CHAPTER VI THE HIGHER K-THEORY OF FIELDS

The problem of computing the higher K-groups of fields has a rich history, beginning with Quillen’s calculation for finite fields (IV.1.13), and Borel’s calculation of K_{∗}(F) ⊗ ℚ for number fields (IV.1.18), Tate’s calculations of the Milnor Kgroups of number fields (III.7.2) and Quillen’s observation that the image of the stable homotopy groups π_{∗}^{s} in K_{∗}(ℤ) contained the image of the J-homomorphism, whose orders are described by Bernoulli numbers. In the early 1970’s, a series of conjectures were made concerning the K-theory of number fields, and the structure of Milnor K-theory. After decades of partial calculations and further conjectures, the broad picture is now in place. The goal of this chapter is to explain what we now know about the K-theory of fields, and especially number fields.

§1. K-theory of algebraically closed fields

We begin by calculating the K-theory of algebraically closed fields. The results in this section are due to Suslin [Su83, Su84].

Let C be a smooth curve over an algebraically closed field k, with function field F. The local ring of C at any closed point c ∈ C is a discrete valuation ring, and we have a specialization map λ_{c} : K_{∗}(F, ℤ/m) → K_{∗}(k, ℤ/m) (see V.6.7 and Ex. V.6.14). If C = ℙ^{1}, we saw in IV, Ex. 6.14 that all of the specialization maps λ_{c} agree. The following result, due to Suslin [Su83], shows that this holds more generally.

Theorem 1.1. (Rigidity) Let C be a smooth curve over an algebraically closed field k, with function field F = k(C). If c_{0}, c_{1} are two closed points of C, the specializations K_{∗}(F, ℤ/m) → K_{∗}(k, ℤ/m) coincide.

Proof. There is no loss of generality in assuming that C is a projective curve. Suppose that f : C → ℙ^{1} is a finite map. Let R_{0} and R′ be the local ring in k(t) at t = 0 and its integral closure in F = k(C), respectively. If s_{c} is a parameter at c and e_{c} is the ramification index at c, so that t = u ∏ s_{c}^{e_{c}} in R′, then for a ∈ K_{∗}(F, ℤ/m) we have ∂_{c}({t, a}) = e_{c}∂_{c}{s_{c}, a} = e_{c}λ_{c}(a). Since k(c) = k for all closed c ∈ C, we see from Chapter V, (6.6.4) that

λ_{0}(N_{F/k(t)}a) = ∂_{0}N_{F/k(t)}({t, a}) = ∑_{f(c)=0} N_{c}∂_{c}({t, a}) = ∑_{f(c)=0} e_{c}λ_{c}(a).

A similar formula holds for any other point of ℙ^{1}. In particular, since λ_{0}(Na) = λ_{∞}(Na) we have the formula

∑_{f(c)=0} e_{c}λ_{c}(a) = ∑_{f(c)=∞} e_{c}λ_{c}(a).

Typeset by AMS-TEX

2 VI. THE HIGHER K-THEORY OF FIELDS

We can assemble this information as follows. Let A denote the abelian group Hom(K_{∗}(F; ℤ/m), K_{∗}(k, ℤ/m)), and recall that Cart(C) denotes the free abelian group on the closed points of C. There is a homomorphism λ : Cart(C) → A sending [c] to the specialization map λ_{c} of IV.6.7. If we regard f as an element of F^{×}, its divisor is ∑_{f(c)=0} e_{c}[c] − ∑_{f(c)=∞} e_{c}[c]. The displayed equation amounts to the formula λ ∘ div = 0. Now the Picard group Pic(C) is the cokernel of the divisor map F^{×} → Cart(C) (see I.5.12), so λ factors through Pic(C). Since A is a group of exponent m, λ factors through Pic(C) ⊗ ℤ/m. However, the kernel J(C) of the degree map Pic(C) → ℤ is a divisible group (I.5.16), so λ is zero on J(C). Since [c_{0}] − [c_{1}] ∈ J(C), this implies that λ_{c_{0}} = λ_{c_{1}}. □

Corollary 1.2. If A is any finitely generated smooth integral k-algebra, and h_{0}, h_{1} : A → k are any k-algebra homomorphisms, then the induced maps h_{i}^{∗} : K_{∗}(A; ℤ/m) → K_{∗}(k; ℤ/m) coincide.

Proof. The kernels of the h_{i} are maximal ideals, and it is known that there is a prime ideal 𝔭 of A contained in their intersection such that A/𝔭 is 1-dimensional. If R is the normalization of A/𝔭 then the h_{i} factor through A → R. Therefore we can replace A by R. Since C = Spec(R) is a smooth curve, the specializations λ_{i} on F = k(C) agree by Theorem 1.1. The result follows, since by Theorem V.6.7, the induced maps h_{i}^{∗} factor as K_{∗}(R; ℤ/m) → K_{∗}(F; ℤ/m) ⟶^{λ_{i}} K_{∗}(k; ℤ/m). □

Theorem 1.3. If k ⊂ F is an inclusion of algebraically closed fields, the maps K_{n}(k; ℤ/m) → K_{n}(F; ℤ/m) are isomorphisms for all m.

Proof. We saw in V.6.7.2 that both K_{n}(k) → K_{n}(F) and K_{n}(k; ℤ/m) → K_{n}(F; ℤ/m) are injections. To see surjectivity, we write F as the union of its finitely generated subalgebras A. Therefore every element of K_{n}(F; ℤ/m) is the image of some element of K_{n}(A; ℤ/m) under the map induced from the inclusion h_{0} : A ↪ F. Since the singular locus of A is closed, some localization A[1/s] is smooth, so we may assume that A is smooth.

But for any maximal ideal 𝔪 of A we have a second map h_{1} : A → A/𝔪 = k ↪ F. Both h_{0} and h_{1} factor through the basechange A → A ⊗_{k} F and the induced maps h_{i} : A ⊗_{k} F → F. Since A ⊗_{k} F is smooth over F, the map h_{0}^{∗} : K_{n}(A; ℤ/m) → K_{n}(F; ℤ/m) coincides with h_{1}^{∗} : K_{n}(A; ℤ/m) → K_{n}(k; ℤ/m) → K_{n}(F; ℤ/m) by Corollary 1.2. This finishes the proof. □

Finite characteristic. For algebraically closed fields of characteristic p > 0, we may take k = 𝔽̄_{p} to determine K_{∗}(F). Recall from IV.1.13 that K_{n}(𝔽̄_{p}) = 0 for even n > 0, and that K_{2i−1}(𝔽̄_{p}) = ∪K_{2i−1}(𝔽_{p^{ν}}) is isomorphic as an abelian group to 𝔽̄_{p}^{×} ≅ ℚ/ℤ[\frac{1}{p}]. In particular, K_{n}(F; ℤ/p) = K_{n}(𝔽̄_{p}; ℤ/p) = 0; this implies that K_{n}(F) is uniquely p-divisible. (We saw in IV.5.6 that this is true more generally for perfect fields of characteristic p.)

We also saw in IV.1.13 that if p ∤ m and β ∈ K_{2}(𝔽̄_{p}; ℤ/m) ≅ μ_{m}(𝔽̄_{p}) is the Bott element (whose Bockstein is a primitive m^{th} root of unity), then K_{∗}(𝔽̄_{p}; ℤ/m) ≅ ℤ/m[β] as a graded ring. The action of the Frobenius automorphism ϕ(x) = x^{p} on F induces multiplication by p^{i} on K_{2i−1}(𝔽̄_{p}); we say that the action is twisted i times. The following corollary to Theorem 1.3 is immediate from these remarks.

Corollary 1.3.1. Let F be an algebraically closed field of characteristic p > 0.

(i) If n is even and n > 0, K_{n}(F) is uniquely divisible. (ii) If n = 2i − 1 is odd, K_{2i−1}(F) is the direct sum of a uniquely divisible group and the torsion group ℚ/ℤ[\frac{1}{p}]. In particular, it is divisible with no p-torsion, and the Frobenius automorphism acts on the torsion subgroup as multiplication by p^{i}. (iii) When p ∤ m, the choice of a Bott element β ∈ K_{2}(F; ℤ/m) determines a graded ring isomorphism K_{∗}(F; ℤ/m) ≅ ℤ/m[β].

Recall that any divisible abelian group is the direct sum of a uniquely divisible group and a divisible torsion group, a divisible torsion group is the sum of its Sylow subgroups, and an ℓ-primary divisible group is a direct sum of copies of ℤ/ℓ^{∞}. Therefore K_{2i−1}(F) is the direct sum of a uniquely divisible group and ⊕_{ℓ≠p}ℤ/ℓ^{∞} ≅ ℚ/ℤ[\frac{1}{p}].

1.3.2. If F is any separably closed field of characteristic p, then K_{n}(F) is noncanonically a summand of K_{n}(F̄) by a transfer argument (as it is uniquely pdivisible for all n > 0, by IV.5.6). Therefore K_{n}(F) also has the structure described in Corollary 1.3.1; see Exercise 1.1.

We now turn to the structure of K_{∗}(F) when F has characteristic zero.

Proposition 1.4. If F is an algebraically closed field of characteristic 0 then for every m > 0 the choice of a Bott element β ∈ K_{2}(F; ℤ/m) determines a graded ring isomorphism K_{∗}(F; ℤ/m) ≅ ℤ/m[β].

Proof. Pick a primitive m^{th} root of unity ζ in ℚ̄, and let β be the corresponding Bott element in K_{2}(ℚ(ζ); ℤ/m). We use this choice to define a Bott element β ∈ K_{2}(E; ℤ/m) for all fields containing ℚ(ζ), natural in E. By Theorem 1.3, it suffices to show that the induced ring map ℤ/m[β] → K_{∗}(F; ℤ/m) is an isomorphism for some algebraically closed field F containing ℚ̄.

Fix an algebraic closure ℚ̄_{p} of ℚ_{p}, where p ∤ m. For each q = p^{ν}, let E_{q} denote the maximal algebraic extension of ℚ_{p} inside ℚ̄_{p} with residue field 𝔽_{q}; ℚ̄_{p} is the union of the E_{q}. For each q ≡ 1 (mod m), we saw in Example V.6.10.2 (which uses Gabber rigidity) that K_{∗}(E_{q}; ℤ/m) = ℤ/m[β]. If q|q′, the map K_{∗}(E_{q}; ℤ/m) → K_{∗}(E_{q′}; ℤ/m) is an isomorphism, by naturality of β. Taking the direct limit over q, we have K_{∗}(ℚ̄_{p}; ℤ/m) = ℤ/m[β], as desired. □

Remark 1.4.1. There is a map K_{∗}(ℂ; ℤ/m) → π_{∗}(BU; ℤ/m) arising from the change of topology; see IV.4.12.3. Suslin proved in [Su84] that this map is an isomorphism. We can formally recover this result from Proposition 1.4, since both rings are polynomial rings in one variable, and the generator β ∈ K_{2}(ℂ; ℤ/m) maps to a generator of π_{2}(BU; ℤ/m) by IV.1.13.2.

To determine the structure of K_{∗}(F) when F has characteristic 0, we need a result of Harris and Segal [HS, 3.1]. Let m = ℓ^{ν} be a prime power and R a ring containing the group μ_{m} of m^{th} roots of unity. The group μ_{m} ≀ Σ_{n} = (μ_{m})^{n} ⋊ Σ_{n} embeds into GL_{n}(R) as the group of invertible n×n matrices with only one nonzero entry in each row and column, every nonzero entry being in μ_{m}. Taking the union over n, the group G = μ_{m} ≀ Σ_{∞} embeds into GL(R).

There is an induced map π_{∗}(BG^{+}) → K_{∗}(R) when μ_{m} ⊂ R. Given a finite field 𝔽_{q} with ℓ ∤ q, transfer maps for R = 𝔽_{q}(ζ_{2ℓ}) give maps π_{∗}(BG^{+}) → K_{∗}(R) → K_{∗}(𝔽_{q}). Let μ_{(ℓ)}(F) denote the ℓ-primary subgroup of μ(F).

Theorem 1.5. (Harris-Segal) If ℓ ∤ q and |μ_{(ℓ)}(𝔽_{q}(ζ_{ℓ}))| = m, then each group π_{2i−1}B(μ_{m} ≀ Σ_{∞})^{+} contains a cyclic summand which maps isomorphically to the ℓ-Sylow subgroup of π_{2i−1}BGL(𝔽_{q})^{+} = K_{2i−1}(𝔽_{q}) ≅ ℤ/(q^{i} − 1).

Proof. Fix i and let N = ℓ^{ν} be the largest power of ℓ dividing q^{i} − 1. Theorem 3.1 of [HS] states that both π_{2i−1}(BG^{+})_{(ℓ)} → K_{2i−1}(𝔽_{q})_{(ℓ)} and π_{2i}(BG^{+}; ℤ/N) → K_{2i}(𝔽_{q}; ℤ/N) ≅ ℤ/N are onto. Since π_{2i}(BG^{+}; ℤ/N) is a ℤ/N-module (IV.2.2), the latter map splits, i.e., there is a ℤ/N summand in π_{2i}(BG^{+}; ℤ/N) and hence in π_{2i−1}(BG^{+}) mapping isomorphically onto K_{2i−1}(𝔽_{q}; ℤ/N) ≅ ℤ/N. □

Corollary 1.5.1. For each prime ℓ ≠ p and each i > 0, K_{2i−1}(ℚ̄_{p}) contains a nonzero torsion ℓ-group.

Proof. Fix a q = p^{ν} such that q ≡ 1 (mod ℓ), and q ≡ 1 (mod 4) if ℓ = 2, and let m be the largest power of ℓ dividing q − 1. We consider the set of all local fields E over ℚ_{p} (contained in a fixed common ℚ̄_{p}) whose ring of integers R has residue field 𝔽_{q}. For each such E, we have K_{2i−1}(R) ≅ K_{2i−1}(E) by V.6.9.2. If R_{q} denotes the union of these R, and E_{q} is the the union E, this implies that K_{2i−1}(R_{q}) ≅ K_{2i−1}(E_{q}).

Because G is a torsion group, the homology groups H_{∗}(G; ℚ) vanish for ∗ > 0, so the groups π_{∗}(BG^{+}) are torsion groups by the Hurewicz theorem. Since G → GL(𝔽_{q}) factors through GL(R_{q}), the surjection π_{∗}(BG^{+}) → π_{∗}BGL(𝔽_{q})^{+} = K_{∗}(𝔽_{q}) factors through π_{∗}BGL(R_{q})^{+} = K_{∗}(R_{q}) and hence through K_{2i−1}(E_{q}). It follows that K_{2i−1}(E_{q}) contains a torsion group mapping onto the ℓ-torsion subgroup of K_{2i−1}(𝔽_{q}) ≅ ℤ/(q − 1). Taking the direct limit over q, it follows that the ℓ-torsion subgroup of K_{2i−1}(ℚ̄_{p}; ℤ/m) maps onto the ℓ-torsion subgroup of K_{2i−1}(𝔽̄_{p}). □

Corollary 1.5.2. If q ≡ 1 (mod ℓ) and m is the order of μ_{(ℓ)}(𝔽_{q}) then each group K_{2i−1}(ℤ[ζ_{m}]) ≅ K_{2i−1}(ℚ(ζ_{m})) contains a cyclic summand mapping isomorphically onto the ℓ-primary component of K_{2i−1}(𝔽_{q}) ≅ ℤ/(q^{i} − 1).

In fact, the summand is the torsion subgroup of K_{2i−1}(ℚ(ζ_{m})), as we shall see in Theorem 8.2.

Proof. We have K_{2i−1}(ℤ[ζ_{m}]) ≅ K_{2i−1}(ℚ(ζ_{m})) by V.6.8, and there is a canonical ring map ℤ[ζ_{m}] → 𝔽_{q}. Since μ_{m} ⊂ ℤ[ζ_{m}], the split surjection of Theorem 1.5 factors through π_{2i−1}(BG^{+}) → K_{2i−1}(ℤ[ζ_{m}]) → K_{2i−1}(𝔽_{q}). □

Remark 1.5.3. Let S denote the symmetric monoidal category of finite free μ_{m}-sets (IV.4.1.1). The space K(S) is ℤ × BG^{+}, and the Barratt-Priddy Theorem identifies it with the zeroth space of the spectrum Σ^{∞}(BG_{+}). As pointed out in IV.4.10.1, the map BG^{+} → GL(R)^{+} arises from the free R-module functor S → P(R), and therefore K(S) → K(R) extends to a map of spectra K(S) → K(R).

If k ⊂ F is an inclusion of algebraically closed fields, K_{∗}(k) → K_{∗}(F) is an injection by V.6.7.2. The following result implies that it is an isomorphism on torsion subgroups, and that K_{n}(F) is divisible for n ≠ 0.

Theorem 1.6. Let F be an algebraically closed field of characteristic 0. Then (i) If n is even and n > 0, K_{n}(F) is uniquely divisible. (ii) If n = 2i − 1 is odd, K_{2i−1}(F) is the direct sum of a uniquely divisible group and a torsion group isomorphic to ℚ/ℤ.

Proof. Fix i > 0. For each prime ℓ, the group K_{2i−1}(F; ℤ/ℓ) is zero by Proposition 1.4. By the universal coefficient sequence IV.2.5, K_{2i−2}(F) has no ℓ-torsion and K_{2i−1}(F)/ℓ = 0. That is, K_{2i−1}(F) is ℓ-divisible for all ℓ and hence divisible, while K_{2i−2}(F) is torsionfree. We now consider the universal coefficient sequence

0 → K_{2i}(F)/ℓ → K_{2i}(F; ℤ/ℓ) → _{ℓ}K_{2i−1}(F) → 0.

The middle group is ℤ/ℓ by Proposition 1.4. By Corollary 1.5.1, the exponent ℓ subgroup of K_{2i−1}(F) is nonzero, and hence cyclic of order ℓ. This implies that K_{2i}(F)/ℓ = 0, i.e., the torsionfree group K_{2i}(F) is divisible.

Since a divisible abelian group is the direct sum of a uniquely divisible group and a divisible torsion group, a divisible torsion group is the sum of its Sylow subgroups, and an ℓ-primary divisible group is a direct sum of copies of ℤ/ℓ^{∞}, it follows that K_{2i−1}(F) is the direct sum of a uniquely divisible group and ⊕_{ℓ}ℤ/ℓ^{∞} ≅ ℚ/ℤ. □

We conclude this section with a description of the torsion module of K_{2i−1}(F) as a representation of the group Aut(F) of field automorphisms of F. For this we need some elementary remarks. If F is algebraically closed, there is a tautological action of Aut(F) on the group μ = μ(F) of roots of unity in F: g ∈ Aut(F) sends ζ to g(ζ). This action gives a surjective homomorphism Aut(F) → Aut(μ), called the cyclotomic representation. To describe Aut(μ), recall that the group μ(F) is isomorphic to either ℚ/ℤ or ℚ/ℤ[\frac{1}{p}], according to the characteristic of F.

Since any endomorphism of ℚ/ℤ induces an endomorphism of its exponent m subgroup ℤ/m, and is equivalent to a compatible family of such, End(ℚ/ℤ) is isomorphic to ℤ̂ = lim ℤ/m. It is easy to see that ℤ̂ is the product over all primes ℓ of the ℓ-adic integers ℤ̂_{ℓ}, so Aut(μ) ≅ ∏ ℤ̂_{ℓ}^{×}. A similar argument, with p ∤ m, shows that End(ℚ/ℤ[\frac{1}{p}]) is isomorphic to ∏_{ℓ≠p} ℤ̂_{ℓ}, and Aut(μ) ≅ ∏_{ℓ≠p} ℤ̂_{ℓ}^{×}.

If char(F) = 0, the subfield of F fixed by the kernel of Aut(F) → Aut(μ) is the infinite cyclotomic extension ℚ(μ) = ∪_{m}ℚ(ζ_{m}), by elementary Galois theory, and Aut(F) surjects onto Aut(ℚ(μ)) = Gal(ℚ(μ)/ℚ) ≅ Aut(μ) ≅ ℤ̂^{×}. If F is algebraically closed of characteristic p > 0, the situation is similar: Aut(F) surjects onto Aut(𝔽̄_{p}) = Gal(𝔽̄_{p}/F_{p}) ≅ Aut(μ); the Frobenius is topologically dense in this group.

Definition 1.7. For all i ∈ ℤ, we shall write μ(i) for the abelian group μ, made into a Aut(F)-module by letting g ∈ Aut(F) act as ζ ↦ g^{i}(ζ). (This modified module structure is called the i^{th} Tate twist of the cyclotomic module μ.) If M is any Aut(F)-submodule of μ, we write M(i) for the abelian group M, considered as a submodule of μ(i). In particular, its Sylow decomposition is μ(i) = ⊕ℤ/ℓ^{∞}(i).

Proposition 1.7.1. If F is algebraically closed and i > 0, the torsion submodule of K_{2i−1}(F) is isomorphic to μ(i) as an Aut(F)-module.

Proof. It suffices to show that the submodule _{m}K_{2i−1}(F) is isomorphic to μ_{m}(i) for all m > 0 prime to the characteristic. Fix a primitive m^{th} root of unity ζ in F, and let β be the corresponding Bott element. Then K_{∗}(F; ℤ/m) ≅ ℤ/m[β], by either 1.3.1 or 1.6; Since _{m}K_{2i−1}(F) ≅ K_{2i}(F; ℤ/m) as Aut(F)-modules, and the abelian group K_{2i}(F; ℤ/m) is isomorphic to ℤ/m on generator β^{i}.

By naturality of the product (IV, 1.10 and 2.8), the group Aut(F) acts on K_{∗}(F; ℤ/m) by ring automorphisms. For each g ∈ Aut(F) there is an a ∈ ℤ/m^{×} such that g(ζ) = ζ^{a}. Thus g sends β to aβ, and g sends β^{i} to (aβ)^{i} = a^{i}β^{i}. Since μ_{m}(i) is isomorphic to the abelian group ℤ/m with g acting as multiplication by a^{i}, we have μ_{m}(i) ≅ K_{2i}(F; ℤ/m). □

EXERCISES

1.1 Show that the conclusion of 1.3.1 holds for any separably closed field of characteristic p: if n > 0 is even then K_{n}(F) is uniquely divisible, while if n is odd then K_{n}(F) is the sum of a uniquely divisible group and (ℚ/ℤ)_{(p)}. Hint: By IV.5.6, K_{n}(F) is uniquely p-divisible for all n > 0. 1.2 (Suslin) Let k ⊂ F be an extension of algebraically closed fields, and let X be an algebraic variety over k. Write X_{F} for the corresponding variety X ⊗_{k} F over F. In this exercise we show that the groups G_{∗}(X; ℤ/m) are independent of k.

(i) If R is the local ring of a smooth curve C at a point c, show that there is a specialization map λ_{c} : G_{∗}(X_{k(C)}; ℤ/m) → G_{∗}(X; ℤ/m).

(ii) (Rigidity) Show that the specialization λ_{c} is independent of the choice of c. (iii) If h_{i} : A→k is as in 1.2, show that maps h_{i}^{∗} : G_{∗}(X_{A}; ℤ/m) → G_{∗}(X; ℤ/m)

exist and coincide.

