1}} ≅ L; the construction works in this context to shows that it is directed.

We now introduce several auxiliary categories. Fix N in A and let E_{N} be the category of pairs (A, h : A → N) for which h̄ is an isomorphism in A/B. By definition, a morphism from (A, h) to (A′, h′) is a morphism A ↞ A′′ ↣ A′ in QA such that the two composites A′′ → N agree. It is easily checked that there is a well defined functor k : E_{N} → QB sending (A, h) to ker(h).

Let E′_{N} denote the full subcategory of E_{N} on the (A, h) with h onto. We will show that E′_{N} → E_{N} and E′_{N} → QB are homotopy equivalences.

Claim 5.1.4: For each N in A, k′ : E′_{N} → QB is a homotopy equivalence.

This will follow from Theorem A once we show that k′/T is contractible for each T in QB. An object of this comma category is a datum (A, h, u), where (A, h) is in E_{N}, and u : ker(h) → T is in QB. The subcategory C of all (A, h, u) with u surjective is contractible because it has (N, 1_{N}, 0 ↞ T) as initial object. And the inclusion of C in k′/T is a homotopy equivalence because it has a left adjoint (IV.3.2), sending (A, h, ker(h) ↣ T_{0} ↞ T) to (A_{0}, h_{0}, T_{0} ↞ T), where A_{0} is the pushout of ker(h) → A along ker(h) ↣ T_{0} and h_{0} is the induced map. The claim follows.

Claim 5.1.5: For each N, E′_{N} ↪ E_{N} is a homotopy equivalence.

Let I_{N} denote the partially ordered set of objects N_{i} of N such that N/N_{i} is in B, and consider the functor E_{N} → I_{N} sending (A, h) to the image h(A). This is a fibered functor (IV.3.6.2); the fiber over i is E′_{N_{i}}, and the base change for N_{j} ⊂ N_{i} sends (A, h) to (h^{−1}(N_{j}), h). Since k_{N}(A, h) = k_{N_{i}}(A, h), it follows from Claim 5.1.4 that the base change maps E′_{N_{i}} → E′_{N_{j}} are homotopy equivalences. By Theorem B (IV.3.7), E′_{N} → E_{N} → I_{N} is a homotopy fibration. Since I_{N} has a final object (N), it is contractible and we get E′_{N} ≃ E_{N} as claimed.

Claim 5.1.6: If g : N → N′ is a map in A which is an isomorphism in A/B, then g_{∗} : E_{N} → E_{N′} is a homotopy equivalence.

Indeed, if (A, h) is in E_{N} then ker(h) ⊆ ker(gh) defines a natural transformation from k : E_{N} → QB to k g_{∗} : E_{N} → E_{N′} → QB. As k is a homotopy equivalence (by Claims 5.1.4 and 5.1.5), so is g_{∗}.

Now fix L in A/B, and let I_{L} be the category of pairs (N, N̄ ⟶^{∼} L); morphisms are maps g : N → N′ in A such that N̄ ≅ N̄′. This category is filtering by II.A.1.2, and there is a functor from I_{L} to categories sending (N, N̄ ⟶^{∼} L) to E_{N} and g to g_{∗}.

Claim 5.1.7: F_{L} is isomorphic to the colimit over I_{L} of the categories E_{N}.

For each n = (N, N̄ ⟶^{∼} L) we have a functor p_{n} : E_{N} → F_{L} sending (A, h) to L ⟶^{∼} N̄ ⟶^{∼} Ā. Since p_{n} = p_{n′}g_{∗} for each morphism g, there is a functor p : colim E_{N} → F_{L}.

Consider the composite functor E_{N} ⟶^{k} QB ⟶^{∼} F_{0} ⟶^{∼} 0\Qloc, sending (A, h) to (ker(h), 0); it is a homotopy equivalence by Claims 5.1.2–5. Using the map i : 0 ↣ L in Q(A/B), we have a second functor E_{N} ⟶^{p} F_{L} ↪ L\Qloc ⟶^{i^{∗}} 0\Qloc, sending (A, h) to (A, 0 ↣ Ā). There is a natural transformation between them, given by the inclusion of ker(h) in A. Hence we have a homotopy commutative diagram:

E_{N} ⟶^{p_{n}} F_{L} ↪ L\Qloc
k↓  ↓_{i^{∗}}
QB ⟶^{∼} F_{0} ↪ 0\Qloc.

From Claims 5.1.6 and 5.1.7, it follows that each p_{n} : E_{N} → F_{L} is a homotopy equivalence. From Claim 5.1.3, F_{L} ↪ L\Qloc is a homotopy equivalence. It follows that i^{∗} : L\Qloc → 0\Qloc is also a homotopy equivalence. This finishes the proof of the Localization Theorem 5.1. □

Taking homotopy groups with coefficients mod ℓ also converts homotopy fibration sequences into long exact sequences (IV.2.1.1), so we immediately obtain a finite coefficient analogue of (5.1.1).

Corollary 5.2. For each ℓ there is also a long exact sequence

⋯ → K_{n+1}(A/B; ℤ/ℓ) ⟶^{∂} K_{n}(B; ℤ/ℓ) → K_{n}(A; ℤ/ℓ) → K_{n}(A/B; ℤ/ℓ) ⟶^{∂} ⋯

Open Problem 5.3. Let B be a Serre subcategory of an abelian category A (II.6.4), and let Ch_{B}^{b}(A) denote the category of all bounded complexes in A whose cohomology lies in B (II, Ex. 9.5). Is K(Ch_{B}^{b}(A)) ≃ K(B)? Such a result would make the Localization Theorem 5.1 for abelian categories an immediate consequence of Proposition 2.5 and Theorem 2.2.

EXERCISES

5.1 Suppose that α : A → A is a morphism in A which is an isomorphism in A/B, and so determines an element [α] of K_{1}(A/B). Show that ∂ : K_{1}(A/B) → K_{0}(B) sends [α] to [coker(α)]−[ker(α)]. Hint: Use the representative of [α] in π_{2}BQ(A/B) given in IV, Ex. 7.9, and IV.6.2. 5.2 Show that the map S^{n} → P^{n+1}(ℤ/ℓ) of IV.2.1.1 applied to the homotopy fibration of Theorem 5.1 yields a commutative diagram comparing the localization sequences of 5.1 and 5.2:

K_{n+1}(A/B; ℤ/ℓ) ⟶^{∂} K_{n}(B; ℤ/ℓ) ⟶ K_{n}(A; ℤ/ℓ) ⟶ K_{n}(A/B; ℤ/ℓ)
↓  ↓  ↓  ↓
K_{n}(A/B)  ⟶^{∂} K_{n−1}(B) ⟶ K_{n−1}(A) ⟶ K_{n−1}(A/B).

5.3 Suppose that there is a biexact functor A × C → A′ which induces biexact functors B × C → B′ and A/B × C → A′/B′. Use Ex. IV.1.23 to show that for x ∈ K_{j}(A/B) and y ∈ K_{n}(C) the element {x, y} ∈ K_{n+i}(A′/B′) satisfies ∂({x, y}) = {∂(x), y} in K_{n+i}(B′).

§6. Applications of the Localization Theorem

In this section, we give two families of applications of the Localization Theorem 5.1, to G(R) and to G(X). All rings and schemes will be noetherian in this section, so that M(R) and M(X) are abelian categories.

Application 6.1. Let S be a central multiplicatively closed set in a noetherian ring R. We saw in II.6.4.1 that the category M_{S}(R) of finitely generated S-torsion modules is a Serre subcategory of M(R) with quotient category M(S^{−1}R). By the Localization Theorem 5.1, there is a homotopy fibration

KM_{S}(R) → G(R) → G(S^{−1}R).

We observed in 4.4 that M_{S}(R) is the colimit over all s ∈ S of the M(R/sR), and that K_{∗}M_{S}(R) ≅ \underrightarrow{lim} G_{∗}(R/sR). Comparing this to the Waldhausen localization sequence 2.6.1, we see that M_{S}(R) → Ch_{S}^{b}M(R) induces a homotopy equivalence

KM_{S}(R) ⟶^{∼} KCh_{S}^{b}M(R) = G(R on S).

The prototype is the case when S = {s^{n}}. Here the maps G(R/sR) → G(R/s^{n}R) are homotopy equivalences by Devissage, so G(R/sR) ≃ KM_{S}(R); see 4.4. By inspection, the map G(R/sR) → G(R) identifying it with the homotopy fiber of G(R) → G(R[1/s]) is the transfer i_{∗} (IV.6.3.3) associated to i : R → R/sR. Thus the long exact Localization sequence (5.1.1) becomes:

(6.1.1) ⋯ → G_{n+1}(R[s^{−1}]) ⟶^{∂} G_{n}(R/sR) ⟶^{i_{∗}} G_{n}(R) → G_{n}(R[s^{−1}]) ⟶^{∂} · · · .

This is a sequence of K_{∗}(R)-modules, because P(R) acts on the sequence of abelian categories M_{S}(R) → M(R) → M(S^{−1}R).

Example 6.1.2. It is useful to observe that any s ∈ S determines an element [s] of K_{1}(R[1/s]) and hence G_{1}(R[1/s]), and that ∂(s) ∈ G_{0}(R/sR) is [R/sR] − [I], where I = {r ∈ R : sr = 0}. This formula is immediate from Ex. 5.1. In particular, when R is a domain we have ∂(s) = [R/sR].

Example 6.1.3. If R = ℤ[s, 1/f(s)] with f(0) = 1, the maps i_{∗} are zero in (6.1.1) and hence the maps G_{n}(R) → G_{n}(R[1/s]) are injections. Indeed, the vanishing of i_{∗} follows from the projection formula (3.5.3): i_{∗}(i^{∗}y) = i_{∗}(1·i^{∗}y) = i_{∗}(1)·y together with the observation in 6.1.2 that i_{∗}([R/sR]) = i_{∗}∂(s) = 0. (Cf. Ex. 3.15.)

For the following result, we adopt the notation that f denotes the inclusion of R into R[s], j is R[s] ↪ R[s, s^{−1}], and (s = 1) denotes the map from either R[s] or R[s, s^{−1}] to R, obtained by sending s to 1. Because R has finite flat dimension over R[s] and R[s, s^{−1}], there are base change maps (s = 1)^{∗} : G(R[s]) → G(R) and (s = 1)^{∗} : G(R[s, s^{−1}]) → G(R).

Fundamental Theorem of G(R) 6.2. Let R be a noetherian ring. Then:

(i) The flat base change f^{∗} : G(R) → G(R[s]) is a homotopy equivalence, split by

(s = 1)^{∗}. Hence

f^{∗} : G_{n}(R) ≅ G_{n}(R[s]) for all n.

(ii) The flat base change j^{∗} : G(R[s]) → G(R[s, s^{−1}]) induces isomorphisms

G_{n}(R[s, s^{−1}]) ≅ G_{n}(R) ⊕ G_{n−1}(R).

Proof. We first observe that (i) implies (ii). Indeed, because (s = 0)_{∗} : G(R) → G(R[s]) is zero by 3.5.1, the localization sequence (6.1.1) for j : R[s] → R[s, s^{−1}] splits into short exact sequences

0 → G_{n}(R[s]) ⟶^{j^{∗}} G_{n}(R[s, s^{−1}]) ⟶^{∂} G_{n−1}(R) → 0.

By (i) we may identify (jf)^{∗} : G_{n}(R) → G_{n}(R[s, s^{−1}]) with j^{∗}. Because (s = 1)∘jf is the identity, (jf)^{∗}(s = 1)^{∗} is homotopic to the identity of G(R). Assertion (ii) follows.

To prove (i), we introduce the graded subring S = R[st, t] of R[s, t] where deg(s) = 0, deg(t) = 1. Let M_{gr}^{b}(S) denote the Serre subcategory of all graded modules M in M_{gr}(S) with only finitely many nonzero M_{n}, i.e., graded t-torsion modules. By Devissage (Ex. 4.4), M_{gr}^{b}(S) has the same K-theory as its subcategory M_{gr}(S/tS). There is also an equivalence of quotient categories

M_{gr}(S)/M_{gr}^{b}(S) ≅ M(R[s])

induced by the exact functor M ↦ M/(t−1)M from M_{gr}(S) to M(R[s]). Note that both S and S/tS = R[st] are flat over R and that R has finite flat dimension over both, so the K-theory of both M_{gr}(S) and M_{gr}^{b}(S) are isomorphic to G(R)[σ, σ^{−1}] by Example 3.5.2. Hence the localization sequence (5.1.1) gives us the following diagram with exact rows:

⋯ → K_{n}M_{gr}(S/tS) ⟶  K_{n}M_{gr}(S)  ⟶ G_{n}(R[s]) → ⋯
≃↑^{3.5.2}  ^{≃}↑^{3.5.2}  ↑
0 → G_{n}(R) ⊗ ℤ[σ, σ^{−1}] ⟶^{h} G_{n}(R) ⊗ ℤ[σ, σ^{−1}] ⟶ G_{n}(R) → 0.

To describe the map h, recall from 3.5.2 that it is induced by the functor from M(R) to KM_{gr}(S) sending M to M ⊗ S/tS. We have an exact sequence of functors from M(R) to M_{gr}(S):

0 → M ⊗ S(−1) ⟶^{t} M ⊗ S → M ⊗ S/tS → 0.

As pointed out in Example 3.5.2, the first two maps induce the maps σ and 1, respectively, from G_{n}(R) to G(R)[σ, σ^{−1}]. Hence h = 1 − σ, proving (i). □

Because R[s] and R[s, s^{−1}] are regular whenever R is, combining Theorems 6.2 and 3.3 yields the following important consequence.

Theorem 6.3. If R is a regular noetherian ring, then the base change K(R) → K(R[s]) is a homotopy equivalence, so K_{n}(R) ≅ K_{n}(R[s]) for all n. In addition,

K_{n}(R[s, s^{−1}]) ≅ K_{n}(R) ⊕ K_{n−1}(R) for all n.

In particular, regular rings are K_{n}-regular for all n (in the sense of III.3.4).

Corollary 6.3.1. For any regular ring R, there is a split exact sequence

0 → K_{n}(R) → K_{n}(R[s]) ⊕ K_{n}(R[s^{−1}]) → K_{n}(R[s, s^{−1}]) ⟶_{←}^{∂} K_{n−1}(R) → 0,

in which the splitting is multiplication by s ∈ K_{1}(ℤ[s, s^{−1}]).

Proof. This is obtained from the direct sum of the split exact sequences (6.1.1) for R[s] → R[s, s^{−1}] and R[s^{−1}] → R[s, s^{−1}], and the isomorphism K_{n}(R) ≅ K_{n}(R[s]). By 6.1, the map ∂ in the localization sequence (6.1.1) is K_{∗}(R)-linear. Thus for x ∈ K_{n−1}(R) and s ∈ K_{1}(R[s, s^{−1}]) we have ∂{s, x} = {∂s, x} = x. □

The technique used in 6.2 to prove that G(R) ≃ G(R[s]) applies more generally to filtered rings. By a filtered ring we mean a ring R equipped with an increasing filtration {F_{i}R} such that 1 ∈ F_{0}R, F_{i}R · F_{j}R ⊆ F_{i+j}R and R = ∪F_{i}R. The associated graded ring is gr(R) = ⊕F_{i}R/F_{i−1}R. It is easy to see that if gr(R) is noetherian then so is R.

For example, any positively graded ring such as R[s] is filtered with F_{i}(R) = R_{0} ⊕ ⋯ ⊕ R_{i}; in this case gr(R) = R.

Theorem 6.4. Let R be a filtered ring such that gr(R) is noetherian and of finite flat dimension d over k = F_{0}R, and k has finite flat dimension over gr(R). Then k ⊂ R induces G(k) ≃ G(R).

Proof. The hypotheses imply that R has flat dimension fd_{k} A ≤ d. Indeed, since each of the F_{i}R/F_{i−1}R has flat dimension at most d it follows by induction that fd_{k} F_{i}R ≤ d and hence that fd_{k} R ≤ d. Thus the map G(k) → G(R) is defined. Let S denote the graded subring ⊕(F_{i}R)t^{i} of R[t]. Then S/tS ≅ gr(R), S is noetherian, fd_{k} S ≤ d and k has finite flat dimension over S. Finally, the category M_{gr}(S) is abelian (as S is noetherian) and we have M_{gr}(S)/M_{gr}^{b}(S) ≅ M(R).

It follows that KM_{gr}(S) and KM_{gr}(S/tS) ≅ KM_{gr}^{b}(S) are both isomorphic to G(k)[σ, σ^{−1}] by Example 3.5.2 (and Ex. 3.4). The rest of the proof is the same as the proof of the Fundamental Theorem 6.2. (See Ex. 6.12). □

Remark 6.4.1. If in addition k is regular, then so is R. (This follows from the fact that for M ∈ M(R), pd_{R}(M) = fd_{R}(M); see [Q341, p. 36].) It follows from Theorem 3.3 that K(k) ≃ K(R).

Example 6.4.2. Let 𝔤 be a finite-dimensional Lie algebra over a field k. Then the universal enveloping algebra R = U(𝔤) is filtered, with F_{1}R = k ⊕ 𝔤, and its associated graded algebra is a polynomial ring (the symmetric algebra of the vector space 𝔤). Thus Theorem 6.4 implies that K(k) ≃ K(U(𝔤)).

Theorem 6.5. (Gersten) Let A = k{X} be a free k-algebra on a set X, where k is noetherian regular. Then K_{∗}(k{X}) ≅ G_{∗}(k{X}) ≅ K_{∗}(k).

Proof. It is known (see [CLL]) that A is a coherent regular ring (3.3), i.e., that the category M(A) of pseudo-coherent modules is abelian. Hence K_{∗}(A) ≅ G_{∗}(A) by Theorem 3.3. Replacing k by k[t], we see that the same is true of A[t].

Now A has the set of all words as a k-basis; we grade A by word length and consider the category of graded pseudo-coherent A-modules. The proof of 6.4 goes through (Exercise 6.13) to prove that G_{∗}(k) ≅ G_{∗}(A), as desired. □

As remarked in IV.1.9(iv), the calculation K_{∗}(ℤ{X}) = K_{∗}(ℤ) was used by Anderson to show that Swan’s definition K^{Sw}(R) of the higher K-theory of R agrees with Quillen’s definitions K_{0}(R) × BGL(R)^{+} and ΩBQP(R) (which we saw were equivalent in IV.7.2).

Dedekind Domains

Suppose that R is a Dedekind domain with fraction field F. Then R and F and regular, as are the residue fields R/𝔭, so K_{∗}(R) ≅ G_{∗}(R), etc. Hence the localization sequence of 6.1 with S = R − {0} becomes the long exact sequence:

(6.6)  ⋯ → K_{n+1}(F) ⟶^{∂} ⊕_{𝔭}K_{n}(R/𝔭) ⟶^{⊕(i_{𝔭})_{∗}} K_{n}(R) → K_{n}(F) ⟶^{∂} · · · .

Here 𝔭 runs over the nonzero prime ideals of R, and the maps (i_{𝔭})_{∗} : K_{n}(R/𝔭) → K_{n}(R) are the transfer maps of 3.3.2.

Writing K_{1}(R) = R^{×} ⊕ SK_{1}(R) (see III.1.1.1), the formula 6.1.2 allows us to identify the ending with the sequence 1 → R^{×} → F^{×} ⟶^{div} D(R) → K_{0}(R) → ℤ → 0 of I.3.6. Therefore we may extract the following exact sequence, which we have already studied in III.6.5:

(6.6.1) ⊕_{𝔭}K_{2}(R/𝔭) → K_{2}(R) → K_{2}(F) ⟶^{∂} ⊕_{𝔭}(R/𝔭)^{×} ⟶^{⊕(i_{𝔭})_{∗}} SK_{1}(R) → 1.

We claim that ∂ is the tame symbol of III.6.3 and that the above continues the sequence of III.6.5. Since the 𝔭-component of ∂ factors through the localization K_{2}(R) → K_{2}(R_{𝔭}) and the localization sequence for R_{𝔭}, we may suppose that R is a DVR with parameter π. In this case, we know that ∂ in K_{∗}(R)-linear, so if u ∈ R^{×} has image ū ∈ R/𝔭 then ∂{π, u} = [ū] in R/𝔭^{×}. Similarly, ∂ sends {π, π} = {π, −1} to {∂π, −1} = [R/π] · [−1], which is the class of the unit −1. Since every element of K_{2}(F) is a product of such terms modulo K_{2}(R), it follows that ∂ is indeed the tame symbol, as claimed.

Here are two special cases of (6.6) which arose in chapter III. First, if all of the residue fields R/𝔭 are finite, then K_{2}(R/𝔭) = 0 and we obtain the exact sequence:

0 → K_{2}(R) → K_{2}(F) ⟶^{∂} ⊕_{𝔭}(R/𝔭)^{×} ⟶^{⊕(i_{𝔭})_{∗}} SK_{1}(R) → 1.

Corollary 6.6.2. If R is any semilocal Dedekind domain, K_{3}(F)⟶^{∂}⊕K_{2}(R/𝔭) is onto, and we obtain the exact sequence

0 → K_{2}(R) → K_{2}(F) ⟶^{∂} ⊕_{𝔭}(R/𝔭)^{×} → 1.

Proof. It suffices to lift a symbol {ā, b̄} ∈ K_{2}(R/𝔭). We can lift ā, b̄ to units a, b of R as R is semilocal. Choose s ∈ R so that R/sR = R/𝔭. Since ∂ is a K_{∗}(R)-module homomorphism, we have:

∂({a, b, s}) = {a, b}∂(s) = {a, b} · [R/sR] = (i_{𝔭})_{∗}{a, b} = {ā, b̄}. □

Suppose that R ⊂ R′ is an inclusion of Dedekind domains, with R′ finitely generated as an R-module. Then the fraction field F′ of R′ is finite over F, so the exact functors M(R′) → M(R) and M(F′) → M(F) inducing the transfer maps (IV.6.3.3) are compatible. Thus we have a homotopy commutative diagram

(6.6.3)

K(MtorsR′) ⟶ G(R′) ⟶ G(F′)
↓  ↓  ↓
K(MtorsR) ⟶ G(R) ⟶ G(F).

Taking homotopy groups yields the morphism of localization sequences (6.6):

(6.6.4) \begin{matrix}K_{n}(R′) & ⟶ & K_{n}(F′) & ⟶^{⊕∂_{𝔭′}} & ⊕_{𝔭′}K_{n−1}(R′/𝔭′) & ⟶ & K_{n−1}(R′) \\ ↓^{N_{R′/R}} & ↓^{N_{F′/F}} & ↓^{N̄=⊕N_{𝔭′/𝔭}} & ↓ \\ K_{n}(R) & ⟶ & K_{n}(F) & ⟶^{∂} & ⊕_{𝔭}K_{n−1}(R/𝔭) & ⟶ & K_{n−1}(R). \end{matrix}

If R is a discrete valuation ring with fraction field F, parameter s and residue field R/sR = k, we define the specialization map λ_{s} : K_{n}(F) → K_{n}(k) by λ_{s}(a) = ∂({s, a}). A different choice of parameter will yield a different specialization map: if u ∈ R^{×} is such that s′ = us then λ_{s′}(a) = λ_{s}(a) + {ū, a}.

Theorem 6.7. If a discrete valuation ring R contains a field k_{0}, and [k : k_{0}] is finite, then the localization sequence (6.6) breaks up into split exact sequences:

0 → K_{n}(R) → K_{n}(F) ⟶_{←}^{∂} K_{n−1}(k) → 0.

Moreover, the canonical map K_{n}(R)→K_{n}(k) factors through the specialization map K_{n}(F) → K_{n}(k). A similar assertion holds for K-theory with coefficients.

Proof. Suppose first that k ⊂ R, so that (6.1) is a sequence of K_{∗}(k)-modules. Consider the map K_{n}(k) → K_{n+1}(F) sending a to {s, a}; we have ∂({s, a}) = {∂(s), a} = [k] · a = a. Hence ∂ is a split surjection, and the result follows.

In general, there is a finite field extension F′ of F so that the integral closure R′ of R contains k, and R → k extends to a map R′ → k with kernel 𝔭 = tR′. Then the map ∂′ : K_{n+1}(R′[1/t]) → K_{n}(k) is a surjection, split by a ↦ {t, a}, by the same argument. Now the composition M(R′/tR′) → M(R′) → M(R) is just M(k) → M(R) so, as in (6.6.4), we have a morphism of localization sequences

K_{n}(R′) ⟶ K_{n}(R′[1/t]) ⟶_{←}^{∂′} K_{n−1}(k)
↓  ↓  ∥
K_{n}(R) ⟶ K_{n}(F) ⟶^{∂} K_{n−1}(k).

