CHAPTER II THE GROTHENDIECK GROUP K_{0}

There are several ways to construct the “Grothendieck group” of a mathematical object. We begin with the group completion version, because it has been the most historically important. After giving the applications to rings and topological spaces, we discuss λ-operations in §4. In sections 6 and 7 we describe the Grothendieck group of an “exact category,” and apply it to the K-theory of schemes in §8. This construction is generalized to the Grothendieck group of a “Waldhausen category” in §9.

§1. The Group Completion of a monoid

Both K_{0}(R) and K^{0}(X) are formed by taking the group completion of an abelian monoid—the monoid P(R) of finitely generated projective R-modules and the monoid VB(X) of vector bundles over X, respectively. We begin with a description of this construction.

Recall that an abelian monoid is a set M together with an associative, commutative operation + and an “additive” identity element 0. A monoid map f: M → N is a set map such that f(0) = 0 and f(m + m′) = f(m) + f(m′). The most famous example of an abelian monoid is ℕ = {0, 1, 2, ...}, the natural numbers with additive identity zero. If A is an abelian group then not only is A an abelian monoid, but so is any additively closed subset of A containing 0.

The group completion of an abelian monoid M is an abelian group M^{−1}M, together with a monoid map [ ]: M → M^{−1}M which is universal in the sense that, for every abelian group A and every monoid map α: M → A, there is a unique abelian group homomorphism α̃: M^{−1}M → A such that α̃([m]) = α(m) for all m ∈ M.

For example, the group completion of ℕ is ℤ. If A is an abelian group then clearly A^{−1}A = A; if M is a submonoid of A (additively closed subset containing 0), then M^{−1}M is the subgroup of A generated by M.

Every abelian monoid M has a group completion. One way to contruct it is to form the free abelian group F (M) on symbols [m], m ∈ M, and then factor out by the subgroup R(M) generated by the relations [m + n] − [m] − [n]. By universality, if M → N is a monoid map, the map M → N → N^{−1}N extends uniquely to a homomorphism from M^{−1}M to N^{−1}N. Thus group completion is a functor from abelian monoids to abelian groups. A little decoding shows that in fact it is left adjoint to the forgetful functor, because of the natural isomorphism

Hom\begin{matrix}abelian \\ monoids \end{matrix}(M, A) ≅ Hom\begin{matrix}abelian \\ groups \end{matrix}(M^{−1}M, A).

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2 II. THE GROTHENDIECK GROUP K_{0}

Proposition 1.1. Let M be an abelian monoid. Then:

(a) Every element of M^{−1}M is of the form [m] − [n] for some m, n ∈ M;

(b) If m, n ∈ M then [m] = [n] in M^{−1}M if and only if m + p = n + p for some p ∈ M;

(c) The monoid map M ×M → M^{−1}M sending (m, n) to [m]−[n] is surjective.

(d) Hence M^{−1}M is the set-theoretic quotient of M × M by the equivalence relation generated by (m, n) ∼ (m + p, n + p).

Proof. Every element of a free abelian group is a difference of sums of generators, and in F (M) we have ([m_{1}] + [m_{2}] + ⋯) ≡ [m_{1} + m_{2} + ⋯] modulo R(M). Hence every element of M^{−1}M is a difference of generators. This establishes (a) and (c). For (b), suppose that [m] − [n] = 0 in M^{−1}M. Then in the free abelian group F (M) we have

[m] − [n] = ∑ ([a_{i} + b_{i}] − [a_{i}] − [b_{i}]) − ∑ ([c_{j} + d_{j}] − [c_{j}] − [d_{j}]) .

Translating negative terms to the other side yields the following equation:

(∗) [m] + ∑([a_{i}] + [b_{i}]) + ∑[c_{j} + d_{j}] = [n] + ∑[a_{i} + b_{i}] + ∑([c_{j}] + [d_{j}]).

Now in a free abelian group two sums of generators ∑[x_{i}] and ∑[y_{j}] can only be equal if they have the same number of terms, and the generators differ by a permutation σ in the sense that y_{i} = x_{σ(i)}. Hence the generators on the left and right of (∗) differ only by a permutation. This means that in M the sum of the terms on the left and right of (∗) are the same, i.e.,

m + ∑(a_{i} + b_{i}) + ∑(c_{j} + d_{j}) = n + ∑(a_{i} + b_{i}) + ∑(c_{j} + d_{j})

in M. This yields (b), and part (d) follows from (a) and (b). □

The two corollaries below are immediate from Proposition 1.1, given the following definitions. An (abelian) cancellation monoid is an abelian monoid M such that for all m, n, p ∈ M, m+p = n+p implies m = n. A submonoid L of an abelian monoid M is called cofinal if for every m ∈ M there is an m′ ∈ M so that m + m′ ∈ L.

Corollary 1.2. M injects into M^{−1}M if and only if M is a cancellation monoid.

Corollary 1.3. If L is cofinal in an abelian monoid M, then:

(a) L^{−1}L is a subgroup of M^{−1}M;

(b) Every element of M^{−1}M is of the form [m] − [ℓ] for some m ∈ M, ℓ ∈ L;

(c) If [m] = [m′] in M^{−1}M then m + ℓ = m′ + ℓ for some ℓ ∈ L.

A semiring is an abelian monoid (M, +), together with an associative product · which distributes over +, and a 2-sided multiplicative identity element 1. That is, a semiring satisfies all the axioms for a ring except for the existence of subtraction. The prototype semiring is ℕ.

The group completion M^{−1}M (with respect to +) of a semiring M is a ring, the product on M^{−1}M being extended from the product on M using 1.1. If M → N is a semiring map, then the induced map M^{−1}M → N^{−1}N is a ring homomorphism. Hence group completion is also a functor from semirings to rings, and from commutative semirings to commutative rings.

Example 1.4. Let X be a topological space. The set [X, ℕ] of continuous maps X → ℕ is a semiring under pointwise + and ·. The group completion of [X, ℕ] is the ring [X, ℤ] of all continuous maps X → ℤ.

If X is (quasi-)compact, [X, ℤ] is a free abelian group. Indeed, [X, ℤ] is a subgroup of the group S of all bounded set functions from X to ℤ, and S is a free abelian group (S is a “Specker group”; see [Fuchs]).

Example 1.5 (Burnside Ring). Let G be a finite group. The set M of (isomorphism classes of) finite G-sets is an abelian monoid under disjoint union, ‘0’ being the empty set ∅. Suppose there are c distinct G-orbits. Since every G-set is a disjoint union of orbits, M is the free abelian monoid ℕ^{c}, a basis of M being the classes of the c distinct orbits of G. Each orbit is isomorphic to a coset G/H, where H is the stabilizer of an element, and G/H ≅ G/H′ if and only if H and H′ are conjugate subgroups of G, so c is the number of conjugacy classes of subgroups of G. Therefore the group completion A(G) of M is the free abelian group ℤ^{c}, a basis being the set of all c coset spaces [G/H].

The direct product of two G-sets is again a G-set, so M is a semiring with ‘1’ the 1-element G-set. Therefore A(G) is a commutative ring; it is called the Burnside ring of G. The forgetful functor from G-sets to sets induces a map M → ℕ and hence an augmentation map ϵ: A(G) → ℤ. For example, if G is cyclic of prime order p, then A(G) is the ring ℤ[x]/(x^{2} = px) and x = [G] has ϵ(x) = p.

Example 1.6. (Representation ring). Let G be a finite group. The set Rep_{ℂ}(G) of finite-dimensional representations ρ: G → GL_{n}ℂ (up to isomorphism) is an abelian monoid under ⊕. By Maschke’s Theorem, ℂG is semisimple and Rep_{ℂ}(G) ≅ ℕ^{r}, where r is the number of conjugacy classes of elements of G. Therefore the group completion R(G) of Rep_{ℂ}(G) is isomorphic to ℤ^{r} as an abelian group.

The tensor product V ⊗_{ℂ} W of two representations is also a representation, so Rep_{ℂ}(G) is a semiring (the element 1 is the 1-dimensional trivial representation). Therefore R(G) is a commutative ring; it is called the Representation ring of G. For example, if G is cyclic of prime order p then R(G) is isomorphic to the group ring ℤ[G], a subring of ℚ[G] = ℚ × ℚ(ζ), ζ^{p} = 1.

Every representation is determined by its character χ: G → ℂ, and irreducible representations have linearly independent characters. Therefore R(G) is isomorphic to the ring of all complex characters χ: G → ℂ, a subring of Map(G, ℂ).

Definition. A (connected) partially ordered abelian group (A, P) is an abelian group A, together with a submonoid P of A which generates A (so A = P^{−1}P) and P ∩ (−P) = {0}. This structure induces a translation-invariant partial ordering ≥ on A: a ≥ b if a − b ∈ P. Conversely, given a translation-invariant partial order on A, let P be {a ∈ A : a ≥ 0}. If a, b ≥ 0 then a + b ≥ a ≥ 0, so P is a submonoid of A. If P generates A then (A, P) is a partially ordered abelian group.

If M is an abelian monoid, M^{−1}M need not be partially ordered (by the image of M), because we may have [a] + [b] = 0 for a, b ∈ M. However, interesting examples are often partially ordered. For example, the Burnside ring A(G) and Representation ring R(G) are partially ordered (by G-sets and representations).

When it exists, the ordering on M^{−1}M is an extra piece of structure. For example, ℤ^{r} is the group completion of both ℕ^{r} and M = {0} ∪ {(n_{1}, ..., n_{r}) ∈ ℕ^{r} : n_{1}, ..., n_{r} > 0}. However, the two partially ordered structures on ℤ^{r} are different.

EXERCISES

1.1 The group completion of a non-abelian monoid M is a group M̂, together with a monoid map M → M̂ which is universal for maps from M to groups. Show that every monoid has a group completion in this sense, and that if M is abelian then M̂ = M^{−1}M. If M is the free monoid on a set X, show that the group completion of M is the free group on the set X. Note: The results in this section fail for non-abelian monoids. Proposition 1.1 fails for the free monoid on X. Corollary 1.2 can also fail: an example of a cancellation monoid M which does not inject into M̂ was given by Mal’cev in 1937. 1.2 If M = M_{1}×M_{2}, show that M^{−1}M is the product group (M_{1}^{−1}M_{1})×(M_{2}^{−1}M_{2}). 1.3 If M is the filtered colimit of abelian monoids M_{α}, show that M^{−1}M is the filtered colimit of the abelian groups M_{α}^{−1}M_{α}. 1.4 Mayer-Vietoris for group completions. Suppose that a sequence L → M_{1} × M_{2} → N of abelian monoids is “exact” in the sense that whenever m_{1} ∈ M_{1} and m_{2} ∈ M_{2} agree in N then m_{1} and m_{2} are the image of a common ℓ ∈ L. If L is cofinal in M_{1}, M_{2} and N, show that there is an exact sequence of groups L^{−1}L → (M_{1}^{−1}M_{1}) ⊕ (M_{2}^{−1}M_{2}) → N^{−1}N, where the first map is the diagonal inclusion and the second map is the difference map (m_{1}, m_{2}) ↦ m̄_{1} − m̄_{2}. 1.5 Classify all abelian monoids which are quotients of ℕ = {0, 1, … } and show that they are all finite. How many quotient monoids M = ℕ/∼ of ℕ have m elements and group completion M̂ = ℤ/nℤ? 1.6 Here is another description of the Burnside ring A(G) of a finite group G. For each subgroup H, and finite G-set X, let χ_{H}(X) denote the cardinality of X^{H}.

(a) Show that χ_{H} defines a ring homomorphism A(G) → ℤ, and ϵ = χ_{1}.

(b) Deduce that the product χ of the χ_{H} (over the c conjugacy classes of subgroups) induces an injection of A(G) into the product ring ∏_{1}^{c} ℤ.

(c) Conclude that A(G) ⊗ ℚ ≅ ∏_{1}^{c} ℚ.

1.7 (T-Y Lam) Let ϕ : G → H be a homomorphism of finite groups. Show that the restriction functor from H-sets to G-sets (gx = ϕ(g)x) induces a ring homomorphism ϕ^{∗} : A(H) → A(G). If X is a G-set, we can form the H-set H ×_{G} X = H × X/{(h, gx) ∼ (hϕ(g), x))}. Show that H×_{G} induces a group homomorphism ϕ_{∗} : A(G) → A(H). If ϕ is an injection, show that the Frobenius Reciprocity formula holds: ϕ_{∗}(ϕ^{∗}(x) · y) = x · ϕ_{∗}(y) for all x ∈ A(H), y ∈ A(G).

§2. K_{0} of a ring

Let R be a ring. The set P(R) of isomorphism classes of finitely generated projective R-modules, together with direct sum ⊕ and identity 0, forms an abelian monoid. The Grothendieck group of R, K_{0}(R), is the group completion P^{−1}P of P(R).

When R is commutative, K_{0}(R) is a commutative ring with 1 = [R], because the monoid P(R) is a commutative semiring with product ⊗_{R}. This follows from the following facts: ⊗ distributes over ⊕; P ⊗_{R} Q ≅ Q ⊗_{R} P and P ⊗_{R} R ≅ P; if P, Q are finitely generated projective modules then so is P ⊗_{R} Q (by Ex. I.2.7).

For example, let k be a field or division ring. Then the abelian monoid P(k) is isomorphic to ℕ = {0, 1, 2, … }, so K_{0}(k) = ℤ. The same argument applies to show that K_{0}(R) = ℤ for every local ring R by (I.2.2), and also for every PID (by the Structure Theorem for modules over a PID). In particular, K_{0}(ℤ) = ℤ.

The Eilenberg Swindle I.2.8 shows why we restrict to finitely generated projectives. If we included R^{∞} (defined in Ex. I.1.7), then the formula P ⊕ R^{∞} ≅ R^{∞} would imply that [P] = 0 for every finitely generated projective R-module, and we would have K_{0}(R) = 0.

K_{0} is a functor from rings to abelian groups, and from commutative rings to commutative rings. To see this, suppose that R → S is a ring homomorphism. The functor ⊗_{R}S: P(R) → P(S) (sending P to P ⊗_{R} S) yields a monoid map P(R) → P(S), hence a group homomorphism K_{0}(R) → K_{0}(S). If R, S are commutative rings then ⊗_{R}S: K_{0}(R) → K_{0}(S) is a ring homomorphism, because ⊗_{R}S: P(R) → P(S) is a semiring map:

(P ⊗_{R} Q) ⊗_{R} S ≅ (P ⊗_{R} S) ⊗_{S} (Q ⊗_{R} S).

The free modules play a special role in understanding K_{0}(R) because they are cofinal in P(R). By Corollary 1.3 every element of K_{0}(R) can be written as [P] − [R^{n}] for some P and n. Moreover, [P] = [Q] in K_{0}(R) if and only if P, Q are stably isomorphic: P ⊕ R^{m} ≅ Q ⊕ R^{m} for some m. In particular, [P] = [R^{n}] if and only if P is stably free. The monoid L of isomorphism classes of free modules is ℕ if and only if R satisfies the Invariant Basis Property of Chapter I, §1. This yields the following information about K_{0}(R).

Lemma 2.1. The monoid map ℕ → P(R) sending n to R^{n} induces a group homomorphism ℤ → K_{0}(R). We have:

(1) ℤ → K_{0}(R) is injective if and only if R satisfies the Invariant Basis Property (IBP);

(2) Suppose that R satisfies the IBP (e.g., R is commutative). Then

K_{0}(R) ≅ ℤ ⟺ every finitely generated projective R-module is stably free.

Example 2.1.1. Suppose that R is commutative, or more generally that there is a ring map R → F to a field F. In this case ℤ is a direct summand of K_{0}(R), because the map K_{0}(R) → K_{0}(F) ≅ ℤ takes [R] to 1. A ring with K_{0}(R) = Q is given in Exercise 2.12 below.

Example 2.1.2 (Simple rings). Consider the matrix ring R = M_{n}(F) over a field F. We saw in Example I.1.1 that every R-module is projective (because it is a sum of copies of the projective module V ≅ F^{n}), and that length is an invariant of finitely generated R-modules. Thus length is an abelian group isomorphism K_{0}(M_{n}(F)) ⟶^{≅} ℤ sending [V] to 1. Since R has length n, the subgroup of K_{0}(R) ≅ ℤ generated by the free modules has index n. In particular, the inclusion ℤ ⊂ K_{0}(R) of Lemma 2.1 does not split.

Example 2.1.3. (Karoubi) We say a ring R is 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 K_{0}(R) = 0. This is because for every P we have a natural isomorphism P ⊕ (P ⊗_{R} M) ≅ P ⊗_{R} (R ⊕ M) ≅ (P ⊗_{R} M).

If R is flasque and the underlying right R-module structure on M is R, we say that R is an infinite sum ring. The right module isomorphism R^{2} ≅ R underlying θ makes R a direct sum ring (Ex. I.1.7). The Cone Rings of Ex. I.1.8, and the rings End_{R}(R^{∞}) of Ex. I.1.7, are examples of infinite sum rings, and hence flasque rings; see Exercise 2.15.

If R = R_{1} × R_{2} then P(R) ≅ P(R_{1}) × P(R_{2}). As in Exercise 1.2, this implies that K_{0}(R) ≅ K_{0}(R_{1}) × K_{0}(R_{2}). Thus K_{0} may be computed componentwise.

Example 2.1.4 (Semisimple rings). Let R be a semisimple ring, with simple modules V_{1}, ..., V_{r} (see Ex. I.1.1). Schur’s Lemma states that each D_{i} = Hom_{R}(V_{i}, V_{i}) is a division ring; the Artin-Wedderburn Theorem states that

R ≅ M_{n_{1}}(D_{1}) × ⋯ × M_{n_{r}}(D_{r}),

where dim_{D_{i}}(V_{i}) = n_{i}. By (2.1.2), K_{0}(R) ≅ ∏ K_{0}(M_{n_{i}}(D_{i})) ≅ ℤ^{r}.

Another way to see that K_{0}(R) ≅ ℤ^{r} is to use the fact that P(R) ≅ ℕ^{r}: the Krull-Schmidt Theorem states that every finitely generated (projective) module M is V_{1}^{ℓ_{1}} × ⋯ × V_{r}^{ℓ_{r}} for well-defined integers ℓ_{1}, ..., ℓ_{r}.

Example 2.1.5 (Von Neumann regular rings). A ring R is said to be von Neumann regular if for every r ∈ R there is an x ∈ R such that rxr = r. Since rxrx = rx, the element e = rx is idempotent, and the ideal rR = eR is a projective module. In fact, every finitely generated right ideal of R is of the form eR for some idempotent, and these form a lattice. Declaring e ≃ e′ if eR = e′R, the equivalence classes of idempotents in R form a lattice: (e_{1} ∧ e_{2}) and (e_{1} ∨ e_{2}) are defined to be the idempotents generating e_{1}R + e_{2}R and e_{1}R ∩ e_{2}R, respectively. Kaplansky proved in [Kap58] that every projective R-module is a direct sum of the modules eR. It follows that K_{0}(R) is determined by the lattice of idempotents (modulo ≃) in R. We will see several examples of von Neumann regular rings in the exercises.

Many von Neumann regular rings do not satisfy the (IBP), the ring End_{F} (F^{∞}) of Ex. I.1.7 being a case in point.

We call a ring R unit-regular if for every r ∈ R there is a unit x ∈ R such that rxr = r. Every unit-regular ring is Von Neumann regular, has stable range 1, and satisfies the (IBP) (Ex. I.1.13). In particular, ℤ ⊆ K_{0}(R). It is unknown whether for every simple unit-regular ring R the group K_{0}(R) is strictly unperforated, meaning that whenever x ∈ K_{0}(R) and nx = [Q] for some Q, then x = [P] for some P.

Goodearl [Gdrl1] has given examples of simple unit-regular rings R in which the group K_{0}(R) is strictly unperforated, but has torsion.

An example of a von Neumann regular ring R having the IBP and stable range 2, and K_{0}(R) = ℤ ⊕ ℤ/n is given in [MM82].

2.1.6. Suppose that R is the direct limit of a filtered system {R_{i}} of rings. Then every finitely generated projective R-module is of the form P_{i} ⊗_{R_{i}} R for some i and some finitely generated projective R_{i}-module P_{i}. Any isomorphism P_{i} ⊗_{R_{i}} R ≅ P′_{i} ⊗_{R_{i}} R may be expressed using finitely many elements of R, and hence P_{i} ⊗_{R_{i}} R_{j} ≅ P′_{i} ⊗_{R_{i}} R_{j} for some j. That is, P(R) is the filtered colimit of the P(R_{i}). By Ex. 1.3 we have

K_{0}(R) ≅ \underrightarrow{lim} K_{0}(R_{i}).

This observation is useful when studying K_{0}(R) of a commutative ring R, because R is the direct limit of its finitely generated subrings. As finitely generated commutative rings are noetherian with finite normalization, properties of K_{0}(R) may be deduced from properties of K_{0} of these nice subrings. If R is integrally closed we may restrict to finitely generated normal subrings, so K_{0}(R) is determined by K_{0} of noetherian integrally closed domains.

Here is another useful reduction; it follows immediately from the observation that if I is nilpotent (or complete) then idempotent lifting (Ex. I.2.2) yields a monoid isomorphism P(R) ≅ P(R/I). Recall that an ideal I is said to be complete if every Cauchy sequence ∑_{n=1}^{∞} x_{n} with x_{n} ∈ I^{n} converges to a unique element of I.

Lemma 2.2. If I is a nilpotent ideal of R, or more generally a complete ideal, then

K_{0}(R) ≅ K_{0}(R/I).

In particular, if R is commutative then K_{0}(R) ≅ K_{0}(R_{red}).

Example 2.2.1 (0-dimensional commutative rings). Let R be a commutative ring. It is elementary that R_{red} is Artinian if and only if Spec(R) is finite and discrete. More generally, it is known (see Ex. I.1.13 and [AM, Ex. 3.11]) that the following are equivalent:

(i) R_{red} is a commutative von Neumann regular ring (2.1.5); (ii) R has Krull dimension 0; (iii) X = Spec(R) is compact, Hausdorff and totally disconnected. (For example, to see that a commutative von Neumann regular R must be reduced, observe that if r^{2} = 0 then r = rxr = 0.)

When R is a commutative von Neumann regular ring, the modules eR are componentwise free; Kaplansky’s result states that every projective module is componentwise free. By I.2, the monoid P(R) is just [X, ℕ], X = Spec(R). By (1.4) this yields K_{0}(R) = [X, ℤ]. By Lemma 2.2, this proves

Pierce’s Theorem 2.2.2. For every 0-dimensional commutative ring R:

K_{0}(R) = [Spec(R), ℤ].

Example 2.2.3 (K_{0} does not commute with infinite products). Let R = ∏ F_{i} be an infinite product of fields. Then R is von Neumann regular, so X = Spec(R) is an uncountable totally disconnected compact Hausdorff space. By Pierce’s Theorem, K_{0}(R) ≅ [X, ℤ]. This is contained in but not equal to the product ∏ K_{0}(F_{i}) ≅ ∏ ℤ.

Rank and H_{0}

Definition. When R is commutative, we write H_{0}(R) for [Spec(R), ℤ], the ring of all continuous maps from Spec(R) to ℤ. Since Spec(R) is quasi-compact, we know by (1.4) that H_{0}(R) is always a free abelian group. If R is a noetherian ring, then Spec(R) has only finitely many (say c) components, and H_{0}(R) ≅ ℤ^{c}. If R is a domain, or more generally if Spec(R) is connected, then H_{0}(R) = ℤ.

H_{0}(R) is a subring of K_{0}(R). To see this, consider the submonoid L of P(R) consisting of componentwise free modules R^{f}. Not only is L cofinal in P(R), but L → P(R) is a semiring map: R^{f} ⊗R^{g} ≅ R^{fg}; by (1.3), L^{−1}L is a subring of K_{0}(R). Finally, L is isomorphic to [Spec(R), ℕ], so as in (1.4) we have L^{−1}L ≅ H_{0}(R). For example, Pierce’s theorem (2.2.2) states that if dim(R) = 0 then K_{0}(R) ≅ H_{0}(R).

Recall from I.2 that the rank of a projective module gives a map from P(R) to [Spec(R), ℕ]. Since rank(P ⊕ Q) = rank(P) + rank(Q) and rank(P ⊗ Q) = rank(P) rank(Q) (by Ex. I.2.7), this is a semiring map. As such it induces a ring map

rank: K_{0}(R) → H_{0}(R).

Since rank(R^{f}) = f for every componentwise free module, the composition H_{0}(R) ⊂ K_{0}(R) → H_{0}(R) is the identity. Thus H_{0}(R) is a direct summand of K_{0}(R).

Definition 2.3. The ideal K̃_{0}(R) of the ring K_{0}(R) is defined as the kernel of the rank map. By the above remarks, there is a natural decomposition

K_{0}(R) ≅ H_{0}(R) ⊕ K̃_{0}(R).

We will see later (in 4.6.1) that K̃_{0}(R) is a nil ideal. Since H_{0}(R) is visibly a reduced ring, K̃_{0}(R) is the nilradical of K_{0}(R).

Lemma 2.3.1. If R is commutative, let P_{n}(R) denote the subset of P(R) consisting of projective modules of constant rank n. There is a map P_{n}(R) → K_{0}(R) sending P to [P]−[R^{n}]. This map is compatible with the stabilization map P_{n}(R) → P_{n+1}(R) sending P to P ⊕ R, and the induced map is an isomorphism:

\underrightarrow{lim} P_{n}(R) ≅ K̃_{0}(R).

Proof. This follows easily from (1.3). □

Corollary 2.3.2. Let R be a commutative noetherian ring of Krull dimension d — or more generally any commutative ring of stable range d + 1 (Ex. I.1.5). For every n > d the above maps are bijections: P_{n}(R) ≅ K̃_{0}(R).

Proof. If P and Q are finitely generated projective modules of rank > d, then by Bass Cancellation (I.2.3b) we may conclude that

[P] = [Q] in K_{0}(R) if and only if P ≅ Q. □

Here is another interpretation of K̃_{0}(R): it is the intersection of the kernels of K_{0}(R) → K_{0}(F) over all maps R → F, F a field. This follows from naturality of rank and the observation that K̃_{0}(F) = 0 for every field F.

This motivates the following definition for a noncommutative ring R: let K̃_{0}(R) denote the intersection of the kernels of K_{0}(R) → K_{0}(S) over all maps R → S, where S is a simple artinian ring. If no such map R → S exists, we set K̃_{0}(R) = K_{0}(R). We define H_{0}(R) to be the quotient of K_{0}(R) by K̃_{0}(R). When R is commutative, this agrees with the above definitions of H_{0} and K̃_{0}, because the maximal commutative subrings of a simple artinian ring S are finite products of 0-dimensional local rings.

H_{0}(R) is a torsionfree abelian group for every ring R. To see this, note that there is a set X of maps R → S_{x} through which every other R → S′ factors. Since each K_{0}(S_{x}) → K_{0}(S′) is an isomorphism, K̃_{0}(R) is the intersection of the kernels of the maps K_{0}(R) → K_{0}(S_{x}), x ∈ X. Hence H_{0}(R) is the image of K_{0}(R) in the torsionfree group ∏_{x∈X} K_{0}(S_{x}) ≅ ∏_{x} ℤ ≅ Map(X, ℤ).

Example 2.4 (Whitehead group W h_{0}). If R is the group ring ℤ[G] of a group G, the (zero-th) Whitehead group W h_{0}(G) is the quotient of K_{0}(ℤ[G]) by the subgroup K_{0}(ℤ) = ℤ. The augmentation map ε: ℤ[G] → ℤ sending G to 1 induces a decomposition K_{0}(ℤ[G]) ≅ ℤ ⊕ W h_{0}(G), and clearly K̃_{0}(ℤ[G]) ⊆ W h_{0}(G). It follows from a theorem of Swan ([Bass, XI(5.2)]) that if G is finite then K̃_{0}(ℤG) = W h_{0}(G) and H_{0}(ℤG) = ℤ. I do not know whether K̃_{0}(ℤG) = W h_{0}(G) for every group.

The group W h_{0}(G) arose in topology via the following result of C.T.C. Wall. We say that a CW complex X is dominated by a complex K if there is a map f: K → X having a right homotopy inverse; this says that X is a retract of K in the homotopy category.

Theorem 2.4.1 (Wall Finiteness Obstruction). Suppose that X is dominated by a finite CW complex, with fundamental group G = π_{1}(X). This data determines an element w(X) of W h_{0}(G) such that w(X) = 0 if and only if X is homotopy equivalent to a finite CW complex.

Hattori-Stallings trace map

For any associative ring R, let [R, R] denote the subgroup of R generated by the elements [r, s] = rs − sr, r, s ∈ R.

For each n, the trace of an n × n matrix provides an additive map from M_{n}(R) to R/[R, R] invariant under conjugation; the inclusion of M_{n}(R) in M_{n+1}(R) via g ↦ \binom{g 0}{0 0} is compatible with the trace map. It is not hard to show that the trace M_{n}(R) → R/[R, R] induces an isomorphism:

M_{n}(R)/[M_{n}(R), M_{n}(R)] ≅ R/[R, R].

If P is a finitely generated projective, choosing an isomorphism P ⊕ Q ≅ R^{n} yields an idempotent e in M_{n}(R) such that P = e(R^{n}) and End(P) = eM_{n}(R)e. By Ex. I.2.3, any other choice yields an e_{1} which is conjugate to e in some larger M_{m}(R). Therefore the trace of an endomorphism of P is a well-defined element of R/[R, R], independent of the choice of e. This gives the trace map End(P) → R/[R, R]. In particular, the trace of the identity map of P is the trace of e; we call it the trace of P.

If P′ is represented by an idempotent matrix f then P ⊕P′ is represented by the idempotent matrix ({e \atop 0} {0 \atop f} ) so the trace of P ⊕P′ is trace(P)+trace(P′). Therefore the trace is an additive map on the monoid P(R). The map K_{0}(R) → R/[R, R] induced by universality is called the Hattori-Stallings trace map, after the two individuals who first studied it.

When R is commutative, we can provide a direct description of the ring map H_{0}(R) → R obtained by restricting the trace map to the subring H_{0}(R) of K_{0}(R). Any continuous map f: Spec(R) → ℤ induces a decomposition R = R_{1} ×⋯×R_{c} by Ex. I.2.4; the coordinate idempotents e_{1}, ..., e_{c} are elements of R. Since trace(e_{i}R) is e_{i}, it follows immediately that trace(f) is ∑ f(i)e_{i}. The identity trace(fg) = trace(f)trace(g) which follows immediately from this formula shows that trace is a ring map.

Proposition 2.5. If R is commutative then the Hattori-Stallings trace factors as

K_{0}(R) ⟶^{rank} H_{0}(R) → R.

Proof. The product over all 𝔭 in Spec(R) yields the commutative diagram:

K_{0}(R) ⟶ ∏ K_{0}(R_{𝔭})
trace↓  ↓^{trace}
R ⟶_{inclusion}^{diagonal} ∏ R_{𝔭}.

The kernel of the top arrow is K̃_{0}(R), so the left arrow factors as claimed. □

Example 2.5.1 (Group rings). Let k be a commutative ring, and suppose that R is the group ring k[G] of a group G. If g and h are conjugate elements of G then h − g ∈ [R, R] because xgx^{−1} − g = [xg, x^{−1}]. From this it is not hard to see that R/[R, R] is isomorphic to the free k-module ⊕k[g] whose basis is the set G/∼ of conjugacy classes of elements of G. Relative to this basis, we may write

trace(P) = ∑ r_{P} (g)[g].

Clearly, the coefficients r_{P} (g) of trace(P) are functions on the set G/∼ for each P.

If G is finite, then any finitely generated projective k[G]-module P is also a projective k-module, and we may also form the trace map End_{k}(P) → k and hence the “character” χ_{P}: G → k by the formula χ_{P} (g) = trace(g). Hattori proved that if Z_{G}(g) denotes the centralizer of g ∈ G then Hattori’s formula holds (see [Bass76, 5.8]):

(2.5.2)  χ_{P} (g) = |Z_{G}(g)| r_{P} (g^{−1}).

Corollary 2.5.3. If G is a finite group, the ring ℤ[G] has no idempotents except 0 and 1.

Proof. Let e be an idempotent element of ℤ[G]. χ_{P} (1) is the rank of the ℤmodule P = eℤ[G], which must be less than the rank |G| of ℤ[G]. Since r_{P} (1) ∈ ℤ, this contradicts Hattori’s formula χ_{P} (1) = |G| r_{P} (1). □

Bass has conjectured that for every group G and every finitely generated projective ℤ[G]-module P we have r_{P} (g) = 0 for g ≠ 1 and r_{P} (1) = rank_{ℤ}(P ⊗_{ℤ[G]} ℤ). For G finite, this follows from Hattori’s formula and Swan’s theorem (cited in 2.4) that K̃_{0} = W h_{0}. See [Bass76].

Example 2.5.4. Suppose that k is a field of characteristic 0 and kG = k[G] is the group ring of a finite group G with c conjugacy classes. By Maschke’s theorem, kG is a product of simple k-algebras: S_{1} × ⋯ × S_{c} so kG/[kG, kG] is k^{c}. By (2.1.4) K_{0}(kG) ≅ ℤ^{c}. Hattori’s formula (and some classical representation theory) shows that the trace map from K_{0}(kG) to kG/[kG, kG] is isomorphic to the natural inclusion of ℤ^{c} in k^{c}.

Determinant

Suppose now that R is a commutative ring. Recall from I.3 that the determinant of a finitely generated projective module P is an element of the Picard group Pic(R).

Proposition 2.6. The determinant induces a surjective group homomorphism

det: K_{0}(R) → Pic(R)

Proof. By the universal property of K_{0}, it suffices to show that det(P ⊕ Q) ≅ det(P) ⊗_{R} det(Q). We may assume that P and Q have constant rank m and n, respectively. Then ∧^{m+n}(P ⊕ Q) is the sum over all i, j such that i + j = m + n of (∧^{i}Q) ⊗ (∧^{j}P). If i > m or j > n we have ∧^{i}P = 0 or ∧^{j}Q = 0, respectively. Hence ∧^{m+n}(P ⊕ Q) = (∧^{m}P) ⊗ (∧^{n}Q), as asserted. □

Definition 2.6.1. Let SK_{0}(R) denote the subset of K_{0}(R) consisting of the classes x = [P] − [R^{m}], where P has constant rank m and ∧^{m}P ≅ R. This is the kernel of det: K̃_{0}(R) → Pic(R), by Lemma 2.3.1 and Proposition 2.6.

SK_{0}(R) is an ideal of K_{0}(R). To see this, we use Ex. I.3.4: if x = [P]−[R^{m}] is in SK_{0}(R) and Q has rank n then det(x · Q) = (det P)^{⊗n}(det Q)^{⊗m}(det Q)^{⊗−m} = R.

Corollary 2.6.2. For every commutative ring R, H_{0}(R) ⊕ Pic(R) is a ring with square-zero ideal Pic(R), and there is a surjective ring homomorphism with kernel SK_{0}(R):

rank ⊕ det: K_{0}(R) → H_{0}(R) ⊕ Pic(R)

Corollary 2.6.3. If R is a 1-dimensional commutative noetherian ring, then the classification of finitely generated projective R-modules in I.3.4 induces an isomorphism:

K_{0}(R) ≅ H_{0}(R) ⊕ Pic(R).

Morita Equivalence We say that two rings R and S are Morita equivalent if mod-R and mod-S are equivalent as abelian categories, that is, if there exist additive functors T and U

mod-R ⟶^{T} ; ⟵_{U} mod-S

such that UT ≅ id_{R} and T U ≅ id_{S}. This implies that T and U preserve filtered colimits. Set P = T (R) and Q = U(S); P is an R-S bimodule and Q is a S–R bimodule via the maps R = End_{R}(R) ⟶^{T} End_{S}(P) and S = End_{S}(S) ⟶^{U} End_{R}(Q). Since T (⊕R) = ⊕P and U(⊕S) = ⊕Q it follows that we have T (M) ≅ M ⊗_{R} P and U(N) ≅ N ⊗_{S} Q for all M, N. Both UT (R) ≅ P ⊗_{S} Q ≅ R and T U(S) ≅ Q⊗_{R} P ≅ S are bimodule isomorphisms. Here is the main structure theorem, taken from [Bass, II.3].

Structure Theorem for Morita Equivalence 2.7. If R and S are Morita equivalent, and P, Q are as above, then:

(a) P and Q are finitely generated projective, both as R-modules and as Smodules;

(b) End_{S}(P) ≅ R ≅ End_{S}(Q)^{op} and End_{R}(Q) ≅ S ≅ End_{R}(P)^{op};

(c) P and Q are dual S-modules: P ≅ Hom_{S}(Q, S) and Q ≅ Hom_{S}(P, S);

(d) T (M) ≅ M ⊗_{R} P and U(N) ≅ N ⊗_{S} Q for every M and N;

(e) P is a “faithful” S-module in the sense that the functor Hom_{S}(P, −) from mod-S to abelian groups is a faithful functor. (If S is commutative then P is faithful if and only if rank(P) ≥ 1.) Similarly, Q is a “faithful” R-module.

Since P and Q are finitely generated projective, the Morita functors T and U also induce an equivalence between the categories P(R) and P(S). This implies the following:

Corollary 2.7.1. If R and S are Morita equivalent then K_{0}(R) ≅ K_{0}(S).

Example 2.7.2. R = M_{n}(S) is always Morita equivalent to S; P is the bimodule S^{n} of “column vectors” and Q is the bimodule (S^{n})^{t} of “row vectors.” More generally suppose that P is a “faithful” finitely generated projective S-module. Then R = End_{S}(P) is Morita equivalent to S, the bimodules being P and Q = Hom_{S}(P, S). By 2.7.1, we see that K_{0}(S) ≅ K_{0}(M_{n}(S)).

Additive Functors 2.8. Any R-S bimodule P which is finitely generated projective as a right S-module, induces an additive (hence exact) functor T (M) = M ⊗_{R} P from P(R) to P(S), and therefore induces a map K_{0}(R) → K_{0}(S). If all we want is an additive functor T from P(R) to P(S), we do not need the full strength of Morita equivalence. Given T, set P = T (R). By additivity we have T (R^{n}) = P^{n} ≅ R^{n} ⊗_{R} P; from this it is not hard to see that T (M) ≅ M ⊗_{R} P for every finitely generated projective M, and that T is isomorphic to –⊗_{R}P. See Ex. 2.14 for more details.

A bimodule map (resp., isomorphism) P → P′ induces an additive natural transformation (resp., isomorphism) T → T′. This is the case, for example, with the bimodule isomorphism R ⊕ M ≅ M defining a flasque ring (2.1.3).

Example 2.8.1 (Base change and Transfer maps). Suppose that f: R → S is a ring map. Then S is an R–S bimodule, and it represents the base change functor f^{∗}: K_{0}(R) → K_{0}(S) sending P to P ⊗_{R} S. If in addition S is finitely generated projective as a right R-module then there is a forgetful functor from P(S) to P(R); it is represented by S as a S–R bimodule because it sends Q to Q ⊗_{S} S. The induced map f_{∗}: K_{0}(S) → K_{0}(R) is called the transfer map. We will return to this point in 7.9 below, explaining why we have selected the contravariant notation f^{∗} and f_{∗}.

Mayer-Vietoris sequences

For any ring R with unit, we can include GL_{n}(R) in GL_{n+1}(R) as the matrices \binom{g 0}{0 1}. The group GL(R) is the union of the groups GL_{n}(R). Now suppose we are given a Milnor square of rings, as in I.2:

R ⟶^{f} S
↓  ↓
R/I ⟶^{f̄} S/I

Define ∂_{n}: GL_{n}(S/I) → K_{0}(R) by Milnor patching: ∂_{n}(g) is [P] − [R^{n}], where P is the projective R-module obtained by patching free modules along g as in (I.2.6). The formulas of Ex. I.2.9 imply that ∂_{n}(g) = ∂_{n+1}({g \atop 0} {0 \atop 1}) and ∂_{n}(g)+∂_{n}(h) = ∂_{n}(gh). Therefore the {∂_{n}} assemble to give a group homomorphism ∂ from GL(S/I) to K_{0}(R). The following result now follows from (I.2.6) and Ex. 1.4.

Theorem 2.9 (Mayer-Vietoris). Given a Milnor square as above, the sequence

GL(S/I) ⟶^{∂} K_{0}(R) ⟶^{Δ} K_{0}(S) ⊕ K_{0}(R/I) ⟶^{±} K_{0}(S/I)

is exact. The image of ∂ is the double coset space

GL(S)\GL(S/I)/GL(R/I) = GL(S/I)/ ∼

where x ∼ gxh for x ∈ GL(S/I), g ∈ GL(S) and h ∈ GL(R/I).

Example 2.9.1. If R is the coordinate ring of the node over a field k (I.3.10.2) then K_{0}(R) ≅ ℤ ⊕ k^{×}. If R is the coordinate ring of the cusp over k (I.3.10.1) then K_{0}(R) ≅ ℤ ⊕ k. Indeed, the coordinate rings of the node and the cusp are 1-dimensional noetherian rings, so 2.6.3 reduces the Mayer-Vietoris sequence to the Units-Pic sequence I.3.10.

We conclude with a useful construction, anticipating several later developments.

Definition 2.10. Let T : P(R) → P(S) be an additive functor, such as the base change or transfer of 2.8.1. P(T) is the category whose objects are triples (P, α, Q), where P, Q ∈ P(R) and α : T (P) → T (Q) is an isomorphism. A morphism (P, α, Q) → (P′, α′, Q′) is a pair of R-module maps p : P → P′, q : Q → Q′ such that α′T (p) = T (q)α. An exact sequence in P(T) is a sequence

(2.10.1)  0 → (P′, α′, Q′) → (P, α, Q) → (P′′, α′′, Q′′) → 0

whose underlying sequences 0 → P′ → P → P′′ → 0 and 0 → Q′ → Q → Q′′ → 0 are exact. We define K_{0}(T) to be the abelian group with generators the objects of P(T) and relations:

(a) [(P, α, Q)] = [(P′, α′, Q′)] + [(P′′, α′′, Q′′)] for every exact sequence (2.10.1);

(b) [(P_{1}, α, P_{2})] + [(P_{2}, β, P_{3})] = [(P_{1}, βα, P_{3})].

If T is the base change f^{∗}, we write K_{0}(f) for K_{0}(T).

It is easy to see that there is a map K_{0}(T) → K_{0}(R) sending [(P, α, Q)] to [P] − [Q]. If T is a base change functor f^{∗} associated to f : R → S, or more generally if the T (R^{n}) are cofinal in P(S), then there is an exact sequence:

(2.10.2)  GL(S) ⟶^{∂} K_{0}(T) → K_{0}(R) → K_{0}(S).

The construction of ∂ and verification of exactness is not hard, but lengthy enough to relegate to exercise 2.17. If f : R → R/I then K_{0}(f^{∗}) is the group K_{0}(I) of Ex. 2.4; see Ex. 2.4(e).

EXERCISES

2.1 Let R be a commutative ring. If A is an R-algebra, show that the functor ⊗_{R}: P(A) × P(R) → P(A) yields a map K_{0}(A) ⊗_{ℤ} K_{0}(R) → K_{0}(A) making K_{0}(A) into a K_{0}(R)-module. If A → B is an algebra map, show that K_{0}(A) → K_{0}(B) is a K_{0}(R)-module homomorphism.

2.2 Projection Formula. Let R be a commutative ring, and A an R-algebra which as an R-module is finitely generated projective of rank n. By Ex. 2.1, K_{0}(A) is a K_{0}(R)-module, and the base change map f^{∗}: K_{0}(R) → K_{0}(A) is a module homomorphism. We shall write x · f^{∗}y for the product in K_{0}(A) of x ∈ K_{0}(A) and y ∈ K_{0}(R); this is an abuse of notation when A is noncommutative.

(a) Show that the transfer map f_{∗}: K_{0}(A) → K_{0}(R) of Example 2.8.1 is a K_{0}(R)-module homomorphism, i.e., that the projection formula holds:

f_{∗}(x · f^{∗}y) = f_{∗}(x) · y for every x ∈ K_{0}(A), y ∈ K_{0}(R).

(b) Show that both compositions f^{∗}f_{∗} and f_{∗}f^{∗} are multiplication by [A].

(c) Show that the kernels of f^{∗}f_{∗} and f_{∗}f^{∗} are annihilated by a power of n.

2.3 Excision for K_{0}. If I is an ideal in a ring R, form the augmented ring R ⊕ I and let K_{0}(I) = K_{0}(R, I) denote the kernel of K_{0}(R ⊕ I) → K_{0}(R).

