Lecture IV The Fundamental Theorems of higher K-theory

Note: References cited as I.x.y, II.x.y and III.x.y refer to chapters I, II and III in the virtual “K-book” located at http://math.rugers.edu/w̃eibel/kbook.html.

In this lecture, we restrict our attention to exact categories and Waldhausen categories, where the extra structure enables us to use the following types of comparison theorems: Additivity, Resolution, Devissage, and Localization. These are the extensions to higher K-theory of the corresponding classical theorems for K_{0}. The highlight of this chapter is the so-called “Fundamental Theorem” of K-theory, which compares K(R) to K(R[t]) and to K(R[t, t^{−1}]).

§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. If B and C are exact categories, and

0 → F′(B) → F (B) → F′′(B) → 0

is an exact sequence in C for every B ∈ B, we say that F′ ↣ F ↠ F is a short exact sequence of exact functors.

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 sequence F′(B) ↣ F (B) ↠ F′′(B) is a cofibration sequence and if for every cofibration A ↣ B in B, F (A) ∪_{F′(A)} F′(B) ↣ F (B) is a cofibration in C.

Regarding exact categories as Waldhausen categories, it is easy to see that the two notions of “short exact sequence of functors” between exact categories are in fact the same.

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 as maps on the homotopy groups K_{i}(B) → K_{i}(C).

Universal Example 1.2.1. Recall from chapter II, 9.3, that the extension category E = E(C) is the category of all 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). The following calculation of K(E) is due to Quillen [Q341] for exact categories and to Waldhausen [W1126] for Waldhausen categories. (Cf. II.9.3.1.)

Typeset by AMS-TEX

Extension Theorem 1.3. The exact functor (s, q) : E = E(C) → C × C induces a homotopy equivalence K(E) → K(C) × K(C). A natural homotopy inverse is given by the coproduct functor ⨿, which sends (A, B) to the sequence A ↣ A ⨿ B ↠ B.

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 functor s∨q: E → C which sends A ↣ B ↠ C to A ⨿ C, because the H-space structure on K(C) is induced from ⨿. The compositions of t and s ∨ q with the coproduct functor ⨿ agree:

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

and hence give the same map on K-theory. Since K(⨿) is a homotopy equivalence, t ≃ s ∨ q, and we are done.

Exercise 1.1. (Additivity for characteristic exact sequences). If

0 → F^{0} → F^{1} → ⋯ → F^{n} → 0

is an exact sequence of exact functors B → C, show that ∑(−1)^{p}F_{∗}^{p} = 0 as maps from K_{i}(B) to K_{i}(C). What definition of “exact sequence” is needed?

Flasque Categories 1.4. We have already indicated one application of the Additivity Theorem in chapter IV, namely to show that the suspension functor (when it exists) is a homotopy inverse on the H-group K(A). Here is another: 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 1.4.1. 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 shows why we restrict to finitely generated modules in defining K(R).

Example 1.4.2 (cone rings). Here is another example of a flasque category, due to Karoubi. Suppose that Λ is a ring (necessarily noncommutative!) such that Λ ≅ Λ⊕Λ as a Λ-bimodule. Then P (Λ) is flasque (∞(M) = M), so the K-theory of Λ is zero. One such ring is the cone ring C(R) = End_{R}(R^{∞}) of all row-and-column-finite infinite matrices over some specified ring R. The finite matrices M_{∞}(R) = ∪M_{n}(R) form an ideal of C(R), and the quotient ring S(R) = C(R)/M_{∞}(R) is called the suspension ring of R.

Negative K-theory 1.5. Karoubi, Gersten and Wagoner used the suspension construction (and variants) to construct “deloopings” of algebraic K-theory. It turns out that K_{i}(SR) ≅ K_{i−1}(R) for i ≥ 1, i.e., that ΩK(SR) ≃ K(R). This provides one possible approach to defining the negative K-theory of a ring: set K_{−1}(R) = K_{0}(SR), and K_{−n}(R) = K_{0}(S^{n}R) in general. The hi-tech way to do this is to define the Ω-spectrum K^{B}(R) to be the sequence of spaces

K(R), K(SR), ⋯, K(S^{n}R), · · · .

With this definition, the homotopy groups of K^{B}(R) are the K-theory groups:

π_{i}K^{B}(R) = K_{i}(R) for all integers i.

§2. Changes in Waldhausen Structure

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

The first result is the Approximation Theorem. Consider the following “approximate lifting property” (App), 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 says that every object and map in B lifts up to weak equivalence to A. The Approximation Theorem is proven by Waldhausen in [W1126, 1.6.7] under the additional hypothesis that the map a be a cofibration. That hypothesis is easily removed using the mapping cylinder T (a).