(iv) Show that the base-change G_{∗}(X; ℤ/m) → G_{∗}(X_{F}; ℤ/m) is an isomorphism. 1.3 Let E be a local field, finite over ℚ_{p} and with residue field 𝔽_{q}. Use Theorem 1.6 and the proof of 1.5.1 to show that K_{2i−1}(E)_{tors} is the direct sum of ℤ/(q^{i} − 1) and a p-group, and that K_{2i−1}(E)_{tors} → K_{2i−1}(ℚ̄_{p}) is an injection modulo p-torsion. 1.4 The Galois group Γ = Gal(ℚ(ζ_{m})/ℚ) acts on μ_{m} and hence on the group G of Theorem 1.5. Show that the induced action of Γ on π_{∗}(BG^{+}) is trivial, and conclude that the summand of K_{2i−1}(ℚ(ζ)) in 1.5.2 is invariant under Γ.

§2. The e-invariant of a field

The odd-indexed K-groups of any field F have a canonical torsion summand, discovered by Harris and Segal in [HS]. It is detected by a map called the e-invariant, which we now define.

Let F̄ be a separably closed field, and μ = μ(F̄) the group of its roots of unity. We saw in Proposition 1.7.1 (and Ex. 1.1) that K_{2i−1}(F̄)_{tors} is isomorphic to the Tate twist μ(i) of μ as an Aut(F̄)-module (see Definition 1.7). The target group μ(i)^{G} is always the direct sum of its ℓ-primary Sylow subgroups μ_{(ℓ)}(i)^{G} ≅ ℤ/ℓ^{∞}(i)^{G}.

Definition 2.1. Let F be a field, with separable closure F̄ and Galois group G = Gal(F̄/F). Since K_{∗}(F) → K_{∗}(F̄) is a homomorphism of G-modules, with G acting trivially on K_{n}(F), it follows that there is a natural map

e : K_{2i−1}(F)_{tors} → K_{2i−1}(F̄)_{tors}^{G} ≅ μ(i)^{G}.

We shall call e the e-invariant.

If μ(i)^{G} is a finite group it is cyclic, and we write w_{i}(F) for its order, so that μ(i)^{G} ≅ ℤ/w_{i}(F). If ℓ is a prime, we write w_{i}^{(ℓ)}(F) for the order of μ_{(ℓ)}(i)^{G}. Thus the target of the e-invariant is ⨁_{ℓ} ℤ/w_{i}^{(ℓ)}(F), and w_{i}(F) = ∏ w_{i}^{(ℓ)}(F).

Example 2.1.1 (finite fields). It is a pleasant exercise to show that w_{i}(𝔽_{q}) = q^{i} − 1 for all i. Since this is the order of K_{2i−1}(𝔽_{q}) by IV.1.13, we see that in this case, the e-invariant is an isomorphism. (See Exercise IV.1.26.)

Example 2.1.2. If i is odd, w_{i}(ℚ) = 2 and w_{i}(ℚ(\sqrt{−1})) = 4. If i is even then w_{i}(ℚ) = w_{i}(ℚ(\sqrt{−1})), and ℓ|w_{i}(ℚ) exactly when (ℓ − 1) divides i. We have: w_{2} = 24, w_{4} = 240, w_{6} = 504 = 2^{3}·3^{2}·7, w_{8} = 480 = 2^{5}·3·5, w_{10} = 1320 = 2^{3}·3·5·11, and w_{12} = 65, 520 = 2^{4}·3^{2}·5·7·13. These formulas may be derived from Propositions 2.2 and 2.3 below.

In [LSz], Lee and Szczarba used a variant of the formula K_{3}(R) = H_{3}(St(R); ℤ) (Ex. IV.1.10) to show that K_{3}(ℤ) ≅ K_{3}(ℚ) ≅ ℤ/48. It follows that the e-invariant K_{3}(ℚ) → ℤ/24 cannot be an injection. (We will see in Remark 2.1.3 that it vanishes on the nonzero symbol {−1, −1, −1}.)

Remark 2.1.3. The complex Adams e-invariant for stable homotopy is a map from π_{2i−1}^{s} to ℤ/w_{i}(ℚ), whence our terminology. Quillen observed in [Qlet] that the Adams e-invariant is the composition π_{2i−1}^{s} → K_{2i−1}(ℚ) ⟶^{e} ℤ/w_{i}(ℚ). (Adams defined his e-invariant using π_{2i}(BU)/w_{i}(ℚ); Quillen’s assertions have been translated using Remark 1.4.1.)

If i is positive and divisible by 4, the real Adams e-invariant coincides with the complex e-invariant. If i ≡ 2 (mod 4), the real (Adams) e-invariant is a map π_{2i−1}^{s} → π_{2i}(BO)/2w_{i}(ℚ) = ℤ/2w_{i}(ℚ). For all even i > 0, Adams proved in 1966 that the real e-invariant restricts to an injection on the image of J : π_{2i−1}O → π_{2i−1}^{s} and induces an isomorphism (imJ)_{2i−1} ≅ ℤ/w_{i}(ℚ). His proof used the “Adams Conjecture,” which was later verified by Quillen. Quillen showed in [Qlet] that the real Adams e-invariant factors through K_{2i−1}(ℤ) = K_{2i−1}(ℚ), so (imJ)_{2i−1} injects into K_{2i−1}(ℤ). In particular, the image {−1, −1, −1} of η^{3} ∈ π_{3}^{s} is nonzero (see Ex. IV.1.12). Since the map from π_{8k+3}(BO) = ℤ to π_{8k+3}(BU) = ℤ has image 2ℤ, it follows that the e-invariant K_{8k+3}(ℚ) → ℤ/w_{4k+2}(ℚ) of Definition 2.1 is not an injection on (imJ)_{8k+3}.

Not all of the image of J injects into K_{∗}(ℤ). If n ≡ 0, 1 (mod 8) then J(π_{n}O) ≅ ℤ/2, but Waldhausen showed (in 1982) that these elements map to zero in K_{n}(ℤ).

Formulas for w_{i}(F)

We now turn to formulas for the numbers w_{i}^{(ℓ)}(F). Let ζ_{m} denote a primitive m^{th} root of unity. For odd ℓ, we have the following simple formula.

Proposition 2.2. Fix a prime ℓ ≠ 2, and let F be a field of characteristic ≠ ℓ. Let a ≤ ∞ be maximal such that F (ζ_{ℓ}) contains a primitive ℓ^{a}th root of unity and set r = [F (ζ_{ℓ}) : F]. If i = cℓ^{b}, where ℓ ∤ c, then the numbers w_{i}^{(ℓ)} = w_{i}^{(ℓ)}(F) are ℓ^{a+b} if r | i, and 1 otherwise. That is:

(a) If ζ_{ℓ} ∈ F then w_{i}^{(ℓ)} = ℓ^{a+b};

(b) If ζ_{ℓ} ∉ F and i ≡ 0 (mod r) then w_{i}^{(ℓ)} = ℓ^{a+b};

(c) If ζ_{ℓ} ∉ F and i ≢ 0 (mod r) then w_{i}^{(ℓ)} = 1.

Proof. Since ℓ is odd, G = Gal(F (ζ_{ℓ^{ν}})/F) is a cyclic group of order rℓ^{ν−a} for all ν ≥ a. If a generator of G acts on μ_{ℓ^{a+ν}} by ζ ↦ ζ^{g} for some g ∈ (ℤ/ℓ^{a+ν})^{×} then it acts on μ^{⊗i} by ζ ↦ ζ^{g^{i}}. Now use the criterion of Lemma 2.2.1; if r | i then Gal(F (ζ_{ℓ^{a+b}})/F) is cyclic of order rℓ^{b}, while if r ∤ i the exponent r of Gal(F (ζ_{ℓ})/F) does not divide i. □

Lemma 2.2.1. w_{i}^{(ℓ)}(F) = max{ℓ^{ν} | Gal(F (ζ_{ℓ^{ν}})/F) has exponent dividing i}

Proof. Set ζ = ζ_{ℓ^{ν}}. Then ζ^{⊗i} is invariant under g ∈ Gal(F̄/F) precisely when g^{i}(ζ) = ζ, and ζ^{⊗i} is invariant under all of G precisely when the group Gal(F (ζ_{ℓ^{ν}})/F) has exponent i. □

Example 2.2.2. Consider F = ℚ(ζ_{p^{a}}). If i = cp^{b} then w_{i}^{(p)}(F) = p^{a+b} (p ≠ 2). If ℓ ≠ 2, p then w_{i}^{(ℓ)}(F) = w_{i}^{(ℓ)}(ℚ) for all i. This number is 1 unless (ℓ − 1) | i; if (ℓ − 1)|i but ℓ ∤ i then w_{i}^{(ℓ)}(F) = ℓ. In particular, if ℓ = 3 and p ≠ 3 then w_{i}^{(3)}(F) = 1 for odd i, and w_{i}^{(3)}(F) = 3 exactly when i ≡ 2, 4 (mod 6).

The situation is more complicated when ℓ = 2, because Aut(μ_{2^{ν}}) = (ℤ/2^{ν})^{×} contains two involutions if ν ≥ 3. We say that a field F is exceptional if char(F) = 0 and the Galois groups Gal(F (ζ_{2^{ν}})/F) are not cyclic for large ν. If F is not exceptional, we say that it is non-exceptional.

Proposition 2.3. (ℓ = 2) Let F be a field of characteristic ≠ 2. Let a be maximal such that F (\sqrt{−1}) contains a primitive 2^{a}th root of unity. If i = c2^{b}, where 2 ∤ c, then the 2-primary numbers w_{i}^{(2)} = w_{i}^{(2)}(F) are:

(a) If \sqrt{−1} ∈ F then w_{i}^{(2)} = 2^{a+b} for all i.

(b) If \sqrt{−1} ∉ F and i is odd then w_{i}^{(2)} = 2.

(c) If \sqrt{−1} ∉ F, F is exceptional and i is even then w_{i}^{(2)} = 2^{a+b}.

(d) If \sqrt{−1} ∉ F, F is non–exceptional and i is even then w_{i}^{(2)} = 2^{a+b−1}.

The proof of Proposition 2.3(a,b,d) is almost identical to that of 2.2 with r = 1. The proof in the exceptional case (c) is relegated to Exercise 2.2.

Both ℝ and ℚ_{2} are exceptional, and so are each of their subfields. In particular, real number fields (like ℚ) are exceptional, and so are some totally imaginary number fields, like ℚ(\sqrt{−7}).

Example 2.3.1 (local fields). Let E be a local field, finite over ℚ_{p} and with residue field 𝔽_{q}. Then w_{i}(E) is w_{i}(𝔽_{q}) = q^{i} − 1 times a power of p. (The precise power of p is given in Exercise 2.3 when E = ℚ_{p}.) This follows from Propositions 2.2 and 2.3, using the observation that (for ℓ ≠ p) the number of ℓ-primary roots of unity in E(ζ_{ℓ}) is the same as in 𝔽_{q}(ζ_{ℓ}).

By Exercise 1.3, the map K_{2i−1}(E)_{tors} ⟶^{e} ℤ/w_{i}(E) is a surjection up to p-torsion, and induces an isomorphism on ℓ-primary torsion subgroups K_{2i−1}(E){ℓ} ≅ ℤ/w_{i}^{(ℓ)} for ℓ ≠ p. We will see in Proposition 7.3 that the torsion subgroup of K_{2i−1}(E) is exactly ℤ/w_{i}(E).

Bernoulli numbers The numbers w_{i}(ℚ) are related to the Bernoulli numbers B_{k}. These were defined by Jacob Bernoulli in 1713 as coefficients in the power series

\frac{t}{e^{t} − 1} = 1 − \frac{t}{2} + ∑_{k=1}^{∞}(−1)^{k+1}B_{k} \frac{t^{2k}}{(2k)!}.

(We use the topologists’ B_{k} from [MSt], all of which are positive. Number theorists would write it as (−1)^{k+1}B_{2k}.) The first few Bernoulli numbers are:

B_{1} = \frac{1}{6}, B_{2} = \frac{1}{30}, B_{3} = \frac{1}{42}, B_{4} = \frac{1}{30}, B_{5} = \frac{5}{66}, B_{6} = \frac{691}{2730}, B_{7} = \frac{7}{6}, B_{8} = \frac{3617}{510}.

The denominator of B_{k} is always squarefree, divisible by 6, and equal to the product of all primes with (p−1)|2k. Moreover, if (p−1) ∤ 2k then p is not in the denominator of B_{k}/k even if p|k; see [MSt]. From this information, it is easy to verify the following identity. Recall from 2.1.2 that w_{i}(ℚ) = 2 when i is odd.

Lemma 2.4. If i = 2k is even then w_{i}(ℚ) is the denominator of B_{k}/4k. The prime ℓ divides w_{i}(ℚ) exactly when (ℓ − 1) divides i.

Although the numerator of B_{k} is difficult to describe, it is related to the notion of irregular primes, which we now define.

Iregular Primes 2.4.1. A prime p is called irregular if p divides the order h_{p} of Pic(ℤ[μ_{p}]); if p is not irregular it is called regular. Iwasawa proved that a prime p is regular if and only if Pic(ℤ[μ_{p^{ν}}]) has no p-torsion for all ν. The smallest irregular primes are 37, 59, 67, 101, 103, 131 and 149. Siegel conjectured that asymptotically about 39% of all primes are irregular, and about 39% of the primes less than 4 million are irregular.

Kummer proved that p is irregular if and only if p divides the numerator of one of the Bernoulli numbers B_{k}, k ≤ (p − 3)/2 (see [Wash, 5.34]). By Kummer’s congruences ([Wash, 5.14]), a regular prime p does not divide the numerator of any B_{k}/k (but 5|B_{5}). Thus only irregular primes can divide the numerator of B_{k}/k.

The historical interest in regular primes is Kummer’s 1847 proof of Fermat’s Last Theorem (case I) for regular primes: x^{p} + y^{p} = z^{p} has no solution in which p ∤ xyz. For us, certain calculations of K-groups become easier at regular primes. (See Example 8.3.2 and Proposition 10.5.)

Remark 2.4.2. Bernoulli numbers also arise as values of the Riemann zeta function. Euler proved (in 1735) that ζ_{ℚ}(2k) = B_{k}(2π)^{2k}/2(2k)!. By the functional equation, we have ζ_{ℚ}(1 − 2k) = (−1)^{k}B_{k}/2k. By Lemma 2.4, the denominator of ζ_{ℚ}(1 − 2k) is \frac{1}{2}w_{2k}(ℚ).

Remark 2.4.3. The Bernoulli numbers are of interest to topologists because if n = 4k − 1 the image of J : π_{n}SO → π_{n}^{s} is cyclic of order equal to the denominator of B_{k}/4k, and the numerator determines the number of exotic (4k − 1)-spheres which bound parallelizable manifolds; see [MSt, App.B].

Harris-Segal Theorem 2.5. Let F be a field with 1/ℓ ∈ F; if ℓ = 2, we also suppose that F is non-exceptional. Set w_{i} = w_{i}^{(ℓ)}(F). Then each K_{2i−1}(F) has a direct summand isomorphic to ℤ/w_{i}(F), detected by the e-invariant.

If F is the field of fractions of an integrally closed domain R then K_{2i−1}(R) also has a direct summand isomorphic to ℤ/w_{i}(F), detected by the e-invariant.

The splitting ℤ/w_{i} → K_{2i−1}(R) is called the Harris-Segal map, and its image is called the (ℓ-primary) Harris-Segal summand of K_{2i−1}(R).

We will see in Theorem 8.2 below that ℤ/w_{i} is the torsion subgroup of K_{2i−1}(ℤ[ζ_{ℓ^{a}}]). It follows that the Harris-Segal map is unique, and hence so is the Harris-Segal summand of K_{2i−1}(R). This uniqueness was originally established by Kahn and others.

Proof. Suppose first that either ℓ ≠ 2 and ζ_{ℓ} ∈ R, or that ℓ = 2 and ζ_{4} ∈ R. If R has m = ℓ^{a} ℓ-primary roots of unity, then w_{i}^{(ℓ)}(ℚ(ζ_{m})) equals w_{i} = w_{i}^{(ℓ)}(F) by 2.2 and 2.3. Thus there is no loss in generality in assuming that R = ℤ[ζ_{m}]. Pick a prime p with p ≢ 1 (mod ℓ^{a+1}). Then ζ_{ℓ^{a+1}} ∉ 𝔽_{p}, and if 𝔭 is any prime ideal of R = ℤ[ζ_{m}] lying over p then the residue field R/𝔭 is 𝔽_{q} = 𝔽_{p}(ζ_{m}). We have w_{i} = w_{i}^{(ℓ)}(𝔽_{q}) by Example 2.2.2.

The quotient map R → R/𝔭 factors through the 𝔭-adic completion R̂_{𝔭}, whose field of fractions is the local field E = ℚ_{p}(ζ_{m}). By Example 2.3.2, w_{i}^{(ℓ)}(E) = w_{i} and K_{2i−1}(E){ℓ} ⟶^{e} ℤ/w_{i} is an isomorphism. Now the e-invariant for the finite group K_{2i−1}(R) is the composite

K_{2i−1}(R)_{(ℓ)} ≅ K_{2i−1}(ℚ(ζ_{m}))_{(ℓ)} ⟶ K_{2i−1}(E){ℓ} ⟶^{e} ℤ/w_{i}.

By Corollary 1.5.2, K_{2i−1}(R) contains a cyclic summand A of order w_{i}, mapping to the summand ℤ/w_{i} of K_{2i−1}(𝔽_{q}) under K_{2i−1}(R) → K_{2i−1}(R̂_{𝔭}) → K_{2i−1}(𝔽_{q}). Therefore A injects into (and is isomorphic to) K_{2i−1}(R̂_{𝔭}){ℓ} ≅ K_{2i−1}(E){ℓ} ≅ ℤ/w_{i}. The theorem now follows in this case.

Suppose now that ζ_{ℓ} ∉ R. By Exercise 2.5, we may assume that F is a subfield of ℚ(ζ_{m}), that ℚ(ζ_{m}) = F (ζ_{ℓ}), and that R is the integral closure of ℤ in F. We may suppose that r = [ℚ(ζ_{m}) : F] divides i since otherwise w_{i} = 1, and set Γ = Gal(ℚ(ζ_{m})/F). By Proposition 2.2, w_{i} = w_{i}^{(ℓ)}(ℚ[ζ_{m}]). We have just seen that there is a summand A of K_{2i−1}(ℚ[ζ_{m}]) mapping isomorphically to ℤ/w_{i} by the e-invariant. By Ex. 1.4, Γ acts trivially on A.

Since f : R → ℤ[ζ_{m}] is Galois, the map f^{∗}f_{∗} is multiplication by ∑_{g∈G} g on K_{2i−1}(ℤ[ζ_{m}]), and hence multiplication by r on A (see Ex. IV.6.13). Since f_{∗}f^{∗} is multiplication by r on f_{∗}(A), we see that f^{∗} : f_{∗}(A) → A is an isomorphism with inverse f_{∗}/r. Hence f_{∗}(A) is a summand of K_{2i−1}(R), and the e-invariant

K_{2i−1}(R) ⟶^{f^{∗}} K_{2i−1}(ℤ[ζ_{m}]) ⟶^{e} ℤ/w_{i} maps f_{∗}(A) isomorphically to ℤ/w_{i}.

When ℓ = 2 and F is non-exceptional but \sqrt{−1} ∉ F, we may again assume by Ex. 2.5 that F is a subfield of index 2 in ℚ(ζ_{m}) = F (\sqrt{−1}). By Proposition 2.3, w_{i}^{(2)}(ℚ(ζ_{m})) = 2w_{i} and there is a summand A of K_{2i−1}(ℚ(ζ_{m})) mapping isomorphically to ℤ/2w_{i} by the e-invariant; by Ex. 1.4, Γ acts trivially on A and we set Ā = f_{∗}(A). Since f^{∗}f_{∗} is multiplication by 2 on A, the image of Ā is 2A. From the diagram

Ā ⊂ K_{2i−1}(F) ⟶^{e} ℤ/w_{i}
f^{∗}↓  ↓^{include}
A ⊂ K_{2i−1}(ℚ(ζ_{m})) ⟶^{e} ℤ/2w_{i}

we see that Ā ≅ 2A ≅ ℤ/w_{i}, as desired. □

Remark 2.5.1. If F is an exceptional field, a transfer argument using F (\sqrt{−1}) shows that there is a cyclic summand in K_{2i−1}(F) whose order is either w_{i}(F), 2w_{i}(F) or w_{i}(F)/2. (Exercise 2.4); we will also call these Harris-Segal summands.

When F is a totally imaginary number field, we will see in Theorem 8.4 below that the Harris-Segal summand always has order w_{i}(F). The following theorem, extracted from Theorem 9.5 below, shows that all possibilities occur for real number fields, i.e., number fields embeddable in ℝ.

Theorem 2.6. Let F be a real number field. Then the Harris-Segal summands in K_{2i−1}(F) and K_{2i−1}(O_{F}) are isomorphic to:

(1) ℤ/w_{i}(F), if i ≡ 0 (mod 4) or i ≡ 1 (mod 4), i.e., 2i − 1 ≡ ±1 (mod 8);

(2) ℤ/2w_{i}(F), if i ≡ 2 (mod 4), i.e., 2i − 1 ≡ 3 (mod 8);

(3) ℤ/\frac{1}{2}w_{i}(F), if i ≡ 3 (mod 4), i.e., 2i − 1 ≡ 5 (mod 8).

Example 2.7. Let F = ℚ(ζ + ζ^{−1}) be the maximal real subfield of the cyclotomic field ℚ(ζ), ζ^{p} = 1 with p odd. Then w_{i}(F) = 2 for odd i, and w_{i}(F) = w_{i}(ℚ(ζ)) for even i > 0 by 2.2 and 2.3 (see Ex. 2.6). Note that p | w_{i}(F (ζ)) for all i, p | w_{i}(F) if and only if i is even, and p | w_{i}(ℚ) only when (p − 1)|i; see 2.2.2.

If n ≡ 3 (mod 4), the groups K_{n}(ℤ[ζ + ζ^{−1}]) = K_{n}(F) are classically finite (see 8.1); the order of their Harris-Segal summands are given by Theorem 2.6. When n ≢ −1 (mod 2p − 2), the group K_{n}(F) has an extra p-primary factor not coming from the image of J (see 2.1.3).

EXERCISES

2.1 For every prime ℓ with 1/ℓ ∈ F, show that the following are equivalent: (i) F (ζ_{ℓ}) has only finitely many ℓ-primary roots of 1; (ii) w_{i}^{(ℓ)}(F) is finite for some i ≡ 0 (mod 2(ℓ − 1)); (iii) w_{i}^{(ℓ)}(F) is finite for all i > 0.

2.2 Prove Proposition 2.3(c), giving the formula w_{i}^{(2)}(F) = 2^{a+b} when i is even and F is exceptional. Hint: Consider μ(i)^{H}, H = Gal(F̄/F (\sqrt{−1})). 2.3 If p is odd, show that w_{i}(ℚ_{p}) = p^{i} − 1 unless (p − 1)|i, and if i = m(p − 1)p^{b} (p ∤ m) then w_{i}(ℚ_{p}) = (p^{i} − 1)p^{1+b}.