(a) If R → S is a ring map sending I isomorphically onto an ideal of S, show that K_{0}(R, I) ≅ K_{0}(S, I). Thus K_{0}(I) is independent of R. Hint. Show that GL(S)/GL(S ⊕ I) = 1.

(b) If I ∩ J = 0, show that K_{0}(I + J) ≅ K_{0}(I) ⊕ K_{0}(J).

(c) Ideal sequence. Show that there is an exact sequence

GL(R) → GL(R/I) ⟶^{∂} K_{0}(I) → K_{0}(R) → K_{0}(R/I).

(d) If R is commutative, use Ex. I.3.6 to show that there is a commutative diagram with exact rows, the vertical maps being determinants:

GL(R) ⟶ GL(R/I) ⟶^{∂} K_{0}(I) ⟶ K_{0}(R) ⟶ K_{0}(R/I)
↓  ↓  ↓  ↓  ↓
R^{×} ⟶ (R/I)^{×} ⟶^{∂} Pic(I) ⟶ Pic(R) ⟶ Pic(R/I).

2.4 K_{0}I. If I is a ring without unit, we define K_{0}(I) as follows. Let R be a ring with unit acting upon I, form the augmented ring R ⊕ I, and let K_{0}(I) be the kernel of K_{0}(R ⊕ I) → K_{0}(R). Thus K_{0}(R ⊕ I) ≅ K_{0}(R) ⊕ K_{0}(I) by definition.

(a) If I has a unit, show that R⊕I ≅ R×I as rings with unit. Since K_{0}(R×I) = K_{0}(R) × K_{0}(I), this shows that the definition of Ex. 2.3 agrees with the usual definition of K_{0}(I).

(b) Show that a map I → J of rings without unit induces a map K_{0}(I) → K_{0}(J)

(c) Let M_{∞}(R) denote the union ∪M_{n}(R) of the matrix groups, where M_{n}(R) is included in M_{n+1}(R) as the matrices ({g \atop 0} {0 \atop 0}). M_{∞}(R) is a ring without unit. Show that the inclusion of R = M_{1}(R) in M_{∞}(R) induces an isomorphism

K_{0}(R) ≅ K_{0}(M_{∞}(R)).

(d) If k is a field, show that R = k ⊕ M_{∞}(k) is a von Neumann regular ring. Then show that H_{0}(R) = ℤ and K_{0}(R) ≅ ℤ ⊕ ℤ.

(e) If f : R → R/I, show that K_{0}(I) is the group K_{0}(f^{∗}) of 2.10. Hint: Use

f_{0} : R ⊕ I → R and Ex. 2.3(c). 2.5 Radical ideals. Let I be a radical ideal in a ring R (see Ex. I.1.12, I.2.1).

(a) Show that K_{0}(I) = 0, and that K_{0}(R) → K_{0}(R/I) is an injection.

(b) If I is a complete ideal, K_{0}(R) ≅ K_{0}(R/I) by Lemma 2.2. If R is a semilocal but not local domain, show that K_{0}(R) → K_{0}(R/I) is not an isomorphism.

2.6 Semilocal rings. A ring R is called semilocal if R/J is semisimple for some radical ideal J. Show that if R is semilocal then K_{0}(R) ≅ ℤ^{n} for some n > 0. 2.7 Show that if f : R → S is a map of commutative rings, then:

ker(f) contains no idempotents(≠ 0) ⇔ H_{0}(R) → H_{0}(S) is an injection.

Conclude that H_{0}(R) = H_{0}(R[t]) = H_{0}(R[t, t^{−1}]). 2.8 Consider the following conditions on a ring R (cf. Ex. I.1.2):

(IBP) R satisfies the Invariant Basis Property (IBP); (PO) K_{0}(R) is a partially ordered abelian group (see §1); (III) For all n, if R^{n} ≅ R^{n} ⊕ P then P = 0.

Show that (III) ⇒ (P O) ⇒ (IBP). This implies that K_{0}(R) is a partially ordered abelian group if R is either commutative or noetherian. (See Ex. I.1.4.) 2.9 Rim squares. Let G be a cyclic group of prime order p, and ζ = e^{2πi/p} a primitive p^{th} root of unity. Show that the map ℤ[G] → ℤ[ζ] sending a generator of G to ζ induces an isomorphism K_{0}(ℤ[G]) ≅ K_{0}(ℤ[ζ]) and hence W h_{0}(G) ≅ Pic(ℤ[ζ]). Hint: Form a Milnor square with ℤG/I = ℤ, ℤ[ζ]/I = 𝔽_{p}, and consider the cyclotomic units u = \frac{ζ^{i}−1}{ζ−1}, 1 ≤ i < p.

2.10 Let R be a commutative ring. Prove that

(a) If rank(x) > 0 for some x ∈ K_{0}(R), then there is an n > 0 and a finitely generated projective module P so that nx = [P]. (This says that the partially ordered group K_{0}(R) is “unperforated” in the sense of [Gdearl].)

(b) If P, Q are finitely generated projectives such that [P] = [Q] in K_{0}(R), then there is an n > 0 such that P ⊕ ⋯ ⊕ P ≅ Q ⊕ ⋯ ⊕ Q (n copies of P, n copies of Q).

Hint: First assume that R is noetherian of Krull dimension d < ∞, and use Bass-Serre Cancellation. In the general case, write R as a direct limit.

2.11 A (normalized) dimension function for a von Neumann regular ring R is a group homomorphism d : K_{0}(R) → ℝ so that d(R^{n}) = n and d(P) > 0 for every nonzero finitely generated projective P.

(a) If P ⊆ Q, show that any dimension function must have d(P) ≤ d(Q)

(b) If R has a dimension function, show that the formula ρ(r) = d(rR) defines a rank function ρ: R → [0, 1] in the sense of Ex. I.1.13. Then show that this gives a 1-1 correspondence between rank functions on R and dimension functions on K_{0}(R).

2.12 Let R be the union of the matrix rings M_{n!}(F) constructed in Ex. I.1.13. Show that the inclusion ℤ ⊂ K_{0}(R) extends to an isomorphism K_{0}(R) ≅ ℚ.

2.13 Let R be the infinite product of the matrix rings M_{i}(ℂ), i = 1, 2, ...

(a) Show that every finitely generated projective R-module P is componentwise trivial in the sense that P ≅ ∏ P_{i}, the P_{i} being finitely generated projective M_{i}(ℂ)-modules.

(b) Show that the map from K_{0}(R) to the group ∏ K_{0}(M_{i}(ℂ)) = ∏ ℤ of infinite sequences (n_{1}, n_{2}, ...) of integers is an injection, and that K_{0}(R) = H_{0}(R) is isomorphic to the group of bounded sequences.

(c) Show that K_{0}(R) is not a free abelian group, even though it is torsionfree. Hint: Consider the subgroup S of sequences (n_{1}, ...) such that the power of 2 dividing n_{i} approaches ∞ as i → ∞; show that S is uncountable but that S/2S is countable.

2.14 Bivariant K_{0}. If R and R′ are rings, let Rep(R, R′) denote the set of isomorphism classes of R–R′ bimodules M such that M is finitely generated projective as a right R′-module. Each M gives a functor ⊗_{R}M from P(R) to P(R′) sending P to P ⊗_{R} M. This induces a monoid map P(R) → P(R′) and hence a homomorphism from K_{0}(R) to K_{0}(R′). For example, if f: R → R′ is a ring homomorphism and R′ is considered as an element of Rep(R, R′), we obtain the map ⊗_{R}R′. Show that:

(a) Every additive functor P(R) → P(R′) is induced from an M in Rep(R, R′);

(b) If K_{0}(R, R′) denotes the group completion of Rep(R, R′), then M ⊗_{R′} N induces a bilinear map from K_{0}(R, R′) ⊗ K_{0}(R′, R′′) to K_{0}(R, R′′);

(c) K_{0}(ℤ, R) is K_{0}(R), and if M ∈ Rep(R, R′) then the map ⊗_{R}M : K_{0}(R) → K_{0}(R′) is induced from the product of (b).

(d) If R and R′ are Morita equivalent, and P is the R-R′ bimodule giving the isomorphism mod-R ≅ mod-R′, the class of P in K_{0}(R, R′) gives the Morita isomorphism K_{0}(R) ≅ K_{0}(R′).

2.15 In this exercise, we connect the definition 2.1.3 of infinite sum ring with a more elementary description due to Wagoner. If R is a direct sum ring, the isomorphism R^{2} ≅ R induces a ring homomorphism ⊕ : R × R ⊂ End_{R}(R^{2}) ≅ End_{R}(R) = R.

(a) Suppose that R is an infinite sum ring with bimodule M, and write r ↦ r^{∞} for the ring homomorphism R → End_{R}(M) ≅ R arising from the left action of R on the right R-module M. Show that r ⊕ r^{∞} = r^{∞} for all r ∈ R.

(b) Conversely, suppose that R is a direct sum ring, and R ⟶^{∞} R is a ring map so that r ⊕ r^{∞} = r^{∞} for all r ∈ R. Show that R is an infinite sum ring.

(c) (Wagoner) Show that the Cone Rings of Ex. I.1.8, and the rings End_{R}(R^{∞}) of Ex. I.1.7, are infinite sum rings. Hint: R^{∞} ≅ ∐_{i=1}^{∞} R^{∞}, so a version of the Eilenberg Swindle I.2.8 applies. 2.16 For any ring R, let J be the (nonunital) subring of E = End_{R}(R^{∞}) of all f such that f(R^{∞}) is finitely generated (Ex. I.1.7). Show that M_{∞}(R) ⊂ J_{n} induces an isomorphism K_{0}(R) ≅ K_{0}(J). Hint: For the projection e_{n} : R^{∞} → R^{n}, J_{n} = e_{n}E maps onto M_{n}(R) = e_{n}Ee_{n} with nilpotent kernel. But J = ∪J_{n}. 2.17 This exercise shows that there is an exact sequence (2.10.2) when T is cofinal.

(a) Show that [(P, α, Q)] + [(Q, −α^{−1}, P)] = 0 and [(P, T (γ), Q)] = 0 in K_{0}(T).

(b) Show that every element of K_{0}(T) has the form [(P, α, R^{n})].

(c) Use cofinality and the maps ∂(α) = [(R^{n}, α, R^{n})] of (2.10.2), from Aut(T R^{n}) to K_{0}(T), to show that there is a homomorphism ∂ : GL(S) → K_{0}(T).

(d) Use (a), (b) and (c) to show that (2.10.2) is exact at K_{0}(T).

(e) Show that (2.10.2) is exact at K_{0}(R).

§3. K(X), KO(X) and KU(X) of a topological space

Let X be a paracompact topological space. The sets VB_{ℝ}(X) and VB_{ℂ}(X) of isomorphism classes of real and complex vector bundles over X are abelian monoids under Whitney sum. By Construction I.4.2, they are commutative semirings under ⊗. Hence the group completions KO(X) of VB_{ℝ}(X) and KU(X) of VB_{ℂ}(X) are commutative rings with identity 1 = [T^{1}]. If the choice of ℝ or ℂ is understood, we will just write K(X) for simplicity.

Similarly, the set VB_{ℍ}(X) is an abelian monoid under ⊕, and we write KSp(X) for its group completion. Although it has no natural ring structure, the construction of Ex. I.4.18 endows KSp(X) with the structure of a module over the ring KO(X).

For example if ∗ denotes a 1-point space then K(∗) = ℤ. If X is contractible, then KO(X) = KU(X) = ℤ by I.4.6.1. More generally, K(X) ≅ K(Y) whenever X and Y are homotopy equivalent by I.4.6.

The functor K(X) is contravariant in X. Indeed, if f: Y → X is continuous, the induced bundle construction E ↦ f^{∗}E yields a function f^{∗}: VB(X) → VB(Y) which is a morphism of monoids and semirings; hence it induces a ring homomorphism f^{∗}: K(X) → K(Y). By the Homotopy Invariance Theorem I.4.5, the map f^{∗} depends only upon the homotopy class of f in [Y, X].

For example, the universal map X → ∗ induces a ring map from ℤ = K(∗) into K(X), sending n > 0 to the class of the trivial bundle T^{n} over X. If X ≠ ∅ then any point of X yields a map ∗ → X splitting the universal map X → ∗. Thus the subring ℤ is a direct summand of K(X) when X ≠ ∅. (But if X = ∅ then K(∅) = 0.) For the rest of this section, we will assume X ≠ ∅ in order to avoid awkward hypotheses.

The trivial vector bundles T^{n} and the componentwise trivial vector bundles T^{f} form sub-semirings of VB(X), naturally isomorphic to ℕ and [X, ℕ], respectively. When X is compact, the semirings ℕ and [X, ℕ] are cofinal in VB(X) by the Subbundle Theorem I.4.1, so by Corollary 1.3 we have subrings

ℤ ⊂ [X, ℤ] ⊂ K(X).

More generally, it follows from Construction I.4.2 that dim: VB(X) → [X, ℕ] is a semiring map splitting the inclusion [X, ℕ] ⊂ VB(X). Passing to Group Completions, we get a natural ring map

dim: K(X) → [X, ℤ]

splitting the inclusion of [X, ℤ] in K(X).

The kernel of dim will be written as K̃(X), or as K̃O(X) or K̃U(X) if we wish to emphasize the choice of ℝ or ℂ. Thus K̃(X) is an ideal in K(X), and there is a natural decomposition

K(X) ≅ K̃(X) ⊕ [X, ℤ].

Warning. If X is not connected, our group K̃(X) differs slightly from the notation in the literature. However, most applications will involve connected spaces, where the notation is the same. This will be clarified by Theorem 3.2 below.

Consider the set map VB_{n}(X) → K̃(X) sending E to [E] − n. This map is compatible with the stabilization map VB_{n}(X) → VB_{n+1}(X) sending E to E ⊕T, giving a map

\underrightarrow{lim} VB_{n}(X) → K̃(X).  (3.1.0)

We can interpret this in terms of maps between the infinite Grassmannian spaces G_{n} (= BO_{n}, BU_{n} or BSp_{n}) as follows. Recall from the Classification Theorem I.4.10 that the set VB_{n}(X) is isomorphic to the set [X, G_{n}] of homotopy classes of maps. Adding a trivial bundle T to the universal bundle E_{n} over G_{n} gives a vector bundle over G_{n}, so again by the Classification Theorem there is a map i_{n}: G_{n} → G_{n+1} such that E_{n} ⊕ T ≅ i_{n}^{∗}(E_{n+1}). By Cellular Approximation there is no harm in assuming i_{n} is cellular. Using I.4.10.1, the map Ωi_{n}: ΩG_{n} → ΩG_{n+1} is homotopic to the standard inclusion O_{n} ↪ O_{n+1} (resp. U_{n} ↪ U_{n+1} or Sp_{n} ↪ Sp_{n+1}), which

sends an n × n matrix g to the n + 1 × n + 1 matrix ( \begin{matrix}g & 0 \\ 0 & 1 \end{matrix} ). By construction,

the resulting map i_{n}: [X, G_{n}] → [X, G_{n+1}] corresponds to the stabilization map. The direct limit \underrightarrow{lim} [X, G_{n}] is then in 1-1 correspondence with the direct limit \underrightarrow{lim} VB_{n}(X) of (3.1.0).

Stabilization Theorem 3.1. Let X be either a compact space or a finite dimensional connected CW complex. Then the map (3.1.0) induces an isomorphism K̃(X) ≅ \underrightarrow{lim} VB(X) ≅ \underrightarrow{lim} [X, G_{n}]. In particular,

K̃O(X) ≅ \underrightarrow{lim} [X, BO_{n}], K̃U(X) ≅ \underrightarrow{lim} [X, BU_{n}] and { \atop KSp}(X) ≅ \underrightarrow{lim} [X, BSp_{n}].

Proof. We argue as in Lemma 2.3.1. Since the monoid of (componentwise) trivial vector bundles T^{f} is cofinal in VB(X) (I.4.1), we see from Corollary 1.3 that every element of K̃(X) is represented by an element [E] − n of some VB_{n}(X), and if [E] − n = [F] − n then E ⊕ T^{ℓ} ≅ F ⊕ T^{ℓ} in some VB_{n+ℓ}(X). Thus K̃(X) ≅ \underrightarrow{lim} VB_{n}(X), as claimed. □

Examples 3.1.1 (Spheres). From I(4.9) we see that KO(S^{1}) ≅ ℤ ⊕ ℤ/2 but KU(S^{1}) = ℤ, KO(S^{2}) = ℤ ⊕ ℤ/2 but KU(S^{2}) = ℤ ⊕ ℤ, KO(S^{3}) = KU(S^{3}) = ℤ and KO(S^{4}) ≅ KU(S^{4}) = ℤ ⊕ ℤ.

By Prop. I.4.8, the n-dimensional (ℝ, ℂ or ℍ) vector bundles on S^{d} are classified by the homotopy groups π_{d−1}(O_{n}), π_{d−1}(U_{n}) and π_{d−1}(Sp_{n}), respectively. By the Stabilization Theorem, K̃O(S^{d}) = lim_{n→∞} π_{d−1}(O_{n}) and K̃U(S^{d}) = lim_{n→∞} π_{d−1}(U_{n}).

Now Bott Periodicity says that the homotopy groups of O_{n}, U_{n} and Sp_{n} stabilize for n ≫ 0. Moreover, if n ≥ d/2 then π_{d−1}(U_{n}) is 0 for d odd and ℤ for d even. Thus KU(S^{d}) = ℤ ⊕ K̃U(S^{d}) is periodic of order 2 in d > 0: the ideal K̃U(S^{d}) is 0 for d odd and ℤ for d even, d ≠ 0.

Similarly, the π_{d−1}(O_{n}) and π_{d−1}(Sp_{n}) stabilize for n ≥ d and n ≥ d/4; both are periodic of order 8. Thus KO(S^{d}) = ℤ⊕K̃O(S^{d}) and KSp(S^{d}) = ℤ⊕{ \atop KSp}(S^{d}) are periodic of order 8 in d > 0, with the groups K̃O(S^{d}) = π_{d−1}(O) and { \atop KSp}(S^{d}) = π_{d−1}(Sp) being tabulated in the following table.

d (mod 8)  1  2  3  4  5  6  7  8^{K̃O(S^{d})  ℤ/2  ℤ/2  0  ℤ  0  0  0  ℤ}^{{ \atop KSp}(S^{d})  0  0  0  ℤ  ℤ/2  ℤ/2  0  ℤ}

Both of the ideals K̃O(S^{d}) and K̃U(S^{d}) are of square zero.

Remark 3.1.2. The complexification maps ℤ ≅ K̃O(S^{4k}) → K̃U(S^{4k}) ≅ ℤ are multiplication by 2 if k is odd, and by 1 if k is even. (The forgetful maps K̃U(S^{4k}) → K̃O(S^{4k}) have the opposite parity in k.) Similarly, the maps ℤ ≅ K̃U(S^{4k}) → { \atop KSp}(S^{4k}) ≅ ℤ are multiplication by 2 if k is odd, and by 1 if k is even. (The forgetful maps { \atop KSp}(S^{4k}) → K̃U(S^{4k}) have the opposite parity in k.) These calculations are taken from [MimTo, IV.5.12 and IV.6.1].

Let BO (resp. BU, BSp) denote the direct limit of the Grassmannians Grass_{n}. As noted after (3.1.0) and in I.4.10.1, the notation reflects the fact that Ω Grass_{n} is O_{n} (resp. U_{n}, Sp_{n}), and the maps in the direct limit correspond to the standard inclusions, so that we have ΩBO ≃ O = ⋃ O_{n}, ΩBU ≃ U = ⋃ U_{n} and ΩBSp ≃ Sp = ⋃ Sp_{n}.

Theorem 3.2. For every compact space X:

KO(X) ≅ [X, ℤ × BO] and K̃O(X) ≅ [X, BO];
KU(X) ≅ [X, ℤ × BU] and K̃U(X) ≅ [X, BU];
KSp(X) ≅ [X, ℤ × BSp] and { \atop KSp}(X) ≅ [X, BSp].

In particular, the homotopy groups π_{n}(BO) = K̃O(S^{n}), π_{n}(BU) = K̃U(S^{n}) and π_{n}(BSp) = { \atop KSp}(S^{n}) are periodic and given in Example 3.1.1.

Proof. If X is compact then we have [X, BO] = \underrightarrow{lim} [X, BO_{n}] and similarly for [X, BU] and [X, BSp]. The result now follows from Theorem 3.1 for connected X. For non-connected compact spaces, we only need to show that the maps [X, BO] → K̃O(X), [X, BU] → K̃U(X) and [X, BSp] → { \atop KSp}(X) of Theorem 3.1 are still isomorphisms.

Since X is compact, every continuous map X → ℤ is bounded. Hence the rank of every vector bundle E is bounded, say rank E ≤ n for some n ∈ ℕ. If f = n−rank E then F = E ⊕ T^{f} has constant rank n, and [E] − rank E = [F] − n. Hence every element of K̃(X) comes from some VB_{n}(X).

To see that these maps are injective, suppose that E, F ∈ VB_{n}(X) are such that [E] − n = [F] − n. By (1.3) we have E ⊕ T^{f} = F ⊕ T^{f} in VB_{n+f}(X) for some f ∈ [X, ℕ]. If f ≤ p, p ∈ ℕ, then adding T^{p−f} yields E ⊕ T^{p} = F ⊕ T^{p}. Hence E and F agree in VB_{n+p}(X). □

Definition 3.2.1 (K^{0}). For every paracompact X we write KO^{0}(X) for [X, ℤ× BO], KU^{0}(X) for [X, ℤ × BU] and KSp^{0}(X) for [X, ℤ × BSp]. By Theorem 3.2, we have KO^{0}(X) ≅ KO(X), KU^{0}(X) ≅ KU(X) and KSp^{0}(X) ≅ KSp(X) for every compact X. Similarly, we shall write K̃O^{0}(X), K̃U^{0}(X) and { \atop KSp}^{0}(X) for [X, BO], [X, BU] and [X, BSp]. When the choice of ℝ, ℂ or ℍ is clear, we will just write K^{0}(X) and K̃^{0}(X).

If Y is a subcomplex of X, we define relative groups K^{0}(X, Y) = K^{0}(X/Y)/ℤ and K̃^{0}(X, Y) = K̃^{0}(X/Y).

When X is paracompact but not compact, K̃^{0}(X) and K̃(X) are connected by stabilization and the map (3.1.0):

K̃O(X) ⟵ \underrightarrow{lim} VB_{n}(X) ≅ \underrightarrow{lim}[X, BO_{n}] ⟶ [X, BO] = K̃O^{0}(X)

and similarly for K̃U(X) and { \atop KSp}(X). We will see in Example 3.7.2 and Ex. 3.2 that the left map need not be an isomorphism. Here are two examples showing that the right map need not be an isomorphism either.

Example 3.2.2. (McGibbon) Let X be the infinite bouquet of odd-dimensional spheres S^{3} ∨ S^{5} ∨ S^{7} ∨ · · ·. By homotopy theory, there is a map f: X → BO_{3} whose restriction to S^{2p+1} is essential of order p for each odd prime p. If E denotes the 3-dimensional vector bundle f^{∗}E_{3} on X, then the class of f in \underrightarrow{lim}[X, BO_{n}] corresponds to [E] − 3 ∈ KO(X). In fact, since X is a suspension, we have \underrightarrow{lim}[X, BO_{n}] ≅ K̃O(X) by Ex. 3.8.

Each (n + 3)-dimensional vector bundle E ⊕ T^{n} is nontrivial, since its restriction to S^{2p+1} is nontrivial whenever 2p > n + 3 (again by homotopy theory). Hence [E] − 3 is a nontrivial element of K̃O(X). However, the corresponding element in K̃O^{0}(X) = [X, BO] is zero, because the homotopy groups of BO have no odd torsion.

Example 3.2.3. If X is the union of compact CW complexes X_{i}, Milnor showed in [M-Axiom] that there is an exact sequence for each n

0 → \underleftarrow{lim}^{1} K^{n−1}(X_{i}) → K^{n}(X) → \underleftarrow{lim} K^{n}(X_{i}) → 0.

In particular, KU^{0}(ℂℙ^{∞}) ≅ ℤ[[x]] by Ex. 3.7.

Proposition 3.3. If Y is a subcomplex of a CW complex X, the following sequences are exact:

K̃^{0}(X/Y) → K̃^{0}(X) → K̃^{0}(Y),
K^{0}(X, Y) → K^{0}(X) → K^{0}(Y).

Proof. Since Y ⊂ X is a cofibration, we have an exact sequence [X/Y, B] → [X, B] → [Y, B] for every connected space B; see III(6.3) in [Wh]. This yields the first sequence (B is BO, BU or BSp). The second follows from this and the classical exact sequence H̃^{0}(X/Y; ℤ) → H^{0}(X; ℤ) → H^{0}(Y; ℤ). □

Change of structure field 3.4. If X is any space, the monoid (or semiring) map VB_{ℝ}(X) → VB_{ℂ}(X) sending [E] to [E ⊗ ℂ] (see Ex. I.4.5) extends by universality to a ring homomorphism KO(X) → KU(X). For example, KO(S^{8n}) → KU(S^{8n}) is an isomorphism but K̃O(S^{8n+4}) ≅ ℤ embeds in K̃U(S^{8n+4}) ≅ ℤ as a subgroup of index 2.

Similarly, the forgetful map VB_{ℂ}(X) → VB_{ℝ}(X) extends to a group homomorphism KU(X) → KO(X). As dim_{ℝ}(V) = 2 · dim_{ℂ}(V), the summand [X, ℤ] of KU(X) embeds as 2[X, ℤ] in the summand [X, ℤ] of KO(X). Since E ⊗ℂ ≅ E ⊕E as real vector bundles (by Ex. I.4.5), the composition KO(X) → KU(X) → KO(X) is multiplication by 2. The composition in the other direction is more complicated; see Exercise 3.1. For example, it is the zero map on K̃U(S^{8n+4}) ≅ ℤ but is multiplication by 2 on K̃U(S^{8n}) ≅ ℤ.

There are analogous maps KU(X) → KSp(X) and KSp(X) → KU(X), whose properties we leave to the exercises.

Higher Topological K-theory

Once we have a representable functor such as K^{0}, standard techniques in infinite loop space theory all us to expand it into a generalized cohomology theory. Rather than get distracted by infinite loop spaces now, we choose to adopt a rather pedestrian approach, ignoring the groups K^{n} for n > 0. For this we use the suspensions S^{n}X of X, which are all connected paracompact spaces.

Definition 3.5. For each integer n>0, define K̃O^{−n}(X) and KO^{−n}(X) by:

K̃O^{−n}(X) = K̃O^{0}(S^{n}X) = [S^{n}X, BO]; KO^{−n}(X) = K̃O^{−n}(X) ⊕ K̃O(S^{n}).

Replacing ‘O’ by ‘U’ yields definitions K̃U^{−n}(X) = K̃U^{0}(S^{n}X) = [S^{n}X, BU] and KU^{−n}(X) = K̃U^{−n}(X) ⊕ K̃U(S^{n}); replacing ‘O’ by ‘Sp’ yields definitions for { \atop KSp}^{−n}(X) and KSp^{−n}(X). When the choice of ℝ, ℂ or ℍ is clear, we shall drop the ‘O,’ ‘U’ and ‘Sp,’, writing simply K̃^{−n}(X) and K^{−n}(X).

We shall also define relative groups as follows. If Y is a subcomplex of X, and n > 0, we set K^{−n}(X, Y) = K̃^{−n}(X/Y).

Based maps 3.5.1. Note that our definitions do not assume X to have a basepoint. If X has a nondegenerate basepoint and Y is an H-space with homotopy inverse (such as BO, BU or BSp), then the group [X, Y] is isomorphic to the group π_{0}(Y) × [X, Y]_{∗}, where the second term denotes homotopy classes of based maps from X to Y; see pp. 100 and 119 of [Wh]. For such spaces X we can interpret the formulas for KO^{−n}(X), KU^{−n}(X) and KSp^{−n}(X) in terms of based maps, as is done in [Atiyah, p.68].

If X_{∗} denotes the disjoint union of X and a basepoint ∗, then we have the usual formula for an unreduced cohomology theory: K^{−n}(X) = K̃(S^{n}(X_{∗})). This easily leads (see Ex. 3.11) to the formulas for n ≥ 1:

KO^{−n}(X) ≅ [X, Ω^{n}BO], KU^{−n}(X) ≅ [X, Ω^{n}BU] and KSp^{−n}(X) ≅ [X, Ω^{n}BSp].

Theorem 3.6. If Y is a subcomplex of a CW complex X, we have the exact sequences (infinite to the left):

⋯→K̃^{−2}(Y)→K̃^{−1}(X/Y)→K̃^{−1}(X)→K̃^{−1}(Y)→K̃^{0}(X/Y)→K̃^{0}(X)→K̃^{0}(Y),

⋯→K^{−2}(Y)→K^{−1}(X, Y)→K^{−1}(X)→K^{−1}(Y)→K^{0}(X, Y)→K^{0}(X)→K^{0}(Y).

Proof. Exactness at K^{0}(X) was proven in Proposition 3.3. The mapping cone cone(i) of i: Y ⊂ X is homotopy equivalent to X/Y, and j: X ⊂ cone(i) induces cone(i)/X ≃ SY. This gives exactness at K^{0}(X, Y). Similarly, cone(j) ≃ SY and cone(j)/cone(i) ≃ SX, giving exactness at K^{−1}(Y). The long exact sequences follows by replacing Y ⊂ X by SY ⊂ SX. □

Characteristic Classes 3.7. The total Stiefel-Whitney class w(E) of a real vector bundle E was defined in chapter I, §4. By (SW3) it satisfies the product formula: w(E ⊕ F) = w(E)w(F). Therefore if we interpret w(E) as an element of the abelian group U of all formal sums 1+a_{1}+⋯ in Ĥ^{∗}(X; ℤ/2) we get a get a group homomorphism w: KO(X) → U. It follows that each Stiefel-Whitney class induces a well-defined set map w_{i}: KO(X) → H^{i}(X; ℤ/2). In fact, since w vanishes on each componentwise trivial bundle T^{f} it follows that w([E] − [T^{f}]) = w(E). Hence each Stiefel-Whitney class w_{i} factors through the projection KO(X) → K̃O(X).

Similarly, the total Chern class c(E) = 1 + c_{1}(E) + ⋯ satisfies c(E ⊕ F) = c(E)c(F), so we may think of it as a group homomorphism from KU(X) to the abelian group U of all formal sums 1 + a_{2} + a_{4} + ⋯ in Ĥ^{∗}(X; ℤ). It follows that the Chern classes c_{i}(E) ∈ H^{2i}(X; ℤ) of a complex vector bundle define set maps c_{i}: KU(X) ⟶ H^{2i}(X; ℤ). Again, since c vanishes on componentwise trivial bundles, each Chern class c_{i} factors through the projection KU(X) → K̃U(X).

Example 3.7.1. For even spheres the Chern class c_{n}: K̃U(S^{2n}) → H^{2n}(S^{n}; ℤ) is an isomorphism. We will return to this point in Ex. 3.6 and in §4.

Example 3.7.2. The map \underrightarrow{lim}[ℝ ℙ^{∞}, BO_{n}] → K̃O(ℝ ℙ^{∞}) of (3.1.0) cannot be onto. To see this, consider the element η = 1 − [E_{1}] of K̃O(ℝ ℙ^{∞}), where E_{1} is the canonical line bundle. Since w(−η) = w(E_{1}) = 1 + x we have w(η) = (1 + x)^{−1} = ∑_{i=0}^{∞} x^{i}, and w_{i}(η) ≠ 0 for every i ≥ 0. Axiom (SW1) implies that η cannot equal [F] − dim(F) for any bundle F.

Similarly, \underrightarrow{lim}[ℂ ℙ^{∞}, BU_{n}] → K̃U(ℂ ℙ^{∞}) cannot be onto; the argument is similar, again using the canonical line bundle: c_{i}(1 − [E_{1}]) ≠ 0 for every i ≥ 0.

EXERCISES

3.1 Let X be a topological space. Show that there is an involution of VB_{ℂ}(X) sending [E] to the complex conjugate bundle [Ē] of Ex. I.4.6. The corresponding involution c on KU(X) can be nontrivial; use I(4.9.2) to show that c is multiplication by −1 on K̃U(S^{2}) ≅ ℤ. (By Bott periodicity, this implies that c is multiplication by (−1)^{k} on K̃U(S^{2}) ≅ ℤ.) Finally, show that the composite KU(X) → KO(X) → KU(X) is the map 1 + c sending [E] to [E] + [Ē].

3.2 If ⨿X_{i} is the disjoint union of spaces X_{i}, show that K(⨿X_{i}) ≅ ∏ K(X_{i}). Then construct a space X such that the map \underrightarrow{lim} VB_{n}(X) → K̃(X) of (3.1.0) is not onto.

3.3 External products. Show that there is a bilinear map K(X_{1}) ⊗ K(X_{2}) → K(X_{1} × X_{2}) for every X_{1} and X_{2}, sending [E_{1}] ⊗ [E_{2}] to [π_{1}^{∗}(E_{1}) ⊗ π_{2}^{∗}(E_{2})], where pi_{i}: X_{1} × X_{2} → X_{i} is the projection. Then show that if X_{1} = X_{2} = X the composition with the diagonal map Δ^{∗}: K(X×X) → K(X) yields the usual product in the ring K(X), sending [E_{1}] ⊗ [E_{2}] to [E_{1} ⊗ E_{2}].

3.4 Recall that the smash product X ∧ Y of two based spaces is the quotient X × Y/X ∨ Y, where X ∨ Y is the union of X × {∗} and {∗} × Y. Show that

K̃^{−n}(X × Y) ≅ K̃^{−n}(X ∧ Y) ⊕ K̃^{−n}(X) ⊕ K̃^{−n}(Y).

3.5 Show that KU^{−2}(∗) ⊗ KU^{−n}(X) → KU^{−n−2}(X) induces a “periodicity” isomorphism β: KU^{−n}(X) ⟶^{∼} KU^{−n−2}(X) for all n. Hint: Let β be a generator of KU^{−2}(∗) ≅ ℤ, and use S^{2} ∧ S^{n}X ≃ S^{n+2}X. 3.6 Let X be a finite CW complex with only even-dimensional cells, such as ℂℙ^{n}. Show that KU(X) is a free abelian group on the set of cells of X, and that KU(SX) = ℤ, so that KU^{−1}(X) = 0. Then use Example 3.7.1 to show that the total Chern class injects the group K̃U(X) into ∏ H^{2i}(X; ℤ). Hint: Use induction on dim(X) and the fact that X_{2n}/X_{2n−2} is a bouquet of 2n-spheres. 3.7 Chern character for ℂℙ^{n}. Let E_{1} be the canonical line bundle on ℂℙ^{n}, and let x denote the class [E_{1}] − 1 in KU(ℂℙ^{n}). Use Chern classes and the previous exercise to show that {1, [E_{1}], [E_{1} ⊗ E_{1}], …, [E_{1}^{⊗n}]}, and hence {1, x, x^{2}, …, x^{n}}, forms a basis of the free abelian group KU(ℂℙ^{n}). Then show that x^{n+1} = 0, so that the ring KU(ℂℙ^{n}) is isomorphic to ℤ[x]/(x^{n+1}). We will see in Ex. 4.11 below that the Chern character ch maps the ring KU(ℂℙ^{n}) isomorphically onto the subring ℤ[t]/(t^{n+1}) of H^{∗}(ℂℙ^{n}; ℚ) generated by t = e^{c_{1}(x)} − 1. 3.8 Consider the suspension X = SY of a paracompact space Y. Use Ex. I.4.16 to show that \underrightarrow{lim}[X, BO_{n}] ≅ K̃O(X). 3.9 If X is a finite CW complex, show by induction on the number of cells that both KO(X) and KU(X) are finitely generated abelian groups. 3.10 Show that KU(ℝℙ^{2n}) = KU(ℝℙ^{2n+1}) = ℤ ⊕ ℤ/2^{n}. Hint: Try the total Stiefel-Whitney class, using 3.3. 3.11 Let X be a compact space with a nondegenerate basepoint. Show that KO^{−n}(X) ≅ [X, Ω^{n}BO] ≅ [X, Ω^{n−1}O] and KU^{−n}(X) ≅ [X, Ω^{n}BU] ≅ [X, Ω^{n−1}U] for all n ≥ 1. In particular, KU^{−1}(X) ≅ [X, U] and KO^{−1}(X) ≅ [X, O]. 3.12 Let X be a compact space with a nondegenerate basepoint. Show that the homotopy groups of the topological groups GL(ℝ^{X}) = Hom(X, GL(ℝ)) and GL(ℂ^{X}) = Hom(X, GL(ℂ)) are (for n > 0):

π_{n−1}GL(ℝ^{X}) = KO^{−n}(X) and π_{n−1}GL(ℂ^{X}) = KU^{−n}(X).

3.13 If E → X is a complex bundle, there is a quaternionic vector bundle E_{ℍ} → X with fibers E_{x} ⊗_{ℂ} ℍ, as in Ex. I.4.5; this induces the map KU(X) → KSp(X) mentioned in 3.4. Show that E_{ℍ} → X, considered as a complex vector bundle, is isomorphic to the Whitney sum E ⊕ E. Deduce that the composition KU(X) → KSp(X) → KU(X) is multiplication by 2. 3.14 Show that ℍ ⊗_{ℂ} ℍ is isomorphic to ℍ ⊕ ℍ as an ℍ-bimodule, on generators 1 ⊗ 1 ± j ⊗ j. This induces a natural isomorphism V ⊗_{ℂ} ℍ ≅ V ⊕ V of vector spaces over ℍ. If E → X is a quaternionic vector bundle, with underlying complex bundle uE → X, show that there is a natural isomorphism (uE)_{ℍ} ≅ E ⊕E. Conclude that the composition KSp(X) → KU(X) → KSp(X) is multiplication by 2. 3.15 Let Ē be the complex conjugate bundle of a complex vector bundle E → X; see Ex. I.4.6. Show that Ē_{ℍ} ≅ E_{ℍ} as quaternionic vector bundles. This shows that KU(X) → KSp(X) commutes with the involution c of Ex. 3.1.

Using exercises 3.1 and 3.14, show that the composition KSp(X) → KO(X) → KSp(X) is multiplication by 4.

§4. Lambda and Adams Operations

A commutative ring K is called a λ-ring if we are given a family of set operations λ^{k}: K → K for k ≥ 0 such that for all x, y ∈ K:

• λ^{0}(x) = 1 and λ^{1}(x) = x for all x ∈ K;

• λ^{k}(x + y) = ∑\begin{matrix}k \\ i=0 \end{matrix} λ^{i}(x)λ^{k−i}(y) = λ^{k}(x) + λ^{k−1}(x)λ^{1}y + ⋯ + λ^{k}(y).

This last condition is equivalent to the assertion that there is a group homomorphism λ_{t} from the additive group of K to the multiplicative group W (K) = 1 + tK[[t]] given by the formula λ_{t}(x) = ∑ λ^{k}(x)t^{k}. A λ-ideal of K is an ideal I with λ^{n}(I) ⊆ I for all n ≥ 1.

Warning: Our notation of λ-ring follows Atiyah; Grothendieck and other authors call this a pre-λ-ring, reserving the term λ-ring for what we call a special λ-ring; see Definition 4.3.1 below.

Example 4.1.1 (Binomial Rings). . The integers ℤ and the rationals ℚ are λ-rings with λ^{k}(n) = \binom{n}{k}. If K is any ℚ-algebra, we define \binom{x}{k} = \frac{x(x−1)···(x−k+1)}{k!} for x ∈ K and k ≥ 1; again the formula λ^{k}(x) = (_{k}^{x}) makes K into a λ-ring.

More generally, a binomial ring is a subring K of a ℚ-algebra K_{ℚ} such that for all x ∈ K and k ≥ 1, \binom{x}{k} ∈ K. We make a binomial ring into a λ-ring by setting λ^{k}(x) = \binom{x}{k}. If K is a binomial ring then formally λ_{t} is given by the formula λ_{t}(x) = (1 + t)^{x}. For example, if X is a topological space, then the ring [X, ℤ] is a λ-ring with λ^{k}(f) = \binom{f}{k}, the function sending x to \binom{f(x)}{k}.

The notion of λ-semiring is very useful in constructing λ-rings. Let M be a semiring (see §1); we know that the group completion M^{−1}M of M is a ring. We call M a λ–semiring if it is equipped with operations λ^{k}: M → M such that λ^{0}(x) = 1, λ^{1}(x) = x and λ^{k}(x + y) = ∑ λ^{i}(x)λ^{k−i}(y).

If M is a λ-semiring then the group completion K = M^{−1}M is a λ-ring. To see this, note that sending x ∈ M to the power series ∑ λ^{k}(x)t^{k} defines a monoid map λ_{t}: M → 1 + tK[[t]]. By universality of K, this extends to a group homomorphism λ_{t} from K to 1 + tK[[t]], and the coefficients of λ_{t}(x) define the operations λ^{k}(x).

Example 4.1.2 (Algebraic K_{0}). Let R be a commutative ring and set K = K_{0}(R). If P is a finitely generated projective R-module, consider the formula λ^{k}(P) = [∧^{k}P]. The decomposition ∧^{k}(P ⊕ Q) ≅ ∑(∧^{i}P) ⊗ (∧^{k−i}Q) given in ch.I, §3 shows that P(R) is a λ-semiring. Hence K_{0}(R) is a λ-ring.

Since rank(∧^{k}P) = \binom{rank P}{k}, it follows that the map rank: K_{0}(R) → H_{0}(R) of 2.3 is a morphism of λ-rings, and hence that K̃_{0}(R) is a λ-ideal of K_{0}(R).

Example 4.1.3 (Topological K^{0}). Let X be a topological space and let K(X) be either KO(X) or KU(X). If E → X is a vector bundle, let λ^{k}(E) be the exterior power bundle ∧^{k}E of Ex. I.4.3. The decomposition of ∧^{k}(E ⊕ F) given in Ex. I.4.3 shows that the monoid VB(X) is a λ-semiring. Hence KO(X) and KU(X) are λ-rings, and KO(X) → KU(X) is a morphism of λ-rings.

Since dim(∧^{k}E) = \binom{dim E}{k}, it follows that KO(X) → [X, ℤ] and KU(X) → [X, ℤ] are λ-ring morphisms, and that K̃O(X) and K̃U(X) are λ-ideals.

Example 4.1.4 (Representation Ring). Let G be a finite group, and consider the complex representation ring R(G), constructed in Example 1.6 as the group completion of Rep_{ℂ}(G), the semiring of finite dimensional representations of G; as an abelian group R(G) ≅ ℤ^{c}, where c is the number of conjugacy classes of elements in G. The exterior powers Λ^{i}(V) of a representation V are also Gmodules, and the decomposition of Λ^{k}(V ⊕ W) as complex vector spaces used in (4.1.2) shows that Rep_{ℂ}(G) is a λ-semiring. Hence R(G) is a λ-ring. (It is true, but harder to show, that R(G) is a special λ-ring; see Ex.4.2.)

If d = dim_{ℂ}(V) then dim_{ℂ}(Λ^{k}V) = \binom{d}{k}, so dim_{ℂ} is a λ-ring map from R(G) to ℤ. The kernel R̃(G) of this map is a λ-ideal of R(G).

Example 4.1.5. Let X be a scheme, or more generally a locally ringed space (Ch. I, §5). We will define a ring K_{0}(X) in §7 below, using the category VB(X). As an abelian group it is generated by the classes of vector bundles on X. We will see in Proposition 8.8 that the operations λ^{k}[E] = [∧^{k}E] are well-defined on K_{0}(X) and make it into a λ-ring. (The formula for λ^{k}(x + y) will follow from Ex. I.5.4.)

Positive structures

Not every λ-ring is well-behaved. In order to avoid pathologies, we introduce a further condition, satisfied by the above examples: the λ-ring K must have a positive structure and satisfy the Splitting Principle.

Definition 4.2.1. By a positive structure on a λ-ring K we mean: 1) a λsubring H^{0} of K which is a binomial ring; 2) a λ-ring surjection ε: K → H^{0} which is the identity on H^{0} (ε is called the augmentation); and 3) a subset P ⊂ K (the positive elements), such that

(1) ℕ = {0, 1, 2, ⋯ } is contained in P.

(2) P is a λ-sub-semiring of K. That is, P is closed under addition, multiplication, and the operations λ^{k}.