It follows that ∂ is also a split surjection. □

Corollary 6.7.1. If k is a field, the localization sequence (6.6) for k[t] ⊂ k(t) breaks up into split short exact sequences:

0 → K_{n}(k) ⟶_{←} K_{n}(k(t)) ⟶^{∂} ⊕_{𝔭}K_{n−1}(k[t]/𝔭) → 0.

Proof. When R = k[t]_{(t)}, 6.7 says that the localization sequence for R ⊂ k(t) breaks up into split exact sequences 0 → K_{n}(R) → K_{n}(k(t)) → K_{n−1}(k) → 0. In addition, the map K_{n}(k) ≅ K_{n}(k[t]) → K_{n}(R) is split by (t = 0)^{∗}, so the localization sequence for k[t] → R also breaks up into the split exact sequences 0 → K_{n}(k) → K_{n}(R) ⟶^{∂} ⊕_{𝔭≠0}K_{n−1}(k[t]/𝔭) → 0. Combining these split exact sequences yields the result. □

We remark that if R contains any field then it always contains a field k_{0} so that k is algebraic over k_{0}. Thus the argument used to prove 6.7 actually proves more.

Corollary 6.7.2. If a discrete valuation ring R contains a field k_{0}, then each K_{n+1}(F) ⟶^{∂} K_{n}(k) is onto, and the localization sequence (6.6) breaks up into short exact sequences. (The sequences may not split.)

Proof. We may assume that k is algebraic over k_{0}, so that every element a ∈ K_{n}(k) comes from a ∈ K_{n}(k′) for some finite field extension k′ of k_{0}. Passing to a finite field extension F′ of F containing k′, such that R → k extends to R′ as above, consider the composite γ : K_{n}(k′) → K_{n+1}(F′) → K_{n+1}(F) sending a′ to the transfer of {t, a′}. As in the proof of Theorem 6.7, ∂γ : K_{n}(k′) → K_{n}(k) is the canonical map, so a = ∂γ(a′). □

Write i and π for the maps R → F and R → k, respectively.

Lemma 6.7.3. The composition of i^{∗} : K_{n}(R) → K_{n}(F) and λ_{s} is the natural map π^{∗} : K_{n}(R) → K_{n}(k). A similar assertion holds for K-theory with coefficients.

Proof. Because the localization sequence (6.6) is a sequence of K_{∗}(R)-modules, for every a ∈ K_{n}(R) we have

λ_{s}(i^{∗}a) = ∂({s, a}) = {∂(s), a} = [k] · a = π^{∗}(a). □

Corollary 6.7.4. If k is an algebraically closed field, and A is any commutative k-algebra, the maps K_{n}(k) → K_{n}(A) and K_{n}(k; ℤ/m) → K_{n}(A; ℤ/m) are injections.

Proof. Choosing a map A → F to a field, we may assume that A = F. Since F is the union of its finitely generated subfields F_{α} and K_{∗}(F) = \underrightarrow{lim} K_{∗}(F_{α}), we may assume that F is finitely generated over k. We may now proceed by induction on the transcendence degree of F over k. It is a standard fact that F is the fraction field of a discrete valuation ring R, with residue field E = R/sR. By Lemma 6.7.3, the composition of K_{∗}(k) → K_{∗}(F) with the specialization map λ_{s} is the natural map K_{n}(k) → K_{n}(E), which is an injection by the inductive hypothesis. It follows that K_{∗}(k) → K_{∗}(F) is an injection. □

To prove that K_{∗}(ℤ) injects into K_{∗}(ℚ) it is useful to generalize to integers in arbitrary global fields. Recall that a global field is either a number field or the function field of a curve over a finite field.

Theorem 6.8. (Soulé) Let R be a Dedekind domain whose field of fractions F is a global field. Then K_{n}(R) ≅ K_{n}(F) for all odd n ≥ 3; for even n ≥ 2 the localization sequence breaks up into exact sequences:

0 → K_{n}(R) → K_{n}(F) → ⨁_{𝔭} K_{n−1}(R/𝔭) → 0.

Proof. Let SK_{n}(R) denote the kernel of K_{n}(R) → K_{n}(F); from (6.6), it suffices to prove that SK_{n}(R) = 0 for n ≥ 1. For n = 1 this is the Bass-Milnor-Serre Theorem III.2.5 (and III.2.5.1). From the computation of K_{n}(𝔽_{q}) in IV.1.13 and the fact that K_{n}(R) is finitely generated (IV.6.9), we see that SK_{n}(R) is 0 for n > 0 even, and is finite for n odd. Fixing n = 2i−1 ≥ 1, we may choose a positive integer ℓ annihilating the (finite) torsion subgroup of K_{n}(R). Thus SK_{n}(R) injects into the subgroup K_{n}(R)/ℓ of K_{n}(R; ℤ/ℓ), which in turn injects into K_{n}(F; ℤ/ℓ) by Proposition 6.8.1 below. Since SK_{n}(R) vanishes in K_{n}(F) and hence in K_{n}(F; ℤ/ℓ), this forces SK_{n}(R) to be zero. □

Proposition 6.8.1. (Soulé) Let R be a Dedekind domain whose field of fractions F is a global field. Then for each ℓ and each even n ≥ 2, the boundary map ∂ : K_{n}(F; ℤ/ℓ) → ⊕K_{n−1}(R/𝔭; ℤ/ℓ) is onto in the localization sequence with coefficients ℤ/ℓ.

The conclusion of 6.8.1 is false for n = 1. Indeed, the kernel of K_{0}(R) → K_{0}(F) is the finite group Pic(R), so K_{1}(F; ℤ/ℓ) → ⊕K_{0}(R/𝔭; ℤ/ℓ) is not onto.

For n = 2, it is easy to see that the map ∂ : K_{2}(F; ℤ/ℓ) → ⊕K_{1}(R/𝔭; ℤ/ℓ) is onto, because K_{1}(R; ℤ/ℓ) injects into K_{1}(F; ℤ/ℓ) by Ex. IV.2.3.

Proof. Since K_{n−1}(R/𝔭)/2ℓ = K_{n−1}(R/𝔭; ℤ/2ℓ) surjects onto K_{n−1}(R/𝔭)/ℓ = K_{n−1}(R/𝔭; ℤ/ℓ) for each 𝔭, we may increase ℓ to assume that ℓ ≢ 2 (mod 4), so that the product (IV.2.8) is defined on K_{∗}(R; ℤ/ℓ).

Suppose first that R contains a primitive ℓ^{th} root of unity, ζ_{ℓ}. If β ∈ K_{2}(R; ℤ/ℓ) is the Bott element (IV.2.5.2), then multiplication by β^{i−1} induces an isomorphism ⊕ℤ/ℓ ≅ ⊕K_{1}(R/𝔭)/ℓ → ⊕K_{2i−1}(R/𝔭) by IV.1.13. That is, every element of ⊕K_{2i−1}(R/𝔭) has the form β^{i−1}a for a in ⊕K_{1}(R/𝔭). Lifting a to s ∈ K_{2}(F), the element x = β^{i−1}s of K_{2i}(F; ℤ/ℓ) satisfies ∂(x) = β^{i−1}∂(x) = β^{i−1}a, as desired.

In the general case, we pass to the integral closure R′ of R in the field F′ = F (ζ_{ℓ}). Every prime ideal 𝔭 of R has a prime ideal 𝔭′ of R′ lying over it, and the transfer maps K_{2i−1}(R′/𝔭′) → K_{2i−1}(R/𝔭) are all onto by IV.1.13. The surjectivity of ∂ follows from the commutative diagram

K_{2i}(F′; ℤ/ℓ) ⟶_{onto}^{∂′} ⊕_{𝔭′}K_{2i−1}(R′/𝔭′; ℤ/ℓ)
↓  ↓^{onto}
K_{2i}(F; ℤ/ℓ) ⟶^{∂} ⊕_{𝔭}K_{2i−1}(R/𝔭; ℤ/ℓ).  □

Wild Kernels 6.8.2. For even n, the map K_{n}(R; ℤ/ℓ) → K_{n}(F; ℤ/ℓ) need not be an injection. In fact the subgroup divK_{n}(F) of elements of K_{n}(F) which map to zero in each quotient K_{n}(F)/ℓ · K_{n}(F) lies in the torsion subgroup T of K_{n}(R) (by 6.8.1) and if ℓ · T = 0 then divK_{n}(F) is the kernel of K_{n}(R; ℤ/ℓ) → K_{n}(F; ℤ/ℓ). Tate observed that divK_{n}(F) can be nonzero even for K_{2}. In fact, divK_{2i}(F) is isomorphic to the wild kernel, defined as the intersection (over all valuations v on F) of the kernels of all maps K_{2i}(F) → K_{2i}(F_{v}). This is proven in [We06].

Gersten’s DVR Conjecture 6.9. Suppose R is a discrete valuation domain with maximal ideal 𝔪 = sR, residue field k = R/𝔪 and field of fractions F = R[s^{−1}]. The localization sequence (6.6) becomes

⋯ ⟶^{∂} K_{n}(k) ⟶^{i_{∗}} K_{n}(R) → K_{n}(F) ⟶^{∂} K_{n−1}(k) ⟶^{i_{∗}} K_{n−1}(R) → · · · .

Gersten conjectured that this sequence splits up (for every discrete valuation ring) into short exact sequences

0 → K_{n}(R) → K_{n}(F) → K_{n−1}(k) → 0.

This conjecture is known for n = 0, 1, 2 (see 6.6.2), when char(F) = char(k) (6.7 ff) or when k is algebraic over 𝔽_{p} (6.9.2 and Ex. 6.11). It is not known in the general mixed characteristic case, i.e., when char(F) = 0 and char(k) = p. However, it does hold for K-theory with coefficients ℤ/ℓ, as we now show.

Theorem 6.9.1. If R is a discrete valuation ring with residue field k, then the localization sequence with coefficients breaks up:

0 → K_{n}(R; ℤ/ℓ) → K_{n}(F; ℤ/ℓ) ⟶^{∂} K_{n−1}(k; ℤ/ℓ) → 0.

Proof. We begin with the case when 1/ℓ ∈ k, which is due to Gillet. The henselization R^{h} of R is a union of localizations R′_{𝔪′} over semilocal Dedekind domains R′ which are étale over R such that R′/𝔪′ ≅ k; see [Milne, I.4.8]. By Gabber Rigidity IV.2.10, K_{∗}(k; ℤ/ℓ) ≅ K_{∗}(R^{h}; ℤ/ℓ) ≅ \underrightarrow{lim} K_{∗}(R′_{𝔪′}; ℤ/ℓ). Hence for each a ∈ K_{n−1}(k; ℤ/ℓ) there is an (R′, 𝔪′) with R′/𝔪′ ≅ k and an a′ ∈ K_{n−1}(R′_{𝔪′}; ℤ/ℓ) mapping to a. If s′ ∈ R′ generates 𝔪′ then F′ = R′[1/s′] is the field of fractions of R′, and the product b = {a′, s′} is an element of K_{n}(F′; ℤ/ℓ). By (6.6.3), the functor M(R′) → M(R) induces a map of localization sequences

K_{n}(R′; ℤ/ℓ) ⟶ K_{n}(F′; ℤ/ℓ) ⟶^{⊕∂_{𝔭}} ⊕_{𝔭}K_{n−1}(R′/𝔭; ℤ/ℓ)
↓^{N_{R′/R}}  ↓^{N_{F′/F}}  ↓^{N̄=⊕N_{(R′/𝔭)/k}}
K_{n}(R; ℤ/ℓ) ⟶ K_{n}(F; ℤ/ℓ) ⟶^{∂} K_{n−1}(k; ℤ/ℓ).

Because the top row is a sequence of K_{∗}(R; ℤ/ℓ)-modules (6.1), we have ∂_{𝔭}{a′, s′} = {a′, ∂_{𝔭}s′} for all 𝔭. Since ∂(s′) = [R′/s′R′] = [k] by III.1.1, we see that ∂_{𝔪′}{a′, s′} = {a′, [k]} = a′ · 1 = a, while if 𝔭 ≠ 𝔪′ we have ∂_{𝔭}{a′, s′} = 0. Hence N̄ ⊕ ∂_{𝔭} sends {a′, s′} to a. By commutativity of the right square in the above diagram, we see that b = N_{F′/F} ({a′, s′}) is an element of K_{n}(F; ℤ/ℓ) with ∂(b) = a. This shows that ∂ is onto, and hence that the sequence breaks up as stated.

Next we consider the case that ℓ = p^{ν} and k has characteristic p, which is due to Geisser and Levine [GeiL, 8.2]. We will need the following facts from VI.4.7 below, which were also proven in op. cit.: the group K_{n−1}(k) has no ℓ-torsion, and K_{n}^{M}(k)/ℓ^{ν} → K_{n}(k; ℤ/ℓ^{ν}) is an isomorphism for all ν. By the Universal Coefficient Theorem IV.2.5, this implies that every element of K_{n}(k, ℓ) is the image of a symbol {a_{1}, …, a_{n}} with a_{i} in k^{×}. If a′_{i} ∈ R^{×} is a lift of a_{i} and s ∈ R is a parameter then ∂ : K_{1}(F) → K_{0}(k) sends s to [k] by III.1.1. It follows that ∂ : K_{n+1}(F; ℤ/ℓ) → K_{n}(k; ℤ/ℓ) is onto, because it sends {s, a′_{1}, …, a′_{n}} to {a_{1}, …, a_{n}}. □

We can now prove Gersten’s DVR Conjecture 6.9 when k is finite; the same proof works if k is algebraic over a finite field (Ex. 6.11).

Corollary 6.9.2. If R is a discrete valuation domain whose residue field k is finite, then for all i > 0: K_{2i−1}(R) ≅ K_{2i−1}(F) and there is a split exact sequence

0 → K_{2i}(R) → K_{2i}(F) ⟶_{←} K_{2i−1}(k) → 0.

Proof. It suffices to show that the map K_{2i}(F) ⟶^{∂} K_{2i−1}(k) is a split surjection in the localization sequence (6.6), because K_{2i}(k) = 0 by IV.1.13. Set ℓ = |k|^{i} − 1 and recall from IV.1.13 that K_{2i−1}(k) ≅ ℤ/ℓ, and hence K_{2i}(k; ℤ/ℓ) ≅ ℤ/ℓ. Pick a generator b of this group; using the surjection K_{2i}(F; ℤ/ℓ) → K_{2i}(F; ℤ/ℓ) of 6.9.1, lift b to an element a of K_{2i}(F; ℤ/ℓ). By compatibility of localization sequences (Ex. 5.2), the image of a under K_{2i}(F; ℤ/ℓ) → K_{2i−1}(F) is sent by ∂ to b. □

We can now determine the K-theory with coefficients of local fields.

Proposition 6.10. Let E be a local field, finite over ℚ_{p}, and let π be a parameter for the ring of integers in E. If the residue field is 𝔽_{q} and q ≡ 1 (mod m), then K_{∗}(E; ℤ/m) is a free ℤ/m[β]-module on generators 1 and z, where β is the Bott element and z is the class of π in K_{1}(E; ℤ/m) = E^{×}/E^{×m}.

Since {π, π} = {−1, π} in K_{2}(E) (III.5.10.2), the ring structure is given by z^{2} = [−1] · z. If q = 2^{ν} or if q ≡ 1 (mod 2m) then [−1] = 0 in K_{1}(E)/m so z^{2} = 0.

Proof. Since the ring R of integers in E is a complete DVR and R/πR ≅ 𝔽_{q}, Gabber rigidity (IV.2.9) implies that K_{∗}(R; ℤ/m) ≅ K_{∗}(𝔽_{q}; ℤ/m) ≅ ℤ/m[β]. By 6.9.2, K_{∗}(E; ℤ/m) is a free ℤ/m[β]-module on generators 1 and z. □

Remark 6.10.1. We will see in VI.7.4.1 that when n = p^{ν} we have

K_{2i−1}(E; ℤ/n) ≅ (ℤ/n)^{d} ⊕ ℤ/m_{i} ⊕ ℤ/m_{i−1},

where d = [E : ℚ_{p}], m_{i} = min{n, w_{i}^{(p)}} and the numbers w_{i}^{(p)} are defined in VI, 2.2 and 2.3. By Moore’s Theorem III.6.2.4, the torsion subgroup of K_{2}(E) is the group of roots of unity in E, cyclic of order w_{1}, so K_{3}(E) must be the direct sum of ℤ/w_{2}^{(p)} and an extension of a p-divisible group by ℤ_{(p)}^{d}.

I do not know how to reconstruct the other groups K_{∗}(E) from the above information; there might be a ℤ_{(p)}^{d} in all odd degrees, or there might be divisible p-torsion in even degrees.

Example 6.10.2. Consider the union E_{q} of the local fields E over ℚ_{p} whose ring of integers R has residue field 𝔽_{q}. For each such E, K_{∗}(E; ℤ/m) is described by Proposition 6.10. If E ⊂ E′ is a finite field extension, with ramification index e, the map K_{∗}(E; ℤ/m) → K_{∗}(E′; ℤ/m) sends z_{E} to e z_{E′}. If m divides e, the map sends z_{E} to 0. Taking the direct limit over all E, we see that K_{∗}(E_{q}; ℤ/m) = ℤ/m[β].

Localization for Schemes

Example 6.11. Let X be a noetherian scheme, j : U ⊂ X an open subscheme, and Z = X − U the closed complement. In this case we take A = M(X) and B = M_{Z}(X) the category of coherent X-modules supported on Z, i.e., modules whose restriction to U is zero. Gabriel has shown that M(X)/M_{Z}(X) is the category M(U) of coherent U-modules; see II.6.4.2. By Devissage (Ex. 4.3), KM_{Z}(X) ≃ G(Z). Hence we have a homotopy fibration G(Z) → G(X) → G(U), and the localization sequence becomes:

⋯ ⟶^{∂} G_{n}(Z) → G_{n}(X) ⟶^{j^{∗}} G_{n}(U) ⟶^{∂} G_{n−1}(Z) → ⋯

ending in the exact sequence G_{0}(Z) → G_{0}(X) → G_{0}(U) → 0 of II.6.4.2. This is a sequence of K_{∗}(X)-modules, because ⊗ is a biexact pairing of VB(X) with the sequence M(Z) → M(X) → M(U).

Example 6.11.1. If R is commutative noetherian, U = Spec(R[s]) is an open subset of the line ℙ_{R}^{1} with complement ∞ : Spec(R) ↪ ℙ_{R}^{1}. Since π : ℙ_{R}^{1} → Spec(R) is flat, the map π^{∗} : G(R) → G(ℙ_{R}^{1}) exists and j^{∗}π^{∗} ≃ p^{∗} is the homotopy equivalence of Theorem 6.2. It follows that π^{∗} splits the localization sequence

G(R) ⟶^{∞_{∗}} G(ℙ_{R}^{1}) ⟶^{j^{∗}} G(R) and hence we have G_{n}(ℙ_{R}^{1}) ≅ G_{n}(R) ⊕ G_{n}(R).

Mayer-Vietoris Sequences 6.11.2. If X = U ∪ V then Z = X − U is contained in V and V − Z = U ∩ V. Comparing the two localization sequences, we see that the square

G(X) ⟶ G(U)
↓  ↓
G(V) ⟶ G(U ∩ V).

is homotopy cartesian, i.e., there is a homotopy fibration sequence G(X) → G(U)× G(V) ⟶^{±} G(U ∩ V) which on homotopy groups yields the long exact “Mayer-Vietoris” sequence

⋯ → G_{n+1}(U ∩ V) ⟶^{∂} G_{n}(X) ⟶^{Δ} G_{n}(U) × G_{n}(V) ⟶^{±} G_{n}(U ∩ V) ⟶^{∂} · · · .

Smooth Curves 6.12. Suppose that X is an irreducible curve over a field k, with function field F. For each closed point x ∈ X, the field k(x) is a finite field extension of k. The category M_{0}(X) of coherent torsion modules (modules of finite length) is a Serre subcategory of M(X), and M(F) ≅ M(X)/M_{0}(X) (II.6.4.2). By Devissage (4.3), K_{∗}M_{0}(X) ≅ ⊕_{x}K_{∗}(k(x)) = ⊕K_{∗}(x). In this case, the Localization sequence (5.1.1) becomes:

⋯ → K_{n+1}(F) ⟶^{∂} ⊕_{x}K_{n}(k(x)) ⟶^{⊕(i_{x})_{∗}} G_{n}(X) → K_{n}(F) ⟶^{∂} · · · .

Here the maps (i_{x})_{∗} : K_{n}(x) → G_{n}(X) are the finite transfer maps of 3.6 associated to the inclusion of x = Spec(k(x)) into X. If X is regular, then K_{n}(X) ≅ G_{n}(X) and this sequence tells us about K_{n}(X).

The residue fields k(x) of X are finite field extensions of k. As such, we have transfer maps N_{k(x)/k} : K_{n}(k(x)) → K_{n}(k). The following result, due to Gillet, generalizes the Weil Reciprocity of III.6.5.3 for symbols {f, g} ∈ K_{2}(F). We write ∂_{x} for the component K_{n+1}(F) → K_{n}(x) of the map ∂ in 6.12.

Weil Reciprocity 6.12.1. Let X be a projective curve over a field k, with function field F. For every a ∈ K_{n+1}(F) we have the following formula in K_{n}(k):

∑_{x∈X} N_{k(x)/k}∂_{x}(a) = 0.

Proof. Consider the proper transfer π_{∗} : G_{n}(X) → G_{n}(k) of 3.7 associated to the structure map π : X → Spec(k). By functoriality, π_{∗}(i_{x})_{∗} = N_{k(x)/k} for each closed point x ∈ X. Because ⊕_{x}(i_{x})_{∗}∂_{x} = (⊕(i_{x})_{∗})∂ = 0, we have:

∑ N_{k(x)/k}∂_{x}(a) = ∑ π_{∗}(i_{x})_{∗}∂_{x}(a) = π_{∗} ∑(i_{x})_{∗}∂_{x}(a) = 0. □

Fundamental Theorem for G(X) 6.13. If X is a noetherian scheme, the flat

maps X[s, s^{−1}] ↪^{j} X[s] →^{p} X induce a homotopy equivalence p^{∗} : G(X) ≃ G(X[s]) and isomorphisms

G_{n}(X[s, s^{−1}]) ≅ G_{n}(X) ⊕ G_{n−1}(X).

Proof. If z : X → X[s] is the zero-section, we saw in 3.6.1 that z_{∗} = 0. Hence the Localization Sequence 6.11 splits into short exact sequences

0 ⟶^{z_{∗}} G_{n}(X[s]) ⟶^{j^{∗}} G_{n}(X[s, s^{−1}]) ⟶^{∂} G_{n−1}(X) ⟶^{z_{∗}} 0.

If f : X → X[s, s^{−1}] is the section s = 1, we saw in 3.6.1 that f defines a map f^{∗} and that f^{∗}j^{∗}p^{∗} is homotopic to the identity of G(X). Hence it suffices to prove that p^{∗} : G(X) ≃ G(X[s]).

We first suppose that X is separated, so that the intersection of affine opens is affine, and proceed by induction on the number of affine opens. If X is affine, this is Theorem 6.2, so suppose that X = U_{1} ∪ U_{2} with U_{2} affine. The inductive hypothesis applies to U_{1} and U_{12} = U_{1} ∩ U_{2}, and we have a map of Mayer-Vietoris sequences (6.11.2):

∗ ⟶ G_{n+1}(U_{12}) ⟶^{∂} G_{n}(X) ⟶ G_{n}(U_{1}) × G_{n}(U_{2}) ⟶ G_{n}(U_{12})
↓^{≃}  ^{≃}↓  ↓  ^{≃}↓  ^{≃}↓
∗ ⟶ G_{n+1}(U_{12}[s]) ⟶^{∂} G_{n}(X[s]) ⟶ G_{n}(U_{1}[s]) × G_{n}(U_{2}[s]) ⟶ G_{n}(U_{12}[s]).

The 5-lemma implies that G(X) ≃ G(X[s]) for X separated. Another induction establishes the result for non-separated X, since any open subscheme of a separated scheme is separated. □

Corollary 6.13.1. If X is noetherian, then X × ℙ^{1} ⟶^{π} X and X ⟶^{∞} X × ℙ^{1} induce homotopy equivalences (π^{∗}, ∞_{∗}) : G(X) ⊕ G(X) ⟶^{≃} G(X × ℙ^{1}).