Approximation Theorem 2.1. Suppose that F : A → B is an exact functor between Waldhausen categories, both of which satisfy the saturation axiom, and such that a morphism f of A is a weak equivalence whenever F (f) is. Suppose that A has a cylinder functor satisfying the cylinder axiom. If the approximate lifting property (App) is satisfied, then F induces homotopy equivalences BwA ≃ BwB and K(A) ≃ K(B). In particular, the K-groups K_{∗}A and K_{∗}B are isomorphic.

Remark 2.1.1. The Approximation Theorem can fail in the absence of a cylinder functor. To see this, considering M^{⊕} ⊂ M, where M is an abelian category.

Changing cofibrations 2.1.2. (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}).

Let A be an exact category, and Ch_{b}(A) the category of bounded chain complexes over A. This is a Waldhausen category with coCh_{b} the degreewise split monics, and weak equivalences are quasi-isomorphisms.

Gillet-Waldhausen Theorem 2.2. If A is an exact category, then

K(A) ≃ KCh_{b}(A), and K_{n}(A) ≅ K_{n}Ch_{b}(A) for every n ≥ 0.

Example 2.2.1 (Thomason). Let Ch_{b} = Ch_{b}(A) be bounded chain complexes over an exact category A. This is a Waldhausen Let co_{1}Ch_{b} be the maps in Ch_{b} which are degreewise admissible monics. Then K(A) ≃ K(Ch_{b}) is equivalent to K(Ch_{b}, co_{1}).

Example 2.2.2 (Perfect complexes). A perfect complex of R-modules is a complex M_{·} which is quasi-isomorphic to a bounded complex of finitely generated projective R-modules, i.e., to a complex in Ch_{b}(P(R)). The perfect R-modules form a Waldhausen subcategory Ch_{perf} of Ch(mod-R). We have that

K(R) ≃ K(Ch_{b}P(R)) ≃ K(Ch_{perf}(R)).

The third result involves a change in the category of weak equivalences, with the same underlying category of cofibrations. As motivation, consider an exact functor F : A → B between two Waldhausen categories. Let v(A) and w(B) denote their respective categories of weak equivalences, 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. If every object and map in B lifts up to weak equivalence to A, and w(B) satisfies the Saturation Axiom (II.9.7.1), then the Approximation Theorem states that K(A, w) ≃ K(B). If we let A^{w} denote the Waldhausen subcategory of A consisting of all objects A such that 0 ⟶^{≃} F (A) in B, then the following result states that there is a long exact sequence:

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

ending in the obvious surjection K_{0}(A) → K_{0}(B).

Waldhausen’s Localization Theorem 2.3. Let A = (A, co(A)) be a category with cofibrations co(A). Suppose that v(A) ⊂ w(A) are categories of weak equivalences such that (A, v(A)) and (A, w(A)) are both Waldhausen categories. Let A^{w} denote the Waldhausen subcategory of A consisting of all A in A for which 0 → A is in w(A). Finally, suppose that A has a cylinder functor such that w(A) satisfies the Cylinder, Saturation and Extension Axioms (I.6.11). Then

K(A^{w}) → K(A, v) → K(A, w)

is a homotopy fibration. 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 obvious surjection K_{0}(A, v) → K_{0}(A, w).

The third result is an Approximation Theorem for Waldhausen categories of chain complexes. Roughly speaking, it says that the K-theory of A only depends on the derived category of A. In order for the statement of this result to make more sense, we first describe just what the derived category of A is.

Derived Categories 2.4. Let A be a full additive subcategory of Ch(M) which is closed under all the shift operators A_{·} ↦ A_{·}[n]. The derived category w^{−1}A of A is the category obtained from A by formally inverting wA, the class of all quasiisomorphisms in A. The following is the key observation that makes it possible for us to better understand the derived category and to even see that it exists.

Lemma 2.4.1. Chain homotopic maps in A are identified in the derived category.

Proof. Write T (A) for the mapping cylinder of the map 1_{A} : A = A, so that T (A)_{n} = A_{n} ⊕ A_{n−1} ⊕ A_{n}, and write j_{1}, j_{2} : A → T (A) for the front and back inclusions. Both j_{1} and j_{2} are right inverses for the “addition” quasi-isomorphism T (A) → A sending the element (a_{1}, b, a_{2}) of T (A)_{n} to a_{1} + a_{2} ∈ A_{n}. Therefore [j_{1}] = [j_{2}] in w^{−1}A. Now if f_{1}, f_{2} : A → B are chain homotopic maps, there is a map F : T (A) → B in A such that f_{i} = F ∘ j_{i}. So [f_{1}] = [f_{2}] in the derived category w^{−1}A.

We can now give a more concrete description of the derived category w^{−1}A. It has the same objects as A. Every map in w^{−1}A from A to B is represented by a formal map fs^{−1} : A ⟵^{s} A′ ⟶^{f} B with s ∈ wA, and also by a formal map t^{−1}g : A ⟶^{g} B′ ⟵^{t} B with t ∈ wA. We identify fs^{−1} and t^{−1}g iff the maps tf and gs from A′ to B′ are chain homotopic. This relation generates an equivalence relation on formal maps, and maps in w^{−1}A are equivalence classes of formal maps.