(3) Every element of the kernel K̃ of ε can be written as p−q for some p, q ∈ P.

(4) If p ∈ P then ε(p) = n ∈ ℕ. Moreover, λ^{i}(p) = 0 for i > n and λ^{n}(p) is a unit of K.

Condition (2) states that the group completion P^{−1}P of P is a λ-subring of K; by (3) we have P^{−1}P = ℤ ⊕ K̃. By (4), ε(p) > 0 for p ≠ 0, so P ∩ (−P) = 0; therefore P^{−1}P is a partially ordered abelian group in the sense of §1. An element ℓ ∈ P with ε(ℓ) = 1 is called a line element; by (4), λ^{1}(ℓ) = ℓ and ℓ is a unit of K. That is, the line elements form a subgroup L of the units of K.

The λ-rings in examples (4.1.2)–(4.1.5) all have positive structures. The λ-ring K_{0}(R) has a positive structure with

H^{0} = H_{0}(R) = [Spec(R), ℤ] and P = {[P] : rank(P) is constant};

the line elements are the classes of line bundles, so L = Pic(R). Similarly, the λ-rings KO(X) and KU(X) have a positive structure in which H^{0} is H^{0}(X, ℤ) = [X, ℤ] and P is {[E] : dim(E) is constant}, as long as we restrict to compact spaces or spaces with π_{0}(X) finite, so that (I.4.1.1) applies. Again, line elements are the classes of line bundles; for KO(X) and KU(X) we have L = H^{1}(X; ℤ/2) and L = H^{2}(X; ℤ), respectively. For R(G), the classes [V] of representations V are the positive elements; H^{0} is ℤ, and L is the set of 1-dimensional representations of G.

Finally, if X is a scheme (or locally ringed space) then in the positive structure on K_{0}(X) we have H^{0} = H^{0}(X; ℤ) and P is {[E] : rank(E) is constant}; see I.5.1. The line bundles are again the line elements, so L = Pic(X) = H^{1}(X, O_{X}^{×}) by I.5.10.1.

There is a natural group homomorphism “det” from K to L, which vanishes on H^{0}. If p ∈ P we define det(p) = λ^{n}(p), where ε(p) = n. The formula for λ^{n}(p + q) and the vanishing of λ^{i}(p) for i > ε(p) imply that det: P → L is a monoid map, i.e., that det(p + q) = det(p) det(q). Thus det extends to a map from P^{−1}P to L. As det(n) = \binom{n}{n} = 1 for every n ≥ 0, det(ℤ) = 1. By (iii), defining det(H^{0}) = 1 extends det to a map from K to L. When K is K_{0}(R) the map det was introduced in §2. For KO(X), det is the first Stiefel-Whitney class; for KU(X), det is the first Chern class.

Having described what we mean by a positive structure on K, we can now state the Splitting Principle.

Definition 4.2.2. The Splitting Principle states that for every positive element p in K there is a extension K ⊂ K′ (of λ-rings with positive structure) such that p is a sum of line elements in K′.

The Splitting Principle for KO(X) and KU(X) holds by Ex. 4.12. Using algebraic geometry, we will show in 8.8 that the Splitting Principle holds for K_{0}(R) as well as K_{0} of a scheme. The Splitting Principle also holds for R(G); see [AT, 1.5]. The importance of the Splitting Principle lies in its relation to “special λ-rings,” a notion we shall define after citing the following motivational result from [FL, ch.I].

Theorem 4.2.3. If K is a λ-ring with a positive structure, and ℕ is cofinal in P, the Splitting Principle holds if and only if K is a special λ-ring.

In order to define special λ-ring, we need the following technical example:

Example 4.3 (Witt Vectors). For every commutative ring R, the abelian group W (R) = 1 + tR[[t]] has the structure of a commutative ring, natural in R; W (R) is called the ring of (big) Witt vectors of R. The multiplicative identity of the ring W (R) is (1 − t), and multiplication ∗ is completely determined by naturality, formal factorization of elements of W (R) as f(t) = ∏_{i=1}^{∞}(1 − r_{i}t^{i}) and the formula:

(1 − rt) ∗ f(t) = f(rt).

It is not hard to see that there are “universal” polynomials P_{n} in 2n variables so:

(∑ a_{i}t^{i}) ∗ (∑ b_{j}t^{j}) = ∑ c_{n}t^{n}, with c_{n} = P_{n}(a_{1}, …, a_{n}; b_{1}, …, b_{n}).

If ℚ ⊆ R there is an isomorphism ∏_{n=1}^{∞} R → W (R), (r_{1}, …) ↦ ∏ exp(1−r_{n}t^{n}/n).

Grothendieck observed that there are operations λ^{k} on W (R) making it into a λ-ring; they are defined by naturality, formal factorization and the formula

λ^{k}(1 − rt) = 0 for all k ≥ 2.

Another way to put it is that there are universal polynomials P_{n,k} such that:

λ^{k}(∑ a_{i}t^{i}) = ∑ b_{n}t^{n}, with b_{n} = P_{n,k}(a_{1}, ..., a_{nk}).

Definition 4.3.1. A special λ-ring is a λ-ring K such that the group homomorphism λ_{t} : K → W (K) is a λ-ring homomorphism. Since λ_{t}(x) = ∑ λ^{k}(x)t^{k}, a special λ-ring is a λ-ring K such that

• λ^{k}(1) = 0 for k ≠ 0, 1

• λ^{k}(xy) is P_{k}(λ^{1}(x), ..., λ^{k}(x); λ^{1}(y), ..., λ^{k}(y)), and

• λ^{n}(λ^{k}(x)) = P_{n,k}(λ^{1}(x), ..., λ^{nk}(x)).

Example 4.3.2. The formula λ^{n}(s_{1}) = s_{n} defines a special λ-ring structure on the polynomial ring U = ℤ[s_{1}, ..., s_{n}, ...]; see [AT, §2]. It is the free special λ-ring on the generator s_{1}, because if x is any element in any special λ-ring K then the map U → K sending s_{n} to λ^{n}(x) is a λ-ring homomorphism. The λ-ring U cannot have a positive structure by Theorem 4.6 below, since U has no nilpotent elements except 0.

Adams operations

For every augmented λ-ring K we can define the Adams operations ψ^{k}: K → K for k ≥ 0 by setting ψ^{0}(x) = ε(x), ψ^{1}(x) = x, ψ^{2}(x) = x^{2} − 2λ^{2}(x) and inductively

ψ^{k}(x) = λ^{1}(x)ψ^{k−1}(x)−λ^{2}(x)ψ^{k−2}(x)+⋯+(−1)^{k}λ^{k−1}(x)ψ^{1}(x)+(−1)^{k−1}kλ^{k}(x).

From this inductive definition we immediately deduce three facts:

• if ℓ is a line element then ψ^{k}(ℓ) = ℓ^{k};

• if I is a λ-ideal with I^{2} = 0 then ψ^{k}(x) = (−1)^{k−1}kλ^{k}(x) for all x ∈ I;

• For every binomial ring H we have ψ^{k} = 1. Indeed, the formal identity

x ∑_{i=0}^{k−1}(−1)^{i}\binom{x}{i} = (−1)^{k+1}k\binom{x}{k} shows that ψ^{k}(x) = x for all x ∈ H. The operations ψ^{k} are named after J.F. Adams, who first introduced them in 1962 in his study of vector fields on spheres.

Here is a slicker, more formal presentation of the Adams operations. Define ψ^{k}(x) to be the coefficient of t^{k} in the power series:

ψ_{t}(x) = ∑ ψ^{k}(x)t^{k} = ε(x) − t\frac{d}{dt} log λ_{−t}(x).

The proof that this agrees with the inductive definition of ψ^{k}(x) is an exercise in formal algebra, which we relegate to Exercise 4.6 below.

Proposition 4.4. Assume K satisfies the Splitting Principle. Each ψ^{k} is a ring endomorphism of K, and ψ^{j}ψ^{k} = ψ^{jk} for all j, k ≥ 0.

Proof. The logarithm in the definition of ψ_{t} implies that ψ_{t}(x + y) = ψ_{t}(x) + ψ_{t}(y), so each ψ^{k} is additive. The Splitting Principle and the formula ψ^{k}(ℓ) = ℓ^{k} for line elements yield the formulas ψ^{k}(pq) = ψ^{k}(p)ψ^{k}(q) and ψ^{j}(ψ^{k}(p)) = ψ^{jk}(p) for positive p. The extension of these formulas to K is clear. □

Example 4.4.1. Consider the λ-ring KU(S^{2n}) = ℤ ⊕ ℤ of 3.1.1. On H^{0} = ℤ, ψ^{k} = 1, but on K̃U(S^{n}) ≅ ℤ, ψ^{k} is multiplication by k^{n/2}. (See [Atiyah, 3.2.2].)

Example 4.4.2. Consider KU(ℝ ℙ^{2n}), which by Ex. 3.10 is ℤ ⊕ ℤ/2^{n}. I claim that for all x ∈ K̃U(X):

ψ^{k}(x) = \{ \begin{matrix}x if k is odd \\ 0 if k is even. \end{matrix}

To see this, note that K̃U(ℝ ℙ^{2n}) ≅ ℤ/2^{n} is additively generated by (ℓ − 1), where ℓ is the nonzero element of L = H^{2}(ℝ ℙ^{2n}; ℤ) = ℤ/2. Since ℓ^{2} = 1, we see that ψ^{k}(ℓ − 1) = (ℓ^{k} − 1) is 0 if k is even and (ℓ − 1) if k is odd. The assertion follows.

γ-operations

Associated to the operations λ^{k} are the operations γ^{k}: K → K. To construct them, we assume that λ^{k}(1) = 0 for k ≠ 0, 1. Note that if we set s = t/(1 − t) then K[[t]] = K[[s]] and t = s/(1 + s). Therefore we can rewrite λ_{s}(x) = ∑ λ^{i}(x)s^{i} as a power series γ_{t}(x) = ∑ γ^{k}(x)t^{k} in t. By definition, γ^{k}(x) is the coefficient of t^{k} in γ_{t}(x). Since γ_{t}(x) = λ_{s}(x) we have γ_{t}(x + y) = γ_{t}(x)γ_{t}(y). In particular γ^{0}(x) = 1, γ^{1}(x) = x and γ^{k}(x + y) = ∑ γ^{i}(x)γ^{k−i}(y). That is, the γ-operations satisfy the axioms for a λ-ring structure on K. An elementary calculation, left to the reader, yields the useful identity:

Formula 4.5. γ^{k}(x) = λ^{k}(x + k − 1). This implies that γ^{2}(x) = λ^{2}(x) + x and

γ^{k}(x) = λ^{k}(x + k − 1) = λ^{k}(x) + \binom{k − 1}{1}λ^{k−1}(x) + ⋯ + \binom{k − 1}{k − 2}λ^{2}(x) + x.

Example 4.5.1. If H is a binomial ring then for all x ∈ H we have

γ^{k}(x) = \binom{x + k − 1}{k} = (−1)^{k}\binom{−x}{k}.

Example 4.5.2. γ^{k}(1) = 1 for all k, because λ_{s}(1) = 1 + s = 1/(1 − t). More generally, if ℓ is a line element then γ^{k}(ℓ) = ℓ for all k ≥ 1.

Lemma 4.5.3. If p ∈ P is a positive element with ε(p) = n, then γ^{k}(p − n) = 0 for all k > n. In particular, if ℓ ∈ K is a line element then γ^{k}(ℓ − 1) = 0 for every k > 1.

Proof. If k > n then q = p+(k −n−1) is a positive element with ε(q) = k −1. Thus γ^{k}(p − n) = λ^{k}(q) = 0. □

If x ∈ K, the γ-dimension dim_{γ}(x) of x is defined to be the largest integer n for which γ^{n}(x − ε(x)) ≠ 0, provided n exists. For example, dim_{γ}(h) = 0 for every h ∈ H^{0} and dim_{γ}(ℓ) = 1 for every line element ℓ (except ℓ = 1 of course). By the above remarks if p ∈ P and n = ε(p) then dim_{γ}(p) = dim_{γ}(p − n) ≤ n. The supremum of the dim_{γ}(x) for x ∈ K is called the γ-dimension of K.

Examples 4.5.4. If R is a commutative noetherian ring, the Serre Cancellation I.2.4 states that every element of K̃_{0}(R) is represented by [P] − n, where P is a finitely generated projective module of rank < dim(R). Hence K_{0}(R) has γdimension at most dim(R).

Suppose that X is a CW complex with finite dimension d. The Real Cancellation Theorem I.4.3 allows us to use the same argument to deduce that KO(X) has γdimension at most d; the Complex Cancellation Theorem I.4.4 shows that KU(X) has γ-dimension at most d/2.

Corollary 4.5.5. If K has a positive structure in which ℕ is cofinal in P, then every element of K̃ has finite γ-dimension.

Proof. Recall that “ℕ is cofinal in P ” means that for every p there is a p′ so that p+p′ = n for some n ∈ ℕ. Therefore every x ∈ K̃ can be written as x = p−m for some p ∈ P with m = ε(p). By Lemma 4.5.3, dim_{γ}(x) ≤ m. □

Theorem 4.6. If every element of K has finite γ-dimension (e.g., K has a positive structure in which ℕ is cofinal in P), then K̃ is a nil ideal. That is, every element of K̃ is nilpotent.

Proof. Fix x ∈ K̃, and set m = dim_{γ}(x), n = dim_{γ}(−x). Then both γ_{t}(x) = 1+xt+γ^{2}(x)t^{2}+⋯+γ^{m}(x)t^{m} and γ_{t}(−x) = 1−xt+⋯+γ^{n}(−x)t^{n} are polynomials in t. Since γ_{t}(x)γ_{t}(−x) = γ_{t}(0) = 1, the polynomials γ_{t}(x) and γ_{t}(−x) are units in the polynomial ring K[t]. By (I.3.12), the coefficients of these polynomials are nilpotent elements of K. □

Corollary 4.6.1. The ideal K̃_{0}(R) is the nilradical of K_{0}(R) for every commutative ring R.

If X is compact then K̃O(X) and K̃U(X) are the nilradicals of the rings KO(X) and KU(X), respectively.

Example 4.6.2. The conclusion of Theorem 4.6 fails for the representation ring R(G) of a cyclic group of order 2. If σ denotes the 1-dimensional sign representation, then L = {1, σ} and R̃(G) ≅ ℤ is generated by (σ − 1). Since (σ − 1)^{2} = (σ^{2} − 2σ + 1) = (−2)(σ − 1), we see that (σ − 1) is not nilpotent, and in fact that R̃(G)^{n} = (2^{n−1})R̃(G) for every n ≥ 1. The hypothesis of Corollary 4.5.5 fails here because σ cannot be a summand of a trivial representation. In fact dim_{γ}(1−σ) = ∞, because γ^{n}(1 − σ) = (1 − σ)^{n} = 2^{n−1}(1 − σ) for all n ≥ 1.

The γ-Filtration

The γ-filtration on K is a descending sequence of ideals:

K = F_{γ}^{0}K ⊃ F_{γ}^{1}K ⊃ ⋯ ⊃ F_{γ}^{n}K ⊃ · · · .

It starts with F_{γ}^{0}K = K and F_{γ}^{1}K = K̃ (the kernel of ε). The first quotient F_{γ}^{0}/F_{γ}^{1} is clearly H^{0} = K/K̃. For n ≥ 2, F_{γ}^{n}K is defined to be the ideal of K generated by the products γ^{k_{1}}(x_{1}) ⋯ γ^{k_{m}}(x_{m}) with x_{i} ∈ K̃ and ∑ k_{i} ≥ n. In particular, F_{γ}^{n}K contains γ^{k}(x) for all x ∈ K̃ and k ≥ n.

It follows immediately from the definition that F_{γ}^{i}F_{γ}^{j} ⊆ F_{γ}^{i+j}. For j = 1, this implies that the quotients F_{γ}^{i}K/F_{γ}^{i+1}K are H^{0}-modules. We will prove that the quotient F_{γ}^{1}/F_{γ}^{2} is the group L of line elements in K:

Theorem 4.7. If K satisfies the Splitting Principle, then the map ℓ ↦ ℓ − 1 induces a group isomorphism, split by the map det:

L ⟶^{≅} F_{γ}^{1}K/F_{γ}^{2}K.

Corollary 4.7.1. For every commutative ring R, the first two ideals in the γfiltration of K_{0}(R) are F_{γ}^{1} = K̃_{0}(R) and F_{γ}^{2} = SK_{0}(R). (See 2.6.2.) In particular,

F_{γ}^{0}/F_{γ}^{1} ≅ H_{0}(R) and F_{γ}^{1}/F_{γ}^{2} ≅ Pic(R).

Corollary 4.7.2. The first two quotients in the γ-filtration of KO(X) are

F_{γ}^{0}/F_{γ}^{1} ≅ [X, ℤ] and F_{γ}^{1}/F_{γ}^{2} ≅ H^{1}(X; ℤ/2).

The first few quotients in the γ-filtration of KU(X) are

F_{γ}^{0}/F_{γ}^{1} ≅ [X, ℤ] and F_{γ}^{1}/F_{γ}^{2} ≅ H^{2}(X; ℤ).

For the proof of Theorem 4.7, we shall need the following consequence of the Splitting Principle. A proof of this principle may be found in [FL, III.1].

Filtered Splitting Principle. Let K be a λ-ring satisfying the Splitting Principle, and let x be an element of F_{γ}^{n}K. Then there exists a λ-ring extension K ⊂ K′ such that F_{γ}^{n}K = K ∩F_{γ}^{n}K′, and x is an H-linear combination of products (ℓ_{1} − 1) ⋯ (ℓ_{m} − 1), where the ℓ_{i} are line elements of K′ and m ≥ n.

Proof of Theorem 4.7. Since (ℓ_{1} − 1)(ℓ_{2} − 1) ∈ F_{γ}^{2}K, the map ℓ ↦ ℓ − 1 is a homomorphism. If ℓ_{1}, ℓ_{2}, ℓ_{3} are line elements of K,

det((ℓ_{1} − 1)(ℓ_{2} − 1)ℓ_{3}) = det(ℓ_{1}ℓ_{2}ℓ_{3}) det(ℓ_{3})/det(ℓ_{1}ℓ_{3}) det(ℓ_{2}ℓ_{3}) = 1.

By Ex. 4.3, the Filtered Splitting Principle implies that every element of F_{γ}^{2}K can be written as a sum of terms (ℓ_{1} − 1)(ℓ_{2} − 1)ℓ_{3} in some extension K′ of K. This shows that det(F_{γ}^{2}) = 1, so det induces a map K̃/F_{γ}^{2}K → L. Now det is the inverse of the map ℓ ↦ ℓ − 1 because for p ∈ P the Splitting Principle shows that p − ε(p) ≡ det(p) − 1 modulo F_{γ}^{2}K. □

Proposition 4.8. If the γ-filtration on K is finite then K̃ is a nilpotent ideal. If K̃ is a nilpotent ideal which is finitely generated as an abelian group, then the γ-filtration on K is finite. That is, F_{γ}^{N}K = 0 for some N.

Proof. The first assertion follows from the fact that K̃^{n} ⊂ F_{γ}^{n}K for all n. If K̃ is additively generated by {x_{1}, ..., x_{s}}, then there is an upper bound on the k for which γ^{k}(x_{i}) ≠ 0; using the sum formula there is an upper bound n on the k for which γ^{k} is nonzero on K̃. If K̃^{m} = 0 then clearly we must have F_{γ}^{mn}K = 0. □

Example 4.8.1. If X is a finite CW complex, both KO(X) and KU(X) are finitely generated abelian groups by Ex. 3.9. Therefore they have finite γ-filtrations.

Example 4.8.2. If R is a commutative noetherian ring of Krull dimension d, then F_{γ}^{d+1}K_{0}(R) = 0 by [FL, V.3.10], even though K_{0}(R) may not be a finitely generated abelian group.

Example 4.8.3. For the representation ring R(G), G cyclic of order 2, we saw in Example 4.6.2 that R̃ is not nilpotent. In fact F_{γ}^{n}R(G) = R̃^{n} = 2^{n−1}R̃ ≠ 0. An even worse example is the λ-ring R_{ℚ} = R(G) ⊗ ℚ, because F_{γ}^{n}R_{ℚ} = R̃_{ℚ} ≅ ℚ for all n ≥ 1.

Remark 4.8.4. Fix x ∈ K̃. It follows from the nilpotence of the γ^{k}(x) that there is an integer N such that x^{N} = 0, and for every k_{1}, …, k_{n} with ∑ k_{i} ≥ N we have

γ^{k_{1}}(x)γ^{k_{2}}(x) ⋯ γ^{k_{n}}(x) = 0.

The best general bound for such an N is N = mn = dim_{γ}(x) dim_{γ}(−x).

Proposition 4.9. Let k, n ≥ 1 be integers. If x ∈ F_{γ}^{n}K then modulo F_{γ}^{n+1}K:

ψ^{k}(x) ≡ k^{n}x;  and  λ^{k}(x) ≡ (−1)^{k}k^{n−1}x.

Proof. If ℓ is a line element then modulo (ℓ − 1)^{2} we have

ψ^{k}(ℓ − 1) = (ℓ^{k−1} + ... + ℓ + 1)(ℓ − 1) ≡ k(ℓ − 1).

Therefore if ℓ_{1}, ⋯, ℓ_{m} are line elements and m ≥ n we have

ψ^{k}((ℓ_{1} − 1) ⋯ (ℓ_{n} − 1)) ≡ k^{n}(ℓ_{1} − 1) ⋯ (ℓ_{n} − 1) modulo F_{γ}^{n+1}K.

The Filtered Splitting Principle implies that ψ^{k}(x) ≡ k^{n}x modulo F_{γ}^{n+1}K for every x ∈ F_{γ}^{n}K. For λ^{k}, we use the inductive definition of ψ^{k} to see that k^{n}x = (−1)^{k−1}kλ^{k}(x) for every x ∈ F_{γ}^{n}K. The Filtered Splitting Principle allows us to consider the universal case W = W_{s} of Exercise 4.4. Since there is no torsion in F_{γ}^{n}W/F_{γ}^{n+1}W, we can divide by k to obtain the formula k^{n−1}x = (−1)^{k−1}λ^{k}(x). □

Theorem 4.10 (Structure of K ⊗ ℚ). Suppose that K has a positive structure in which every element has finite γ-dimension e.g., if ℕ is cofinal in P). Then:

(1) The eigenvalues of ψ^{k} on K_{ℚ} = K ⊗ ℚ are a subset of {1, k, k^{2}, k^{3}, ...} for each k;

(2) The subspace K_{ℚ}^{(n)} = K_{ℚ}^{(n,k)} of eigenvectors for ψ^{k} = k^{n} is independent of k;

(3) K_{ℚ}^{(n)} is isomorphic to F_{γ}^{n}K_{ℚ}/F_{γ}^{n+1}K_{ℚ} ≅ (F_{γ}^{n}K/F_{γ}^{n+1}K) ⊗ ℚ;

(4) K_{ℚ}^{(0)} ≅ H^{0} ⊗ ℚ and K_{ℚ}^{(1)} ≅ L ⊗ ℚ;

(5) The ring K⊗ℚ is isomorphic to the graded ring K_{ℚ}^{(0)}⊕K_{ℚ}^{(1)}⊕⋯⊕K_{ℚ}^{(n)}⊕· · ·.

Proof. For every positive p, consider the universal λ-ring U_{ℚ} = ℚ[s_{1}, ...] of Example 4.3.2, and the map U_{ℚ} → K_{ℚ} sending s_{1} to p and s_{k} to λ^{k}(p). If ε(p) = n then s_{i} maps to zero for i > n and each s_{i} − \binom{n}{i} maps to a nilpotent element by Theorem 4.6. The image A of this map is a λ-ring which is finite-dimensional over ℚ, so A is an artinian ring. Clearly F_{γ}^{N}A = 0 for some large N. Consider the linear operation ∏_{n=0}^{N}(ψ^{k} − k^{n}) on A; by Proposition 4.9 it is trivial on each F_{γ}^{n}/F_{γ}^{n+1}, so it must be zero. Therefore the characteristic polynomial of ψ^{k} on A divides Π(t − k^{n}), and has distinct integer eigenvalues. This proves (1) and that K_{ℚ} is the direct sum of the eigenspaces K_{ℚ}^{(n,k)} for ψ^{k}. As ψ^{k} preserves products, Proposition 4.9 now implies (3) and (4). The rest is immediate from Theorem 4.7. □

Chern class homomorphisms

The formalism in §3 for the Chern classes c_{i}: KU(X) → H^{2i}(X; ℤ) extends to the current setting. Suppose we are given a λ-ring K with a positive structure and a commutative graded ring A = A^{0} ⊕ A^{1} ⊕ · · ·. Chern classes on K with values in A are set maps c_{n}: K → A^{n} for n ≥ 0 with c_{0}(x) = 1, satisfying the following axioms: (CC0) The c_{n} send H^{0} to zero (for n ≥ 1): c_{n}(h) = 0 for every h ∈ H^{0}. (CC1) Dimension. c_{n}(p) = 0 whenever p is positive and n ≥ ε(p). (CC2) Sum Formula. For every x, y in K and every n:

c_{n}(x + y) = ∑_{_{i}=0}^{n} c_{i}(x)c_{n−i}(y).

(CC3) Normalization. c_{1}: L → A^{1} is a group homomorphism. That is, for ℓ, ℓ′: c_{1}(ℓℓ′) = c_{1}(ℓ) + c_{1}(ℓ′).

The total Chern class of x is the element c(x) = ∑ c_{i}(x) of the completion Â = ∏ A^{i} of A. In terms of the total Chern class, (CC2) becomes the product formula

c(x + y) = c(x)c(y).

Example 4.11.1. The Stiefel-Whitney classes w_{i}: KO(X) → A^{i} = H^{i}(X; ℤ/2) and the Chern classes c_{i}: KU(X) → A^{i} = H^{2i}(X; ℤ) are Chern classes in this sense.

Example 4.11.2. Associated to the γ-filtration on K we have the associated graded ring Gr_{γ}^{•}K with Gr_{γ}^{i}K = F_{γ}^{i}/F_{γ}^{i+1}. For a positive element p in K, define c_{i}(p) to be γ^{i}(p − ε(p)) modulo F_{γ}^{i+1}. The multiplicative formula for γ_{t} implies that c_{i}(p + q) = c_{i}(p) + c_{i}(q), so that the c_{i} extend to classes c_{i}: K → Gr_{γ}^{•}K. The total Chern class c: K → Gr_{γ}^{•}K is a group homomorphism with torsion kernel and cokernel, because by Theorem 4.10 and Ex. 4.10 the induced map c_{n}: K_{ℚ}^{(n)} → Gr_{γ}^{n}K_{ℚ} ≅ K_{ℚ}^{(n)} is multiplication by (−1)^{n}(n − 1)!.

The Splitting Principle implies the following Splitting Principle (see [FL, I.3.1]).

Chern Splitting Principle. Given a finite set {p_{i}} of positive elements of K, there is a λ-ring extension K ⊂ K′ in which each p_{i} splits as a sum of line elements, and a graded extension A ⊂ A′ such that the c_{i} extend to maps c_{i}: K′ → (A′)^{i} satisfying (CC1) and (CC2).

The existence of “Chern roots” is an important consequence of this Splitting Principle. Suppose that p ∈ K is positive, and that in an extension K′ of K we can write p = ℓ_{1} + ⋯ + ℓ_{n}, n = ε(p). The Chern roots of p are the elements a_{i} = c_{1}(ℓ_{i}) in (A′)^{1}; they determine the c_{k}(p) in A^{k}. Indeed, because c(p) is the product of the c(ℓ_{i}) = 1 + a_{i}, we see that c_{k}(p) is the k^{th} elementary symmetric polynomial σ_{k}(a_{1}, ..., a_{n}) of the a_{i} in the larger ring A′. In particular, the first Chern class is c_{1}(p) = ∑ a_{i} and the “top” Chern class is c_{n}(p) = ∏ a_{i}.

A famous theorem of Isaac Newton states that every symmetric polynomial in n variables t_{1}, ..., t_{n} is in fact a polynomial in the symmetric polynomials σ_{k} = σ_{k}(t_{1}, ..., t_{n}), k = 1, 2, · · ·. Therefore every symmetric polynomial in the Chern roots of p is also a polynomial in the Chern classes c_{k}(p), and as such belongs to the subring A of A′. Here is an elementary application of these ideas.

Proposition 4.11.3. Suppose that K satisfies the Splitting Principle. Then c_{n}(ψ^{k}x) = k^{n}c_{n}(x) for all x ∈ K. That is, the following diagram commutes:

K ⟶^{c_{n}} A^{n}
↓_{ψ^{k}}  ↓_{k^{n}}
K ⟶^{c_{n}} A^{n}.

Corollary 4.11.4. If ℚ ⊂ A then c_{n} vanishes on K_{ℚ}^{(i)}, i ≠ n.

Chern character

As an application of the notion of Chern roots, suppose given Chern classes c_{i}: K → A^{i}, where for simplicity A is an algebra over ℚ. If p ∈ K is a positive element, with Chern roots a_{i}, define ch(p) to be the formal expansion

ch(p) = ∑_{i=0}^{n} exp(a_{i}) = ∑_{k=0}^{∞} \frac{1}{k!} (∑_{i=0}^{n} a_{i}^{k})

of terms in A′. The k^{th} term \frac{1}{k!} ∑ a_{i}^{k} is symmetric in the Chern roots, so it is a polynomial in the Chern classes c_{1}(p), ..., c_{k}(p) and hence belongs to A^{k}. Therefore ch(p) is a formal expansion of terms in A, i.e., an element of Â = ∏ A^{k}. For example, if ℓ is a line element of K then ch(ℓ) is just exp(c_{1}(ℓ)). From the definition, it is immediate that ch(p + q) = ch(p) + ch(q), so ch extends to a map from P^{−1}P to Â. Since ch(1) = 1, this is compatible with the given map H^{0} → A^{0}, and so it defines a map ch: K → Â, called the Chern character on K. The first few terms in the expansion of the Chern character are

ch(x) = ε(x) + c_{1}(x) + \frac{1}{2}[c_{1}(x)^{2} − c_{2}(x)] + \frac{1}{6}[c_{1}(x)^{3} − 3c_{1}(x)c_{2}(x) + 3c_{3}(x)] + ⋯

An inductive formula for the term in ch(x) is given in Exercise 4.14.

Proposition 4.12. If ℚ ⊂ A then the Chern character is a ring homomorphism

ch: K → Â.

Proof. By the Splitting Principle, it suffices to verify that ch(pq) = ch(p)ch(q) when p and q are sums of line elements. Suppose that p = ∑ ℓ_{i} and q = ∑ m_{j} have Chern roots a_{i} = c_{1}(ℓ_{i}) and b_{j} = c_{1}(m_{j}), respectively. Since pq = ∑ ℓ_{i}m_{j}, the Chern roots of pq are the c_{1}(ℓ_{i}m_{j}) = c_{1}(ℓ_{i}) + c_{1}(m_{j}) = a_{i} + b_{j}. Hence

ch(pq) = ∑ ch(ℓ_{i}m_{j}) = ∑ exp(a_{i} + b_{j}) = ∑ exp(a_{i}) exp(b_{j}) = ch(p)ch(q). □

Corollary 4.12.1. Suppose that K has a positive structure in which every x ∈ K has finite γ-dimension (e.g., ℕ is cofinal in P). Then the Chern character lands in A, and the induced map from K_{ℚ} = ⊕K_{ℚ}^{(n)} to A is a graded ring map. That is, the n^{th} term ch_{n}: K_{ℚ} → A^{n} vanishes on K_{ℚ}^{(i)} for i ≠ n.

Example 4.12.2. The universal Chern character ch: K_{ℚ} → K_{ℚ} is the identity map. Indeed, by Ex. 4.10(b) and Ex. 4.14 we see that ch_{n} is the identity map on each K_{ℚ}^{(n)}.

The following result was proven by M. Karoubi in [Kar63]. (See Exercise 4.11 for the proof when X is a finite CW complex.)

Theorem 4.13. If X is a compact topological space and Ȟ denotes Čech cohomology, then the Chern character is an isomorphism of graded rings.

ch: KU(X) ⊗ ℚ ≅ ⨁ Ȟ^{2i}(X; ℚ)

Example 4.13.1 (Spheres). For each even sphere, we know by Example 3.7.1 that c_{n} maps K̃U(S^{2n}) isomorphically onto H^{2n}(S^{2n}; ℤ) = ℤ. The inductive formula for ch_{n} shows that in this case ch(x) = dim(x) + (−1)^{n}c_{n}(x)/(n − 1)! for all x ∈ KU(X). In this case it is easy to see directly that ch: KU(S^{2n}) ⊗ ℚ ≅ H^{2∗}(S^{2n}; ℚ)

EXERCISES 4.1 Show that in K_{0}(R) or K^{0}(X) we have

λ^{k}([P] − n) = ∑(−1)^{i}\binom{n + i − 1}{i}[∧^{k−i}P].

4.2 For every group G and every commutative ring A, let R_{A}(G) denote the group K_{0}(AG, A) of Ex. 2.14, i.e., the group completion of the monoid Rep(AG, A) of all AG-modules which are finitely generated projective as A-modules. Show that the ∧^{k} make R_{A}(G) into a λ-ring with a positive structure given by Rep(AG, A).

If A = ℂ, show that R_{ℂ}(G) satisfies the Splitting Principle and hence is a special λ-ring (by 4.2.3); the line elements are the characters. Swan proved in [Swan70] that R_{A}(G) satisfies the Splitting Principle for every A. (Another proof is in [SGA6], VI(3.3).) This proves that R_{A}(G) is a special λ-ring for every A.

When p = 0 in A, show that ψ^{p} = Φ^{∗} in R_{A}(G), where Φ : A → A is the Frobenius Φ(a) = a^{p}. To do this, reduce to the case in which χ is a character and show that ψ^{k}χ(g) = χ(g^{p}) = χ(g)^{p}. 4.3 Suppose that a λ-ring K is generated as an H-algebra by line elements. Show that F_{γ}^{n} = K̃^{n} for all n, so the γ-filtration is the adic filtration defined by the ideal K̃. Then show that if K is any λ-ring satisfying the Splitting Principle every element x of F_{γ}^{n}K can be written in an extension K′ of K as a product

x = (ℓ_{1} − 1) ⋯ (ℓ_{m} − 1)

of line elements with m ≥ n. In particular, show that every x ∈ F_{γ}^{2} can be written as a sum of terms (ℓ_{i} − 1)(ℓ_{j} − 1)ℓ in K′. 4.4 Universal special λ-ring. Let W_{s} denote the Laurent polynomial ring ℤ[u_{1}, u_{1}^{−1}, ..., u_{s}, u_{s}^{−1}], and ε: W_{s} → ℤ the augmentation defined by ε(u_{i}) = 1.

(a) Show that W_{s} is a λ-ring with a positive structure, the line elements being the monomials u^{α} = ∏ u_{i}^{n_{i}}. This implies that W_{s} is generated by the group L ≅ ℤ^{s} of line elements, so by Exercise 4.3 the ideal F_{γ}^{n}W_{s} is W̃^{n}.

(b) Show that each F_{γ}^{n}W/F_{γ}^{n+1}W is a torsionfree abelian group.

(c) If K is a special λ-ring show that any family {ℓ_{1}, ..., ℓ_{s}} of line elements determines a λ-ring map W_{s} → K sending u_{i} to ℓ_{i}.

(d) (Splitting Principle for the free λ-ring) Let U → W_{s} be the λ-ring homomorphism sending s_{1} to ∑ u_{i} (see 4.3.2). Show that U injects into \underleftarrow{lim} W_{s}.

4.5 A line element ℓ is called ample for K if for every x ∈ K̃ there is an integer N = N(x) such that for every n ≥ N there is a positive element p_{n} so that ℓ^{n}x = p_{n} − ε(p_{n}). (The terminology comes from Algebraic Geometry; see 8.8.4 below.) If K has an ample line element, show that every element of K̃ is nilpotent. 4.6 Verify that the inductive definition of ψ^{k} and the ψ_{t} definition of ψ^{k} agree. 4.7 If p is prime, use the Splitting Principle to verify that ψ^{p}(x) ≡ x^{p} modulo p for every x ∈ K. 4.8 Adams e-invariant. Suppose given a map f: S^{2m−1} → S^{2n}. The mapping cone

C(f) fits into a cofibration sequence S^{2n} ↣^{i} C(f) ↠^{j} S^{2m}. Associated to this is the exact sequence:

0 → K̃U(S^{2m})⟶^{j^{∗}}K̃U(C)⟶^{i^{∗}}K̃U(S^{2n}) → 0.

Choose x, y ∈ K̃U(C) so that i^{∗}(x) generates K̃U(S^{2n}) ≅ ℤ and y is the image of a generator of K̃U(S^{2m}) ≅ ℤ. Since j^{∗} is a ring map, y^{2} = 0.

(a) Show by applying ψ^{k} that xy = 0, and that if m ≠ 2n then x^{2} = 0. (When m = 2n, x^{2} defines the Hopf invariant of f; see the next exercise.)

(b) Show that ψ^{k}(x) = k^{n}x + a_{k}y for appropriate integers a_{k}. Then show (for fixed x and y) that the rational number

e(f) = \frac{a_{k}}{k^{m} − k^{n}}

is independent of the choice of k.

(c) Show that a different choice of x only changes e(f) by an integer, so that e(f) is a well-defined element of ℚ/ℤ; e(f) is called the Adams e-invariant of f.

(d) If f and f′ are homotopic maps, it follows from the homotopy equivalence between C(f) and C(f′) that e(f) = e(f′). By considering the mapping cone of f ∨ g, show that the well-defined set map e: π_{2m−1}(S^{2n}) → ℚ/ℤ is a group homomorphism. J.F. Adams used this e-invariant to detect an important cyclic subgroup of π_{2m−1}(S^{2n}), namely the “image of J.”

4.9 Hopf Invariant One. Given a continuous map f: S^{4n−1} → S^{2n}, define an integer H(f) as follows. Let C(f) be the mapping cone of f. As in the previous exercise, we have an exact sequence:

0 → K̃U(S^{4n})⟶^{j^{∗}}K̃U(C(f))⟶^{i^{∗}}K̃U(S^{2n}) → 0.

Choose x, y ∈ K̃U(C(f)) so that i^{∗}(x) generates K̃U(S^{2n}) ≅ ℤ and y is the image of a generator of K̃U(S^{4n}) ≅ ℤ. Since i^{∗}(x^{2}) = 0, we can write x^{2} = Hy for some integer H; this integer H = H(f) is called the Hopf invariant of f.

(a) Show that H(f) is well-defined, up to ± sign.

(b) If H(f) is odd, show that n is 1, 2, or 4. Hint: Use Ex. 4.7 to show that the integer a_{2} of the previous exercise is odd. Considering e(f), show that 2^{n} divides p^{n} − 1 for every odd p.

It turns out that the classical “Hopf maps” S^{3} → S^{2}, S^{7} → S^{4} and S^{15} → S^{8} all have Hopf invariant H(f) = 1. In contrast, for every even integer H there is a map S^{4n−1} → S^{2n} with Hopf invariant H. 4.10 Operations. A natural operation τ on λ-rings is a map τ: K → K defined for every λ-ring K such that fτ = τf for every λ-ring map f: K → K′. The operations λ_{k}, γ_{k}, and ψ_{k} are all natural operations on λ-rings.

(a) If K satisfies the Splitting Principle, generalize Proposition 4.9 to show that every natural operation τ preserves the γ-filtration of K and that there are integers ω_{n} = ω_{n}(τ), independent of K, such that for every x ∈ F_{γ}^{n}K

τ(x) ≡ ω_{n}x modulo F_{γ}^{n+1}K.

(b) Show that for τ = γ^{k} and x ∈ F_{γ}^{n} we have:

ω_{n}(γ^{k}) = \{ \begin{matrix}0 if n < k \\ (−1)^{k−1}(k − 1)! if n = k \\ ω_{n} ≠ 0 if n > k \end{matrix}

(c) Show that s_{k} ↦ λ^{k} and τ ↦ τ(s_{1}) give λ-ring isomorphisms from the freeλring U = ℤ[s_{1}, s_{2}, ⋯] of 4.3.2 to the ring of all natural operations on λ-rings. (See [Atiyah, 3.1.7].)

4.11 By Example 4.13.1, the Chern character ch: KU(S^{n}) ⊗ ℚ → H^{2∗}(S^{n}; ℚ) is an isomorphism for every sphere S^{n}. Use this to show that ch: KU(X) ⊗ ℚ → H^{2∗}(X; ℚ) is an isomorphism for every finite CW complex X. 4.12 Let K be a λ-ring. Given a K-module M, construct the ring K ⊕ M in which M^{2} = 0. Given a sequence of K-linear endomorphisms φ_{k} of M with φ_{1}(x) = x, show that the formulae λ^{k}(x) = φ_{k}(x) extend the λ-ring structure on K to a λ-ring structure on K ⊕ M. Then show that K ⊕ M has a positive structure if K does, and that K ⊕ M satisfies the Splitting Principle whenever K does. (The elements in 1 + M are to be the new line elements.) 4.13 Hirzebruch characters. Suppose that A is an H^{0}-algebra and we fix a power series α(t) = 1+α_{1}t+α_{2}t^{2} +⋯ in A^{0}[[t]]. Suppose given Chern classes c_{i}: K → A^{i}. If p ∈ K is a positive element, with Chern roots a_{i}, define ch_{α}(p) to be the formal expansion

ch_{α}(p) = ∑_{i=0}^{n} α(a_{i}) ∑_{k=0}^{∞} α_{k} (∑_{i=0}^{n} a_{i}^{k})

of terms in A′. Show that ch_{α}(p) belongs to the formal completion Â of A, and that it defines a group homomorphism ch_{α}: K → Â. This map is called the Hirzebruch character for α. 4.14 Establish the following inductive formula for the n^{th} term ch_{n} in the Chern character:

ch_{n} − \frac{1}{n}c_{1}ch_{n−1} + ⋯ ± \frac{1}{i!\binom{n}{i}}c_{i}ch_{n−i} + ⋯ + \frac{(−1)^{n}}{(n − 1)!}c_{n} = 0.

To do this, set x = −t_{i} in the identity ∏(x + a_{i}) = x^{n} + c_{1}x^{n−1} + ⋯ + c_{n}.

§5. K_{0} of a Symmetric Monoidal Category

The idea of group completion in §1 can be applied to more categories than just the categories P(R) in §2 and VB(X) in §3. It applies to any category with a “direct sum”, or more generally any natural product □ making the isomorphism classes of objects into an abelian monoid. This leads us to the notion of a symmetric monoidal category.

Definition 5.1. A symmetric monoidal category is a category S, equipped with a functor □: S × S → S, a distinguished object e and four basic natural isomorphisms:

e□s ≅ s, s□e ≅ s, s□(t□u) ≅ (s□t)□u, and s□t ≅ t□s.

These basic isomorphisms must be “coherent” in the sense that two natural isomorphisms of products of s_{1}, …, s_{n} built up from the four basic ones are the same whenever they have the same source and target. (We refer the reader to [Mac] for the technical details needed to make this definition of “coherent” precise.) Coherence permits us to write expressions without parentheses like s_{1}□ ⋯ □s_{n} unambiguously (up to natural isomorphism).

Example 5.1.1. Any category with a direct sum ⊕ is symmetric monoidal; this includes additive categories like P(R) and VB(X) as we have mentioned. More generally, a category with finite coproducts is symmetric monoidal with s□t = s⨿t. Dually, any category with finite products is symmetric monoidal with s□t = s × t.

Definition 5.1.2 (K_{0}S). Suppose that the isomorphism classes of objects of S form a set, which we call S^{iso}. If S is symmetric monoidal, this set S^{iso} is an abelian monoid with product □ and identity e. The group completion of this abelian monoid is called the Grothendieck group of S, and is written as K_{0}^{□}(S), or simply as K_{0}(S) if □ is understood.

From §1 we see that K_{0}^{□}(S) may be presented with one generator [s] for each isomorphism class of objects, with relations that [s□t] = [s] + [t] for each s and t. From Proposition 1.1 we see that every element of K_{0}^{□}(S) may be written as a difference [s] − [t] for some objects s and t.

Examples 5.2. (1) The category P(R) of finitely generated projective modules over a ring R is symmetric monoidal under direct sum. Since the above definition is identical to that in §2, we see that we have K_{0}(R) = K_{0}^{⊕}(P(R)).