Proof. As in 6.11.1, the complement of ∞(X) is X × 𝔸^{1} and the map π^{∗} is split by j^{∗} : G(X × ℙ^{1}) → G(X × 𝔸^{1}) ≃ G(X). Hence π^{∗} splits the localization

sequence G(X) ⟶^{∞_{∗}} G(X × ℙ^{1}) ⟶^{j^{∗}} G(X). □

Corollary 6.13.2. If X is regular noetherian then for all n: K_{n}(X)≅K_{n}(X[s]), K_{n}(X[s, s^{−1}]) ≅ K_{n}(X) ⊕ K_{n−1}(X) and K_{n}(X × ℙ^{1}) ≅ K_{n}(X) ⊕ K_{n}(X).

We now generalize 6.11.1 and 6.13.1 from ℙ^{1} to ℙ^{r}; the G_{0} version of the following result was given in II, Ex. 6.14. If X is regular, Theorem 6.14 and Ex. 6.7 provide another proof of the Projective Bundle Theorem 1.5 and 1.5.1.

Theorem 6.14. If X is a noetherian scheme, the functors M(X)⟶^{O(i)}M(ℙ_{X}^{r}) sending M to M ⊗ O(i) induce an isomorphism ∑ O(i): ∐_{i=0}^{r} G(X) ⟶^{≃} G(ℙ_{X}^{r}). That is, G_{n}(ℙ_{X}^{r}) ≅ G_{n}(X) ⊗ K_{0}(ℙ_{ℤ}^{r}).

Proof. By localization, we may assume X = Spec(R); set S = R[x_{0}, ⋯, x_{r}]. The standard description of coherent sheaves on ℙ_{R}^{r} ([Hart, Ex. II.5.9]) amounts to the equivalence of quotient categories

M(ℙ_{R}^{r}) ≅ M_{gr}(S)/M_{gr}^{b}(S),

where M_{gr}^{b}(S) is the Serre subcategory of graded modules M with M_{m} = 0 for all but finitely many m. By Devissage (Ex. 4.4), M_{gr}^{b}(S) has the same K-theory as its subcategory M_{gr}(R), i.e., G_{∗}(R)[σ, σ^{−1}]. Thus the localization sequence (5.1.1) and Example 3.5.2 give us the following diagram with exact rows

⋯ ⟶ K_{n}(M_{gr}(R)) ⟶ K_{n}(M_{gr}(R[x_{0}, ⋯, x_{r}])) ⟶ G_{n}(ℙ_{R}^{r}) ⟶ ⋯
≃↑  ^{≃}↑  ↑
0 ⟶ G_{n}(R) ⊗ ℤ[σ, σ^{−1}] ⟶^{h} G_{n}(R) ⊗ ℤ[σ, σ^{−1}] ⟶ G_{n}(R) ⟶ 0.

Now σ is induced by ⊗_{S}S(−1). Setting E = S^{r+1}, consider the Koszul resolution for R (I.5.3):

0 → ∧^{r+1}E(−r − 1) → ⋯ → ∧^{2}E(−2) → E(−1) → S → R → 0.

The Additivity Theorem for ⊗_{S}S : M_{gr}(S) → M_{gr}(S) shows that h is multiplication by the map ∑ \binom{r+1}{i}(−σ)^{i} = (1 − σ)^{r+1}. Since this is an injection with the prescribed cokernel, we have proven the first assertion. The second is immediate from II.8.6. □

EXERCISES

6.1 Suppose that R is a 1-dimensional commutative noetherian domain with fraction field F. Show that there is a long exact sequence

⋯ → K_{n+1}(F) ⟶^{∂} ⊕_{𝔭}K_{n}(R/𝔭) → G_{n}(R) → K_{n}(F) ⟶^{∂} · · · .

ending in the Heller-Reiner sequence of Ex. II.6.8. If R is regular, i.e., a Dedekind domain, this is the sequence (6.6).

Now let R be the local ring of ℝ + xℂ[x] at the maximal ideal (x, ix). Show that G_{0}(R) ≅ ℤ⊕ℤ/2, and conclude that the localization sequence (6.1.1) does not split up into short exact sequences. 6.2 If S is the seminormalization of R (I, Ex. 3.15–16), and S is finite over R, show that the transfer G(S) → G(R) is an equivalence. 6.3 If R is noetherian, show that the splitting in Theorem 6.2 is given by the map G_{n}(R) → G_{n+1}(R[s, s^{−1}]) which is multiplication by s ∈ K_{1}(ℤ[s, s^{−1}]). Hint: Use Ex. 5.3 with C = M(R). 6.4 Let φ be an automorphism of a commutative noetherian ring R, and let R_{φ}[t] be the twisted polynomial ring of all polynomials ∑ r_{i}t^{i} with multiplication determined by tr = φ(r)t. It is well known that R_{φ}[t] is a regular ring when R is. Show that the hypotheses of Theorem 6.4 are satisfied, so that G(R) ≃ G(R_{φ}[t]). Then show that {t^{n}} is a multiplicative system (II.A.1.1), so that the localization R_{φ}[t, 1/t] is a well defined ring. Then show that there is an exact sequence:

⋯ → G_{n}(R) ⟶^{φ_{∗}} G_{n}(R) → G_{n}(R_{φ}[t, 1/t]) → · · · .

6.5 Let R be commutative noetherian, and S = Sym(P) the symmetric algebra (I.5.8) of a finitely generated projective R-module P. Show that R ⊂ S induces an equivalence G(R) ≃ G(S). Hint: Locally, S ≅ R[s_{1}, ..., s_{n}].

6.6 Jouanolou’s Trick. Let X be a noetherian scheme. If p : E → X is a vector bundle, show that p^{∗} : G(X) → G(E) is an equivalence. More generally, show that p^{∗} : G(X) → G(E) is an equivalence for any flat map p : E → X whose fibers are affine spaces (such as a torsor under a vector bundle E).

Jouanolou proved that if X is quasi-projective over a field then there is an affine scheme E = Spec(R) and a torsor E → X under a vector bundle. It follows that G(X) ≃ G(R). This trick reduces the study of G(X) for quasi-projective X to the study of G(R). 6.7 (Quillen) If E is any vector bundle over X of rank r + 1, we can form the projective bundle ℙ(E) = Proj(Sym(E)). Modify the proof of Theorem 6.14 to show that once again the maps M ↦ M ⊗ O(i) induce an isomorphism

G_{n}(ℙ(E)) ≅ K_{0}(ℙ(E)) ⊗ G_{n}(X) ≅ ∐_{i=0}^{r} G_{n}(X).

6.8 Let X be the affine plane with a double origin over a field k, the standard example of a (quasi-compact) scheme which is not separated. Generalize II.8.2.4 by showing that G(X) ≃ G(k)×G(k). Then show that every vector bundle is trivial and VB(X) ≅ VB(𝔸^{2}), so KVB(X) ≃ K(k). Since G(X) ≃ K(X) (by Ex. 3.9), this shows that KVB(X) ≠ K(X). Note that G(X) = KVB(X) for separated regular schemes by Resolution (see 3.4.2). 6.9 (Roberts) If π : X′ → X is a finite birational map, there is a closed Z ⊂ X (nowhere dense) such that π : X′ − Z′ ≅ X − Z, where Z′ = Z ×_{X} X′.

(a) Show that there is a fibration G(Z′) → G(Z) × G(X′) → G(X). This yields a Mayer-Vietoris sequence

G_{n+1}(X) ⟶^{∂} G_{n}(Z′) ⟶^{Δ} G_{n}(Z) × G_{n}(X′) ⟶^{±} G_{n}(X) ⟶^{∂} G_{n−1}(Z′).

(b) If X is the node Spec(k[x, y]/(y^{2}−x^{3}−x)), show that G_{n}(X) ≅ G_{n}(k)⊕G_{n−1}(k) for all n. This contrasts with K_{0}(X) ≅ K_{0}(k) ⊕ K_{1}(k); see II.2.9.1. 6.10 Suppose that a curve X is a union of n affine lines, meeting in a set Z of m rational points (isomorphic to Spec(k)), and that no three lines meet in a point. Show that G_{n}(X) ≅ G_{n}(k)^{n} ⊕ G_{n−1}(k)^{m}. 6.11 Let R be a DVR whose residue field k is infinite, but algebraic over 𝔽_{p}. Modify the proof of 6.9.2 to show that Gersten’s DVR conjecture 6.9 holds for R. 6.12 Complete the proof of Theorem 6.4 for filtered rings. 6.13 Let R = k ⊕ R_{1} ⊕ ⋯ be a graded ring such that R and R[t] are coherent (3.3) and of finite flat dimension over k, and k has finite flat dimension over R. Modify the proof of Theorem 6.4 to show that k ⊂ R induces G(k) ≃ G(R). Hint: M_{gr}(S) is still an abelian category. 6.14 The formula λ_{s}(a) = ∂({s, a}) defines a specialization map for K-theory with coefficients ℤ/m. Now suppose that k is algebraically closed, and show that: (i) the specialization λ_{s}: K_{n}(F; ℤ/m) → K_{n}(k; ℤ/m) is independent of the choice of s ∈ R; (ii) if F = k(t) and then the specialization λ_{s} : K_{n}(F; ℤ/m) → K_{n}(k; ℤ/m) is independent of R and s. Hint: If f ∈ F then λ(f) = 0 as K_{1}(k; ℤ/m) = 0.

§7. Localization for K_{∗}(R) and K_{∗}(X).

When R is a ring and s ∈ R is a central element, we would like to say something about the localization map K(R) → K(R[\frac{1}{s}]). If R is regular, we know the third term in the long exact sequence from the localization theorem for G_{∗}:

⋯ ⟶^{∂} G_{n}(R/sR) ⟶^{i_{∗}} K_{n}(R) → K_{n}(R[1/s]) ⟶^{∂} · · · .

Because R is regular, every R/sR-module has finite projective dimension over R, and i_{∗} is induced from the inclusion M(R/sR) ⊂ M(R) = H(R). More generally, suppose that S is a central multiplicatively closed set of nonzerodivisors in R, and consider the category H_{S}(R) of all S-torsion R-modules M in H(R).

We saw in II.7.7.4 that the sequence K_{0}H_{S}(R) → K_{0}(R) → K_{0}(S^{−1}R) is exact. This K_{0} exact sequence has the following extension to higher K-theory.

Theorem 7.1. (Localization for Nonzerodivisors). Let S be a central multiplicatively closed subset of R consisting of nonzerodivisors. Then

KH_{S}(R) → K(R) → K(S^{−1}R)

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

⋯ → K_{n+1}(S^{−1}R) ⟶^{∂} K_{n}H_{S}(R) → K_{n}(R) → K_{n}(S^{−1}R) ⟶^{∂} ⋯

ending in K_{0}H_{S}(R) → K_{0}(R) → K_{0}(S^{−1}R), the sequence of II.7.7.4.

In this section, we will give a direct proof of Theorem 7.1, following Quillen. We note that an indirect proof is given in Exercise 3.14, using Theorem 2.6.3. In addition, we saw in Ex. 2.9 that the nonzerodivisor hypothesis is necessary. In the general case, the third term K(R on S) is more complicated; see Theorem 2.6.3.

Caveat 7.1.1. The map K_{0}(R) → K_{0}(S^{−1}R) is not onto. Instead, the sequence continues with K_{−1}H_{S}(R) etc., using negative K-groups. In order to get a spectrum-level fibration, therefore, one needs to use the nonconnective spectra K^{B}(R) of IV.10.4 to get nontrivial negative homotopy groups, or else replace K(S^{−1}R) with K(P), as we shall now do (and did in the proofs of II.9.8 and 2.6.3).

Definition 7.2. Let P denote the exact subcategory of P(S^{−1}R) consisting of projective modules which are localizations of projective R-modules.

By Cofinality (IV.6.4.1), K_{n}P ≅ K_{n}(S^{−1}R) for n > 0, and K_{0}(R) → K_{0}(S^{−1}R) factors as a surjection K_{0}(R) → K_{0}P followed by an inclusion K_{0}P → K_{0}(S^{−1}R). Hence we may replace K(S^{−1}R) by KP in Theorem 7.1.

Let E denote the category of admissible exact sequences in P, as defined in IV.7.3; morphisms are described in (IV.7.3.1), and the target of A ↣ B ↠ C yields a functor t : E → QP. We write F for the pullback category QP(R) ×_{QP} E whose objects are pairs (P, A ↣ B ↠ S^{−1}P) and whose morphisms are pairs of compatible morphisms. Since t is a fibered functor (by Ex. IV.7.2), so is F → QP(R).

The monoidal category T = iso P(R) acts fiberwise on E via the inclusion of T in E given by IV.7.4.1. Hence T also acts on F, and we may localize at T (IV.4.7.1).

Lemma 7.2.1. The sequence T^{−1}F → QP(R) → QP is a homotopy fibration.

Proof. By IV, Ex. 3.11, the fibers of T^{−1}F → QP(R) and T^{−1}E → P are equivalent. By Quillen’s Theorem B (IV.3.8.1), we have a homotopy cartesian square:

T^{−1}F ⟶ T^{−1}E (contractible)
↓  ↓^{t}
QP(R) ⟶ QP.

We saw in the proof of Theorem IV.7.1 that T^{−1}E is contractible (because E is, by Ex. IV.7.3), whence the result. □

To prove Theorem 7.1, we need to identify T^{−1}F and KH_{S}(R). We first observe that we may work with the exact subcategory H_{1,S} of H_{S}(R) consisting of S-torsion modules of projective dimension ≤ 1. Indeed, the Resolution Theorem (see 3.2.1) implies that K_{∗}H_{S}(R) ≅ K_{∗}H_{1,S}.

Next, we construct a diagram of categories on which T acts, of the form

QH_{1,S} ⟵^{h} G ⟶^{f} F

and whose localization is QH_{1,S} ⟵^{T^{−1}h} T^{−1}G ⟶^{T^{−1}f} T^{−1}F.

Definition 7.3. Let G denote the category whose objects are exact sequences 0 → K ↣ P ↠ M ⊕ Q with P, Q in P(R) and M in H_{1,S}. The morphisms in G are isomorphism classes of diagrams (in which the maps in the bottom row are direct sums of maps):

K′  ↢  K  =  K
∨ ↓  ∨ ↓  ∨ ↓
P′  =  P′  ↣  P
↓ ↓  ↓ ↓  ↓ ↓
M′ ⊕ Q′ ↞ M_{1} ⊕ Q_{1} ↣ M ⊕ Q.

We let T = iso P(R) act on G by T □(P ↠ M ⊕ Q) = T ⊕ P ↠ P ↠ M ⊕ Q.

The functor f : G → F sends K ↣ P ↠ M ⊕ Q to (Q, T −1P ↠ T^{−1}M). By inspection, f is compatible with the action of T, so T^{−1}f is defined.

The functor h : G → QH_{1,S} sends K ↣ P ↠ M ⊕ Q to M. The action of T is fiberwise for h (IV, Ex. 4.11) so h induces a functor T^{−1}h : T^{−1}G → QP.

Proof of Theorem 7.1. The maps QH_{1,S} ⟵^{≃} T^{−1}G ⟶^{≃} T^{−1}F are homotopy equivalences by Lemmas 7.3.1–2. Let g be the composite of these equivalences with T^{−1}F → QP(R), so that QH_{1,S} ⟶^{g} QP(R) → QP is a homotopy fibration by 7.2.1. In Lemma 7.4, we identify −g with the canonical map, proving the theorem. □

Lemma 7.3.1. The functor h : G → QH_{1,S} is a homotopy equivalence.

It follows that G → T^{−1}G and T^{−1}G → QH_{1,S} are also homotopy equivalences.

Proof. For each M, let G_{M} denote the category whose objects are surjections P ↠ M, and whose morphisms are admissible monics P′ ↣ P in P(R) compatible with the maps to M. Choosing a basepoint P_{0} ↠ M, P ×_{M} P′ is projective and there are natural transformations (P ↠ M) ← (P ×_{M} P′ ↠ M) → (P_{0} ↠ M) giving a contracting homotopy for each G_{M}.

The Segal subdivision Sub(G_{M}) of G_{M} (IV, Ex. 3.9) is equivalent to the fiber h^{−1}(M) (by Ex. 7.2), so the fibers of h are contractible. Since h is fibered (by Ex. 7.1), Quillen’s Theorem A (variation IV.3.7.4) applies to show that h is a homotopy equivalence. In particular, since T acts trivially on QH_{1,S} it acts invertibly on G in the sense of IV.4.7. It follows from Ex. IV.4.6 that G → T^{−1}G is also a homotopy equivalence. □

For the following lemma, we need the following easily checked fact: if P is a projective R-module, then P is a submodule of S^{−1}P, because S consists of central nonzerodivisors. In addition, S^{−1}P is the union of the submodules \frac{1}{s}P, s ∈ S.

Lemma 7.3.2. The functor f : G → F is a homotopy equivalence, and hence T^{−1}f : T^{−1}G → T^{−1}F is too.

Proof. We will show that for each Q in P(R) the map (pf)^{−1}(Q) → p^{−1}(Q) is a homotopy equivalence of fibers. Since p : F → QP(R) and pf : G → QP(R) are fibered (Ex. 7.1), the result will follow from Quillen’s Theorem B (IV.3.8.1).

Fix Q and let T_{Q} be the category whose objects are maps P ↠ Q with P in P(R), and whose morphisms are module injections P′ ↣ P over Q whose cokernel is S-torsion. Then the functor Sub(T_{Q}) → (pf)^{−1}(Q) which sends P′ ↣ P over Q to P ↠ Q ⊕ (P/P′) is an equivalence of categories (by Ex. 7.3).

Set W = S^{−1}Q, so that p^{−1}(Q) = E_{W}. Then Sub(T_{Q}) ⟶^{∼} (pf)^{−1}(Q) → E_{W}

sends P′ ↣ P over Q to S^{−1}P → W; this factors through the target functor Sub(T_{Q}) → T_{Q} (which is a homotopy equivalence by Ex. IV.3.9), and the functor w : T_{Q} → E_{W} sending P ↠ Q to S^{−1}P ↠ W. Thus we are reduced to showing that w is an equivalence. By Quillen’s Theorem A (IV.3.7), it suffices to fix V in P and E : V ↠ W in E_{W} and show that w/E is contractible.

Consider the poset Λ of projective R-submodules P of V such that S^{−1}P ≅ V, and whose image under V → W is Q. The evident map from Λ to w/E sending P to (P, S^{−1}P ↠ W) is an equivalence, so we are reduced to showing that Λ is contractible. Fix P and P′ in Λ and let K, K′ denote the kernels of P ↠ Q and P′ ↠ Q. Then S^{−1}K = S^{−1}K′ and S^{−1}K + P = S^{−1}K′ + P′ as submodules of V. Thus for some s ∈ S, both P and P′ are contained in the submodule P′′ = P + \frac{1}{s}K of V. But P′′ is easily seen to be in Λ, proving that Λ is a filtering poset. Thus Λ and hence w/E are contractible. □

Lemma 7.4. The canonical map QH_{1,S} → QP(R) is homotopic to the additive

inverse of g : QH_{1,S} ⟵^{T^{−1}h} T^{−1}G ⟶^{T^{−1}f} T^{−1}F ⟶^{p} QP(R).

Proof. Recall from II.7.7 that H_{1} = H_{1}(R) denotes the category of R-modules M having a resolution 0 → P_{1} → P_{0} → M → 0 with the P_{i} in P(R). By the Resolution Theorem 3.1.1, K(R) ≃ KH_{1} and QP(R) ≃ QH_{1}. We claim that the following diagram commutes up to sign.

G  ⟶^{p∘f} QP(R)
h↓  ↓^{≃}
QH_{1,S}(R) ⟶ QH_{1}.

The two functors from G to QH_{1} in this diagram send π = (K ↣ P ↠ M ⊕ Q) to M and Q, respectively. By Additivity 1.2, their sum is homotopic to the target functor t. It suffices to show that t is null homotopic.

Let s be the functor G → QP(R) sending π to P; since s maps all morphisms to injections, 0 ↣ P defines a natural transformation 0 ⇒ s. Since P ↠ M ⊕ Q is a natural transformation t ⇒ s, we have a homotopy t ∼ s ∼ 0, as desired. □

The following result generalizes Exercises III.3.10 and III.4.11.

Proposition 7.5. (Karoubi) Let f : A → B be a ring homomorphism and S a central multiplicatively closed set of nonzerodivisors in A such that f(S) is a central set of nonzerodivisors in B. Assume that f : A/sA ≅ B/sB for all s ∈ S. Then:

(1) the functor H_{S}(A) → H_{S}(B) is an equivalence;

(2) The square

K^{B}(A) ⟶ K^{B}(B)
↓  ↓
K^{B}(S^{−1}A) ⟶ K^{B}(S^{−1}B)

is homotopy cartesian, and there is a Mayer-Vietoris sequence

⋯ → K_{n+1}(S^{−1}B) ⟶^{∂} K_{n}(A) ⟶^{Δ} K_{n}(B) × K_{n}(S^{−1}A) ⟶^{±} K_{n}(S^{−1}B) ⟶^{∂} · · · .

Proof. The assumption implies that Â = \underleftarrow{lim} A/sA and B̂ = \underleftarrow{lim} B/sB are isomorphic. Therefore, in order to prove part (1) we may assume that B = Â. Then B is faithfully flat over A, and ⊗_{A}B is a faithful exact functor from H_{S}(A) to H_{S}(B). To show that H_{S}(A) → H_{S}(B) is an equivalence, it suffices to show that every M in H_{S}(B) is isomorphic to a module coming from H_{S}(A).

Choose a B-module resolution 0 → Q_{1} ⟶^{f} Q_{0} → M → 0, and s ∈ S so that sM = 0. Then the inclusion sQ_{0} ⊂ Q_{0} factors through a map f′ : sQ_{0} → Q_{1}. The cokernel M′ = Q_{1}/sQ_{0} is also in H_{S}(B), and if we choose Q′ so that Q_{0}⊕Q_{1}⊕Q′ ≅ B^{n} then we have a resolution

0 → B^{n} ⟶^{γ=(f,f′,1)} B^{n} → M ⊕ M′ → 0.

Replacing M by M ⊕ M′, we may assume that Q_{1} = Q_{0} = B^{n}. Then there is a matrix γ′ over B such that γγ′ = sI. Now the assumption that A/sA ≅ B/sB implies that there is a matrix α : A^{n} → A^{n} of the form α = γ + s^{2}β. Since α = γ(I + sγ′β) and I + sγ′β is invertible over B = Â, we see that the A-module coker(α) is in H_{S}(A) and that coker(α) ⊗_{A} B ≅ coker(γ) = M.

Part (2) follows from Part(1) and Theorem 7.1; the Mayer-Vietoris sequence follows formally from the square (as in 6.11.1). □

Example 7.5.1. The proposition applies to S = {p^{n}} and the rings ℤ → ℤ_{(p)} → ℤ̂_{p} (p-adics). If G is a group, it also applies to ℤ[G] → ℤ_{(p)}[G] → ℤ̂_{p}[G]. That is, K_{∗}(ℤ[G]) fits into a Mayer-Vietoris sequence involving K_{∗}(ℤ_{(p)}[G]), K_{∗}(ℤ[\frac{1}{p}][G]) and K_{∗}(ℚ[G]), as well as a Mayer-Vietoris sequence of the form:

K_{n+1}(ℚ̂_{p}[G]) ⟶^{∂} K_{n}(ℤ[G]) → K_{n}(ℤ̂_{p}[G]) ⊕ K_{n}(ℤ[\frac{1}{p}][G]) → K_{n}(ℚ̂_{p}[G]) ⟶^{∂} · · · .

More generally, let S ⊂ ℤ be generated by a finite set P of primes; Proposition 7.5 applies to ℤ[G] → ∏_{p∈P} ℤ̂_{p}[G], and there is a similar Mayer-Vietoris sequence relating K_{∗}(ℤ[G]) to the product of the K_{∗}(ℤ̂_{p}[G]). This sequence is particularly useful when G is finite and P is the set of primes dividing |G|, because the rings S^{−1}ℤ[G] and ℚ̂_{p}[G] are semisimple.

Localization for vector bundles

Here is the analogue of Theorem 7.1 for vector bundles on a scheme X, which generalizes the exact sequence in Ex. II.8.1. As discussed in II.8.3, we assume that X is quasi-compact in order that every module in H(X) has a finite resolution by vector bundles. We define K(X on Z) to be the K-theory space of the Waldhausen category Ch_{perf,Z}(X) of perfect complexes on X which are exact on U.

