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THE FUNDAMENTAL THEOREMS of HIGHER K-THEORY We Now Restrict Our Attention to Exact Categories and Waldhausen Categories, Where T

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THE FUNDAMENTAL THEOREMS of HIGHER K-THEORY We Now Restrict Our Attention to Exact Categories and Waldhausen Categories, Where T 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 the- orems: 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 1 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∗ : Ki(B) → Ki(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: ` t C×C −→` E ⇉ C, s∨q 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 → An →···→ A0 → 0 is admissibly exact if each map decomposes as Ai+1 ։ Bi ֌ Ai, and each Bi ֌ Ai ։ Bi−1 is an exact sequence. Corollary 1.2.1. (Additivity for characteristic exact sequences). If 0 → F 0 → F 1 →···→ F n → 0 p p is an admissibly exact sequence of exact functors B→C, then (−1) F∗ = 0 as P maps from Ki(B) to Ki(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 K0 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. V. THE FUNDAMENTAL THEOREMS OF HIGHER K-THEORY 3 w Proof for Waldhausen categories. Let Cm denote the category of se- quences A −→≃ B −→·≃ · · of m weak equivalences; this is a category with cofibrations w w (defined termwise). The set snCm of sequences of n cofibrations in Cm (IV.8.5.2) A0 ֌ A1 ֌ ··· ֌ An ↓∼ ↓∼ ↓∼ B0 ֌ B1 ֌ ··· ֌ Bn ↓∼ ↓∼ ↓∼ ··· ··· ··· is naturally isomorphic to the m-simplices in the nerve of the category wSnC. That w w is, the bisimplicial sets wS.C and s.C. are isomorphic. By Ex.IV.8.10 applied to Cm, w w w each of the maps s.E(Cm) → s.Cm × s.Cm is a homotopy equivalence. As m varies, w w w we get a bisimplicial map s.E(C. ) → s.C. × s.C. , 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 A0 ֌ B0 ։ C0 and both u : A0 → A and v : C0 → C are morphisms in QC. We will compare T to its subcategories TA and TC , 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. u v T : A ←− A0 ֌ B0 ։ C0 −→ C ∨ ∨ ηT ↓ || i↓ ↓ || || j v p(T ): A ։ A1 ֌ B ։ C0 −→ C πp(T ) ↑ || ||↑ ↑ || qp(T ): A ։ A1 ֌ B1 ։ C1 ֌ C ↓ ↑ ↑ || ↓ ↑ ∧ || 0 A ։ 0 ֌ 0 ։ 0 ֌ C i j Given a triple T , choose a factorization of u as A0֌A1ևA, and let B be the pushout of A1 and B0 along A0; B is in C and p(E) : A1 ֌ B ։ C0 is an exact sequence by II, Ex.7.8(2). Thus p(T )=(j, p(E),v) is in the subcategory TA. The construction shows that p is a functor from T to TA, and that there is a natural transformation ηT : T → p(T ). This provides a homotopy Bη between the identity of BT and the map Bp : BT → BTA ⊂ BT . By duality, if we choose a factorization of v as C0 և C1 ֌ C and let B1 be the pullback of B0 and C1 along C0, then we obtain a functor q : T →TC , and a natural transformation πT : q(T ) → T . This provides a homotopy Bπ between Bq : BT → BTC ⊂ 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(TA ∩TC ) ⊂ BT . Finally, the category 4 V. THE FUNDAMENTAL THEOREMS OF HIGHER K-THEORY (TA ∩TC ) 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 Wald- hausen 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.
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