(2) Similarly, the category VB(X) of (real or complex) vector bundles over a topological space X is symmetric monoidal, with □ being the Whitney sum ⊕. From the definition we see that we also have K(X) = K_{0}^{⊕}(VB(X)), or more explicitly:

KO(X) = K_{0}^{⊕}(VB_{ℝ}(X)), KU(X) = K_{0}^{⊕}(VB_{ℂ}(X)).

(3) If R is a commutative ring, let Pic(R) denote the category of algebraic line bundles L over R and their isomorphisms (§I.3). This is a symmetric monoidal category with □ = ⊗_{R}, and the isomorphism classes of objects already form a group, so K_{0}Pic(R) = Pic(R).

Finite Sets 5.2.1. Let Sets_{fin} denote the category of finite sets. It has a coproduct, the disjoint sum ⨿, and it is not hard to see that K_{0}^{⨿}(Sets_{fin}) = ℤ.

Another monoidal operation on Sets_{fin} is the product (×). However, since the empty set satisfies ∅ = ∅ × X for all X we have K_{0}^{×}(Sets_{fin}) = 0.

The category Sets_{fin}^{×} of nonempty finite sets has for its isomorphism classes the set ℕ_{>0} = {1, 2, … } of positive integers, and the product of finite sets corresponds to multiplication. Since the group completion of (ℕ_{>0}, ×) is the multiplicative monoid ℚ_{>0}^{×} of positive rational numbers, we have K_{0}^{×}(Sets_{fin}^{×}) ≅ ℚ_{>0}^{×}.

Burnside Ring 5.2.2. Suppose that G is a finite group, and let G-Sets_{fin} denote the category of finite G-sets. It is a symmetric monoidal category under disjoint union. We saw in Example 1.5 that K_{0}(G-Sets_{fin}) is the Burnside Ring A(G) ≅ ℤ^{c}, where c is the number of conjugacy classes of subgroups of G.

Representation ring 5.2.3. Similarly, the finite-dimensional complex representations of a finite group G form a category Rep_{ℂ}(G). It is symmetric monoidal under ⊕. We saw in Example 1.6 that K_{0}Rep_{ℂ}(G) is the representation ring R(G) of G, which is a free abelian group on the classes [V_{1}], ..., [V_{r}] of the irreducible representations of G.

Cofinality

Let T be a full subcategory of a symmetric monoidal category S. If T contains e and is closed under finite products, then T is also symmetric monoidal. We say that T is cofinal in S if for every s in S there is an s′ in S such that s□s′ is isomorphic to an element in T, i.e., if the abelian monoid T^{iso} is cofinal in S^{iso} in the sense of §1. When this happens, we may restate Corollary 1.3 as follows.

Cofinality Theorem 5.3. Let T be cofinal in a symmetric monoidal category S. Then (assuming S^{iso} is a set):

(1) K_{0}(T) is a subgroup of K_{0}(S);

(2) Every element of K_{0}(S) is of the form [s] − [t] for some s in S and t in T;

(3) If [s] = [s′] in K_{0}(S) then s□t ≅ s′□t for some t in T.

Example 5.4.1 (Free modules). Let R be a ring. The category Free(R) of finitely generated free R-modules is cofinal (for □ = ⊕) in the category P(R) of finitely generated projective modules. Hence K_{0}Free(R) is a subgroup of K_{0}(R). In fact K_{0}Free(R) is is a cyclic abelian group, and equals ℤ whenever R satisfies the Invariant Basis Property. Moreover, the subgroup K_{0}Free(R) of K_{0}(R) = K_{0}P(R) is the image of the map ℤ → K_{0}(R) described in Lemma 2.1.

Free(R) is also cofinal in the smaller category P^{st.free}(R) of finitely generated stably free modules. Since every stably free module P satisfies P ⊕ R^{m} ≅ R^{n} for some m and n, the Cofinality Theorem yields K_{0}Free(R) = K_{0}P^{st.free}(R).

Example 5.4.2. Let R be a commutative ring. A finitely generated projective R-module is called faithfully projective if its rank is never zero. The tensor product of faithfully projective modules is again faithfully projective by Ex. 2.7. Hence the category FP(R) of faithfully projective R-modules is a symmetric monoidal category under the tensor product ⊗_{R}. For example, if R is a field then the monoid FP^{iso} is the multiplicative monoid (ℕ_{>0}, ×) of Example 5.2.1, so in this case we have K_{0}^{⊗}FP(R) ≅ ℚ_{>0}^{×}. We will describe the group K_{0}^{⊗}FP(R) in the exercises below.

Example 5.4.3 (Brauer groups). Suppose first that F is a field, and let Az(F) denote the category of central simple F -algebras. This is a symmetric monoidal category with product ⊗_{F}, because if A and B are central simple then so is A ⊗_{F} B. The matrix rings M_{n}(F) form a cofinal subcategory, with M_{m}(F) ⊗_{F} M_{n}(F) ≅ M_{mn}(F). From the previous example we see that the Grothendieck group of this subcategory is ℚ_{>0}^{×}. The classical Brauer group Br(F) of the field F is the quotient of K_{0}Az(F) by this subgroup. That is, Br(F) is generated by classes [A] of central simple algebras with two families of relations: [A ⊗_{F} B] = [A] + [B] and [M_{n}(F)] = 0.

More generally, suppose that R is a commutative ring. Recall (from [Milne, IV]) that an R-algebra A is is called an Azumaya algebra if there is another R-algebra B such that A ⊗_{R} B ≅ M_{n}(R) for some n. The category Az(R) of Azumaya Ralgebras is thus symmetric monoidal with product ⊗_{R}. If P is a faithfully projective R-module, End_{R}(P) is an Azumaya algebra. Since End_{R}(P ⊗_{R} P′) ≅ End_{R}(P) ⊗_{R} End_{R}(P′), there is a monoidal functor End_{R} from FP(R) to Az(R), and a group homomorphism K_{0}FP(R) → K_{0}Az(R). The cokernel Br(R) of this map is called the Brauer group of R. That is, Br(R) is generated by classes [A] of Azumaya algebras with two families of relations: [A ⊗_{R} B] = [A] + [B] and [End_{R}(P)] = 0.

G-bundles and equivariant K-theory

The following discussion is taken from Atiyah’s very readable book [Atiyah]. Suppose that G is a finite group and that X is a topological space on which G acts continuously. A (complex) vector bundle E over X is called a G-vector bundle if G acts continuously on E, the map E → X commutes with the action of G, and for each g ∈ G and x ∈ X the map E_{x} → E_{gx} is a vector space homomorphism. The category VB_{G}(X) of G-vector bundles over X is symmetric monoidal under the usual Whitney sum, and we write K_{G}^{0}(X) for the Grothendieck group K_{0}^{⊕}VB_{G}(X). For example, if X is a point then VB_{G}(X) = Rep_{ℂ}(G), so we have K_{G}^{0}(point) = R(G). More generally, if x is a fixed point of X, then E ↦ E_{x} defines a monoidal functor from VB_{G}(X) to Rep_{ℂ}(G), and hence a group map K_{G}^{0}(X) → R(G).

If G acts trivially on X, every vector bundle E on X can be considered as a G-bundle with trivial action, and the tensor product E ⊗ V with a representation V of G is another G-bundle. The following result is proven on p. 38 of [Atiyah].

Proposition 5.5 (Krull-Schmidt Theorem). Let V_{1}, ..., V_{r} be a complete set of irreducible G-modules, and suppose that G acts trivially on X. Then for every G-bundle F over X there are unique vector bundles E_{i} = Hom_{G}(V_{i}, F) so that

F ≅ (E_{1} ⊗ V_{1}) ⊕ ⋯ ⊕ (E_{r} ⊗ V_{r}).

Corollary 5.5.1. If G acts trivially on X then K_{G}^{0}(X) ≅ KU(X) ⊗_{ℤ} R(G).

The Witt ring W (F) of a field

5.6. Symmetric bilinear forms over a field F provide another classical application of the K_{0} construction. The following discussion is largely taken from Milnor’s pretty book [M-SBF], and the reader is encouraged to look there for the connections with other branches of mathematics.

A symmetric inner product space (V, B) is a finite dimensional vector space V, equipped with a nondegenerate symmetric bilinear form B: V ⊗ V → F. The category SBil(F) of symmetric inner product spaces and form-preserving maps is symmetric monoidal, where the operation □ is the orthogonal sum (V, B)⊕(V′, B′), defined as the vector space V ⊕V′, equipped with the bilinear form β(v⊕v′, w⊕w′) = B(v, w) + B′(v′, w′).

A crucial role is played by the hyperbolic plane H, which is V = F^{2} equipped with the bilinear form B represented by the symmetric matrix ({0 \atop 1} {1 \atop 0}). An inner product space is called hyperbolic if it is isometric to an orthogonal sum of hyperbolic planes.

Let (V, B)⊗(V′, B′) denote the tensor product V ⊗V′, equipped with the bilinear form β(v ⊗ v′, w ⊗ w′) = B(v, w)B′(v′, w′); this is also a symmetric inner product space, and the isometry classes of inner product spaces forms a semiring under ⊕ and ⊗ (see Ex. 5.10). Thus the Grothendieck group GW (F) = K_{0}SBil(F) is a commutative ring with unit 1 = ⟨1⟩; it is called the Grothendieck-Witt ring of F. The forgetful functor SBil(F) → P(F) sending (V, B) to V induces a ring augmentation GW (F) ⟶^{ε} K_{0}(F) ≅ ℤ. We write Î for the augmentation ideal of GW (F).

Example 5.6.1. For each a ∈ F^{×}, we write ⟨a⟩ for the inner product space with V = F and B(v, w) = avw. Clearly ⟨a⟩ ⊗ ⟨b⟩ ≅ ⟨ab⟩. Note that a change of basis 1 ↦ b of F induces an isometry ⟨a⟩ ≅ ⟨ab^{2}⟩ for every unit b, so the inner product space only determines a up to a square.

If char(F) ≠ 2, it is well known that every symmetric bilinear form is diagonalizable. Thus every symmetric inner product space is isometric to an orthogonal sum ⟨a_{1}⟩ ⊕ ⋯ ⊕ ⟨a_{n}⟩. For example, it is easy to see that H ≅ ⟨1⟩ ⊕ ⟨−1⟩. This also implies that Î is additively generated by the elements ⟨a⟩ − 1.

If char(F) = 2, every symmetric inner product space is isomorphic to ⟨a_{1}⟩⊕⋯⊕ ⟨a_{n}⟩ ⊕ N, where N is hyperbolic; see [M-SBF, I.3]. In this case Î has the extra generator H − 2.

If char(F) ≠ 2, there is a Cancellation Theorem due to Witt: if X, Y, Z are inner product spaces, then X ⊕ Y ≅ X ⊕ Z implies that Y ≅ Z. For a proof, we refer the reader to [M-SBF]. We remark that cancellation fails if char(F) = 2; see Ex. 5.11(d). The following definition is due to Knebusch.

Definition 5.6.2. Suppose that char(F) ≠ 2. The Witt ring W (F) of F is defined to be the quotient of the ring GW (F) by the subgroup {nH} generated by the hyperbolic plane H. This subgroup is an ideal by Ex. 5.11, so W (F) is also a commutative ring.

Since the augmentation GW (F) → ℤ has ε(H) = 2, it induces an augmentation ε: W (F) → ℤ/2. We write I for the augmentation ideal ker(ε) of W (F).

When char(F) = 2, W (F) is defined similarly, as the quotient of GW (F) by the subgroup of “split” spaces; see Ex. 5.11. In this case we have 2 = 0 in the Witt ring W (F), because the inner product space ⟨1⟩ ⊕ ⟨1⟩ is split (Ex. 5.11(d)).

When char(F) ≠ 2, the augmentation ideals of GW (F) and W (F) are isomorphic: Î ≅ I. This is because ε(nH) = 2n, so that {nH} ∩ Î = 0 in GW (F).

Since (V, B) + (V, −B) = 0 in W (F) by Ex. 5.11, every element of W (F) is represented by an inner product space. In particular, I is additively generated by the classes ⟨a⟩ + ⟨−1⟩, even if char(F) = 2. The powers I^{n} of I form a decreasing chain of ideals W (F) ⊃ I ⊃ I^{2} ⊃ · · ·. We shall describe I/I^{2} now, and return to this topic in chapter III, §7.

The discriminant of an inner product space (V, B) is a classical invariant with values in F^{×}/F^{×2}, where F^{×2} denotes {a^{2}|a ∈ F^{×}}. For each basis of V, there is a matrix M representing B, and the determinant of M is a unit of F. A change of basis replaces M by A^{t}MA, and det(A^{t}MA) = det(M) det(A)^{2}, so w_{1}(V, B) = det(M) is a well defined element in F^{×}/F^{×2}, called the first Stiefel-Whitney class of (V, B). Since w_{1}(H) = −1, we have to modify the definition slightly in order to get an invariant on the Witt ring.

Definition 5.6.3. If dim(V) = r, the discriminant of (V, B) is defined to be the element d(V, B) = (−1)^{r(r−1)/2} det(M) of F^{×}/F^{×2}.

For example, we have d(H) = d(1) = 1 but d(2) = −1. It is easy to verify that the discriminant of (V, B) ⊕ (V′, B′) is (−1)^{rr′}d(V, B)d(V′, B′), where r = dim(V) and r′ = dim(V′). In particular, (V, B) and (V, B)⊕H have the same discriminant. It follows that the discriminant is a well-defined map from W (F) to F^{×}/F^{×2}, and its restriction to I is additive.

Theorem 5.6.4. (Pfister) The discriminant induces an isomorphism between I/I^{2} and F^{×}/F^{×2}.

Proof. Since the discriminant of ⟨a⟩ ⊕ ⟨−1⟩ is a, the map d: I → F^{×}/F^{×2} is onto. This homomorphism annihilates I^{2} because I^{2} is additively generated by products of the form

(⟨a⟩ − 1)(⟨b⟩ − 1) = ⟨ab⟩ + ⟨−a⟩ + ⟨−b⟩ + 1,

and these have discriminant 1. Setting these products equal to zero, the identity ⟨a⟩ + ⟨−a⟩ = 0 yields the congruence

(5.6.5)  (⟨a⟩ − 1) + (⟨b⟩ − 1) ≡ ⟨ab⟩ − 1  mod I^{2}.

Hence the formula s(a) = ⟨a⟩−1 defines a surjective homomorphism s: F^{×} → I/I^{2}. Since ds(a) = a, it follows that s is an isomorphism with inverse induced by d. □

Corollary 5.6.6. W (F) contains ℤ/2 as a subring (i.e., 2 = 0) if and only if −1 is a square in F.

Classical Examples 5.6.7. If F is an algebraically closed field, or more generally every element of F is a square, then ⟨a⟩ ≅ ⟨1⟩ and W (F) = ℤ/2.

If F = ℝ, every bilinear form is classified by its rank and signature. For example, ⟨1⟩ has signature 1 but H has signature 0, with H ⊗ H ≅ H ⊕ H. Thus GW (ℝ) ≅ ℤ[H]/(H^{2} − 2H) and the signature induces a ring isomorphism W (ℝ) ≅ ℤ.

If F = 𝔽_{q} is a finite field with q odd, then I/I^{2} ≅ ℤ/2, and an elementary argument due to Steinberg shows that the ideal I^{2} is zero. The structure of the ring W (F) now follows from 5.6.6: if q ≡ 3 (mod 4) then W (F) = ℤ/4; if q ≡ 1 (mod 4), W (𝔽_{q}) = ℤ/2[η]/(η^{2}), where η = ⟨a⟩ − 1 for some a ∈ F.

If F is a finite field extension of the p-adic rationals, then I^{3} = 0 and I^{2} is cyclic of order 2. If p is odd and the residue field is 𝔽_{q}, then W (F) contains ℤ/2 as a subring if q ≡ 1 (mod 4) and contains ℤ/4 if q ≡ 3 (mod 4). If p = 2 then W (F) contains ℤ/2 as a subring if and only if \sqrt{−1} ∈ F. Otherwise W (F) contains ℤ/4 or ℤ/8, according to whether −1 is a sum of two squares, an issue which is somewhat subtle.

If F = ℚ, the ring map W (ℚ) → W (ℝ) = ℤ is onto, with kernel N satisfying N^{3} = 0. Since I/I^{2} = ℚ^{×}/ℚ^{×2}, the kernel is infinite but under control.

Quadratic Forms

The theory of symmetric bilinear forms is closely related to the theory of quadratic forms, which we now sketch.

Definition 5.7. Let V be a vector space over a field F. A quadratic form on V is a function q: V → F such that q(av) = a^{2} q(v) for every a ∈ F and v ∈ V, and such that the formula B_{q}(v, w) = q(v + w) − q(v) − q(w) defines a symmetric bilinear form B_{q} on V. We call (V, q) a quadratic space if B_{q} is nondegenerate, and call (V, B_{q}) the underlying symmetric inner product space. We write Quad(F) for the category of quadratic spaces and form-preserving maps.

The orthogonal sum (V, q)⊕(V′, q′) of two quadratic spaces is defined to be V ⊕V′ equipped with the quadratic form v ⊕ v′ ↦ q(v) + q′(v′). This is a quadratic space, whose underlying symmetric inner product space is the orthogonal sum (V, B_{q}) ⊕ (V′, B_{q′}). Thus Quad(F) is a symmetric monoidal category, and the underlying space functor Quad(F) → SBil(F) sending (V, q) to (V, B_{q}) is monoidal.

Here is one source of quadratic spaces. Suppose that β is a (possibly nonsymmetric) bilinear form on V. The function q(v) = β(v, v) is visibly quadratic, with associated symmetric bilinear form B_{q}(v, w) = β(v, w) + β(w, v). By choosing an ordered basis of V, it is easy to see that every quadratic form arises in this way. Note that when β is symmetric we have B_{q} = 2β; if char(F) ≠ 2 this shows that β ↦ \frac{1}{2}q defines a monoidal functor SBil(F) → Quad(F) inverse to the underlying functor, and proves the following result.

Lemma 5.7.1. If char(F) ≠ 2 then the underlying space functor Quad(F) → SBil(F) is an equivalence of monoidal categories.

A quadratic space (V, q) is said to be split if it contains a subspace N so that q(N) = 0 and dim(V) = 2 dim(N). For example, the quadratic forms q(x, y) = xy + cy^{2} on V = F^{2} are split.

Definition 5.7.2. The group W Q(F) is defined to be the quotient of the group K_{0}Quad(F) by the subgroup of all split quadratic spaces.

It follows from Ex. 5.11 that the underlying space functor defines a homomorphism W Q(F) → W (F). By Lemma 5.7.1, this is an isomorphism when char(F) ≠ 2.

When char(F) = 2, the underlying symmetric inner product space of a quadratic space (V, q) is always hyperbolic, and V is always even-dimensional; see Ex. 5.13. In particular, W Q(F) → W (F) is the zero map when char(F) = 2. By Ex. 5.13, W Q(F) is a W (F)-module with W Q(F)/I ·W Q(F) given by the Arf invariant. We will describe the rest of the filtration I^{n} · W Q(F) in III.7.10.4.

EXERCISES

5.1 Let R be a ring and let P^{∞}(R) denote the category of all countably generated projective R-modules. Show that K_{0}^{⊕}P^{∞}(R) = 0. 5.2 Suppose that the Krull-Schmidt Theorem holds in an additive category C, i.e., that every object of C can be written as a finite direct sum of indecomposable objects, in a way that is unique up to permutation. Show that K_{0}^{⊕}(C) is the free abelian group on the set of isomorphism classes of indecomposable objects. 5.3 Use Ex. 5.2 to prove Corollary 5.5.1. 5.4 Let R be a commutative ring, and let H^{0}(Spec R, ℚ_{>0}^{×}) denote the free abelian group of all continuous maps Spec(R) → ℚ_{>0}^{×}. Show that [P] ↦ rank(P) induces a split surjection from K_{0}FP(R) onto H^{0}(Spec R, ℚ_{>0}^{×}). In the next two exercises, we shall show that the kernel of this map is isomorphic to K̃_{0}(R) ⊗ ℚ. 5.5 Let R be a commutative ring, and let U_{+} denote the subset of the ring K_{0}(R)⊗ℚ consisting of all x such that rank(x) takes only positive values.

(a) Use the fact that the ideal K̃_{0}(R) is nilpotent to show that U_{+} is an abelian group under multiplication, and that there is a split exact sequence

0 → K̃_{0}(R) ⊗ ℚ ⟶^{exp} U_{+} ⟶^{rank} H^{0}(Spec R, ℚ_{>0}^{×}) → 0.

(b) Show that P ↦ [P] ⊗ 1 is an additive function from FP(R) to the multiplicative group U_{+}, and that it induces a map K_{0}FP(R) → U_{+}.

5.6 (Bass) Let R be a commutative ring. Show that the map K_{0}FP(R) → U_{+} of the previous exercise is an isomorphism. Hint: The map is onto by Ex. 2.10. Conversely, if [P] ⊗ 1 = [Q] ⊗ 1 in U_{+}, show that P ⊗ R^{n} ≅ Q ⊗ R^{n} for some n. 5.7 Suppose that a finite group G acts freely on X, and let X/G denote the orbit space. Show that VB_{G}(X) is equivalent to the category VB(X/G), and conclude that K_{G}^{0}(X) ≅ KU(X/G). 5.8 Let R be a commutative ring. Show that the determinant of a projective module induces a monoidal functor det: P(R) → Pic(R), and that the resulting map K_{0}(det): K_{0}P(R) → K_{0}Pic(R) is the determinant map K_{0}(R) → Pic(R) of Proposition 2.6. 5.9 Let G be a finite group. Given a finite G-set X and a ℤ[G]-module M, the abelian group X ×M carries a ℤ[G]-module structure by g(x, m) = (gx, gm). Show that X × − induces an additive functor from P(ℤ[G]) to itself (2.8). Then show that the pairing (X, M) ↦ X ×M makes K_{0}(ℤ[G]) into a module over the Burnside ring A(G). 5.10 If X = (V, B) and X′ = (V′, B′) are two inner product spaces, show that there is a nondegenerate bilinear form β on V ⊗ V′ satisfying β(v ⊗ v′, w ⊗ w′) = B(v, w)B′(v′, w′) for all v, w ∈ V and v′, w′ ∈ V′. Writing X ⊗ X′ for this inner product space, show that X ⊗ X′ ≅ X′ ⊗ X and (X_{1} ⊕ X_{2}) ⊗ X′ ≅ (X_{1} ⊗ X′) ⊕ (X_{2} ⊗ X′). Then show that X ⊗ H ≅ H ⊕ ⋯ ⊕ H. 5.11 A symmetric inner product space S = (V, B) is called split if it has a basis so that B is represented by a matrix ({0 \atop I} {I \atop A}). Note that the sum of split spaces is also split, and that the hyperbolic plane is split. We define W (F) to be the quotient of GW (F) by the subgroup of classes [S] of split spaces.

(a) If char(F) ≠ 2, show that every split space S is hyperbolic. Conclude that this definition of W (F) agrees with the definition given in 5.6.2.

(b) For any a ∈ F^{×}, show that ⟨a⟩ ⊕ ⟨−a⟩ is split.

(c) If S is split, show that each (V, B)⊗S is split. In particular, (V, B)⊕(V, −B) = (V, B) ⊗ (⟨1⟩ ⊕ ⟨−1⟩) is split. Conclude that W (F) is also a ring when char(F) = 2.

(d) If char(F) = 2, show that the split space S = ⟨1⟩ ⊕ ⟨1⟩ is not hyperbolic, yet ⟨1⟩ ⊕ S ≅ ⟨1⟩ ⊕ H. This shows that Witt Cancellation fails if char(F) = 2. Hint: consider the associated quadratic forms. Then consider the basis (1, 1, 1), (1, 0, 1), (1, 1, 0) of ⟨1⟩ ⊕ S.

5.12 If a + b = 1 in F, show that ⟨a⟩ ⊕ ⟨b⟩ ≅ ⟨ab⟩ ⊕ ⟨1⟩. Conclude that in both GW (F) and W (F) we have the Steinberg identity (⟨a⟩ − 1)(⟨b⟩ − 1) = 0. 5.13 Suppose that char(F) = 2 and that (V, q) is a quadratic form.

(a) Show that B_{q}(v, v) = 0 for every v ∈ V.

(b) Show that the underlying inner product space (V, B_{q}) is hyperbolic, hence split in the sense of Ex. 5.11. This shows that dim(V) is even, and that the map W Q(F) → W (F) is zero. Hint: Find two elements x, y in V so that B_{q}(x, y) = 1, and show that they span an orthogonal summand of V.

(c) If (W, β) is a symmetric inner product space, show that there is a unique quadratic form q′ on V′ = V ⊗W satisfying q′(v⊕w) = q(v)β(w, w), such that the underlying bilinear form satisfies B_{q′}(v ⊗ w, v′ ⊗ w′) = B_{q}(v, v′)β(w, w′). Show that this product makes W Q(F) into a module over W (F).

(d) (Arf invariant) Let ℘: F → F denote the additive map ℘(a) = a^{2} + a. By (b), we may choose a basis x_{1}, …, x_{n}, y_{1} …, y_{n} of V so that each x_{i}, y_{i} span a hyperbolic plane. Show that the element Δ(V, q) = ∑ q(x_{i})q(y_{i}) of F/℘(F) is independent of the choice of basis, called the Arf invariant of the quadratic space (after C. Arf, who discovered it in 1941). Then show that Δ is an additive surjection. H. Sah showed that the Arf invariant and the module structure in (c) induces an isomorphism W Q(F)/I · W Q(F) ≅ F/℘(F).

(e) Consider the quadratic forms q(a, b) = a^{2} + ab + b^{2} and q′(a, b) = ab on V = F^{2}. Show that they are isometric if and only if F contains the field 𝔽_{4}.

5.14 (Kato) If char(F) = 2, show that there is a ring homomorphism W (F) → F ⊗_{F^{p}} F sending ⟨a⟩ to a^{−1} ⊗ a.

§6. K_{0} of an Abelian Category

Another important situation in which we can define Grothendieck groups is when we have a (skeletally) small abelian category. This is due to the natural notion of exact sequence in an abelian category. We begin by quickly reminding the reader what an abelian category is, defining K_{0} and then making a set-theoretic remark.

It helps to read the definitions below with some examples in mind. The reader should remember that the prototype abelian category is the category mod-R of right modules over a ring R, the morphisms being R-module homomorphisms. The full subcategory with objects the free R-modules {0, R, R^{2}, … } is additive, and so is the slightly larger full subcategory P(R) of finitely generated projective Rmodules (this observation was already made in chapter I). For more information on abelian categories, see textbooks like [Mac] or [WHomo].

Definitions 6.1. (1) An additive category is a category containing a zero object ‘0’ (an object which is both initial and terminal), having all products A × B, and such that every set Hom(A, B) is given the structure of an abelian group in such a way that composition is bilinear. In an additive category the product A × B is also the coproduct A ⨿ B of A and B; we call it the direct sum and write it as A ⊕ B to remind ourselves of this fact.

(2) An abelian category A is an additive category in which (i) every morphism f: B → C has a kernel and a cokernel, and (ii) every monic arrow is a kernel, and every epi is a cokernel. (Recall that f: B → C is called monic if fe_{1} ≠ fe_{2} for every e_{1} ≠ e_{2}: A → B; it is called epi if g_{1}f ≠ g_{2}f for every g_{1} ≠ g_{2}: C → D.)

(3) In an abelian category, we call a sequence A ⟶^{f} B ⟶^{g} C exact if ker(g) equals im(f) ≡ ker{B → coker(f)}. A longer sequence is exact if it is exact at all places. By the phrase short exact sequence in an abelian category A we mean an exact sequence of the form:

0 → A′ → A → A′′ → 0.  (∗)

Definition 6.1.1 (K_{0}A). Let A be an abelian category. Its Grothendieck group K_{0}(A) is the abelian group presented as having one generator [A] for each object A of A, with one relation [A] = [A′] + [A′′] for every short exact sequence (∗) in A.

Here are some useful identities which hold in K_{0}(A).

a) [0] = 0 (take A = A′).

b) if A ≅ A′ then [A] = [A′] (take A′′ = 0).

c) [A′ ⊕ A′′] = [A′] + [A′′] (take A = A′ ⊕ A′′).

If two abelian categories are equivalent, their Grothendieck groups are naturally isomorphic, as b) implies they have the same presentation. By c), the group K_{0}(A) is a quotient of the group K_{0}^{⊕}(A) defined in §5 by considering A as a symmetric monoidal category.

Universal Property 6.1.2. An additive function from A to an abelian group Γ is a function f from the objects of A to Γ such that f(A) = f(A′)+f(A′′) for every short exact sequence (∗) in A. By construction, the function A ↦ [A] defines an additive function from A to K_{0}(A). This has the following universal property: any additive function f from A to Γ induces a unique group homomorphism f: K_{0}(A) → Γ, with f([A]) = f(A) for every A.

For example, the direct sum A_{1} ⊕ A_{2} of two abelian categories is also abelian. Using the universal property of K_{0} it is clear that K_{0}(A_{1}⊕A_{2}) ≅ K_{0}(A_{1})⊕K_{0}(A_{2}). More generally, an arbitrary direct sum ⨁ A_{i} of abelian categories is abelian, and we have K_{0}(⨁ A_{i}) ≅ ⨁ K_{0}(A_{i}).

Set-theoretic Considerations 6.1.3. There is an obvious set-theoretic difficulty in defining K_{0}A when A is not small; recall that a category A is called small if the class of objects of A forms a set.

We will always implicitly assume that our abelian category A is skeletally small, i.e., it is equivalent to a small abelian category A′. In this case we define K_{0}(A) to be K_{0}(A′). Since any other small abelian category equivalent to A will also be equivalent to A′, the definition of K_{0}(A) is independent of this choice.

Example 6.1.4 (All R-modules). We cannot take the Grothendieck group of the abelian category mod-R because it is not skeletally small. To finesse this difficulty, fix an infinite cardinal number κ and let mod_{κ}-R denote the full subcategory of mod-R consisting of all R-modules of cardinality < κ. As long as κ ≥ |R|, mod_{κ}-R is an abelian subcategory of mod-R having a set of isomorphism classes of objects. The Eilenberg Swindle I.2.8 applies to give K_{0}(mod_{κ}-R) = 0. In effect, the formula M ⊕ M^{∞} ≅ M^{∞} implies that [M] = 0 for every module M.

6.1.5. The natural type of functor F: A → B between two abelian categories is an additive functor; this is a functor for which all the maps Hom(A, A′) → Hom(F A, F A′) are group homomorphisms. However, not all additive functors induce homomorphisms K_{0}(A) → K_{0}(B).

We say that an additive functor F is exact if it preserves exact sequences—that is, for every exact sequence (∗) in A, the sequence 0 → F (A′) → F (A) → F (A′′) → 0 is exact in B. The presentation of K_{0} implies that any exact functor F defines a group homomorphism K_{0}(A) → K_{0}(B) by the formula [A] ↦ [F (A)].

Suppose given an inclusion A ⊂ B of abelian categories, with A a full subcategory of B. If the inclusion is an exact functor, we say that A is an exact abelian subcategory of B. As with all exact functors, the inclusion induces a natural map K_{0}(A) → K_{0}(B).

Definition 6.2 (G_{0}R). If R is a (right) noetherian ring, let M(R) denote the subcategory of mod-R consisting of all finitely generated R-modules. The noetherian hypothesis implies that M(R) is an abelian category, and we write G_{0}(R) for K_{0}M(R). (We will give a definition of M(R) and G_{0}(R) for non-noetherian rings in Example 7.1.4 below.)

The presentation of K_{0}(R) in §2 shows that there is a natural map K_{0}(R) → G_{0}(R), which is called the Cartan homomorphism (send [P] to [P]).

Associated to a ring homomorphism f: R → S are two possible maps on G_{0}: the contravariant transfer map and the covariant base change map.

When S is finitely generated as an R-module (e.g., S = R/I), there is a “transfer” homomorphism f_{∗}: G_{0}(S) → G_{0}(R). It is induced from the forgetful functor f_{∗}: M(S) → M(R), which is exact.

Whenever S is flat as an R-module, there is a “base change” homomorphism f^{∗}: G_{0}(R) → G_{0}(S). Indeed, the base change functor f^{∗}: M(R) → M(S), f^{∗}(M) = M ⊗_{R} S, is exact if and only if S is flat over R. We will extend the definition of f^{∗} in §7 to the case in which S has a finite resolution by flat R-modules using Serre’s Formula (7.9.3): f^{∗}([M]) = ∑(−1)^{i}[Tor_{i}^{R}(M, S)].

If F is a field then every exact sequence in M(F) splits, and it is easy to see that G_{0}(F) ≅ K_{0}(F) ≅ ℤ. In particular, if R is an integral domain with field of fractions F, then there is a natural map G_{0}(R) → G_{0}(F) = ℤ, sending [M] to the integer dim_{F} (M ⊗_{R} F).

Example 6.2.1 (Abelian groups). When R = ℤ the Cartan homomorphism is an isomorphism: K_{0}(ℤ) ≅ G_{0}(ℤ) ≅ ℤ. To see this, first observe that the sequences

0 → ℤ ⟶^{n} ℤ → ℤ/nℤ → 0

imply that [ℤ/nℤ] = [ℤ] − [ℤ] = 0 in G_{0}(ℤ) for every n. By the Fundamental Theorem of finitely generated Abelian Groups, every finitely generated abelian group M is a finite sum of copies of the groups ℤ and ℤ/n, n ≥ 2. Hence G_{0}(ℤ) is generated by [ℤ]. To see that G_{0}(ℤ) ≅ ℤ, observe that since ℚ is a flat ℤ-module there is a homomorphism from G_{0}(ℤ) to G_{0}(ℚ) ≅ ℤ sending [M] to r(M) = dim_{ℚ}(M ⊗ ℚ). In effect, r(M) is an additive function; as such it induces a homomorphism r: G_{0}(ℤ) → ℤ. As r(ℤ) = 1, r is an isomorphism.

More generally, the Cartan homomorphism is an isomorphism whenever R is a principal ideal domain, and K_{0}(R) ≅ G_{0}(R) ≅ ℤ. The proof is identical.

Example 6.2.2 (p-groups). Let Ab_{p} denote the abelian category of all finite abelian p-groups for some prime p. Then K_{0}(Ab_{p}) ≅ ℤ on generator [ℤ/p]. To see this, we observe that the length ℓ(M) of a composition series for a finite p-group M is well-defined by the Jordan-Hölder Theorem. Moreover ℓ is an additive function, and defines a homomorphism K_{0}(Ab_{p}) → ℤ with ℓ(ℤ/p) = 1. To finish we need only observe that ℤ/p generates K_{0}(Ab_{p}); this follows by induction on the length of a p-group, once we observe that any L ⊂ M yields [M] = [L] + [M/L] in K_{0}(Ab_{p}).

Example 6.2.3. The category Ab_{fin} of all finite abelian groups is the direct sum of the categories Ab_{p} of Example 6.2.2. Therefore K_{0}(Ab_{fin}) = ⨁ K_{0}(Ab_{p}) is the free abelian group on the set {[ℤ/p], p prime}.

Example 6.2.4. The category M(ℤ/p^{n}) of all finite ℤ/p^{n}-modules is an exact abelian subcategory of Ab_{p}, and the argument above applies verbatim to prove that the simple module [ℤ/p] generates the group G_{0}(ℤ/p^{n}) ≅ ℤ. In particular, the canonical maps from G_{0}(ℤ/p^{n}) = K_{0}M(ℤ/p^{n}) to K_{0}(Ab_{p}) are all isomorphisms.

Recall from Lemma 2.2 that K_{0}(ℤ/p^{n}) ≅ ℤ on [ℤ/p^{n}]. The Cartan homomorphism from K_{0} ≅ ℤ to G_{0} ≅ ℤ is not an isomorphism; it sends [ℤ/p^{n}] to n[ℤ/p].

Definition 6.2.5 (G_{0}(X)). Let X be a noetherian scheme. The category M(X) of all coherent O_{X}-modules is an abelian category. (See [Hart, II.5.7].) We write G_{0}(X) for K_{0}M(X). When X = Spec(R) this agrees with Definition 6.2: G_{0}(X) ≅ G_{0}(R), because of the equivalence of M(X) and M(R).

If f: X → Y is a morphism of schemes, there is a base change functor f^{∗}: M(Y) → M(X) sending F to f^{∗}F = F ⊗_{O_{Y}} O_{X}; see I.5.2. When f is flat, the base change f^{∗} is exact and therefore the formula f^{∗}([F]) = [f^{∗}F] defines a homomorphism f^{∗}: G_{0}(Y) → G_{0}(X). Thus G_{0} is contravariant for flat maps.

If f: X → Y is a finite morphism, the direct image f_{∗}F of a coherent sheaf F is coherent, and f_{∗}: M(X) → M(Y) is an exact functor [EGA, I(1.7.8)]. In this case the formula f_{∗}([F]) = [f_{∗}F] defines a “transfer” map f_{∗}: G_{0}(X) → G_{0}(Y).

If f: X → Y is a proper morphism, the direct image f_{∗}F of a coherent sheaf F is coherent, and so are its higher direct images R^{i}f_{∗}F. (This is Serre’s “Theorem B”; see I.5.2 or [EGA, III(3.2.1)].) The functor f_{∗}: M(X) → M(Y) is not usually exact (unless f is finite). Instead we have:

Lemma 6.2.6. If f: X → Y is a proper morphism of noetherian schemes, there is a “transfer” homomorphism f_{∗}: G_{0}(X) → G_{0}(Y). It is defined by the formula f_{∗}([F]) = ∑(−1)^{i}[R^{i}f_{∗}F]. The transfer homomorphism makes G_{0} functorial for proper maps.

Proof. For each coherent F the R^{i}f_{∗}F vanish for large i, so the sum is finite. By 6.2.1 it suffices to show that the formula gives an additive function. But if 0 → F′ → F → F′′ → 0 is a short exact sequence in M(X) there is a finite long exact sequence in M(Y):

0 → f_{∗}F′ → f_{∗}F → f_{∗}F′′ → R^{1}f_{∗}F′ → R^{1}f_{∗}F → R^{1}f_{∗}F′′ → R^{2}f_{∗}F′ → ⋯

and the alternating sum of the terms is f_{∗}[F′]−f_{∗}[F]+f_{∗}[F′′]. This alternating sum must be zero by Proposition 6.6 below, so f_{∗} is additive as desired. (Functoriality is relegated to Ex. 6.15.) □

The next lemma follows by inspection of the definition of the direct limit (or filtered colimit) A = \underrightarrow{lim} A_{i} of a filtered system of small categories; the objects and morphisms of A are the direct limits of the object and morphisms of the A_{i}.

Lemma 6.2.7 (Filtered colimits). Suppose that {A_{i}}_{i∈I} is a filtered family of small abelian categories and exact functors. Then the direct limit A = \underrightarrow{lim} A_{i} is also an abelian category, and

K_{0}(A) = \underrightarrow{lim} K_{0}(A_{i}).

Example 6.2.8 (S-torsion modules). Suppose that S is a multiplicatively closed set of elements in a noetherian ring R. Let M_{S}(R) be the subcategory of M(R) consisting of all finitely generated R-modules M such that Ms = 0 for some s ∈ S. For example, if S = {p^{n}} then M_{S}(ℤ) = Ab_{p} was discussed in Example 6.2.2. In general M_{S}(R) is not only the union of the M(R/RsR), but is also the union of the M(R/I) as I ranges over the ideals of R with I ∩ S ≠ ϕ. By 6.2.7,

K_{0}M_{S}(R) = \underrightarrow{lim}_{I∩S≠ϕ} G_{0}(R/I) = \underrightarrow{lim}_{s∈S} G_{0}(R/RsR).

Devissage

The method behind the computation in Example 6.2.4 that G_{0}(ℤ/p^{n}) ≅ K_{0}Ab_{p} is called Devissage, a French word referring to the “unscrewing” of the composition series. Here is a formal statement of the process, due to Alex Heller.

Devissage Theorem 6.3. Let B ⊂ A be small abelian categories. Suppose that (a) B is an exact abelian subcategory of A, closed in A under subobjects and

quotient objects; and

(b) Every object A of A has a finite filtration A = A_{0} ⊃ A_{1} ⊃ ⋯ ⊃ A_{n} = 0 with all quotients A_{i}/A_{i+1} in B.

Then the inclusion functor B ⊂ A is exact and induces an isomorphism

K_{0}(B) ≅ K_{0}(A).

Proof. Let i_{∗}: K_{0}(B) → K_{0}(A) denote the canonical homomorphism. To see that i_{∗} is onto, observe that every filtration A = A_{0} ⊃ A_{1} ⊃ ⋯ ⊃ A_{n} = 0 yields [A] = ∑[A_{i}/A_{i+1}] in K_{0}(A). This follows by induction on n, using the observation that [A_{i}] = [A_{i+1}]+[A_{i}/A_{i+1}]. Since by (b) such a filtration exists with the A_{i}/A_{i+1} in B, this shows that the canonical i_{∗} is onto.

For each A in A, fix a filtration A = A_{0} ⊃ A_{1} ⊃ ⋯ ⊃ A_{n} = 0 with each A_{i}/A_{i+1} in B, and define f(A) to be the element ∑[A_{i}/A_{i+1}] of K_{0}(B). We claim that f(A) is independent of the choice of filtration. Because any two filtrations have equivalent refinements (Ex. 6.2), we only need check refinements of our given filtration. By induction we need only check for one insertion, say changing A_{i} ⊃ A_{i+1} to A_{i} ⊃ A′ ⊃ A_{i+1}. Appealing to the exact sequence

0 → A′/A_{i+1} → A_{i}/A_{i+1} → A_{i}/A′ → 0,

we see that [A_{i}/A_{i+1}] = [A_{i}/A′] + [A′/A_{i+1}] in K_{0}(B), as claimed.

Given a short exact sequence 0 → A′ → A → A′′ → 0, we may construct a filtration {A_{i}} on A by combining our chosen filtration for A′ with the inverse image in A of our chosen filtration for A′′. For this filtration we have ∑[A_{i}/A_{i+1}] = f(A′) + f(A′′). Therefore f is an additive function, and defines a map K_{0}(A) → K_{0}(B). By inspection, f is the inverse of the canonical map i_{∗}. □

Corollary 6.3.1. Let I be a nilpotent ideal of a noetherian ring R. Then the inclusion mod-(R/I) ⊂ mod-R induces an isomorphism

G_{0}(R/I) ≅ G_{0}(R).

Proof. To apply Devissage, we need to observe that if M is a finitely generated R-module, the filtration M ⊇ MI ⊇ MI^{2} ⊇ ⋯ ⊇ MI^{n} = 0 is finite, and all the quotients MI^{n}/MI^{n+1} are finitely generated R/I-modules. □

Notice that this also proves the scheme version:

Corollary 6.3.2. Let X be a noetherian scheme, and X_{red} the associated reduced scheme. Then G_{0}(X) ≅ G_{0}(X_{red}).

Application 6.3.3 (R-modules with support). Example 6.2.2 can be generalized as follows. Given a central element s in a ring R, let M_{s}(R) denote the abelian subcategory of M(R) consisting of all finitely generated R-modules M such that Ms^{n} = 0 for some n. That is, modules such that M ⊃ Ms ⊃ Ms^{2} ⊃ ⋯ is a finite filtration. By Devissage,

K_{0}M_{s}(R) ≅ G_{0}(R/sR).

More generally, suppose we are given an ideal I of R. Let M_{I}(R) be the (exact) abelian subcategory of M(R) consisting of all finitely generated R-modules M such that the filtration M ⊃ MI ⊃ MI^{2} ⊃ ⋯ is finite, i.e., such that MI^{n} = 0 for some n. By Devissage,

K_{0}M_{I}(R) ≅ K_{0}M(R/I) = G_{0}(R/I).

Example 6.3.4. Let X be a noetherian scheme, and i: Z ⊂ X the inclusion of a closed subscheme. Let M_{Z}(X) denote the abelian category of coherent O_{X}- modules Z supported on Z, and I the ideal sheaf in O_{X} such that O_{X}/I ≅ O_{Z}. Via the direct image i_{∗}: M(Z) ⊂ M(X), we can consider M(Z) as the subcategory of all modules M in M_{Z}(X) such that IM = 0. Every M in M_{Z}(X) has a finite filtration M ⊃ MI ⊃ MI^{2} ⊃ ⋯ with quotients in M(Z), so by Devissage:

K_{0}M_{Z}(X) ≅ K_{0}M(Z) = G_{0}(Z).