Theorem 7.6 (Thomason-Trobaugh [TT, 5.1]). Let X be a quasi-compact, quasi-separated scheme, and U a quasi-compact open in X with complement Z. Then K(X on Z) → K(X) ⟶^{j^{∗}} K(U) is a homotopy fibration, and there is a long exact sequence

⋯ → K_{n+1}(U) ⟶^{∂} K_{n}(X on Z) → K_{n}(X) → K_{n}(U) ⟶^{∂} ⋯

ending in K_{0}(X on Z) → K_{0}(X) → K_{0}(U), the sequence of Ex. II.8.1.

Recall that H_{Z}(X) denotes the category of modules in H(X) supported on Z.

Corollary 7.6.1. If X is quasiprojective and Z is defined by an invertible ideal, then H_{Z}(X) ⊂ Ch_{perf,Z}(X) induces an equivalence on K-theory. Thus there is a long exact sequence

⋯ → K_{n+1}(U) ⟶^{∂} K_{n}H_{Z}(X) → K_{n}(X) → K_{n}(U) ⟶^{∂} · · · .

Proof. (Thomason) The Approximation Theorem implies that KH_{Z}(X) ≃ K(X on Z), as we saw in Ex. 3.16 □

Remark 7.6.2. Since the model KCh_{perf,Z}(X) for the fiber K(X on Z) of K(X) → K(U) is complicated, it would be nice to have a simpler model. A naïve guess for such a model would be the K-theory of the category H_{Z}(X). This is correct if X is regular by the localization sequence 6.11 for G-theory, if Z is a divisor (7.6.1), and even if Z is locally a complete intersection in X [TT, 5.7]. Exercise 7.4 shows that this cannot be right in general, even if X = Spec(A).

Definition 7.6.3. Let us define the (non-connective) spectrum K^{B}(X on Z) to be the homotopy fiber of the morphism K^{B}(X) → K^{B}(X − Z). Then Theorem 7.6 states that the K-theory spectrum K(X on Z) of Ch_{perf,Z}(X) is the (−1)-connected cover of K^{B}(X on Z). In particular, K_{n}(X on Z) is π_{n}K^{B}(X on Z) for all n ≥ 0. For n < 0 we define K_{n}(X on Z) = π_{n}K^{B}(X on Z); since K_{n}(X) = π_{n}K^{B}(X) in this range (by IV.10.6), the sequence of Theorem 7.6 may be continued:

K_{0}(X) → K_{0}(U) → K_{−1}(X on Z) → K_{−1}(X) → K_{−1}(U) → · · · .

Proof of 7.6. (Thomason) For simplicity, we write A for the Waldhausen category Ch_{perf}(X) of II.9.2, in which weak equivalences are quasi-isomorphisms. Let w be the class of weak equivalences such that F ⟶^{∼}_{w} G if and only if F|_{U} ⟶^{∼} G|_{U}. By the Waldhausen Localization Theorem 2.1, we have a fibration K(X on Z) → K(X) → K(wA). Let G be the cokernel of K_{0}(X) → K_{0}(U) and let B denote the full Waldhausen category of all perfect complexes F on U with [F] = 0 in G. By the Cofinality Theorem 2.3, K_{n}(B) ≃ K_{n}(U) for all n > 0 and K_{0}(B) is the image of K_{0}(X) → K_{0}(U). Thus the proof reduces to showing that K(wA) → K(B) is a homotopy equivalence. By the Approximation Theorem 2.4, this reduces to showing that A → B induces an equivalence of derived categories. This is the conclusion of the following theorem of Thomason and Trobaugh. □

Theorem 7.7. w^{−1}Ch_{perf}(X) → w^{−1}B is an equivalence of triangulated categories. In more detail:

(a) For every perfect complex F on U with [F] in the image of K_{0}(X) → K_{0}(U), there is a perfect complex E on X such that E|_{U} ≃ F in D(U).

(b) Given perfect complexes E, E′ on X and a map b : E|_{U} → E′|_{U} in D(U), there is

a diagram E ⟵^{a} E′′ ⟶^{a′} E′ of perfect complexes on X so that a|_{U} is an isomorphism and b = a′|_{U}(a|_{U})^{−1} in D(U).

(c) If a : E → E′ is a map of perfect complexes on X which is 0 in D(U), there is a perfect complex E′′ and a map s : E′′ → E which is a quasi-isomorphism on U such that a s = 0 in D(X).

Proof. The proof of this theorem is quite deep, and beyond the level of this book; we quote [TT, 5.2.2–4] for the proof. As a side-note, it is part (a) that was suggested to Thomason by the Trobaugh simulacrum. □

7.8 We shall now give an elementary proof (due to Quillen) of Corollary 7.6.1 in the easier case when Z is a divisor and U is affine. We need assume that X is quasi-projective (over a commutative ring R), in order to use the definition K(X) = KVB(X) (IV.6.3.4). We write H_{1,Z}(X) for H_{Z}(X) ∩ H_{1}(X), the (exact) subcategory of modules in H(X) with a length one resolution by vector bundles, which are supported on Z.

For our proof of 7.6.1, let P denote the exact category of vector bundles on U of the form j^{∗}(P), where P is a vector bundle on X. The assumption that U is affine guarantees P is cofinal in VB(U). Therefore, as with the P of 8.2, we may replace K(U) by KP. We also need the following analogue of the easily checked fact about the relation between P and S^{−1}P. If P is a vector bundle on X then j^{∗}(P) is a vector bundle on U, and its direct image j_{∗}j^{∗}(P) is a quasicoherent module on X. If X = Spec(R) and I = sR then j_{∗}j^{∗}(P) is the R-module P [1/s].

Lemma 7.8.1. If P is a vector bundle on X, then P is a submodule of j_{∗}j^{∗}(P), and j_{∗}j^{∗}(P) is the union of its submodules I^{−n}P.

Proof of Corollary 7.6.1. The proof of Theorem 7.1 goes through formally, replacing P(R) by VB(X), S^{−1}R by U and H_{1,S} by H_{1,Z}(X), the (exact) subcategory of modules in H(X) with a length one resolution by vector bundles, which are supported on Z. The definitions of P, F and G make sense in this context, and every exact sequence in P splits because U is affine. Now everything goes through immediately except for the proof that Λ is a filtering poset. That proof used the structure of S^{−1}P; the argument still goes through formally if we use the corresponding structure of j_{∗}j^{∗}(P) which is given by Lemma 7.8.1. □

Proposition 7.9. (Excision) Let X and Z be as in Theorem 7.6, and let i : V ⊂ X be the inclusion of an open subscheme containing Z. Then the restriction i^{∗} : Ch_{perf,Z}(X) → Ch_{perf,Z}(V) is exact and induces a homotopy equivalence

K^{B}(X on Z) ⟶^{≃} K^{B}(V on Z).

Proof. ([TT, 3.19]) Let T (F) denote the functorial Godement resolution of F, and represent the functor Ri_{∗} as i_{∗} ∘ T : Ch_{perf,Z}(V) → Ch_{perf,Z}(X). This is an exact functor, taking values in the Waldhausen subcategory A of perfect complexes which are strictly zero on U = X − Z. The left exact functor Γ_{Z} (subsheaf with supports in Z) induces an exact functor Γ_{Z} ∘ T from Ch_{perf,Z}(X) to A which is an inverse of the inclusion up to natural quasi-isomorphism. Thus we have a homotopy equivalence KA ≃ KCh_{perf,Z}(X).

If F is a perfect complex on V, the natural map i^{∗}Ri_{∗}F → F is a quasiisomorphism since F → T (F) is. If G is a perfect complex on X, strictly zero on U, then G → Ri_{∗}(i^{∗}G) is a quasi-isomorphism because it is an isomorphism at points of Z and a quasi-isomorphism at points of U. Thus Ri_{∗} : KCh_{perf,Z}(V) → K(A) is a homotopy inverse of i^{∗}. This proves that K(X on Z) ⟶^{≃} K(V on Z). Given the Fundamental Theorem for schemes (8.3 below), it is a routine matter of bookkeeping to deduce the result for the non-connective Bass spectra K^{B}. □

Corollary 7.10. (Mayer-Vietoris) Let X be a quasi-compact, quasi-separated scheme, and U, V quasi-compact open subschemes with X = U ∪ V. Then the square

K^{B}(X) ⟶ K^{B}(U)
↓  ↓
K^{B}(V) ⟶ K^{B}(U ∩ V)

is homotopy cartesian, i.e., there is a homotopy fibration sequence

K^{B}(X) → K^{B}(U) × K^{B}(V) ⟶^{±} K^{B}(U ∩ V).

On homotopy groups, this yields the long exact “Mayer-Vietoris” sequence:

⋯ → K_{n+1}(U ∩ V) ⟶^{∂} K_{n}(X) ⟶^{Δ} K_{n}(U) × K_{n}(V) ⟶^{±} K_{n}(U ∩ V) ⟶^{∂} · · · .

Proof. Take Z = X − U in Theorem 7.6, and apply Proposition 7.9. □

Corollary 7.11. Let X be a quasi-compact, quasi-separated scheme, and U, V quasi-compact open subschemes with X = U ∪ V. Then the square

KH(X) ⟶ KH(U)
↓  ↓
KH(V) ⟶ KH(U ∩ V)

is homotopy cartesian. On homotopy groups, this yields the long exact “Mayer-Vietoris” sequence:

⋯ → KH_{n+1}(U ∩V) ⟶^{∂} KH_{n}(X) ⟶^{Δ} KH_{n}(U)×KH_{n}(V) ⟶^{±} KH_{n}(U ∩V) ⟶^{∂} · · · .

Proof. Recall from IV.12.7 that KH(X) is defined to be the realization of K^{B}(X × Δ^{·}). The square of simplicial spectra is degreewise homotopy cartesian by 7.10, and hence homotopy cartesian. □

EXERCISES

7.1 Show that the functors h : G → QP and pf : G → QP(R) of Definition 7.3 are fibered, with base change ϕ^{∗} constructed as in Lemma IV.6.7. (The proofs for h and pf are the same.) 7.2 Show that there is a functor from the Segal subdivision Sub(G_{M}) of G_{M} (IV, Ex. 3.9) to the fiber h^{−1}(M), sending P′ ↣ P to P → (P/P′) ⊕ M. Then show that it is an equivalence of categories. 7.3 Show that the functor Sub(T) → (pf)^{−1}(Q) which sends P′ ↣ P ↠ Q to P ↠ Q ⊕ (P/P′) is an equivalence of categories. (See IV, Ex. 7.3.)

7.4 (Gersten) Let A be the homogeneous coordinate ring of a smooth projective curve X over an algebraically closed field. Then A is a 2-dimensional graded domain such that the “punctured spectrum” U = Spec(A) − {𝔪} is regular, where 𝔪 is the maximal ideal at the origin. The blowup of Spec(A) at the origin is a line bundle over X, and U is isomorphic to the complement of the zero-section of this bundle.

(a) Show that K_{n}(U) ≅ K_{n}(X) ⊕ K_{n−1}(X).

(b) Suppose now that A is not a normal domain. We saw in II, Ex. 8.1 that H_{𝔪}(A) ≅ 0. Show that the image of K_{1}(A) → K_{1}(U) is k^{×}, and conclude that K_{∗}H_{𝔪}(A) cannot be the third term K_{∗}(Spec(A) on U) in the localization sequence for K_{∗}(A) → K_{∗}(U).

7.5 Let ℙ_{R}^{1} denote the projective line over an associative ring R, as in 1.5.4, let H_{1} denote the subcategory of modules which have a resolution of length 1 by vector bundles, as in Ex. 3.13, and let H_{1,t} denote the subcategory H_{1} consisting of modules F = (M, 0, 0).

(a) Show that H_{1,T} (R[t]) → H_{1,t}, M ↦ (M, 0, 0), is an equivalence of categories.

(b) Show that there is an exact functor VB(ℙ_{R}^{1}) ⟶^{j^{∗}} P(R[1/t]), j^{∗}(F) = M_{−}.

(c) Using Ex. 3.13, show that the inclusion H_{1,t} ⊂ H_{1} induces a homotopy fibration

sequence KH_{T} (R[t]) → K(ℙ_{R}^{1}) ⟶^{i^{∗}} K(R[s]).

(d) If R is commutative, show that the fibration sequence in (c) is the same as the sequence in 7.6.1 with Z the origin of X = ℙ_{R}^{1}.

For the next few exercises, we fix an automorphism φ of a ring R, and form the twisted polynomial ring R_{φ}[t] and its localization R_{φ}[t, 1/t], as in Ex. 6.4. Similarly, there is a twisted polynomial ring R_{φ^{−1}}[s] and an isomorphism R_{φ}[t, 1/t] ≅ R_{φ^{−1}}[s, 1/s] obtained by identifying s = 1/t. Let H_{1,t}(R_{φ}[t]) denote the exact category of t-torsion modules in H_{1}(R[t]).

7.6 Show that KH_{1,t}(R_{φ}[t]) → K(R_{φ}[t]) → K(R_{φ}[t, 1/t]) is a homotopy fibration. This generalizes Theorem 7.1 and parallels Ex. 6.4. (This result is due to Grayson; the induced K_{1}–K_{0} sequence was discovered by Farrell and Hsiang.) 7.7 (Twisted projective line ℙ_{R,φ}^{1}) We define mod- ℙ_{R,φ}^{1} to be the abelian category of triples F = (M_{t}, M_{s}, α), where M_{t} (resp., M_{s}) is a module over R_{φ}[t] (resp. R_{φ^{−1}}[s]) and α is is an isomorphism M_{t}[1/t] ⟶^{≃} M_{s}[1/s]. It has a full (exact) subcategory VB(ℙ_{R,φ}^{1}) consisting of triples where M_{t} and M_{s} are finitely generated projective modules, and we write K(ℙ_{R,φ}^{1}) for KVB(ℙ_{R,φ}^{1}).

(a) Show that F(−1) = (M_{t}, φ^{∗}M_{s}, αt) is a vector bundle whenever F is.

(b) Generalize Theorem 1.5.4 to show that K(R) × K(R) ≃ K(ℙ_{R,φ}^{1}).

(c) Show that there is an exact functor VB(ℙ_{R,φ}^{1}) ⟶^{j^{∗}} P(R_{φ^{−1}}[s]), j^{∗}(F) = M_{s}.

(d) Show that H_{1,t}(R_{φ}[t]) is equivalent to the category of modules F having a resolution of length 1 by vector bundles, and with j^{∗}(F) = 0. (Cf. Ex. 7.5(a).)

(e) Generalize Ex. 7.5(c) to show that there is a homotopy fibration

KH_{T} (R[t]) → K(ℙ_{R,φ}^{1}) ⟶^{j^{∗}} K(R[s]).

§8. The Fundamental Theorem for K_{∗}(R) and K_{∗}(X).

The main goal of this section is to prove the Fundamental Theorem, which gives a decomposition of K_{∗}(R[t, 1/t]). For regular rings, the decomposition simplifies to the formulas NK_{n}(R) = 0 and K_{n}(R[t, 1/t]) ≅ K_{n}(R) ⊕ K_{n−1}(R) of Theorem 6.3.

Let R be a ring, set T = {t^{n}} ⊂ R[t], and consider the category H_{1,T} of ttorsion R[t]-modules M in H_{1}(R[t]). On the one hand, we know from II.7.8.2 that H_{1,T} is equivalent to the category Nil(R) of nilpotent endomorphisms of projective R-modules (II.7.4.4). We also saw in IV.6.7 that the forgetful functor induces a decomposition KNil(R) ≃ K(R) × Nil(R).

Theorem 8.1. For every R and every n, Nil_{n}(R) ≅ NK_{n+1}(R)

Theorem 8.1 was used in IV.6.7 to derive several properties of the group NK_{∗}(R), including: if R is a ℤ/pℤ-algebra then each NK_{n}(R) is a p-group, and if ℚ ⊂ R then NK_{n}(R) is a uniquely divisible abelian group.

Proof. We know from Ex. 7.5 that Nil(R) is equivalent to the category H_{1,t} of modules F on the projective line ℙ_{R}^{1} with j^{∗}F = 0, and which have a length 1 resolution by vector bundles. Substituting this into Corollary 7.6.1 (or Ex. 7.5 if R is not commutative) yields the exact sequence

(8.1.1) ⋯ K_{n+1}(R[1/t]) → K_{n}(R) ⊕ Nil_{n}(R) ⟶ K_{n}(ℙ_{R}^{1}) ⟶^{j^{∗}} K_{n}(R[1/t]) → · · · .

Now the composition P(R) ⊂ Nil(R) → H(ℙ_{R}^{1}) sends P to (P, 0, 0), and there is an exact sequence u_{1}(P) ↣ u_{0}(P) ↠ (P, 0, 0) obtained by tensoring P with the standard resolution 0 → O(−1) → O → (R, 0, 0) → 0. By Additivity 1.2, the corresponding map K(R) → K(ℙ_{R}^{1}) in (8.1.1) is u_{0} − u_{1}. By the Projective Bundle Theorem 1.5 (or 1.5.4 if R is not commutative), the map (u_{0}, u_{0} − u_{1}) : K(R) × K(R) → K(ℙ_{R}^{1}) is an equivalence. Since j^{∗}u_{0}(P) ≅ P ⊗_{R} R[1/t], the composition j^{∗} ∘ u_{0} : K(R) → K(R[1/t]) is the standard base change map inducing the decomposition K_{n}(R[1/t]) ≅ K_{n}(R)⊕NK_{n}(R). Thus (8.1.1) splits into the split extension 0 → K_{n}(R) ⟶^{u_{0}−u_{1}} K_{n}(ℙ_{R}^{1}) → K_{n}(R) → 0 and the desired isomorphism

NK_{n+1}(R) ≅ K_{n+1}(R[t])/K_{n+1}(R) ⟶^{≃} Nil_{n}(R). □

Theorem 8.2. There is a canonically split exact sequence

0 → K_{n}(R) ⟶^{Δ} K_{n}(R[t]) ⊕ K_{n}(R[1/t]) ⟶^{±} K_{n}(R[t, 1/t]) ⟶_{←}^{∂} K_{n−1}(R) → 0.

in which the splitting of ∂ is given by multiplication by t ∈ K_{1}(ℤ[t, t^{−1}]).

Proof. Because the base change mod- ℙ_{R}^{1} → mod-R[t] is exact, there is a map between the localization sequences for T in Theorem 7.1 and (8.1.1), yielding the commutative diagram:

⋯ → K_{n}H_{1,T} ⟶ K_{n}(ℙ_{R}^{1}) ⟶^{j^{∗}} K_{n}(R[1/t]) ⟶ K_{n−1}H_{1,T} → ⋯
∥  ↓  ↓  ∥
⋯ → K_{n}H_{1,T} ⟶^{f_{∗}⊕0} K_{n}(R[t]) ⟶^{j^{∗}} K_{n}(R[t, 1/t]) ⟶ K_{n−1}H_{1,T} → · · · .

Now K_{n}(R) → K_{n}Nil(R) → K_{n}(R[t]) is the transfer map f_{∗} of (3.3.2), induced from the ring map f : R[t] → R, and Nil_{n}(R) → K_{n}(ℙ_{R}^{1}) is zero by the proof of Theorem 8.1, Since f_{∗} is zero by 3.5.1, the diagram yields the exact sequence Seq(K_{n}, R) displayed in the Theorem, for n ≥ 1. The exact sequence Seq(K_{n}, R) for n ≤ 0 was constructed in III.4.1.2.

To see that Seq(K_{n}, R) is split exact, we only need to show that ∂ is split by the cup product with [t] ∈ K_{1}(R[t, 1/t]). But the maps in the localization sequence commute with multiplication by K_{∗}(R), by Exercise 8.1 (or Ex. IV.1.23). Hence we have the formula: ∂({t, x}) = ∂(t) · x = [R[t]/tR[t]] · x. This shows that x ↦ {t, x} is a right inverse to ∂; the maps t ↦ 1 from K_{n}(R[t]) and K_{n}(R[1/t]) to K_{n}(R) yield the rest of the splitting. □

There is of course a variant of the Fundamental Theorem 8.2 for schemes. For every scheme X, let X[t] and X[t, t^{−1}] denote the schemes X × Spec(ℤ[t]) and X × Spec(ℤ[t, t^{−1}]) respectively.

Theorem 8.3. For every quasi-projective scheme X we have canonically split exact sequences for all n, where the splitting of ∂ is by multiplication by t.

0 → K_{n}(X) ⟶^{Δ} K_{n}(X[t]) ⊕ K_{n}(X[1/t]) ⟶^{±} K_{n}(X[t, 1/t]) ⟶_{←}^{∂} K_{n−1}(X) → 0.

in which the splitting of ∂ is given by multiplication by t ∈ K_{1}(ℤ[t, t^{−1}]).

Proof. Consider the closed subscheme X_{0} = X × 0 of ℙ_{X}^{1}. The open inclusion X[t] ↪ ℙ_{X}^{1} is a flat map, and induces a morphism of homotopy fibration sequences for n ≥ 1 from Corollary 7.6.1:

(8.3.1) \begin{matrix}K_{n}H_{X_{0}}(ℙ_{X}^{1}) & ⟶ & K_{n}(ℙ_{X}^{1}) & ⟶^{j^{∗}} & K_{n}(X[1/t]) & ⟶^{∂} & K_{n−1}H_{X_{0}}(ℙ_{X}^{1}) \\ ∥ & & ↓ & & ↓ & & ∥ \\ K_{n}H_{X_{0}}(X[t]) & ⟶ & K_{n}(X[t]) & ⟶^{j^{∗}} & K_{n}(X[t, 1/t]) & ⟶^{∂} & K_{n−1}H_{X_{0}}(X[t]). \end{matrix}

As in the proof of Theorem 8.1, there is an exact sequence u_{1} ↣ u_{0} ↠ i_{∗} of functors from VB(X) to H(ℙ_{X}^{1}), where u_{i}(E) = E ⊗ O(i) and i_{∗} is the restriction of scalars associated to i : X_{0} ↪ ℙ_{X}^{1}. By Additivity 1.2, i_{∗} = u_{0} − u_{1}. Since j^{∗}u_{0} is the base change, we see that the top row splits in the same way that (8.1.1) does. Since the map K_{n}(X) → K_{n}H_{X_{0}}(X[t]) → K_{n}(X[t]) is zero by 3.6.1, the bottom left map is zero on homotopy groups. Via a diagram chase on (8.3.1), using the Projective Bundle Theorem 1.5, the exact sequence of Theorem 8.3 follows formally for n ≥ 1, and for n = 0 provided we define K_{−1}(X) to be the cokernel of the displayed map ‘±’.

Since the maps in (8.3.1) are K_{∗}(X)-module maps (by Ex. 8.2) we have ∂({t, x})= ∂(t) · x. But the base change X[t] ⟶^{π} Spec(ℤ[t]) induces a morphism of localization sequences, and t lifts to K_{1}(ℤ[t, 1/t]), so by naturality we have ∂_{X}(t) = π^{∗}∂_{ℤ}(t) = π^{∗}([ℤ[t]/t]) = 1 in the subgroup K_{0}(X) of K_{0}H_{X_{0}}(X[t]). Hence ∂({t, x}) = x, regarded as an element of the subgroup K_{∗}(X) of K_{∗}H_{X_{0}}(X[t]).

Finally, given the result for n = 1, the result follows formally for n ≤ 0, where K_{n−1}(X) is given by Definition IV.10.6; see Ex. 8.3. □

Remark 8.3.2. K_{n}(X) was defined for n ≤ −1 to be π_{n}K^{B}(X) in Definition IV.10.6, where K^{B}(X) was defined using Theorem 8.4 below. Unravelling that definition, we see that the groups K_{n}(X) may also be inductively defined to be the cokernel LK_{n+1}(X) of K_{n+1}(X[t])⊕K_{n+1}(X[1/t]) → K_{n+1}(R[t, 1/t]). As with Definition III.4.1.1, this definition is concocted so that Theorem 8.3 remains true for n = 0 and also for all negative n; see Ex. 8.3.

Theorems 8.2 and 8.3 have versions which involve the (connective) spectra K(R) and K(X) associated to K(R) and K(X). Recall from IV.10.1 that the spectrum ΛK(R) is defined so that we have a cofibration sequence

ΛK(R) → K(R[t]) ∨_{K(R)} K(R[1/t]) ⟶^{f_{0}} K(R[t, 1/t]) ⟶^{∂} Ω^{−1}ΛK(R).