For p = 2, show that w_{i}(ℚ_{2}) = 2(2^{i} − 1) for i odd; if i is even, say i = 2^{b}m with m odd, show that w_{i}(ℚ_{2}) = (2^{i} − 1)2^{2+b}. 2.4 Let f : F → E be a field extension of degree 2, and suppose x ∈ K_{∗}(E) is fixed by Gal(E/F). If x generates a direct summand of order 2m, show that f_{∗}(x) is contained in a cyclic summand of K_{∗}(F) of order either m, 2m or 4m. 2.5 Let F be a field of characteristic 0. If ℓ ≠ 2 and a < ∞ is as in 2.2, show that there is a subfield F_{0} of ℚ(ζ_{ℓ^{a}}) such that w_{i}^{(ℓ)}(F_{0}) = w_{i}^{(ℓ)}(F). If ℓ = 2 and a < ∞ is as in 2.3, show that there is a subfield F_{0} of ℚ(ζ_{2^{a}}) such that w_{i}^{(2)}(F_{0}) = w_{i}^{(2)}(F), and that F_{0} is exceptional (resp., non-exceptional) if F is. 2.6 Let ℓ be an odd prime, and F = ℚ(ζ_{ℓ} +ζ_{ℓ}^{−1}) the maximal real subfield of ℚ(ζ_{ℓ}). Show that w_{i}(F) = 2 for odd i, and that w_{i}(F) = w_{i}(ℚ(ζ_{ℓ})) for even i > 0. In particular, ℓ|w_{i}(ℚ(ζ_{ℓ})) for all i, but ℓ|w_{i}(F) if and only if i is even.

§3. The K-theory of ℝ

In this section, we describe the algebraic K-theory of the real numbers ℝ, or rather the torsion subgroup K_{n}(ℝ)_{tors} of K_{n}(ℝ). Here is the punchline:

Theorem 3.1. (Suslin) For all n ≥ 1,

(a) K_{n}(ℝ) is the direct sum of a uniquely divisible group and K_{n}(ℝ)_{tors}.

(b) The torsion groups and K_{n}(ℝ)_{tors} → K_{n}(ℂ)_{tors} are given by Table 3.1.1.

(c) The map K_{n}(ℝ; ℤ/m) → K̃O(S^{n}; ℤ/m) = π_{n}(BO; ℤ/m) is an isomorphism.

i (mod 8)  1  2  3 4  5  6  7 8
K_{i}(ℝ)  ℤ/2 ℤ/2  ℚ/ℤ 0  0  0  ℚ/ℤ 0
↓  injects  0 a ↦ 2a 0  0  0  ≅ 0
K_{i}(ℂ)  ℚ/ℤ  0  ℚ/ℤ 0 ℚ/ℤ  0  ℚ/ℤ 0
↓  0  0  ≅ 0  0  0 a ↦ 2a 0
K_{i}(ℍ)  0  0  ℚ/ℤ 0 ℤ/2 ℤ/2  ℚ/ℤ 0

Table 3.1.1. The torsion subgroups of K_{n}(ℝ), K_{n}(ℂ) and K_{n}(ℍ), n > 0.

Remark 3.1.2. Table 3.1.1 shows that K_{n}(ℝ)_{tors} ≅ π_{n+1}(BO; ℚ/ℤ) is the Harris-Segal summand of K_{n}(ℝ) (in the sense of 2.5.1) for all n odd; the e-invariant (Definition 2.1) is the map from K_{n}(ℝ)_{tors} to K_{n}(ℂ)_{tors}. When n is 8k + 1, the ℤ/2-summand in K_{n}(ℝ) is generated by the image of Adams’ element μ_{n} ∈ π_{n}^{s}; Adams showed that μ_{n} is detected by the complex Adams e-invariant. Adams also showed that the elements μ_{8k+2} = η · μ_{8k+1} and μ_{8k+3} = η^{2} · μ_{8k+1} are nonzero and detected by the real Adams e-invariant which, as Quillen showed in [Qlet], maps π_{n}^{s} to K_{n}(ℝ)_{tors} ≅ π_{n+1}(BO; ℚ/ℤ). (See Remark 2.1.3.) It follows that when n = 8k + 3, the kernel of the e-invariant K_{n}(ℝ)_{tors} → ℚ/ℤ is isomorphic to ℤ/2, and is generated by the nonzero element {−1, −1, μ_{8k+1}}. When n = 8k + 5, the Harris-Segal summand is zero even though the target of the e-invariant is ℤ/2.

A similar calculation for the quaternions ℍ is also due to Suslin [Su86, 3.5]. The proof uses the algebraic group SL_{n}(ℍ) in place of SL_{n}(ℝ).

Theorem 3.2. (Suslin) For all n ≥ 1,

(a) K_{n}(ℍ) is the direct sum of a uniquely divisible group and K_{n}(ℍ)_{tors}.

(b) The torsion groups and K_{n}(ℂ)_{tors} → K_{n}(ℍ)_{tors} are given by Table 3.1.1.

(c) The map K_{n}(ℍ; ℤ/m) → K̃Sp(S^{n}; ℤ/m) = π_{n}(BSp; ℤ/m) is an isomorphism.

The method of proof uses a universal homotopy construction which is of independent interest, and also gives an alternative calculation of K_{∗}(ℂ) to the one we gave in Proposition 1.4. As observed in 1.4.1, the punchline of that calculation is that, for all n ≥ 1, K_{n}(ℂ; ℤ/m) → K̃U(S^{n}; ℤ/m) = π_{n}(BU; ℤ/m) is an isomorphism.

We begin with some general remarks. If G is a topological group, then it is important to distinguish between the classifying space BG^{δ} of the discrete group G, which is the simplicial set of IV.3.4.1, and the classifying space BG^{top} of the topological group G^{top}, which is discussed in IV.3.9. For example, the homotopy groups of BGL(ℂ)^{δ} are zero, except for the fundamental group, while the homotopy groups of BGL(ℂ)^{top} ≃ BU are given by Bott Periodicity II.3.1.1.

§3. THE K-THEORY OF ℝ 13

Next, suppose that G is a Lie group having finitely many components, equipped with a left invariant Riemannian metric. Given ε > 0, let G_{ε} denote the ε–ball about 1. If ε is small, then G_{ε} is geodesically convex; the geodesic between any two points lies in G_{ε}.

Definition 3.3. Let BG_{ε} denote the simplicial subset of BG^{δ} whose p-simplices are the p-tuples [g_{1}, …, g_{p}] such that there is a point in the intersection of G_{ε} with all the translates g_{1} ⋯ g_{i}G_{ε}, i ≤ p. (This condition is preserved by the face and degeneracy maps of BG^{δ}.)

Suslin proved the following result in [Su84, 4.1].

Theorem 3.4. (Suslin) Let G be a Lie group. If ε is small enough so that G_{ε} is geodesically convex, then BG_{ε} → BG^{δ} → BG^{top} is a homotopy fibration.

The next step is the construction of a universal chain homotopy. Given a commutative ring R, the algebraic group GL_{n} is Spec(H), where H is the Hopf algebra R[{x_{ij}}_{i,j=1}^{n}][det(X)^{−1}], where det(X) is the determinant of the universal matrix X = (x_{ij}) in GL_{n}(H). For every commutative R-algebra B there is a bijection Hom_{R}(H, B) → GL_{n}(B) sending f to the matrix f(X); the counit structure map H → R corresponds to the identity matrix of GL_{n}(R).

For each positive integer p, let A^{p} denote the henselization of the p-fold tensor product H^{⊗p} along the kernel of the evident structure map H^{⊗p} → R, so that (A^{p}, I^{p}) is a hensel pair, where I^{p} is the kernel of A^{p} → R. For i = 1, …, p, the coordinates pr_{i} : H → H^{⊗p} → A^{p} determine matrices X^{i} = pr_{i}(X) in GL_{n}(A^{p}), and since X^{i} is congruent to the identity modulo I^{p} we even have X^{i} ∈ GL_{n}(I^{p}).

Recall that for any discrete group G and integer m, the homology H_{∗}(G, ℤ/m) of G is the homology of a standard complex, which we will write as C_{∗}(G), whose degree p piece is ℤ/m[G^{p}]; see [WHomo, 6.5]. We write u^{p} for the p-chain [X^{1}, ..., X^{p}] in C_{p}(GL_{n}(I^{p})). The differential d sends u^{p} to

[X^{2}, ..., X^{p}] + ∑_{i=1}^{p−1}(−1)^{i}[…, X^{i}X^{i+1}, …] + (−1)^{p−1}[X^{1}, ..., X^{p−1}].

Now the A^{p} fit together to form a cosimplicial R-algebra A^{•}, whose cofaces ∂^{j} : A^{p} → A^{p+1} are induced by the comultiplication Δ : H → H ⊗ H. Applying GL_{n} yields cosimplicial groups GL_{n}(A^{•}) and GL_{n}(I^{•}). We are interested in the cosimplicial chain complex C_{∗}(GL_{n}(I^{•})), which we may regard as a third quadrant double chain complex, with C_{p}(GL_{n}(I^{−q})) in the (p, q) spot. Thus (u^{p}) = (0, u^{1}, u^{2}, …) is an element of total degree 0 in the associated product total complex, i.e., in ∏_{p=0}^{∞} C_{p}(GL_{n}(I^{p})) (see [WHomo, 1.2.6]). By construction, d(u^{p}) = ∑(−1)^{j}∂^{j}(u^{p−1}), so u^{p} is a cycle in this total complex.

Proposition 3.5. For each n, the image of the cycle (u^{p}) in ∏_{p=0}^{∞} C_{p}(GL(I^{p})) is a boundary in the product total complex of C_{∗}(GL(I^{•})). That is, there are chains c^{p} ∈ C_{p+1}(GL(I^{p})) so that d(c^{p}) + ∑(−1)^{j}∂^{j}(c^{p−1}) = u^{p} for all p ≥ 1.

Proof. Since the reduced complex C̃_{∗}(G) is the subcomplex of C_{∗}(G) obtained by setting C_{0} = 0, and u^{p} ∈ C̃_{p}, it suffices to show that (u^{p}) is a boundary in the total complex T_{∗} of C̃_{∗}(GL(I^{•})). By Gabber Rigidity IV.2.10, the reduced homology H̃_{∗}(GL(I^{t}), ℤ/m) is zero for each t. Thus the rows C̃_{∗}(GL(I^{−q})) of the double complex are exact. By the Acyclic Assembly Lemma [WHomo, 2.7.3], the product total complex T_{∗} is exact, so every cycle is a boundary. □

Lemma 3.6. For G = GL_{n}(ℝ), if ε is small enough then the embedding BG_{ε} → BG^{δ} → BGL(ℝ) induces the zero map on H̃_{∗}(−, ℤ/m).

Proof. Let B^{p} denote the ring of germs of continuous ℝ-valued functions on the topological space G^{p} = G × ⋯ × G defined in some neighborhood of (1, …, 1); B^{p} is a hensel ring. The coordinate functions give a canonical map H^{⊗p} → B^{p}, whose henselization is a map A^{p} → B^{p}, and we write c_{ctn}^{p} for the image of c^{p} in C_{p+1}(GL(B^{p})). Since GL(B^{p}) is the group of germs of continuous GL(ℝ)-valued functions on G^{p}, u^{p} is the germ of the function G^{p} → GL(ℝ)^{p} sending (g_{1}, …, g_{p}) to (g_{1}, …, g_{p}), and c_{ctn}^{p} is a ℤ/m-linear combination of germs of continuous functions γ : G^{p} → GL(ℝ)^{p+1}. That is, we may regard each c_{ctn}^{p} as a continuous map of some neighborhood of (1, …, 1) to C_{p+1}(GL(ℝ)^{top}).

If N is a fixed integer, there is an ε > 0 so that the c_{ctn}^{p} are defined on (G_{ε})^{p} for all p ≤ N. Extending c_{ctn}^{p} by linearity, we get homomorphisms s^{p} : C_{p}(BG_{ε}) → C_{p+1}(BSL(ℝ)^{top}). It is clear from Proposition 3.5 that s is a chain contraction for the canonical embedding C̃_{∗}(BG_{ε}) → C̃_{∗}(BGL(ℝ)^{top}), defined in degrees at most N. This proves the Lemma. □

Proposition 3.7. For G = SL_{n}(ℝ), if ε is small enough then

(a) the embedding BG_{ε} → BG^{δ} → BSL(ℝ) induces the zero map on H̃_{∗}(−, ℤ/m).

(b) H̃_{i}(BG_{ε}, ℤ/m) = 0 for all i ≤ (n − 1)/2.

(c) H_{i}(BG^{δ}, ℤ/m) → H_{i}(BG^{top}, ℤ/m) is an isomorphism for all i ≤ (n − 1)/2.

Proof. Since H_{∗}(BSL(ℝ)) → H_{∗}(BGL(ℝ)) is a split injection (see Ex. 3.1), part (a) follows from Lemma 3.6. For (b) and (c), let q be the smallest integer such that H̃_{q}(BG_{ε}, ℤ/m) is nonzero; we will show that q > (n − 1)/2. Since BG_{ε} has only one 0-simplex, q ≥ 1. Consider the Serre spectral sequence associated to 3.4:

E_{p,q}^{2} = H_{p}(BG^{top}, H_{q}(BG_{ε}, ℤ/m)) ⇒ H_{∗}(BG^{δ}, ℤ/m).

Then H_{i}(BG^{δ}, ℤ/m) → H_{i}(BG^{top}, ℤ/m) is an isomorphism for i < q and the exact sequence of low degree terms for the spectral sequence is

H_{q+1}(BG^{δ}, ℤ/m) ⟶^{onto} H_{q+1}(BG^{top}, ℤ/m) ⟶^{d^{q}} H_{q}(BG_{ε}, ℤ/m) → H_{q}(BG^{δ}, ℤ/m).

Milnor proved in [M-Lie, Thm. 1] that the left-hand map is a split surjection; it follows that the right-hand map is an injection. By Homological Stability IV.1.15, H_{i}(BSL_{n}(ℝ)^{δ}) → H_{i}(BSL(ℝ)^{δ}) is an isomorphism for i ≤ (n − 1)/2; part (a) implies that q > (n − 1)/2. This proves parts (b) and (c). □

Corollary 3.8. BSL(ℝ)^{δ} → BSL(ℝ)^{top} and BGL(ℝ)^{δ} → BGL(ℝ)^{top} induce isomorphisms on H̃_{∗}(−, ℤ/m).

Proof. Set G = SL_{n}(ℝ). Passing to the limit as n → ∞ in 3.7 proves the assertion for SL. □

§4. RELATION TO MOTIVIC COHOMOLOGY 15

Proof of Theorem 3.1. Since BSL(ℝ)^{+} and BSL(ℝ)^{top} ≃ BSO are simply connected, Corollary 3.8 implies that π_{n}(BSL(ℝ)^{+}; ℤ/m) → π_{n}(BSO; ℤ/m) is an isomorphism for all n. We saw in Chapter IV, Ex. 1.8 that π_{n}(BSL(ℝ)^{+}; ℤ/m) → K_{n}(ℝ; ℤ/m) is an isomorphism for n ≥ 3, and an injection for n = 2 with cokernel ℤ/2; the same is true for π_{n}(BSO; ℤ/m) → π_{n}(BO; ℤ/m). This proves part (c) for n ≥ 3; the result for n = 1, 2 is classical (see III.1.5.4 and IV.2.5.1, or [Milnor, p. 61]).

Now consider the action of complex conjugation c on K_{∗}(ℂ). The image of i^{∗} : K_{n}(ℝ) → K_{n}(ℂ) lands in the invariant subgroup K_{n}(ℂ)^{c}, which by Theorem 1.6 is the direct sum of a uniquely divisible group and a torsion group (which is one of 0, ℤ/2, 0 or ℚ/ℤ depending on n modulo 4). The transfer i_{∗} satisfies i_{∗}i^{∗} = 2 and i^{∗}i_{∗} = 1 + c; see Ex. IV.6.13. Hence (for n ≥ 1) K_{n}(ℝ) is the direct sum of its torsion submodule K_{n}(ℝ)_{tors} and the uniquely divisible abelian group K_{n}(ℂ)^{c} ⊗ ℚ, and the kernel and cokernel of K_{n}(ℝ)_{tors} → K_{n}(ℂ)_{tors}^{c} are elementary abelian 2groups. By Ex. IV.2.6, we have K_{n}(ℝ)_{tors} ≅ K_{n+1}(ℝ; ℚ/ℤ) ≅ π_{n+1}(BO, ℚ/Z). These torsion groups may be read off from Bott Periodicity II.3.1.1. □

EXERCISES

3.1 For any commutative ring R and ideal I, show that R^{×} acts trivially on the homology of SL(R) and GL(R), while (1 + I)^{×} acts trivially on the homology of SL(I) and GL(I). Conclude that SL(R) → GL(R) and SL(I) → GL(I) are split injections on homology. 3.2 Using Exercise 3.1, show that there is a universal homotopy construction for SL_{n} parallel to the one in Proposition 3.5 for GL_{n}. Use this to prove Proposition 3.7 directly, modifying the proof of Lemma 3.6. 3.3 Check that 3.6, 3.7 and 3.8 go through with ℝ replaced by ℂ. Using these, prove the analogue of Theorem 3.1 for ℂ and compare it to Theorem 1.5. 3.4 If F is any formally real field, such as ℝ ∩ ℚ̄, show that K_{∗}(F; ℤ/m) ≅ K_{∗}(ℝ; ℤ/m) for all m.

§4. Relation to motivic cohomology

Motivic cohomology theory, developed by Voevodsky, is intimately related to algebraic K-theory. For every abelian group A and every (n, i), the motivic cohomology of a smooth scheme X over a field consists of groups H^{n}(X, A(i)), defined as the hypercohomology of a certain chain complex A(i) of Nisnevich sheaves. An introduction to motivic cohomology is beyond the scope of this book, and we refer the reader to [MVW] for the definitions and properties of motivic cohomology.

When A = ℚ, we have isomorphisms H^{n}(X, ℚ(i)) ≅ K_{2i−n}^{(i)}(X), where the right side refers to the eigenspace of K_{2i−n}(X) ⊗ ℚ on which the Adams operations ψ^{k} act as multiplication by k^{i}, described in IV, Theorem 5.11. This fact is due to Bloch [Bl86], and follows from 4.1 and 4.9 below; see Ex. 4.5.

Here is the fundamental structure theorem for motivic cohomology with finite coefficients, due to Rost and Voevodsky. Since the proof is scattered over 15-20 research papers, we refer the reader to the book [HW] for the proof.

For any smooth X, there is a natural map H^{n}(X, ℤ/m(i)) → H_{et}^{n}(X, μ_{m}^{⊗i}) from motivic to étale cohomology. It arises from the forgetful functor a_{∗} from étale sheaves to Nisnevich sheaves on X, via the isomorphism H_{et}^{n}(X, μ_{m}^{⊗i}) ≅ H_{nis}^{n}(X, Ra_{∗}μ_{m}^{⊗i}) and a natural map ℤ/m(i) → Ra_{∗}μ_{m}^{⊗i}.

Norm Residue Theorem 4.1 (Rost-Voevodsky). If k is a field containing 1/m, the natural map induces isomorphisms

H^{n}(k, ℤ/m(i)) ≅ \{ \begin{matrix}H_{et}^{n}(k, μ_{m}^{⊗i}) & n ≤ i \\ 0 & n > i. \end{matrix}

If X is a smooth scheme over k, the natural map H^{n}(X, ℤ/m(i)) → H_{et}^{n}(X, μ_{m}^{⊗i}) is an isomorphism for n ≤ i. For n > i, the map identifies H^{n}(X, ℤ/m(i)) with the Zariski hypercohomology on X of the truncated direct image complex τ^{≤i}Ra_{∗}(μ_{m}^{⊗i}).

A result of Totaro, and Nesterenko-Suslin [NS, Tot] states that the i^{th} Galois symbol (III.7.11) factors through an isomorphism K_{i}^{M}(k) ⟶^{≃} H^{i}(k, ℤ(i)), compatibly with multiplication. This means that the Milnor K-theory ring K_{∗}^{M}(k) (III.7.1) is isomorphic to the ring ⊕H^{i}(k, ℤ(i)). Since H^{i}(k, ℤ(i))/m ≅ H^{i}(k, ℤ/m(i)), we deduce the following special case of Theorem 4.1. This special case was once called the Bloch-Kato conjecture, and is in fact equivalent to Theorem 4.1; see [SV00],

[GL01] or [HW] for a proof of this equivalence.

Corollary 4.1.1. If k is a field containing 1/m, the Galois symbols are isomorphisms for all n: K_{i}^{M}(k)/m ⟶^{≃} H_{et}^{i}(k, μ_{m}^{⊗i}). They induce a ring isomorphism:

⊕K_{i}^{M}(k)/m ⟶^{≃} ⊕H^{i}(k, ℤ(i))/m ≅ ⊕H^{i}(k, ℤ/m(i)) ≅ ⊕H_{et}^{i}(k, μ_{m}^{⊗i}).

The Merkurjev-Suslin isomorphism K_{2}(k)/m ≅ H_{et}^{2}(k, μ_{m}^{⊗2}) of [MS], mentioned in III.6.10.4, is the case i = 2 of 4.1.1. The isomorphism K_{3}^{M}(k)/m ≅ H_{et}^{3}(k, μ_{m}^{⊗3}) for m = 2^{ν} was established by Rost and Merkurjev-Suslin; see [MS2].

The key technical tool which allows us to use Theorem 4.1 in order to make calculations is the motivic-to-K-theory spectral sequence, so-named because it goes from motivic cohomology to algebraic K-theory. The construction of this spectral sequence is given in the references cited in the Historical Remark 4.4; the final assertion in 4.2 is immediate from the first assertion and Theorem 4.1.

Theorem 4.2. For any coefficient group A, and any smooth scheme X over a field k, there is a spectral sequence, natural in X and A:

E_{2}^{p,q} = H^{p−q}(X, A(−q)) ⇒ K_{−p−q}(X; A).

If X = Spec(k) and A = ℤ/m, where 1/m ∈ k, then the E_{2} terms are just the étale cohomology groups of k, truncated to lie in the octant q ≤ p ≤ 0.

Addendum 4.2.1. If A = ℤ/m and m ≢ 2 (mod 4), the spectral sequence has a multiplicative structure which is the product in motivic cohomology on the E_{2} page and the K-theory product (IV.8) on the abutment.

For any coefficients ℤ/m, there is a pairing between the spectral sequence 4.2 for ℤ coefficients and the spectral sequence 4.2 with coefficients ℤ/m. On the E_{2} page, it is the product in motivic cohomology and on the abutment it is the pairing K_{∗}(X) ⊗ K_{∗}(X; ℤ/m) → K_{∗}(X; ℤ/m) of IV, Ex. 2.5.

Remark 4.2.2. The spectral sequence (4.2) has an analogue for non-smooth schemes over k, in which the motivic cohomology groups are replaced by higher Chow groups CH^{i}(X, n). It is established in [FS, 13.12] and [L01, 8.9]. For any equidimensional quasi-projective scheme X, there is a convergent spectral sequence

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

If X is smooth and k is perfect, then H^{n}(X, ℤ(i)) ≅ CH^{i}(X, 2i − n); see [MVW, 19.1]. This identifies the present spectral sequence with (4.2). Since CH^{i}(X, n) is the same as the Borel-Moore homology group H_{2i+n}^{BM}(X, ℤ(i)), this spectral sequence is sometimes cited as a homology spectral sequence with E_{p,q}^{2} = H_{p−q}^{BM}(X, ℤ(−q)).