The Localization Theorem

Let A be an abelian category. A Serre subcategory of A is an abelian subcategory B which is closed under subobjects, quotients and extensions. That is, if 0 → B → C → D → 0 is exact in A then

C ∈ B ⇔ B, D ∈ B.

Now assume for simplicity that A is small. If B is a Serre subcategory of A, we can form a quotient abelian category A/B as follows. Call a morphism f in A a B-iso if ker(f) and coker(f) are in B. The objects of A/B are the objects of A, and morphisms A_{1} → A_{2} are equivalence classes of diagrams in A:

A_{1} ⟵^{f} A′ ⟶^{g} A_{2}, f a B-iso.

Such a morphism is equivalent to A_{1} ← A′′ → A_{2} if and only if there is a commutative diagram:

\begin{aligned}& & A′ \\ & ↙ & ↑ & ↘ \\ A_{1} & ← & A & → & A_{2} \\ & ↖ & ↓ & ↗ \\ & & A′′ \end{aligned} where A′ ← A → A′′ are B-isos.

The composition with A_{2} ⟵^{f′} A′′ ⟶^{h} A_{3} is A_{1} ⟵^{f} A′ ⟵ A ⟶ A′′ ⟶^{h} A_{3}, where A

is the pullback of A′ and A′′ over A_{2}. The proof that A/B is abelian, and that the quotient functor loc: A → A/B is exact, may be found in [Swan, p.44ff] or [Gabriel]. (See the appendix to this chapter.)

It is immediate from the construction of A/B that loc(A) ≅ 0 if and only if A is an object of B, and that for a morphism f: A → A′ in A, loc(f) is an isomorphism if and only if f is a B-iso. In fact A/B solves a universal problem (see op. cit.): if T: A → C is an exact functor such that T (B) ≅ 0 for all B in B, then there is a unique exact functor T′: A/B → C so that T = T′ ∘ loc.

Localization Theorem 6.4. (Heller) Let A be a small abelian category, and B a Serre subcategory of A. Then the following sequence is exact:

K_{0}(B) → K_{0}(A) \underrightarrow{loc} K_{0}(A/B) → 0.

Proof. By the construction of A/B, K_{0}(A) maps onto K_{0}(A/B) and the composition K_{0}(B) → K_{0}(A/B) is zero. Hence if Γ denotes the cokernel of the map K_{0}(B) → K_{0}(A) there is a natural surjection Γ → K_{0}(A/B); to prove the theorem it suffices to give an inverse. For this it suffices to show that γ(loc(A)) = [A] defines an additive function from A/B to Γ, because the induced map γ: K_{0}(A/B) → Γ will be inverse to the natural surjection Γ → K_{0}(A/B).

Since loc: A → A/B is a bijection on objects, γ is well-defined. We claim that if loc(A_{1}) ≅ loc(A_{2}) in A/B then [A_{1}] = [A_{2}] in Γ. To do this, represent the isomorphism by a diagram A_{1} ⟵^{f} A ⟶^{g} A_{2} with f a B-iso. As loc(g) is an isomorphism in A/B, g is also a B-iso. In K_{0}(A) we have

[A] = [A_{1}] + [ker(f)] − [coker(f)] = [A_{2}] + [ker(g)] − [coker(g)].

Hence [A] = [A_{1}] = [A_{2}] in Γ, as claimed.

To see that γ is additive, suppose given an exact sequence in A/B of the form:

0 → loc(A_{0}) ⟶^{i} loc(A_{1}) ⟶^{j} loc(A_{2}) → 0;

we have to show that [A_{1}] = [A_{0}]+[A_{2}] in Γ. Represent j by a diagram A_{1} ⟵^{f} A ⟶^{g} A_{2} with f a B-iso. Since [A] = [A_{1}] + [ker(f)] − [coker(f)] in K_{0}(A), [A] = [A_{1}] in Γ. Applying the exact functor loc to

0 → ker(g) → A ⟶^{g} A_{2} → coker(g) → 0,

we see that coker(g) is in B and that loc(ker(g)) ≅ loc(A_{0}) in A/B. Hence [ker(g)] ≡ [A_{0}] in Γ, and in Γ we have

[A_{1}] = [A] = [A_{2}] + [ker(g)] − [coker(g)] ≡ [A_{0}] + [A_{2}]

proving that γ is additive, and finishing the proof of the Localization Theorem. □

Application 6.4.1. Let S be a central multiplicative set in a ring R, and let mod_{S}(R) denote the Serre subcategory of mod-R consisting of S-torsion modules, i.e., those R-modules M such that every m ∈ M has ms = 0 for some s ∈ S. Then there is a natural equivalence between mod-(S^{−1}R) and the quotient category mod-R/mod_{S}(R). If R is noetherian and M_{S}(R) denotes the Serre subcategory of M(R) consisting of finitely generated S-torsion modules, then M(S^{−1}R) is equivalent to M(R)/M_{S}(R). The Localization exact sequence becomes:

K_{0}M_{S}(R) → G_{0}(R) → G_{0}(S^{−1}R) → 0.

In particular, if S = {s^{n}} for some s then by Application 6.3.3 we have an exact sequence

G_{0}(R/sR) → G_{0}(R) → G_{0}(R[\frac{1}{s}]) → 0.

More generally, if I is an ideal of a noetherian ring R, we can consider the Serre subcategory M_{I}(R) of modules with some MI^{n} = 0 discussed in Application 6.3.3. The quotient category M(R)/M_{I}(R) is known to be isomorphic to the category M(U) of coherent O_{U}-modules, where U is the open subset of Spec(R) defined by I. The composition of the isomorphism K_{0}M(R/I) ≅ K_{0}M_{I}(R) of 6.3.3 with K_{0}M_{I}(R) → K_{0}M(R) is evidently the transfer map i_{∗}: G_{0}(R/I) → G_{0}(R). Hence the Localization Sequence becomes the exact sequence

G_{0}(R/I) ⟶^{i_{∗}} G_{0}(R) → G_{0}(U) → 0

Application 6.4.2. Let X be a scheme, and i: Z ⊂ X a closed subscheme with complement j: U ⊂ X. Let mod_{Z}(X) denote the Serre subcategory of O_{X}-mod consisting of all O_{X}-modules F with support in Z, i.e., such that F|_{U} = 0. Gabriel proved [Gabriel] that j^{∗} induces an equivalence: O_{U}-mod ≅ O_{X}-mod/mod_{Z}(X).

Morover, if X is noetherian and M_{Z}(X) denotes the category of coherent sheaves supported in Z, then M(X)/M_{Z}(X) ≅ M(U). The inclusion i: Z ⊂ X induces an exact functor i_{∗}: M(Z) ⊂ M(X), and G_{0}(Z) ≅ K_{0}M_{Z}(X) by Example 6.3.4. Therefore the Localization sequence becomes:

G_{0}(Z) ⟶^{i_{∗}} G_{0}(X) ⟶^{j^{∗}} G_{0}(U) → 0.

For example, if X = Spec(R) and Z = Spec(R/I), we recover the exact sequence in the previous application.

Application 6.4.3 (Higher Divisor Class Groups). Given a commutative noetherian ring R, let D^{i}(R) denote the free abelian group on the set of prime ideals of height exactly i; this is generalizes the group of Weil divisors in Ch.I, §3. Let M^{i}(R) denote the category of finitely generated R-modules M whose associated prime ideals all have height ≥ i. Each M^{i}(R) is a Serre subcategory of M(R); see Ex. 6.9. Let F^{i}G_{0}(R) denote the image of K_{0}M^{i}(R) in G_{0}(R) = K_{0}M(R). These subgroups form a filtration ⋯ ⊂ F^{2} ⊂ F^{1} ⊂ F^{0} = G_{0}(R), called the coniveau filtration of G_{0}(R).

It turns out that there is an equivalence M^{i}/M^{i+1}(R) ≅ ⨁ M_{𝔭}(R_{𝔭}), ht(𝔭) = i. By Application 6.3.3 of Devissage, K_{0}M_{𝔭}(R_{𝔭}) ≅ G_{0}(R_{𝔭}/𝔭R_{𝔭}) ≅ ℤ, so there is an isomorphism D^{i}(R) ⟶^{≅} K_{0}M^{i}/M^{i+1}(R), [𝔭] ↦ [R/𝔭]. By the Localization Theorem, we have an exact sequence

K_{0}M^{i+1}(R) → K_{0}M^{i}(R) → D^{i}(R) → 0.

Thus G_{0}(R)/F^{1} ≅ D^{0}(R), and each subquotient F^{i}/F^{i+1} is a quotient of D^{i}(R).

For i ≥ 1, the generalized Weil divisor class group CH^{i}(R) is defined to be the subgroup of K_{0}M^{i−1}/M^{i+1}(R) generated by the classes [R/𝔭], ht(𝔭) ≥ i. This definition is due to L. Claborn and R. Fossum; the notation reflects a theorem (in V.9 below) that the kernel of D^{i}(R) → CH^{i}(R) is generated by rational equivalence. For example, we will see in Ex. 6.9 that if R is a Krull domain then CH^{1}(R) is the usual divisor class group Cl(R), and G_{0}(R)/F^{2} ≅ ℤ ⊕ Cl(R).

Similarly, if X is a noetherian scheme, there is a coniveau filtration on G_{0}(X). Let M^{i}(X) denote the subcategory of M(X) consisting of coherent modules whose support has codimension ≥ i, and let D^{i}(X) denote the free abelian group on the set of points of X having codimension i. Then each M^{i}(X) is a Serre subcategory and M^{i}/M^{i+1}(X) ≅ ⨁ M_{x}(O_{X,x}), where x runs over all points of codimension i in X. Again by Devissage, there is an isomorphism K_{0}M^{i}/M^{i+1}(X) ≅ D^{i}(X) and hence G_{0}(X)/F^{1} ≅ D^{0}(X). For i ≥ 1, the generalized Weil divisor class group CH^{i}(X) is defined to be the subgroup of K_{0}M^{i−1}/M^{i+1}(X) generated by the classes [O_{Z}], codim_{X}(Z) = i. We will see later on (in V.9.4.1) that CH^{i}(X) is the usual Chow group of codimension i cycles on X modulo rational equivalence, as defined in [Fulton]. The verification that CH^{1}(X) = Cl(X) is left to Ex. 6.10.

We now turn to a clasical application of the Localization Theorem: the Fundamental Theorem for G_{0} of a noetherian ring R. Via the ring map π: R[t] → R sending t to zero, we have an inclusion M(R) ⊂ M(R[t]) and hence a transfer map π_{∗}: G_{0}(R) → G_{0}(R[t]). By 6.4.1 there is an exact localization sequence

G_{0}(R) ⟶^{π_{∗}} G_{0}(R[t]) ⟶^{j^{∗}} G_{0}(R[t, t^{−1}]) → 0.  (6.4.4)

Given an R-module M, the exact sequence of R[t]-modules

0 → M[t] ⟶^{t} M[t] → M → 0

shows that in G_{0}(R[t]) we have

π_{∗}[M] = [M] = [M[t]] − [M[t]] = 0.

Thus π_{∗} = 0, because every generator [M] of G_{0}(R) becomes zero in G_{0}(R[t]). From the Localization sequence (6.4.4) it follows that j^{∗} is an isomorphism. This proves the easy part of the following result.

Fundamental Theorem for G_{0}-theory of Rings 6.5. For every noether-

ian ring R, the inclusions R ↪^{i} R[t] ↪^{j} R[t, t^{−1}] induce isomorphisms

G_{0}(R) ≅ G_{0}(R[t]) ≅ G_{0}(R[t, t^{−1}]).

Proof. The ring inclusions are flat, so they induce maps i^{∗}: G_{0}(R) → G_{0}(R[t]) and j^{∗}: G_{0}(R[t]) → G_{0}(R[t, t^{−1}]). We have already seen that j^{∗} is an isomorphism; it remains to show that i^{∗} is an isomorphism.

Because R = R[t]/tR[t], Serre’s formula defines a map π^{∗}: G_{0}(R[t]) → G_{0}(R) by the formula: π^{∗}[M] = [M/Mt] − [ann_{M}(t)], where ann_{M}(t) = {x ∈ M : xt = 0}. (See Ex. 6.6 or 7.9.3 below.) Since π^{∗}i^{∗}[M] = π^{∗}[M[t]] = [M], i^{∗} is an injection split by π^{∗}.

We shall present Grothendieck’s proof that i^{∗}: G_{0}(R) → G_{0}(R[t]) is onto, which assumes that R is a commutative ring. A proof in the non-commutative case (due to Serre) will be sketched in Ex. 6.13.

If G_{0}(R) ≠ G_{0}(R[t]), we proceed by noetherian induction to a contradiction. Among all ideals J for which G_{0}(R/J) ≠ G_{0}(R/J[t]), there is a maximal one. Replacing R by R/J, we may assume that G_{0}(R/I) = G_{0}(R/I[t]) for each I ≠ 0 in R. Such a ring R must be reduced by Corollary 6.3.1. Let S be the set of non-zero divisors in R; by elementary ring theory S^{−1}R is a finite product ∏ F_{i} of fields F_{i}, so G_{0}(S^{−1}R) ≅ ⊕G_{0}(F_{i}). Similarly S^{−1}R[t] = ∏ F_{i}[t] and G_{0}(S^{−1}R[t]) ≅ ⊕G_{0}(F_{i}[t]). By Application 6.4.1 and Example 6.2.8 we have a diagram with exact rows:

\underrightarrow{lim} G_{0}(R/sR) ⟶ G_{0}(R) ⟶ ⊕G_{0}(F_{i}) ⟶ 0
≅↓_{i^{∗}}  ↓_{i^{∗}}  ↓
\underrightarrow{lim} G_{0}(R/sR[t]) ⟶ G_{0}(R[t]) ⟶ ⊕G_{0}(F_{i}[t]) ⟶ 0.

Since the direct limits are taken over all s ∈ S, the left vertical arrow is an isomorphism by induction. Because each F_{i}[t] is a principal ideal domain, (2.6.3) and Example 6.2.1 imply that the right vertical arrow is the sum of the isomorphisms

G_{0}(F_{i}) ≅ K_{0}(F_{i}) ≅ ℤ ≅ K_{0}(F_{i}[t]) ≅ G_{0}(F_{i}[t]).

By the 5-lemma, the middle vertical arrow is onto, hence an isomorphism. □

We can generalize the Fundamental Theorem from rings to schemes by a slight modification of the proof. 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. Thus if X = Spec(R) we have X[t] = Spec(R[t]) and X[t, t^{−1}] = Spec(R[t, t^{−1}]). Now suppose that X is noetherian. Via the map π: X → X[t] defined by t = 0, we have an inclusion M(X) ⊂ M(X[t]) and hence a transfer map π_{∗}: G_{0}(X) → G_{0}(X[t]) as before. The argument we gave after (6.4.4) above goes through to show that π_{∗} = 0 here too, because any generator [F] of G_{0}(X) becomes zero in G_{0}(X[t]). By 6.4.2 we have an exact sequence

G_{0}(X) ⟶^{π_{∗}} G_{0}(X[t]) → G_{0}(X[t, t^{−1}]) → 0

and therefore G_{0}(X[t]) ≅ G_{0}(X[t, t^{−1}]).

Fundamental Theorem for G_{0}-theory of Schemes 6.5.1. If X is a noe-

therian scheme then the flat maps X[t, t^{−1}] ↪^{j} X[t] →^{i} X induce isomorphisms:

G_{0}(X) ≅ G_{0}(X[t]) ≅ G_{0}(X[t, t^{−1}]).

Proof. We have already seen that j^{∗} is an isomorphism. By Ex. 6.7 there is a map π^{∗}: G_{0}(X[t]) → G_{0}(X) sending [F] to [F/tF] − [ann_{F}(t)]. Since π^{∗}i^{∗}[F] = (iπ)^{∗}[F] = [F], we again see that i^{∗} is an injection, split by π^{∗}.

It suffices to show that i^{∗} is a surjection for all X. By noetherian induction, we may suppose that the result is true for all proper closed subschemes Z of X. In particular, if Z is the complement of an affine open subscheme U = Spec(R) of X, we have a commutative diagram whose rows are exact by Application 6.4.2.

G_{0}(Z) ⟶ G_{0}(X) ⟶ G_{0}(R) ⟶ 0
≅↓_{i^{∗}}  ↓_{i^{∗}}  ≅↓_{i^{∗}}
G_{0}(Z[t]) ⟶ G_{0}(X[t]) ⟶ G_{0}(R[t]) ⟶ 0

The outside vertical arrows are isomorphisms, by induction and Theorem 6.5. By the 5-lemma, G_{0}(X) ⟶^{i^{∗}} G_{0}(X[t]) is onto, and hence an isomorphism. □

Euler Characteristics

Suppose that C_{•}: 0 → C_{m} → ⋯ → C_{n} → 0 is a bounded chain complex of objects in an abelian category A. We define the Euler characteristic χ(C_{•}) of C_{•} to be the following element of K_{0}(A):

χ(C_{•}) = ∑(−1)^{i}[C_{i}].

Proposition 6.6. If C_{•} is a bounded complex of objects in A, the element χ(C_{•}) depends only upon the homology of C_{•} :

χ(C_{•}) = ∑(−1)^{i}[H_{i}(C_{•})].

In particular, if C_{•} is acyclic (exact as a sequence) then χ(C_{•}) = 0.

Proof. Write Z_{i} and B_{i−1} for the kernel and image of the map C_{i} → C_{i−1}, respectively. Since B_{i−1} = C_{i}/Z_{i} and H_{i}(C_{•}) = Z_{i}/B_{i}, we compute in K_{0}(A):

∑(−1)^{i}[H_{i}(C_{•})] = ∑(−1)^{i}[Z_{i}] − ∑(−1)^{i}[B_{i}]
= ∑(−1)^{i}[Z_{i}] + ∑(−1)^{i}[B_{i−1}]
= ∑(−1)^{i}[C_{i}] = χ(C_{•}). □

Let Ch^{hb}(A) denote the abelian category of (possibly unbounded) chain complexes of objects in A having only finitely many nonzero homology groups. We call such complexes homologically bounded.

Corollary 6.6.1. There is a natural surjection χ_{H}: K_{0}(Ch^{hb}) → K_{0}(A) sending C_{•} to ∑(−1)^{i}[H_{i}(C_{•})]. In particular, if 0 → A_{•} → B_{•} → C_{•} → 0 is a exact sequence of homologically bounded complexes then:

χ_{H}(B_{•}) = χ_{H}(A_{•}) + χ_{H}(C_{•}).

EXERCISES

6.1 Let R be a ring and mod_{fl}(R) the abelian category of R-modules with finite length. Show that K_{0}mod_{fl}(R) is the free abelian group ⨁_{𝔪} ℤ, a basis being {[R/𝔪], 𝔪 a maximal right ideal of R}. Hint: Use the Jordan-Hölder Theorem for modules of finite length. 6.2 Schreier Refinement Theorem. Let A = A_{0} ⊇ A_{1} ⊇ ⋯ ⊇ A_{r} = 0 and A = A′_{0} ⊇ A′_{1} ⊇ ⋯ ⊇ A′_{s} = 0 be two filtrations of an object A in an abelian category A. Show that the subobjects A_{i,j} = (A_{i} ∩ A′_{j}) + A_{i+1}, ordered lexicographically, form a filtration of A which refines the filtration {A_{i}}. By symmetry, there is also a filtration by the A′_{j,i} = (A_{i} ∩ A′_{j}) + A′_{j+1} which refines the filtration {A′_{j}}.

Prove Zassenhaus’ Lemma, that A_{i,j}/A_{i,j+1} ≅ A′_{j,i}/A′_{j,i+1}. This shows that the factors in the two refined filtrations are isomorphic up to a permutation; the slogan is that “any two filtrations have equivalent refinements.”

6.3 Jordan-Hölder Theorem in A. An object A in an abelian category A is called simple if it has no proper subobjects. We say that an object A has finite length if it has a composition series A = A_{0} ⊃ ⋯ ⊃ A_{s} = 0 in which all the quotients A_{i}/A_{i+1} are simple. By Ex. 6.2, the Jordan-Hölder Theorem holds in A_{fl}: the simple factors in any composition series of A are unique up to permutation and isomorphism. Let A_{fl} denote the subcategory of objects in A of finite length. Show that A_{fl} is a Serre subcategory of A, and that K_{0}(A_{fl}) is the free abelian group on the set of isomorphism classes of simple objects. 6.4 Let A be a small abelian category. If [A_{1}] = [A_{2}] in K_{0}(A), show that there are short exact sequences in A

0 → C′ → C_{1} → C′′ → 0, 0 → C′ → C_{2} → C′′ → 0

such that A_{1} ⊕ C_{1} ≅ A_{2} ⊕ C_{2}. Hint: First find sequences 0 → D′_{i} → D_{i} → D′′_{i} → 0 such that A_{1} ⊕ D′_{1} ⊕ D′′_{1} ⊕ D_{2} ≅ A_{2} ⊕ D′_{2} ⊕ D′′_{2} ⊕ D_{1}, and set C_{i} = D′_{i} ⊕ D′′_{i} ⊕ D_{j}. 6.5 Resolution. Suppose that R is a regular noetherian ring, i.e., that every Rmodule has a finite projective resolution. Show that the Cartan homomorphism K_{0}(R) → G_{0}(R) is onto. (We will see in Theorem 7.8 that it is an isomorphism.) 6.6 Serre’s Formula. (Cf. 7.9.3) If s is a central element of a ring R, show that there is a map π^{∗}: G_{0}(R) → G_{0}(R/sR) sending [M] to [M/Ms] − [ann_{M}(s)], where ann_{M}(s) = {x ∈ M : xs = 0}. Theorem 6.5 gives an example where π^{∗} is onto, and if s is nilpotent the map is zero by Devissage 6.3.1. Hint: Use the map M ⟶^{s} M.

6.7 Let Y be a noetherian scheme over the ring ℤ[t], and let X ↪^{π} Y be the closed subscheme defined by t = 0. If F is an O_{Y} -module, let ann_{F}(t) denote the submodule of F annihilated by t. Show that there is a map π^{∗}: G_{0}(Y) → G_{0}(X) sending [F] to [F/tF] − [ann_{F}(t)]. 6.8 (Heller-Reiner) Let R be a commutative domain with field of fractions F. If S = R − {0}, show that there is a well-defined map Δ : F^{×} → K_{0}M_{S}(R) sending the fraction r/s ∈ F^{×} to [R/rR] − [R/sR]. Then use Ex. 6.4 to show that the localization sequence extends to the exact sequence

1 → R^{×} → F^{×} ⟶^{Δ} K_{0}M_{S}(R) → G_{0}(R) → ℤ → 0.

6.9 Weil Divisor Class groups. Let R be a commutative noetherian ring.

(a) Show that each M^{i}(R) is a Serre subcategory of M(R).

(b) Show that K_{0}M^{i−1}/M^{i+1}(R) ≅ CH^{i}(R) ⊕ D^{i−1}(R). In particular, if R is a 1-dimensional domain then G_{0}(R) = ℤ ⊕ CH^{1}(R).

(c) Show that each F^{i}G_{0}(R)/F^{i+1}G_{0}(R) is a quotient of the group CH^{i}(R).

(d) Suppose that R is a domain with field of fractions F. As in Ex. 6.8, show that there is an exact sequence generalizing Proposition I.3.6:

0 → R^{×} → F^{×} ⟶^{Δ} D^{1}(R) → CH^{1}(R) → 0.

In particular, if R is a Krull domain, conclude that CH^{1}(R) ≅ Cl(R) and G_{0}(R)/F^{2} ≅ ℤ ⊕ Cl(R).

(e) If (R, 𝔪) is a 1-dimensional local domain and k_{1}, …, k_{n} are the residue fields of the normalization of R over k = R/𝔪, show that CH^{1}(R) ≅ ℤ/ gcd{[k_{i} : k]}.

6.10 Generalize the preceding exercise to a noetherian scheme X, as indicated in Application 6.4.3. Hint: F becomes the function field of X, and (d) becomes I.5.12. 6.11 If S is a multiplicatively closed set of central elements in a noetherian ring R, show that

K_{0}M_{S}(R) ≅ K_{0}M_{S}(R)[t]) ≅ K_{0}M_{S}(R[t, t^{−1}]).

6.12 Graded modules. When S = R ⊕ S_{1} ⊕ S_{2} ⊕ ⋯ is a noetherian graded ring, let M_{gr}(S) denote the abelian category of finitely generated graded S-modules. Write σ for the shift automorphism M ↦ M(−1) of the category M_{gr}(S). Show that:

(a) K_{0}M_{gr}(S) is a module over the ring ℤ[σ, σ^{−1}]

(b) If S is flat over R, there is a map from the direct sum G_{0}(R)[σ, σ^{−1}] = ⨁_{n∈ℤ} G_{0}(R)σ^{n} to K_{0}M_{gr}(S) sending [M]σ^{n} to [σ^{n}(M ⊗ S)].

(c) If S = R, the map in (b) is an isomorphism: K_{0}M_{gr}(R) ≅ G_{0}(R)[σ, σ^{−1}].

(d) If S = R[x_{1}, ⋯, x_{m}] with x_{1}, ⋯, x_{m} in S_{1}, the map is surjective, i.e., K_{0}M_{gr}(S) is generated by the classes [σ^{n}M[x_{1}, ..., x_{m}]]. We will see in Ex. 7.14 that the map in (b) is an isomorphism for S = R[x_{1}, ⋯, x_{m}].

(e) Let B be the subcategory of M_{gr}(R[x, y]) of modules on which y is nilpotent. Show that B is a Serre subcategory, and that

K_{0}B ≅ K_{0}M_{gr}(R) ≅ G_{0}(R)[σ, σ^{−1}]

6.13 In this exercise we sketch Serre’s proof of the Fundamental Theorem 6.5 when R is a non-commutative ring. We assume the results of the previous exercise. Show that the formula j(M) = M/(y − 1)M defines an exact functor j: M_{gr}(R[x, y]) → M(R[x]), sending B to zero. In fact, j induces an equivalence

M_{gr}(R[x, y])/B ≅ M(R[x]).

Then use this equivalence to show that the map i^{∗}: G_{0}(R) → G_{0}(R[x]) is onto. 6.14 G_{0} of projective space. Let k be a field and set S = k[x_{0}, …, x_{m}], with X = ℙ_{k}^{m}. Using the notation of Exercises 6.3 and 6.12, let M_{gr}^{b}(S) denote the Serre subcategory of M_{gr}(S) consisting of graded modules of finite length. It is well-known (see [Hart, II.5.15]) that every coherent O_{X}-module is of the form M̃ for some M in M_{gr}(S), i.e., that the associated sheaf functor M_{gr}(S) → M(X) is onto, and that if M has finite length then M̃ = 0. In fact, there is an equivalence

M_{gr}(S)/M_{gr}^{b}(S) ≅ M(ℙ_{k}^{m}).

(See [Hart, Ex. II.5.9(c)].) Under this equivalence σ^{i}(S) represents O_{X}(−i).

(a) Let F denote the graded S-module S^{m+1}, whose basis lies in degree 0. Use the Koszul exact sequence of (I.5.4):

0 → σ^{m+1}(⋀^{m+1} F) → ⋯ → σ^{2}(⋀^{2} F) → σ F ⟶^{x_{0},...} S → k → 0

to show that in K_{0}M_{gr}(S) every finitely generated k-module M satisfies

[M] = ∑_{i=0}^{m+1}(−1)^{i}\binom{m + 1}{i}σ^{i} [M ⊗_{k} S] = (1 − σ)^{m+1}[M ⊗_{k} S].

(b) Show that in G_{0}(ℙ_{k}^{m}) every [O_{X}(n)] is a linear combination of the classes [O_{X}], [O_{X}(−1)], ⋯, [O_{X}(−m)], and that

∑_{i=0}^{m+1}(−1)^{i}\binom{m + 1}{i}[O(−i)] = 0 in G_{0} ℙ_{k}^{m} .

(c) We will see in Ex. 7.14 that the map in Ex. 6.12(b) is an isomorphism:

K_{0}M_{gr}(S) ≅ G_{0}(R)[σ, σ^{−1}].

Assume this calculation, and show that

G_{0}(ℙ_{k}^{m}) ≅ ℤ^{m} on generators [O_{X}], [O_{X}(−1)], ⋯, [O_{X}(−m)].

6.15 Naturality of f_{∗}. Suppose that X ⟶^{f} Y ⟶^{g} Z are proper morphisms between noetherian schemes. Show that (gf)_{∗} = g_{∗} f_{∗} as maps G_{0}(X) → G_{0}(Z). 6.16 Let R be a noetherian ring, and r ∈ R. If r is a nonzerodivisor on modules M_{j} whose associated primes all have height i, and 0 = ∑ ±[M_{j}] in K_{0}M^{i}(R), show that 0 = ∑ ±[M_{j}/rM_{j}] in K_{0}M^{i+1}(R). Hint: By Devissage, the formula holds in K_{0}(R/I) for some product I of height i primes. Modify r to be a nonzerodivisor on R/I without changing the M_{j}/rM_{j} and use f_{∗} : G_{0}(R/I) → G_{0}(R/(I + rR)).

§7. K_{0} of an Exact Category

If C is an additive subcategory of an abelian category A, we may still talk about exact sequences: an exact sequence in C is a sequence of objects (and maps) in C which is exact as a sequence in A. With hindsight, we know that it helps to require C to be closed under extensions. Thus we formulate the following definitions.

Definition 7.0 (Exact categories). An exact category is a pair (C, E), where C is an additive category and E is a family of sequences in C of the form

0 → B ⟶^{i} C ⟶^{j} D → 0,  (†)

satisfying the following condition: there is an embedding of C as a full subcategory of an abelian category A so that

(1) E is the class of all sequences (†) in C which are exact in A;

(2) C is closed under extensions in A in the sense that if (†) is an exact sequence

in A with B, D ∈ C then C is isomorphic to an object in C. The sequences in E are called the short exact sequences of C. We will often abuse notation and just say that C is an exact category when the class E is clear. We call a map in C an admissible monomorphism (resp. an admissible epimorphism) if it occurs as the monomorphism i (resp. as the epi j) in some sequence (†) in E.

The following hypothesis is commonly satisfied in applications, and is needed for Euler characteristics and the Resolution Theorem 7.6 below. (7.0.1) We say that C is closed under kernels of surjections in A provided that whenever a map f: B → C in C is a surjection in A then ker(f) ∈ C. The well-read reader will observe that the definition of exact category in [Bass] is what we call an exact category closed under kernels of surjections.

An exact functor F: B → C between exact categories is an additive functor F carrying short exact sequences in B to exact sequences in C. If B is a full subcategory of C, and the exact sequences in B are precisely the sequences (†) in B which are exact in C, we call B an exact subcategory of C. This is consistent with the notion of an exact abelian subcategory in 6.1.5.

Definition 7.1 (K_{0}). Let C be a small exact category. K_{0}(C) is the abelian group having generators [C], one for each object C of C, and relations [C] = [B]+[D] for every short exact sequence 0 → B → C → D → 0 in C.

As in 6.1.1, we have [0] = 0, [B ⊕ D] = [B] + [D] and [B] = [C] if B and C are isomorphic. As before, we could actually define K_{0}(C) when C is only skeletally small, but we shall not dwell on these set-theoretic intricacies. Clearly, K_{0}(C) satisfies the universal property 6.1.2 for additive functions from C to abelian groups.

Example 7.1.1. The category P(R) of finitely generated projective R-modules is exact by virtue of its embedding in mod-R. As every exact sequence of projective modules splits, we have K_{0}P(R) = K_{0}(R).

Any additive category is a symmetric monoidal category under ⊕, and the above remarks show that K_{0}(C) is a quotient of the group K_{0}^{⊕}(C) of §5. Since abelian categories are exact, Examples 6.2.1–4 show that these groups are not identical.

Example 7.1.2 (Split exact categories). A split exact category C is an exact category in which every short exact sequence in E is split (i.e., isomorphic to 0 → B → B ⊕ D → D → 0). In this case we have K_{0}(C) = K_{0}^{⊕}(C) by definition. For example, the category P(R) is split exact.

If X is a topological space, embedding VB(X) in the abelian category of families of vector spaces over X makes VB(X) into an exact category. By the Subbundle Theorem I.4.1, VB(X) is a split exact category, so that K^{0}(X) = K_{0}(VB(X)).

We will see in Exercise 7.7 that any additive category C may be made into a split exact category by equipping it with the class E_{split} of sequences isomorphic to 0 → B → B ⊕ D → D → 0

Warning. Every abelian category A has a natural exact category structure, but it also has the split exact structure. These will yield different K_{0} groups in general, unless something like a Krull-Schmidt Theorem holds in A. We will always use the natural exact structure unless otherwise indicated.

Example 7.1.3 (K_{0} of a scheme). Let X be a scheme (or more generally a ringed space). The category VB(X) of algebraic vector bundles on X, introduced in (I.5), is an exact category by virtue of its being an additive subcategory of the abelian category O_{X}-mod of all O_{X}-modules. If X is quasi-projective over a commutative ring, we write K_{0}(X) for K_{0}VB(X). If X is noetherian, the inclusion VB(X) ⊂ M(X) yields a Cartan homomorphism K_{0}(X) → G_{0}(X). We saw in (I.5.3) that exact sequences in VB(X) do not always split, so VB(X) is not always a split exact category.

Example 7.1.4 (G_{0} of non-noetherian rings). If R is a non-noetherian ring, the category mod_{fg}(R) of all finitely generated R-modules will not be abelian, because R → R/I has no kernel inside this category. However, it is still an exact subcategory of mod-R, so once again we might try to consider the group K_{0}mod_{fg}(R). However, it turns out that this definition does not have good properties (see Ex. 7.3 and 7.4).

Here is a more suitable definition, based upon [SGA6, I.2.9]. An R-module M is called pseudo-coherent if it has an infinite resolution ⋯ → P_{1} → P_{0} → M → 0 by finitely generated projective R-modules. Pseudo-coherent modules are clearly finitely presented, and if R is right noetherian then every finitely generated module is pseudo-coherent. Let M(R) denote the category of all pseudo-coherent R-modules. The “Horseshoe Lemma” [WHomo, 2.2.8] shows that M(R) is closed under extensions in mod-R, so it is an exact category. (It is also closed under kernels of surjections, and cokernels of injections in mod-R, as can be seen using the mapping cone.)

Now we define G_{0}(R) = K_{0}M(R). Note that if R is right noetherian then M(R) is the usual category of §6, and we have recovered the definition of G_{0}(R) in 6.2.

Example 7.1.5. The opposite category C^{op} has an obvious notion of exact sequence: turn the arrows around in the exact sequences of C. Formally, this arises from the inclusion of C^{op} in A^{op}. Clearly K_{0}(C) ≅ K_{0}(C^{op}).

Example 7.1.6. The direct sum C_{1}⊕C_{2} of two exact categories is also exact, the ambient abelian category being A_{1} ⊕ A_{2}. Clearly K_{0}(C_{1} ⊕ C_{2}) ≅ K_{0}(C_{1}) ⊕ K_{0}(C_{2}).

More generally, the direct sum ⨁ C_{i} of exact categories is an exact category (inside the abelian category ⊕A_{i}), and as in 6.1.2 this yields K_{0}(⊕C_{i}) ≅ ⊕K_{0}(C_{i}).

Example 7.1.7 (Filtered colimits). Suppose that {C_{i}} is a filtered family of exact subcategories of a fixed abelian category A. Then C = ∪C_{i} is also an exact subcategory of A, and by inspection of the definition we see that

K_{0}(⋃ C_{i}) = \underrightarrow{lim} K_{0}(C_{i}).

The ambient A is unnecessary: if {C_{i}} is a filtered family of exact categories and exact functors, then K_{0}(\underrightarrow{lim} C_{i}) = \underrightarrow{lim} K_{0}(C_{i}); see Ex. 7.9. As a case in point, if a ring R is the union of subrings R_{α} then P(R) is the direct limit of the P(R_{α}), and we have K_{0}(R) = \underrightarrow{lim} K_{0}(R_{α}), as in 2.1.6.

Cofinality Lemma 7.2. Let B be an exact subcategory of C which is closed under extensions in C, and which is cofinal in the sense that for every C in C there is a C′ in C so that C ⊕ C′ is in B. Then K_{0}B is a subgroup of K_{0}C.

Proof. By (1.3) we know that K_{0}^{⊕}B is a subgroup of K_{0}^{⊕}C. Given a short exact sequence 0 → C_{0} → C_{1} → C_{2} → 0 in C, choose C′_{0} and C′_{2} in C so that B_{0} = C_{0}⊕C′_{0} and B_{2} = C_{2} ⊕ C′_{2} are in B. Setting B_{1} = C_{1} ⊕ C′_{0} ⊕ C′_{2}, we have the short exact sequence 0 → B_{0} → B_{1} → B_{2} → 0 in C. As B is closed under extensions in C, B_{1} ∈ B. Therefore in K_{0}^{⊕}C:

[C_{1}] − [C_{0}] − [C_{2}] = [B_{1}] − [B_{0}] − [B_{2}].

Thus the kernel of K_{0}^{⊕}C → K_{0}C equals the kernel of K_{0}^{⊕}B → K_{0}B, which implies that K_{0}B → K_{0}C is an injection. □

Remark 7.2.1. The proof shows that K_{0}(C)/K_{0}(B) ≅ K_{0}^{⊕}C/K_{0}^{⊕}B, and that every element of K_{0}(C) has the form [C] − [B] for some B in B and C in C.

Idempotent completion.

7.3. A category C is called idempotent complete if every idempotent endomorphism e of an object C factors as C → B → C with the composite B → C → B being the identity. Given C, we can form a new category Ĉ whose objects are pairs (C, e) with e an idempotent endomorphism of an object C of C; a morphism from (C, e) to (C′, e′) is a map f: C → C′ in C such that f = e′fe. The category Ĉ is idempotent complete, since an idempotent endomorphism f of (C, e) factors through the object (C, efe).

Ĉ is called the idempotent completion of C. To see why, consider the natural embedding of C into Ĉ sending C to (C, id). It is easy to see that any functor from C to an idempotent complete category D must factor through a functor Ĉ → D that is unique up to natural equivalence. In particular, if C is idempotent then C ≅ Ĉ.

If C is an additive subcategory of an abelian category A, then Ĉ is equivalent to a larger additive subcategory C′ of A (see Ex. 7.6). Moreover, C is cofinal in Ĉ, because (C, e) is a summand of C in A. By the Cofinality Lemma 7.2, we see that K_{0}(C) is a subgroup of K_{0}(Ĉ).

Example 7.3.1. Consider the full subcategory Free(R) of M(R) consisting of finitely generated free R-modules. The idempotent completion of Free(R) is the category P(R) of finitely generated projective modules. Thus the cyclic group K_{0}Free(R) is a subgroup of K_{0}(R). If R satisfies the Invariant Basis Property (IBP), then K_{0}Free(R) ≅ ℤ and we have recovered the conclusion of Lemma 2.1.

Example 7.3.2. Let R → S be a ring homomorphism, and let B denote the full subcategory of P(S) on the modules of the form P ⊗_{R} S for P in P(R). Since it contains all the free modules S^{n}, B is cofinal in P(S), so K_{0}B is a subgroup of K_{0}(S). Indeed, K_{0}B is the image of the natural map K_{0}(R) → K_{0}(S).

Products

Let A, B and C be exact categories. A functor F : A × B → C is called biexact if F (A, −) and F (−, B) are exact functors for every A in A and B in B, and F (0, −) = F (−, 0) = 0. (The last condition, not needed in this chapter, can always be arranged by replacing C by an equivalent category.) The following result is completely elementary.

Lemma 7.4. A biexact functor F: A × B → C induces a bilinear map

K_{0}A ⊗ K_{0}B → K_{0}C.
[A] ⊗ [B] ↦ [F (A, B)]

Application 7.4.1. Let R be a commutative ring. The tensor product ⊗_{A} defines a biexact functor P(R) × P(R) → P(R), as well as a biexact functor P(R) × M(R) → M(R). The former defines the product [P][Q] = [P ⊗ Q] in the commutative ring K_{0}(R), as we saw in §2. The latter defines an action of K_{0}(R) on G_{0}(R), making G_{0}(R) into a K_{0}(R)-module.

Application 7.4.2. Let X be a scheme (or more generally a locally ringed space) The tensor product of vector bundles defines a biexact functor VB(X) × VB(X) → VB(X) (see I.5.3). This defines a product on K_{0}(X) satisfying [E][F] = [E ⊗ F]. This product is clearly commutative and associative, so it makes K_{0}(X) into a commutative ring. We will discuss this ring further in the next section.

If X is noetherian, recall from 6.2.5 that G_{0}(X) denotes K_{0}M(X). Since the tensor product of a vector bundle and a coherent module is coherent, we have a functor VB(X) × M(X) → M(X). It is biexact (why?), so it defines an action of K_{0}(X) on G_{0}(X), making G_{0}(X) into a K_{0}(X)-module.

Application 7.4.3 (Almkvist). If R is a ring, let End(R) denote the exact category whose objects (P, α) are pairs, where P is a finitely generated projective R-module and α is an endomorphism of P. A morphism (P, α) → (Q, β) in End(R) is a morphism f: P → Q in P(R) such that fα = βf, and exactness in End(R) is determined by exactness in P(R).

If R is commutative, the tensor product of modules gives a biexact functor

⊗_{R} : End(R) × End(R) → End(R),
((P, α), (Q, β)) ↦ (P ⊗_{R} Q, α ⊗_{R} β).

As ⊗_{R} is associative and symmetric up to isomorphism, the induced product makes K_{0}End(R) into a commutative ring with unit [(R, 1)]. The inclusion of P(R) in End(R) by α = 0 is split by the forgetful functor, and the kernel End_{0}(R) of K_{0}End(R) → K_{0}(R) is not only an ideal but a commutative ring with unit 1 = [(R, 1)] − [(R, 0)]. Almkvist proved that (P, α) ↦ det(1 − αt) defines an isomorphism of End_{0}(R) with the subgroup of the multiplicative group W (R) = 1 + tR[[t]] consisting of all quotients f(t)/g(t) of polynomials in 1 + tR[t] (see Ex. 7.18). Almkvist also proved that End_{0}(R) is a subring of W (R) under the ring structure of 4.3.

If A is an R-algebra, ⊗_{R} is also a pairing End(R)×End(A) → End(A), making End_{0}(A) into an End_{0}(R)-module. We leave the routine details to the reader.

Example 7.4.4. If R is a ring, let Nil(R) denote the category whose objects (P, ν) are pairs, where P is a finitely generated projective R-module and ν is a nilpotent endomorphism of P. This is an exact subcategory of End(R). The forgetful functor Nil(R) → P(R) sending (P, ν) to P is exact, and is split by the exact functor P(R) → Nil(R) sending P to (P, 0). Therefore K_{0}(R) = K_{0}P(R) is a direct summand of K_{0}Nil(R). We write Nil_{0}(R) for the kernel of K_{0}Nil(R) → P(R), so that there is a direct sum decomposition K_{0}Nil(R) = K_{0}(R) ⊕ Nil_{0}(R). Since [P, ν] = [P ⊕ Q, ν ⊕ 0] − [Q, 0] in K_{0}Nil(R), we see that Nil_{0}(R) is generated by elements of the form [(R^{n}, ν)] − n[(R, 0)] for some n and some nilpotent matrix ν.

If A is an R-algebra, then the tensor product pairing on End restricts to a biexact functor F: End(R) × Nil(A) → Nil(A). The resulting bilinear map K_{0}End(R) × K_{0}Nil(A) → K_{0}Nil(A) is associative, and makes Nil_{0}(A) into a module over the ring End_{0}(R), and makes Nil_{0}(A) → End_{0}(A) an End_{0}(R)-module map.

Any additive functor T : P(A) → P(B) induces an exact functor Nil(A) → Nil(B) and a homomorphism Nil_{0}(A) → Nil_{0}(B). If A and B are R-algebras and T is R-linear, Nil_{0}(A) → Nil_{0}(B) is an End_{0}(R)-module homomorphism. (Exercise!)

Example 7.4.5. If R is a commutative regular ring, and A = R[x]/(x^{N}), we will see in III.3.8.1 that Nil_{0}(A) → End_{0}(A) is an injection, identifying Nil_{0}(A) with the ideal (1 + xtA[t])^{×} of End_{0}(A), and identifying [(A, x)] with 1 − xt.

This isomorphism End_{0}(A) ≅ (1 + xtA[t])^{×} is universal in the following sense. If B is an R-algebra and (P, ν) is in Nil(B), with ν^{N} = 0, we may regard P as an A–B bimodule. By 2.8, this yields an R-linear functor Nil_{0}(A) → Nil_{0}(B) sending (A, x) to (P, ν). By 7.4.4, there is an End_{0}(R)-module homomorphism (1 + xtA[t])^{×} → Nil_{0}(B) sending 1 − xt to [(P, ν)].