Here is the description of the derived category given by the standard references [H20] and [V569]. Let A^{ch} denote the quotient category of A obtained by identifying chain homotopic maps. In the terminology of op. cit., A^{ch} is a triangulated category, so the wA maps in A^{ch} admit a calculus of 2-sided fractions (this is the analogue of the ∅re condition used to localize noncommutative rings), so we can form w^{−1}(A^{ch}) by taking the usual equivalence classes of formal maps fs^{−1} = t^{−1}g. The above Lemma implies that w^{−1}A ≅ w^{−1}A^{ch}, i.e., that w^{−1}A^{ch} is the derived category. In fact, the concrete description we gave above for w^{−1}A was based on this approach.

We are now ready for the third main theorem, due to Thomason and Trobaugh [TT], which I think of as a modified Approximation Theorem. Fix an abelian category M and recall that Ch(M) is the category of all chain complexes over M.

Thomason-Trobaugh Approximation Theorem 2.5. Let A ⊂ B be full Waldhausen subcategories of Ch(M), closed under all shift maps A_{.} ↦ A_{.}[n]. If the derived categories of A and B are isomorphic, then KA ≃ KB.

Sketch of Proof. (See [TT], 1.9.8.) Let A^{+} be the comma category of pairs (A, A ⟶^{∼} B); the forgetful functor A^{+} → A is right adjoint to the functor A ↦ (A, A = A), so K(A) ≃ K(A^{+}). We now apply the Approximation Theorem (or, rather, its dual) to the exact functor A^{+} → B sending (A, A ⟶^{∼} B) to B. The Approximation property (App) for A^{+} → B is a consequence of the hypothesis that A and B have the same derived category.

Remark 2.5.1. The derived categories w^{−1}A and w^{−1}B are isomorphic if every complex in B is quasi-isomorphic to a complex in A. For example, the derived category of Ch_{b}(M) is the same as the derived category of the category of all complexes in Ch(M) which are merely quasi-isomorphic to a bounded complex.

Exercise 2.5.2. Fix a ring R, and let Ch_{bh}^{+}(M(R)) be the category of all bounded above chain complexes in Ch(mod-R) which are quasi-isomorphic to a (bounded) complex in Ch_{b}(M(R)). Let I ⊂ Ch_{bh}^{+}(M(R)) denote the subcategory of chain complexes of injective R-modules. For example, injective resolutions of fin. gen. R-modules belong to I. Show that the derived categories of I, Ch_{bh}^{+}(M(R)) and Ch_{b}(M(R)) are isomorphic. Conclude that

K(I) ≃ Ch_{bh}^{+}(M(R)) ≃ K(Ch_{b}(M(R)) ≃ G(R).

§3. The Resolution Theorem

If R is a ring, let H(R) be the exact category of all fin. gen. R-modules M having a finite P (R)-resolution, i.e., a resolution

(∗)  0 → P_{n} → ⋯ → P_{1} → P_{0} → M → 0

by fin. gen. projective R-modules. In K_{0}(H(R)) we clearly have:

[M] = [P_{0}] − [P_{1}] + [P_{2}] + ⋯ + (−1)^{n}[P_{n}].

Thus the map K_{0}P (R) → K_{0}(H(R)) is onto; in fact, it is an isomorphism.

Original Resolution Theorem 3.1 (for P(R) ⊂ H(R)). For every ring R, K(R) = K(P(R)) ≃ K(H(R)), so that K_{i}(R) = K_{i}P(R) ≅ K_{i}H(R) for all i.

Proof. Consider the diagram

KCh_{b}(P) ⟶^{≃} KCh_{b}(H)
≃↑  ↑^{≃}
KP ⟶ KH.

The vertical maps are homotopy equivalences by the Gillet-Waldhausen theorem2.2. The derived categories of Ch_{b}(P) and Ch_{b}(H) are isomorphic, because (*) allows us to replace bounded complexes in H with bounded complexes in P. The Thomason-Trobaugh Approximation theorem shows that KCh_{b}(P) ≃ KCh_{b}(H).

Application 3.2. A regular ring R is a noetherian ring such that every R-module has a finite projective resolution, i.e., finite projective dimension. In this case H(R) is the category M(R) of all fin. gen R-modules, and so the Resolution Theorem gives

K_{∗}(R) ≅ G_{∗}(R) for every regular ring R.

Similarly, a regular scheme X is a noetherian scheme such that H(X) is M(X), the category of all coherent X-modules, so again we have K_{∗}(X) ≅ K_{∗}(H(X)) ≅ G_{∗}(X).

The above theorem is a special case of a more general result, which was originally proven in [Q341] by reduction to the special case in which objects of H have a Presolution of length one; shorter proofs are given in [Gra87] and [Staff].