Edge map 4.3. Let k be a field. The edge map K_{2i}(k; ℤ/m) → H_{et}^{0}(k, μ_{m}^{⊗i}) in (4.2) is the e-invariant of 2.1, and is an isomorphism for the algebraic closure k̄ of k; the details are given in Example 4.5(ii) below.

We now consider the other edge map, from E_{2}^{p,−n} = H^{n}(k, ℤ(n)) ≅ K_{n}^{M}(k) to K_{n}(k). Since the ring K_{∗}^{M}(k) is generated by its degree 1 terms, and the low degree terms of (4.2) yield isomorphisms H^{1}(k, ℤ(1)) ≅ K_{1}(k), and H^{1}(k, ℤ/m(1)) ≅ K_{1}(k)/m, the multiplicative structure described in 4.2.1 implies that the edge maps in the spectral sequence are canonically identified with the maps K_{∗}^{M}(k) → K_{∗}(k) and K_{∗}^{M}(k)/m → K_{∗}(k; ℤ/m) described in IV.1.10.1. (This was first observed in [GeilL, 3.3] and later in [FS, 15.5].)

By V.11.13, the kernel of the edge map K_{n}^{M}(k) → K_{n}(k) is a torsion group of exponent (n − 1)!. This is not best possible; we will see in 4.3.2 that the edge map K_{3}^{M}(k) → K_{3}(k) is an injection.

Since {−1, −1, −1, −1} is nonzero in K_{4}^{M}(ℚ) and K_{4}^{M}(ℝ) but zero in K_{4}(ℚ) (by Ex. IV.1.12), the edge map K_{4}^{M}(k) → K_{4}(k) is not an injection for subfields of ℝ. This means that the differential d_{2} : H^{1}(ℚ, ℤ(3)) → K_{4}^{M}(ℚ) is nonzero.

Similarly, using the étale Chern class c_{3,3} : K_{3}(k, ℤ/m) → H_{et}^{n}(k, μ_{m}^{⊗n}) of V.11.10, we see that the kernel of the edge map K_{n}^{M}(k)/m → K_{n}(k; ℤ/m) has exponent (n−1)!. (The composition of c_{3,3} with the isomorphism H_{et}^{n}(k, μ_{m}^{⊗n}) ≅ K_{n}^{M}(k)/m of Corollary 4.1.1 satisfies c_{3,3}(x) = −2x for all x ∈ K_{3}^{M}(k)/m.) Since K_{3}^{M}(ℚ) ≅ ℤ/2 on {−1, −1, −1} (Remark 2.1.3), and this element dies in K_{3}(ℚ)/8 ≅ ℤ/8 and hence K_{3}(ℚ; ℤ/8), the edge map K_{3}^{M}(ℚ)/8 → K_{3}(ℚ; ℤ/8) is not an injection.

Low degree terms 4.3.1. When k is a field, the edge map K_{2}^{M}(k) → K_{2}(k) is an isomorphism by Matsumoto’s Theorem III.6.1, so the low degree sequence 0 → K_{2}^{M}(k)/m → K_{2}(k; ℤ/m) → μ_{m}(k) → 0 of 4.2 may be identified with the Universal Coefficient sequence IV.2.5. This yields the Merkurjev-Suslin formula K_{2}(k)/m ≅ H_{et}^{2}(k, μ_{m}^{⊗2}) of III(6.10.4). Since H^{n}(X, ℤ(0)) = 0 for n < 0 and H^{n}(X, ℤ(1)) = 0 for n ≤ 0, by [MVW, 4.2], we also obtain the exact sequences

K_{4}(k) → H^{0}(k, ℤ(2)) ⟶^{d_{2}} K_{3}^{M}(k) → K_{3}(k) → H^{1}(k, ℤ(2)) → 0,
K_{4}(k; ℤ/m) → H_{et}^{0}(k, μ_{m}^{⊗2}) ⟶^{d_{2}} K_{3}^{M}(k)/m → K_{3}(k; ℤ/m) → H_{et}^{1}(k, μ_{m}^{⊗2}) → 0.

Since the kernel of K_{3}^{M}(k)/m → K_{3}(k; ℤ/m) is nonzero for k = ℚ, the differential d_{2} can be nontrivial with finite coefficients. The integral d_{2} is always zero:

Proposition 4.3.2. The map K_{3}^{M}(k)→K_{3}(k) is an injection for every field k.

Proof. We have seen that the kernel of K_{3}^{M}(k) → K_{3}(k) has exponent 2. If char(k) = 2, then K_{3}^{M}(k) has no 2-torsion (Izhboldin’s Theorem III.7.8) and the result holds, so we may suppose that char(k) ≠ 2. Consider the motivic group H(k) = H^{0}(k, ℤ(2))/2. Since the differential d_{2} : H^{0}(k, ℤ(2)) → K_{3}^{M}(k) in 4.3.1 factors through H(k), it suffices to show that H(k) = 0. By universal coefficients, H(k) is a subgroup of H^{0}(k, ℤ/2(2)) ≅ H_{et}^{0}(k, ℤ/2) = ℤ/2; by naturality this implies that H(k) ⊆ H(k′) ⊂ ℤ/2 for any field extension k′ of k. Thus we may suppose that k is algebraically closed. In this case, K_{4}(k) is divisible (by 1.6) and K_{3}^{M}(k) is uniquely divisible (by III.7.2), so it follows from 4.3.1 that H^{0}(k, ℤ(2)) is divisible and hence H(k) = 0. □

Historical Remark 4.4. This spectral sequence 4.2 has an awkward history. In 1972, Lichtenbaum [Li2] made several conjectures relating the K-theory of integers in number fields to étale cohomology and (via this) to values of Zeta functions at negative integers (see 6.10 below). Expanding on these conjectures, Quillen speculated that there should be a spectral sequence like (4.2) (with finite coefficients) at the 1974 Vancouver ICM, and Beilinson suggested in 1982 that one might exist with coefficients ℤ.

The existence of such a spectral sequence was claimed by Bloch and Lichtenbaum in their 1994 preprint [BL], which was heavily cited for a decade, but there is a gap in their proof. Friedlander and Suslin showed in [FS] that one could start with the construction of [BL] to get a spectral sequence for all smooth schemes, together with the multiplicative structure of 4.2.1. The spectral sequence in [BL] was also used to construct the Borel-Moore spectral sequence in 4.2.3 for quasi-projective X in [FS, 13.12] and [L01, 8.9]

Also in the early 1990s, Grayson constructed a spectral sequence in [Gr95], following suggestions of Goodwillie and Lichtenbaum. Although it converged to the K-theory of regular rings, it was not clear what the E_{2} terms were until 2001, when Suslin showed (in [Su03]) that the E_{2} terms in Grayson’s spectral sequence agreed with motivic cohomology for fields. Using the machinery of [FS], Suslin then constructed the spectral sequence of Theorem 4.2 for all smooth varieties over a field, and also established the multiplicative structure of 4.2.1.

In 2000–1, Voevodsky observed (in [VV02a, VV02b]) that the slice filtration for the motivic spectrum representing K-theory (of smooth varieties) gave rise to a spectral sequence, and showed that it had the form given in Theorem 4.2 modulo two conjectures about motivic homotopy theory (since verified). Yet a third construction was given by Levine in [Le01] [Le08]; a proof that these three spectral sequences agree is also given in [Le08].

Remark 4.4.1. A similar motivic spectral sequence was established by Levine in [Le01, (8.8)] over a Dedekind domain, in which the group H_{M}^{n}(X, A(i)) is defined to be the (2i − n)th hypercohomology on X of the complex of higher Chow group sheaves z^{i} ⊗ A.

We now give several examples in which the motivic-to-K-theory spectral sequence degenerates at the E_{2} page, quickly yielding the K-groups.

Examples 4.5. (i) When k is a separably closed field, H_{et}^{n}(k, −) = 0 for n > 0 and the spectral sequence degenerates along the line p = q to yield K_{2i}(k; ℤ/m) ≅ ℤ/m, K_{2i−1}(k; ℤ/m) = 0. This recovers the calculations of 1.3.1 and 1.4 above. In particular, the Bott element β ∈ K_{2}(k, ℤ/m) (for a fixed choice of ζ) corresponds to the canonical element ζ in E_{2}^{−1,−1} = H_{et}^{0}(k, μ_{m}).

(ii) If k is any field containing 1/m, and G = Gal(k̄/k), then H_{et}^{0}(k, μ_{m}^{⊗i}) is the subgroup of μ_{m}^{⊗i} invariant under G; by Definition 2.1 it is isomorphic to ℤ/(m, w_{i}(k)). By naturality in k and (i), the edge map of 4.2 (followed by the inclusion) is the composition K_{2i}(k; ℤ/m) → K_{2i}(k̄; ℤ/m) → μ_{m}^{⊗i}. Therefore the edge map vanishes on K_{2i}(k)/m and (by the Universal Coefficient Sequence of IV.2.5) induces the e-invariant _{m}K_{2i−1}(k) → Hom(ℤ/m, ℤ/w_{i}(k)) = ℤ/(m, w_{i}(k)) of 2.1.

(iii) For a finite field 𝔽_{q} with m prime to q, we have H_{et}^{n}(𝔽_{q}, −) = 0 for n > 1 [Shatz, p. 69]. There is also a duality isomorphism H_{et}^{1}(𝔽_{q}, μ_{m}^{⊗i}) ≅ H_{et}^{0}(𝔽_{q}, μ_{m}^{⊗i}). Thus each diagonal p + q = −n in the spectral sequence 4.2 has only one nonzero entry, so K_{2i}(𝔽_{q}; ℤ/m) and K_{2i−1}(𝔽_{q}; ℤ/m) are both isomorphic to ℤ/(m, w_{i}(k)). This recovers the computation for finite fields given in 2.1.1 and IV.1.13.1.

(iv) Let F be the function field of a curve over a separably closed field containing 1/m. Then H_{et}^{n}(F, −) = 0 for n > 1 (see [Shatz, p. 119]) and H_{et}^{0}(F, μ_{m}^{⊗i}) ≅ ℤ/m as in (i). By Kummer theory,

H_{et}^{1}(F, μ_{m}^{⊗i}) ≅ H_{et}^{1}(F, μ_{m}) ⊗ μ_{m}^{⊗i−1} ≅ F^{×}/F^{×m} ⊗ μ_{m}^{⊗i−1}.

(The twist by μ_{m}^{⊗i−1} is to keep track of the action of the Galois group.) As in (iii), the spectral sequence degenerates to yield K_{2i}(F; ℤ/m) ≅ ℤ/m, K_{2i−1}(F; ℤ/m) ≅ F^{×}/F^{×m}. Since the spectral sequence is multiplicative, it follows that the map F^{×}/F^{×m} → K_{2i+1}(F; ℤ/m) sending u to {β^{i}, u} is an isomorphism because it corresponds to the isomorphism E_{2}^{0,−1} → E_{2}^{−i,−i−1} obtained by multiplication by the element ζ^{⊗i} of E_{2}^{−i,−i}. Thus

K_{n}(F; ℤ/m) ≅ \{ \begin{matrix}ℤ/m on β^{i}, & n = 2i, \\ F^{×}/F^{×m} on \{β^{i}, u\} & n = 2i + 1. \end{matrix}

When X has dimension d > 0, the spectral sequence (4.2) extends to the fourth quadrant, with terms only in columns ≤ d. This is because H^{n}(X, A(i)) = 0 for n > i + d; see [MVW, 3.6]. To illustrate this, we consider the case d = 1, i.e., when X is a curve.

Example 4.6. Let X be a smooth projective curve over a field k containing 1/m, with function field F. By Theorem 4.1, E_{2}^{p,q} = H^{p−q}(X, ℤ/m(−q)) is H_{et}^{p−q}(X, μ_{m}^{⊗q}) for p ≤ 0, and E_{2}^{p,q} = 0 for p ≥ 2 by the above remarks. That is, the E_{2}-terms in the third quadrant of (4.2) are étale cohomology groups, but there are also modified terms in the column p = +1. To determine these, we note that a comparison of the localization sequences for Spec(F) → X in motivic cohomology [MVW, 14.5] and étale cohomology yields an exact sequence

0 → H^{i+1}(X, ℤ/m(i)) → H_{et}^{i+1}(X, μ_{m}^{⊗i}) → H_{et}^{i+1}(F, μ_{m}^{⊗i}).

In particular, E_{2}^{1,0} = 0 and E_{2}^{1,−1} = E_{∞}^{1,−1} = Pic(X)/m. In this case, we can identify the group K_{0}(X; ℤ/m) = ℤ/m⊕Pic(X)/m (see II.8.2.1) with the abutment of (4.2) in total degree 0.

Now suppose that k is separably closed and m = ℓ^{ν}. Then X has (ℓ-primary) étale cohomological dimension 2, and it is well known that H_{et}^{1}(X, μ_{m}) ≅ _{m}Pic(X) and H_{et}^{2}(X, μ_{m}) ≅ ℤ/m; see [Milne, pp. 126, 175]. Thus the spectral sequence has only three diagonals (p − q = 0, 1, 2) with terms ℤ/m, _{m}Pic(X) ≅ (ℤ/m)^{2g} and Pic(X)/m ≅ ℤ/m (see I.5.15); the only nonzero term in the column p = +1 is Pic(X)/m ≅ ℤ/m. By 4.5(ii), there is simply no room for any differentials, so the spectral sequence degenerates at E_{2}. Since the e-invariant maps K_{2i}(k; ℤ/m) isomorphically onto μ_{m}^{⊗i} ≅ E_{∞}^{−i,−i}, the extensions split and we obtain

Proposition 4.6.1. Let X be a smooth projective curve over a separably closed field containing 1/m. Then

K_{n}(X; ℤ/m) = \{ \begin{matrix}ℤ/m ⊕ ℤ/m, & n = 2i, n ≥ 0, \\ _{m}Pic(X) ≅ (ℤ/m)^{2g}, & n = 2i − 1, n > 0. \end{matrix}

The multiplicative structure of K_{∗}(X; ℤ/m) is given in Exercise 4.3. When k = 𝔽̄_{p}, the structure of K_{∗}(X) is given in Theorem 6.4 below.

Geisser and Levine proved in [GeiL] that if k is a field of characteristic p > 0 then the motivic cohomology groups H^{n,i}(X, ℤ/p^{ν}) vanish for all i ≠ n. This allows us to clarify the relationship between K_{∗}^{M}(k) and K_{∗}(k) at the prime p. Part (b) should be compared with Izhboldin’s Theorem III.7.8 that K_{n}^{M}(k) has no p-torsion.

Theorem 4.7. Let k be a field of characteristic p. Then for all n ≥ 0,

(a) for all ν > 0, the map K_{n}^{M}(k)/p^{ν} → K_{n}(k; ℤ/p^{ν}) is an isomorphism;

(b) K_{n}(k) has no p-torsion;

(c) the kernel and cokernel of K_{n}^{M}(k) → K_{n}(k) are uniquely p-divisible groups.

proof. The Geisser-Levine result implies that the spectral sequence (4.2) with coefficients ℤ/p^{ν} collapses at E_{2}, with all terms zero except for E_{2}^{0,q} = K_{−q}^{M}(k)/p^{ν}. Hence the edge maps of 4.2.2 are isomorphisms. This yields (a). Since the surjection K_{n}^{M}(k) → K_{n}^{M}(k)/p ≅ K_{n}(k; ℤ/p) factors through K_{n}(k)/p, the Universal Coefficient sequence of IV.2.5 implies (b), that K_{n−1}(k) has no p-torsion. Finally (c) follows from the 5-lemma applied to the diagram

0 ⟶ K_{n}^{M}(k) ⟶^{p} K_{n}^{M}(k) ⟶ K_{n}^{M}(k)/p ⟶ 0
↓  ↓  ↓≅
0 ⟶ K_{n}(k) ⟶^{p} K_{n}(k) ⟶ K_{n}(k; ℤ/p) ⟶ 0.  □

4.8 Periodicity for ℓ > 2. Let β denote the Bott element in K_{2}(ℤ[ζ_{ℓ}]; ℤ/ℓ) corresponding to the primitive ℓ^{th} root of unity ζ_{ℓ}, and let b ∈ K_{2(ℓ−1)}(ℤ; ℤ/ℓ) denote the image of −β^{ℓ−1} under the transfer map i_{∗}. Since i^{∗}(b) = −(ℓ − 1)β^{ℓ−1} = β^{ℓ−1}, the e-invariant of b is the canonical generator ζ^{⊗ℓ−1} of H^{0}(ℤ[1/ℓ], μ_{ℓ}^{ℓ−1}) by naturality. If X is any smooth variety over a field containing 1/ℓ, multiplication by b gives a map K_{n}(X; ℤ/ℓ) → K_{n+2(ℓ−1)}(X; ℤ/ℓ); we refer to this as a periodicity map. Indeed, the multiplicative pairing in Addendum 4.2.1, of b with the spectral sequence converging to K_{∗}(X; ℤ/ℓ), gives a morphism of spectral sequences E_{r}^{p,q} → E_{r}^{p+1−ℓ,q+1−ℓ} from (4.2) to a shift of itself. On the E_{2} page, these maps are isomorphisms for p ≤ 0, induced by μ_{ℓ}^{⊗i} ≅ μ_{ℓ}^{⊗i+ℓ−1}.

The term ‘periodicity map’ comes from the fact that the periodicity map is an isomorphism K_{n}(X; ℤ/ℓ) ⟶^{≃} K_{n+2(ℓ−1)}(X; ℤ/ℓ) for all n > dim(X) + cd_{ℓ}(X), cd_{ℓ}(X) being the étale cohomological dimension of X for ℓ-primary sheaves. This follows from the comparison theorem for the morphism ∪b of the spectral sequence (4.2) to itself.

4.8.1 Periodicity for ℓ = 2. Pick a generator v_{1}^{4} of π^{s}(S^{8}; ℤ/16) ≅ ℤ/16; it defines a generator of K_{8}(ℤ[1/2]; ℤ/16) and, by the edge map in (4.2), a canonical element of H_{et}^{0}(ℤ[1/2]; μ_{16}^{⊗4}) which we shall also call v_{1}^{4}. If X is any scheme, smooth over ℤ[1/2], the multiplicative pairing of v_{1}^{4} (see Addendum 4.2.1) with the spectral sequence converging to K_{∗}(X; ℤ/2) gives a morphism of spectral sequences E_{r}^{p,q} → E_{r}^{p−4,q−4} from (4.2) to itself. For p ≤ 0 these maps are isomorphisms, induced by E_{2}^{p,q} ≅ H_{et}^{p−q}(X, ℤ/2); we shall refer to these isomorphisms as periodicity isomorphisms.

Adams Operations 4.9. The Adams operations ψ^{k} act on the spectral sequence (4.2), commuting with the differentials and converging to the action of ψ^{k} on K_{∗}(k) and K_{∗}(k; ℤ/m) (IV.5), with ψ^{k} = k^{i} on the row q = −i. This was proven by Soule; see [GiS, 7.1]. Since (k^{i} − k^{i+r−1}) d_{r}(x) = d_{r}(ψ^{k}x) − ψ^{k}(d_{r}x) = 0 for all x in row −i, we see that the image of the differentials d_{r} are groups of bounded exponent. That is, the spectral sequence (4.2) degenerates modulo bounded torsion.

EXERCISES

4.1 If cd_{ℓ}(k) = d and μ_{ℓ^{ν}} ⊂ k, show that ∪β : K_{n}(k; ℤ/ℓ^{ν}) → K_{n+2}(k; ℤ/ℓ^{ν}) is an isomorphism for all n ≥ d. This is a strong form of periodicity. 4.2 (Browder) Show that the periodicity maps K_{n}(𝔽_{q}; ℤ/ℓ) → K_{n+2(ℓ−1)}(𝔽_{q}; ℤ/ℓ) of 4.8 are isomorphisms for finite fields 𝔽_{q} (with ℓ ∤ q) for all n ≥ 0. 4.3 Let X be a smooth projective curve over an algebraically closed field k, and [x] ∈ Pic(X) the class of a closed point x. By I.5.15, Pic(X)/m ≅ ℤ/m on [x], and by II.8.2.1 we have K_{0}(X)/m ≅ ℤ/m ⊕ ℤ/m with basis {1, [x]}. In this exercise, we clarify (4.2.1), assuming 1/m ∈ k.

(i) Show that multiplication by the Bott element β^{i} induces an isomorphism

K_{0}(X)/m ⟶^{≃} K_{2i}(X; ℤ/m).

(ii) Show that K_{1}(X) is divisible, so that the map K_{1}(X; ℤ/m) → _{m}Pic(X) in the Universal Coefficient sequence is an isomorphism.

(iii) Show that multiplication by the Bott element β^{i} induces an isomorphism

K_{1}(X; ℤ/m) ⟶^{≃} K_{2i+1}(X; ℤ/m).

(iv) Conclude that the ring K_{∗}(X; ℤ/m) is ℤ/m[β] ⊗ ℤ/m[M], where M = Pic(X)/m ⊕_{m}Pic(X) is a graded ideal of square zero. 4.4 Use the formula H^{n}(ℙ_{k}^{1}, A(i)) ≅ H^{n}(k, A(i)) ⊕ H^{n−2}(k, A(i − 1)) (see [MVW, 15.12]) to show that the spectral sequence (4.2) for ℙ_{k}^{1} is the direct sum of two copies of the spectral sequence for k, on generators 1 ∈ E_{2}^{0,0} and [L] ∈ E_{2}^{1,−1}. Using this, re-derive the calculation of V.6.14 that K_{n}(ℙ_{k}^{1}) ≅ K_{n}(k) ⊗ K_{0}(ℙ^{1}).

4.5 Use (4.2) and 4.9 to recover Bloch’s isomorphism H^{n}(X, ℚ(i)) ≅ K_{2i−n}^{(i)}(X).

4.6 The Vanishing Conjecture in K-theory states that K_{n}^{(i)}(X) vanishes whenever i ≤ n/2, n > 0. (See [Sou85, p. 501].) Using the Universal Coefficient sequence

0 → H^{j}(X, ℤ(i))/m → H^{j}(X, ℤ/m(i)) → _{m}H^{j+1}(X, ℤ(i)) → 0,

(a) show that H^{j}(X, ℤ(i)) is uniquely divisible for j ≤ 0, and (b) conclude that the Vanishing Conjecture is equivalent to the assertion that H^{j}(X, ℤ(i)) vanishes for all j ≤ 0 (i ≠ 0). This and Exercise 4.5 show that the Vanishing Conjecture holds for any field k whose groups K_{n}(k) are finitely generated, such as number fields.

§5. K_{3} OF A FIELD 23

§5. K_{3} of a field

In this section, we study the group K_{3}(F) of a field F. By Proposition 4.3.2, K_{3}^{M}(F) injects into K_{3}(F). By IV.1.20, the map K_{3}(F) → H_{3}(SL(F)) is onto, and its kernel is the subgroup of K_{3}^{M}(F) generated by the symbols {−1, a, b}. Assuming that K_{3}^{M}(F) is known, we may use homological techniques. The focus of this section will be to relate the group K_{3}^{ind}(F) := K_{3}(F)/K_{3}^{M}(F) to Bloch’s group B(F) of a field F, which we now define.

For any abelian group A, let ˜∧^{2}A denote the quotient of the group A ⊗ A by the subgroup generated by all a ⊗ b + b ⊗ a. The exterior product ∧^{2}A is the quotient of ˜∧^{2}A by the subgroup (isomorphic to A/2A) of all symbols x ∧ x.