Replacing R by X in Definition IV.10.1 yields a spectrum ΛK(X) fitting into the cofibration sequence obtained from this one by replacing R with X throughout. Fixing a spectrum map S^{1} → K(ℤ[t, 1/t]) representing [x] ∈ [S^{1}, K(ℤ[t, 1/t])] = K_{1}(ℤ[t, 1/t]), the product yields a map K(R) to K(R[t, 1/t]); composed with Ω∂, it yields morphisms of spectra K(R) → ΛK(R), and K(X) → ΛK(X). The following result was used in section IV.10 to define the non-commutative “Bass K-theory spectra” K^{B}(R) and K^{B}(X) as the homotopy colimit of the iterates Λ^{k}K(R) and Λ^{k}K(X).

Theorem 8.4. 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.

Similarly, for any quasi-projective scheme X, the map K(X) → ΛK(X) induces a homotopy equivalence between K(X) and the (−1)-connective cover of the spectrum ΛK(X). In particular, K_{n}(X) ≅ π_{n}ΛK(X) for all n ≥ 0.

Proof. We give the proof for R; the proof for X is the same. By Theorem 8.3, there is a morphism K(R) → ΛK(R) which is an isomorphism on π_{n} for all n ≥ 0, and the only other nonzero homotopy group is π_{−1}ΛK(R) = K_{−1}(R). The theorem is immediate. □

Corollary 8.4.1. For every ring R, the spectrum K^{B}(R[t]) decomposes as K^{B}(R) ∨ NK^{B}(R) and the spectrum K^{B}(R[t, 1/t]) splits as

K^{B}(R[t, 1/t]) ≃ K^{B}(R) ∨ NK^{B}(R) ∨ NK^{B}(R) ∨ Ω^{−1}K^{B}(R).

K_{n}-regularity

The following material is due to Vorst [Vo] and van der Kallen. Let s be a central nonzerodivisor in R, and write [s] : R[x] → R[x] for the substitution f(x) ↦ f(sx). Let NK_{∗}(R)_{[s]} denote the colimit of the directed system

NK_{∗}(R) ⟶^{[s]} NK_{∗}(R) ⟶^{[s]} NK_{∗}(R) ⟶^{[s]} · · · .

Lemma 8.5. For any central nonzerodivisor s, NK_{∗}(R)_{[s]} ⟶^{≃}NK_{∗}(R[1/s]). In particular, if NK_{n}(R) = 0 then NK_{n}(R[1/s]) = 0.

Proof. Let C be the colimit of the directed system R[x] ⟶^{[s]} R[x] ⟶^{[s]} R[x] ⋯ of ring homomorphisms. There is an evident map C → R splitting the inclusion, and the kernel is the ideal I = xR[1/s][x] ⊂ R[1/s][x]. By IV.6.4, K_{∗}(C, I) is the filtered colimit of the groups NK_{n}(R) = K_{n}(R[x], x) along the maps [s], i.e., NK_{∗}(R)_{[s]}. Moreover, C[1/s] ≅ R[1/s][x].

We can apply Proposition 7.5 to R → C, since I/sI = 0 implies R/s^{i}R ≅ C/s^{i}C, to obtain the Mayer-Vietoris sequence

⋯ → K_{n}(R) ⟶^{Δ} K_{n}(C) × K_{n}(R[1/s]) ⟶^{±} K_{n}(R[1/s][x]) ⟶^{∂} · · · .

Splitting off K_{n}(R) from K_{n}(C) and K_{n}(R[1/s]) from K_{n}(R[1/s][x]) yields the desired isomorphism of NK_{n}(R)_{[s]} ≅ K_{n}(C, I) with NK_{n}(R[1/s]). □

Example 8.5.1. If R is commutative and reduced, then NK_{n}(R) = 0 implies that NK_{n}(R_{P}) = 0 for every prime ideal P of R. Vorst [Vo] also proved the converse: if NK_{n}(R_{𝔪}) = 0 for every maximal ideal 𝔪 then NK_{n}(R) = 0. Van der Kallen proved the stronger result that NK_{n} is a Zariski sheaf on Spec(R).

Recall from III.3.4 that R is called K_{n}-regular if K_{n}(R) ≅ K_{n}(R[t_{1}, …, t_{m}]) for all m. By Theorem 6.3, every noetherian regular ring is K_{n}-regular for all n.

Theorem 8.6. If R is K_{n}-regular, then it is K_{n−1}-regular.

More generally, if K_{n}(R) ≅ K_{n}(R[s, t]) then K_{n−1}(R) ≅ K_{n−1}(R[s]).

Proof. It suffices to suppose that K_{n}(R) ≅ K_{n}(R[s, t]), so that NK_{n}(R[s]) = 0, and prove that NK_{n−1}(R) = 0. By 8.5, NK_{n}(R[s, 1/s]) = 0. But NK_{n−1}(R) is a summand of NK_{n}(R[s, 1/s]) by the Fundamental Theorem 8.2. □

The following partial converse was proven in [CHWW].

Theorem 8.7. Let R be a commutative ring containing ℚ.

If R is K_{n−1}-regular and NK_{n}(R) = 0, then R is K_{n}-regular.

Remark 8.7.1. There are rings R for which NK_{n}(R) = 0 but R is not K_{n}- regular. For example, the ring R = ℚ[x, y, z]/(z^{2} + y^{3} + x^{10} + x^{7}y) has K_{0}(R) ≅ K_{0}(R[t]) but K_{0}(R) ≅ K̸_{0}(R[s, t]), and K_{−1}(R) ≅ K̸_{−1}(R[t]); see [CHWW].

EXERCISES

8.1 For any central nonzerodivisor s ∈ R, multiplication by [s] ∈ K_{1}(R[1/s]) yields a map K_{n}(R) → K_{n+1}(R[1/s]). Show that the boundary map ∂ : K_{n+1}(R[1/s]) → K_{n}H_{s}(R) satisfies ∂({s, x}) = x̄ for every x ∈ K_{n}(R), where x̄ is the image of x under the natural map K_{n}(R) → K_{n}(R/sR) → K_{n}H_{s}(R). Hint: Use Exercises 5.3 and IV.1.23; the ring map ℤ[t] → R, t ↦ s, induces compatible pairings P(ℤ[t]) × P(R) → P(R) and P(ℤ[t, 1/t]) × P(R) → P(R[1/s]). 8.2 Show that VB(X) acts on the terms in the localization sequence of Theorem 8.7 in the sense of IV.6.6.4. Deduce that the map ∂ : K_{n}(U) → K_{n−1}H_{Z}(X) satisfies ∂({u, x}) = ∂(u) · x for u ∈ K_{∗}(U) and x ∈ K_{∗}(X). Hint: use Ex. IV.1.23 and mimick Ex. 5.3. 8.3 Given a scheme X, show that F_{n}(R) = K_{n}(X × Spec R) is a contracted functor in the sense of III.4.1.1. Then show that the functor K_{−n}(X) of 8.3.1 is the contracted functor L^{n}F_{0}(ℤ). 8.4 Twisted Nil groups, Let φ be an automorphism of a ring R, and consider the category Nil(φ) whose objects are pairs (P, ν), where P is a finitely generated projective R-module and ν is a nilpotent endomorphism of P which is semi-linear in the sense that ν(xr) = ν(x)φ(r). If φ = id_{R}, this is the category Nil(R) of II.7.4.4. As in loc. cit., we define Nil_{n}(φ) to be the kernel of the map K_{n}Nil(φ) → K_{n}(R) induced by (P, ν) ↦ P. Similarly, we define NK_{n}(φ) to be the cokernel of the natural map K_{n}(R) → K_{n}(R_{φ}[t]).

(a) Show that K_{n}Nil_{φ}(R)≅K_{n}(R) ⊕ Nil_{n}(φ), and K_{n}(R_{φ}[t])≅K_{n}(R) ⊕ NK_{n}(φ).

(b) Show that Nil_{φ}(R) is equivalent to the category H_{1,t}(R_{φ}[t]) of Ex. 7.6.

(c) Show that NK_{n}(φ) ≅ NK_{n}(φ^{−1}). Hint: P(R^{op}) ≅ P(R)^{op} by IV.6.4.

(d) Prove the twisted analogue of Theorem 8.1: Nil_{n}(φ) ≅ NK_{n+1}(φ) for all n.

8.5 (Grayson) Let K^{φ}(R) be the homotopy fiber of K(R) ⟶^{1−φ^{∗}} K(R), and set K_{n}^{φ}(R) = π_{n}K^{φ}(R). If R is regular, use Ex. 6.4 to show that K_{∗}(R_{φ}[t, 1/t]) ≅ K_{∗}^{φ}(R). Then show that there is a canonical isomorphism for any R:

K_{n}(R_{φ}[t, 1/t]) ≅ K_{n}^{φ}(R) ⊕ Nil_{n−1}(φ) ⊕ Nil_{n−1}(φ^{−1}).

§9. The coniveau spectral sequence of Gersten and Quillen

In this section we give another application of the Localization Theorem 5.1, which reduces the calculation of G_{n}(X) to a knowledge of the K-theory of fields, up to extensions. The prototype of the extension problem is illustrated by the exact sequences in 6.6 and 6.12, for Dedekind domains and smooth curves.

Suppose first that R is a finite-dimensional commutative noetherian ring. We let M^{i}(R) denote the subcategory of M(R) consisting of those R-modules M whose associated prime ideals all have height ≥ i. We saw in II.6.4.3 and Ex. II.6.9 that each M^{i}(R) is a Serre subcategory of M(R) and that M^{i}/M^{i+1}(R) ≅ ⨁_{ht(𝔭)=i} M_{𝔭}(R_{𝔭}), where M_{𝔭}(R_{𝔭}) is the category of R_{𝔭}-modules of finite length.

By Devissage (Application 4.4) we have KM_{𝔭}(R_{𝔭}) ≃ G(k(𝔭)) ≃ K(k(𝔭)), where k(𝔭) = R_{𝔭}/𝔭R_{𝔭}, so K_{∗}M^{i}/M^{i+1}(R) ≅ ⊕_{ht(𝔭)=i}G_{∗}(k(𝔭)). The Localization Theorem yields long exact sequences

⟶^{∂} K_{n}M^{i+1}(R) → K_{n}M^{i}(R) → ⨁_{ht(𝔭)=i}K_{n}(k(𝔭)) ⟶^{∂} K_{n−1}M^{i+1}(R) → ⋯

and, writing M^{i−1}/M^{i+1} for M^{i−1}(R)/M^{i+1}(R),

(9.1) ⨁_{ht(𝔭)=i−1} K_{n+1}(k(𝔭)) ⟶^{∂} ⨁_{ht(𝔭)=i} K_{n}(k(𝔭)) → K_{n}(M^{i−1}/M^{i+1}) → ⨁_{ht(𝔭)=i−1} K_{n}(k(𝔭))⟶^{∂}

ending in an extension to K_{1} of the K_{0} sequence of II.6.4.3:

⨁_{ht(𝔭)=i−1} k(𝔭)^{×} ⟶^{Δ} D^{i}(R) → K_{0}(M^{i−1}/M^{i+1}) → D^{i−1}(R) → 0.

Here D^{i}(R) is the free abelian group on the height i primes, and Δ sends r/s ∈ k(𝔭)^{×} to [R/(r, 𝔭)]−[R/(s, 𝔭)] by Example 6.1.2. (This is the formula of II, Ex. 6.8.)

Recall from II.6.4.3 that the generalized Weil divisor class group CH^{i}(R) is defined to be the image of D^{i}(R) → K_{0}M^{i−1}/M^{i+1}(R). As the kernel of this map is the image of Δ, this immediately gives the interpretation, promised in II.6.4.3:

Lemma 9.1.1. CH^{i}(R) is the quotient of D^{i}(R) by the relations that Δ(r/s) = 0 for each r/s ∈ k(𝔭)^{×} and each prime ideal 𝔭 of height i − 1.

This equivalence relation, that the length of R_{𝔭}/(r, 𝔭) is zero in D^{i}(R) for each r ∈ R and each prime 𝔭 of height i − 1, is called rational equivalence; see 9.4.1.

For general R, the localization sequences (9.1) cannot break up. Indeed, we saw in I.3.6 and II.6 that even the map ⊕k(y)^{×} ⟶^{Δ} D^{i}(X) can be nonzero. Instead, the sequences assemble to form a spectral sequence converging to G_{∗}(R).

Proposition 9.2. If R is noetherian and dim(R) < ∞, there is a convergent 4^{th} quadrant cohomological spectral sequence

E_{1}^{p,q} = ⨁_{ht(𝔭)=p} K_{−p−q}(k(𝔭)) ⇒ G_{−p−q}(R).

The edge maps G_{n}(R) → E_{1}^{0,−n} = ⊕G_{n}(k(𝔭)) associated to the minimal primes 𝔭 of R are induced by the localizations R → R_{𝔭} followed by G(k(𝔭)) ≃ G(R_{𝔭}) of 4.2.1.

Along the line p + q = 0, we have E_{1}^{p,−p} ≅ D^{p}(R) and E_{2}^{p,−p} ≅ CH^{p}(R).

\overline{\begin{aligned}ℤ & 0 & 0 & 0 \\ F^{×} & → D^{1}(R) & 0 & 0 \\ K_{2}(F) & → ⊕k(x_{1})^{×} & → D^{2}(R) & 0 \\ K_{3}(F) & → ⊕K_{2}(x_{1}) & → ⊕k(x_{2})^{×} & → D^{3}(R) \end{aligned}}

The E_{1} page of the spectral sequence

Proof. Setting D_{1}^{p,q} = ⊕_{i}K_{−p−q}M^{i}(R), the localization sequences (9.1) yield an exact couple (D_{1}, E_{1}). Because M(R) = M^{0}(R) and M^{p}(R) = ∅ for p > dim(R), the resulting spectral sequence is bounded and converges to K_{∗}(R) (see [WHomo, 5.9.7]). Since E_{1}^{p,−p} is the divisor group D^{p}(R), and d_{1}^{p−1,p} is the map ⊕k(x)^{×} ⟶^{Δ} D^{p}(R) of 9.1, the group E_{2}^{p,−p} is isomorphic to CH^{p}(R) by 9.1.1. Finally, the edge maps are given by G_{n}(R) = K_{n}M(R) → K_{n}M(R)/M^{1}(R) ≅ ⊕K_{n}(k(η)); the component maps are induced by M(R) → M(k(η)). □

Coniveau Filtration 9.2.1. The p^{th} term in the filtration on the abutment G_{n}(R) is defined to be the image of K_{n}M^{p}(R) → K_{n}M(R). In particular, the filtration on G_{0}(R) is the coniveau filtration of II.6.4.3.

Remark 9.2.2. If f : R → S is flat then ⊗_{R}S sends M^{i}(R) to M^{i}(S). It follows that the spectral sequence 9.2 is covariant for flat maps.

Remark 9.2.3. If dim(R) = ∞, it follows from [WHomo, Ex. 5.9.2] that the spectral sequence 9.2 converges to \underleftarrow{lim} K_{∗}M(R)/M^{i}(R).

Motivated by his work with Brown, Gersten made the following conjecture for regular local rings. (If dim(R) = 1, this is Gersten’s conjecture 6.9.) His conjecture was extended to semilocal rings by Quillen in [Q341], who then established the important special case when R is essentially of finite type over a field (9.6 below).

Gersten-Quillen Conjecture 9.3. If R is a semilocal regular ring, the maps K_{n}M^{i+1}(R) → K_{n}M^{i}(R) are zero for every n and i.

This conjecture implies that, for R a regular semilocal domain, (9.1) breaks into short exact sequences 0 → K_{n}M^{i}(R) → ⊕_{ht=i}K_{n}(k(𝔭)) → K_{n−1}M^{i+1}(R) → 0, which splice to yield exact sequences (with F the field of fractions of R):

(9.3.1) 0 → K_{n}(R) → K_{n}(F) → ⨁_{ht(𝔭)=1}K_{n−1}(k(𝔭)) → ⋯ → ⨁_{ht(𝔭)=i}K_{n−i}(k(𝔭)) → ⋯

It follows of course that the spectral sequence 9.2 collapses at E_{2}, with E_{2}^{p,q} = 0 for p > 0. Hence the Gersten-Quillen conjecture implies that K_{n}(R) is the kernel of K_{n}(F) → ⊕K_{n−1}(k(𝔭)) for all n.

The coniveau filtration for schemes

Of course, the above discussion extends to modules over a noetherian scheme X. Here we let M^{i}(X) denote the category of coherent O_{X}-modules whose support has codimension ≥ i. If i < j we write M^{i}/M^{j} for the quotient abelian category M^{i}(X)/M^{j}(X). Then M^{i}/M^{i+1} is equivalent to the direct sum, over all points x of codimension i in X, of the M_{x}(O_{X,x}); by Devissage this category has the same K-theory as its subcategory M(k(x)). Thus the Localization Theorem yields a long exact sequence

(9.4) ⨁_{y} K_{n+1}(k(y)) ⟶^{∂} ⨁_{x} K_{n}(k(x)) → K(M^{i−1}/M^{i+1}) → ⨁_{y} K_{n}(k(y)),

where y runs over all points of codimension i − 1 and x runs over all points of codimension i. This ends in the K_{1}–K_{0} sequence of II.6.4.3:

⨁_{codim(y)=i−1} k(y)^{×} ⟶^{Δ} D^{i}(X) → K_{0}(M^{i−1}/M^{i+1}) → D^{i−1}(X) → 0,

where D^{i}(X) denotes the free abelian group on the set of points of X having codimension i (see II.6.4.3). If r/s ∈ k(𝔭)^{×}, the formula for Δ on k(y)^{×} is determined by the formula in 9.1: choose an affine open Spec(R) ⊂ X containing y; then Δ(r/s) is [R/(r, 𝔭)] − [R/(s, 𝔭)] in D^{i}(R) ⊂ D^{i}(X). This gives a presentation for the Weil divisor class group CH^{i}(X) of II.6.4.3, defined as the image of D^{i}(X) in K_{0}M^{i−1}/M^{i+1}: it is the cokernel of Δ.

Now the usual Chow group A^{i}(X) of codimension i cycles on X modulo rational equivalence, as defined in [Fulton], is the quotient of D^{i}(X) by the following relation: for every irreducible subvariety W of X × ℙ^{1} having codimension i, meeting X × {0, ∞} properly, the cycle [W ∩ X × 0] is equivalent to [W ∩ X × ∞]. We have the following identification, which was promised in II.6.4.3.

Lemma 9.4.1. CH^{i}(X) is the usual Chow group A^{i}(X).

Proof. The projection W → ℙ^{1} defines a rational function, i.e., an element t ∈ k(W). Let Y denote the image of the projection W → X, and y its generic point; the proper intersection condition implies that Y has codimension i + 1 and W is finite over Y. Hence the norm f ∈ k(y) of t exists. Let x ∈ X be a point of codimension i; it defines a discrete valuation ν on k(y), and it is well known [Fulton] that the multiplicity of x in the cycle [W ∩ X × 0] − [W ∩ X × ∞] is ν(f). □

As observed in 9.1 (and II.6.6), the sequences in (9.4) do not break up for general X. The proof of Proposition 9.2 generalizes to this context to prove the following:

Proposition 9.5 (Gersten). If X is noetherian and dim(X) < ∞, there is a convergent 4^{th} quadrant cohomological spectral sequence (zero unless p + q ≤ 0):

E_{1}^{p,q} = ⨁_{codim(x)=p} K_{−p−q}(k(x)) ⇒ G_{−p−q}(X).

If X is reduced, the components G_{n}(X) → G_{n}(k(η)) of the edge maps G_{n}(X) → E_{1}^{0,−n} associated to the generic points η of X are induced by the flat inclusions Spec(k(η)) → X.

Along the line p + q = 0, we have E_{2}^{p,−p} ≅ CH^{p}(X).

Coniveau Filtration 9.5.1. The p^{th} term in the filtration on the abutment G_{n}(X) is defined to be the image of K_{n}M^{p}(X) → K_{n}M(X). In particular, the filtration on G_{0}(X) is the coniveau filtration of II.6.4.3.

Remark 9.5.2. The spectral sequence in 9.5 is covariant for flat morphisms, for the reasons given in Remark 9.2.2. If X is noetherian but not finite-dimensional, it converges to \underleftarrow{lim} K_{∗}M(X)/M^{i}(X).

We shall now prove the Gersten-Quillen Conjecture 9.3 for algebras over a field.

Theorem 9.6. (Quillen) Let R be an algebra of finite type over a field, and let A = S^{−1}R be the semilocal ring of R at a finite set of prime ideals. Then Conjecture 9.3 holds for A: for each i the map KM^{i+1}(A) → KM^{i}(A) is zero.

Proof. We may replace R by R[1/f], f ∈ S, to assume that R is smooth. Because M^{i+1}(S^{−1}R) is the direct limit (over s ∈ S) of the M^{i+1}(R[1/s]), which in turn is the direct limit (over nonzerodivisors t) of the M^{i}(R[1/s]/tR[1/s]), it suffices to show that for every nonzerodivisor t of R that there is an s ∈ S so that the functor M^{i}(R/tR) → M^{i}(R[1/s]) is null homotopic on K-theory spaces. This is the conclusion of Proposition 9.6.1 below. □

Proposition 9.6.1. (Quillen) Let R be a smooth domain over a field and S ⊂ R a multiplicative set so that S^{−1}R is semilocal. Then for each t ≠ 0 in R with t ∉ S there is an s ∈ S so that each base change M^{i}(R/tR) → M^{i}(R[1/s]) induces a null-homotopic map on K-spaces.

Proof. Suppose first that R contains a subring B mapping isomorphically onto R/tR, and that R is smooth over B. We claim that the kernel I of R → R/tR is locally principal. To see this, we may assume that B is local, and even (by Nakayama’s Lemma) that B is a field. But then R is a Dedekind domain and every ideal is locally principal, whence the claim. We choose s so that I[1/s] ≅ R[1/s].

Now for any B-module M we have the characteristic split exact sequence of R[1/s]-modules:

0 → I[1/s] ⊗_{B} M → R[1/s] ⊗_{B} M → M[1/s] → 0.

Since R is flat, if M is in M^{i}(B) then this is an exact sequence in M^{i}(R[1/s]). That is, we have a short exact sequence of exact functors M^{i}(B) → M^{i}(R[1/s]). By the Additivity Theorem 1.2, we see that KM^{i}(B) → KM^{i}(R[1/s]) is null homotopic.

For the general case of 9.6.1, we need the following algebraic lemma, due to Quillen, and we refer the reader to [Q341, 5.12] for the proof.

Lemma 9.6.2. Suppose that X = Spec(R), for a finitely generated ring R over an infinite field k. If Z ⊂ X is closed of dimension r and T ⊂ X is a finite set of closed points, then there is a projection X → Spec(k[t_{1}, ..., t_{r}]) which is finite on Z and smooth at each point of T.

Resuming the proof of 9.6, we next assume that k is an infinite field. Let A = k[t_{1}, ..., t_{r}] be the polynomial subalgebra of R given by Lemma 9.6.1 so that B = R/tR is finite over A and R is smooth over A at the primes of R not meeting S. Set R′ = R ⊗_{A} B; then R′/R is finite and hence S^{−1}R′ is also semilocal, and R′ is smooth over B at the primes of R′ not meeting S. In particular, there is an s ∈ S so that R′[1/s] is smooth (hence flat) over B. For such s, each M^{i}(B) → M^{i}(R[1/s]) factors through exact functors M^{i}(B) → M^{i}(R′[1/s]) → M^{i}(R[1/s]). But KM^{i}(B) → KM^{i}(R′[1/s]) is zero by the first part of the proof. It follows that KM^{i}(B) → KM^{i}(R[1/s]) is zero as well.

Finally, if k is a finite field we invoke the following standard transfer argument. Given x ∈ K_{n}M^{i}(R/tR) and a prime p, let k′′ denote the infinite p-primary algebraic extension of k; as the result is true for R ⊗_{k} k′′ then there is a finite subextension k′ with [k′ : k] = p^{r} so that p^{r}x maps to zero over R ⊗_{k} k′. Applying the transfer from K_{n}M^{i}(R ⊗ k′) to K_{n}M^{i}(R), it follows that F_{∗}^{i}(x) is killed by a power of p. Since this is true for all p, we must have F_{∗}^{i}(x) = 0. □

Pure exactness 9.6.3. A subgroup B of an abelian group A is said to be pure if B/nB → A/nA is an injection for every n. More generally, an exact sequence A_{∗} of abelian groups is pure exact if the image of each A_{n+1} is a pure subgroup of A_{n}; it follows that each sequence A_{∗}/nA_{∗} is exact. If A_{∗}^{i} is a filtered family of pure exact sequences, it is easy to see that colim A_{∗}^{i} is pure exact.