Euler characteristics can be useful in exact categories as well as in abelian categories, as the following analogue of Proposition 6.6 shows.

Proposition 7.5. Suppose that C is closed under kernels of surjections in an abelian category A. If C_{•} is a bounded chain complex in C whose homology H_{i}(C_{•}) is also in C then in K_{0}(C):

χ(C_{•}) = ∑(−1)^{i}[C_{i}] equals ∑(−1)^{i}[H_{i}(C_{•})].

In particular, if C_{•} is any exact sequence in C then χ(C_{•}) = 0.

Proof. The proof we gave in 6.6 for abelian categories will go through, provided that the Z_{i} and B_{i} are objects of C. Consider the exact sequences:

0 → Z_{i} →C_{i} → B_{i} → 0
0 → B_{i} →Z_{i} → H_{i}(C_{•}) → 0.

Since B_{i} = 0 for i << 0, the following inductive argument shows that all the B_{i} and Z_{i} belong to C. If B_{i−1} ∈ C then the first sequence shows that Z_{i} ∈ C; since H_{i}(C_{•}) is in C, the second sequence shows that B_{i} ∈ C. □

Corollary 7.5.1. Suppose C is closed under kernels of surjections in A. If f: C′_{•} → C_{•} is a morphism of bounded complexes in C, inducing an isomorphism on homology, then

χ(C′_{•}) = χ(C_{•}).

Proof. Form cone(f), the mapping cone of f, which has C_{n} ⊕ C′_{n−1} in degree n. By inspection, χ(cone(f)) = χ(C_{•}) − χ(C′_{•}). But cone(f) is an exact complex because f is a homology isomorphism, so χ(cone(f)) = 0. □

The Resolution Theorem

We need a definition in order to state our next result. Suppose that P is an additive subcategory of an abelian category A. A P-resolution P_{•} → C of an object C of A is an exact sequence in A

⋯ → P_{n} → ⋯ → P_{1} → P_{0} → C → 0

in which all the P_{i} are in P. The P-dimension of C is the minimum n (if it exists) such that there is a resolution P_{•} → C with P_{i} = 0 for i > n.

Resolution Theorem 7.6. Let P ⊂ C ⊂ A be an inclusion of additive categories with A abelian (A gives the notion of exact sequence to P and C). Assume that:

(a) Every object C of C has finite P-dimension; and

(b) C is closed under kernels of surjections in A. Then the inclusion P ⊂ C induces an isomorphism K_{0}(P) ≅ K_{0}(C).

Proof. To see that K_{0}(P) maps onto K_{0}(C), observe that if P_{•} → C is a finite P-resolution, then the exact sequence

0 → P_{n} → ⋯ → P_{0} → C → 0

has χ = 0 by 7.5, so [C] = ∑(−1)^{i}[P] = χ(P_{•}) in K_{0}(C). To see that K_{0}(P) ≅ K_{0}(C), we will show that the formula χ(C) = χ(P_{•}) defines an additive function from C to K_{0}(P). For this, we need the following lemma, due to Grothendieck.

Lemma 7.6.1. Given a map f: C′ → C in C and a finite P-resolution P_{•} → C, there is a finite P-resolution P ′_{•} → C′ and a commutative diagram

0 ⟶ P′_{m} ⟶ ⋯ ⟶ P′_{n} ⟶ ⋯ ⟶ P′_{1} ⟶ P′_{0} ⟶ C′ ⟶ 0
↓  ↓  ↓  ^{f}↓
0 ⟶ P_{n} ⟶ ⋯ ⟶ P_{1} ⟶ P_{0} ⟶ C ⟶ 0

We will prove this lemma in a moment. First we shall use it to finish the proof of Theorem 7.6. Suppose given two finite P-resolutions P_{•} → C and P ′_{•} → C of an object C. Applying the lemma to the diagonal map C → C⊕C and P_{•}⊕P ′_{•} → C⊕C, we get a P-resolution P ′′_{•} → C and a map P ′′_{•} → P_{•} ⊕ P ′_{•} of complexes. Since the maps P_{•} ← P ′′_{•} → P ′_{•} are quasi-isomorphisms, Corollary 7.5.1 implies that χ(P_{•}) = χ(P ′′_{•}) = χ(P ′_{•}). Hence χ(C) = χ(P_{•}) is independent of the choice of P-resolution.

Given a short exact sequence 0 → C′ → C → C′′ → 0 in C and a P-resolution P_{•} → C, the lemma provides a P-resolution P ′_{•} → C′ and a map f: P ′_{•} → P_{•}. Form the mapping cone complex cone(f), which has P_{n} ⊕P′_{n}[−1] in degree n, and observe that χ(cone(f)) = χ(P_{•}) − χ(P ′_{•}). The homology exact sequence

H_{i}(P′) → H_{i}(P) → H_{i}(cone(f)) → H_{i−1}(P′) → H_{i−1}(P)

shows that H_{i}cone(f) = 0 for i ≠ 0, and H_{0}(cone(f)) = C′′. Thus cone(f) → C′′ is a finite P-resolution, and so

χ(C′′) = χ(cone(f)) = χ(P_{•}) − χ(P ′_{•}) = χ(C) − χ(C′).

This proves that χ is an additive function, so it induces a map χ: K_{0}C → K_{0}(P). If P is in P then evidently χ(P) = [P], so χ is the inverse isomorphism to the map K_{0}(P) → K_{0}(C). This finishes the proof of the Resolution Theorem 7.6. □

Proof of Lemma 7.6.1. We proceed by induction on the length n of P_{•}. If n = 0, we may choose any P-resolution of C′; the only nonzero map P′_{n} → P_{n} is P′_{0} → C′ → C ≅ P_{0}. If n ≥ 1, let Z denote the kernel (in A) of ε: P_{0} → C and let B denote the kernel (in A) of (ε, −f): P_{0} ⊕ C′ → C. As C is closed under kernels, both Z and B are in C. Moreover, the sequence

0 → Z → B → C′ → 0

is exact in C (because it is exact in A). Choose a surjection P′_{0} → B with P′_{0} in P, let f_{0} be the composition P′_{0} → B → P_{0} and let Y denote the kernel of the surjection P′_{0} → B → C′. By induction applied to the induced map Y → Z, we can find a P-resolution P ′_{•}[+1] of Y and maps f_{i}: P′_{i} → P_{i} making the following diagram commute (the rows are not exact at Y and Z):

⋯ ⟶ P′_{2} ⟶ P′_{1} ⟶ Y ⟶^{monic} P′_{0} ⟶ C′ ⟶ 0
↓  ↓  ↓  ↓^{f_{0}}  ↓^{f}
⋯ ⟶ P_{2} ⟶ P_{1} ⟶ Z ⟶^{monic} P_{0} ⟶ C ⟶ 0

Splicing the rows by deleting Y and Z yields the desired P-resolution of C′. □

Definition 7.7 (H(R)). Given a ring R, let H(R) denote the category of all R-modules M having a finite resolution by finitely generated projective modules, and let H_{n}(R) denote the subcategory in which the resolutions have length ≤ n.

By the Horseshoe Lemma [WHomo, 2.2.8], both H(R) and H_{n}(R) are exact subcategories of mod-R. The following Lemma shows that they are also closed under kernels of surjections in mod-R.

Lemma 7.7.1. If 0 → L → M ⟶^{f} N → 0 is a short exact sequence of modules, with M in H_{m}(R) and N in H_{n}(R), then L is in H_{ℓ}(R), where ℓ = min{m, n − 1}.

Proof. If P_{•} → M and Q_{•} → N are projective resolutions, and P_{•} → Q_{•} lifts f, then the kernel P′_{0} of the surjection P_{0} ⊕ Q_{1} → Q_{0} is finitely generated projective, and the truncated mapping cone ⋯ → P_{1} ⊕ Q_{2} → P′_{0} is a resolution of L. □

Corollary 7.7.2. K_{0}(R) ≅ K_{0}H(R) ≅ K_{0}H_{n}(R) for all n ≥ 1.

Proof. Apply the Resolution Theorem to P(R) ⊂ H(R). □

Here is a useful variant of the above construction. Let S be a multiplicatively closed set of central nonzerodivisors in a ring R. We say a module M is S-torsion if Ms = 0 for some s ∈ S (cf. Example 6.2.8), and write H_{S}(R) for the exact subcategory H(R) ∩ M_{S}(R) of S-torsion modules M in H(R). Similarly, we write H_{n,S}(R) for the S-torsion modules in H_{n}(R). Note that H_{0,S}(R) = 0, and that the modules R/sR belong to H_{1,S}(R).

Corollary 7.7.3. K_{0}H_{S}(R) ≅ K_{0}H_{n,S}(R) ≅ K_{0}H_{1,S}(R) for all n ≥ 1.

Proof. We apply the Resolution Theorem with P = H_{1,S}(R). By Lemma 7.7.1, each H_{n,S}(R) is closed under kernels of surjections. Every N in H_{n,S}(R) is finitely generated, so if Ns = 0 there is an exact sequence 0 → L → (R/sR)^{m} → N → 0. If n ≥ 2 then L is in H_{n−1,S}(R) by Lemma 7.7.1. By induction, L and hence N has a P-resolution. □

Corollary 7.7.4. If S is a multiplicatively closed set of central nonzerodivisors in a ring R, the sequence K_{0}H_{S}(R) → K_{0}(R) → K_{0}(S^{−1}R) is exact.

Proof. If [P] − [R^{n}] ∈ K_{0}(R) vanishes in K_{0}(S^{−1}R), S^{−1}P is stably free (Cor. 1.3). Hence there is an isomorphism (S^{−1}R)^{m+n} → S^{−1}P ⊕(S^{−1}R)^{m}. Clearing denominators yields a map f: R^{m+n} → P ⊕ R^{m} whose kernel and cokernel are S-torsion. But ker(f) = 0 because S consists of nonzerodivisors, and therefore M = coker(f) is in H_{1,S}(R). But the map K_{0}H_{S}(R) → K_{0}H(R) = K_{0}(R) sends [M] to [M] = [P] − [R^{n}]. □

Let R be a regular noetherian ring. Since every module has finite projective dimension, H(R) is the abelian category M(R) discussed in §6. Combining Corollary 7.7.2 with the Fundamental Theorem for G_{0} 6.5, we have:

Fundamental Theorem for K_{0} of regular rings 7.8. If R is a regular noetherian ring, then K_{0}(R) ≅ G_{0}(R). Moreover,

K_{0}(R) ≅ K_{0}(R[t]) ≅ K_{0}(R[t, t^{−1}]).

If R is not regular, we can still use the localization sequence 7.7.4 to get a partial result, which will be considerably strengthened by the Fundamental Theorem for K_{0} in chapter III.

Proposition 7.8.1. K_{0}(R[t]) → K_{0}(R[t, t^{−1}]) is injective for every ring R.

To prove this, we need the following lemma. Recall from Example 7.4.4 that Nil(R) is the category of pairs (P, ν) with ν a nilpotent endomorphism of P ∈ P(R).

Lemma 7.8.2. Let S be the multiplicative set {t^{n}} in the polynomial ring R[t]. Then Nil(R) is equivalent to the category H_{1,S}(R[t]) of t-torsion R[t]-modules M in H_{1}(R[t]).

Proof. If (P, ν) is in Nil(R), let P_{ν} denote the R[t]-module P on which t acts as ν. It is a t-torsion module because t^{n}P_{ν} = ν^{n}P = 0 for large n. A projective resolution of P_{ν} is given by the “characteristic sequence” of ν:

(7.8.3)  0 → P [t] ⟶^{t−ν} P [t] → P_{ν} → 0,

Thus P_{ν} is an object of H_{1,S}(R[t]). Conversely, each M in H_{1,S}(R[t]) has a projective resolution 0 → P → Q → M → 0 by finitely generated projective R[t]-modules, and M is killed by some power t^{n} of t. From the exact sequence

0 → Tor_{1}^{R[t]}(M, R[t]/(t^{n})) → P/t^{n}P → Q/t^{n}Q → M → 0

and the identification of the first term with M we obtain the exact sequence 0 → M ⟶^{t^{n}} P/t^{n}P → P/t^{n}Q → 0. Since P/t^{n}P is a projective R-module and pd_{R}(P/t^{n}Q) ≤ 1, we see that M must be a projective R-module. Thus (M, t) is an object of Nil(R). □

Combining Lemma 7.8.2 with Corollary 7.7.3 yields:

Corollary 7.8.4. K_{0}Nil(R) ≅ K_{0}H_{S}(R[t]).

Proof of Proposition 7.8.1. By Corollaries 7.7.4 and 7.8.4, we have an exact sequence

K_{0}Nil(R) ⟶ K_{0}(R[t]) → K_{0}(R[t, t^{−1}]).

The result will follow once we calculate that the left map is zero. This map is induced by the forgetful functor Nil(R) → H(R[t]) sending (P, ν) to P. Since the characteristic sequence (7.8.3) of ν shows that [P] = 0 in K_{0}(R[t]), we are done. □

Base change and Transfer Maps for Rings

7.9. Let f: R → S be a ring homomorphism. We have already seen that the base change ⊗_{R}S: P(R) → P(S) is an exact functor, inducing f^{∗}: K_{0}(R) → K_{0}(S). If S ∈ P(R), we observed in (2.8.1) that the forgetful functor P(S) → P(R) is exact, inducing the transfer map f_{∗}: K_{0}(S) → K_{0}(R).

Using the Resolution Theorem, we can also define a transfer map f_{∗} if S ∈ H(R). In this case every finitely generated projective S-module is in H(R), because if P ⊕ Q = S^{n} then pd(P) ≤ pd(S^{n}) = pd(S) < ∞. Hence there is an (exact) forgetful functor P(S) → H(R), and we define the transfer map to be the induced map

f_{∗}: K_{0}(S) = K_{0}P(S) → K_{0}H(R) ≅ K_{0}(R).  (7.9.1)

A similar trick works to construct base change maps for the groups G_{0}. We saw in 6.2 that if S is flat as an R-module then ⊗_{R}S is an exact functor M(R) → M(S) and we obtained a map f^{∗}: G_{0}(R) → G_{0}(S). More generally, suppose that S has finite flat dimension fd_{R}(S) = n as a left R-module, i.e., that there is an exact sequence

0 → F_{n} → ⋯ → F_{1} → F_{0} → S → 0

of R-modules, with the F_{i} flat. Let F denote the full subcategory of M(R) consisting of all finitely generated R-modules M with Tor_{i}^{R}(M, S) = 0 for i ≠ 0; F is an exact category concocted so that ⊗_{R}S defines an exact functor from F to M(S). Not only does F contain P(R), but from homological algebra one knows that (if R is noetherian) every finitely generated R-module has a finite resolution by objects in F: for any projective resolution P_{•} → M the kernel of P_{n} → P_{n−1} (the n^{th} syzygy) of any projective resolution will be in F. The long exact Tor sequence shows that F is closed under kernels, so the Resolution Theorem applies to yield K_{0}(F) ≅ K_{0}(M(R)) = G_{0}(R). Therefore if R is noetherian and fd_{R}(S) < ∞ we can define the base change map f^{∗}: G_{0}(R) → G_{0}(S) as the composite

G_{0}(R) ≅ K_{0}(F) ⟶^{⊗} K_{0}M(S) = G_{0}(S).  (7.9.2)

The following formula for f^{∗} was used in §6 to show that G_{0}(R) ≅ G_{0}(R[x]).

Serre’s Formula 7.9.3. Let f: R → S be a map between noetherian rings with fd_{R}(S) < ∞. Then the base change map f^{∗}: G_{0}(R) → G_{0}(S) of (7.9.2) satisfies:

f^{∗}([M]) = ∑(−1)^{i} [Tor_{i}^{R}(M, S)] .

Proof. Choose an F-resolution L_{•} → M (by R-modules L_{i} in F):

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

From homological algebra, we know that Tor_{i}^{R}(M, S) is the i^{th} homology of the chain complex L_{•} ⊗_{R} S. By Prop. 7.5, the right-hand side of (7.9.3) equals

χ(L_{•} ⊗_{R} S) = ∑(−1)^{i}[L_{i} ⊗_{R} S] = f^{∗}(∑(−1)^{i}[L_{i}]) = f^{∗}([M]).

EXERCISES

7.1 Suppose that P is an exact subcategory of an abelian category A, closed under kernels of surjections in A. Suppose further that every object of A is a quotient of an object of P (as in Corollary 7.7.2). Let P_{n} ⊂ A be the full subcategory of objects having P-dimension ≤ n. Show that each P_{n} is an exact category closed under kernels of surjections, so that by the Resolution Theorem K_{0}(P) ≅ K_{0}(P_{n}). Hint. If 0 → L → P → M → 0 is exact with P ∈ P and M ∈ P_{1}, show that L ∈ P. 7.2 Let A be a small exact category. If [A_{1}] = [A_{2}] in K_{0}(A), show that there are short exact sequences in A

0 → C′ → C_{1} → C′′ → 0, 0 → C′ → C_{2} → C′′ → 0

such that A_{1} ⊕ C_{1} ≅ A_{2} ⊕ C_{2}. (Cf. Ex. 6.4.) 7.3 This exercise shows why the noetherian hypothesis was needed for G_{0} in Corollary 6.3.1, and motivates the definition of G_{0}(R) in 7.1.4. Let R be the ring k ⊕ I, where I is an infinite-dimensional vector space over a field k, with multiplication given by I^{2} = 0.

(a) (Swan) Show that K_{0}mod_{fg}(R) = 0 but K_{0}mod_{fg}(R/I) = G_{0}(R/I) = ℤ.

(b) Show that every pseudo-coherent R-module is isomorphic to R^{n} for some n. Conclude that G_{0}(R) = ℤ.

7.4 The groups G_{0}(ℤ[G]) and K_{0}mod_{fg}(ℤ[G]) are very different for the free group G on two generators x and y. Let I be the two-sided ideal of ℤ[G] generated by y, so that ℤ[G]/I = ℤ[x, x^{−1}]. As a right module, ℤ[G]/I is not finitely presented.

(a) (Lück) Construct resolutions 0 → ℤ[G]^{2} → ℤ[G] → ℤ → 0 and 0 → ℤ[G]/I → ℤ[G]/I → ℤ → 0, and conclude that K_{0}mod_{fg}(ℤ[G]) = 0

(b) Gersten proved in [Ger74] that K_{0}(ℤ[G]) = ℤ by showing that every finitely presented ℤ[G]-module is in H(ℤ[G]), i.e., has a finite resolution by finitely generated projective modules. Show that G_{0}(ℤ[G]) ≅ K_{0}(ℤ[G]) ≅ ℤ.

7.5 Naturality of base change. Let R ⟶^{f} S ⟶^{g} T be maps between noetherian rings, with fd_{R}(S) and fd_{S}(T) finite. Show that g^{∗}f^{∗} = (gf)^{∗} as maps G_{0}(R) → G_{0}(T). 7.6 Idempotent completion. Suppose that (C, E) is an exact category. Show that there is a natural way to make the idempotent completion Ĉ of C into an exact category, with C an exact subcategory. As noted in 7.3, this proves that K_{0}(C) is a subgroup of K_{0}(Ĉ).

7.7 Let C be a small additive category, and A = Ab^{C^{op}} the (abelian) category of all additive contravariant functors from C to Ab. The Yoneda embedding h: C → A, defined by h(C) = Hom_{C}(−, C), embeds C as a full subcategory of A. Show that every object of C is a projective object in A. Then conclude that this embedding makes C into a split exact category (see 7.1.2). 7.8 (Quillen). Let C be an exact category, with the family E of short exact sequences (and admissible monics i and admissible epis j)

0 → B ⟶^{i} C ⟶^{j} D → 0  (†)

as in Definition 7.0. Show that the following three conditions hold:

(1) Any sequence in C isomorphic to a sequence in E is in E. If (†) is a sequence in E then i is a kernel for j (resp. j is a cokernel for i) in C. The class E contains all of the sequences

0 → B ⟶^{(1,0)} B ⊕ D ⟶^{(0,1)} D → 0.

(2) The class of admissible epimorphisms (resp. monomorphisms) is closed under composition. If (†) is in E and B → B′′, D′ → D are maps in C then the base change sequence 0 → B → (C ×_{D} D′) → D′ → 0 and the cobase change sequence 0 → B′′ → (B′′ ⨿_{B} C) → D → 0 are in E.

(3) If C → D is a map in C possessing a kernel, and there is a map C′ → C in C so that C′ → D is an admissible epimorphism, then C → D is an admissible epimorphism. Dually, if B → C has a cokernel and some B → C → C′′ is admissible monomorphism, then so is B → C.

Keller [Ke90, App. A] has proven that (1) and (2) imply (3).

Quillen observed that a converse is true: let C be an additive category, equipped with a family E of sequences of the form (†). If conditions (1) and (2) hold, then C is an exact category in the sense of definition 7.0. The ambient abelian category used in 7.0 is the category L of contravariant left exact functors: additive functors F: C → Ab which carry each (†) to a “left” exact sequence

0 → F (D) → F (C) → F (B),

and the embedding C ⊂ L is the Yoneda embedding.

We refer the reader to Appendix A of [TT] for a detailed proof that E is the class of sequences in C which are exact in L, as well as the following useful result: If C is idempotent complete then it is closed under kernels of surjections in L. 7.9 Let {C_{i}} be a filtered system of exact categories and exact functors. Use Ex. 7.8 to generalize Example 7.1.7, showing that C = \underrightarrow{lim} C_{i} is an exact category and that K_{0}(C) = \underrightarrow{lim} K_{0}(C_{i}). 7.10 Projection Formula for rings. Suppose that R is a commutative ring, and A is an R-algebra which as an R-module is in H(R). By Ex. 2.1, ⊗_{R} makes K_{0}(A) into a K_{0}(R)-module. Generalize Ex. 2.2 to show that the transfer map f_{∗}: K_{0}(A) → K_{0}(R) is a K_{0}(R)-module map, i.e., that the projection formula holds:

f_{∗}(x · f^{∗}y) = f_{∗}(x) · y for every x ∈ K_{0}(A), y ∈ K_{0}(R).

7.11 For a localization f : R → S^{−1}R at a central set of nonzerodivisors, every α : S^{−1}P → S^{−1}Q has the form α = γ/s for some γ ∈ Hom_{R}(P, Q) and s ∈ S. Show that [(P, γ/s, Q)] ↦ [Q/γ(P)]−[Q/sQ] defines an isomorphism K_{0}(f) → K_{0}H_{S}(R) identifying the sequences (2.10.2) and 7.7.4. 7.12 This exercise generalizes the Localization Theorem 6.4. Let C be an exact subcategory of an abelian category A, closed under extensions and kernels of surjections, and suppose that C contains a Serre subcategory B of A. Let C/B denote the full subcategory of A/B on the objects of C. Considering B-isos A → C with C in C, show that the following sequence is exact:

K_{0}(B) → K_{0}(C) \underrightarrow{loc} K_{0}(C/B) → 0.

7.13 δ-functors. Let T = {T_{i}: C → A, i ≥ 0} be a bounded homological δ-functor from an exact category C to an abelian category A, i.e., for every exact sequence (†) in C we have a finite long exact sequence in A:

0 → T_{n}(B) → T_{n}(C) → ⋯ → T_{1}(D) ⟶^{δ} T_{0}(B) → T_{0}(C) → T_{0}(D) → 0.

Let F denote the category of all C in C such that T_{i}(C) = 0 for all i > 0, and assume that every C in C is a quotient of some object of F.

(a) Show that K_{0}(F) ≅ K_{0}(C), and that T defines a map K_{0}(C) → K_{0}(A) sending [C] to ∑(−1)^{i}[T_{i}C]. (Cf. Ex. 6.6 and 8.4 below.)

(b) Suppose that f: X → Y is a map of noetherian schemes, and that O_{X} has finite flat dimension over f^{−1}O_{Y}. Show that there is a base change map f^{∗}: G_{0}(Y) → G_{0}(X) satisfying f^{∗}g^{∗} = (gf)^{∗}, generalizing (7.9.2) and Ex. 7.5.

7.14 This exercise is a refined version of Ex. 6.12. Consider S = R[x_{0}, ⋯, x_{m}] as a graded ring with x_{1}, ⋯, x_{n} in S_{1}, and let M_{gr}(S) denote the exact category of finitely generated graded S-modules.

(a) Use Ex. 7.13 with T_{i} = Tor_{i}^{S}(−, R) to show that K_{0}M_{gr}(S) ≅ G_{0}(R)[σ, σ^{−1}].

(b) Use (a) and Ex. 6.12(e) to obtain an exact sequence

G_{0}(R)[σ, σ^{−1}] ⟶^{i} G_{0}(R)[σ, σ^{−1}] → G_{0}(R[x]) → 0.

Then show that the map i sends α to α − σα.

(c) Conclude that G_{0}(R) ≅ G_{0}(R[x]). 7.15 Let R be a noetherian ring. Show that the groups K_{0}M_{i}(R) of Application 6.4.3 are all K_{0}(R)-modules, and that the subgroups F^{i} in the coniveau filtration of G_{0}(R) are K_{0}(R)-submodules. Conclude that if R is regular then the F^{i} are ideals in the ring K_{0}(R). 7.16 (Grayson) Show that the operations λ^{n}(P, α) = (∧^{n}P, ∧^{n}α) make K_{0}End(R) and End_{0}(R) into λ-rings. Then show that the ring map End_{0}(R) → W (R) (of 7.4.3) is a λ-ring injection, where W (R) is the ring of big Witt vectors of R (see Example 4.3). Conclude that End_{0}(R) is a special λ-ring (4.3.1).

The exact endofunctors F_{m} : (P, α) ↦ (P, α^{m}) and V_{m} : (P, α) ↦ (P [t]/t^{m}−α, t) on End(R) induce operators F_{m} and V_{m} on End_{0}(R). Show that they agree with the classical Frobenius and Verschiebung operators, respectively. 7.17 This exercise is a refinement of 7.4.4. Let F_{n}Nil(R) denote the full subcategory of Nil(R) on the (P, ν) with ν^{n} = 0. Show that F_{n}Nil(R) is an exact subcategory of Nil(R). If R is an algebra over a commutative ring k, show that the kernel F_{n} Nil_{0}(R) of K_{0}F_{n}Nil(R) → K_{0}P(R) is an End_{0}(k)-module, and F_{n} Nil_{0}(R) → Nil_{0}(R) is a module map.

The exact endofunctor F_{m} : (P, ν) ↦ (P, ν^{m}) on Nil(R) is zero on F_{n}Nil(R). For α ∈ End_{0}(k) and (P, ν) ∈ Nil_{0}(R), show that (V_{m}α) · (P, ν) = V_{m}(α · F_{m}(P, ν)), and conclude that V_{m} End_{0}(k) acts trivially on the image of F_{m} Nil_{0}(A) in Nil_{0}(A). 7.18 Let α_{n} = α_{n}(a_{1}, ..., a_{n}) denote the n × n matrix over a commutative ring R:

α_{n}(a_{1}, ..., a_{n}) = ( \begin{matrix}0 & & & −a_{n} \\ 1 & 0 & & −a_{n−1} \\ & ... & ... & ⋮ \\ & & 1 & −a_{1} \end{matrix} ) .

(a) Show that [(R^{n}, α_{n})] = 1 + a_{1}t + ⋯ + a_{n}t^{n} in W (R). Conclude that the image of the map End_{0}(R) → W (R) in 7.4.3 is indeed the subgroup of all quotients f(t)/g(t) of polynomials in 1 + tR[t].

(b) Let A be an R-algebra. Recall that (R^{n}, α_{n}) ∗ [(P, ν)] = [(P^{n}, α_{n}ν)] in the End_{0}(R)-module Nil_{0}(A) (see 7.4.4). Show that (R^{n+1}, α_{n+1}(a_{1}, …, a_{n}, 0)) ∗ [(P, ν)] = (R^{n}, α_{n}) ∗ [(P, ν)].

(c) Use 7.4.5 with R = ℤ[a_{1}, ..., a_{n}] to show that (R^{n}, α_{n}) ∗ [(P, ν)] = [(P^{n}, β)], β = α_{n}(a_{1}ν, ..., a_{n}ν^{n}). If ν^{N} = 0, this is clearly independent of the a_{i} for i ≥ N.

(d) Conclude that the End_{0}(R)-module structure on Nil_{0}(A) extends to a W (R)- module structure by the formula

(1 + ∑ a_{i}t^{i}) ∗ [(P, ν)] = (R^{n}, α_{n}(a_{1}, ..., a_{n})) ∗ [(P, ν)], n ≫ 0.

7.19 (Lam) If R is a commutative ring, and Λ is an R-algebra, we write G_{0}^{R}(Λ) for K_{0}Rep_{R}(Λ), where Rep_{R}(Λ) denotes the full subcategory of mod-Λ consisting of modules M which are finitely generated and projective as R-modules. If Λ = R[G] is the group ring of a group G, the tensor product M ⊗_{R} N of two R[G]-modules is again an R[G]-module where g ∈ G acts by (m ⊗ n)g = mg ⊗ ng. Show that:

(a) ⊗_{R} makes G_{0}^{R}(R[G]) an associative, commutative ring with identity [R].

(b) G_{0}^{R}(R[G]) is an algebra over the ring K_{0}(R), and K_{0}(R[G]) is a G_{0}^{R}(R[G])- module.

(c) If R is a regular ring and Λ is finitely generated projective as an R-module, G_{0}^{R}(Λ) ≅ G_{0}(Λ).

(d) If R is regular and G is finite, then G_{0}(R[G]) is a commutative K_{0}(R)-algebra, and that K_{0}(R[G]) is a module over G_{0}(R[G]).

7.20 (Deligne) A filtered object in an abelian category A is an object A together with a finite filtration ⋯ ⊆ W_{n}A ⊆ W_{n+1}A ⊆ ⋯; if A and B are filtered, a morphism f : A → B in A is filtered if F (W_{n}A) ⊆ W_{n}B for all n. The category Afilt of filtered objects in A is additive but not abelian (because images and coimages can differ). Let E denote the family of all sequences 0 → A → B → C → 0 in Afilt such that each sequence 0 → gr_{n}^{W} A → gr_{n}^{W} B → gr_{n}^{W} C → 0 is exact in A.

(a) Show that (Afilt, E) is an exact category. (See [BBD, 1.1.4].)

(b) Show that K_{0}(Afilt) ≅ ℤ × K_{0}(A).

7.21 Replete exact categories. A sequence 0 → B ⟶^{i} C ⟶^{j} D → 0 in an additive category C is called replete if i is the categorical kernel of j, and j is the categorical cokernel of i. Let E_{rep} denote the class of all replete sequences, and show that (C, E_{rep}) is an exact category. 7.22 Fix a prime p, let Ab_{p} be the category of all finite abelian p-groups (6.2.2), and let C denote the full subcategory of all groups in Ab_{p} whose cyclic summands have even length (e.g., ℤ/p^{2i}). Show that C is an additive category, but not an exact subcategory of Ab_{p}. Let E be the sequences in C which are exact in Ab_{p}; is (C, E) an exact category? 7.23 Give an example of a cofinal exact subcategory B of an exact category C, such that the map K_{0}B → K_{0}C is not an injection (see 7.2). 7.24 Suppose that C_{i} are exact categories. Show that the product category ∏ C_{i} is an exact category. Need K_{0}(∏ C_{i}) → ∏ K_{0}(C_{i}) be an isomorphism? 7.25 (Claborn-Fossum). Set R_{n} = ℂ[x_{0}, ⋯, x_{n}]/(∑ x_{i}^{2} = 1). This is the complex coordinate ring of the n-sphere; it is a regular ring for every n, and R_{1} ≅ ℂ[z, z^{−1}]. In this exercise, we show that

K̃_{0}(R_{n}) ≅ K̃U(S^{n}) ≅ \{ \begin{matrix}0 & if n is odd \\ ℤ & if n is even, (n ≠ 0) \end{matrix} .

(a) Set z = x_{0} + ix_{1} and z̄ = x_{0} − ix_{1}, so that zz̄ = x_{0}^{2} + x_{1}^{2}. Show that

R_{n}[z^{−1}] ≅ ℂ[z, z^{−1}, x_{2}, …, x_{n}]
R_{n}/zR_{n} ≅ R_{n−2}[z̄], n ≥ 2.

(b) Use (a) to show that K̃_{0}(R_{n}) = 0 for n odd, and that if n is even there is a surjection β: K_{0}(R_{n−2}) → K̃_{0}(R_{n}).

(c) If n is even, show that β sends [R_{n−2}] to zero, and conclude that there is a surjection ℤ → K̃_{0}(R_{n}).

Fossum produced a finitely generated projective R_{2n}-module P_{n} such that the map K̃_{0}(R_{2n}) → K̃U(S^{2n}) ≅ ℤ sends [P_{n}] to the generator. (See [Foss].)

(d) Use the existence of P_{n} to finish the calculation of K_{0}(R_{n}). 7.26 (Keller) Recall that any exact category C embeds into the abelian category L of left exact functors C → Ab, and is closed under extensions (see Ex. 7.8). The countable envelope C^{e} of C is the full subcategory of L consisting of all colimits of sequences A_{0} ↣ A_{1} ↣ ⋯ of admissible monics in C. Show that countable direct sums exist in C^{e}. Then use the Eilenberg Swindle (I.2.8) to show that K_{0}(C^{e}) = 0.

§8. K_{0} of Schemes and Varieties

We have already introduced the Grothendieck group K_{0}(X) of a scheme X in Example 7.1.3. By definition, it is K_{0}VB(X), where VB(X) denotes the (exact) category of vector bundles on X. The tensor product of vector bundles makes K_{0}(X) into a commutative ring, as we saw in 7.4.2. This ring structure is natural in X: K_{0} is a contravariant functor from schemes to commutative rings. Indeed, we saw in I.5.2 that a morphism of schemes f: X → Y induces an exact base change functor f^{∗}: VB(Y) → VB(X), preserving tensor products, and such an exact functor induces a (ring) homomorphism f^{∗}: K_{0}(Y) → K_{0}(X).

In this section we shall study K_{0}(X) in more depth. Such a study requires that the reader has somewhat more familiarity with algebraic geometry than we assumed in the previous section, which is why this study has been isolated in its own section. We begin with two general invariants: the rank and determinant of a vector bundle.

The ring of continuous functions X → ℤ is isomorphic to the global sections of the constant sheaf ℤ, i.e., to the cohomology group H^{0}(X; ℤ); see [Hart, I.1.0.3]. We saw in I.5.1 that the rank of a vector bundle F is a continuous function, so rank(F) ∈ H^{0}(X; ℤ). Similarly, we saw in I.5.3 that the determinant of F is a line bundle on X, i.e., det(F) ∈ Pic(X).

Theorem 8.1. Let X be a scheme. Then H^{0}(X; ℤ) is isomorphic to a subring of K_{0}(X), and the rank of a vector bundle induces a split surjection of rings

rank: K_{0}(X) → H^{0}(X; ℤ).

Similarly, the determinant of a vector bundle induces a surjection of abelian groups

det: K_{0}(X) → Pic(X).

Their sum rank ⊕ det: K_{0}(X) → H^{0}(X; ℤ) ⊕ Pic(X) is a surjective ring map.

The ring structure on H^{0}(X; ℤ)⊕Pic(X) is (a_{1}, L_{1})·(a_{2}, L_{2}) = (a_{1}a_{2}, L_{1}^{a_{2}} ⊗L_{2}^{a_{1}}).

Proof. Let 0 → E → F → G → 0 be a short exact sequence of vector bundles on X. At any point x of X we have an isomorphism of free O_{x}-modules F_{x} ≅ E_{x} ⊕ G_{x}, so rank_{x}(F) = rank_{x}(E) + rank_{x}(G). Hence each rank_{x} is an additive function on VB(X). As x varies rank becomes an additive function with values in H^{0}(X; ℤ), so by 6.1.2 it induces a map rank: K_{0}(X) → H^{0}(X; ℤ). This is a ring map, since the formula rank(E ⊗ F) = rank(E) · rank(F) may be checked at each point x. If f: X → ℕ is continuous, the componentwise free module O_{X}^{f} has rank f. It follows that rank is onto. Since the class of componentwise free O_{X}-modules are closed under ⊕ and ⊗, the elements [O_{X}^{f}] − [O_{X}^{g}] in K_{0}(X) form a subring isomorphic to H^{0}(X; ℤ).

Similarly, det is an additive function, because we have det(F) ≅ det(E) ⊗ det(G) by Ex. I.5.4. Hence det induces a map K_{0}(X) → Pic(X) by 6.1.2. If L is a line bundle on X, then the element [L] − [O_{X}] of K_{0}(X) has rank zero and determinant L. Hence rank ⊕ det is onto; the proof that it is a ring map is given in Ex. 8.5. □

Definition 8.1.1. As in 2.3 and 2.6.1, the ideal K̃_{0}(X) of K_{0}(X) is defined to be the kernel of the rank map, so that K_{0}(X) = H^{0}(X; ℤ) ⊕ K̃_{0}(X) as an abelian group. In addition, we let SK_{0}(X) denote the kernel of rank ⊕ det. By Theorem 8.1, these are both ideals of the ring K_{0}(X). In fact, they form the beginning of the γ-filtration.

Regular Noetherian Schemes and the Cartan Map

Historically, the group K_{0}(X) first arose when X is a smooth projective variety, in Grothendieck’s proof of the Riemann-Roch Theorem (see [BoSe]). The following theorem was central to that proof.

Recall from §6 that G_{0}(X) is the Grothendieck group of the category M(X) of coherent O_{X}-modules. The inclusion VB(X) ⊂ M(X) induces a natural map K_{0}(X) → G_{0}(X), called the Cartan homomorphism (see 7.1.3).

Theorem 8.2. If X is a separated regular noetherian scheme, then every coherent O_{X}-module has a finite resolution by vector bundles, and the Cartan homomorphism is an isomorphism:

K_{0}(X) ⟶^{≅} G_{0}(X).

Proof. The first assertion is [SGA6, II, 2.2.3 and 2.2.7.1]. It implies that the Resolution Theorem 7.6 applies to the inclusion VB(X) ⊂ M(X). □

Proposition 8.2.1 (Nonsingular Curves). Let X be a 1-dimensional separated regular noetherian scheme, such as a nonsingular curve. Then SK_{0}(X) = 0, and

K_{0}(X) = H^{0}(X; ℤ) ⊕ Pic(X).

Proof. Given Theorem 8.2, this does follow from Ex. 6.10 (see Example 8.2.2 below). However, we shall give a slightly different proof here.

Without loss of generality, we may assume that X is irreducible. If X is affine, this is just Corollary 2.6.3. Otherwise, choose any closed point P on X. By [Hart, Ex. IV.1.3] the complement U = X − P is affine, say U = Spec(R). Under the isomorphism Pic(X) ≅ Cl(X) of I.5.14, the line bundles L(P) correspond to the class of the Weil divisor [P]. Hence the right-hand square commutes in the following diagram

G_{0}(P) ⟶^{i_{∗}}  K̃_{0}(X) ⟶ K̃_{0}(R) ⟶ 0
⋮ ; ∨  ↓ det  ≅ ↓ det
0 →  ℤ  ⟶^{{L(P)}} Pic(X) ⟶ Pic(R) ⟶ 0.

The top row is exact by 6.4.2 (and 8.2), and the bottom row is exact by I.5.14 and Ex. I.5.12. The right vertical map is an isomorphism by 2.6.2.

Now G_{0}(P) ≅ ℤ on the class [O_{P}]. From the exact sequence 0 → L(−P) → O_{X} → O_{P} → 0 we see that i_{∗}[O_{P}] = [O_{X}] − [L(−P)] in K_{0}(X), and det(i_{∗}[O_{P}]) = det L(−P)^{−1} in Pic(X). Hence the isomorphism G_{0}(P) ≅ ℤ is compatible with the above diagram. A diagram chase yields K̃_{0}(X) ≅ Pic(X). □

Example 8.2.2 (Classes of subschemes). Let X be a separated noetherian regular scheme. Given a subscheme Z of X, it is convenient to write [Z] for the element [O_{Z}] ∈ K_{0}M(X) = K_{0}(X). By Ex. 6.10(d) we see that SK_{0}(X) is the subgroup of K_{0}(X) generated by the classes [Z] as Z runs through the irreducible subschemes of codimension ≥ 2. In particular, if dim(X) = 2 then SK_{0}(X) is generated by the classes [P] of closed points (of codimension 2).

Transfer for finite and proper maps to regular schemes 8.2.3. Let f: X → Y be a finite morphism of separated noetherian schemes with Y regular. As pointed out in 6.2.5, the direct image f_{∗} is an exact functor M(X) → M(Y). In this case we have a transfer map f_{∗} on K_{0} sending [F] to [f_{∗}F]: K_{0}(X) → G_{0}(X) → G_{0}(Y) ≅ K_{0}(Y).

If f: X → Y is a proper morphism of separated noetherian schemes with Y regular, we can use the transfer G_{0}(X) → G_{0}(Y) of Lemma 6.2.6 to get a functorial transfer map f_{∗}: K_{0}(X) → K_{0}(Y), this time sending [F] to ∑(−1)^{i}[R^{i}f_{∗}F].

A Non-separated Example 8.2.4. Here is an example of a regular but nonseparated scheme X with K_{0}VB(X) ≠ G_{0}(X). Let X be “affine n-space with a double origin” over a field k, where n ≥ 2. This scheme is the union of two copies of 𝔸^{n} = Spec(k[x_{1}, …, x_{n}]) along 𝔸^{n} − {0}. Using the localization sequence for either origin and the Fundamental Theorem 6.5, one can show that G_{0}(X) = ℤ ⊕ ℤ. However the inclusion 𝔸^{n} ⊂ X is known to induce an equivalence VB(X) ≅ VB(𝔸^{n}) (see [EGA, IV(5.9)]), so by Theorem 7.8 we have K_{0}VB(X) ≅ K_{0}(k[x_{1}, …, x_{n}]) ≅ ℤ.

Definition 8.3. Let H(X) denote the category consisting of all quasi-coherent O_{X}-modules F such that, for each affine open subscheme U = Spec(R) of X, F|_{U} has a finite resolution by vector bundles. Since F|_{U} is defined by the finitely generated R-module M = F(U) this condition just means that M is in H(R).

If X is regular and separated, then we saw in Theorem 8.2 that H(X) = M(X). If X = Spec(R), it is easy to see that H(X) is equivalent to H(R).

H(X) is an exact subcategory of O_{X}-mod, closed under kernels of surjections, because each H(R) is closed under extensions and kernels of surjections in R-mod.

To say much more about the relation between H(X) and K_{0}(X), we need to restrict our attention to quasi-compact schemes such that every F in H(X) is a quotient of a vector bundle E_{0}. This implies that every module F ∈ H(X) has a finite resolution 0 → E_{d} → ⋯ → E_{1} → E_{0} → F → 0 by vector bundles. Indeed, the kernel F′ of a quotient map E_{0} → F is always locally of lower projective dimension than F, and X has a finite affine cover by U_{i} = Spec(R_{i}), it follows that the d^{th} syzygy is a vector bundle, where d = max{pd_{R_{i}}M_{i}}, M_{i} = F(U_{i}).

For this condition to hold, it is easiest to assume that X is quasi-projective (over a commutative ring k), i.e., a locally closed subscheme of some projective space ℙ_{k}^{n}. By [EGA II, 4.5.5 and 4.5.10], this implies that every quasicoherent O_{X}-module of finite type F is a quotient of some vector bundle E_{0} of the form E_{0} = ⨁ O_{X}(n_{i}).

Proposition 8.3.1. If X is quasi-projective (over a commutative ring), then K_{0}(X) ≅ K_{0}H(X).

Proof. Because H(X) is closed under kernels of surjections in O_{X}-mod, and every object in H(X) has a finite resolution by vector bundles, the Resolution Theorem 7.6 applies to VB(X) ⊂ H(X). □

Technical remark 8.3.2. Another assumption that guarantees that every F in H(X) is a quotient of a vector bundle is that X be quasi-separated and quasicompact with an ample family of line bundles. Such schemes are called divisorial in [SGA6, II.2.2.4]. For such schemes, the proof of 8.3.1 goes through to show that we again have K_{0}VB(X) ≅ K_{0}H(X).