Definition 5.1. For any field F, let P(F) denote the abelian group presented with generators symbols [x] for x ∈ F − {0}, with relations [1] = 0 and

[x] − [y] + [y/x] − [\frac{1 − x^{−1}}{1 − y^{−1}} ] + [\frac{1 − x}{1 − y}] = 0, x ≠ y in F − \{0, 1\}.

There is a canonical map P(F) → ˜∧^{2}F^{×} sending [1] to 0 and [x] to x ∧ (1 − x) for x ≠ 1, and Bloch’s group B(F) is defined to be its kernel. Thus we have an exact sequence

0 → B(F) → P(F) → ˜∧^{2}F^{×} → K_{2}(F) → 0.

Remark 5.1.1. Since the cases B(𝔽_{2}) = 0 and B(𝔽_{3}) = ℤ are pathological, we will tacitly assume that |F | ≥ 4 in this section. Theorem 5.2 below implies that if q > 3 is odd then B(𝔽_{q}) is cyclic of order (q + 1)/2, while if q > 3 is even then B(𝔽_{q}) is cyclic of order q + 1. This is easy to check for small values of q.

Remark 5.1.2. The group P(F) is closely related to the scissors congruence group for polyhedra in hyperbolic 3-space H^{3} with vertices in H^{3} or ∂H^{3}, and has its origins in Hilbert’s Third Problem. It was first studied for ℂ by Wigner, Bloch and Thurston and later by Dupont and Sah; see [DSah, 4.10].

For any finite cyclic abelian group A of even order m, there is a unique nontrivial extension Ã of A by ℤ/2. If A is cyclic of odd order, we set Ã = A. Since the group μ(F) of roots of unity is a union of finite cyclic groups, we may define μ̃(F) as the union of the μ̃_{n}(F). Here is the main result of this section.

Theorem 5.2. (Suslin) For any infinite field F, there is an exact sequence

0 → μ̃(F) → K_{3}^{ind}(F) → B(F) → 0.

The proof is taken from [Su91, 5.2], and will be given at the end of this section. To prepare for the proof, we introduce the element c in Lemma 5.4 and construct a map ψ : H_{3}(GL(F), ℤ) → B(F) in Theorem 5.7. In Theorem 5.16 we connect ψ to the group M of monomials matrices, and the group μ̃(F) appears in 5.20 as part of the calculation of π_{3}(BM^{+}). The proof of Theorem 5.2 is obtained by collating all this information.

Remark 5.2.1. In fact, B(ℚ) ≅ ℤ/6. This follows from Theorem 5.2 and the calculations that K_{3}(ℚ) ≅ ℤ/48 (2.1.2), K_{3}^{M}(ℚ) ≅ ℤ/2 (III.7.2.d) and μ̃(ℚ) ≅ ℤ/4. In fact, the element c = [2]+[−1] has order exactly 6 in both B(ℚ) and B(ℝ). This may be proven using the Rogers L-function, which is built from the dilogarithm function. See [Su91, pp. 219–220].

As an application, we compute K_{3} of a number field F. Let r_{1} and r_{2} denote the number of real and complex embeddings, i.e., the number of factors of ℝ and ℂ in the ℝ-algebra F ⊗_{ℚ} ℝ. Then K_{3}^{M}(F) ≅ (ℤ/2)^{r_{1}} by III.7.2(d), and K_{3}(F) is finitely generated by IV.6.9 and V.6.8. By Borel’s Theorem IV.1.18, K_{3}(F) is the sum of ℤ^{r_{2}} and a finite group. We can make this precise.

Corollary 5.3. Let F be a number field, with r_{1} real embeddings and r_{2} complex embeddings, and set w = w_{2}(F). Then K_{3}^{ind}(F) ≅ ℤ^{r_{2}} ⊕ ℤ/w, and:

(a) If F is totally imaginary then K_{3}(F) ≅ ℤ^{r_{2}} ⊕ ℤ/w;

(b) If F has r_{1} > 0 embeddings into ℝ then K_{3}(F) ≅ ℤ^{r_{2}} ⊕ ℤ/(2w) ⊕ (ℤ/2)^{r_{1}−1}.

Proof. By 4.3.1 and Proposition 4.3.2, there is an exact sequence

0 → K_{3}^{M}(F) → K_{3}(F) → H^{1}(F, ℤ(2)) → 0.

Therefore H^{1}(F, ℤ(2)) ≅ K_{3}^{ind}(F) is the direct sum of ℤ^{r_{2}} and a finite group, say of order m′. Choose m divisible by m′ and w. Because H^{0}(F, ℤ(2)) is divisible by Ex. 4.6(a), the map H^{0}(F, μ_{m}^{⊗2}) → H^{0}(F, ℤ(2))_{tors} is an isomorphism. But H^{0}(F, μ_{m}^{⊗2}) ≅ ℤ/w, so K_{3}^{ind}(F) ≅ ℤ/w. This establishes the result when F is totally imaginary, since in that case K_{3}^{M}(F) = 0.

When F = ℚ then w = 24 and K_{3}(ℚ) ≅ ℤ/48 is a nontrivial extension of ℤ/w by ℤ/2; K_{3}(ℚ) embeds in K_{3}(ℝ) by Theorem 3.1. When F has a real embedding, it follows that K_{3}(ℚ) ⊆ K_{3}(F) so {−1, −1, −1} is a nonzero element of 2K_{3}(F). Hence the extension is nontrivial, as claimed. □

Rigidity Conjecture 5.3.1. (Suslin [Su86, 5.4]) Let F_{0} denote the algebraic closure of the prime field in F. The Rigidity Conjecture states that K_{3}^{ind}(F_{0}) → K_{3}^{ind}(F) is an isomorphism. If char(F) > 0 then K_{3}^{ind}(F_{0}) is ℤ/w_{2}(F); if char(F) = 0, K_{3}^{ind}(F_{0}) is given by Corollary 5.3.

The element c of B(F)

The elements c = [x] + [1 − x] and ⟨x⟩ = [x] + [x^{−1}] of B(F) play an important role, as illustrated by the following calculations.

Lemma 5.4. Assuming that |F | > 3,

(a) c = [x] + [1 − x] is independent of the choice of x ∈ F − {0, 1}.

(b) For each x in F − {0, 1}, 2⟨x⟩ = 0.

(c) There is a homomorphism F^{×} → B(F) sending x to ⟨x⟩.

(d) 3c = ⟨−1⟩ and hence 6c = 0 in B(F).

Proof. Given x ≠ y, we have the relations in P(F):

[1 − y] − [1 − x] + [\frac{1 − x}{1 − y}] − [\frac{1 − x^{−1}}{1 − y^{−1}} ] + [y/x] = 0;
[x^{−1}] − [y^{−1}] + [x/y] − [\frac{1 − x}{1 − y}] + [\frac{1 − x^{−1}}{1 − y^{−1}} ] = 0.

Subtracting the first from the relation in 5.1 yields [x] + [1 − x] − [y] − [1 − y] = 0, whence (a) holds. Adding the second to the relation in 5.1 yields ⟨y⟩ − ⟨x⟩ = ⟨y/x⟩. Interchanging x and y, and using ⟨y/x⟩ = ⟨x/y⟩, we obtain 2⟨x⟩ = 0. Because |F | ≥ 4, any z ∈ F − {0, 1} has the form z = zx/x for x ≠ 1 and hence 2⟨z⟩ = 0. For (d), we compute using (b) and (c):

3c =[x] + [1 − x] + [x^{−1}] + [1 − x^{−1}] + [(1 − x)^{−1}] + [1 − (1 − x)^{−1}]
=⟨x⟩ + ⟨1 − x⟩ + ⟨1 − x^{−1}⟩ = ⟨−(1 − x)^{2}⟩ = ⟨−1⟩.  □

Corollary 5.4.1. If char(F) = 2 or \sqrt{−1} ∈ F then 3c = 0 in B(F); if char(F) = 3 or \sqrt[3]{−1} ∈ F then 2c = 0 in B(F).

The map ψ : H_{3}(GL_{2}) → B(F)

We will now construct a canonical map H_{3}(GL_{2}(F), ℤ) → B(F); see Theorem 5.7. To do this, we use the group hyperhomology of GL_{2}(F) with coefficients in the chain complex arising from the following construction (for a suitable X).

Definition 5.5. If X is any set, let C_{∗}(X) denote the “configuration” chain complex in which C_{n} is the free abelian group on the set of (n+1)-tuples (x_{0}, …, x_{n}) of distinct points in X, with differential

d(x_{0}, …, x_{n}) = ∑_{i=0}^{n}(−1)^{i}(x_{0}, …, x̂_{i}, …, x_{n}).

There is a natural augmentation C_{0}(X) → ℤ sending each (x) to 1.

If a group G acts on X, then C_{∗}(X) is a complex of G-modules, and we may form its hyperhomology ℍ_{n}(G, C_{∗}(X)); see [WHomo, 6.1.15]. There is a canonical map from ℍ_{n}(G, C_{∗}(X)) to H_{n}(C_{G}), where C_{G} denotes C_{∗}(X) ⊗_{G} ℤ.

Lemma 5.5.1. If X is infinite then C_{∗}(X) → ℤ is a quasi-isomorphism.

Proof. If X_{0} is a proper subset of X and z ∈ X − X_{0} then s_{n}(x_{0}, …) = (z, x_{0}, …) defines a chain homotopy s_{n} : C_{n}(X_{0}) → C_{n+1}(X) from the inclusion C_{∗}(X_{0}) → C_{∗}(X) to the projection C_{∗}(X_{0}) → ℤ → C_{∗}(X), where the last map sends 1 to (z). □

If X is finite and |X| > n + 1, we still have H_{n}C_{∗}(X) = 0, by Exercise 5.1.

Corollary 5.5.2. If a group G acts on X, and X is infinite (or |X| > n + 1), then ℍ_{n}(G, C_{∗}(X)) ≅ H_{n}(G, ℤ).

The case of most interest to us is the action of the group G = GL_{2}(F) on X = ℙ^{1}(F). If n ≤ 2 then G acts transitively on the basis of C_{n}(X), and C_{n}(X)⊗_{G}ℤ is an induced module from the stablizer subgroup G_{x} of an element x. By Shapiro’s Lemma [WH, 6.3.2] we have H_{q}(C_{n}(X) ⊗_{G} ℤ) = H_{q}(G_{x}, ℤ).

Lemma 5.6. H_{0}(C_{G}) = ℤ, H_{n}(C_{G}) = 0 for n = 1, 2 and H_{3}(C_{G}) ≅ P(F).

Proof. By right exactness of ⊗_{G}, we have H_{0}(C_{∗}(X) ⊗_{G} ℤ) = ℤ. The differential from C_{2} ⊗_{G} ℤ ≅ ℤ to C_{1} ⊗_{G} ℤ ≅ ℤ is an isomorphism, since d(0, 1, ∞) ≡ (0, 1). For n = 3 we write [x] for (0, ∞, 1, x); C_{3} is a free ℤ[G]-module on the set {[x], x ∈ F − {0, 1}}. Similarly, C_{4} is a free ℤ[G]-module of the set of all 5-tuples (0, ∞, 1, x, y) and we have

d(0, ∞, 1, x, y) =(∞, 1, x, y) − (0, 1, x, y) + (0, ∞, x, y) − (0, ∞, 1, y) + (0, ∞, 1, x)
= [\frac{1 − x}{1 − y}] − [\frac{1 − x^{−1}}{1 − y^{−1}} ] + [y/x] − [y] + [x].

Thus the cokernel H_{3}(C_{G}) of d : C_{4} ⊗_{G} ℤ → C_{3} ⊗_{G} ℤ is P(F). □

Remark 5.6.1. The proof of Lemma 5.6 goes through for all finite fields, since | ℙ^{1}(F)| ≥ | ℙ^{1}(𝔽_{2})| = 7. Hence H_{n}(C_{∗}(X)) = 0 for n ≤ 3 by Exercise 5.1.

Let T_{2} denote the diagonal subgroup (isomorphic to F^{×} × F^{×}) of GL_{2}(F); the semidirect product T_{2} ⋊ Σ_{2} is the subgroup of M_{2} of monomial matrices in GL_{2}(F) (matrices with only one nonzero term in every row and column).

Theorem 5.7. For all F, H_{2}(GL_{2}(F), ℤ) = F^{×}, and there is an exact sequence

H_{3}(M_{2}, ℤ) → H_{3}(GL_{2}(F), ℤ) ⟶^{ψ} B(F) → 0.

To prove Theorem 5.7, we consider the hyperhomology spectral sequence

(5.8)  E_{p,q}^{1} = H_{q}(G, C_{p}(X)) ⇒ ℍ_{p+q}(G, C_{∗}(X)) ≅ H_{p+q}(G, ℤ)

with G = GL_{2}(F); see [WHomo, 6.1.15]. By Lemma 5.6, the edge map H_{3}(G, ℤ) → E_{3,0}^{∞} lands in a subset of E_{3,0}^{1} = H_{3}(C_{G}) ≅ P(F), which we must show is B(F).

It is not hard to determine all 10 nonzero terms of total degree at most 4 in (5.8). Indeed, the stabilizer of 0 ∈ X is the group B of upper triangular matrices, so E_{0,q}^{1} = H_{q}(G, C_{0}) = H_{q}(B, ℤ); the stabilizer of (0, ∞) ∈ X^{2} is the diagonal subgroup T_{2} = F^{×} × F^{×}, so E_{1,q}^{1} = H_{q}(G, C_{1}) = H_{q}(T_{2}, ℤ), and the stabilizer of (0, ∞, 1) ∈ X^{3} is the subgroup Δ = {(a, a^{−1})} of T_{2}, isomorphic to F^{×}, so E_{2,q}^{1} = H_{q}(Δ, ℤ). By [Su84, §3], the inclusion T_{2} ⊂ B induces an isomorphism on homology.

It is not hard to see that the differential d^{1} : H_{q}(Δ) → H_{q}(T_{2}) is induced by the inclusion Δ ⊂ T_{2}. Since the inclusion is split (by projection onto the first component of T_{2}), the map d_{1} : H_{q}(Δ) → H_{q}(T_{2}) is a split injection, and hence E_{2,q}^{2} = 0 for all q. The following lemma is proven in Exercise 5.2.

Lemma 5.8.1. Let σ : T_{2} → T_{2} be the involution σ(a, b) = (b, a). Then the differential d^{1} : H_{q}(T_{2}) → H_{q}(B) ≅ H_{q}(T_{2}) is induced by 1 − σ.

Thus E_{0,q}^{2} = H_{q}(T_{2})_{σ} and E_{1,q}^{2} = H_{q}(T_{2})^{σ}/H_{q}(F^{×}).

Proof of Theorem 5.7. ([Su91, Thm. 2.1]) By Lemma 5.8.1, the row q = 1 in (5.8) has E_{0,1}^{2} = (T_{2})_{σ} = F^{×} and E_{1,1}^{2} = 0 (because T_{2}^{σ} = F^{×}). Writing (H_{n})_{σ} for H_{n}(T_{2}, ℤ)_{σ}, the low degree terms of the E^{2} page are depicted in Figure 5.8.2.

(H_{3})_{σ}

(H_{2})_{σ} (H_{2})^{σ}  0
F^{×}  0  0  E_{3,1}^{2}
ℤ  0  0  P(F)

Figure 5.8.2. The E^{2} page of (5.8)

By the Künneth formula [WH, 6.1.13], H_{∗}(T_{2}) ≅ H_{∗}(F^{×}) ⊗ H_{∗}(F^{×}), with σ interchanging the factors; if x, y ∈ H_{i}(F^{×}) then σ(x ⊗ y) = y ⊗ x = (−1)^{i}x ⊗ y. Since H_{2}(F^{×}) = ∧^{2}F^{×}, the group H_{2}(T_{2})_{σ} is the direct sum of ∧^{2}F^{×} and ˜∧^{2}(F^{×}). A routine but tedious calculation shows that the differential d^{3} : P(F) → H_{2}(T_{2})_{σ} is the canonical map P(F) → ˜∧^{2}(F^{×}) of 5.1 followed by the split inclusion of ˜∧^{2}(F^{×}) into H_{2}(T_{2})_{σ}; see [Su91, 2.4]. Thus the cokernel of d^{3} is E_{0,2}^{3} ≅ ∧^{2}F^{×} ⊕ K_{2}(F). In particular, we have E_{3,0}^{3} = B(F) and H_{2}(GL_{2}(F), ℤ) ≅ E_{0,2}^{3} ≅ ∧^{2}F^{×} ⊕ K_{2}(F).

Let K denote the kernel of the edge map H_{3}(GL_{2}(F), ℤ) → B(F). From (5.8.2) we see that K is an extension of a quotient Q_{2} of H_{2}(T_{2})^{σ} by a quotient Q_{3} of H_{3}(T_{2})_{σ}. It follows that H_{3}(GL_{2}(F), ℤ) is an extension of B(F) by K.

Recall that M_{2} ≅ T_{2} ⋊ Σ_{2}. Since H_{p}(Σ_{2}, T_{2}) = 0 for p ≠ 0, the Hochschild-Serre spectral sequence H_{p}(Σ_{2}, H_{q}T_{2}) ⇒ H_{p+q}(M_{2}) degenerates enough to show that the cokernel of H_{3}(T_{2}) ⊕ H_{3}(Σ_{2}) → H_{3}(M_{2}) is a quotient of H_{2}(T_{2})_{σ}. Analyzing the subquotient Q_{2} in (5.8), Suslin showed in [Su91, p. 223] that K is the image of H_{3}(M_{2}, ℤ) → H_{3}(GL_{2}(F), ℤ). The result follows. □

In order to extend the map H_{3}(GL_{2}, ℤ) ⟶^{ψ} B(F) of Theorem 5.7 to a map ψ : H_{3}(GL_{3}, ℤ) → P(F), we need a small digression.

A cyclic homology construction

Recall that under the Dold-Kan correspondence [WHomo, 8.4], a nonnegative chain complex C_{∗} (i.e., one with C_{n} = 0 if n < 0) corresponds to to a simplicial abelian group {C̃_{n}}. Conversely, given a simplicial abelian group {C̃_{n}}, C_{∗} is the associated reduced chain complex.

For example, the chain complex C_{∗}(X) of Definition 5.5 corresponds to a simplicial abelian group; C̃_{n}(X) is the free abelian group on the set X^{n+1} of all (n + 1)- tuples (x_{0}, …, x_{n}) in X, including duplication. In fact, C̃_{n}(X) is a cyclic abelian group in the following sense. (These assertions are relegated to Ex. 5.6.)

Definition 5.9. ([WHomo, 9.6]) A cyclic abelian group is a simplicial abelian group {C̃_{n}} together with an automorphism t_{n} of each C̃_{n} satisfying: t_{n}^{n+1} = 1;

∂_{i}t_{n} = t_{n}∂_{i−1} and σ_{i}t_{n} = t_{n}σ_{i} for i ≠ 0; ∂_{0}t_{n} = ∂_{n} and σ_{0}t_{n} = t_{n+1}^{2}σ_{n}.

The associated acyclic complex (C_{∗}^{a}, d^{a}) is the complex obtained from the reduced complex C_{∗} by omitting the last face operator; C_{∗}^{a} is acyclic, and there is a chain map N : C_{∗} → C_{∗}^{a} defined by N = ∑_{i=0}^{n}(−1)^{i}t_{n}^{i} on C_{n}. The mapping cone of N has C_{n−1} ⊕ C_{n}^{a} in degree n, and (b, c) ↦ b defines a natural quasi-isomorphism cone(N)[1] ⟶^{≃} C_{∗} (see [WHomo, 1.5]). In fact, cone(N)[1] is a reduced form of two columns of Tsygan’s double complex [WHomo, 9.6.6].

Since N_{0} : C_{0} → C_{0}^{a} is the natural identification isomorphism, we may truncate the zero terms to get a morphism of chain complexes

(5.9.1)  \begin{aligned}0 & ⟵ & C_{1} & ⟵^{d} & C_{2} & ⟵^{d} & ⋯ & ⟵^{d} & C_{n} & ⟵^{d} \\ & & ^{N}↓ & & ^{N}↓ & & ^{N}↓ & & ^{N}↓ \\ 0 & ⟵ & C_{1}^{a} & ⟵^{d^{a}} & C_{2}^{a} & ⟵^{d^{a}} & ⋯ & ⟵^{d^{a}} & C_{n}^{a} & ⟵^{d^{a}} & . \end{aligned}

We write D_{∗} for the the associated mapping cone of this morphism. Thus, D_{0} = C_{1}^{a}, and D_{n} = C_{n+1}^{a} ⊕ C_{n} for n > 0 with differential (x, y) ↦ (d^{a}x − Ny, −dy). Then cone(N)[1] → D_{∗} is a quasi-isomorphism.

Example 5.9.2. When X = ℙ^{2}(F), let C_{n} denote the subgroup of C_{n}(X) generated by the (n + 1)-tuples of points (x_{0}, …, x_{n}) for which no three x_{i} are collinear. Since C_{n} is closed under the operator t_{n}, the associated simplicial abelian subgroup {C̃_{n}} of {C̃_{n}(X)} has the structure of a cyclic abelian subgroup. The proof of Lemma 5.5.1 goes through to show that if X is infinite then C_{∗} → ℤ and hence C_{∗} → C_{∗}(X) are quasi-isomorphisms. It follows that the map ε : D_{0} → ℤ sending (x, y) to 1 induces a quasi-isomorphism D_{∗} → ℤ.

Under the canonical action of the group GL_{3} = GL_{3}(F) on X = ℙ^{2}(F), GL_{3} sends the subcomplex C_{∗} of C_{∗}(X) to itself, so GL_{3} acts on C_{∗} and D_{∗}.

The map ψ : H_{3}(GL_{3}) → B(F)

We shall now construct a map H_{3}(GL_{3}, ℤ) ⟶^{ψ} P(F) whose image is B(F). We will relate it to the map of Theorem 5.7 in Lemma 5.10.

Let C_{∗} be the subcomplex of Example 5.9.2 for X = ℙ^{2}(F). By 5.5 and Lemma 5.5.1, the hyperhomology ℍ_{n}(GL_{3}, C_{∗}) of GL_{3} = GL_{3}(F) is just H_{n}(GL_{3}, ℤ) when F is infinite (or |F | > n + 1), and there is a canonical map from H_{n}(GL_{3}, ℤ) = ℍ_{n}(GL_{3}, D_{∗}) to H_{n}(D_{G}), where D_{G} denotes D_{∗} ⊗_{GL_{3}} ℤ.

The four points p_{1} = (1 : 0 : 0), p_{2} = (0 : 1 : 0), p_{3} = (0 : 0 : 1) and q = (1 : 1 : 1) play a useful role in any analysis of the action of GL_{3} on ℙ^{2}(F). For example, the ℤ[GL_{3}]-module C_{3} is generated by P = (p_{1}, p_{2}, q, p_{3}). Just as in Lemma 5.6, C_{n} ⊗_{G} ℤ = ℤ for n ≤ 3, while C_{4} and C_{5} are free ℤ[G]-modules on the set of all 5-tuples and 6-tuples

\left[{a \atop x}\right] := (p_{1}, p_{2}, q, (1 : a : x), p_{3})

[{a \atop x} {b \atop y} ] := (p_{1}, p_{2}, q, (1 : a : x), (1 : b, y), p_{3}) ,

where a ≠ x, b ≠ y, a ≠ b, x ≠ y and ay ≠ bx. By inspection of D_{G}, d(\left[{a \atop x}\right]) = P, d^{a}(\left[{a \atop x}\right]) = N(\left[{a \atop x}\right]) = 0, and d : (D_{G})_{3} → (D_{G})_{2} is zero. Thus the terms n ≤ 4 of the complex D_{G} have the form:

⊕ℤ \left[{a \atop x}\right] ⊕ ℤ [{a \atop x} {b \atop y} ] \underrightarrow{( \begin{matrix}−1 & 0 \\ −N & d^{a} \end{matrix} )} ℤ P ⊕ ⨁ ℤ \left[{a \atop x}\right] ⟶^{0} ℤ^{2} ⟶ ℤ^{2} ⟶ ℤ → 0.