Corollary 9.6.4. If R is a semilocal ring, essentially of finite type over a field, then the sequence (9.3.1) is also pure exact.

In particular, we have exact sequences (see Ex. 9.4):

0 → K_{n}(R)/ℓ → K_{n}(F)/ℓ → ⨁_{ht(𝔭)=1}K_{n−1}(k(𝔭))/ℓ → ⋯,
0 → _{ℓ}K_{n}(R) → _{ℓ}K_{n}(F) → ⨁_{ht(𝔭)=1}_{ℓ}K_{n−1}(k(𝔭)) → · · · .

Proof. (Grayson) Because the maps in 9.6.1 are null homotopic, we see from 1.2.2 that if F_{s,t}^{i} is the homotopy fiber of M^{i}(R/tR) → M^{i}(R[1/s]) the the exact sequences 0 → K_{n+1}M^{i}(R[1/s]) → π_{n}F_{s,t}^{i} → K_{n}M^{i}(R/tR) → 0 splits. Taking the direct limit, we see that the sequences 0 → K_{n+1}M^{i}(S^{−1}R) → π_{n}F^{i} → K_{n}M^{i+1}(R) → 0 are pure exact. Splicing these sequences together yields the assertion. □

In mixed characteristic, Gillet and Levine proved the following result; we refer the reader to [GL] for the proof. Let Λ be a discrete valuation domain with parameter π, residue field k = Λ/π of characteristic p > 0 and field of fractions of characteristic 0.

Theorem 9.7. Let A be a smooth algebra of finite type over Λ and S ⊂ A a multiplicative set so that R = S^{−1}A is semilocal. If t ∈ A and A/tA is flat over Λ then every base change KM^{i}(A/tA) → KM^{i}(R) is null-homotopic.

Now the set T = {t ∈ A : R/tR is flat over Λ} is multiplicatively closed, generated by the height 1 primes other than πR. Hence the localization D = T^{−1}R is the discrete valuation ring D = R_{(πR)} whose residue field is the quotient field of R/πR. By Theorem 9.7 with i = 0, K_{n}(R) → K_{n}(D) is an injection.

Corollary 9.7.1. Let R be a regular semilocal Λ-algebra, as in 9.7. Then:

(a) For i ≥ 1, the transfer map KM^{i+1}(R) → KM^{i}(R) is null homotopic.

(b) Each K_{0}M^{i}(R) is generated by the classes [R/xR], where x = (x_{1}, …, x_{i}) is a regular sequence in R of length i.

(c) Sequence (9.3.1) is exact except possibly at K_{n}(R) and ⊕_{ht(𝔭)=1}K_{n−1}(k(𝔭)).

(d) If Gersten’s DVR conjecture 6.9 is true for D, then the Gersten-Quillen conjecture 9.3 is true for R.

Proof. (a) As i+1 > 1 = dim(D), each module in KM^{i+1}(R) vanishes over D, so it is killed by a t for which R/tR is flat. Thus KM^{i+1}(R) is the direct limit over t ∈ T and s ∈ S of the KM^{i}(A[1/s]/t); since the maps KM^{i}(A[1/s]/t) → KM^{i}(R) are zero by Theorem 9.7, (a) follows. We prove (b) by induction on i, the case i = 1 being immediate from the localization sequence

R^{×} → frac(R)^{×} ⟶^{Δ} K_{0}M^{1}(R) → 0,

together with the formula Δ(r/s) = [R/rR] − [R/sR] of 6.1.2. For i > 1 the localization sequence (9.1), together with (a), yield an exact sequence

⊕_{ht(𝔭)=i−1}k(𝔭)^{×} ⟶^{∂} K_{0}M^{i}(R) ⟶^{0} K_{0}(R) ⟶^{≅} K_{0}(frac(R)) → 0.

If 0 ≠ r, s ∈ R/𝔭 then the formula (Ex. 5.1) for ∂_{𝔭} : k(𝔭)^{×} → K_{0}M^{i}(R) yields ∂_{𝔭}(r/s) = [R/𝔭 + rR] − [R/𝔭 + sR]. Thus it suffices to show that each [R/𝔭 + rR] is a sum of terms R/xR. Since the height 1 primes of R are principal we have 𝔭 = xR and R/𝔭 + aR = R/(x, r)R when i = 2. By induction there are regular sequences x of length i − 1 such that [R/𝔭] = ∑[R/xR] in K_{0}M^{i−1}(R). Given r ∈ R − 𝔭 we can choose (by prime avoidance) an r′ so that 𝔭 + rR = 𝔭 + r′R and (x, r′) is a regular sequence in R. Because R/x is Cohen-Macaulay, all its associated primes have height i−1. Hence we have the desired result (by Ex. II.6.16): [R/𝔭 + rR] = ∑[R/(x, r′)R] in K_{0}M^{i}(R).

Part (c) is immediate from exactness of K_{n}(R) → K_{n}(E) → K_{n−1}M^{1}(R), where F is the field of fractions of R, and part (a), which yields the exact sequence:

0 → K_{n}M^{1}(R) → ⨁_{ht(𝔭)=1} K_{n−1}(k(𝔭)) → ⋯ → ⨁_{ht(𝔭)=i}K_{n−i}(k(𝔭)) → ⋯

It follows that (9.3.1) is exact for R (the Gersten-Quillen conjecture 9.3 holds for R) if and only if each K_{n}(R) → K_{n}(F) is an injection. Suppose now that K_{∗}(D) → K_{∗}(F) is an injection (conjecture 6.9 holds for D). Then the composition G_{n}(R) ↪ G_{n}(D) ↪ G_{n}(F) is an injection, proving (d). □

K-cohomology

Fix a noetherian scheme X. For each point x, let us write i_{x} for the inclusion of x in X. Then we obtain a skyscraper sheaf (i_{x})_{∗}A on X (for the Zariski topology) for each abelian group A. If we view the coniveau spectral sequence 9.5 as a presheaf on X and sheafify, the rows of the E_{1} page assemble to form the following chain complexes of sheaves on X:

(9.8)  0 → K_{n} → ⊕_{cd=1}(i_{x})_{∗}K_{n}(x) → ⊕_{cd=2}(i_{y})_{∗}K_{n}(y) → ⋯ → 0.

Here K_{n} is the sheaf associated to the presheaf U ↦ K_{n}(U); its stalk at x ∈ X is K_{n}(O_{X,x}). Since the stalk sequence of (9.8) at x ∈ X is the row q = −n of the coniveau spectral sequence for R = O_{X,x}, we see that if (9.3.1) is exact for every local ring of X then (9.8) is an exact sequence of sheaves. Since each (i_{y})_{∗}K_{n}(y) is a flasque sheaf, (9.8) is a flasque resolution of the sheaf K_{n}. In summary, we have proven:

Proposition 9.8.1. Assume that X is a regular quasi-projective scheme, or more generally that the Gersten-Quillen conjecture 9.3 holds for the local rings of X. Then (9.8) is a flasque resolution of K_{n}, and the E_{2} page of the coniveau spectral sequence (9.5) is

E_{2}^{p,q} ≅ H^{p}(X, K_{−q}).

In addition, we have H^{p}(X, K_{p}) ≅ CH^{p}(X) for all p > 0.

The isomorphism H^{p}(X, K_{p}) ≅ CH^{p}(X) is often referred to as Bloch’s formula, since it was first discovered for p = 2 by Spencer Bloch in [Bl74].

Example 9.8.2. When X is the projective line ℙ_{k}^{1} over a field k, the direct image π_{∗} : K_{n}(ℙ_{k}^{1}) → K_{n}(k) is compatible with a morphism of spectral sequences (see Ex. 9.2); the only nontrivial maps are from E_{1}^{1,−n−1}(ℙ_{k}^{1}) = ⊕K_{n}(k(x)) to E_{1}^{0,−n}(k) = K_{n}(k). As these are the transfer maps for the field extensions k(x)/k, this is a split surjection. Comparing with Corollary 1.5.1, we see that

H^{0}(ℙ_{k}^{1}, K_{n}) ≅ K_{n}(k), H^{1}(ℙ_{k}^{1}, K_{n}) ≅ K_{n−1}(k).

EXERCISES

9.1 Let k be a field and S a multiplicative set in a domain R = k[x_{1}, …, x_{m}]/J so that S^{−1}R is regular. Modify the proof of Proposition 9.6 to show that for each t ≠ 0 in R there is an s ∈ S so that each M^{i}(R/tR) → M^{i}(R[1/s]) induces the zero map on K-groups. 9.2 (Quillen) Let R = k[[x_{1}, ..., x_{n}]] be a power series ring over a field. Modify the proof of Theorem 9.6 to show that the maps KM^{i+1}(R) → KM^{i}(R) are zero, so that (9.3.1) is exact.

If k is complete with respect to a nontrivial valuation, show that the above holds for the subring A of convergent power series in R. 9.3 (Gillet) Let f : X → Y be a proper map of relative dimension d. Show that f_{∗} : M^{i}(X) → M^{i−d}(Y) for all i, and deduce that there is a homomorphism of spectral sequences f_{∗} : E_{r}^{p,q}(X) → E_{r}^{p−d,q+d}(Y), compatible with the proper transfer map f_{∗} : G_{∗}(X) → G_{∗}(Y) of 3.7, and the pushforward maps CH^{i}(X) → CH^{i−d}(Y). 9.4 Let F : Ab → Ab be any additive functor which commutes with filtered direct limits. Show that F sends pure exact sequences to pure exact sequences. In particular, this applies to F (A) = _{ℓ}A = {a ∈ A : ℓa = 0} and F (A) = A ⊗ B.

§10. Descent and Mayer-Vietoris properties

In recent decades, the fact that K-theory has the Mayer-Vietoris property with respect to special cartesian squares of schemes has played an important role in understanding its structure. For cartesian squares describing open covers (see 10.1 below), it is equivalent to the assertion that K-theory satisfies “Zariski descent,” and this in turn is related to features such as the coniveau spectral sequence of Proposition 9.2. These notions generalize from K-theory to other presheaves, either of simplicial sets or of spectra, on a finite-dimensional noetherian scheme X.

As in 10.6 below, a presheaf F is said to satisfy descent if the fibrant replacement F (U) → ℍ_{zar}(U, F) is a weak equivalence for all U, where “fibrant replacement” is with respect to the local injective model structure (defined in 10.5 below). From a practical viewpoint, the recognition criterion for descent is the Mayer-Vietoris property.

Definition 10.1. Let F be a presheaf of simplicial sets (or spectra) on a scheme X. We say that F has the Mayer-Vietoris property (for the Zariski topology on X) if for every pair of open subschemes U and V the following square is homotopy cartesian.

F (U ∪ V) ⟶ F (U)
↓  ↓
F (V) ⟶ F (U ∩ V).

Before defining Zariski descent, we motivate it by stating the following theorem, due to Brown and Gersten. We postpone the proof of this theorem until later in this section.

Theorem 10.2. Let F be a presheaf of simplicial sets (or spectra) on X. Then F satisfies Zariski descent if and only if it has the Mayer-Vietoris property.

Examples 10.3. Recall that G(X) denotes the K-theory space for the category of coherent sheaves on X. Since restriction of sheaves is functorial, U ↦ G(U) is a presheaf on X for the Zariski topology. Theorem 6.11.2 states that G has the Mayer-Vietoris property on any noetherian scheme X. This was the original example of a presheaf satisfying Zariski descent, discovered in 1972 by Brown and Gersten in [BG].

On the other hand, let K^{B} denote the non-connective K-theory spectrum of vector bundles (IV.10.6). As above, U ↦ K^{B}(U) is a presheaf on X. Using big vector bundles IV.10.5, we may even arrange that it is a presheaf on schemes of finite type over X. Corollary 7.10 states that K^{B} has the Mayer-Vietoris property on any noetherian scheme X. (This fails for the connective spectrum K because the map K_{0}(U) ⊕ K_{0}(V) → K_{0}(U ∩ V) may not be onto. Corollary 7.11 states that KH has the Mayer-Vietoris property on any noetherian scheme X.

Model categories

In order to define Zariski descent in the category of presheaves on X, we need to consider two model structures on this category. Recall that a map i : A → B is said to have the left lifting property relative to a class F of morphisms if for every commutative diagram

\begin{matrix}A & ⟶^{a} & X \\ ^{i}↓ & & ^{p}↓ \\ B & ⟶^{b} & Y \end{matrix}

with p in F there is a lift h : B → X such that hi = a and ph = b.

Definition 10.4. Let C be a complete and cocomplete category. A model structure on C consists of three subcategories whose morphisms are called weak equivalences, fibrations and cofibrations, together with two functorial factorizations of each morphism, satisfying the following properties.

(1) (2-out-of-3) If f and g are composable morphisms in C and two of f, g, fg are weak equivalences, so is the third.

(2) (Retracts) If f is a retract of g, and g is a weak equivalence (resp., a fibration, resp., a cofibration) then so is f.

A map which is both a cofibration and a weak equivalence is called a trivial cofibration, and a map which is both a fibration and a weak equivalence is called a trivial fibration. These notions are used in the next two axioms.

(3) (Lifting) A trivial cofibration has the left lifting property with respect to fibrations. Similarly, cofibrations have the left lifting property with respect to trivial fibrations.

(4) (Factorization) One of the two functorial factorizations of a morphism is as a trivial cofibration followed by a fibration; the other is as a cofibration followed by a trivial fibration.

A model category is a category C with a model structure. The homotopy category of C, Ho C, is defined to be the localization of C with respect to the class of weak equivalences. (This kind of localization is described in the Appendix to Chapter II.)

We refer the reader to [Hovey] for more information about model categories. The main point is that a model structure provides us with a calculus of fractions for the homotopy category of C.

Examples 10.4.1. The standard example is the model structure on the category of simplicial sets, in which cofibrations are injections, fibrations are Kan fibrations; a map X → Y is a weak equivalence if its geometric realization |X| → |Y | is a homotopy equivalence. (|X| is defined in IV.3.1.4.)

There is also a model structure on the category of spectra. Since the details depend on the exact definition of spectrum used, we will not dwell on this point. The model category of symmetric spectra is introduced and studied in [HSS].

Definition 10.5. A morphism A → B of presheaves (of either simplicial sets or spectra) is called a global weak equivalence if each A(U) → B(U) is a weak equivalence; it is called a (Zariski) local weak equivalence if it induces an isomorphism on (Zariski) sheaves of homotopy groups (resp., stable homotopy groups).

We say that A(U) → B(U) is a cofibration if each A(U) → B(U) is a cofibration; a local injective fibration is a map which has the right lifting property with respect to cofibrations which are local weak equivalences. Jardine showed that the Zariskilocal weak equivalences, cofibrations and local injective fibrations determine model structures on the categories of presheaves of simplicial sets, and of spectra (see [J, 2.3]). We shall call these the local injective model structures, to distinguish them from other model structures in the literature.

In any model structure, an object B is called fibrant if the terminal map B → ∗ is a fibration. A fibrant replacement of an object A is a trivial cofibration A → B with B fibrant. By the Factorization axiom, there is a functorial fibrant replacement. For the local injective model structure on presheaves, we write the fibrant replacement as A → ℍ_{zar}(−, A), and write ℍ_{zar}^{n}(X, A) for π_{−n}ℍ_{zar}(X, A).

Definition 10.6. Let A be a presheaf of either simplicial presheaves or spectra. We say that A satisfies Zariski descent on a scheme X if the fibrant replacement A → ℍ_{zar}(−, A) is a global weak equivalence. That is, if A(U) ⟶^{≃} ℍ_{zar}(U, A) is a weak equivalence for every open U in X.

Note that although inverse homotopy equivalences ℍ_{zar}(U, A) → A(U) exist, they are only natural in U up to homotopy (unless A is fibrant).

Given any abelian group A and a positive integer n, the Eilenberg-Mac Lane spectrum K(A, n) is a functorial spectrum satisfying π_{n}K(A, n) = A, and π_{i}K(A, n) = 0 for i ≠ n. More generally, given a chain complex A_{∗} of abelian groups, the Eilenberg-Mac Lane spectrum K(A_{∗}, n) is a functorial spectrum satisfying π_{k}K(A_{∗}, n) = H_{k−n}(A_{∗}). See [WHomo, 8.4.1, 10.9.19 and Ex. 8.4.4] for one construction.

Example 10.6.1. Let A be a Zariski sheaf on X. The Eilenberg-Mac Lane spectrum K(A, n) is the sheaf U ↦ K(A(U), n). If A → I is an injective resolution, then ℍ(−, A) is weak equivalent to the sheaf of Eilenberg-Mac Lane spectra K(I, 0), so ℍ^{n}(X, A) = π_{−n}ℍ(X, A) ≅ π_{−n}K(I, 0)(X) = H^{n}I(X) is the usual sheaf cohomology group H^{n}(X, A). This observation, that the fibrant replacement is the analogue of an injective resolution, is due to Brown-Gersten [BG] and dubbed the “Great Enlightenment” by Thomason.

Remark 10.6.2. Given any Grothendieck topology t, we obtain the notion of t-local weak equivalence of presheaves by replacing Zariski sheaves by t-sheaves in the above definition. Jardine also showed that the t-local weak equivalences and cofibrations determine a model structure on the category of presheaves, where the t-local injective fibrations are defined by the right lifting property. Thus we have other fibrant replacements A → ℍ_{t}(−, A).

We say that A satisfies t-descent (on the appropriate site) if the fibrant replacement A → ℍ_{t}(−, A) is a global weak equivalence. The most commonly used versions are: Zariski descent, étale descent, Nisnevich descent and cdh descent.

If A is a presheaf of simplicial sets, we can sheafify it to form a Zariski sheaf, a_{zar}A. Since A → a_{zar}A is a weak equivalence, their fibrant replacements are weak equivalent. This is the idea behind our next lemma.

Lemma 10.7. Let A, B be presheaves of simplicial sets on schemes. If B satisfies Zariski descent, then any natural transformation η_{R} : A(Spec R) → B(Spec R) (from commutative rings to simplicial sets) extends to a natural transformation η_{U} : A(U) → ℍ(U, B).

The composite A(U) → B(U) with a homotopy equivalence ℍ(U, B) ⟶^{≃} B(U) exists, but (unless B is fibrant) is only well defined up to weak equivalence.

Proof. Define the presheaf A_{aff} by A_{aff}(U) = A(Spec O(U)), and define B_{aff} similarly; the natural map A_{aff} → A is a weak equivalence because it is so locally. Clearly η extends to η_{aff} : A_{aff} → B_{aff}. The composite map in the diagram

A(U) ⟶ ℍ_{zar}(U, A) ⟵_{→}^{≃} ℍ_{zar}(U, A_{aff})
↓^{η_{aff}}
B(U) ⟶^{≃} ℍ_{zar}(U, B) ⟵^{≃} ℍ_{zar}(U, B_{aff})

is the natural transformation we desire. Note that the top right horizontal map is a trivial fibration, evaluated at U; the existence of a natural inverse, i.e., a presheaf map ℍ_{zar}(−, A) → ℍ_{zar}(−, A_{aff}) splitting it, follows from the lifting axiom 10.4(3). Thus the map A(U) → ℍ_{zar}(U, B) is natural in U. □

Remark 10.7.1. In Lemma 10.7, η only needs to be defined on a category of commutative rings containing the rings O(U) and maps O(U) → O(V).

Zariski descent and the Mayer-Vietoris property

If F is a presheaf of spectra which satisfies Zariski descent, then F has the Mayer-Vietoris property, by Exercise 10.5. In order to show that the converse holds, we need the following result, originally proven for simplicial presheaves in [BG, Thm. 1′].

Proposition 10.8. Let X be a finite-dimensional noetherian space, and F a presheaf of spectra on X which has the Mayer-Vietoris property (for the Zariski topology). If all the presheaves π_{q}F have zero as associated sheaves, then π_{q}F (X) = 0 for all q.

Proof. It suffices to prove the following assertion for all d ≥ 0: for all open X′ ⊆ X, all q and all a ∈ π_{q}F (X′), there exists an open U ⊆ X′ with a|_{U} = 0 and codim_{X}(X − U) ≥ d. Indeed, when d > dim(X) we must have U = X′ and hence a|_{X′} = 0 as required.

The assertion is clear for d = 0 (take U = ∅), so suppose that it holds for d and consider a ∈ π_{q}F (X′) as in the assertion. By induction, a|_{U} = 0 for an open U whose complement Z has codimension ≥ d. Let x_{1}, …, x_{n} be the generic points of Z of codimension d. By assumption, there is a neighborhood V of these points such that a|_{V} = 0, and X − V has codimension at least d. The Mayer-Vietoris property gives an exact sequence

π_{q+1}F (U ∩ V) ⟶^{∂} π_{q}F (U ∪ V) → π_{q}F (U) ⊕ π_{q}F (V).

If a|_{U∪V} vanishes we are done. If not, there is a z ∈ π_{q+1}F (U ∩ V) with a = ∂(z). By induction, with X′′ = U ∩ V, there is an open W in X′′ whose complement has codimension ≥ d and such that z|_{W} = 0. Let y_{1}, … be the generic points of X′′ − W, Y the closure of these points in V and set V′ = V − Y. Looking at codimensions, we see that V′ is also a neighborhood of the x_{i} and that U ∩V′ = W. Mapping the above sequence to the exact sequence

π_{q+1}F (W) ⟶^{∂} π_{q}F (U ∪ V′) → π_{q}F (U) ⊕ π_{q}F (V′),

we see that a|_{U∪V′} = ∂(z_{|}W) = 0. As U ∪V′ contains all the x_{i}, its complement has codimension > d. This completes the inductive step, proving the desired result. □

Proof of Theorem 10.2. We have already noted that the ‘only if’ direction holds by Ex. 10.5. In particular, since ℍ(−, E) is fibrant, it always has the Mayer-Vietoris property. Now assume that E has the Mayer-Vietoris property, and let F (U) denote the homotopy fiber of E(U) → ℍ(U, E). Then F is a presheaf, and F has the Mayer-Vietoris property by Exercise 10.1. Since E → ℍ(−, E) is a local weak equivalence, the stalks of the presheaves π_{q}F are zero. By Proposition 10.8, π_{q}F (U) = 0 for all U. From the long exact homotopy sequence of a fibration, this implies that π_{∗}E(U) ≅ π_{∗}ℍ(U, E) for all U, i.e., E → ℍ(−, E) is a global weak equivalence. □

Nisnevich descent

To discuss descent for the Nisnevich topology, we need to introduce some terminology. A commutative square of schemes of the form

U_{Y} ⟶ Y
↓  ↓^{f}
U ⟶^{i} X

is called upper distinguished if U_{Y} = U ×_{X} Y, U is open in X, f is étale and (Y −U_{Y}) → (X−U) is an isomorphism of the underlying reduced closed subschemes. The Nisnevich topology on the category of schemes of finite type over X is the Grothendieck topology generated by coverings {U → X, Y → X} for the upper distinguished squares. In fact, a presheaf F is a sheaf if and only if F takes upper distinguished squares to cartesian (i.e., pullback) squares; see [MVW, 12.7].

For each point x ∈ X, there is a canonical map from the hensel local scheme Spec O_{X,x}^{h} to X; these form a conservative family of points for the Nisnevich topology. Thus a sheaf F is zero if and only if its stalks are zero at these points, and if a ∈ F(X) is zero then for every point x there is a point y in an étale Y → X such that k(x) ≅ k(y) and a|_{Y} = 0.

Definition 10.9. A presheaf of simplicial sets on X is said to have the Mayer-Vietoris property for the Nisnevich topology if it sends upper distinguished squares to homotopy cartesian squares. (Taking Y = V and X = U ∪ V, this implies that it also has the Mayer-Vietoris property for the Zariski topology.)

Theorem 10.10. A presheaf has the Mayer-Vietoris property for the the Nisnevich topology if and only if it satisfies Nisnevich descent.

Proof. (Thomason) The proof of Theorem 10.2 goes through, replacing Ex. 10.5 by Ex. 10.6 and Proposition 10.8 by Ex. 10.7 □

Examples 10.10.1. As in 10.3, the localization sequence 6.11 for G(X) → G(U) and devissage (Ex. 4.2) show that G satisfies Nisnevich descent.

The functor K^{B} satisfies Nisnevich descent, by [TT, 10.8]. It follows formally from this, and the definition of KH (IV.12.1) that homotopy K-theory KH also satisfies Nsnevich descent.