Restricting Bundles 8.3.3. Given an open subscheme U of a quasi-projective scheme X, let B denote the full subcategory of VB(U) consisting of vector bundles F whose class in K_{0}(U) is in the image of j^{∗}: K_{0}(X) → K_{0}(U). We claim that the category B is cofinal in VB(U), so that K_{0}B is a subgroup of K_{0}(U) by the Cofinality Lemma 7.2. To see this, note that each vector bundle F on U fits into an exact sequence 0 → F′ → E_{0} → F → 0, where E_{0} = ⨁ O_{U}(n_{i}). But then F ⊕ F′ is in B, because in K_{0}(U)

[F ⊕ F′] = [F] + [F′] = [E_{0}] = ∑ j^{∗}[O_{X}(n_{i})].

Transfer Maps for Schemes

8.4 We can define a transfer map f_{∗}: K_{0}(X) → K_{0}(Y) with (gf)_{∗} = g_{∗}f_{∗} associated to various morphisms f: X → Y. If Y is regular, we have already done this in 8.2.3.

Suppose first that f is a finite map. In this case, the inverse image of any affine open U = Spec(R) of Y is an affine open f^{−1}U = Spec(S) of X, S is finitely generated as an R-module, and the direct image sheaf f_{∗}O_{X} satisfies f_{∗}O(U) = S. Thus the direct image functor f_{∗} is an exact functor from VB(X) to O_{Y} -modules (as pointed out in 6.2.5).

If f is finite and f_{∗}O_{X} is a vector bundle then f_{∗} is an exact functor from VB(X) to VB(Y). Indeed, locally it sends each finitely generated projective S-module to a finitely generated projective R-module, as described in Example 2.8.1. Thus there is a canonical transfer map f_{∗}: K_{0}(X) → K_{0}(Y) sending [F] to [f_{∗}F].

If f is finite and f_{∗}O_{X} is in H(X) then f_{∗} sends VB(X) into H(X), because locally it is the forgetful functor P(S) → H(R) of (7.9.1). Therefore f_{∗} defines a homomorphism K_{0}(X) → K_{0}H(Y). If Y is quasi-projective then composition with K_{0}H(Y) ≅ K_{0}(Y) yields a “finite” transfer map K_{0}(X) → K_{0}(Y).

Now suppose that f: X → Y is a proper map between quasi-projective noetherian schemes. The transfer homomorphism f_{∗}: G_{0}(X) → G_{0}(Y) was constructed in Lemma 6.2.6, with f_{∗}[F] = ∑(−1)^{i}[R^{i}f_{∗}F].

If in addition f has finite Tor-dimension, then we can also define a transfer map f_{∗}: K_{0}(X) → K_{0}(Y), following [SGA 6, IV.2.12.3]. Recall that an O_{X}-module F is called f_{∗}-acyclic if R^{q}f_{∗}F = 0 for all q > 0. Let P(f) denote the category of all vector bundles F on X such that F(n) is f_{∗}-acyclic for all n ≥ 0. By the usual yoga of homological algebra, P(f) is an exact category, closed under cokernels of injections, and f_{∗} is an exact functor from P(f) to H(Y). Hence the following lemma allows us to define the transfer map as

K_{0}(X) ⟵^{≅} K_{0}P(f) ⟶^{f_{∗}} K_{0}H(Y) ⟵^{≅} K_{0}(Y)  (8.4.1)

Lemma 8.4.2. Every vector bundle F on a quasi-projective X has a finite resolution

0 → F → P_{0} → ⋯ → P_{m} → 0

by vector bundles in P(f). Hence by the Resolution Theorem K_{0}P(f) ≅ K_{0}(X).

Proof. For n ≥ 0 the vector bundle O_{X}(n) is generated by global sections. Dualizing the resulting surjection O_{X}^{r} → O_{X}(n) and twisting n times yields a short exact sequence of vector bundles 0 → O_{X} → O_{X}(n)^{r} → E → 0. Hence for every vector bundle F on X we have a short exact sequence of vector bundles 0 → F → F(n)^{r} → E ⊗ F → 0. For all large n, the sheaf F(n) is f_{∗}-acyclic (see [EGA, III.3.2.1] or [Hart, III.8.8]), and F(n) is in P(f). Repeating this process with E ⊗ F in place of F, we obtain the desired resolution of F. □

Like the transfer map for rings, the transfer map f_{∗} is a K_{0}(Y)-module homomorphism. (This is the projection formula; see Ex. 7.10 and Ex. 8.3.)

Projective Bundles

Let E be a vector bundle of rank r + 1 over a quasi-compact scheme X, and let ℙ = ℙ(E) denote the projective space bundle of Example I.5.8. (If E|_{U} is free over U ⊆ X then ℙ |_{U} is the usual projective space ℙ_{U}^{r}.) Via the structural map π: ℙ → X, the base change map is a ring homomorphism π^{∗}: K_{0}(X) → K_{0}(ℙ), sending [M] to [f^{∗}M], where f^{∗}M = O_{ℙ} ⊗_{X} M. In this section we give Quillen’s proof [Q341, §8] of the following result, originally due to Berthelot [SGA6, VI.1.1].

Projective Bundle Theorem 8.5. Let ℙ be the projective bundle of E over a quasi-compact scheme X. Then K_{0}(ℙ) is a free K_{0}(X)-module with basis the twisting line bundles {1 = [O_{ℙ}], [O_{ℙ}(−1)], …, [O_{ℙ}(−r)]}.

Corollary 8.6. As a ring, K_{0}(ℙ_{ℤ}^{r}) = ℤ[z]/(z^{r+1}), where z = [ℙ^{r−1}] = 1 − [O(−1)]. (The relation z^{r+1} = 0 is Ex. 6.14(b).)

Hence K_{0}(ℙ_{X}^{r}) ≅ K_{0}(X) ⊗ K_{0}(ℙ_{ℤ}^{r}) = K_{0}(X)[z]/(z^{r+1}).

To prove Theorem 8.5, we would like to apply the direct image functor π_{∗} to a vector bundle F and get a vector bundle. This requires a vanishing condition. The proof of this result rests upon the following notion, which is originally due to Castelnuovo. It is named after David Mumford, who exploited it in [Mum].

Definition 8.7.1. A quasicoherent O_{ℙ}-module F is called Mumford-regular if for all q > 0 the higher derived sheaves R^{q}π_{∗}(F(−q)) vanish. Here F(n) is F ⊗ O_{ℙ}(n), as in Example I.5.3.1. We write MR for the additive category of all Mumford-regular vector bundles, and abbreviate ⊗_{X} for ⊗_{O_{X}}.

Examples 8.7.2. If N is a quasicoherent O_{X}-module then the standard cohomology calculations on projective spaces show that π^{∗}N = O_{ℙ} ⊗_{X} N is Mumfordregular, with π_{∗}π^{∗}N = N. More generally, if n ≥ 0 then π^{∗}N (n) is Mumfordregular, with π_{∗}π^{∗}N (n) = Sym_{n}E ⊗_{X} N. For n < 0 we have π_{∗}π^{∗}N (n) = 0. In particular, O_{ℙ}(n) = π^{∗}O_{X}(n) is Mumford-regular for all n ≥ 0.

If X is noetherian and F is coherent, then for n ≫ 0 the twists F(n) are Mumford-regular, because the higher derived functors R^{q}π_{∗}F(n) vanish for large n and also for q > r (see [Hart, III.8.8]).

The following facts were discovered by Castelnuovo when X = Spec(ℂ), and proven in [Mum, Lecture 14] as well as [Q341, §8]:

Proposition 8.7.3. If F is Mumford-regular, then

(1) The twists F(n) are Mumford-regular for all n ≥ 0;

(2) Mumford-regular modules are π_{∗}-acyclic, and in fact R^{q}π_{∗}F(n) = 0 for all q > 0 and n ≥ −q;

(3) The canonical map ε: π^{∗}π_{∗}(F) → F is onto.

Remark. Suppose that X is affine. Since π^{∗}π_{∗}(F) = O_{ℙ} ⊗_{X} π_{∗}F, and π_{∗}F is quasicoherent, item (3) states that Mumford-regular sheaves are generated by their global sections.

Lemma 8.7.4. Mumford-regular modules form an exact subcategory of O_{ℙ}-mod, and π_{∗} is an exact functor from Mumford-regular modules to O_{X}-modules.

Proof. Suppose that 0 → F′ → F → F′′ → 0 is a short exact sequence of O_{ℙ}-modules with both F′ and F′′ Mumford-regular. From the long exact sequence

R^{q}π_{∗}F′(−q) → R^{q}π_{∗}F(−q) → R^{q}π_{∗}F′′(−q)

we see that F is also Mumford-regular. Thus Mumford-regular modules are closed under extensions, i.e., they form is an exact subcategory of O_{ℙ}-mod. Since F′(1) is Mumford-regular, R^{1}π_{∗}F′ = 0, and so we have an exact sequence

0 → π_{∗}F′ → π_{∗}F → π_{∗}F′′ → 0.

This proves that π_{∗} is an exact functor. □

The following results were proven by Quillen in [Q341, §8].

Lemma 8.7.5. Let F be a vector bundle on ℙ.

(1) F(n) is a Mumford-regular vector bundle on ℙ for all large enough n;

(2) If F(n) is π_{∗}-acyclic for all n ≥ 0 then π_{∗}F is a vector bundle on X.

(3) Hence by 8.7.3, if F is Mumford-regular then π_{∗}F is a vector bundle on X.

(4) π^{∗}N ⊗_{ℙ} F(n) is Mumford-regular for all large enough n, and all quasicoherent O_{X}-modules N.

Definition 8.7.6 (T_{n}). Given a Mumford-regular O_{ℙ}-module F, we define a natural sequence of O_{X}-modules T_{n} = T_{n}F and O_{ℙ}-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 ε: π^{∗}π_{∗}F → F of Proposition 8.7.3. Inductively, we define T_{n}F = π_{∗}Z_{n−1}(n) and define Z_{n} to be ker(ε)(−n), where ε is the canonical map from π^{∗}T_{n} = π^{∗}π_{∗}Z_{n−1}(n) to Z_{n−1}(n).

Thus we have sequences (exact except possibly at Z_{n−1}(n))

0 → Z_{n}(n) → π^{∗}(T_{n}F) ⟶^{ε} Z_{n−1}(n) → 0  (8.7.7)

whose twists fit together into the sequence of the following theorem.

Quillen Resolution Theorem 8.7.8. Let F be a bundle on ℙ(E), rank(E) = r + 1. If F is Mumford-regular then Z_{r} = 0, and the sequences (8.7.7) are exact for n ≥ 0, so there is an exact sequence

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

Moreover, each F ↦ T_{i}F is an exact functor from Mumford-regular modules to O_{X}-modules.

Proof. We first prove by induction on n ≥ 0 that (a) the module Z_{n−1}(n) is Mumford-regular, (b) π_{∗}Z_{n}(n) = 0 and (c) the canonical map ε: π^{∗}T_{n} → Z_{n−1}(n) is onto, i.e., that (8.7.7) is exact for n.

We are given that (a) holds for n = 0, so we suppose that (a) holds for n. This implies part (c) for n by Proposition 8.7.3. Inductively then, we are given that (8.7.7) is exact, so π_{∗}Z_{n}(n) = 0 and the module Z_{n}(n + 1) is Mumford-regular by Ex. 8.6. That is, (b) holds for n and (a) holds for n+1. This finishes the first proof by induction.

Using (8.7.7), another induction on n shows that (d) each F ↦ Z_{n−1}F(n) is an exact functor from Mumford-regular modules to itself, and (e) each F ↦ T_{n}F is an exact functor from Mumford-regular modules to O_{X}-modules. Note that (d) implies (e) by Lemma 8.7.4, since T_{n} = π_{∗}Z_{n−1}(n).

Since the canonical resolution is obtained by splicing the exact sequences (8.7.7) together for n = 0, ..., r, all that remains is to prove that Z_{r} = 0, or equivalently, that Z_{r}(r) = 0. From (8.7.7) we get the exact sequence

R^{q−1}π_{∗}Z_{n+q−1}(n) → R^{q}π_{∗}Z_{n+q}(n) → R^{q}π_{∗}(π^{∗}T_{n}(−q))

which allows us to conclude, starting from (b) and 8.7.2, that R^{q}π_{∗}(Z_{n+q}) = 0 for all n, q ≥ 0. Since R^{q}π_{∗} = 0 for all q > r, this shows that Z_{r}(r) is Mumford-regular. Since π^{∗}π_{∗}Z_{r}(r) = 0 by (b), we see from Proposition 8.7.3(3) that Z_{r}(r) = 0 as well. □

Corollary 8.7.9. If F is Mumford-regular, each T_{i}F is a vector bundle on X.

Proof. For every n ≥ 0, the n^{th} twist of the Quillen resolution 8.7.8 yields exact sequences of π_{∗}-acyclic modules. Thus applying π_{∗} yields an exact sequence of O_{X}-modules, which by 8.7.2 is

0 → T_{n} → E ⊗ T_{n−1} → ⋯ → Sym_{n−i}E ⊗ T_{i} → ⋯ → π_{∗}F(n) → 0.

The result follows from this sequence and induction on i, since π_{∗}F(n) is a vector bundle by Lemma 8.7.5(3). □

Let MR(n) denote the n^{th} twist of MR; it is the full subcategory of VB(ℙ) consisting of vector bundles F such that F(−n) is Mumford-regular. Since twisting is an exact functor, each MR(n) is an exact category. By Lemma 8.7.3 we have

MR = MR(0) ⊆ MR(−1) ⊆ ⋯ ⊆ MR(n) ⊆ MR(n − 1) ⊆ · · · .

Proposition 8.7.10. The inclusions MR(n) ⊂ VB(ℙ) induce isomorphisms K_{0}MR ≅ K_{0}MR(n) ≅ K_{0}(ℙ).

Proof. The union of the MR(n) is VB(ℙ) by Lemma 8.7.5(1). By Example 7.1.7 we have K_{0}VB(ℙ) = \underrightarrow{lim} K_{0}MR(n), so it suffices to show that each inclusion MR(n) ⊂ MR(n − 1) induces an isomorphism on K_{0}. For i > 0, let u_{i}: MR(n − 1) → MR(n) be the exact functor F ↦ F(i) ⊗_{X} ∧^{i}E. It induces a homomorphism u_{i}: K_{0}MR(n − 1) → K_{0}MR(n). By Proposition 7.5 (Additivity), applied to the Koszul resolution

0 → F → F(1) ⊗_{X} E → ⋯ → F(r + 1) ⊗_{X} ∧^{r+1}E → 0

we see that the map ∑_{i>0}(−1)^{i−1}u_{i} is an inverse to the map ι_{n}: K_{0}MR(n) → K_{0}MR(n−1) induced by the inclusion. Hence ι_{n} is an isomorphism, as desired. □

Proof of Projective Bundle Theorem 8.5. Each T_{n} is an exact functor from MR to VB(X) by Theorem 8.7.8 and 8.7.9. Hence we have a homomorphism

t: K_{0}MR → K_{0}(X)^{r+1}, [F] ↦ ([T_{0}F], −[T_{1}F], …, (−1)^{r}[T_{r}F]).

This fits into the diagram

K_{0}(ℙ) ⟵^{≅} K_{0}MR ⟶^{t} K_{0}(X)^{r+1} ⟶^{u} K_{0}(ℙ) ⟵^{≅} K_{0}MR ⟶^{v} K_{0}(X)^{r+1}

where u(a_{0}, ..., a_{r}) = π^{∗}a_{0} + π^{∗}a_{1} · [O_{ℙ}(−1)] + ⋯ + π^{∗}a_{r} · [O_{ℙ}(−r)] and v[F] = ([π_{∗}F], [π_{∗}F(1)], …, [π_{∗}F(r)]). The composition ut sends [F] to the alternating sum of the [(π^{∗}T_{i}F)(−i)], which equals [F] by Quillen’s Resolution Theorem. Hence u is a surjection.

Since the (i, j) component of vu sends N_{j} to π_{∗}(π^{∗}N_{j}(i − j)) = Sym_{i−j}E ⊗_{X} N_{j} by Example 8.7.2, it follows that the composition vu is given by a lower triangular matrix with ones on the diagonal. Therefore vu is an isomorphism, so u is injective. □

λ-operations in K_{0}(X)

The following result was promised in Example 4.1.5.

Proposition 8.8. The operations λ^{k}[F] = [∧^{k}F] are well-defined on K_{0}(X), and make K_{0}(X) into a λ-ring.

Proof. It suffices to show that the formula λ_{t}(F) = ∑[∧^{k}F]t^{k} defines an additive homomorphism from VB(X) to the multiplicative group 1 + tK_{0}(X)[[t]]. Note that the constant term in λ_{t}(F) is 1 because ∧^{0}F = O_{X}. Suppose given an exact sequence of vector bundles 0 → F′ → F → F′′ → 0. By Ex. I.5.4, each ∧^{k}F has a finite filtration whose associated quotient modules are the ∧^{i}F′ ⊗ ∧^{k−i}F′′, so in K_{0}(X) we have

[∧^{k}F] = ∑[∧^{i}F′ ⊗ ∧^{k−i}F′′] = ∑[∧^{i}F′] · [∧^{k−i}F′′].

Assembling these equations yields the formula λ_{t}(F) = λ_{t}(F′)λ_{t}(F′′) in the group 1 + tK_{0}(X)[[t]], proving that λ_{t} is additive. Hence λ_{t} (and each coefficient λ^{k}) is well-defined on K_{0}(X). □

Splitting Principle 8.8.1 (see 4.2.2). Let f: 𝔽(E) → X be the flag bundle of a vector bundle E over a quasi-compact scheme X. Then K_{0}(𝔽(E)) is a free module over the ring K_{0}(X), and f^{∗}[E] is a sum of line bundles ∑[L_{i}].

Proof. Let f: 𝔽(E) → X be the flag bundle of E; by Theorem I.5.9 the bundle f^{∗}E has a filtration by sub-vector bundles whose successive quotients L_{i} are line bundles. Hence f^{∗}[E] = ∑[L_{i}] in K_{0}(𝔽(E)). Moreover, we saw in I.5.8 that the flag bundle is obtained from X by a sequence of projective bundle extensions, beginning with ℙ(E). By the Projective Bundle Theorem 8.5, K_{0}(𝔽(E)) is obtained from K_{0}(X) by a sequence of finite free extensions. □

The λ-ring K_{0}(X) has a positive structure in the sense of Definition 4.2.1. The “positive elements” are the classes [F] of vector bundles, and the augmentation ε: K_{0}(X) → H^{0}(X; ℤ) is given by Theorem 8.1. In this vocabulary, the “line elements” are the classes [L] of line bundles on X, and the subgroup L of units in K_{0}(X) is just Pic(X). The following corollary now follows from Theorems 4.2.3 and 4.7.

Corollary 8.8.2. K_{0}(X) is a special λ-ring. Consequently, the first two ideals in the γ-filtration of K_{0}(X) are F_{γ}^{1} = K̃_{0}(X) and F_{γ}^{2} = SK_{0}(X). In particular,

F_{γ}^{0}/F_{γ}^{1} ≅ H^{0}(X; ℤ) and F_{γ}^{1}/F_{γ}^{2} ≅ Pic(X).

Corollary 8.8.3. For every commutative ring R, K_{0}(R) is a special λ-ring.

Proposition 8.8.4. If X is quasi-projective, or more generally if X has an ample line bundle L then every element of K̃_{0}(X) is nilpotent. Hence K̃_{0}(X) is a nil ideal of K_{0}(X).

Proof. By Ex. 4.5, it suffices to show that ℓ = [L] is an ample line element. Given x = [E]−[F] in K̃_{0}(X), the fact that L is ample implies that F(n) = F ⊗L^{⊗n} is generated by global sections for all large n. Hence there are short exact sequences

0 → G_{n} → O_{X}^{r_{n}} → F(n) → 0

and therefore in K_{0}(X) we have the required equation:

ℓ^{n}x = [E(n)] − [O_{X}^{r_{n}}] + [G_{n}] = [E(n) ⊕ G_{n}] − r_{n}. □

Remark 8.8.5 (Nilpotence). If X is noetherian and quasiprojective of dimension d, then K̃_{0}(X)^{d+1} = 0, because it lies inside F_{γ}^{d+1}, which vanishes by [SGA6, VI.6.6] or [FL, V.3.10]. (See Example 4.8.2.)

If X is a nonsingular algebraic variety, the Chow groups CH^{i}(X) are defined to be the quotient of D^{i}(X), the free group on the integral codimension i subvarieties, by rational equivalence; see 6.4.3. If E is a locally free O_{X}-module of rank n, form the projective bundle ℙ(E) and flag bundle 𝔽(E) of E; see I.4.10. The Projective Bundle Theorem (see [Fulton]) states that the Chow group CH^{∗}(ℙ(E)) is a free graded module over CH^{∗}(X) with basis {1, ξ, …, ξ^{n−1}}, where ξ ∈ CH^{1}(ℙ(E)) is the class of a divisor corresponding to O(1). We define the Chern classes c_{i}(E) in CH^{i}(X) to be (−1)^{i} times the coefficients of ξ^{n} relative to this basis, with c_{i}(E) = 0 for i > n, with c_{0}(E) = 1. Thus we have the equation in CH^{n}(ℙ(E)):

ξ^{n} − c_{1}ξ^{n−1} + ⋯ + (−1)^{i}c_{i}ξ^{n−i} + ⋯ + (−1)^{n}c_{n} = 0.

If E is trivial then ξ^{n} = 0, and all the c_{i} vanish except c_{0}; if E has rank 1 then there is a divisor D with E = L(D) then ξ = [D] and c_{1}(E) = ξ = [D].

Proposition 8.9. (Grothendieck, 1957) The classes c_{i}(E) define Chern classes on K_{0}(X) with values in CH^{∗}(X).

Proof. We have already established axioms (CC0) and (CC1); the Normalization axiom (CC3) follows from the observation that

c_{1}(L(D_{1}) ⊗ L(D_{2})) = [D_{1}] + [D_{2}] = c_{1}(L(D_{1})) + c_{1}(L(D_{2})).

We now invoke the Splitting Principle, that we may assume that E has a filtration with invertible sheaves L_{j} as quotients; this is because CH^{∗}(ℙ(E)) embeds into CH^{∗}(𝔽(E)), where such a filtration exists; see I.5.9. Since ∏(ξ − c_{1}(L_{j})) = 0 in CH^{n}(𝔽(E)), expanding the product gives the coefficients c_{i}(E); the coefficients of each ξ^{k} establish the Sum Formula (CC2). □

Corollary 8.9.1. If X is nonsingular, the Chern classes induce isomorphisms c_{i} : K_{0}^{(i)}(X) ≅ CH^{i}(X) ⊗ ℚ, and the Chern character induces a ring isomorphism ch : K_{0}(X) ⊗ ℚ ≅ CH^{∗}(X) ⊗ ℚ.

Proof. By 4.12 the Chern character K_{0}(X) → CH^{∗}(X)⊗ℚ is a ring homomorphism. By 4.11.4, c_{n} vanishes on K_{0}^{(i)}(X) for i ≠ n, and by 4.12.1 it is a graded ring map, where K_{0}(X) ⊗ ℚ is given the γ-filtration. Let F^{r}K_{0}(X) denote the image of K_{0}M(X) → K_{0}(X); it is well known (see [FL, V.3] that F_{γ}^{r}K_{0}(X) ⊆ F^{r}K_{0}(X). We will prove by induction on r that F^{r}K_{0}(X) ≅ ⊕_{i≥r}CH^{i}(X). By 6.4.3, there is a canonical surjection CH^{i}(X) → F^{i}K_{0}(X)/F^{i+1}K_{0}(X) sending [Z] to the class of [O_{Z}]; removing a closed subvariety of Z, we can assume that Z is a complete intersection. In that case, c_{i}(O_{Z}) ≅ (−1)^{i}(i − 1)![Z] by Ex. 8.7. □

We cite the following result from [Fulton]. For any smooth X, the Todd class td(X) is defined to be the Hirzebruch character (Ex. 4.13) of the tangent bundle of X for the power series x/(1 − e^{−x}). If a = a_{0} + ⋯ + a_{d} ∈ CH^{∗}(X) with a_{i} ∈ CH^{i}(X) and d = dim(X), we write deg(a) for the image of a_{d} under the degree map CH^{d}(X) → ℤ.

Riemann-Roch Theorem 8.10. Let X be a nonsingular projective variety over a field k, and let E be a locally free sheaf of rank n on X. Then the Euler characteristic χ(E) = ∑(−1)^{i} dim H^{i}(X, E) equals deg(ch(E) · td(X)).

More generally, if f : X → Y is a smooth projective morphism, then the pushforward f_{∗} : K_{0}(X) → K_{0}(Y) satisfies ch(f_{∗}x) = f_{∗}(ch(x) · td(T_{f})), where T_{f} is the relative tangent sheaf of f.

Limits of schemes 8.11. Here is the analogue for schemes of the fact that every commutative ring is the filtered union of its finitely generated (noetherian) subrings. By [EGA, IV.8.2.3], every quasi-compact separated scheme X is the inverse limit of a filtered inverse system i ↦ X_{i} of noetherian schemes, each finitely presented over ℤ, with affine transition maps.

Let i ↦ X_{i} be any filtered inverse system of schemes such that the transition morphisms X_{i} → X_{j} are affine, and let X be the inverse limit scheme \underleftarrow{lim} X_{i}. This scheme exists by [EGA, IV.8.2]. In fact, over an affine open subset Spec(R_{j}) of any X_{j} we have affine open subsets Spec(R_{i}) of each X_{i}, and the corresponding affine open of X is Spec(\underrightarrow{lim} R_{i}). By [EGA, IV.8.5] every vector bundle on X comes from a bundle on some X_{j}, and two bundles on X_{j} are isomorphic over X just in case they are isomorphic over some X_{i}. Thus the filtered system of groups K_{0}(X_{i}) has the property that

K_{0}(X) = \underrightarrow{lim} K_{0}(X_{i}).

EXERCISES

8.1 Suppose that Z is a closed subscheme of a quasi-projective scheme X, with complement U. Let H_{Z}(X) denote the subcategory of H(X) consisting of modules supported on Z.

(a) Suppose that U = Spec(R) for some ring R, and that Z is locally defined by a nonzerodivisor. (The ideal I_{Z} is invertible; see I.5.12.) As in Cor. 7.7.4, show that there is an exact sequence: K_{0}H_{Z}(X) → K_{0}(X) → K_{0}(U).

(b) Suppose that Z is contained in an open subset V of X which is regular. Show that H_{Z}(X) is the abelian category M_{Z}(X) of 6.4.2, so that K_{0}H_{Z}(X) ≅ G_{0}(Z). Then apply Ex. 7.12 to show that there is an exact sequence

G_{0}(Z) → K_{0}(X) → K_{0}(U) → 0.

(c) (Deligne) Let R be a 2-dimensional noetherian domain which is not Cohen-Macaulay. If M is in H(A) then pd(M) = 1, by the Auslander-Buchsbaum equality [WHomo, 4.4.15]. Setting X = Spec(R) and Z = {𝔪}, show that H_{Z}(X) = 0. Hint: If M is the cokernel of f : R^{m} → R^{m}, show that det(f) must lie in a height 1 prime 𝔭 and conclude that M_{𝔭} ≠ 0.

There are 2-dimensional normal domains where K_{0}(R) → G_{0}(R) is not into [We80]; for these R the sequence K_{0}H_{Z}(X) → K_{0}(X) → K_{0}(U) is not exact. 8.2 Let X be a curve over an algebraically closed field. By Ex. I.5.7, K_{0}(X) is generated by classes of line bundles. Show that K_{0}(X) = H^{0}(X; ℤ) ⊕ Pic(X). 8.3 Projection Formula for schemes. Suppose that f: X → Y is a proper map between quasi-projective schemes, both of which have finite Tor-dimension.

(a) Given E in VB(X), consider the subcategory L(f) of M(Y) consisting of coherent O_{Y} -modules which are Tor-independent of both f_{∗}E and f_{∗}O_{X}. Show that G_{0}(Y) ≅ K_{0}L(f).

(b) Set x = [E] ∈ K_{0}(X). By (8.4.1), we can regard f_{∗}x as an element of K_{0}(Y). Show that f_{∗}(x · f^{∗}y) = f_{∗}(x) · y for every y ∈ G_{0}(Y).

(c) Using 7.4.2 and the ring map f^{∗}: K_{0}(Y) → K_{0}(X), both K_{0}(X) and G_{0}(X) are K_{0}(Y)-modules. Show that the transfer maps f_{∗}: G_{0}(X) → G_{0}(Y) of Lemma 6.2.6 and f_{∗}: K_{0}(X) → K_{0}(Y) of (8.4.1) are K_{0}(Y)-module homomorphisms, i.e., that the projection formula holds for every y ∈ K_{0}(Y):

f_{∗}(x · f^{∗}y) = f_{∗}(x) · y for every x ∈ K_{0}(X) or x ∈ G_{0}(X).

8.4 Suppose given a commutative square of quasi-projective schemes

X′ ⟶^{g′} X
f′↓  ↓^{f}
Y′ ⟶^{g} Y

with X′ = X ×_{Y} Y′ and f proper. Assume that g has finite flat dimension, and that X and Y′ are Tor-independent over Y, i.e., for q > 0 and all x ∈ X, y′ ∈ Y′ and y ∈ Y with y = f(x) = g(y′) we have

Tor_{q}^{O_{Y,y}}(O_{X,x}, O_{Y′,y′}) = 0.

Show that g^{∗}f_{∗} = f′_{∗}g′^{∗} as maps G_{0}(X) → G_{0}(Y′). 8.5 Let F_{1} and F_{2} be vector bundles of ranks r_{1} and r_{2}, respectively. Modify Ex. I.2.7 to show that det(F_{1} ⊗ F_{2}) ≅ (det F_{1})^{r_{2}} ⊗ (det F_{2})^{r_{1}}. Conclude that K_{0}(X) → H^{0}(X; ℤ) ⊕ Pic(X) is a ring map. 8.6 Let π: ℙ → X be a projective bundle as in 8.5, and let F be a Mumford-regular O_{ℙ}-module. Let N denote the kernel of the canonical map ε: π^{∗}π_{∗}F → F. Show that N (1) is Mumford-regular, and that π_{∗}N = 0. 8.7 Suppose that Z is a codimension i subvariety of a nonsingular X, with conormal bundle E; E → O_{X} → O_{Z} → 0 is exact. Show that c_{i}([O_{Z}]) = (−1)^{i}(i − 1)![Z] in CH^{i}(X). Hint: Passing to a flag bundle of E, show that [O_{Z}] is a product of terms 1 − L_{j}, for line bundles L_{j} = [O_{D_{j}}] of divisors D_{j}. Now use the product formula for the total Chern class and c_{1}(L_{j}) = [D_{j}]. Since this is a formal computation, it suffices to compute ln c([O_{Z}]).

§9. K_{0} of a Waldhausen category

It is useful to be able to define the Grothendieck group K_{0}(C) of a more general type of category than exact categories, by adding a notion of weak equivalence. A structure that generalizes well to higher K-theory is that of a category of cofibrations and weak equivalences, which we shall call a “Waldhausen category” for brevity. The definitions we shall use are due to Friedhelm Waldhausen, although the ideas for K_{0} are due to Grothendieck and were used in [SGA6].

We need to consider two families of distinguished morphisms in a category C, the cofibrations and the weak equivalences. For this we use the following device. Suppose that we are given a family F of distinguished morphisms in a category C. We assume that these distinguished morphisms are closed under composition, and contain every identity. It is convenient to regard these distinguished morphisms as the morphisms of a subcategory of C, which by abuse of notation we also call F.

Definition 9.1. Let C be a category equipped with a subcategory co = co(C) of morphisms in a category C, called “cofibrations” (and indicated with feathered arrows ↣). The pair (C, co) is called a category with cofibrations if the following axioms are satisfied: (W0) Every isomorphism in C is a cofibration; (W1) There is a distinguished zero object ‘0’ in C, and the unique map 0 ↣ A in C

is a cofibration for every A in C; (W2) If A ↣ B is a cofibration, and A → C is any morphism in C, then the pushout

B ∪_{A} C of these two maps exists in C, and moreover the map C ↣ B ∪_{A} C is a cofibration.

A ↣  B
↓  ↓
C ↣ B ∪_{A} C

These axioms imply that two constructions make sense in C: (1) the coproduct B ⨿ C of any two objects exists in C (it is the pushout B ∪_{0} C), and (2) every cofibration i: A ↣ B in C has a cokernel B/A (this is the pushout B ∪_{A} 0 of i along A → 0). We refer to A ↣ B ↠ B/A as a cofibration sequence in C.

For example, any abelian category is naturally a category with cofibrations: the cofibrations are the monomorphisms. More generally, we can regard any exact category as a category with cofibrations by letting the cofibrations be the admissible monics; axiom (W2) follows from Ex. 7.8(2). In an exact category, the cofibration sequences are exactly the admissible exact sequences.

Definition 9.1.1. A Waldhausen category C is a category with cofibrations, together with a family w(C) of morphisms in C called “weak equivalences” (abbreviated ‘w.e.’ and indicated with decorated arrows ⟶^{∼}). Every isomorphism in C is to be a weak equivalence, and weak equivalences are to be closed under composition (so we may regard w(C) as a subcategory of C). In addition, the following “Glueing axiom” must be satisfied:

(W3) Glueing for weak equivalences. For every commutative diagram of the form

C ← A ↣ B
↓∼ ↓∼ ↓∼
C′ ← A′ ↣ B′

(in which the vertical maps are weak equivalences and the two right horizontal maps are cofibrations), the induced map

B ∪_{A} C → B′ ∪_{A′} C′

is also a weak equivalence. Although a Waldhausen category is really a triple (C, co, w), we will usually drop the (co, w) from the notation and just write C. We say that C (or just wC) is saturated if: whenever f, g are composable maps and fg is a weak equivalence, f is a weak equivalence if and only if g is.

Definition 9.1.2 (K_{0}C). Let C be a Waldhausen category. K_{0}(C) is the abelian group presented as having one generator [C] for each object C of C, subject to the relations

(1) [C] = [C′] if there is a weak equivalence C ⟶^{∼} C′

(2) [C] = [B] + [C/B] for every cofibration sequence B ↣ C ↠ C/B. Of course, in order for this to be set-theoretically meaningful, we must assume that the weak equivalence classes of objects form a set. We shall occasionally use the notation K_{0}(wC) for K_{0}(C) to emphasize the choice of wC as weak equivalences.

These relations imply that [0] = 0 and [B ⨿ C] = [B] + [C], as they did in §6 for abelian categories. Because pushouts preserve cokernels, we also have [B ∪_{A} C] = [B] + [C] − [A]. However, weak equivalences add a new feature: [C] = 0 whenever 0 ≃ C.

Example 9.1.3. Any exact category A becomes a Waldhausen category, with cofibrations being admissible monics and weak equivalences being isomorphisms. By construction, the Waldhausen definition of K_{0}(A) agrees with the exact category definition of K_{0}(A) given in §7.

More generally, any category with cofibrations (C, co) may be considered as a Waldhausen category in which the category of weak equivalences is the category iso C of all isomorphisms. In this case K_{0}(C) = K_{0}(iso C) has only the relation (2). We could of course have developed this theory in §7 as an easy generalization of the preceding paragraph.

Topological Example 9.1.4. To show that we need not have additive categories, we give a topological example due to Waldhausen. Let R = R(∗) be the category of based CW complexes with countably many cells (we need a bound on the cardinality of the cells for set-theoretic reasons). Morphisms are cellular maps, and R_{f} = R_{f}(∗) is the subcategory of finite based CW complexes. Both are Waldhausen categories: “cofibration” is a cellular inclusion, and “weak equivalence” means weak homotopy equivalence (isomorphism on homotopy groups). The coproduct B ∨C is obtained from the disjoint union of B and C by identifying their basepoints.

The Eilenberg Swindle shows that K_{0}R = 0. In effect, the infinite coproduct C^{∞} of copies of a fixed complex C exists in R, and equals C ∨ C^{∞}. In contrast, the finite complexes have interesting K-theory:

Proposition 9.1.5. K_{0}R_{f} ≅ ℤ.

Proof. The inclusion of S^{n−1} in the n-disk D^{n} has D^{n}/S^{n−1} ≅ S^{n}, so [S^{n−1}]+ [S^{n}] = [D^{n}] = 0. Hence [S^{n}] = (−1)^{n}[S^{0}]. If C is obtained from B by attaching an n-cell, C/B ≅ S^{n} and [C] = [B] + [S^{n}]. Hence K_{0}R_{f} is generated by [S^{0}]. Finally, the reduced Euler characteristic χ(C) = ∑(−1)^{i} dim H̃^{i}(X; ℚ) defines a surjection from K_{0}R_{f} onto ℤ, which must therefore be an isomorphism. □

BiWaldhausen Categories 9.1.6. In general, the opposite C^{op} need not be a Waldhausen category, because the quotients B ↠ B/A need not be closed under composition: the family quot(C) of these quotient maps need not be a subcategory of C^{op}. We call C a category with bifibrations if C is a category with cofibrations, C^{op} is a category with cofibrations, co(C^{op}) = quot(C), the canonical map A⨿B → A×B is always an isomorphism, and A is the kernel of each quotient map B ↠ B/A. We call C a biWaldhausen category if C is a category with bifibrations, having a subcategory w(C) so that both (C, co, w) and (C^{op}, quot, w^{op}) are Waldhausen categories. The notions of bifibrations and biWaldhausen category are self-dual, so we have:

Lemma 9.1.7. K_{0}(C) ≅ K_{0}(C^{op}) for every biWaldhausen category.

Example 9.1.3 shows that exact categories are biWaldhausen categories. We will see in 9.2 below that chain complexes form another important family of biWaldhausen categories.

Exact Functors 9.1.8. A functor F: C → D between Waldhausen categories is called an exact functor if it preserves all the relevant structure: zero, cofibrations, weak equivalences and pushouts along a cofibration. The last condition means that the canonical map F B ∪_{F A} F C → F (B ∪_{A} C) is an isomorphism for every cofibration A ↣ B. Clearly, an exact functor induces a group homomorphism K_{0}(F): K_{0}C → K_{0}D.

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

For example, suppose that C and D are exact categories (in the sense of §7), considered as Waldhausen categories. A functor F: C → D is exact in the above sense if and only if F is additive and preserves short exact sequences, i.e., F is an exact functor between exact categories in the sense of §7. The routine verification of this assertion is left to the reader.

Here is an elementary consequence of the definition of exact functor. Let A and C be Waldhausen categories and F, F′, F′′ three exact functors from A to C. Suppose moreover that there are natural transformations F′ ⇒ F ⇒ F′′ so that for all A in A

F′A ↣ F A ↠ F′′A  (9.1.8)

is a cofibration sequence in C. Then [F A] = [F′A] + [F′′A] in K_{0}C, so as maps from K_{0}A to K_{0}C we have K_{0}(F) = K_{0}(F′) + K_{0}(F′′).

Chain complexes

9.2 Historically, one of the most important families of Waldhausen categories are those arising from chain complexes. The definition of K_{0} for a category of (co)chain complexes dates to the 1960’s, being used in [SGA6] to study the Riemann-Roch Theorem. We will work with chain complexes here, although by reindexing we could equally well work with cochain complexes.

Given a small abelian category A, let Ch = Ch(A) denote the category of all chain complexes in A, and let Ch^{b} denote the full subcategory of all bounded complexes. The following structure makes Ch into a Waldhausen category, with Ch^{b}(A) as a Waldhausen subcategory. We will show below that K_{0}Ch(A) = 0 but that K_{0}Ch^{b}(A) ≅ K_{0}(A).

A cofibration C → D is a chain map such that every map C_{n} ↣ D_{n} is monic in A. Thus a cofibration sequence is just a short exact sequence of chain complexes. A weak equivalence C ⟶^{∼} D is a quasi-isomorphism, i.e., a chain map inducing isomorphisms on homology.

Here is a slightly more general construction, taken from [SGA6, IV(1.5.2)]. Suppose that B is an exact category, embedded in an abelian category A. Let Ch(B), resp. Ch^{b}(B), denote the category of all (resp. all bounded) chain complexes in B. A cofibration A_{•} ↣ B_{•} in Ch(B) (resp. Ch^{b}B) is a map which is a degreewise admissible monomorphism, i.e., such that each C_{n} = B_{n}/A_{n} is in B, yielding short exact sequences A_{n} ↣ B_{n} ↠ C_{n} in B. To define the weak equivalences, we use the notion of homology in the ambient abelian category A: let wCh(B) denote the family of all chain maps in Ch(B) which are quasi-isomorphisms of complexes in Ch(A). With this structure, both Ch(B) and Ch^{b}(B) become Waldhausen subcategories of Ch(A).

Subtraction in K_{0}Ch and K_{0}Ch^{b} is given by shifting indices on complexes. To see this, recall from [WHomo, 1.2.8] that the n^{th} translate of C is defined to be the chain complex C[n] which has C_{i+n} in degree i. (If we work with cochain complexes then C^{i−n} is in degree i.) Moreover, the mapping cone complex cone(f) of a chain complex map f: B → C fits into a short exact sequence of complexes:

0 → C → cone(f) → B[−1] → 0.

Therefore in K_{0} we have [C]+[B[−1]] = [cone(f)]. In particular, if f is the identity map on C, the cone complex is exact and hence w.e. to 0. Thus we have [C] + [C[−1]] = [cone(id)] = 0. We record this observation as follows.

Lemma 9.2.1. Let C be any Waldhausen subcategory of Ch(A) closed under translates and the formation of mapping cones. Then [C[n]] = (−1)^{n}[C] in K_{0}(C). In particular, this is true in K_{0}Ch(B) and K_{0}Ch^{b}(B) for every exact subcategory B of A.

A chain complex C is called bounded below (resp. bounded above) if C_{n} = 0 for all n ≪ 0 (resp. all n ≫ 0). If C is bounded above, then each infinite direct sum C_{n} ⊕ C_{n+2} ⊕ ⋯ is finite, so the infinite direct sum of shifts

B = C ⊕ C[2] ⊕ C[4] ⊕ ⋯ ⊕ C[2n] ⊕ ⋯

is defined in Ch. From the exact sequence 0 → B[2] → B → C → 0, we see that in K_{0}Ch we have the Eilenberg swindle: [C] = [B] − [B[2]] = [B] − [B] = 0. A similar argument shows that [C] = 0 if C is bounded below. But every chain complex C fits into a short exact sequence

0 → B → C → D → 0

in which B is bounded above and D is bounded below. (For example, take B_{n} = 0 for n > 0 and B_{n} = C_{n} otherwise.) Hence [C] = [B] + [D] = 0 in K_{0}Ch. This shows that K_{0}Ch = 0, as asserted.

If B is any exact category, the natural inclusion of B into Ch^{b}(B) as the chain complexes concentrated in degree zero is an exact functor. Hence it induces a homomorphism K_{0}(B) → K_{0}Ch^{b}(B).

Theorem 9.2.2 ([SGA6, I.6.4]). Let A be an abelian category. Then

K_{0}(A) ≅ K_{0}Ch^{b}(A),

and the class [C] of a chain complex C in K_{0}A is the same as its Euler characteristic, namely χ(C) = ∑(−1)^{i}[C_{i}].

Similarly, if B is an exact category closed under kernels of surjections in an abelian category (in the sense of 7.0.1), then K_{0}(B) ≅ K_{0}Ch^{b}(B), and again we have χ(C) = ∑(−1)^{i}[C_{i}] in K_{0}(B).

Proof. We give the proof for A; the proof for B is the same, except one cites 7.5 in place of 6.6. As in Proposition 6.6 (or 7.5), the Euler characteristic χ(C) of a bounded complex is the element ∑(−1)^{i}[C_{i}] of K_{0}(A). We saw in 6.6 (and 7.5.1) that χ(B) = χ(C) if B → C is a weak equivalence (quasi-isomorphism). If B ↣ C ↠ D is a cofibration sequence in Ch^{b}, then from the short exact sequences 0 → B_{n} → C_{n} → D_{n} → 0 in A we obtain χ(C) = χ(B) + χ(C/B) by inspection (as in 7.5.1). Hence χ satisfies the relations needed to define a homomorphism χ from K_{0}(Ch^{b}) to K_{0}(A). If C is concentrated in degree 0 then χ(C) = [C_{0}], so the composite map K_{0}(A) → K_{0}(Ch^{b}) → K_{0}(A) is the identity.

It remains to show that [C] = χ(C) in K_{0}Ch^{b} for every complex

C: 0 → C_{m} → ⋯ → C_{n} → 0.