Resolution Theorem 3.3. Let P ⊂ H be full exact subcategories of an abelian category M, both closed under extensions and inheriting their exact structure from M. Suppose that:

(i) Every object M of H has a finite P-resolution (*), and (ii) P is closed under kernels in H, i.e., if L ↣ M ↠ N is an exact sequence in

H with M, N in P, then L is also in P. Then KP ≃ KH (and thus K_{i}P ≅ K_{i}H for all i).

Problem 3.3.1. Generalize the Resolution Theorem to Waldhausen categories.

Application 3.4. Let f : A → B be a homomorphism of noetherian rings such that B has finite Tor-dimension as a right A-module. Let P ⊂ M(A) be the full subcategory of all A-modules M with

T or_{n}^{A}(B, M) = 0 for n ≠ 0.

The usual properties of Tor show that every fin. gen. A-module M has a finite resolution by objects of P, that P is an exact subcategory, and that M ↦ M ⊗_{A} B is an exact functor from P to M(B). Hence there is a natural map

G(A) ⟵^{≃} K(P) → G(B),

giving maps f^{∗}: G_{∗}(A) → G_{∗}(B).

Example 3.4.1. If f : Y → X is a morphism of noetherian schemes such that f^{∗}O_{X} has finite Tor-dimension over Y, there is a map f^{∗} from G(X) to G(Y).

Note that if the ring B is finite over A then the usual restriction of scalars is an exact functor M(B) → M(A); this induces a map f_{∗} : G(B) → G(A). The seemingly strange notation (f^{∗} and f_{∗}) is chosen with an eye towards schemes: if X = Spec(A) and Y = Spec(B) then f maps Y to X, so f^{∗} is contravariant and f_{∗} is covariant.

Example 3.4.2. Let f : B[x] → B be the map f(x) = 0. The sequence

0 → A ⟶^{x} A ⟶ B → 0

shows that T or_{n}^{A}(B, −) = 0 for n ≥ 2, so there is a map f^{∗}: G(B[x]) → G(B). Similarly, the inclusion i: B → B[x] induces a map i^{∗}: G(B) → G(B[x]). It is easy to show that the composite f^{∗}i^{∗} : G_{∗}(B) → G_{∗}(B[x]) → G_{∗}(B) is the identity map. Then using Additivity, it follows that f_{∗} : G_{∗}(B[x]) → G_{∗}(B) is the zero map.

Example 3.5. Suppose B = k ⊕ B_{1} ⊕ B_{2} ⊕ ⋯ is a graded noetherian ring, and let grM(B) be the abelian category of fin. gen. graded B-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. Now assume that B is flat over k, so that tensoring with B gives a functor from M(k) to grM(B), and hence a ℤ[σ, σ^{−1}]-module map

β: G_{i}(k) ⊗ ℤ[σ, σ^{−1}] → K_{i}(grM(B)).

In the special case B = k, grM(k) is just a coproduct of copies of M(k), and the map β is the isomorphism G_{∗}(k)[σ, σ^{−1}] ≅ K_{∗}(grM(k)). If k has finite Tordimension over B (via B → k), then the method of 3.4 induces a map

K_{i}(grM(B)(→ K_{i}(grM(k)) ≅ G_{i}(k) ⊗ ℤ[σ, σ^{−1}]

which is evidently a left inverse to β.

Exercise 3.5.1. Show that β is an isomorphism. To do this, let F_{m}(M) be the submodule of M generated by M_{−m} ⊕ ⋯ ⊕ M_{m}, and consider the subcategory N_{m}(B) of graded B-modules M with M = F_{m}(M). Show that K_{i}[N_{m}(k)] is a sum of copies of G_{i}(k) and use the characteristic exact filtration

0 ⊂ F_{0}(M) ⊂ ⋯ ⊂ F_{m}(M) = M

of modules in N_{m}(B) to show that K_{i}N_{m}(B) ≅ K_{i}N_{m}(k) for all m.

§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 taken from [Q341].

Devissage Theorem 4.1. Let A ⊂ B be an inclusion of abelian categories such that 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_{i}(A) ≅ K_{i}(B).

Example 4.2. 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 A = M(k) is an exact subcategory of B = M(R) and every R-module M has the characteristic filtration

0 = 𝔪^{r}M ⊂ 𝔪^{r−1}M ⊂ ⋯ ⊂ 𝔪M ⊂ M.

The Devissage Theorem tells us that G_{∗}(R) ≅ G_{∗}(k) = K_{∗}(k).

Open Problem 4.2.1. 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). If char(k) ≠ 0, this can be done using topological cyclic homology (McCarthy’s Theorem), but concretely we only know K_{0}, K_{1}, K_{2} and sometimes K_{3} at present.