The descent spectral sequence

The descent spectral sequence, due to Brown and Gersten [BG], is a method for computing the homotopy groups of a presheaf F of spectra on X. It is convenient to write F_{n} for the presheaf of abelian groups U ↦ π_{n}F (U), and let F_{n} denote the Zariski sheaf associated to the presheaf F_{n}.

Theorem 10.11. Let X be a noetherian scheme with dim(X) < ∞. Then for every presheaf of spectra F having the Mayer-Vietoris property (for the Zariski topology on X) there is a spectral sequence:

E_{2}^{pq} = H_{zar}^{p}(X, F_{−q}) ⇒ F_{−p−q}(X).

Proof. For each U, we have a Postnikov tower ⋯ P_{n}F (U) ⟶^{p_{n}} P_{n−1}F (U) ⋯ for F (U); each p_{n} is a fibration, π_{q}P_{n}F (U) = π_{q}F (U) for n ≤ q and π_{q}P_{n}F (U) = 0 for all n > q. It follows that the fiber of p_{n} is an Eilenberg-Mac Lane space K(π_{n}F (U), n). Since the Postnikov tower is functorial, we get a tower of presheaves of spectra. Since the fibrant replacement is also functorial, we have a tower of

spectra ⋯ ℍ(−, P_{n}F) ⟶^{p′_{n}} ℍ(−, P_{n−1}F) · · ·. By Ex. 10.2, the homotopy fiber of p′_{n} is K(π_{n}F, n). Thus we have an exact couple with D_{pq}^{1} = ⊕π_{p+q}ℍ(X, P_{p}F) and E_{pq}^{1} = ⊕π_{p+q}K(π_{p}F, p)(X) = ⊕H^{−q}(X, F_{p}). Since the spectral sequence is bounded, it converges to \underrightarrow{lim} D_{∗}^{1} = π_{∗}F (X); see [WHomo, 5.9.7]. Re-indexing as in [WHomo, 5.4.3], and rewriting E^{2} as E_{2} yields the desired cohomological spectral sequence. □

Remark 10.11.1. The proof of Theorem 10.11 goes through for the Nisnevich topology. That is, if F has the Mayer-Vietoris property for the Nisnevich topology, there is a spectral sequence:

E_{2}^{pq} = H_{nis}^{p}(X, F_{−q}) ⇒ F_{−p−q}(X).

Example 10.12. As in Example 10.2, the G-theory presheaf G satisfies the Mayer-Vietoris property, so there is a fourth quadrant spectral sequence with E_{2}^{p,q} = H^{p}(X, G_{−q}) converging to G_{∗}(X). Here G_{n} is the sheaf associated to the presheaf G_{n}. This is the original Brown-Gersten spectral sequence of [BG].

Similarly, the presheaf U ↦ K^{B} satisfies the Mayer-Vietoris property, so there is a spectral sequence with E_{2}^{p,q} = H^{p}(X, K_{−q}) converging to K_{∗}(X). This spectral sequence lives mostly in the fourth quadrant. For example for F = K we have K_{0} = ℤ and K_{1} = O_{X}^{×} (see Section II.2 and III.1.4), so the rows q = 0 and q = −1 are the cohomology groups H^{p}(X, ℤ) and H^{p}(X, O_{X}^{×}). This spectral sequence first appeared in [TT].

If X is regular, of finite type over a field, then Gillet and Soulé proved in [GiS, 2.2.4] that the descent spectral sequence 10.11 is isomorphic to the spectral sequence of Proposition 9.8.1, which arises from the coniveau spectral sequence of 9.2. This assertion holds more generally if X is regular and the Gersten-Quillen Conjecture 9.3 holds for the local rings of X.

EXERCISES

10.1 Let F → E → B be a sequence of presheaves which is a homotopy fibration sequence when evaluated at any U. If E and B have the Mayer-Vietoris property, for either the Zariski or Nisnevich topology, show that F does too. 10.2 Suppose that F → E → B is a sequence of presheaves such that each F (U) → E(U) → B(U) is a fibration sequence. Show that each ℍ(U, F) → ℍ(U, E) → ℍ(U, B) is a homotopy fibration sequence. 10.3 Let F be a simplicial presheaf which is fibrant for the local injective model structure 10.5. Show that F (V) → F (U) is a Kan fibration for every U ⊂ V. 10.4 Let F be a simplicial presheaf which is fibrant for the local injective model structure 10.5. Show that F has the Mayer-Vietoris property. Hint: ([MV, 3.3.1]) If F is fibrant, hom(−, F) takes homotopy cocartesian squares to homotopy cartesian squares of simplicial sets. Apply this to the square involving U, V and U ∩ V. 10.5 (Jardine) Let F be a presheaf of spectra which satisfies Zariski descent on X. Show that F has the Mayer-Vietoris property. Hint: We may assume that F is fibrant for the local injective model structure of 10.5. In that case, F is a sequence of simplicial presheaves F^{n}, fibrant for the local injective model structure on simplicial presheaves, such that all the bonding maps F^{n} → ΩF^{n+1} are global weak equivalences; see [J]. 10.6 Let F be a presheaf of spectra satisfying Nisnevich descent. Show that F has the Mayer-Vietoris property for the Nisnevich topology. 10.7 Show that the Nisnevich analogue of Proposition 10.8 holds: if F has the Mayer-Vietoris property for the Nisnevich topology, and the Nisnevich sheaves associated to the π_{q}F are zero, then π_{q}F (X) = 0 for all q. Hint: Consider the f^{−1}(x_{i}) ⊂ Y. 10.8 (Gillet) Let G be a sheaf of groups on a scheme X, and ρ : G → Aut(F) a representation of G on a locally free O_{X}-module F of rank n. Show that ρ determines a homotopy class of maps from B_{•}G to B_{•}GL_{n}(O_{X}). Hint: Use the Mayer-Vietoris property for ℍ(−, B_{•}GL_{n}(O_{X})) and induction on the number of open subschemes in a cover trivializing F. 10.9 Suppose that X is a 1-dimensional noetherian scheme, with singular points x_{1}, …, x_{s}. If A_{i} is the local ring of X at x_{i}, show that K_{−1}(X) ≅ ⊕K_{−1}(A_{i}). (Note that K_{n}(X) = K_{n}(A_{i}) = 0 for all n ≤ −2, by Ex. III.4.4.)

Using the fact (III.4.4.3) that K_{−1} vanishes on hensel local rings, show that K_{−1}(X) ≅ H_{nis}^{1}(X, ℤ). This group is isomorphic to H_{et}^{1}(X, ℤ); see III.4.1.4. 10.10 Suppose that X is an irreducible 2-dimensional noetherian scheme, with isolated singular points x_{1}, …, x_{s}. If A_{i} is the local ring of X at x_{i}, show that there is an exact sequence

0 → H_{zar}^{2}(X, O_{X}^{×}) → K_{−1}(X) → ⊕K_{−1}(A_{i}) → 0.

V. THE FUNDAMENTAL THEOREMS OF HIGHER K-THEORY 79

If X is normal, it is well known that H_{nis}^{1}(X, ℤ) = 0. In this case, use III.4.4.2 to show that K_{−1}(X) ≅ H_{nis}^{2}(X, O_{X}^{×}). (This can be nonzero, as Ex. III.4.13 shows.)

§11. Chern classes

The machinery involved in defining Chern classes on higher K-theory can be overwhelming, so we begin with two simple constructions over rings.

Dennis Trace map 11.1. Let R be a ring. It is a classical fact that H_{∗}(G; R) is a direct summand of HH_{∗}(R[G]) for any group G; see [WHomo, 9.1.2]. Using the ring maps R[GL_{m}(R)] → M_{m}(R) and Morita invariance of Hochschild homology [WHomo, 9.5], we have natural maps

HH_{∗}(R[GL_{m}(R)]) → HH_{∗}(M_{m}(R)) ⟶^{≃} HH_{∗}(R).

Via the Hurewicz map π_{n}BGL(R)^{+} ⟶^{h} H_{n}(BGL(R)^{+}, ℤ), this yields homomorphisms:

K_{n}(R) ⟶^{h} H_{n}(GL(R), ℤ) = \underrightarrow{lim}_{m} H_{n}(GL_{m}(R), ℤ) → HH_{n}(R).

They are called the Dennis trace maps, having been discovered by R. K. Dennis around 1975. In fact, they are ring homomorphisms; see [Igusa, p. 133].

Given any representation ρ : G → GL_{m}(R), there is a natural map

H_{∗}(G, ℤ) → HH_{∗}(R[G]) ⟶^{ρ} HH_{∗}(R[GL_{m}(R)]) → HH_{∗}(M_{m}(R)) ⟶^{≃} HH_{∗}(R).

This map, regarded as an element of Hom(H_{∗}(G), HH_{∗}(R)), naturally lifts to an element c_{1}(ρ) of ℍ^{0}(G, HH). The verification that the c_{1}(ρ) form Chern classes (in the sense of 11.2 below) is left to Exercise 11.4.

Let Ω_{R}^{∗} denote the exterior algebra of Kähler differentials of R over ℤ [WHomo, 9.4.2]. If ℚ ⊂ R, there is a projection HH_{n}(R) → Ω_{R}^{n} sending r_{0} ⊗ ⋯ ⊗ r_{n} to (r_{0}/n!) dr_{1} ∧ ⋯ ∧ dr_{n}. Gersten showed that, up to the factor (−1)^{n−1}n, this yields Chern classes c_{n} : K_{n}(R) → Ω_{R}^{n}. We encountered them briefly in Chapter III (Ex.6.10 and 7.7): if x, y are units of R then c_{2}(\{x, y\}) = −\frac{dx}{x} ∧ \frac{dy}{y}.

Chern classes for rings

Suppose that A = A(0)⊕A(1)⊕⋯ is a graded-commutative ring. Then for any group G, the group cohomology ⊕H^{i}(G, A(i)) is a graded-commutative ring (see [WHomo, 6.7.11]). More generally we may suppose that each A(i) is a chain complex, so that A = A(0)⊕A(1)⊕⋯ is a graded dg ring; the group hypercohomology ⊕H^{i}(G, A(i)) is still a graded-commutative ring

Definition 11.2. A theory of Chern classes for a ring R with coefficients A is a rule assigning to every group G and every representation G ⟶^{ρ} Aut_{R}(P) with P in P(R) elements c_{i}(ρ) ∈ H^{i}(G, A(i)) with c_{0}(ρ) = 1 and satisfying the following axioms. The formal power series c_{t}(ρ) = ∑ c_{i}(ρ)t^{i} in 1 + ∏ H^{i}(G, A(i)) is called the total Chern class of ρ.

(1) Functoriality. For each homomorphism ϕ : H → G, c_{i}(ρ ∘ ϕ) = ϕ^{∗}c_{i}(ρ).

(2) Triviality. The trivial representation ε of G on R has c_{i}(ε) = 0 for i > 0. If ρ : G → Aut(P) then c_{i}(ρ) = 0 for i > rank(P).

(3) Sum Formula. For all ρ_{1} and ρ_{2}, c_{n}(ρ_{1} ⊕ ρ_{2}) = ∑_{i+j=n} c_{i}(ρ_{1})c_{j}(ρ_{2}). Alternatively, c_{t}(ρ_{1} ⊕ ρ_{2}) = c_{t}(ρ_{1})c_{t}(ρ_{2}).

(4) Multiplicativity. For all ρ_{1} and ρ_{2}, c_{t}(ρ_{1} ⊗ ρ_{2}) = c_{t}(ρ) ∗ c_{t}(ρ_{2}). Here ∗ denotes the product in the (non-unital) λ-subring 1+ ∏_{i>0} H^{i}(G, A(i)) of W (H^{∗}(G, A));

see II.4.3. (We ignore this axiom if R is non-commutative.)

Example 11.2.1. Taking G = 1, the Chern classes c_{i}(ρ) belong to the cohomology H^{i}A(i) of the complex A(i). Since ρ : 1 → Aut(P) depends only on P, we write c_{i}(P) for c_{i}(ρ). The Sum Formula for c_{t}(P ⊕ Q) shows that the total Chern class c_{t} factors through K_{0}(R), with c_{t}([P]) = c_{t}(P). Comparing the above axioms to the axioms given in II.4.11, we see that a theory of Chern classes for R induces Chern classes c_{i} : K_{0}(R) → H^{i}A(i) in the sense of II.4.11.

Variant 11.2.2. A common trick is to replace A(i) by its cohomological shift B(i) = A(i)[i]. Since H^{n}B(i) = H^{n}A(i)[i] = H^{i+n}A(i), a theory of Chern classes for B is a rule assigning elements c_{i}(ρ) ∈ H^{2i}(G, A(i)).

Universal elements 11.2.3. The tautological representations id_{n}: GL_{n}(R)→ Aut(R^{n}) play a distinguished role in any such theory. They are interconnected by the natural inclusions ρ_{mn} : GL_{n}(R) ↪ GL_{m}(R) for m ≥ n, and if m ≥ n then c_{n}(id_{n}) = ρ_{mn}^{∗}c_{n}(id_{m}). These elements form an inverse system, and we set

c_{n}(id) = \underleftarrow{lim}_{m} c_{n}(id_{m}) ∈ \underleftarrow{lim}_{m} H^{n}(GL_{m}(R), A(n)).

The elements c_{n}(id) are universal in the sense that, if P ⊕ Q ≅ R^{m} and we extend G ⟶^{ρ} Aut(P) to ρ ⊕ 1_{Q} : G → GL_{m}(R), then we have c_{n}(ρ) = (ρ ⊕ 1_{Q})^{∗}c_{n}(id_{m}).

To get Chern classes, we need to recall how the cohomology H^{∗}(G, A) is computed. Let B_{∗} denote the bar resolution of G (it is a projective ℤ[G]-module resolution of ℤ), and set C = B_{∗} ⊗_{ℤ[G]} ℤ, so that H_{∗}(G, ℤ) = H_{∗}C. Since G acts trivially on A, H^{∗}(G, A) is the cohomology of Hom_{ℤ[G]}(B_{∗}, A) = Hom_{ℤ}(C, A). The i^{th} cohomology of this Hom complex is the group of chain homotopy equivalence classes of maps C → A[i], and each such class determines a map from H_{n}(G, ℤ) = H_{n}(C) to H_{n−i}A = H^{i−n}A; see [WHomo, 2.7.5].

When A is a complex of ℤ/m-modules, we could replace the bar construction B_{∗} with B_{∗}/mB_{∗} in the above: H^{∗}(G, A) is the cohomology of Hom(C/mC, A). Since H_{∗}(G, ℤ/m) = H_{∗}(C/mC), we get maps H_{∗}(G, ℤ/m) → H^{i−n}A.

Lemma 11.3. A theory of Chern classes for a ring R with coefficients A yields homomorphisms for all n, i > 0, called Chern classes:

c_{i,n} : K_{n}(R) → H_{n−i}A(i) = H^{i−n}A(i).

If A is a graded ring, the maps are zero for i ≠ n and we simply have classes

c_{i} : K_{i}(R) → A(i).

When R is commutative, we have the product rule: if a_{m} ∈ K_{m}(R) and a_{n} ∈ K_{n}(R) then

c_{m+n}(a_{m}a_{n}) = \frac{−(m + n − 1)!}{(m − 1)!(n − 1)!}c_{m}(a_{m})c_{n}(a_{n}).

In particular, c_{n}({x_{1}, …, x_{n}}) = (−1)^{n−1}(n − 1)!c_{1}(x_{1}) ⋯ c_{1}(x_{n}) for all x_{1}, …, x_{n} in K_{1}(R).

Proof. By the above remarks, each element c_{i}(id_{m}) determines a homomorphism H_{∗}(GL_{m}(R), ℤ) → H_{∗}A(i). By functoriality, these maps are compatible as m varies and yield a homomorphism c_{i} from H_{∗}(GL(R), ℤ) = \underrightarrow{lim} H_{∗}(GL_{m}(R), ℤ) to H_{∗}A(i). Via the Hurewicz map h, this yields homomorphisms for n > 0:

c_{i,n} : K_{n}(R) ⟶^{h} H_{n}(GL(R), ℤ) ⟶^{c_{i}} H^{i−n}A(i).

In the special case that A is a ring, the target vanishes for i ≠ n. The product rule follows from Exercise 11.3, which states that the right side is c_{m}(a_{m}) ∗ c_{n}(a_{n}). □

Construction 11.3.1. Here is a homotopy-theoretic construction for the maps c_{i,n} of Lemma 11.3, due to Quillen. Recall that the Dold-Kan construction applied to the chain complex A(i)[i] produces an H-space K(A(i), i), called a generalized Eilenberg-Mac Lane space, with π_{n}K(A(i), i) = H_{n−i}A(i). It classifies cohomology in the sense that for any topological space X, elements of H^{n}(X, A(i)) are in 1–1 correspondence with homotopy classes of maps X → K(A(i), n); for X = BG, elements of H^{n}(G, A(i)) = H^{n}(BG, A(i)) correspond to maps BG → K(A(i), n). See [WHomo, 6.10.5 and 8.6.4].

In this way, each c_{i}(id) determines a system of maps BGL_{m}(R) → K(A(i), i), compatible up to homotopy. As K(A(i), i) is an H-space, these maps factor through the spaces BGL_{m}(R)^{+}; via the telescope construction, they determine a map BGL(R)^{+} → K(A(i), i). The Chern classes of 11.3 may also be defined as:

c_{i,n} : K_{n}(R) = π_{n}BGL(R)^{+} → π_{n}K(A(i), i) = H^{i−n}A(i).

The map BGL(R)^{+} → K(A(i), i) can be made functorial in R by the following trick. Apply the integral completion functor ℤ_{∞} (IV.1.9.ii) to get a functorial map ℤ_{∞}BGL(R) → ℤ_{∞}K(A(i), i), and choose a homotopy inverse for the homotopy equivalence K(A(i), i) → ℤ_{∞}K(A(i), i).

Finite coefficients 11.3.2. If A is a complex of ℤ/m-modules, then by the remarks before 11.3, the c_{i}(id_{m}) determine a homomorphism from H_{∗}(GL(R), ℤ/m) to H_{∗}A(i). Via the Hurewicz map h (IV.2.4), this yields homomorphisms for n > 0:

c_{i,n} : K_{n}(R; ℤ/m) ⟶^{h} H_{n}(GL(R), ℤ/m) ⟶^{c_{i}} H^{i−n}A(i).

The product rule holds in this case if m is odd or if 8|m; see IV.2.8.

Example 11.4. Suppose that A is a complex of Zariski sheaves on X = Spec(R). If A → I is an injective resolution (or the total complex of a Cartan-Eilenberg resolution if A is a bounded below complex), then H^{∗}I(X) = H^{∗}(X, A). If A = ⊕A(i), a theory of Chern classes for R with coefficients A = ⊕I(i)(X) yields Chern classes c_{i,n} : K_{n}(R) → H^{i−n}(X, A(i)). We will see many examples of this construction below.

Chern classes for schemes

One way to get a theory of Chern classes for a scheme X is to consider equivariant cohomology groups H^{∗}(X, G, A). The functors F ↦ H^{n}(X, G, F) are defined to be the right derived functors of the functor H^{0}(X, G, F) = Hom_{X,G}(ℤ_{X}, F) = H^{0}(X, F^{G}), where F belongs to the abelian category of sheaves of G-modules on X. Since it is useful to include open subschemes of X, we want to extend the definition in a natural way over a category of schemes including X.

Definition 11.5. Let V be a category of schemes, and A = ⊕A(i) a graded complex of Zariski sheaves of abelian groups on V, forming a graded sheaf of dg rings. A theory of Chern classes on V is a rule associating to every X in V, every sheaf G of groups on X and every representation ρ of G in a locally free O_{X}- module F, elements c_{i}(ρ) ∈ H^{i}(X, G, A(i)) for i ≥ 0, with c_{0}(ρ) = 1, satisfying the following axioms (resembling those of 11.2). Here c_{t}(ρ) denotes the total Chern class ∑ c_{i}(ρ)t^{i} in 1 + ∏ H^{i}(X, G, A(i)).

(1) Functoriality. For each compatible system ϕ of morphisms X ⟶^{f} X′, G → f^{∗}G′ and F → f^{∗}F′ we have c_{i}(ϕ^{∗}ρ) = ϕ^{∗}c_{i}(ρ).

(2) Triviality. The trivial representation ε of G on O_{X} has c_{i}(ε) = 0 for i > 0. If ρ : G → Aut(F) then c_{i}(ρ) = 0 for i > rank(F).

(3) Whitney sum formula. If 0 → ρ′ → ρ → ρ′′ → 0 is a short exact sequence of representations of G in locally free O_{X}-modules, then c_{t}(ρ) = c_{t}(ρ′)c_{t}(ρ′′).

(4) Multiplicativity. Given representations ρ_{1} and ρ_{2} of G on locally free O_{X}- modules, c_{t}(ρ_{1} ⊗ ρ_{2}) = c_{t}(ρ) ∗ c_{t}(ρ_{2}). Here ∗ denotes the product in the non-unital λ-subring ∏_{i>0} H^{i}(X, G, A(i)) of W (H^{∗}(X, G, A)); see II.4.3.

Example 11.5.1. Taking G = 1, the Chern classes c_{i}(ρ) belong to the cohomology H^{i}(X, A(i)) of the complex A(i). By the Whitney sum formula, the total Chern class c_{t} factors through K_{0}(X); cf. 11.2.1. Comparing the above axioms to the axioms given in II.4.11, we see that a theory of Chern classes for X induces Chern classes c_{i} : K_{0}(X) → H^{i}(X, A(i)) in the sense of II.4.11.

Lemma 11.6. Let G be a sheaf of groups and A a complex of Zariski sheaves on a scheme X, with injective resolution A → I. Then there are natural maps

H^{n}(X, G, A) → H^{n}(G, I(X)) → ⨁_{i+j=n} Hom (H_{i}(G, ℤ), H^{j}(X, A)) .

Proof. Since G acts trivially on A, we can compute the equivariant cohomology of A as hyperExt groups. Take an injective sheaf resolution A → I and let B_{∗} denote the bar resolution of G, with C = B_{∗} ⊗_{ℤ[G]} ℤ as before. Then

H^{n}(X, G, A) = H^{n} Hom_{ℤ[G]}(B_{∗}, ℤ) ⊗ I(X) = H^{n} Hom_{ℤ}(C, ℤ) ⊗ I(X).

The cohomology of the natural map Hom(C, ℤ) ⊗ I(X) → Hom(C, I(X)) is the desired map H^{n}(X, G, A) → H^{n}(G, I(X)). The second map in the Lemma comes from the discussion before Lemma 11.3, since H^{∗}I(X) = H^{∗}(X, A). □

Corollary 11.7. A theory of Chern classes for Spec(R) with coefficients A (in the sense of Definition 11.5) determines a theory of Chern classes for the ring R with coefficients I(X) (in the sense of Definition 11.2), and hence (by 11.3) Chern class homomorphisms

c_{i,n} : K_{n}(R) → H^{i−n}(Spec(R), A(i)).

Proposition 11.8. A theory of Chern classes for X with coefficients A determines Chern class homomorphisms

c_{i,n} : K_{n}(X) → H^{i−n}(X, A(i)).

Proof. By 11.7, there is a theory of Chern classes for rings, and by Construction 11.3.1 there are natural transformations c_{i} : K_{0}(R) × BGL(R)^{+} → K(A(i)(R), i)). Let K(O_{X}) denote the presheaf sending X to K_{0}(R) × BGL(R)^{+}, where R = O(X); the map K → K(O_{X}) is a local weak equivalence (10.5). By construction, the presheaf ℍ(−, K(A(i), i)) satisfies Zariski descent (10.6). By Lemma 10.7, each of the c_{i} extends to a morphism of presheaves c_{i} : ℍ_{zar}(−, K(O_{X})) → ℍ(−, K(A(i), i)). Composing with K(X) → ℍ_{zar}(X, K) → ℍ_{zar}(X, K(O_{X})) and taking homotopy groups yields the desired Chern classes on X. □

Example 11.9. Let K_{i} be the Zariski sheaf on a scheme X associated to the presheaf U ↦ K_{i}(U). (This sheaf was discussed in 9.8 and 10.12.) Gillet showed in his 1978 thesis (see [Gillet, §8]) that there is a theory of Chern classes with coefficients ⊕K_{i}, defined on the category of (smooth) varieties of finite type over a field. By 11.7 and 11.8, this yields Chern classes for each algebra, c_{i,n} : K_{n}(R) → H^{i−n}(Spec(R), K_{i}), and more generally Chern classes K_{n}(X) → H^{i−n}(X, K_{i}). Via the isomorphism Pic(X) ≅ H^{1}(X, O_{X}^{×}) of I.5.10, the Chern class c_{1,0} : K_{0}(X) → Pic(X) is the determinant II.2.6. The Splitting Principle II.8.8 implies that the Chern classes c_{i,0} : K_{0}(X) → H^{i}(X, K_{i}) ≅ CH^{i}(X) are the same as the Chern classes of II.8.9, discovered by Grothendieck (where the last isomorphism is Bloch’s formula 9.8.1).