If m = n, then C = C_{n}[−n] is the object C_{n} of A concentrated in degree n; we have already observed that [C] = (−1)^{n}[C_{n}[0]] = (−1)^{n}[C_{n}] in this case. If m > n, let B denote the subcomplex consisting of C_{n} in degree n, and zero elsewhere. Then B ↣ C is a cofibration whose cokernel C/B has shorter length than C. By induction, we have the desired relation in K_{0}Ch^{b}, finishing the proof:

[C] = [B] + [C/B] = χ(B) + χ(C/B) = χ(C). □

Remark 9.2.3 (K_{0} and derived categories). Let B be an exact category. Theorem 9.2.2 states that the group K_{0}Ch^{b}(B) is independent of the choice of ambient abelian category A, as long as B is closed under kernels of surjections in A. This is the group k(B) introduced in [SGA6], Expose IV(1.5.2). (The context of [SGA6] was triangulated categories, and the main observation in op. cit. is that this definition only depends upon the derived category D_{B}^{b}(A). See Ex. 9.5 below.)

We warn the reader that if B is not closed under kernels of surjections in A, then K_{0}Ch^{b}(B) can differ from K_{0}(B). (See Ex. 9.11).

If A is an abelian category, or even an exact category, the category Ch^{b} = Ch^{b}(A) has another Waldhausen structure with the same weak equivalences: we redefine cofibration so that B → C is a cofibration if and only if each B_{i} → C_{i} is a split injection in A. If split Ch^{b} denotes Ch^{b} with this new Waldhausen structure, then the inclusion split Ch^{b} → Ch^{b} is an exact functor, so it induces a surjection K_{0}(split Ch^{b}) → K_{0}(Ch^{b}).

Lemma 9.2.4. If A is an exact category then

K_{0}(split Ch^{b}) ≅ K_{0}(Ch^{b}) ≅ K_{0}(A).

Proof. Lemma 9.2.1 and enough of the proof of 9.2.2 go through to prove that [C[n]] = (−1)^{n}[C] and [C] = ∑(−1)^{n}[C_{n}] in K_{0}(split Ch^{b}). Hence it suffices to show that A ↦ [A] defines an additive function from A to K_{0}(split Ch^{b}). If A is an object of A, let [A] denote the class in K_{0}(split Ch^{b}) of the complex which is A concentrated in degree zero. Any short exact sequence E: 0 → A → B → C → 0 in A may be regarded as an (exact) chain complex concentrated in degrees 0, 1 and 2 so:

[E] = [A] − [B] + [C]

in K_{0}(split Ch^{b}). But E is weakly equivalent to zero, so [E] = 0. Hence A ↦ [A] is an additive function, defining a map K_{0}(A) → K_{0}(split Ch^{b}). □

Extension Categories 9.3. If B is a category with cofibrations, the cofibration sequences A ↣ B ↠ C in B form the objects of a category E = E(B). A morphism E → E′ in E is a commutative diagram:

E :  A ↣ B ↠ C
↓ ↓ ↓ ↓
E′ :  A′ ↣ B′ ↠ C′

We make E into a category with cofibrations by declaring that a morphism E → E′ in E is a cofibration if A → A′, C → C′ and A′∪_{A}B → B′ are cofibrations in B. This is required by axiom (W2), and implies that the composite B ↣ A′ ∪_{A} B ↣ B′ is a cofibration too. If B is a Waldhausen category then so is E(B): a weak equivalence in E is a morphism whose component maps A → A′, B → B′, C → C′ are weak equivalences in B.

Here is a useful variant. If A and C are Waldhausen subcategories of B, the extension category E(A, B, C) of C by A is the Waldhausen subcategory of the extension category of B consisting of cofibration sequences A ↣ B ↠ C with A in A and C in C. Clearly, E(B) = E(B, B, B).

There is an exact functor ⨿: A × C → B, sending (A, C) to A ↣ A ⨿ C ↠ C. Conversely, there are three exact functors (s, t and q) from E to A, B and C, which send A ↣ B ↠ C to A, B and C, respectively. By the above remarks, if A = B = C then t_{∗} = s_{∗} + q_{∗} as maps K_{0}(E) → K_{0}(B).

Proposition 9.3.1. K_{0}(E(A, B, C)) ≅ K_{0}(A) × K_{0}(C).

Proof. Since (s, q) is a left inverse to ⨿, ⨿_{∗} is a split injection from K_{0}(A) × K_{0}(C) to K_{0}(E). Thus it suffices to show that for every E: A ↣ B ↠ C in E we have [E] = [⨿(A, 0)] + [⨿(0, C)] in K_{0}(E). This relation follows from the fundamental relation 9.1.2(2) of K_{0}, given that

⨿(A, 0) :  A ↣^{=} A ↠ 0
∨ ↓  ||  ∨ ↓  ↓
E :  A ↣ B ↠ C

is a cofibration in E with cokernel ⨿(0, C) : 0 ↣ C ↠ C. □

Example 9.3.2 (Higher Extension categories). Here is a generalization of the extension category E = E_{2} constructed above. Let E_{n} be the category whose objects are sequences of n cofibrations in a Waldhausen category C:

A : 0 = A_{0} ↣ A_{1} ↣ ⋯ ↣ A_{n}.

A morphism A → B in E_{n} is a natural transformation of sequences, and is a weak equivalence if each component A_{i} → B_{i} is a w.e. in C. It is a cofibration when for each 0 ≤ i < j < k ≤ n the map of cofibration sequences

A_{j}/A_{i} ↣ A_{k}/A_{i} ↠ A_{k}/A_{j}
↓ ↓ ↓
B_{j}/B_{i} ↣ B_{k}/B_{i} ↠ B_{k}/B_{j}

is a cofibration in E. The reader is encouraged in Ex. 9.4 to check that E_{n} is a Waldhausen category, and to compute K_{0}(E_{n}).

Cofinality Theorem 9.4. Let B be a Waldhausen subcategory of C closed under extensions. If B is cofinal in C (in the sense that for all C in C there is a C′ in C so that C ⨿ C′ is in B), then K_{0}(B) is a subgroup of K_{0}(C).

Proof. Considering B and C as symmetric monoidal categories with product ⨿, we have K_{0}^{⨿}(B) ⊂ K_{0}^{⨿}(C) by (1.3). The proof of cofinality for exact categories (Lemma 7.2) goes through verbatim to prove that K_{0}(B) ⊂ K_{0}(C). □

Remark 9.4.1. The proof shows that K_{0}(C)/K_{0}(B) ≅ K_{0}^{⨿}C/K_{0}^{⨿}B, and that every element of K_{0}(C) has the form [C] − [B] for some B in B and C in C.

Products

9.5 Our discussion in 7.4 about products in exact categories carries over to the Waldhausen setting. Let A, B and C be Waldhausen categories, and suppose given a functor F: A × B → C. The following result is completely elementary:

Lemma 9.5.1. If each F (A, −): B → C and F (−, B): A → C is an exact functor, then F: A × B → C induces a bilinear map

K_{0}A ⊗ K_{0}B → K_{0}C
[A] ⊗ [B] ↦ [F (A, B)].

Note that the 3 × 3 diagram in C determined by F (A ↣ A′, B ↣ B′) yields the following relation in K_{0}(C).

[F (A′, B′)] = [F (A, B)] + [F (A′/A, B)] + [F (A, B′/B)] + [F (A′/A, B′/B)]

Higher K-theory will need this relation to follow from more symmetric considerations, viz. that F (A ↣ A′, B ↣ B′) should represent a cofibration in the category E of all cofibration sequences in C. With this in mind, we introduce the following definition.

Definition 9.5.2. A functor F: A × B ↣ C between Waldhausen categories is called biexact if each F (A, −) and F (−, B) is exact, and the following condition is satisfied:

For every pair of cofibrations (A ↣ A′ in A, B ↣ B′ in B) the following map must be a cofibration in C:

F (A′, B) ∪_{F (A,B)} F (A, B′) ↣ F (A′, B′).

Our next result requires some notation. Suppose that a category with cofibrations C has two notions of weak equivalence, a weak one v and a stronger one w. (Every map in v belongs to w.) We write vC and wC for the two Waldhausen categories (C, co, v) and (C, co, w). The identity on C is an exact functor vC → wC.

Let C^{w} denote the full subcategory of all w-acyclic objects in C, i.e., those C for which 0 ↣ C is in w(C); C^{w} is a Waldhausen subcategory (9.1.8) of vC, i.e., of the category C with the v-notion of weak equivalence.

Recall from 9.1.1 that wC is called saturated if, whenever f, g are composable maps and fg is a weak equivalence, f is a weak equivalence if and only if g is.

Localization Theorem 9.6. Suppose that C is a category with cofibrations, endowed with two notions (v ⊂ w) of weak equivalence, with wC saturated, and that C^{w} is defined as above.

Assume in addition that every map f: C_{1} → C_{2} in C factors as the composition of a cofibration C_{1} ↣ C and an equivalence C ⟶^{∼} C_{2} in v(C).

Then the exact inclusions C^{w} → vC → wC induce an exact sequence

K_{0}(C^{w}) → K_{0}(vC) → K_{0}(wC) → 0.

Proof. Our proof of this is similar to the proof of the Localization Theorem 6.4 for abelian categories. Clearly K_{0}(vC) maps onto K_{0}(wC) and K_{0}(C^{w}) maps to zero. Let L denote the cokernel of K_{0}(C^{w}) → K_{0}(vC); we will prove the theorem by showing that λ(C) = [C] induces a map K_{0}(wC) → L inverse to the natural surjection L → K_{0}(wC). As vC and wC have the same notion of cofibration, it suffices to show that [C_{1}] = [C_{2}] in L for every equivalence f: C_{1} → C_{2} in wC. Our hypothesis that f factors as C_{1} ↣ C ⟶^{∼} C_{2} implies that in K_{0}(vC) we have [C_{2}] = [C] = [C_{1}] + [C/C_{1}]. Since wC is saturated, it contains C_{1} ↣ C. The following lemma implies that C/C_{1} is in C^{w}, so that [C_{2}] = [C_{1}] in L. This is the relation we needed to have λ define a map K_{0}(wC) → L, proving the theorem. □

Lemma 9.6.1. If B ↣^{∼} C is both a cofibration and a weak equivalence in a Waldhausen category, then 0 ↣ C/B is also a weak equivalence.

Proof. Apply the Glueing Axiom (W3) to the diagram:

0 ← B  = B
|| ∥ ∼ ↓∼
0 ← B ↣ C. □

Here is a simple application of the Localization Theorem. Let (C, co, v) be a Waldhausen category, and G an abelian group. Given a surjective homomorphism π: K_{0}(C) → G, we let C^{π} denote the Waldhausen subcategory of C consisting of all objects C such that π([C]) = 0.

Proposition 9.6.2. Assume that every morphism in a Waldhausen category C factors as the composition of a cofibration and a weak equivalence. There is a short exact sequence

0 → K_{0}(C^{π}) → K_{0}(C) ⟶^{π} G → 0.

Proof. Define wC to be the family of all morphisms A → B in C with π([A]) = π([B]). This satisfies axiom (W3) because [C ∪_{A} B] = [B] + [C] − [A], and the factorization hypothesis ensures that the Localization Theorem 9.6 applies to v ⊆ w. Since C^{π} is the category C^{w} of w-acyclic objects, this yields exactness at K_{0}(C). Exactness at K_{0}(C^{π}) will follow from the Cofinality Theorem 9.4, provided we show that C^{π} is cofinal. Given an object C, factor the map C → 0 as a cofibration C ↣ C′′ followed by a weak equivalence C′′ ⟶^{∼} 0. If C′ denotes C′′/C, we compute in G that

π([C ⨿ C′]) = π([C]) + π([C′]) = π([C] + [C′]) = π([C′′]) = 0.

Hence C ⨿ C′ is in C^{π}, and C^{π} is cofinal in C. □

Approximation Theorem 9.7. Let F: A → B be an exact functor between two Waldhausen categories. Suppose also that F satisfies the following conditions:

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

(b) Given any map b: F (A) → B in B, there is a cofibration a: A ↣ A′ in A and a weak equivalence b′: F (A′) ⟶^{∼} B in B so that b = b′ ∘ F (a).

(c) If b is a weak equivalence, we may choose a to be a weak equivalence in A. Then F induces an isomorphism K_{0}A ≅ K_{0}B.

Proof. Applying (b) to 0 ↣ B, we see that for every B in B there is a weak equivalence F (A′) ⟶^{∼} B. If F (A) ⟶^{∼} B is a weak equivalence, so is A ⟶^{∼} A′ by (c). Therefore not only is K_{0}A → K_{0}B onto, but the set W of weak equivalence classes of objects of A is isomorphic to the set of w.e. classes of objects in B.

Now K_{0}B is obtained from the free abelian group ℤ[W] on the set W by modding out by the relations [C] = [B] + [C/B] corresponding to the cofibrations B ↣ C in B. Given F (A) ⟶^{∼} B, hypothesis (b) yields A ↣ A′ in A and a weak equivalence F (A′) ⟶^{∼} C in B. Finally, the Glueing Axiom (W3) applied to

0 ← F (A) ↣ F (A′)
|| ↓∼ ↓∼
0 ← B ↣ C

implies that the map F (A′/A) → C/B is a weak equivalence. Therefore the relation [C] = [B]+[C/B] is equivalent to the relation [A′] = [A]+[A′/A] in the free abelian group ℤ[W], and already holds in K_{0}A. This yields K_{0}A ≅ K_{0}B, as asserted. □

Approximation for saturated categories 9.7.1. If B is saturated (9.1.1), then condition (c) is redundant in the Approximation Theorem, because F (a) is a weak equivalence by (b) and hence by (a) the map a is a w.e. in A.

Example 9.7.2. Recall from Example 9.1.4 that the category R(∗) of based CW complexes is a Waldhausen category. Let R_{hf}(∗) denote the Waldhausen subcategory of all based CW-complexes weakly homotopic to a finite CW complex. The Approximation Theorem applies to the inclusion of R_{f}(∗) into R_{hf}(∗); this may be seen by using the Whitehead Theorem and elementary obstruction theory. Hence

K_{0}R_{hf}(∗) ≅ K_{0}R_{f}(∗) ≅ ℤ.

Example 9.7.3. If A is an exact category, the Approximation Theorem applies to the inclusion split Ch^{b} ⊂ Ch^{b} = Ch^{b}(A) of Lemma 9.2.4, yielding a more elegant proof that K_{0}(split Ch^{b}) = K_{0}(Ch^{b}). To see this, observe that any chain complex map f: A → B factors through the mapping cylinder complex cyl(f) as the composite A ↣ cyl(f) ⟶^{∼} B, and that split Ch^{b} is saturated (9.1.1).

Example 9.7.4 (Homologically bounded complexes). Fix an abelian category A, and consider the Waldhausen category Ch(A) of all chain complexes over A, as in (9.2). We call a complex C_{•} homologically bounded if it is exact almost everywhere, i.e., if only finitely many of the H_{i}(C) are nonzero. Let Ch^{hb}(A) denote the Waldhausen subcategory of Ch(A) consisting of the homologically bounded complexes, and let Ch_{−}^{hb}(A) ⊂ Ch^{hb}(A) denote the Waldhausen subcategory of all bounded above, homologically bounded chain complexes 0 → C_{n} → C_{n−1} → · · ·. These are all saturated biWaldhausen categories (see 9.1.1 and 9.1.6). We will prove that

K_{0}Ch^{hb}(A) ≅ K_{0}Ch_{−}^{hb}(A) ≅ K_{0}Ch^{b}(A) ≅ K_{0}(A),

the first isomorphism being Theorem 9.2.2. From this and Proposition 6.6 it follows that if C is homologically bounded then

[C] = ∑(−1)^{i}[H_{i}(A)] in K_{0}A.

We first claim that the Approximation Theorem 9.7 applies to Ch^{b} ⊂ Ch_{−}^{hb}, yielding K_{0}Ch^{b} ≅ K_{0}Ch_{−}^{hb}. If C_{•} is bounded above then each good truncation τ_{≥n}C = (⋯ C_{n+1} → Z_{n} → 0) of C is a bounded subcomplex of C such that H_{i}(τ_{≥n}C) is H_{i}(C) for i ≥ n, and 0 for i < n. (See [WHomo, 1.2.7].) Therefore τ_{≥n}C ⟶^{∼} C is a quasi-isomorphism for small n (n ≪ 0). If B is a bounded complex, any map f: B → C factors through τ_{≥n}C for small n; let A denote the mapping cylinder of B → τ_{≥n}C (see [WHomo, 1.5.8]). Then A is bounded and f factors as the cofibration B ↣ A composed with the weak equivalence A ⟶^{∼} τ_{≥n}C ⟶^{∼} C. Thus we may apply the Approximation Theorem, as claimed.

The Approximation Theorem does not apply to Ch_{−}^{hb} ⊂ Ch^{hb}, but rather to Ch_{+}^{hb} ⊂ Ch^{hb}, where the “+” indicates bounded below chain complexes. The argument for this is the same as for Ch^{b} ⊂ Ch_{−}^{hb}. Since these are biWaldhausen categories, we can apply 9.1.7 to Ch_{−}^{hb}(A)^{op} = Ch_{+}^{hb}(A^{op}) and Ch^{hb}(A)^{op} = Ch^{hb}(A^{op}) to get

K_{0}Ch_{−}^{hb}(A) = K_{0}Ch_{+}^{hb}(A^{op}) ≅ K_{0}Ch^{hb}(A^{op}) = K_{0}Ch^{hb}(A).

This completes our calculation that K_{0}(A) ≅ K_{0}Ch^{hb}(A).

Example 9.7.5 (K_{0} and Perfect Complexes). Let R be a ring. A chain complex M_{•} of R-modules is called perfect if there is a quasi-isomorphism P_{•} ⟶^{∼} M_{•}, where P_{•} is a bounded complex of finitely generated projective R-modules, i.e., P_{•} is a complex in Ch^{b}(P(R)). The perfect complexes form a Waldhausen subcategory Ch_{perf}(R) of Ch(mod-R). We claim that the Approximation Theorem applies to Ch^{b}(P(R)) ⊂ Ch_{perf}(R), so that

K_{0}Ch_{perf}(R) ≅ K_{0}Ch^{b}P(R) ≅ K_{0}(R).

To see this, consider the intermediate Waldhausen category Ch_{perf}^{b} of bounded perfect complexes. The argument of Example 9.7.4 applies to show that K_{0}Ch_{perf}^{b} ≅ K_{0}Ch_{perf}(R), so it suffices to show that the Approximation Theorem applies to Ch^{b}P(R) ⊂ Ch_{perf}^{b}. This is an elementary application of the projective lifting property, which we relegate to Exercise 9.2.

Example 9.7.6 (G_{0} and Pseudo-coherent Complexes). Let R be a ring. A complex M_{•} of R-modules is called pseudo-coherent if there exists a quasiisomorphism P_{•} ⟶^{∼} M_{•}, where P_{•} is a bounded below complex ⋯ → P_{n+1} → P_{n} → 0 of finitely generated projective R-modules, i.e., P_{•} is a complex in Ch_{+}(P(R)). For example, if R is noetherian we can consider any finitely generated module M as a pseudo-coherent complex concentrated in degree zero. Even if R is not noetherian, it follows from Example 7.1.4 that M is pseudo-coherent as an R-module if and only if it is pseudo-coherent as a chain complex. (See [SGA6, I.2.9].)

The pseudo-coherent complexes form a Waldhausen subcategory Ch_{pcoh}(R) of Ch(mod-R), and the subcategory Ch_{pcoh}^{hb}(R) of homologically bounded pseudocoherent complexes is also Waldhausen. Moreover, the above remarks show that M(R) is a Waldhausen subcategory of both of them. We will see in Ex. 9.7 that the Approximation Theorem applies to the inclusions of Ch^{b}M(R) and Ch_{+}^{hb}P(R) in Ch_{pcoh}^{hb}(R), so that in particular we have

G_{0}(R) ≅ K_{0}Ch_{+}^{hb}P(R) ≅ K_{0}Ch_{pcoh}^{hb}(R) ≅ G_{0}(R).

Chain complexes with support Suppose that S is a multiplicatively closed set of central elements in a ring R. Let Ch_{S}^{b}P(R) denote the Waldhausen subcategory of C = Ch^{b}P(R) consisting of complexes E such that S^{−1}E is exact, and write K_{0}(R on S) for K_{0}Ch_{S}^{b}P(R).

The category Ch_{S}^{b}P(R) is the category C^{w} of the Localization Theorem 9.6, where w is the family of all morphisms P → Q in C such that S^{−1}P → S^{−1}Q is a quasi-isomorphism. By Theorem 9.2.2 we have K_{0}(C) = K_{0}(R). Hence there is an exact sequence

K_{0}(R on S) → K_{0}(R) → K_{0}(wC) → 0.

Theorem 9.8. The localization wC → Ch^{b}P(S^{−1}R) induces an injection on K_{0}, so there is an exact sequence

K_{0}(R on S) → K_{0}(R) → K_{0}(S^{−1}R).

Proof. Let B denote the category of S^{−1}R-modules of the form S^{−1}P for P in P(R). By Example 7.3.2 and Theorem 9.2.2, K_{0}Ch^{b}(B) = K_{0}(B) is a subgroup of K_{0}(S^{−1}R). Therefore the result follows from the following Proposition. □

Proposition 9.8.1. The Approximation Theorem 9.7 applies to wC → Ch^{b}(B).

Proof. Let P be a complex in Ch^{b}P(R) and b: S^{−1}P → B a map in B. Because each B_{n} has the form S^{−1}Q_{n} and each B_{n} → B_{n−1} is s_{n}^{−1}d_{n} for some s_{n} ∈ S and d_{n}: Q_{n} → Q_{n−1} such that d_{n}d_{n−1} = 0, B is isomorphic to the localization S^{−1}Q of a bounded complex Q in P(R), and some sb is the localization of a map f: P → Q in Ch^{b}P(R). Hence f factors as P ↣ cyl(f) ⟶^{∼} Q. Since b is the localization of f, followed by an isomorphism S^{−1}Q ≅ B in B, it factors as desired. □

EXERCISES

9.1 Retracts of a space. Fix a CW complex X and let R(X) be the category of CW complexes Y obtained from X by attaching cells, and having a retraction Y → X. Let R_{f}(X) be the subcategory of those Y obtained by attaching only finitely many cells. Let R_{fd}(X) be the subcategory of those Y which are finitely dominated, i.e., are retracts up to homotopy of spaces in R_{f}(X). Show that K_{0}R_{f}(X) ≅ ℤ and K_{0}R_{fd}(X) ≅ K_{0}(ℤ[π_{1}X]). Hint: The cellular chain complex of the universal covering space Ỹ is a chain complex of free ℤ[π_{1}X]-modules. 9.2 Let R be a ring. Use the projective lifting property to show that the Approximation Theorem applies to the inclusion Ch^{b}P(R) ⊂ Ch_{perf}^{b} of Example 9.7.5. Conclude that K_{0}(R) = K_{0}Ch_{perf}(R).

If S is a multiplicatively closed set of central elements of R, show that the Approximation Theorem also applies to the inclusion of Ch_{S}^{b}P(R) in Ch_{perf,S}(R), and conclude that K_{0}(R on S) ≅ K_{0}Ch_{perf,S}(R).

9.3 Consider the category Ch^{b} = Ch^{b}(A) of Theorem 9.2.2 as a Waldhausen category in which the weak equivalences are the isomorphisms, iso Ch^{b}, as in Example 9.1.3. Let Ch_{acyc}^{b} denote the subcategory of complexes whose differentials are all zero. Show that Ch_{acyc}^{b} is equivalent to the category ⊕_{n∈ℤ}A, and that the inclusion in Ch^{b} induces an isomorphism

K_{0}(iso Ch^{b}) ≅ ⨁_{n∈ℤ} K_{0}(A).

9.4 Higher Extension categories. Consider the category E_{n} constructed in Example 9.3.2, whose objects are sequences of n cofibrations in C. Show that E_{n} is a category with cofibrations, that E_{n} is a Waldhausen category when C is, and in that case

K_{0}(E_{n}) ≅ ⨁_{i=1}^{n} K_{0}(C).

9.5 ([SGA6, IV(1.6)]) Let B be a Serre subcategory of an abelian category A, or more generally any exact subcategory of A closed under extensions and kernels of surjections. Let Ch_{B}^{b}(A) denote the Waldhausen subcategory of Ch^{b}(A) of bounded complexes C with H_{i}(C) in B for all i. Show that

K_{0}B ≅ K_{0}Ch^{b}(B) ≅ K_{0}Ch_{B}^{b}(A).

9.6 Perfect injective complexes. Let R be a ring and let Ch_{inj}^{+}(R) denote the Waldhausen subcategory of Ch(mod-R) consisting of perfect bounded below cochain complexes of injective R-modules 0 → I^{m} → I^{m+1} · · ·. (Recall from Example 9.7.5 that I^{•} is called perfect if it is quasi-isomorphic to a bounded complex P^{•} of finitely generated projective modules.) Show that

K_{0}Ch_{inj}^{+}(R) ≅ K_{0}(R).

9.7 Pseudo-coherent complexes and G_{0}(R). Let R be a ring. Recall from Example 9.7.6 that Ch_{pcoh}^{hb}(R) denotes the Waldhausen category of all homologically bounded pseudo-coherent chain complexes of R-modules. Show that:

(a) The category M(R) is a Waldhausen subcategory of Ch_{pcoh}^{hb}(R).

(b) K_{0}Ch_{pcoh}(R) = K_{0}Ch_{+}P(R) = 0

(c) The Approximation Theorem applies to the inclusions of both Ch_{+}^{hb}M(R) and Ch_{+}^{hb}P(R) in Ch_{pcoh}^{hb}(R), and Ch^{b}M(R) ⊂ Ch_{−}^{hb}M(R). Hint: See 9.7.4.

This shows that G_{0}(R) ≅ K_{0}Ch_{+}^{hb}P(R) ≅ K_{0}Ch_{pcoh}^{hb}(R).

9.8 Pseudo-coherent complexes and G_{0}^{der}. Let X be a scheme. A cochain complex E^{•} of O_{X}-modules is called strictly pseudo-coherent if it is a bounded above complex of vector bundles, and pseudo-coherent if it is locally quasi-isomorphic to a strictly pseudo-coherent complex, i.e., if every point x ∈ X has a neighborhood U, a strictly pseudo-coherent complex P^{•} on U and a quasi-isomorphism P^{•} → E^{•}|_{U}. Let Ch_{pcoh}^{hb}(X) denote the Waldhausen category of all pseudo-coherent complexes E^{•} which are homologically bounded, and set G_{0}^{der}(X) = K_{0}Ch_{pcoh}^{hb}(X); this is the definition used in [SGA6, Expose IV(2.2)].

(a) If X is a noetherian scheme, show that every coherent O_{X}-module is a pseudocoherent complex concentrated in degree zero, so that we may consider M(X) as a Waldhausen subcategory of Ch_{pcoh}^{hb}(X). Then show that a complex E^{•} is pseudo-coherent if and only if is it homologically bounded above and all the homology sheaves of E^{•} are coherent O_{X}-modules.

(b) If X is a noetherian scheme, show that G_{0}(X) ≅ G_{0}^{der}(X).

(c) If X = Spec(R) for a ring R, show that G_{0}^{der}(X) is isomorphic to the group K_{0}Ch_{pcoh}^{hb}(R) of the previous exercise.

9.9 Let Z be a closed subscheme of X. Let Ch_{pcoh,Z}^{hb}(X) denote the subcategory of complexes in Ch_{pcoh}^{hb}(X) which are acyclic on X − Z, and define G_{0}(X on Z) to be K_{0}Ch_{pcoh}^{hb}(X).

(a) If X is a noetherian scheme, show that G_{0}(Z) ≅ G_{0}(X on Z).

(b) Show that there is an exact sequence G_{0}(X on Z) → G_{0}^{der}(X) → G_{0}^{der}(X−Z). 9.10 Perfect complexes and K_{0}^{der}. Let X be a scheme. A complex E^{•} of O_{X}- modules is called strictly perfect if it is a bounded complex of vector bundles, i.e., a complex in Ch^{b}VB(X). A complex is called perfect if it is locally quasi-isomorphic to a strictly perfect complex, i.e., if every point x ∈ X has a neighborhood U, a strictly perfect complex P^{•} on U and a quasi-isomorphic P^{•} → E^{•}|_{U}. Write Ch_{perf}(X) for the Waldhausen category of all perfect complexes, and K_{0}^{der}(X) for K_{0}Ch_{perf}(X); this is the definition used in [SGA6], Expose IV(2.2).

(a) If X = Spec(R), show that K_{0}(R) ≅ K_{0}^{der}(X). Hint: show that the Approximation Theorem 9.7 applies to Ch_{perf}(R) ⊂ Ch_{perf}(X).

(b) If X is noetherian, show that the category C = Ch_{perf}^{qc} of perfect complexes of quasi-coherent O_{X}-modules also has K_{0}(C) = K_{0}^{der}(X).

(c) If X is a regular noetherian scheme, show that a homologically bounded complex is perfect if and only if it is pseudo-coherent, and conclude that K_{0}^{der}(X) ≅ G_{0}(X).

(d) Let X be the affine plane with a double origin over a field k, obtained by glueing two copies of 𝔸^{2} = Spec(k[x, y]) together; X is a regular noetherian scheme. Show that K_{0}VB(X) = ℤ but K_{0}^{der}(X) = ℤ ⊕ ℤ. Hint. Use the fact that 𝔸^{2} → X induces an isomorphism VB(X) ≅ VB(𝔸^{2}) and the identification of K_{0}^{der}(X) with G_{0}(X) from part (c).

9.11 Give an example of an exact subcategory B of an abelian category A in which K_{0}(B) ≠ K_{0}Ch^{b}(B). Here Ch^{b}(B) is the Waldhausen category defined in 9.2. Note that B cannot be closed under kernels of surjections, by Theorem 9.2.2. 9.12 Finitely dominated complexes. Let C be a small exact category, closed under extensions and kernels of surjections in an ambient abelian category A (Definition 7.0.1). A bounded below complex C_{•} of objects in C is called finitely dominated if there is a bounded complex B_{•} and two maps C_{•} → B_{•} → C_{•} whose composite C_{•} → C_{•} is chain homotopic to the identity. Let Ch_{+}^{fd}(C) denote the category of finitely dominated chain complexes of objects in C. (If C is abelian, this is the category Ch_{+}^{hb}(C) of Example 9.7.4.)

(a) Let e be an idempotent endomorphism of an object C, and let tel(e) denote the nonnegative complex

⋯ ⟶^{e} C ⟶^{1−e} C ⟶^{e} C → 0.

Show that tel(e) is finitely dominated.

(b) Let Ĉ denote the idempotent completion 7.3 of C. Show that there is a map from K_{0}(Ĉ) to K_{0}Ch_{+}^{fd}(C) sending [(C, e)] to [tel(e)].

(c) Show that the map in (b) induces an isomorphism K_{0}(Ĉ) ≅ K_{0}Ch_{+}^{fd}(C). 9.13 Let S be a multiplicatively closed set of central nonzerodivisors in a ring R. Show that K_{0}H_{S}(R) ≅ K_{0}(R on S), and compare Cor. 7.7.4 to Theorem 9.8.

9.14 (Grayson’s Trick) Let B be a Waldhausen subcategory of C closed under extensions. Suppose that B is cofinal in C, so that K_{0}(B) ⊆ K_{0}(C) by Cofinality 9.4. Define an equivalence relation ∼ on objects of C by C ∼ C′ if there are B, B′ in B with C ⨿ B ≅ C′ ⨿ B′.

(a) Given a cofibration sequence C′ ↣ C ↠ C′′ in C, use the proof of Cofinality 7.2 to show that C ∼ C′ ⨿ C′′.

(b) Conclude that C ∼ C′ if and only if [C]−[C′] is in K_{0}(B) ⊆ K_{0}(C). (See 9.4.1.)

(c) For each sequence C_{1}, ..., C_{n} of objects in C such that [C_{1}] = ⋯ = [C_{n}] in K_{0}(C)/K_{0}(B), show that there is a C′ in C so that each C_{i} ⨿ C′ is in B.

(d) If K_{0}(B) = K_{0}(C), show that B is strictly cofinal in C, meaning that for every C in C there is a B in B so that C ⨿ B is in B.

9.15 Triangulated Categories. If C is a triangulated category, the Grothendieck group k(C) is the free abelian group on the objects, modulo the relation that [A] − [B] + [C] = 0 for every triangle A → B → C → A[1]. (a) If B is an additive category, regarded as a split exact category (7.1.2), show that K_{0}(B) is isomorphic to k(K^{b}B). (b) If B is an exact subcategory of an abelian category, closed under kernels, show that K_{0}(B) is isomorphic to k(K^{b}B). Hint. See 9.2.2. (c) If C has a bounded t-structure with heart A [BBD], show that K_{0}(A) ≅ k(C).

Appendix. Localizing by calculus of fractions

If C is a category and S is a collection of morphisms in C, then the localization of C with respect to S is a category C_{S}, together with a functor loc: C → C_{S} such that

(1) For every s ∈ S, loc(s) is an isomorphism

(2) If F: C → D is any functor sending S to isomorphisms in D, then F factors uniquely through loc: C → C_{S}.

Example. We may consider any ring R as an additive category R with one object. If S is a central multiplicative subset of R, there is a ring S^{−1}R obtained by localizing R at S, and the corresponding category is R_{S}. The useful fact that every element of the ring S^{−1}R may be written in standard form s^{−1}r = rs^{−1} generalizes to morphisms in a localization C_{S}, provided that S is a “locally small multiplicative system” in the following sense.

Definition A.1. A collection S of morphisms in C is called a multiplicative system if it satisfies the following three self-dual axioms: (FR1) S is closed under composition and contains the identity morphisms 1_{X} of all

objects X of C. That is, S forms a subcategory of C with the same objects. (FR2) (Ore condition) (a) If t : Z → Y is in S, then for every g : X → Y in C

there is a commutative diagram in C with s ∈ S:

\begin{matrix}W & ⟶^{f} & Z \\ ^{s}↓ & & ↓^{t} \\ X & ⟶^{g} & Y. \end{matrix}

(The slogan is “t^{−1}g = fs^{−1} for some f and s.”) (b) The dual statement (whose slogan is “fs^{−1} = t^{−1}g for some t and g”) is also valid.

(FR3) (Cancellation) If f, g: X → Y are parallel morphisms in C, then the following

two conditions are equivalent:

(a) sf = sg for some s: Y → Z in S

(b) ft = gt for some t: W → X in S.

We say that S is a right multiplicative system if it satisfies (FR1) and (FR2a), and if (FR3a) implies (FR3b) Left multiplicative systems are defined dually.

Example A.1.1. If S is a multiplicatively closed subset of a ring R, then S forms a multiplicative system if and only if S is a “2–sided denominator set.” (Onesided denominator sets (left and right) correspond to left and right multiplicative systems.) The localization of rings at denominator sets was the original application of Øystein Ore.

Example A.1.2 (Gabriel). Let B be a Serre subcategory (see §6) of an abelian category A, and let S be the collection of all B-isos, i.e., those maps f such that ker(f) and coker(f) is in B. Then S is a multiplicative system in A; the verification of axioms (FR2), (FR3) is a pleasant exercise in diagram chasing. In this case, A_{S} is the quotient abelian category A/B discussed in the Localization Theorem 6.4.

We would like to say that every morphism X → Z in C_{S} is of the form fs^{−1}. However, the issue of whether or this construction makes sense (in our universe) involves delicate set-theoretic questions. The following notion is designed to avoid these set-theoretic issues.

We say that S is locally small (on the left) if for each X in C there is a set S_{X} of morphisms X′ ⟶^{s} X in S such that every map Y → X in S factors as Y → X′ ⟶^{s} X for some s ∈ S_{X}.

Definition A.2 (Fractions). A (left) fraction between X and Y is a chain in C of the form:

fs^{−1} : X ⟵^{s} X_{1} ⟶^{f} Y, s ∈ S.

Call fs^{−1} equivalent to X ← X_{2} → Y just in case there is a chain X ← X_{3} → Y fitting into a commutative diagram in C:

X_{1}
↙ ↑ ↘
X ← X_{3} → Y
↖ ↓ ↗
X_{2}

It is easy to see that this is an equivalence relation. Write Hom_{S}(X, Y) for the equivalence classes of such fractions between X and Y. (Hom_{S}(X, Y) is a set when S is locally small.)

We cite the following theorem without proof from [WHomo, 10.3.7], relegating its routine proof to Exercises A.1 and A.2.

Gabriel-Zisman Theorem A.3. Let S be a locally small multiplicative system of morphisms in a category C. Then the localization C_{S} of C exists, and may be constructed by the following “calculus” of left fractions.

C_{S} has the same objects as C, but Hom_{C_{S}}(X, Y) is the set of equivalence classes of chains X ← X′ → Y with X′ → X in S, and composition is given by the Ore condition. The functor loc: C → C_{S} sends X → Y to the chain X ⟵^{=} X → Y, and if s: X → Y is in S its inverse is represented by Y ← X ⟶^{=} X.

Corollary A.3.1. Two parallel arrows f, g: X → Y become identified in C_{S} if and only if the conditions of (FR3) hold.

Corollary A.3.2. Suppose that C has a zero object, and that S is a multiplicative system in C. Assume that S is saturated in the sense that if s and st are in S then so is t. Then for every X in C:

loc(X) ≅ 0 ⇔ The zero map X ⟶^{0} X is in S.

Proof. Since loc(0) is a zero object in C_{S}, loc(X) ≅ 0 if and only if the parallel maps 0, 1: X → X become identified in C_{S}. □

Now let A be an abelian category, and C a full subcategory of the category Ch(A) of chain complexes over A, closed under translation and the formation of mapping cones. Let K be the quotient category of C, obtained by identifying chain homotopic maps in C. Let Q denote the family of (chain homotopy equivalence classes of) quasi-isomorphisms in C. The following result states that Q forms a multiplicative system in K, so that we can form the localization K_{Q} of K with respect to Q by the calculus of fractions.

Lemma A.4. The family Q of quasi-isomorphisms in the chain homotopy category K forms a multiplicative system.

Proof. (FR1) is trivial. To prove (FR2), consider a diagram X ⟶^{u} Y ⟵^{s} Z with s ∈ Q. Set C = cone(s), and observe that C is acyclic. If f: Y → C is the natural map, set W = cone(fu), so that the natural map t: W → X[−1] is a quasiisomorphism. Now the natural projections from each W_{n} = Z_{n−1} ⊕ Y_{n} ⊕ X_{n−1} to Z_{n−1} form a morphism v: W → Z of chain complexes making the following diagram commute:

X ⟶^{fu} C ⟶ W  ⟶^{t} X[−1]
u↓  ||  ↓^{v}  ↓
Z ⟶^{s} Y ⟶^{f} C ⟶ Z[−1] ⟶^{s[−1]} Y [−1].

Applying X ↦ X[1] to the right square gives the first part of (FR2); the second part is dual and is proven similarly.

To prove (FR3), we suppose given a quasi-isomorphism s: Y → Y′ and set C = cone(s); from the long exact sequence in homology we see that C is acyclic. Moreover, if v denotes the map C[1] → Y then there is an exact sequence:

Hom_{K}(X, C[1]) ⟶^{v} Hom_{K}(X, Y) ⟶^{s} Hom_{K}(X, Y′)

(see [WHomo, 10.2.8]). Given f and g, set h = f − g. If sh = 0 in K, there is a map w: X → C[1] such that h = vw. Setting X′ = cone(w)[1], the natural map X′ ⟶^{t} X must be a quasi-isomorphism because C is acyclic. Moreover, wt = 0, so we have ht = vwt = 0, i.e., ft = gt. □

Definition A.5. Let C ⊂ Ch(A) be a full subcategory closed under translation and the formation of mapping cones. The derived category of C, D(C), is defined to be the localization K_{Q} of the chain homotopy category K at the multiplicative system Q of quasi-isomorphisms. The derived category of A is D(A) = D(Ch(A)).

Another application of calculus of fractions is Verdier’s formation of quotient triangulated categories by thick subcategories. We will use Rickard’s definition of thickness, which is equivalent to Verdier’s.

Definition A.6. Let K be any triangulated category (see [WHomo, 10.2.1]). A full additive subcategory E of K is called thick if:

(1) In any distinguished triangle A → B → C → A[1], if two out of A, B, C are in E then so is the third.

(2) if A ⊕ B is in E then both A and B are in E.

If E is a thick subcategory of K, we can form a quotient triangulated category K/E, parallel to Gabriel’s construction of a quotient abelian category in A.1.2. That is, K/E is defined to be S^{−1}K, where S is the family of maps whose cone is in E. By Ex. A.6, S is a saturated multiplicative system of morphisms, so S^{−1}K can be constructed by the calculus of fractions (theorem A.3).

To justify this definition, note that because S is saturated it follows from A.3.2 and A.6(2) that: (a) X ≅ 0 in K/E if and only if X is in E, and (b) a morphism f : X → Y in K becomes an isomorphism in K/E if and only if f is in S.

We conclude with a more recent application, due to M. Schlichting [Schl].

Definition A.7. Let A ⊂ B be exact categories, with A closed under extensions, admissible subobjects and admissible quotients in B. We say that A is right filtering in B if every map from an object B of B to an object of A factors through an admissible epi B ↠ A with A in A.

A morphism of B is called a weak isomorphism if it is a finite composition of admissible monics with cokernel in A and admissible epis with kernel in A. We write B/A for the localization of B with respect to the weak isomorphisms.

Proposition A.7.1. If A is right filtering in B, then the class Σ of weak isomorphisms is a right multiplicative system. By the Gabriel-Zisman Theorem A.3, B/A may be constructed using a calculus of right fractions.

Proof. By construction, weak isomorphisms are closed under composition, so (FR1) holds. Given an admissible t : Z ↠ Y in B with kernel in A and g : X → Y, the base change s : Z ×_{Y} X ↠ X is an admissible epi in Σ and the canonical map Z ×_{Y} X → Z → Y equals gs. Given an admissible monic t : Z ↣ Y with kernel A′ in A, the map X → A′ factors throught an admissible epi q : X ↠ A with A in A because A ⊂ B is right filtering; the kernel W ↣ X of q is in Σ and W → X → Y factors through a universal map W → Z. An arbitrary t in Σ is a finite composition of these two types, so by induction on the length of t, we see that Σ satisfies (FR2a).

Finally, suppose that sf = sg for some weak isomorphism s : Y → Z and f, g : X → Y. If s is an admissible monic, then f = g already. If s is an admissible epi, f − g factors through the kernel A ↣ Y of s. Because A is right filtering in B, there is an admissible exact sequence W ↣^{t} X ↠ A with A in A, such that f − g factors through A. Hence t is a weak equivalence and ft = gt. As before, induction shows that (a) implies (b) in axiom (FR3). □

EXERCISES

A.1 Show that the construction of the Gabriel-Zisman Theorem A.3 makes C_{S} into a category by showing that composition is well-defined and associative. A.2 If F: C → D is a functor sending S to isomorphisms, show that F factors uniquely through the Gabriel-Zisman category C_{S} of the previous exercise as C → C_{S} → D. This proves the Gabriel-Zisman Theorem A.3, that C_{S} is indeed the localization of C with respect to S. A.3 Let B be a full subcategory of C, and let S be a multiplicative system in C such that S ∩ B is a multiplicative system in B. Assume furthermore that one of the following two conditions holds:

(a) Whenever s: C → B is in S with B in B, there is a morphism f: B′ → C with B′ in B such that sf ∈ S

(b) Condition (a) with the arrows reversed, for s: B → C. Show that the natural functor B_{S} → C_{S} is fully faithful, so that B_{S} can be identified with a full subcategory of C_{S}. A.4 Let F: A → A′ be an exact functor between two abelian categories, and let S be the family of morphisms s in Ch(A) such that F (s) is a quasi-isomorphism. Show that S is a multiplicative system in ChA.

A.5 Suppose that C is a subcategory of Ch(A) closed under translation and the formation of mapping cones, and let Σ be the family of all chain homotopy equivalences in C. Show that the localization C_{Σ} is the quotient category K of C described before Lemma A.4. Conclude that the derived category D(C) is the localization of C at the family of all quasi-isomorphisms. Hint: If two maps f_{1}, f_{2}: X → Y are chain homotopic then they factor through a common map f: cyl(X) → Y out of the mapping cylinder of X. A.6 Let E be a thick subcategory of a triangulated category K, and S the morphisms whose cone is in E, as in A.6. Show that S is a multiplicative system of morphisms. Then show that S is saturated in the sense of A.3.2.