Example 4.3. Let R be a local noetherian ring with maximal ideal 𝔪, and let M_{𝔪}(R) denote the category of all R modules of finite length. This is the union as r → ∞ of the categories mod-(R/𝔪^{r}), so by example 4.2 we have

KM_{𝔪}(R) ≃ KM(R/𝔪) = G(R/𝔪).

Exercise 4.3.1. Let Z be a closed subscheme of a scheme X, and let M_{Z}(X) be the exact category of all coherent X-modules supported on Z. Show that

K(M_{Z}(X)) ≃ K(M(Z)) = G(Z)

Hint: Let B_{n} be the category of X-modules killed by I^{n}, I being the ideal of Z, apply Devissage to B_{n}, and pass to the limit as n → ∞.

Problem 4.4. Generalize the Devissage Theorem to Waldhausen categories.

§5. Localization Sequences for G_{∗}

In classical ring theory, one localizes a ring R using a central multiplicatively closed set S to form a new ring S^{−1}R. More modern treatments (since the mid-1960’s) focus on the fact that the abelian category mod- − S^{−1}R is obtained from mod- − R by “modding out by” the subcategory T = mod-_{S} − R of S-torsion Rmodules, i.e., those M such that for every element m ∈ M some s ∈ S has sm = 0. We write this as

mod-S^{−1}R = (mod-R)/T .

Definition 5.1. (Gabriel, Goldman) An abelian subcategory T of an abelian category M is called a Serre subcategory if it is a full subcategory which is closed under subobjects, quotients and extensions. That is, whenever A ↣ B ↠ C is an exact sequence in M,

B ∈ T ⇔ both A and C are in T .

In this case we can construct a quotient abelian category M/T by making every object of T isomorphic to zero. (M/T has the same objects as M but Hom_{M}(A, B) is replaced by an appropriate filtered colimit of hom-sets.)

If M is big enough that all small colimits exist and filtered colimits are exact (such as M = mod-R), then M/T may be constructed as a subcategory of M. In this case T determines a torsion theory and there is a substantial literature about localization from this viewpoint. Such an M is too big for our purposes, because it is flasque (consider M^{∞} = M ⊕ M ⊕ ⋯) and thus K(M) = 0. However, if M_{0} ⊂ M and M is big enough, then M_{0}/T will lie inside M/T and hence M. For example, the quotient M(S^{−1}R) of M(R) ⊂ mod-R may be considered as a subcategory of mod-R.

Abelian Localization Theorem 5.2. (Quillen [Q341]) Let T be a Serre subcategory of a (small) abelian category M. Then

K(T ) → K(M) → K(M/T )

is a homotopy fibration sequence with K_{0}(M) mapping onto K_{0}(M/T ). Thus there is a long exact sequence of homotopy groups

⋯ K_{i+1}(M/T ) → K_{i}(T ) → K_{i}(M) → K_{i}(M/T ) → K_{i−1}(T ) → ⋯

ending in → K_{0}(M) → 0.

Example 5.3. Let S be a central multiplicatively closed set in a noetherian ring R. Then M(R) is abelian and M_{S}(R) = M(R) ∩ mod-R is a Serre subcategory with quotient M(S^{−1}R). Thus

K(M_{S}(R)) → G(R) → G(S^{−1}R)

is a homotopy fibration. Note that K(M_{S}(R)) colim_{s∈S} G(R/sR) because

M_{S}(R) = ⋃_{s∈S} M(R/sR).

Now suppose that S = {s^{n}}. The maps G(R/sR) → G(R/s^{n}R) are homotopy equivalences by Devissage, so G(R/sR) ≃ KM_{S}(R). The long exact sequence (5.2) is:

(5.3.1)  ⋯ G_{i+1}(R[s^{−1}]) → G_{i}(R/sR) → G_{i}(R) → G_{i}(R[s^{−1}]) · · · .

An even more special case occurs when R = k[s] :

Fundamental Theorem of G-theory 5.4 (Quillen). Let k be noetherian. Then (i) G_{i}(k[s]) ≅ G_{i}(k) via the maps k → k[s] ⟶^{s=1} k. (ii) G_{i}(k[s, s^{−1}]).

Proof. We first observe that (i) implies (ii), because the localization sequence

⟶^{∂} G_{i}(k) → G_{i}(k[s])  →  G_{i}(k[s, s^{−1}]) ⟶^{∂} G_{i−1}(k) → ⋯
↑ ≃ s = 1 ↙
G_{i}(k)

splits into (split) short exact sequences by (i). Note that k has Tor-dimension one over both k[s] and k[s, s^{−1}] so the oblique arrow is defined, as is the map G(k[s]) → G(k) of (i).