The map ψ′ : (D_{G})_{3} → P(F) defined by ψ′(P) = 2c, ψ′(\left[{a \atop x}\right]) = a vanishes on the image of (D_{G})_{4}, by a straightforward calculation done in [Su91, 3.3], so it induces a homomorphism ψ : H_{3}(GL_{3}, ℤ) → H_{3}(D_{G}) → P(F).

Lemma 5.10. The composition H_{3}(GL_{2}, ℤ) → H_{3}(GL_{3}, ℤ) ⟶^{ψ} P(F) is the homomorphism H_{3}(GL_{2}, ℤ) → B(F) ⊂ P(F) of Theorem 5.7.

Proof. ([Su91, 3.4]) The subgroup GL_{2} = GL_{2}(F) also acts on X = ℙ^{2}(F) and fixes the origin p_{3} = (0 : 0 : 1), so it acts on X_{0} = X − {p_{3}}. The maps f_{n} : C_{n}(X_{0}) → C_{n+1}(X) = C_{n+1}^{a}(X) ⊂ D_{n} sending (x_{0}, …, x_{n}) to (x_{0}, …, x_{n}, p_{3}) satisfy fd = d^{a}f, so they form a GL_{2}-equivariant chain map f : C_{∗}(X_{0}) → C_{∗}^{a}(X)/C_{0}(X)[1].

If C′_{n} denotes the subgroup f_{n}^{−1}(C_{n+1}) of C_{n}(X_{0}), then the restriction of f defines a GL_{2}-equivariant chain map C′_{∗} → C_{∗}^{a}/C_{0}^{a}[1] ⊂ D_{∗}. Therefore the composition of Lemma 5.10 factors as

H_{3}(GL_{2}, ℤ) → H_{3}(C′_{∗}_{GL_{2}}) ⟶^{f} H_{4}(C_{G}^{a}) → H_{3}(D_{G}) ⟶^{ψ′} P(F).

The projection from ℙ^{2}(F) − {p_{3}} to ℙ^{1}(F) with center p_{3} is GL_{2}-equivariant and determines a homomorphism from C′_{∗} to C_{∗}^{1} = C_{∗}(ℙ^{1}(F)) over ℤ. By inspection, the composition H_{3}(GL_{2}) → H_{3}(C′_{∗}_{GL_{2}}) → H_{3}(C_{∗}^{1}_{GL_{2}}) ≅ P(F) is the inclusion of Theorem 5.7. □

Lemma 5.11. Let T_{3} denote the diagonal subgroup F^{×} × F^{×} × F^{×} of GL_{3}(F). If F is infinite, H_{n}(T_{3}, ℤ) → H_{n}(D_{∗}_{T_{3}}) is zero for n > 0.

Proof. Since p_{1} = (1 : 0 : 0) and p_{2} = (0 : 0 : 1) are fixed by T_{3}, the augmentation D_{0} → ℤ has a T_{3}-equivariant section sending 1 to (p_{1}, p_{2}). Therefore D_{∗} ≅ ℤ ⊕ D′_{∗} as T_{3}-modules, and D′_{∗} is acyclic. Therefore if n > 0 we have ℍ_{n}(T_{3}, D′_{∗}) = 0. Since H_{n}(D_{∗}_{T_{3}}) = H_{n}(D′_{∗}_{T_{3}}) for n > 0, the map H_{n}(T_{3}, ℤ) → H_{n}(D_{∗}_{T_{3}}) factors through ℍ_{n}(T_{3}, D′_{∗}) = 0. □

Proposition 5.12. The image of ψ is B(F), and there is an exact sequence

H_{3}(M_{2}, ℤ) ⊕ H_{3}(T_{3}, ℤ) → H_{3}(GL_{3}(F), ℤ) ⟶^{ψ} B(F) → 0.

Proof. By [Su-KM, 3.4], H_{3}(T_{3}, ℤ) and H_{3}(GL_{2}, ℤ) generate H_{3}(GL_{3}, ℤ). The restriction of ψ to H_{3}(T_{3}, ℤ) is zero by Lemma 5.11, since it factors as

H_{3}(T_{3}, ℤ) → H_{3}(D_{T_{3}}) → H_{3}(D_{G}) ⟶^{ψ′} P(F).

The proposition now follows from Theorem 5.7 and Lemma 5.10. □

Remark 5.12.1. The map ψ extends to a map defined on H_{3}(GL(F), ℤ), because of the stability result H_{3}(GL_{3}(F), ℤ) ≅ H_{3}(GL(F), ℤ); see IV.1.15.

We now consider the image of H_{3}(Σ_{∞}, ℤ) in B(F), where we regard Σ_{∞} as the subgroup of permutation matrices in GL(F), i.e., as the direct limit of the permutation embeddings ι_{n} : Σ_{n} ⊂ GL_{n}(F). We will use the following trick.

Let S be the p-Sylow subgroup of a finite group G. Then the transfer-corestriction composition H_{n}(G) → H_{n}(S) → H_{n}(G) is multiplication by [G : S], which is prime to p. Therefore the p-primary torsion in H_{n}(G, ℤ) is the image of H_{n}(S, ℤ) when n > 0.

Proposition 5.13. The image of H_{3}(Σ_{∞}, ℤ) ⟶^{ι_{∗}} H_{3}(GL(F), ℤ) ⟶^{ψ} B(F) is the subgroup generated by 2c, which is trivial or cyclic of order 3.

Proof. We saw in Ex. IV.1.13 that H_{3}(Σ_{∞}, ℤ) ≅ ℤ/12 ⊕ (ℤ/2)^{2}, so the image of H_{3}(Σ_{∞}, ℤ) → H_{3}(GL(F), ℤ) has at most 2- and 3-primary torsion. Nakaoka proved in [Nak] that H_{3}(Σ_{6}, ℤ) ≅ H_{3}(Σ_{∞}, ℤ).

Using the Sylow 2-subgroup of Σ_{6}, Suslin proves [Su-KM, 4.4.1] that the 2primary subgroup of H_{3}(Σ_{6}, ℤ) maps to zero in B(F). We omit the details.

Now the 3-primary component of H_{3}(Σ_{6}, ℤ) is the image of H_{3}(S), where S is a Sylow 3-subgroup of Σ_{6}. We may take S = A_{3} × σA_{3}σ^{−1}, generated by the 3cycles (123) and (456), where σ = (14)(25)(36). Since H_{2}(A_{3}, ℤ) = 0, the Künneth formula yields H_{3}(S, ℤ) = H_{3}(A_{3}, ℤ) ⊕ H_{3}(σA_{3}σ^{−1}, ℤ). By [WHomo, 6.7.8], these two summands have the same image in H_{3}(Σ_{6}, ℤ). By Lemma 5.14, the 3-primary component of the image of H_{3}(S, ℤ) in B(F) is generated by 2c, as desired. □

We are reduced to the alternating group A_{3}, which is cyclic of order 3, embedded in GL_{3}(F) as the subgroup of even permutation matrices. We need to analyze the homomorphism H_{3}(A_{3}, ℤ) → H_{3}(GL_{3}(F), ℤ) ⟶^{ψ} B(F).

Lemma 5.14. If |F |≥4, the image of ℤ/3≅H_{3}(A_{3}) in B(F) is generated by 2c.

Proof. If char(F) = 3 then the permutation representation of A_{3} is conjugate to an upper triangular representation, so H_{∗}(A_{3}, ℤ) → H_{∗}(GL_{3}(F), ℤ) is trivial (see [S82]). Since 2c = 0 by Corollary 5.4.1, the result follows. Thus we may assume that char(F) ≠ 3.

When char(F) ≠ 3, the permutation representation is conjugate to the representation A_{3} → GL_{2}(F) with generator λ = ({0 \atop 1} {−1 \atop −1}). For example, if char(F) = 2, this representation identifies A_{3} with a subgroup of Σ_{3} ≅ GL_{2}(𝔽_{2}). By the Comparison Theorem [WHomo, 2.2.6], there is a morphism between ℤ[A_{3}]-module resolutions of ℤ, from the standard periodic free resolution (see [WHomo, 6.2.1]) to C_{∗}:

(5.14.1) \begin{matrix}ℤ[A_{3}] & ⟵^{1−λ} & ℤ[A_{3}] & ⟵^{1+λ+λ^{2}} & ℤ[A_{3}] & ⟵^{1−λ} & ℤ[A_{3}] & ⟵^{1+λ+λ^{2}} & ⋯ \\ ^{f_{0}}↓ & & ^{f_{1}}↓ & & ^{f_{2}}↓ & & ^{f_{3}}↓ \\ C_{0} & ⟵^{d} & C_{1} & ⟵^{d} & C_{2} & ⟵^{d} & C_{3} & ⟵^{d} & · · · . \end{matrix}

We can build the morphisms f_{n} by induction on n, starting with f_{0}(1) = (0), f_{1}(1) = (∞, 0) and f_{2}(1) = (∞, 0, x) + (1, ∞, x) = (0, 1, x) for any x ∈ F (x ≠ 0, 1). If we set w = 1 − 1/x and choose y ≠ ∞, 0, 1, x, w (which is possible when |F | ≥ 4) then we may also take

f_{3}(1) = − (∞, 0, x, y) − (1, ∞, x, y) − (0, 1, x, y)
+ (1, ∞, w, y) + (0, 1, w, y) + (∞, 0, w, y).

Here we have regarded F as embedded in ℙ^{1}(F) via x ↦ \binom{1}{x}. Taking coinvariants in (5.14.1), the generator 1 of H_{3}(A_{3}) in the periodic complex maps to f_{3}(1), representing an element of H_{3}(C_{∗}_{G}). Applying ψ sends this element to

− [\frac{x}{y}] − [\frac{y − 1}{x − 1}] − [\frac{y(x − 1)}{x(y − 1)}] + [x(1 − y)] + [\frac{1 − x}{xy}] + [\frac{y}{(1 − x)(1 − y)}]

in B(F). As |F | ≥ 4, we can take x ≠ −1 and y = x^{−1}, so this expression becomes −[x^{2}] − 2[−x^{2}] + 2[x − 1] + [1/x^{2}]. This equals −2c, by Exercise 5.5. □

Monomial matrices

By definition, a monomial matrix in GL_{n}(F) is one which has only one nonzero entry in each row and column. We write M_{n} for the group of all monomial matrices in GL_{n}(F) and M for the union of the M_{n} in GL(F); M_{n} is isomorphic to the wreath products F^{×} ≀ Σ_{n} = (F^{×})^{n} ⋊ Σ_{n} and M ≅ F^{×} ≀ Σ_{∞}. We encountered M_{2} in Theorem 5.7, and the subgroup μ(F) ≀ Σ_{n} of M_{n} in Theorem 1.5.

Proposition 5.15. Let ι : Σ_{∞} → GL(F) be the inclusion. Then there is an exact sequence:

H_{3}(M, ℤ) ⟶ H_{3}(GL(F), ℤ) ⊕ H_{3}(Σ_{∞}, ℤ) ⟶^{(ψ,−ψι_{∗})} B(F) → 0.

Proof. (Cf. [Su91, 4.3]) Let H_{3}^{0}M_{n} denote the kernel of H_{3}(M_{n}, ℤ)→H_{3}(Σ_{n}, ℤ); it suffices to show that the image of H_{3}^{0}M_{n} → H_{3}(GL(F), ℤ) is the kernel of ψ for large n. The image contains the kernel of ψ because, by Proposition 5.12, the kernel of ψ comes from the image of H_{3}(T_{3}, ℤ), which is in H_{3}^{0}M_{n}, and H_{3}(M_{2}, ℤ), which is in H_{3}^{0}M_{n} by Ex. 5.10(d) since the image of H_{3}(Σ_{2}, ℤ) is in the image of H_{3}([M, M], ℤ) by Ex. 5.12.

The group M_{n} = T_{n} ⋊ Σ_{n} contains M_{2} × Σ_{n−2} as a subgroup. Let S denote the group Σ_{2} × Σ_{n−2}, and write A for the kernel of the split surjection H_{3}(M_{2} × Σ_{n−2}, ℤ) → H_{3}(S, ℤ). By Ex. 5.10(e), H_{3}(T_{3}, ℤ) + A maps onto H_{3}^{0}M_{n}.

By the Künneth formula for M_{2} × Σ_{n−2}, A is the direct sum of H_{3}^{0}(M_{2}, ℤ), H_{2}^{0}(M_{2}, ℤ) ⊗ H_{1}(Σ_{n−2}, ℤ) and F^{×} ⊗ H_{2}(Σ_{n−2}, ℤ). Suslin proved in [Su-KM, 4.2] that the images of the latter two summands in H_{3}(M_{n}) are contained in the image of H_{3}(T_{n}, ℤ) for large n. It follows that the image of H_{3}^{0}M_{n} is the kernel of ψ. □

In Example IV.4.10.1 and Ex. IV.1.27 we saw that the homotopy groups of K(F^{×}-Sets_{fin}) ≃ ℤ × BM^{+} form a graded-commutative ring with π_{1}(BM^{+}) ≅ F^{×} × π_{1}^{s}. By Ex. IV.4.12 , K_{∗}(F^{×}-Sets_{fin}) → K_{∗}(F) is a ring homomorphism. By Matsumoto’s Theorem III.6.1, it follows that π_{2}(BM^{+}) → K_{2}(F) is onto; in fact, π_{2}(BM^{+}) = π_{2}^{s} ⊕ ˜∧^{2}F^{×} by Exercise 5.11. Multiplying by π_{1}(BM^{+}), this implies that K_{3}^{M}(F) lies in the image of π_{3}(BM^{+}) → K_{3}(F).

We saw in Theorem 5.7 that the kernel of the map H_{3}(GL_{2}) → B(F) is the image of H_{3}(M_{2}). The monomial matrices M_{3} in GL_{3}(F) contain T_{3}, so Proposition 5.12 shows that the kernel of ψ is contained in the image of H_{3}(M_{3}).

Theorem 5.16. ( [Su91, Lemma 5.4]) The cokernel of π_{3}BM^{+} → K_{3}(F) is B(F)/(2c), and there is an exact sequence:

π_{3}(BM^{+}) → K_{3}(F) ⊕ π_{3}^{s} ⟶^{(ψ,−ψι_{∗})} B(F) → 0.

Proof. Write H_{n}G for H_{n}(G, ℤ), and let P denote the commutator subgroup [M, M]; we saw in IV, Ex. 1.27 that P is perfect. By IV.1.19–20, π_{3}^{s} maps onto H_{3}A_{∞} and there is a commutative diagram

π_{2}(BP^{+}) ⟶^{∘η} π_{3}(BP^{+}) ⟶^{h} H_{3}P → 0
↓  ↓  ↓
K_{2}(F) ⟶^{[−1]} K_{3}(F) ⟶^{h} H_{3}SL(F) → 0.

Since π_{n}BP^{+} ≅ π_{n}BM^{+} for n ≥ 2 (Ex. IV.1.8), and π_{2}(BM^{+}) → K_{2}(F) is onto (as we noted above), a diagram chase shows that K_{3}(F)/π_{3}(BM^{+}) ≅ H_{3}SL(F)/H_{3}P.

Let SM denote the kernel of det : M → F^{×}. We saw in Chapter IV, Ex. 1.27 that BM^{+} ≃ BP^{+} × B(F^{×}) × BΣ_{2}; it follows that BSM^{+} ≃ BP^{+} × BΣ_{2}. By the Künneth formula, H_{3}SM ≅ H_{3}P ⊕ (H_{2}P ⊗ H_{1}Σ_{2}) ⊕ H_{3}Σ_{2}. Under the map H_{3}SM → H_{3}SL(F), the final term lands in the image of H_{3}P by Ex. 5.12, and the middle term factors through H_{2}SL(F) ⊗ H_{1}SL(F), which is zero as SL(F) is perfect. Hence H_{3}SL(F)/H_{3}P = H_{3}SL(F)/H_{3}SM.

This explains the top half of the following diagram.

H_{3}(P) ⟶ H_{3}(SL) ⟶ K_{3}(F)/π_{3}(BM^{+}) → 0
↓  ↓  ↓≅
H_{3}(SM) ⟶ H_{3}(SL) ⟶ H_{3}(SL)/H_{3}(SM) → 0
↓  ↓  ↓≅
H_{3}(M) ⟶ H_{3}(GL) ⟶ B(F)/(2c) → 0.

Similarly, Ex. IV.1.8 implies that H_{n}GL(F) ≅ ⊕_{i+j=n}H_{i}SL(F) ⊗ H_{j}F^{×}. There is a compatible splitting H_{n}M ≅ ⊕_{i+j=n}H_{i}SM ⊗ H_{j}F^{×} by Ex. IV.1.27. This implies that H_{n}SL(F)/H_{n}SM → H_{n}GL(F)/H_{n}M is injective for all n. For n = 3, the summands H_{3}F^{×} and H_{2}SL(F) ⊗ F^{×} of H_{3}GL(F) are in the image of H_{3}SM (because π_{2}BSM^{+} →K_{2}(F) ≅H_{2}SL(F) is onto). Since H_{1}SL(F) = 0, we conclude that H_{3}SL(F)/H_{3}SM → H_{3}GL(F)/H_{3}M is onto and hence an isomorphism.

Finally, combining Propositions 5.12 and 5.13 with Exercise 5.10 and Lemma 5.14, we see that H_{3}SL(F) → B(F) is onto, and the cokernel of H_{3}M → H_{3}GL(F) is B(F)/(2c). Concatenating the isomorphisms yields the first assertion. The second assertion follows from this by the argument in the proof of Proposition 5.15. □

We define π_{3}^{ind}(BM^{+}) to be the quotient of π_{3}(BM^{+}) by all products from π_{1}(BM^{+}) ⊗ π_{2}(BM^{+}). There is a natural map π_{3}^{ind}(BM^{+}) → π_{3}^{s}/(η^{3}) ≅ ℤ/12. Since the products map to K_{3}^{M}(F) in K_{3}(F), we have the following reformulation.

Corollary 5.16.1. The sequence of Theorem 5.16 induces an exact sequence

π_{3}^{ind}(BM^{+}) → K_{3}^{ind}(F) ⊕ ℤ/12 ⟶ B(F) → 0.

Thus to prove Theorem 5.2, we need to study π_{3}^{ind}(BM^{+}).

Remark 5.16.2. The diagram in the proof of Theorem 5.16 shows that the map π_{3}^{ind}(BM^{+}) → K_{3}^{ind}(F) is a quotient of the map H_{3}(P, ℤ) → H_{3}(SL(R), ℤ).

If E is any homology theory and X any pointed topological space, the Atiyah-Hirzebruch spectral sequence converging to E_{∗}(X) has E_{p,q}^{2} = H_{p}(X, E_{q}(∗)). For stable homotopy we have E_{∗}(X) = π_{∗}^{s}(X). When X = BG_{+}, the Barratt-Priddy Theorem (IV.4.10.1) states that π_{∗}^{s}(BG_{+}) = π_{∗}(ℤ × B(G ≀ Σ_{∞})^{+}).

Proposition 5.17. There is an exact sequence

0 → μ_{2}(F) → π_{3}^{ind}(BM^{+}) ⟶^{γ} μ(F) ⊕ ℤ/12 → 0.

Proof. ([Su91, §5]) We analyze the Atiyah-Hirzebruch spectral sequence

E_{p,q}^{2} = H_{p}(F^{×}, π_{q}^{s}) ⇒ π_{p+q}(ℤ × BM^{+})

which has a module structure over the stable homotopy ring π_{∗}^{s}. Note that the yaxis E_{0,∗}^{2} = π_{∗}^{s} is a canonical summand of π_{∗}(ℤ × BM^{+}), so it survives to E^{∞}. By Ex. 5.11, π_{2}BM^{+}/π_{2}^{s} ≅ ˜∧^{2}F^{×}. It is easy to see that the map from E_{1,1}^{2} = F^{×}/F^{×2} to E_{1,1}^{∞} = ˜∧^{2}F^{×} is injective, as it sends x to x ⊗ x. It follows that the differential from E_{3,0}^{2} = H_{3}(F^{×}, ℤ) to E_{1,1}^{2} is zero, so E_{3,0}^{∞} = H_{3}(F^{×}, ℤ); see Figure 5.17.1.

π_{3}^{s}
π_{2}^{s} F^{×}/F^{×2}
π_{1}^{s} F^{×}/F^{×2}  E_{2,1}^{2}
ℤ  F^{×}  ∧^{2}F^{×}  H_{3}(F^{×})  H_{4}(F^{×})
Figure 5.17.1. The E^{2} page converging to π_{∗}(ℤ × BM^{+})

The universal coefficient sequence expresses E_{2,1}^{2} = H_{2}(F^{×}, ℤ/2) as an extension:

0 → (∧_{2}F^{×})/2 → H_{2}(F^{×}, ℤ/2) → μ_{2}(F) → 0.

The differential E_{4,0}^{2} ⟶ E_{2,1}^{2} lands in (∧_{2}F^{×})/2 because the composite to μ_{2}(F) is zero: by naturality in F, it factors through the divisible group H_{4}(F̄^{×}, ℤ).

The cokernel of the product maps from F^{×} ⊗ π_{2}^{2} and ∧^{2}F^{×} ⊗ π_{1}^{s} → H_{3}(F^{×}, ℤ) is therefore the direct sum of π_{3}^{s}/(η^{3}) and an extension of H_{3}(F^{×}, ℤ) by μ_{2}(F). Modding out by the products from ∧^{2}F^{×}⊗F^{×} replaces H_{3}(F^{×}, ℤ) by H_{3}(F^{×}, ℤ)/∧^{3}F^{×}, which by Ex. 5.9 is isomorphic to μ(F). □

Since H_{3}(P, ℤ) ≅ π_{3}(BM^{+})/η∘π_{2}(BM^{+}) (see IV.1.19), there is a natural surjection H_{3}(P, ℤ) → π_{3}^{ind}(BM^{+}). Consider the homomorphism δ : μ(F) → P sending x to ({x \atop 0} {0 \atop x^{−1}} ). The composition with the map γ of Proposition 5.17 is:

μ(F) ≅ H_{3}(μ(F), ℤ) ⟶^{δ} H_{3}(P, ℤ) ⟶^{onto} π_{3}^{ind}(BM^{+}) ⟶^{γ} μ(F).

Lemma 5.18. The composition μ(F) ⟶^{δ} π_{3}^{ind}(BM^{+}) ⟶^{γ} μ(F) sends x to x^{2}.