If x is a unit of R, the map c_{1,0} : R^{×} → H^{0}(R, K_{1}) = R^{×} is the natural identification, so by the product rule the map c_{n,n} : K_{n}(R) → H^{0}(R, K_{n}) satisfies

c_{n,n}({x_{1}, …, x_{n}}) = (−1)^{n−1}(n − 1)!{x_{1}, …, x_{n}}.

Étale Chern classes

Example 11.10. In [GDix, p. 245], Grothendieck constructed a theory of étale Chern classes c_{i}(ρ) ∈ H_{et}^{2i}(X, G, μ_{m}^{⊗i}) for every X over Spec(ℤ[1/m]). This is the case of Definition 11.5 in which A(i) = Rπ_{∗}μ_{m}^{⊗i}[i] (using the shift trick of 11.2.2), where Rπ_{∗} is the direct image functor from étale sheaves to Zariski sheaves: H^{i−n}(X, A(i)) = H_{et}^{2i−n}(X, μ_{m}^{⊗i}). This yields a theory of étale Chern classes and hence (by 11.8) Chern class maps

c_{i,n} : K_{n}(X; ℤ/m) → H^{2i−n}(X, Rπ_{∗}μ_{m}^{⊗i}) = H_{et}^{2i−n}(X, μ_{m}^{⊗i}).

Soulé introduced the étale Chern classes K_{n}(R; ℤ/m) ⟶^{c_{i,n}} H_{et}^{2i−n}(Spec(R), μ_{m}^{⊗i}) in [Sou], using them to detect many of the elements of K_{n}(ℤ) described in Chapter VI. Suslin used the étale class c_{2,2} to describe K_{3}(F) in [Su91]; see VI.5.19 below.

By construction, c_{1}(ρ) is the boundary of the element in H_{et}^{1}(X, G, 𝔾_{m}) classifying the G-line bundle of det(ρ). Letting G be R^{×} and ρ : G → Aut(R) the canonical isomorphism, we see that c_{1,1} : R^{×} → H_{et}^{1}(Spec(R), μ_{m}) is the Kummer map R^{×}/R^{×n} ⊂ H_{et}^{1}(Spec(R), μ_{m}). Therefore the Chern class c_{1,1} : K_{1}(R) → H_{et}^{1}(Spec(R), μ_{m}) is the determinant K_{1}(R) → R^{×} followed by the Kummer map.

Lemma 11.10.1. If R contains a primitive m^{th} root of unity ζ, with corresponding Bott element β ∈ K_{2}(R; ℤ/m), then c_{1,2} : K_{2}(R; ℤ/m) → H_{et}^{0}(R, μ_{m}) sends K_{2}(R)/m to 0 and satisfies c_{1,2}(β) = ζ.

Proof. Recall that H_{et}^{1}(Spec(R), 𝔾_{m}) ≅ R^{×}. We claim that the left square in the following diagram is commutative. (The right square clearly commutes.)

K_{2}(R) ⟶ K_{2}(R; ℤ/m) ⟶^{Bockstein} K_{1}(R) ⟶^{m} K_{1}(R)
c_{1,2}↓  ↓^{det}  ↓^{det}
0 → H_{et}^{0}(R, μ_{m}) ⟶ H_{et}^{0}(R, 𝔾_{m}) ⟶^{m} H_{et}^{0}(R, 𝔾_{m}).

Since the Bockstein sends β to ζ, and K_{2}(R) to 0, the lemma will follow.

To see the claim, we fix injective resolutions 𝔾_{m} → J and μ_{m} → I; we have a distinguished triangle

I → J ⟶^{m} J ⟶^{δ} I[1].

If C is the standard chain complex for G introduced before Lemma 11.3, then δ induces a map Hom(C, ℤ) ⊗ J (X) → Hom(C, ℤ) ⊗ I(X)[1]. By construction, the boundary H_{et}^{1}(X, G, 𝔾_{m}) → H_{et}^{2}(X, G, μ_{m}) is the cohomology of this map. By Lemma 11.6, this induces a map

Hom(H_{1}(G, ℤ), H_{et}^{0}(X, 𝔾_{m})) → Hom(H_{2}(G, ℤ/m), H_{et}^{0}(X, μ_{m})).

For G = GL_{n}(R), c_{1,2} is the image of det, and the result follows. □

Motivic Chern classes

Let ℤ(i), i ≥ 0 denote the motivic complexes (of Zariski sheaves), defined on the category V of smooth quasi-projective varieties over a fixed field k, as in [MVW]. The motivic cohomology of X, H^{n,i}(X, ℤ), is defined to be H^{n}(X, ℤ(i)). Here is our main result, due to S. Bloch [Bl86]:

Theorem 11.11. There is a theory of Chern classes with coefficients in motivic cohomology. The associated Chern classes are c_{i,n} : K_{n}(X) → H^{2i−n,i}(X).

Remark. The Chern classes c_{i,0} : K_{0}(X) → H^{2i,i}(X) ≅ CH^{i}(X) are the same as the Chern classes of II.8.9, discovered by Grothendieck in 1957. The class c_{1,1} : K_{1}(R) → H^{1,1}(Spec(R)) ≅ R^{×} is the same as the determinant map of III.1.1.1.

Before proving Theorem 11.11, we use it to generate many other applications.

Examples 11.12. A. Huber has constructed natural multiplicative transformations from motivic cohomology to other cohomology theories; see [Hu]. Applying them to the motivic classes c_{i}(ρ) yields a theory of Chern classes, and hence Chern classes, for these other cohomology theories. Here is an enumeration.

(1) de Rham. If k is a field of characteristic 0, and X is smooth, the de Rham cohomology groups H_{dR}^{∗}(X) are defined using the de Rham complex Ω_{X}^{∗}. Thus we have Chern classes c_{i} : K_{n}(X) → H_{dR}^{2i−n}(X).

(2) Betti. If X is a smooth variety over ℂ, there is a topological space X(ℂ) and its Betti (=topological) cohomology is H_{top}^{∗}(X(ℂ), ℤ). The Betti Chern classes are c_{i} : K_{n}(X) → H_{top}^{2i−n}(X(ℂ), ℤ), arising from the realization H^{n,i}(X) → H_{top}^{n}(X(ℂ), ℤ). Beilinson has pointed out that the image of the Betti Chern classes in H_{top}^{∗}(X(ℂ), ℂ) are compatible with the de Rham Chern classes, via the Hodge structure on H_{top}^{∗}(X(ℂ), ℂ).

Of course, if R is a ℂ-algebra the natural maps GL(R) → GL(R)^{top} (described in IV.3.9.2) induce the transformations H_{∗}(GL(R)) → H_{∗}^{top}(GL(R)^{top}), as well as H^{n,i}(X, GL_{n}(R)) → H_{top}^{n}(X(ℂ), GL_{n}(R)^{top}). It should come as no surprise that these Chern classes are nothing more than the maps K_{n}(X) → KU^{−n}(X(ℂ)) followed by the topological Chern classes c_{i}:KU^{−n}(X(ℂ))→H_{top}^{2i−n}(X(ℂ)) of II.3.7.

(3) Deligne-Beilinson. If X is a smooth variety defined over ℂ, the Deligne-Beilinson cohomology groups H_{D}^{∗}(X, ℤ(∗)) are defined using truncations of the augmented de Rham complex ℤ → Ω_{X}^{∗}. The map H^{n,i}(X) → H_{D}^{n}(X, ℤ(i)) induces Chern classes K_{n}(X) → H_{D}^{2i−n}(X, ℤ(i)). These were first described by Beilinson.

(4) ℓ-adic. If k is a field of characteristic ≠ ℓ, the ℓ-adic cohomology of a variety makes sense and there are maps H^{n,i}(X) → H_{ℓ}^{n}(X, ℤ_{ℓ}(i)). This gives ℓ-adic Chern classes

c_{i} : K_{n}(X) → H_{et}^{2i−n}(X, ℤ_{ℓ}(i)).

The projection to H_{et}^{2i−n}(X, μ_{ℓ^{ν}}^{⊗i}) recovers the étale Chern classes of 11.9.

We shall now use the motivic Chern classes to prove that K_{∗}^{M}(F) → K_{∗}(F) is an injection modulo torsion for any field F. For this we use the ring map K_{∗}^{M}(F) ⟶^{≃} ⊕_{i}H^{i,i}(Spec(F)), induced by the isomorphism K_{1}^{M}(F) = F^{×} ≅ H^{1,1}(F) and the presentation of K_{∗}^{M}(F). It is an isomorphism, by a theorem of Totaro and Nesterenko-Suslin [NS].

Lemma 11.13. Let F be a field. Then the composition of K_{i}^{M}(F) → K_{i}(F) and c_{i,i} : K_{i}(F) → H^{i,i}(Spec(F)) ≅ K_{i}^{M}(F) is multiplication by (−1)^{i−1}(i − 1)!.

In particular, K_{∗}^{M}(F) → K_{∗}(F) is an injection modulo torsion.

Proof. Since c_{1,1}(a) = a, the product rule (11.3) implies that c_{i,i} : K_{i}(F) → K_{i}^{M}(F) takes {a_{1}, …, a_{i}} to (−1)^{i−1}(i − 1)!{a_{1}, …, a_{i}}. □

The outlines of the following proof are due to Grothendieck, and use only formal properties of motivic cohomology; these properties will be axiomatized in 11.14.

Proof of Theorem 11.11. Let B_{•}G denote the simplicial group scheme

Spec(k) ⇇ G ← ; ← ; ← G × G ⇇ ⇇ ⋯,

and similarly for E_{•}G. Given a representation ρ of G in a locally free O_{X}-module F, G acts on the geometric vector bundle 𝔸 of rank n associated to F, and we can construct the projective bundle ℙ = ℙ(𝔸) ×_{G} E_{•}G over B_{•}G. There is a canonical line bundle L on ℙ, and an element ξ ∈ H^{2,1}(ℙ) ≅ H^{1}(ℙ, O^{×}) = Pic(ℙ) associated to L. By the Projective Bundle Theorem [MVW, 15.12], H^{∗,∗}(ℙ) is a free H^{∗,∗}(B_{•}G)-module with basis the elements ξ^{i} ∈ H^{2i,i}(ℙ) (0 ≤ i < n). That is, we have an isomorphism

⊕_{i=0}^{n−1}H^{p−2i,q−i}(B_{•}G) ⟶^{(1,ξ,...,ξ^{n−1})} H^{p,q}(ℙ).

The Chern classes c_{i} = c_{i}^{G}(ρ) ∈ H^{2i,i}(B_{•}G) are defined by the coefficients of x^{n} ∈ H^{2n,n}(ℙ) relative to this basis, by the following equation in H^{2n,n}(ℙ):

ξ^{n} − c_{1}ξ^{n−1} + ⋯ + (−1)^{i}c_{i}ξ^{n−i} + ⋯ + (−1)^{n}c_{n} = 0.

By definition, we have c_{i}^{G}(ρ) = 0 for i > n. For each n, this construction is natural in G and ρ. Moreover, if ϵ : G → GL_{n} is the trivial representation, then ℙ is ℙ^{n−1}×B_{•}G; since ξ^{n} = 0 in H^{2n,n}(ℙ^{n−1}), we have c_{i}^{G}(ϵ) = 0 for all i > 0.

We now fix n and X. For every sheaf A and sheaf G of groups on X, there is a canonical map H^{∗}(B_{•}G, A) → H^{∗}(X, G, A); setting A = ℤ(i), we define the classes c_{i}(ρ) ∈ H^{2i}(X, G, ℤ(i)) to be the images of the c_{i} ∈ H^{2i,i}(B_{•}G) under this map.

It remains to verify the axioms in Definition 11.5. The Functoriality and Triviality axioms have been already checked. For the Whitney sum formula, suppose that we are given representations ρ′ and ρ′′ in locally free sheaves F′, F′′ of rank n′ and n′′, respectively, and a representation ρ on an extension F. The projective space bundle ℙ(F) contains ℙ(F′), and ℙ(F) − ℙ(F′) is a vector bundle over ℙ(F′′). The localization exact sequence of [MVW, (14.5.5)] becomes:

H^{2n−2n′′,n′}(ℙ(F′′)) ⟶^{i_{∗}} H^{2n,n}(ℙ(F)) ⟶^{j^{∗}} H^{2n,n}(ℙ(F′′)).

Now both i_{∗}(1) and f′′ = ξ^{n′′} − c_{1}(ρ′′) + ⋯ ± c_{n′′} generate the kernel of j^{∗}. By the projection formula, f′ = ξ^{n′} − c_{1}(ρ′) + ⋯ ± c_{n′} satisfies 0 = i_{∗}(i^{∗}f′) = i_{∗}(1)f′. Hence 0 = f′′f′ in H^{2n,n}(ℙ(F)). Since this is a monic polynomial in ξ, we must have

ξ^{n}− c_{1}(ρ) + ⋯ + ±c_{n} = (ξ^{n′′} − c_{1}(ρ′′) + ⋯ ± c_{n′′})(ξ^{n′} − c_{1}(ρ′) + ⋯ ± c_{n′}).

Equating coefficients give the Whitney sum formula.

For the Multiplicativity axiom, we may use the splitting principle to assume both sheaves have filtrations with line bundles as quotients. By the Whitney sum formula, we may further reduce to the case in which F_{1} and F_{2} are lines bundles. In the case the formula is just the isomorphism Pic(X) ≅ H^{2,1}(X), L ↦ c_{1}(L); see [MVW, 4.2]. □

Twisted duality theory

Although many cohomology theories (such as those listed in 11.12) come equipped with a natural map from motivic cohomology, and thus have induced Chern classes, it is possible to axiomatize the proof of Theorem 11.11 to avoid the need from such a natural map. This was done by Gillet in [Gillet], using the notion of a twisted duality theory. Fix a category V of schemes over a base S, and a graded sheaf ⊕A(i) of dg rings, as in 11.5. We assume that A satisfies the homotopy invariance property: H^{∗}(X, A(∗)) ≅ H^{∗}(X × 𝔸^{1}, A(∗)).

Definition 11.14. A twisted duality theory on V with coefficients A consists of a covariant bigraded “homology” functor H_{n}(−, A(i)) on the subcategory of proper maps in V, equipped with contravariant maps j^{∗} for open immersions and (for X flat over S of relative dimension d) a distinguished element η_{X} of H_{2d}(X, A(d)) called the fundamental classes of X, subject to the following axioms.

(i) For every open immersion j : U ↪ X and every proper map p : Y → X, the following square commutes:

H_{n}(Y, A(i)) ⟶^{j_{Y}^{∗}} H_{n}(U ×_{X} Y, A(i))
↓^{p_{!}}  ↓^{p_{U !}}
H_{n}(X, A(i)) ⟶^{j^{∗}} H_{n}(U, A(i)).

(ii) If j : U → X is an open immersion with closed complement ι : Z → X, there is a long exact sequence (natural for proper maps)

⋯ H_{n}(Z, A(i)) ⟶^{ι_{!}} H_{n}(X, A(i)) ⟶^{j^{∗}} H_{n}(U, A(i)) ⟶^{∂} H_{n−1}(Z, A(i)) ⋯

(iii) For each (X, U, Z) as in (ii), there is a cap product

H_{p}(X, A(r)) ⊗ H_{Z}^{q}(X, A(s)) ⟶^{∩} H_{p−q}(Z, A(r − s)),

which is a pairing of presheaves on each X, and such that for each proper map p : Y → X the projection formula holds:

p_{!}(y) ∩ z = p_{Z !}(y ∩ p^{∗}(z)), y ∈ H_{p}(Y, A(r)), z ∈ H_{Z}^{q}(X, A(s)).

(iv) If X is smooth over S of relative dimension d, and Z ↪ X is closed, the cap product with η_{X} is an isomorphism: H_{Z}^{2d−n}(X, A(d − s)) ⟶^{≃} H_{n}(Z, A(s)). (This axiom determines H_{∗}(Z, A(∗)).) In addition:

• the isomorphism H^{0}(X, A(0)) ⟶^{η_{X}∩}H_{2d}(X, A(d)) sends 1 to η_{X};

• If Z has codimension 1, the fundamental class η_{Z} corresponds to an element [Z] of H_{Z}^{2}(X, A(1)), i.e., η_{X} ∩ [Z] = η_{Z}. Writing cycle(Z) for the image of [Z] in H^{2}(X, A(1)), we require that these cycle classes extend to a natural transformation Pic(X) → H^{2}(X, A(1)).

(v) If Z ⟶^{ι} X is a closed immersion of smooth schemes (of codimension c) then the isomorphism H^{n}(Z, A(r)) ≅ H_{Z}^{n+2c}(X, A(r + c)) in (iv) is induced by a map ι_{!} : A(r) ⟶^{≃} Rι^{!}A(r + c)[2c] in the derived category of Z, where ι^{!} is the “sections with support” functor. The projection formula of (iii) for Z → X is represented in the derived category of X by the commutative diagram

Rι_{!}A(r) ⊗^{𝕃} A(s)  ⟶^{Rι_{!}(1⊗ι^{!})} Rι_{!} (A(r) ⊗^{𝕃} A(s)) ≅ Rι_{!}A(r + s)
≃↓^{ι_{!}⊗1}  ^{≃}↓^{ι_{!}}
Rι^{!}A(r + c)[2c] ⊗^{𝕃} A(s) ⟶ Rι^{!}A(r + s + c)[2c].

(vi) For all n ≥ 1 and X in V, let ξ ∈ H^{2}(ℙ_{X}^{n}, A(1)) be the cycle class of a hyperplane. Then the map π^{∗} : H_{∗}(X, A(∗)) → H^{∗}(ℙ_{X}^{n}, A(∗)), composed with the cap product with powers of ξ, is onto, and the cap product with powers of ξ induces an isomorphism:

⨁_{i=0}^{n} H_{n+2i}(X, A(q + i)) ⟶^{(1,ξ,...,ξ^{n})∩π^{∗}} H_{n}(ℙ_{X}^{n}, A(q)).

Theorem 11.14.1. [Gillet, 2.2] If there is a twisted duality theory on V with coefficients A, then there is a theory of Chern classes on V with coefficients A.

EXERCISES

11.1 Suppose given a theory of Chern classes for a ring A with coefficients in a cochain complex C, as in 11.2, and let R_{A}(G) denote the representation ring of G over A (see Ex. II.4.2). Show that there are well defined functions c_{n} : R_{A}(G) → H^{i}(G, C(i)) forming Chern classes on the λ-ring R_{A}(G) in the sense of II.4.11. If K is the space K(C) = ∏ K(C(n), n), show that they determine a natural transformation R_{A}(G) → [BG, K(C)]. By the universal property of the +-construction (IV.5.7), this yields a unique element of [BGL(A)^{+}, K(C)]. Compare this with the construction in 11.3. 11.2 Consider the unital λ-ring ℤ×(1+∏ H^{n}(G, A(n))) with multiplication (a, f)∗ (b, g) = (ab, f^{b}g^{a}(f ∗ g)). Show that (rank, c_{t}) : K_{∗}(X) → ℤ × (1 + ∏ H^{n}(G, A(n))) is a homomorphism of unital λ-rings. 11.3 (Grothendieck) Let H = ⊕H^{n} be a graded ring and write W for the non-unital subalgebra 1 + ∏ H^{n} of W (H). Show that for x ∈ H^{m}, y ∈ H^{n} we have

(1 − x) ∗ (1 − y) = 1 − \frac{−(m + n − 1)!}{(m − 1)!(n − 1)!}xy + · · · .

Hint: In the universal case H = ℤ[x, y], W (H) embeds in W (H ⊗ ℚ). Now use the isomorphism W (H ⊗ ℚ) ≅ ∏ H ⊗ ℚ and compute in the mn coordinate. 11.4 Dennis trace as a Chern class. Let C_{∗} denote the standard chain complex with H_{n}(C_{∗}) = HH_{n}(R); see [WHomo, 9.1]. In the notation of 11.2.3, there is a natural chain map B_{∗} ⊗ ℤ → C_{∗}(ℤ[G]), representing an element of H^{0} Hom(B_{∗} ⊗ ℤ, C_{∗}(ℤ[G])), which on homology is the H_{∗}(G, ℤ) → HH_{∗}(ℤ[G]) of 11.1; see [WHomo, 9.7.5]. Since the C_{∗}(M_{m}(R)) → C_{∗}(R) are chain maps [WHomo, 9.5.7], any representation ρ determines a chain map and an associated element c_{1}(ρ) of H^{0}(G, C_{∗}(R)) = H^{0} Hom(B_{∗} ⊗ ℤ, C_{∗}(R)).

Setting A(0) = ℤ, A(1) = C_{∗}[−1] and A(n) = 0 for n > 1, we may regard c_{1}(ρ) as an element of H^{1}(G, A(1)). Show that c_{1}(ρ) determines a theory of Chern classes with c_{i} = 0 for i ≥ 2. Hint: the product c_{1}(ρ_{1})c_{1}(ρ_{2}) is in H^{2}(G, A(2)) = 0, and the method of 11.2.3 applies. 11.5 Suppose that R contains both 1/m and the group μ_{m} of primitive m^{th} roots of unity. Let ρ : μ_{m} → GL_{2}(R) be the representation ρ(ζ)(x, y) = (ζx, ζ^{−1}y). Show that the étale Chern class c_{2,2}(ρ) : H_{4}(μ_{m}, ℤ/m) → H_{et}^{0}(R, μ_{m}^{⊗2}) ≅ ℤ/m is an isomorphism. Hint: if λ : μ_{m} → R^{×} is the tautological representation, show that c_{2,2} = c_{1,1}(λ)^{2}.

11.6 Show that a theory of Chern classes with coefficients A(i) yields Chern classes with coefficients B(i) = A(i) ⊗^{𝕃} ℤ/m, and that there is a commutative diagram

K_{n}(X)  ⟶ K_{n}(X; ℤ/m) ⟶  K_{n−1}(R)
↓^{c_{i}}  ↓^{c_{i}}  ↓^{c_{i}}
H^{i−n}(X, A(i)) ⟶ H^{i−n}(X, B(i)) ⟶ H^{i+1−n}(X, A(i)).

Hint: There is a triangle Hom(C_{∗}, A)[1] → Hom(C_{∗}, A⊗^{𝕃} ℤ/m) → Hom(C_{∗}, A) →.

As an application of this, let A(i) be the complex for Deligne-Beilinson cohomology of a complex variety; B(i) computes étale cohomology, and we have a commutative diagram

K_{n}(X)  ⟶ K_{n}(X; ℤ/m) ⟶  K_{n−1}(R)
↓^{c_{i}}  ↓^{c_{i}}  ↓^{c_{i}}
H_{D}^{2i−n}(X, ℤ(i)) ⟶ H_{et}^{2i−n}(X, μ_{m}^{⊗i}) ⟶ H_{D}^{2i+1−n}(X, ℤ(i)).

11.7 Recall from Kummer theory that H_{et}^{1}(X, μ_{m}) ≅ _{m} Pic(X) ⊕ U(X)/mU(X), where U(X) = O(X)^{×}. Modifying Example 11.10, show that the Chern class c_{1,1} : K_{1}(X; ℤ/m) → H_{et}^{1}(X, μ_{m}) sends K_{1}(X)/m to U(X)/mU(X) by the determinant, and that the induced quotient map c_{1,1} : _{m}K_{0}(X) → _{m} Pic(X) may be identified with the canonical map of II.8.1. 11.8 (Suslin) Suppose given a theory of Chern classes for a field F with coefficients A. Show that the Chern class c_{n} : K_{n}(F) → H^{0}A(n) factors through Suslin’s map K_{n}(F) → K_{n}^{M}(F) described in IV.1.15. In particular, the K-cohomology Chern class c_{n} : K_{n}(F) → H^{0}(Spec(F), K_{n}) ≅ K_{n}(F) factors through K_{n}(F) → K_{n}^{M}(F). Hint: Use K_{n}^{M}(F) ≅ H_{n}(GL_{n}(F))/H_{n}(GL_{n−1}(F)) to reduce to the Triviality axiom 11.2(2) that c_{n}(id_{n−1}) = 0.