To prove (i), we introduce the subring A = k[st, t] of k[s, t] and the abelian category gr M(A) of finitely generated graded A-modules, where A is graded by deg(s) = 0, deg(t) = 1. There is an equivalence of quotient categories

gr M(A)/gr M_{t}(A) ≅ M(k[s])

induced by the exact functor M ↦ M/(t − 1)M from gr M(A) to M(k[s]). By Devissage, the Serre subcategory gr M_{t}(A) of t-torsion modules has the same Ktheory as its subcategory gr M(A/tA). Note that both A and A/tA = k[st] are flat over k and that k has finite Tor-dimension over either. Hence the localization sequence and (3.5.1) give us the following diagram with exact rows, which proves (i):

⋯ →K_{i}(gr M(A/tA)) ⟶ K_{i}(gr M(A)) ⟶ G_{i}(k[s])→ ⋯
≃↑  ^{≃}↑  ↑
0 →G_{i}(k) ⊗ ℤ[σ, σ^{−1}] ⟶^{1−σ} G_{i}(k) ⊗ ℤ[σ, σ^{−1}] ⟶ G_{i}(k)→ 0.

To see that the diagram commutes, apply the Additivity Theorem to the exact sequence

0 → M ⊗ A(−1) ⟶^{t} M ⊗ A → M ⊗ A/tA → 0

of functors from M(k) to gr M(A), and consult the construction of the vertical isomorphisms in (3.5).

Theorem 5.5. Let k be a noetherian ring. The maps M(k) → M(ℙ_{k}^{n}) sending M to M ⊗ O(d) induce an isomorphism ∑ ⊗O(d): ∐_{d=0}^{n} G_{i}(k) ⟶^{≃} G_{i}(ℙ_{k}^{n}) for every n.

Proof. The standard description of coherent sheaves on ℙ^{n} amounts to the equivalence of quotient categories

M(ℙ^{n}) ≅ gr M(k[x_{0}, ⋯, x_{n}])/T ,

where T is the Serre subcategory of graded modules M with M_{n} = 0 for m >> 0. By Devissage, T has the same K-theory as its subcategory gr M(k). Thus the localization sequence and (3.5.1) give us the following diagram with exact rows

⋯ ⟶ K_{i}(gr M(k))  ⟶ K_{i}(gr M(k[x_{0}, ⋯, x_{n}])) ⟶ G_{i}(ℙ_{k}^{n}]) ⟶ ⋯
≃↑  ^{≃}↑  ↑
0 ⟶ G_{i}(k) ⊗ ℤ[σ, σ^{−1}] ⟶^{1−σ}  G_{i}(k) ⊗ ℤ[σ, σ^{−1}]  ⟶ G_{i}(k) ⟶ 0.

Setting A = k[x_{0}, ⋯, x_{n}] and E = A^{n+1}, consider the Koszul resulution

0 → ∧^{n+1}E(−n − 1) → ⋯ → ∧^{2}E(−2) → E(−1) → A → k → 0.

The Additivity Theorem shows that h is multiplication by the map ∑(_{d}^{n+1})(−σ)^{d} = (1 − σ)^{n+1}. Since this is an injection with the prescribed cokernel, we have proven the theorem.

Exercise 5.5.1. If E is any projective k-module of rank n + 1, we can form the associated symmetric algebra A = k ⊕ E ⊕ ⋯ and its projective bundle ℙ(E) = Proj(A). Modify the above proof to show that once again the maps M ↦ M ⊗O(d) induce an isomorphism

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

Example 5.6. 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 becomes

⋯ K_{n}(k) → K_{n}(R) → K_{n}(F) → K_{n−1}(k) → k_{n−1}(R) · · · .

Gersten’s conjecture is that this sequence splits up (for every D.V.R.) 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 or when k is algebraic over 𝔽_{p}, or when char(F) = char(k). It is not known if char(F) = 0 and char(k) = p in general.

Example 5.7. Suppose that R is a 1-dimensional commutative noetherian domain. Let A = M(R) and T = fin. gen. torsion R-modules. Now T is the direct sum of the categories T_{𝔪},

T_{𝔪} = ⋃_{n=1}^{∞} M(R/𝔪^{n}) = fin. gen. R-modules supported at 𝔪,

as 𝔪 runs over the maximal ideals of R. By Devissage,

K_{i}(T ) = ⊕_{𝔪}K_{i}(T_{𝔪}) = ⊕_{𝔪} lim_{n→∞} G_{∗}(R/𝔪^{n})
= ⊕_{𝔪}G_{∗}(R/𝔪) = ⊕_{𝔪}K_{∗}(R/𝔪).

On the other hand, M(R)/T is M(F), where F is the quotient field of R. Since G_{∗}(F) = K_{∗}(F), we have a long exact localization sequence

⋯ ⊕_{𝔪} K_{i}(R/𝔪) → G_{i}(R) → K_{i}(F) ⟶^{∂} ⊕_{𝔪}K_{i−1}(R/𝔪) → G_{i−1}(R) · · · .