Proof. Let us write μ for μ(F) and H_{n}G for H_{n}(G, ℤ). The homomorphism μ ⟶^{δ} P factors through D = μ ⋊ Σ_{2} (where Σ_{2} → A_{∞} ⊂ P is given by (12)(34)), and the map μ ≅ H_{3}(μ) → H_{3}(D) lands in the subgroup H_{3}(μ)_{Σ_{2}} ≅ μ/{±1}. By Exercise 5.13, the composition of μ → μ/{±1} with H_{3}(D) → H_{3}(P) ⟶^{γ} μ sends x ∈ μ to x^{2}. □

Recall that the e-invariant maps K_{3}^{ind}(F) to H^{0}(F, μ^{⊗2}) ⊂ μ(F̄) (see Definition 2.1). We use it to detect the image of μ(F) in π_{3}^{ind}(BM^{+}).

Lemma 5.19. The following composition is an injection:

μ(F) ⟶^{δ} π_{3}^{ind}(BM^{+}) → K_{3}^{ind}(F) ⟶^{e} H^{0}(F, μ^{⊗2}).

If F is algebraically closed it is an isomorphism.

Proof. If F ⊂ F′, μ(F) ⊆ μ(F′). Therefore we may enlarge F to assume that it is algebraically closed. In this case H^{0}(F, μ^{⊗2}) = μ(F) and K_{3}^{M}(F) is uniquely divisible (III.7.2), so it is a summand of K_{3}(F) (by V.11.13) and therefore mK_{3}(F) ≅ _{m}K_{3}^{ind}(F).

Now consider the étale Chern class c_{2,4} : K_{4}(F; ℤ/m) → H^{0}(F, μ_{m}^{⊗2}), where 1/m ∈ F and m ≢ 2 (mod 4). Since K_{4}(F; ℤ/m) ≅ ℤ/m on generator β^{2} (1.4) and c_{1}(β) = ζ by V.11.10.1, the product rule yields c_{2}(β^{2}) = ζ^{−1} ⊗ ζ. Since the Bockstein is an isomorphism: K_{4}(F; ℤ/m) ≅ _{m}K_{3}(F), this implies that the e-invariant is −c_{2,4} on _{m}K_{3}(F).

The isomorphism K_{4}(F; ℤ/m) ≅ _{m}K_{3}^{ind}(F) factors through the Hurewicz map, the map H_{4}(SL(F), ℤ/m) → _{m}H_{3}(SL(F), ℤ) and the quotient _{m}H_{3}(SL(F), ℤ) → mK_{3}^{ind}(F) of IV.1.20. Therefore we have a commutative diagram:

H_{4}(μ, ℤ/m) ⟶^{δ} H_{4}(P, ℤ/m) ⟶ H_{4}(SL(F), ℤ/m) ⟶^{c_{2,4}} H_{et}^{0}(F, μ_{m}^{⊗2})
↓≅  ↓  ↓  ∥
μ_{m}  ⟶^{δ} _{m}π_{3}^{ind}(BM^{+}) ⟶  _{m}K_{3}^{ind}(F)  ⟶^{−e} μ_{m} ⊗ μ_{m}.

Since the top composition is an isomorphism by Ex. V.11.5, the result follows. □

Corollary 5.20. π_{3}^{ind}(BM^{+}) ≅ μ̃(F) ⊕ ℤ/12.

Proof. If char(F) = 2, the result is immediate from Proposition 5.17, so we may suppose that char(F) ≠ 2. If F ⊂ E then μ(F) ⊆ μ(E) and therefore, by naturality of Proposition 5.17 in F, the map on π_{3}(BM^{+}) groups is an injection. We may therefore assume that F is algebraically closed. In this case, it follows from 5.18 and 5.19 that (δ, i) : μ(F) ⊕ ℤ/12 → π_{3}(BM^{+}) is an isomorphism, because γδ is onto and δ(−1) is a nonzero element of the kernel μ_{2}(F) of γ. □

Proof of Theorem 5.2. (Suslin) By Corollaries 5.16.1 and 5.20, the kernel of K_{3}^{ind}(F) → B(F) is the image of μ̃(F). Thus it suffices to show that the summand μ̃(F) of π_{3}(BM^{+}) (given by Corollary 5.20) injects into K_{3}^{ind}(F). When F is algebraically closed, this is given by Lemma 5.19. Since μ̃(F) → μ̃(E) is an injection for all field extensions F ⊂ E, the general case follows. □

EXERCISES

5.1 (Hutchinson) If |X| = d, C_{n}(X) = 0 for n ≥ d by 5.5. Show that H_{n}(C_{∗}(X)) = 0 for all n ≠ 0, d − 1, and that H_{n}(C′_{∗}(X)) = 0 for all n ≠ d − 1. (C′_{∗} is defined in 5.10.) Conclude that if |X| ≥ 5 then H_{3}(G, ℤ) ≅ ℍ_{3}(G, C_{∗}(X)) and H_{4}(G, C′_{∗}(X)) = 0.

5.2 In this exercise we prove Lemma 5.8.1. In the hypercohomology spectral sequence (5.8), the differential d^{1} from H_{q}(G, C_{1}) to H_{q}(G, C_{0}) ≅ H_{q}(B, ℤ) ≅ H_{q}(T_{2}, ℤ) is induced by the map d_{0}^{1} − d_{1}^{1} : C_{1} → C_{0}, where d_{0}^{1}(x_{0}, x_{1}) = x_{1} and d_{1}^{1}(x_{0}, x_{1}) = x_{0}. Let T_{2} be the diagonal subgroup, and B the upper triangular subgroup of GL_{2}(F). We saw after (5.8) that the inclusion of T_{2} and B induces isomorphisms ι : H_{q}(T_{2}, ℤ) ⟶^{≅} H_{q}(G, C_{1}) and H_{q}(B, ℤ) ⟶^{≅} H_{q}(G, C_{0}). Show that:

(i) d_{0}^{1} ι = σ(d_{1}^{1} ι), where σ is the involution σ(a, b) = (b, a) of F^{2}; (ii) The composition d_{0}^{1} ι is the natural map H_{q}(T_{2}, ℤ) → H_{q}(B, ℤ) ≅ H_{q}(G, C_{0}); (iii) The fact that H_{q}(T_{2}, ℤ) → H_{q}(B, ℤ) is an isomorphism ([Su84, §3]) implies that d^{1} = 1 − σ.

5.3 Show that B(𝔽_{5}) ≅ ℤ/3 on generator c = [2] = 2[3], with [−1] = 0. Then show that B(𝔽_{7}) ≅ ℤ/4 on generator [−1] = 2[3] with c = [4] = 2[−1]. (In both cases,

[3] ∉ B(𝔽).)

5.4 Show that c = 0 in B(𝔽_{q}) if either: (a) char(𝔽_{q}) = 2 and q ≡ 1 (mod 3); (b) char(𝔽_{q}) = 3 and q ≡ 1 (mod 4); or (c) char(𝔽_{q}) > 3 and q ≡ 1 (mod 6);

5.5 (Dupont-Sah) Show that [x^{2}] = 2 ([x] + [−x] + [−1]) in P(F) for all x ≠ ±1. Using this, show that [x^{2}] + 2[−x^{2}] − 2[x − 1] − [1/x^{2}] = 2c.

5.6 Given an arbitrary set X, let C̃_{n}(X) be the free abelian group on the set X^{n+1} of all (n + 1)-tuples (x_{0}, …, x_{n}) in X, including duplication.

a) Show that C̃_{n}(X) is a simplicial abelian group, whose degeneracy operators σ_{i} are duplication of the i^{th} entry. Under the Dold-Kan correspondence, the simplicial abelian group C̃_{n}(X) corresponds to the chain complex C_{∗}(X) of Definition 5.5.

b) Show that the rotations t_{n}(x_{0}, …, x_{n}) = (x_{n}, x_{0}, …, x_{n−1}) satisfy t_{n}^{n+1} = 1, ∂_{i}t_{n} = t_{n}∂_{i−1} and σ_{i}t_{n} = t_{n}σ_{i} for i ≠ 0, ∂_{0}t_{n} = ∂_{n} and σ_{0}t_{n} = t_{n+1}^{2}σ_{n}. This shows that C̃_{n}(X) is a cyclic abelian group [WHomo, 9.6.1].

5.7 Transfer maps. Let F ⊂ E be a finite field extension. If the transfer map K_{3}(E) → K_{3}(F) induces a map N_{E/F} : B(E) → B(F) (via Theorem 5.2), we call N_{E/F} a transfer map. If N_{E/F} exists, the composition B(F) → B(E) → B(F) must be multiplication by [E : F].

(a) Conclude that there is no transfer map N_{E/F} : B(E) → B(F) defined for all F ⊂ E. Hint: Consider 𝔽_{5} ⊂ 𝔽_{25} or ℝ ⊂ ℂ.

(b) Show that a transfer map B(E) → B(F) exists if μ(F) = μ(E), or more generally if E has an F -basis such that μ(E) is represented by monomial matrices over F.

5.8 If F_{0} is the field of constants in F, show that B(F)/B(F_{0}) is the cokernel of K_{3}^{M}(F)⊕K_{3}(F_{0}) → K_{3}(F). (The Rigidity Conjecture 5.3.1 implies that it is zero.) Conclude that B(F (t)) ≅ B(F), using V.6.7.1.

5.9 If A is an abelian group, H_{∗}(A, ℤ) is a graded-commutative ring [WHomo, 6.5.14]. Since H_{1}(A, ℤ) ≅ A, there is a ring map ∧^{∗}A → H_{∗}(A, ℤ). The Künneth formula [WHomo, 6.1.13] for H_{∗}(A × A) and the diagonal provide a natural map H_{3}(A, ℤ) → H_{3}(A × A, ℤ) → Tor(A, A), whose image is invariant under the transposition involution τ on A × A.

(a) Show that ∧^{2}A → H_{2}(A, ℤ) is an isomorphism, and that ∧^{3}A → H_{3}(A, ℤ) is an injection whose cokernel H_{3}^{ind}(A) is canonically isomorphic to Tor(A, A)^{τ}. Hint: It is true for cyclic groups; use the Künneth formula to check it for finitely generated A.

(b) If A is a finite cyclic group and σ is an automorphism of A, show that the composite

A ≅ H_{3}^{ind}(A) ⟶^{σ} H_{3}^{ind}(A) ≅ A

sends a ∈ A to σ^{2}(a). In particular, σ acts trivially when σ^{2} = 1. 5.10 In this exercise we analyze H_{3} of the group M_{n} = T_{n} ⋊ Σ_{n} of monomial matrices in GL_{n}(F), using the Hochschild-Serre spectral sequence. Here F is a field and T_{n} = (F^{×})^{n} denotes the subgroup of diagonal matrices. In this exercise we write H_{i}(G) for H_{i}(G, ℤ).

(a) Show that the images of H_{3}(T_{3}) and H_{3}(T_{n}) in H_{3}(M_{n}(F)) are the same.

(b) Show that H_{∗}(Σ_{n}, T_{n}) ≅ H_{∗}(Σ_{n−1}, F^{×}). (Σ_{n−1} is the stabilizer of T_{1} = F^{×}.)

(c) Show that H_{∗}(Σ_{n}, ∧^{2}T_{n}) ≅ H_{∗}(Σ_{n−1}, ∧^{2}F^{×}) ⊕ H_{∗}(Σ_{2} × Σ_{n−2}, F^{×} ⊗ F^{×}).

(d) When n = 2, conclude that H_{3}(M_{2}) ≅ H_{3}(Σ_{2}) ⊕ H_{3}(T_{2})_{Σ_{2}} ⊕ ˜∧^{2}F^{×}.

(e) Let A denote the kernel of H_{3}(M_{2} ×Σ_{n−2}, ℤ) → H_{3}(Σ_{2} ×Σ_{n−2}, ℤ). Conclude that H_{3}(T_{3}) + A → H_{3}(M_{n}) → H_{3}(Σ_{n}, ℤ) is exact for n ≥ 5. 5.11 Show that H_{2}(P, ℤ) ≅ π_{2}BM^{+} equals π_{2}^{s} ⊕ ˜∧^{2}F^{×}, where P = [M, M]. Hint: Recall from IV, Ex. 1.13 that BM^{+} ≃ BP^{+} × B(F^{×}) × BΣ_{2}. Thus it suffices to compute H_{2}(M, ℤ). Now use Exercise 5.10(b,c). 5.12 Show that the linear transformation α(x_{1}, x_{2}, x_{3}) = (x_{2}, x_{1}, −x_{3}) is in SM, inducing a decomposition SM ≅ P ⋊ Σ_{2}. If char(F) ≠ 2, show that α is conjugate in GL_{3}(F) to the matrix diag(−1, −1, +1) in P, and hence that the image of H_{∗}(Σ_{2}, ℤ) → H_{∗}(SM, ℤ) → H_{∗}(GL(F), ℤ) lies in the image of H_{∗}(P, ℤ). If char(F) = 2, show that α is conjugate to an upper triangular matrix in GL_{3}(F) and hence that the map H_{∗}(Σ, ℤ) → H_{∗}(GL(F), ℤ) is trivial. 5.13 Let T′ denote the group of diagonal matrices in SL(F).

(a) Show that P ≅ T′ ⋊ A_{∞}, and that H_{3}(T′, ℤ)_{A_{∞}} ≅ H_{3}(F^{×}, ℤ).

(b) Use the proof of Proposition 5.17 to show that μ(F) ≅ H_{3}(F^{×}, ℤ)/ ∧^{3} F^{×} is the image of H_{3}(T′, ℤ) in H_{3}(P, ℤ)/ ∧^{3} F^{×} ≅ π_{3}(BM^{+}).

(c) If D = μ(F) ⋊ Σ_{2}, show that the map from H_{3}(D, ℤ) to π_{3}(BM^{+}) sends μ(F)/{±1} to μ(F).

§6. GLOBAL FIELDS OF FINITE CHARACTERISTIC 37

§6. Global fields of finite characteristic

A global field of finite characteristic p is a field F which is finitely generated of transcendence degree one over 𝔽_{p}; the algebraic closure of 𝔽_{p} in F is a finite field 𝔽_{q} of characteristic p. It is classical (see [Hart, I.6]) that there is a unique smooth projective curve X over 𝔽_{q} whose function field is F. If S is any nonempty set of closed points of X, then X\S is affine; we call the coordinate ring R of X\S the ring of S-integers in F. In this section, we discuss the K-theory of F, X and the rings of S-integers of F.

Any discussion of the K-theory of F must involve the K-theory of X. For example, K_{1}(F) is related to the Picard group Pic(X) by the Units-Pic sequence I.5.12:

1 → 𝔽_{q}^{×} → F^{×} → ⊕_{x∈X}ℤ → Pic(X) → 0.

Recall that K_{0}(X) = ℤ ⊕ Pic(X), and that Pic(X) ≅ ℤ ⊕ J(X), where J(X) is a finite group (see I.5.16).

Since K_{2} vanishes on finite fields (III.6.1.1), the localization sequence V.6.12 for X ends in the exact sequence

0 → K_{2}(X) → K_{2}(F) ⟶^{∂} ⊕_{x∈X}k(x)^{×} → K_{1}(X) → 𝔽_{q}^{×} → 0.

By classical Weil reciprocity (V.6.12.1), the cokernel of ∂ is 𝔽_{q}^{×}, so K_{1}(X) ≅ 𝔽_{q}^{×} × 𝔽_{q}^{×}. By III.2.5.1, if R is the coordinate ring of any affine open in X then SK_{1}(R) = 0. A diagram chase shows that the image of K_{1}(X) in K_{1}(F) = F^{×} is 𝔽_{q}^{×}.

By III.7.2(a) (due to Bass and Tate), the kernel K_{2}(X) of ∂ is finite of order prime to p. This establishes the low dimensional cases of the following theorem, first proven by Harder [Har], using the method pioneered by Borel [Bo].

Harder’s Theorem 6.1. Let X be a smooth projective curve over a finite field of characteristic p. For n ≥ 1, each K_{n}(X) is a finite group of order prime to p.

Parshin has conjectured that if X is any smooth projective variety over a finite field then K_{n}(X) is a torsion group for n ≥ 1. Harder’s Theorem (6.1) shows that Parshin’s conjecture holds for curves.

Proof. By III.7.2(a), K_{n}^{M}(F) = 0 for all n ≥ 3. By Geisser and Levine’s Theorem 4.7, the Quillen groups K_{n}(F) are uniquely p-divisible for n ≥ 3. For every closed point x ∈ X, the groups K_{n}(x) are finite of order prime to p (n > 0) because k(x) is a finite field extension of 𝔽_{q}. From the localization sequence

⊕_{x∈X}K_{n}(x) → K_{n}(X) → K_{n}(F) → ⊕_{x∈X}K_{n−1}(x),

a diagram chase shows that K_{n}(X) is uniquely p-divisible. By IV.6.9 (due to Quillen), the abelian groups K_{n}(X) are finitely generated. As any finitely generated p-divisible abelian group A is finite with p ∤ |A|, this is true for each K_{n}(X). □

Corollary 6.2. If R is the ring of S-integers in F = 𝔽_{q}(X) (and S ≠ ∅) then:

a) K_{1}(R) ≅ R^{×} ≅ 𝔽_{q}^{×} × ℤ^{s}, |S| = s − 1;

b) For n ≥ 2, K_{n}(R) is a finite group of order prime to p.

Proof. Recall (III.1.1.1) that K_{1}(R) = R^{×} ⊕ SK_{1}(R). We saw in III.2.5.1 that SK_{1}(R) = 0, and the units of R were determined in I.5.16, whence (a). The rest follows from the localization sequence K_{n}(X) → K_{n}(R) → ⊕_{x∈S}K_{n−1}(x). □

The e-invariant 6.3. The targets of the e-invariant of X and F are the same groups as for 𝔽_{q}, because every root of unity is algebraic over 𝔽_{q}. Hence the inclusions of K_{2i−1}(𝔽_{q}) ≅ ℤ/(q^{i} − 1) in K_{2i−1}(X) and K_{2i−1}(F) are split by the e-invariant, and this group is the Harris-Segal summand.

The inverse limit of the finite curves X_{ν} = X × Spec(𝔽_{q^{ν}}) is the curve X̄ = X ⊗_{𝔽_{q}} 𝔽̄_{q} over the algebraic closure 𝔽̄_{q}. To understand K_{n}(X) for n > 1, it is useful to know not only what the groups K_{n}(X̄) are, but how the (geometric) Frobenius φ : x ↦ x^{q} acts on them.

By II.8.2.1 and I.5.15, K_{0}(X̄) = ℤ ⊕ ℤ ⊕ J(X̄), where J(X̄) is the group of points on the Jacobian variety over 𝔽̄_{q}; it is a divisible torsion group. If ℓ ≠ p, the ℓ-primary torsion subgroup J(X̄)_{ℓ} of J(X̄) is isomorphic to the abelian group (ℤ/ℓ^{∞})^{2g}. The group J(X̄) may or may not have p-torsion. For example, if X is an elliptic curve then the p-torsion in J(X̄) is either 0 or ℤ/p^{∞}, depending on whether X is supersingular (see [Hart, Ex. IV.4.15]). Note that the localization J(X̄)[1/p] is the direct sum over all ℓ ≠ p of the ℓ-primary groups J(X̄)_{ℓ}.

Next, recall that the group of units 𝔽̄_{q}^{×} may be identified with the group μ of all roots of unity in 𝔽̄_{q}; its underlying abelian group is isomorphic to ℚ/ℤ[1/p]. Passing to the direct limit of the K_{1}(X_{ν}) yields K_{1}(X̄) ≅ μ ⊕ μ.

For n ≥ 1, the groups K_{n}(X̄) are all torsion groups, of order prime to p, because this is true of each K_{n}(X_{ν}) by 6.1. We can now determine the abelian group structure of the K_{n}(X̄) as well as the action of the Galois group on them. Recall from Definition 1.7 that M(i) denotes the i^{th} Tate twist of a Galois module M.

Theorem 6.4. Let X be a smooth projective curve over 𝔽_{q}, and set X̄ = X ⊗_{𝔽_{q}} 𝔽̄_{q}. Then for all n ≥ 0 we have isomorphisms of Gal(𝔽̄_{q}/𝔽_{q})-modules:

K_{n}(X̄) ≅ \{ \begin{matrix}ℤ ⊕ ℤ ⊕ J(X̄), & n = 0 \\ μ(i) ⊕ μ(i), & n = 2i − 1 > 0 \\ J(X̄)[1/p](i), & n = 2i > 0. \end{matrix}

For ℓ ≠ p, the ℓ-primary subgroup of K_{n−1}(X) is isomorphic to K_{n}(X̄; ℤ/ℓ^{∞}), n > 0, whose Galois module structure is given by:

K_{n}(X̄; ℤ/ℓ^{∞}) ≅ \{ \begin{matrix}ℤ/ℓ^{∞}(i) ⊕ ℤ/ℓ^{∞}(i), & n = 2i ≥ 0 \\ J(X̄)_{ℓ}(i − 1), & n = 2i − 1 > 0. \end{matrix}

Proof. Since the groups K_{n}(X̄) are torsion for all n > 0, the universal coefficient theorem (Ex. IV.2.6) shows that K_{n}(X̄; ℤ/ℓ^{∞}) is isomorphic to the ℓprimary subgroup of K_{n−1}(X̄). Thus we only need to determine the Galois modules K_{n}(X̄; ℤ/ℓ^{∞}). For n = 0, 1, 2 they may be read off from the above discussion. For n > 2 we consider the motivic spectral sequence (4.2); by Theorem 4.1, the terms E_{2}^{p,q} vanish unless p = q, q + 1, q + 2. There is no room for differentials, so the spectral sequence degenerates at E_{2} to yield the groups K_{n}(X̄; ℤ/ℓ^{∞}). There are no extension issues because the edge maps are the e-invariants K_{2i}(X; ℤ/ℓ^{∞}) → H_{et}^{0}(X̄, ℤ/ℓ^{∞}(i)) = ℤ/ℓ^{∞}(i) of 6.3, and are therefore split surjections. Finally, we note that as Galois modules we have H_{et}^{1}(X̄, ℤ/ℓ^{∞}(i)) ≅ J(X̄)_{ℓ}(i − 1), and (by Poincaré Duality [Milne2, V.2]) H_{et}^{2}(X̄, ℤ/ℓ^{∞}(i + 1)) ≅ ℤ/ℓ^{∞}(i). □

Passing to invariants under the group G = Gal(𝔽̄_{q}/𝔽_{q}), there is a natural map from K_{n}(X) to K_{n}(X̄)^{G}. For odd n, we see from 6.4 and 2.3 that K_{2i−1}(X̄)^{G} ≅ ℤ/(q^{i} −1)⊕ℤ/(q^{i} −1); for even n, we have the less concrete description K_{2i}(X̄)^{G} ≅ J(X̄)[1/p](i)^{G}. One way of studying this group is to consider the action of the Frobenius on H_{et}^{∗}(X̄, ℚ_{ℓ}(i)) and use the identity H_{et}^{∗}(X, ℚ_{ℓ}(i)) = H_{et}^{∗}(X̄, ℚ_{ℓ}(i))^{G}, which follows from the spectral sequence H^{p}(G, H_{et}^{∗}(X̄, ℚ_{ℓ}(i))) ⇒ H_{et}^{∗}(X, ℚ_{ℓ}(i)) since H^{p}(G, −) is torsion for p > 0; see [WHomo, 6.11.14].