If R is regular, i.e., a Dedekind domain, then K_{i}(R) ≅ G_{i}(R) and this sequence ends in the exact sequences

K_{2}(R) → K_{2}(F) ⟶_{symbols}^{tame} ⊕_{m}(R/𝔪)∗ → SK_{1}(R) → 0,

K_{1}(R) = SK_{1}(R) ⊕ R∗, the Class group sequence

1 → R^{×} → F^{×} → ⊕_{𝔪}ℤ → Pic(R) → 0,

and K_{0}(R) = ℤ ⊕ P ic(R).

Exercise 5.7.1. Modify the above discussion to describe G_{∗}(X) for a curve X in terms of K_{∗}(F), F = k(X) and the K-theory of the closed points of X.

Example 5.8. Let X be a noetherian scheme, U an open subscheme, and Z = X − U the closed complement. In this case we take A = M(X) and T the coherent X-modules supported on Z, i.e., modules whose restriction to U is zero. Gabriel has shown that A/T is the category M(U) of coherent U-modules. By Devissage again, K(T ) ≃ G(Z). Hence the localization sequence becomes the sequence

⋯ ⟶^{∂} G_{i}(Z) → G_{i}(X) → G_{i}(U) ⟶^{∂} G_{i−1}(Z)

ending in G_{0}(Z) → G_{0}(X) → G_{0}(U) → 0.

Exercise 5.8.1. Let X be the affine line with a double origin, the standard example of a scheme which is not separated. Show that G_{i}(X) ≅ G_{i}(k) × G_{i}(k) for all i.

§6. The Brown-Gersten-Quillen spectral sequence.

Suppose that a scheme X is the union of two open subschemes U and V. Since X − U = V − (U ∩ V) is closed in V, we can compare the localization sequences (example 5.8), part of which is

G_{n+1}(U) ⟶ G_{n}(X − U) ⟶ G_{n}(X) ⟶ G_{n}(U) ⟶ G_{n−1}(X − U)
↓  ∥  ↓  ↓  ∥
G_{n+1}(U ∩ V) ⟶ G_{n}(X − U) ⟶ G_{n}(V) ⟶ G_{n}(U ∩ V) ⟶ G_{n−1}(X − U)

to get (via a diagram chase) a Mayer-Vietoris sequence:

G_{n+1}(U ∩ V) ⟶^{∂} G_{n}(X) ⟶ G_{n}(U) ⊕ G_{n}(V) ⟶ G_{n}(U ∩ V) ⟶^{∂} · · · .

Brown and Gersten [BG] realized that the formalism of presheaves of spectra allowed them to promote this into a spectral sequence, which converges if X is noetherian and finite dimensional. To state this result, we introduce the notation K_{n} and G_{n} for the Zariski sheaves associated to the presheaves U ↦ K_{n}(U) and U ↦ G_{n}(U), respectively.

Brown-Gersten Theorem 6.1. If X is noetherian and finite dimensional, there is a convergent 4th quadant spectral sequence

E_{2}^{pq} = H_{zar}^{p}(X, G_{−q}) ⇒ G_{−p−q}(X).

A pretty interpretation of this, along with a conceptually simpler proof, was given by Quillen [Q341]. Let M_{p} = M_{p}(X) denote the Serre subcategory of M(X) consisting of the coherent sheaves whose support Z has codimension ≥ p; M_{p} is the union of the categories M_{Z}(X), so by Ex. 4.3.1 we have G_{∗}(X) = \underrightarrow{lim}_{Z} G_{∗}(Z). If X_{p} denotes the set of all points of X of codimension p in X, then the quotient categories are easy to describe:

M_{p}/M_{p+1} ≅ ∐_{x∈X_{p}} M_{p}(O_{X,x}).

But R = O_{X,x} is a p-dimensional local ring, so M_{p}(O_{X,x}) is the category of finite length R-modules. By example 4.3 we have K(M_{p}/M_{p+1}) ≅ ∐_{x∈X_{p}} K(k(x)), where k(x) is the residue field of O_{X,x}. By the localization theorem 5.2 we get exact sequences

⋯ K_{n}M_{p+1} → K_{n}M_{p} → ∐_{x∈X_{p}} G_{n}(k(x)) → K_{n−1}M_{p+1} · · · .

Since M_{0} = M(X) and M_{p} = 0 if p > dim(X), these form a bounded exact couple, which gives rise to a spectral sequence in the usual way. That spectral sequence is:

Quillen’s spectral sequence 6.2. If X has finite Krull dimension, there is a convergent spectral sequence, lying in the octant p ≥ 0, p + q ≤ 0:

E_{1}^{pq} = ∐_{x∈X_{p}} K_{−p−q}k(x) ⇒ G_{−p−q}(X).

Suppose for example that X is an integral scheme, with field of fractions F = k(X). Then the row for n = −q is:

(6.2.1) 0 → K_{n}(F) → ∐_{x∈X_{1}} K_{n−1}k(x) ⟶^{d_{1}} ⋯ ∐_{x∈X_{n−1}} k(x)^{×} ⟶^{d_{1}} ∐_{x∈X_{n}} ℤ → 0.

Quillen showed that the last map in the sequence (6.2.1) is the usual divisor map defining the Chow group CH^{n}(X) of codimension n cycles on X modulo rational equivalence, as defined in [Fulton]. (It is also the generalized Weil divisor class group of II.6.4.3.) It follows that

(6.2.2)  E_{2}^{p,−p} ≅ CH^{p}(X)

Example 6.2.3. 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 spectral sequence has two rows, encoding the localization sequence for R → F (example 5.6). Recall from 5.6 that Gersten’s conjecture is that this sequence splits up into short exact sequences

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

Gersten also conjectured that whenever X = Spec(R), R a regular local ring, the sequence (6.2.4)

0 → K_{n}(R) → K_{n}(F) → ∐_{x∈X_{1}} K_{n−1}k(x) ⟶^{d_{1}} ⋯ ∐_{x∈X_{n−1}} k(x)^{×} ⟶^{d_{1}} ∐_{x∈X_{n}} ℤ → 0.

should be exact. That is, the spectral sequence for R should degenerate at E_{2}. Although this appears to be more general, it is equivalent to the DVR version of the conjecture.

Theorem 6.3 (Quillen [Q341]). Let R be a regular semilocal ring, essentially of finite type over a field k, with field of fractions F. Then the sequence (6.2.4) is exact.

Theorem 6.4. If X is a regular scheme of finite type over a field, the term E_{2}^{p,−qq} is canonically isomorphic to the Zariski sheaf cohomology H_{zar}^{p}(X, K_{q}). Hence the spectral sequence may be written

E_{2}^{pq} = H_{zar}^{p}(X, K_{−q}) ⇒ K_{−p−q}(X).

Proof. If we view the sequence (6.2.1) as a sequence of presheaves on X and sheafify, we get the sequence of sheaves

(6.4.1) 0 → K_{n} → K_{n}(F) → ∐_{x∈X_{1}}(i_{x})_{∗}K_{n−1}k(x) ⟶ ⋯ ⟶ ∐_{x∈X_{n}}(i_{x})_{∗}ℤ → 0.

where K_{n}(F) is a constant sheaf, and i_{x}: Spec k(x) → X is the canonical inclusion. The sequence of stalks at x in (6.4.1) is the sequence (6.3.4) for R = O_{X,x}, and is exact by theorem 6.3. Hence (6.4.1) is an exact sequence of sheaves, and may be viewed as a flasque resolution of the Zariski sheaf K_{n}. Hence H^{p}(X, K_{n}) is the pth cohomlogy of the sequence of global sections, which is the exact sequence (6.2.1).

Corollary 6.4.2 (Bloch’s Formula). If X is a regular scheme of finite type over a field, then

H^{p}(X, K_{p}) ≅ CH^{p}(X).

This formula is called Bloch’s formula because the case p = 2 was discovered by Bloch.

§7. Localization for K_{∗}.

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) → K_{n}(R) → K_{n}(R[\frac{1}{s}]) ⟶^{∂} · · · .

If R isn’t regular, there is a more complicated third term.

Definition. Given a central multiplicatively closed subset S of R, let H_{S}(R) denote the exact category of all fin. gen. R-modules M such that:

(i) M is in H(R), i.e., M has a finite resolution by fin. gen. projective R-modules; (ii) M is S-torsion, i.e., for each m ∈ M there is an s ∈ S such that sm = 0. If S = {s^{n}, n ≥ 0}, we often write H_{s} for H_{S}.

If R is regular then (i) is automatic, and we know by Devissage that K(H_{s}(R)) = K(M_{s}(R)) ≃ G(R/s), the third term in the above sequence. In general, however, K(H_{s}(R)) is hard to compute.

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

K(H_{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) ⟶^{∂} · · · .

Caveat. 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 spectrumlevel fibration, therefore, one needs to use nonconnective spectra, i.e., spectra with nontrivial negative homotopy groups.

The technical restriction that S contain no zerodivisors can be quite annoying. We can remove it, but at the cost of introducing more definitions. Recall that a chain complex M_{·} of R-modules is called a perfect complex if it is quasi-isomorphic to a bounded complex of finitely generated projective modules; the perfect complexes form a Waldhausen subcategory Ch_{perf}(R) of Ch(mod − R) and by the Approximation Theorem (2.3) we have K(R) ≃ K(Ch_{b}(P (R)) ≃ K(Ch_{perf}(R)).

Theorem 7.2 (K_{∗} Localization). Let S be a central multiplicatively closed subset of R. Let Ch_{perf}(R, S) denote the Waldhausen subcategory of Ch_{perf}(R) consisting of all perfect complexes of R-modules M_{·} such that S^{−1}M_{·} is an exact complex. Then

K(Ch_{perf}(R, S)) → 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}(Ch_{perf}(R, S)) → K_{n}(R) → K_{n}(S^{−1}R) ⟶^{∂} · · · .