Example 6.5. φ^{∗} acts trivially on H_{et}^{0}(X̄, ℚ_{ℓ}) = ℚ_{ℓ} and H_{et}^{2}(X̄, ℚ_{ℓ}(1)) = ℚ_{ℓ}. It acts as q^{−i} on the twisted groups H_{et}^{0}(X̄, ℚ_{ℓ}(i)) and H_{et}^{2}(X̄, ℚ_{ℓ}(i + 1). Taking G-invariants yields H_{et}^{0}(X, ℚ_{ℓ}(i)) = 0 for i ≠ 0 and H_{et}^{2}(X, ℚ_{ℓ}(i)) = 0 for i ≠ 1.

Weil’s 1948 proof of the Riemann Hypothesis for Curves implies that the eigenvalues of φ^{∗} acting on H_{et}^{1}(X̄, ℚ_{ℓ}(i)) have absolute value q^{1/2−i}. Taking G-invariants yields H_{et}^{1}(X, ℚ_{ℓ}(i)) = 0 for all i.

For any G-module M, we have an exact sequence [WH, 6.1.4]

(6.5.1)  0 → M^{G} → M ⟶^{φ^{∗}−1} M → H^{1}(G, M) → 0.

The case i = 1 of the following result reproduces Weil’s theorem that the ℓ-primary torsion part of the Picard group of X is J(X̄)_{ℓ}^{G}.

Lemma 6.6. For a smooth projective curve X over 𝔽_{q}, ℓ ∤ q and i ≥ 2 we have:

(1) H_{et}^{n+1}(X, ℤ_{ℓ}(i)) ≅ H_{et}^{n}(X, ℤ/ℓ^{∞}(i)) ≅ H_{et}^{n}(X̄, ℤ/ℓ^{∞}(i))^{G} for all n;

(2) H_{et}^{0}(X, ℤ/ℓ^{∞}(i)) ≅ ℤ/w_{i}^{(ℓ)}(F);

(3) H_{et}^{1}(X, ℤ/ℓ^{∞}(i)) ≅ J(X̄)_{ℓ}(i − 1)^{G};

(4) H_{et}^{2}(X, ℤ/ℓ^{∞}(i)) ≅ ℤ/w_{i−1}^{(ℓ)}(F); and

(5) H_{et}^{n}(X, ℤ/ℓ^{∞}(i)) = 0 for all n ≥ 3.

Proof. Since i ≥ 2, we see from 6.5 that H_{et}^{n}(X, ℚ_{ℓ}(i)) = 0 for all n. Since ℚ_{ℓ}/ℤ_{ℓ} = ℤ/ℓ^{∞}, this yields H_{et}^{n}(X, ℤ/ℓ^{∞}(i)) ≅ H_{et}^{n+1}(X, ℤ_{ℓ}(i)) for all n.

Since each H^{n} = H_{et}^{n}(X̄, ℤ/ℓ^{∞}(i)) is a quotient of H_{et}^{n}(X̄, ℚ_{ℓ}(i)), φ^{∗} − 1 is a surjection, i.e., H^{1}(G, H^{n}) = 0. Since H^{n}(G, −) = 0 for n > 1, the Leray spectral sequence E_{2}^{p,q} = H^{p}(G, H_{et}^{q}(X̄, ℤ/ℓ^{∞}(i))) ⇒ H_{et}^{p+q}(X, ℤ/ℓ^{∞}(i)) for X → Spec(𝔽_{q}) [Milne, III.1.18], collapses for i > 1 to yield exact sequences

0 → H_{et}^{n}(X, ℤ/ℓ^{∞}(i)) → H_{et}^{n}(X̄, ℤ/ℓ^{∞}(i)) ⟶^{φ^{∗}−1} H_{et}^{n}(X̄, ℤ/ℓ^{∞}(i)) → 0.

In particular, H_{et}^{n}(X, ℤ/ℓ^{∞}(i)) = 0 for n>2. As in the proof of 6.4, H_{et}^{1}(X, ℤ/ℓ^{∞}(i)) is J(X̄)_{ℓ}(i − 1), so H_{et}^{1}(X, ℤ/ℓ^{∞}(i)) ≅ J(X̄)_{ℓ}(i − 1)^{G}, and H_{et}^{2}(X̄, ℤ/ℓ^{∞}(i)) is ℤ/ℓ^{∞}(i − 1) by duality, so H_{et}^{2}(X, ℤ/ℓ^{∞}(i)) is ℤ/ℓ^{∞}(i − 1)^{G} ≅ ℤ/w_{i−1}^{(ℓ)}. □

Given the calculation of K_{n}(X̄)^{G} in 6.4 and the calculation of H_{et}^{n}(X, ℤ/ℓ^{∞}(i)) in 6.6, we see that the natural map K_{n}(X) → K_{n}(X̄)^{G} is a surjection. Thus the real content of the following theorem is that K_{n}(X) → K_{n}(X̄)^{G} is an isomorphism.

Theorem 6.7. Let X be the smooth projective curve corresponding to a global field F over 𝔽_{q}. Then K_{0}(X) = ℤ⊕Pic(X), and the finite groups K_{n}(X) for n > 0 are given by:

K_{n}(X) ≅ K_{n}(X̄)^{G} ≅ \{ \begin{matrix}K_{n}(𝔽_{q}) ⊕ K_{n}(𝔽_{q}), & n odd, \\ ⨁_{ℓ≠p} J(X̄)_{ℓ}(i)^{G}, & n = 2i even. \end{matrix}

Proof. We may assume that n ≠ 0, so that the groups K_{n}(X) are finite by 6.1. It suffices to calculate the ℓ-primary part K_{n+1}(X; ℤ/ℓ^{∞}) of K_{n}(X). But this follows from the motivic spectral sequence (4.2), which degenerates by 6.6. □

Theorem 6.8. If F is the function field of a smooth projective curve X over 𝔽_{q}, then for all i ≥ 1: 𝔽_{q} ⊂ F induces an isomorphism K_{2i+1}(𝔽_{q}) ≅ K_{2i+1}(F), and there is an exact reciprocity sequence (generalizing V.6.12.1):

0 → K_{2i}(X) → K_{2i}(F) ⟶^{⊕∂_{x}} ⊕_{x∈X}K_{2i−1}(𝔽_{q}(x)) ⟶^{N} K_{2i−1}(𝔽_{q}) → 0.

Proof. The calculation of K_{2}(F) was carried out at the beginning of this section, so we restrict attention to K_{n}(F) for n ≥ 3. Because K_{2i}(𝔽_{q}) = 0, the localization sequence V.6.12 breaks up into exact sequences for i ≥ 2:

0 → K_{2i}(X) → K_{2i}(F) ⟶^{⊕∂_{x}} ⊕_{x∈X}K_{2i−1}(𝔽_{q}(x)) → K_{2i−1}(X) → K_{2i−1}(F) → 0.

Let SK_{2i−1}(X) denote the kernel of K_{2i−1}(X) → K_{2i−1}(F). Since 𝔽_{q} ⊂ F ⊂ F̄ induces an injection from the subgroup K_{2i−1}(𝔽_{q}) of K_{2i−1}(𝔽̄_{q}) into K_{2i−1}(F̄) by V.6.7.2, we see from Theorem 6.7 that |SK_{2i−1}(X)| is at most |K_{2i−1}(𝔽_{q})|. As in V.6.12, for each closed point x the composition of the transfer K_{2i−1}(𝔽_{q}(x)) → SK_{2i−1}(X) with the proper transfer π_{∗} : K_{2i−1}(X) → K_{2i−1}(𝔽_{q}) is the transfer associated to 𝔽_{q} ⊂ 𝔽_{q}(x), i.e., the transfer map K_{2i−1}(𝔽_{q}(x)) → K_{2i−1}(𝔽_{q}); each of these transfer maps are onto by IV.1.13. It follows that π_{∗} : SK_{2i−1}(X) → K_{2i−1}(𝔽_{q}) is an isomorphism. The theorem now follows. □

The Zeta Function 6.9. We can relate the orders of the K-groups of the curve X to values of the zeta function ζ_{X}(s). By definition, ζ_{X}(s) = Z(X, q^{−s}), where

Z(X, t) = exp( ∑_{n=1}^{∞} |X(𝔽_{q^{n}})| \frac{t^{n}}{n} ).

Weil proved that Z(X, t) = P (t)/(1 − t)(1 − qt) for every smooth projective curve X, where P (t) ∈ ℤ[t] is a polynomial of degree 2 · genus(X) with all roots of absolute value 1/\sqrt{q}. This formula is a restatement of Weil’s proof of the Riemann Hypothesis for X (6.5 above), given Grothendieck’s formula P (t) = det(1 − φ^{∗}t), where φ^{∗} is regarded as an endomorphism of H_{et}^{1}(X̄; ℚ_{ℓ}). Note that by 6.5 the action of φ^{∗} on H_{et}^{0}(X̄; ℚ_{ℓ}) has det(1−φ^{∗}t) = (1−t), and the action on H_{et}^{2}(X̄; ℚ_{ℓ}) has det(1 − φ^{∗}t) = (1 − qt).

Here is application of Theorem 6.7, which was conjectured by Lichtenbaum in [Li3] and proven by Thomason in [Th, (4.7)]. For legibility, let #A denote the order of a finite abelian group A.

Proposition 6.10. If X is a smooth projective curve over 𝔽_{q} then for all i ≥ 2,

\frac{#K_{2i−2}(X) · #K_{2i−3}(𝔽_{q})}{#K_{2i−1}(𝔽_{q}) · #K_{2i−3}(X)} = ∏_{ℓ} \frac{#H_{et}^{2}(X; ℤ_{ℓ}(i))}{#H_{et}^{1}(X; ℤ_{ℓ}(i)) · #H_{et}^{3}(X; ℤ_{ℓ}(i))} = |ζ_{X}(1 − i)|.

Proof. We have seen that all the groups appearing in this formula are finite. The first equality follows from 6.6 and 6.7. The second equality follows from the formula for ζ_{X}(1 − i) in 6.9. □

EXERCISES

6.1 Let X be the projective line ℙ_{𝔽_{q}}^{1} over 𝔽_{q}. Use Theorem 6.7 to recover the calculation of K_{∗}(X) in IV.1.5. Show directly that Z(X, t) = 1/(1 − t)(1 − qt) and use this to verify the formula in 6.9 for ζ_{X}(i − 1). 6.2 Let R be the ring of S-integers in a global field F = 𝔽_{q}(X) of finite characteristic. Show that K_{n}(R) → K_{n}(F) is an injection for all n ≥ 1, and that K_{2i−1}(R) → K_{2i−1}(F) is an isomorphism for all i > 1. (This generalizes the Bass-Milnor-Serre Theorem III.2.5, and provides another proof of 6.2(a).) 6.3 Let F = 𝔽_{q}(X) be a global field, of degree d over a function field 𝔽_{q}(t). For i > 0, show that the transfer K_{2i}(F) → K_{2i}(𝔽_{q}(t)) is onto, and that the transfer K_{2i−1}(F) → K_{2i−1}(𝔽_{q}(t)) is multiplication by d (under the identification of both groups with K_{2i−1}(𝔽_{q}) in 6.8).

§7. Local Fields

A local field is a field E which is complete under a discrete valuation v, and whose residue field k_{v} is finite. The subring V of elements of positive valuation is a complete valuation domain. It is classical that every local field is either a finite extension of the p-adic rationals ℚ̂_{p} or isomorphic to 𝔽_{q}((t)). (See [S-LF].)

We saw in II.2 and III.1.4 that K_{0}(V) = K_{0}(E) = ℤ and K_{1}(V) = V^{×}, K_{1}(E) = E^{×} ≅ (V^{×})×ℤ, where the factor ℤ is identified with the powers {π^{m}} of a parameter π of V. It is well known that V^{×} ≅ μ(E) × U_{1}, where μ(E) is the group of roots of unity in E (or V), and that U_{1} is a torsionfree ℤ_{p}-module.

We also saw in Moore’s Theorem (Chapter III, Theorem 6.2.4 and Ex. 6.11) that K_{2}(E) ≅ U_{2} ⊕ 𝔽_{q}^{×}, where U_{2} is an uncountable, uniquely divisible abelian group. Since K_{2}(E) ≅ K_{2}(V) ⊕ 𝔽_{q}^{×} by V.6.9.2, this implies that K_{2}(V) ≅ U_{2}.

Proposition 7.1. Let E be a local field. For n ≥ 3, K_{n}^{M}(E) is an uncountable, uniquely divisible group. The group K_{2}(E) is the sum of 𝔽_{q}^{×} and an uncountable, uniquely divisible group.

Proof. The group is uncountable by Ex. III.7.14, and divisibility follows easily from Moore’s Theorem (see III, Ex. 7.4). We give a proof that it is uniquely divisible using the isomorphism K_{n}^{M}(E) ≅ H^{n}(E, ℤ(n)). If char(E) = p, K_{n}^{M}(E) has no p-torsion by Izhboldin’s Theorem III.7.8, so we consider m-torsion when 1/m ∈ E. The long exact sequence in motivic cohomology associated to the coefficient sequence 0 → ℤ(n) ⟶^{m} ℤ(n) → ℤ/m(n) → 0 yields the exact sequence for m:

(7.1.1)  H_{et}^{n−1}(E, μ_{m}^{⊗n}) → K_{n}^{M}(E) ⟶^{m} K_{n}^{M}(E) → H_{et}^{n}(E, μ_{m}^{⊗n}).

Since H_{et}^{n}(E, −)=0 for n ≥ 3, this immediately implies that K_{n}^{M}(E) is uniquely mdivisible for n > 3 (and m-divisible for n = 3). Moreover, the m-torsion subgroup of K_{3}^{M}(E) is a quotient of the group H_{et}^{2}(E, μ_{m}^{⊗3}), which by duality is ℤ/(w_{2}, m), where w_{2} = w_{2}(E) is q^{2} − 1 by 2.3.1. Thus the torsion subgroup of K_{3}^{M}(E) is a quotient of w_{2}. We may therefore assume that w_{2} divides m. Now map the sequence (7.1.1) for m^{2} to the sequence (7.1.1) for m; the map from ℤ/w_{i} = H_{et}^{2}(E, μ_{m^{2}}^{⊗3}) to ℤ/w_{i} = H_{et}^{2}(E, μ_{m}^{⊗3}) is the identity but the map from the image _{m^{2}}K_{3}^{M}(E) to mK_{3}^{M}(E) is multiplication by m and thus zero, as required. □

Equicharacteristic local fields

We first dispose of the equi-characteristic case, where E = 𝔽_{q}((t)), V ≅ 𝔽_{q}[[π]] and char(E) = p. In this case, μ(E) = 𝔽_{q}^{×}, and U_{1} = 1 + π𝔽_{q}[[π]] is isomorphic to the big Witt vectors of 𝔽_{q} (II.4.3), which is the product of a countably infinite number of copies of ℤ_{p} (see Ex. 7.2). (In fact, it is a countably infinite product of copies of the ring ℤ_{p}[ζ_{q−1}] of Witt vectors over 𝔽_{q}.)

Here is a description of the abelian group structure on K_{n}(V) for n ≥ 2.

Theorem 7.2. Let V = 𝔽_{q}[[π]] be the ring of integers in the local field E = 𝔽_{q}((π)). For n ≥ 2 there are uncountable, uniquely divisible abelian groups U_{n} and canonical isomorphisms:

K_{n}(E) ≅ K_{n}(V) ⊕ K_{n−1}(𝔽_{q}), K_{n}(V) ≅ K_{n}(𝔽_{q}) ⊕ U_{n}.

§7. LOCAL FIELDS 43

Proof. The splitting K_{n}(E) ≅ K_{n}(V) ⊕ K_{n−1}(𝔽_{q}) was established in V.6.9.2. Now let U_{n} denote the kernel of the canonical map K_{n}(V) → K_{n}(𝔽_{q}). Since V → 𝔽_{q} splits, naturality yields K_{n}(V) = U_{n} ⊕ K_{n}(𝔽_{q}). By Gabber rigidity (IV.2.10), U_{n} is uniquely ℓ-divisible for all ℓ ≠ p and n > 0. It suffices to show that U_{n} is uncountable and uniquely p-divisible when n ≥ 2; this holds for n = 2 by 7.1.

Now the Milnor groups K_{n}^{M}(E) are uncountable, uniquely divisible abelian groups for n ≥ 3, by Proposition 7.1. The group K_{n}^{M}(E) is a summand of the Quillen K-group K_{n}(E) by 4.3 or 4.9. On the other hand, the Geisser-Levine Theorem 4.7 shows that the complementary summand is uniquely p-divisible. □

p-adic local fields

In the mixed characteristic case, when char(E) = 0, even the structure of V^{×} is quite interesting. The torsionfree part U_{1} is a free ℤ_{p}-module of rank [E : ℚ_{p}]; it is contained in (1 + πV)^{×} and injects into E as a lattice by the convergent power series for x ↦ ln(x).

The quotient V^{×} → 𝔽_{q}^{×} splits, and the subgroup of V^{×} isomorphic to 𝔽_{q}^{×} is called the group of Teichmüller units. Thus K_{1}(V) = V^{×} is a product U_{1} × 𝔽_{q}^{×} × μ_{p^{∞}}(E), where μ_{p^{∞}}(E) is the finite cyclic group of p-primary roots of unity in V. There seems to be no simple formula for the order of the cyclic p-group μ_{p^{∞}}(E).

To understand the groups K_{n}(E) for n ≥ 3, recall from Proposition 7.1 that K_{n}^{M}(E) is an uncountable, uniquely divisible abelian group. From Example 4.3, this is a direct summand of K_{n}(E); since K_{n}(E) ≅ K_{n}(V) ⊕ K_{n−1}(𝔽_{q}) by V.6.9.2, it is also a summand of K_{n}(V). Thus, as in the equicharacteristic case, K_{n}(E) will contain an uncountable uniquely divisible summand about which we can say very little.

Before stating our next result, we need an étale calculation. Since E has étale cohomological dimension 2, we may ignore H_{et}^{n}(E, −) for n > 2. By Tate–Poitou duality [Milne2, I.2.3], H_{et}^{2}(E, μ_{m}^{⊗i+1}) is isomorphic to H_{et}^{0}(E, μ_{m}^{⊗i}). We shall assume that i > 0 and m is divisible by w_{i}(E), so that these groups are isomorphic to ℤ/w_{i}(E). Now consider the change of coefficients μ_{m}^{⊗i} ⊂ μ_{m^{2}}^{⊗i}. The induced endomorphisms of ℤ/w_{i}(E) = H_{et}^{0}(E, μ_{m}^{⊗i}) and ℤ/w_{i}(E) = H_{et}^{2}(E, μ_{m}^{⊗i+1}) are the identity map and the zero map, respectively. Since μ = ∪μ_{m}, passing to the limit over m yields:

H_{et}^{0}(E, μ^{⊗i}) ≅ ℤ/w_{i}(E) and H_{et}^{2}(E, μ^{⊗i+1}) = 0, i > 0.

Recall that an abelian group which is uniquely ℓ-divisible for all ℓ ≠ p is the same thing as a ℤ_{(p)}-module.

Proposition 7.3. For n > 0 we have K_{n}(E) ≅ K_{n}(V) ⊕ K_{n−1}(𝔽_{q}), and the groups K_{n}(V) are ℤ_{(p)}-modules.

When n = 2i−1, K_{n}(V) ≅ K_{n}(E) is the direct sum of a torsionfree ℤ_{(p)}-module and the Harris-Segal summand (see 2.5), which is isomorphic to ℤ/w_{i}(E).

When n = 2i, K_{n}(V) is the direct sum of ℤ/w_{i}^{(p)}(E) and a divisible ℤ_{(p)}-module.

Proof. The decomposition K_{n}(E) ≅ K_{n}(V) ⊕ K_{n−1}(𝔽_{q}) was established in V.6.9.2. In particular, K_{2i−1}(V) ≅ K_{2i−1}(E).

To see that K_{2i−1}(E) has a cyclic summand of order w_{i}(E), consider the spectral sequence (4.2) with coefficients ℚ/ℤ. By the above remarks, it degenerates completely to yield K_{2i}(E; ℚ/ℤ) ≅ ℤ/w_{i}(E). Since this injects into K_{2i}(Ē; ℚ/ℤ) ≅ ℚ/ℤ (by Theorem 1.6, Ē being the algebraic closure of E), this implies that the Adams e-invariant is an isomorphism: K_{2i}(E; ℚ/ℤ) ⟶^{≃} ℤ/w_{i}(E). By Ex. IV.2.5, this implies that K_{2i−1}(E) ≅ T_{i} ⊕ ℤ/w_{i}(E) where T_{i} is torsionfree.

To see that K_{2i}(E) is the sum of a divisible group and ℤ/w_{i}(E), fix i and suppose that w_{i}(E) divides m. Since H_{et}^{n}(E, −) = 0 for n > 2, the spectral sequence (4.2) with coefficients ℤ/m degenerates completely and describes K_{2i}(E; ℤ/m) as an extension of H_{et}^{0}(E, μ_{m}^{⊗i}) ≅ ℤ/w_{i}(E) by H_{et}^{2}(E, μ_{m}^{⊗i+1}) ≅ ℤ/w_{i}(E). By the previous paragraph, the quotient ℤ/w_{i}(E) is identified with the m-torsion in K_{2i−1}(E), so the kernel ℤ/w_{i}(E) is identified with K_{2i}(E)/m. Setting m = m′w_{i}(E), it follows that the subgroup D_{i} = w_{i}(E)K_{2i}(E) of K_{2i}(E) is a divisible group. Thus K_{2i}(E) ≅ D_{i} ⊕ ℤ/w_{i}(E), and K_{2i}(V) ≅ D_{i} ⊕ ℤ/w_{i}^{(p)}(E).

It remains to show that T_{i} and D_{i} are uniquely ℓ-divisible for ℓ ≠ p and i > 0, i.e., that T_{i}/ℓT_{i} = _{ℓ}(D_{i}) = 0. By Universal Coefficients IV.2.5 and the calculations above, K_{2i−1}(V; ℤ/ℓ^{ν}) is isomorphic to ℤ/w_{i}^{(ℓ)}(E) ⊕ T_{i}/ℓ^{ν}T_{i} ⊕_{ℓ^{ν}} (D_{i}) for large ν. By Gabber Rigidity IV.2.10, K_{2i−1}(V; ℤ/ℓ^{ν}) and K_{2i−1}(𝔽_{q}; ℤ/ℓ^{ν}) ≅ w_{i}^{(ℓ)}(𝔽_{q}) are isomorphic. Since w_{i}^{(ℓ)}(E) = w_{i}^{(ℓ)}(𝔽_{q}) by 2.3.1, we have T_{i}/ℓ^{ν}T_{i} = _{ℓ^{ν}} (D_{i}) = 0, as required. □

Remark 7.3.1. Either T_{i} fails to be p-divisible, or else D_{i−1} has p-torsion. This follows from Corollary 7.4.1 below: dim(T_{i}/pT_{i}) + dim(_{p}D_{i−1}) = [E : ℚ_{p}].

We now consider the p-adic K-groups K_{∗}(E; ℤ_{p}) of E, as in IV.2.9. This result was first proved in [RW, 3.7] for p = 2, and in [HM, thm. A] for p > 2.

Theorem 7.4. Let E be a local field, of degree d over ℚ_{p}, with ring of integers V. Then for n ≥ 2 we have:

K_{n}(V; ℤ_{p}) ≅ K_{n}(E; ℤ_{p}) ≅ \{ \begin{matrix}ℤ/w_{i}^{(p)}(E), & n = 2i, \\ (ℤ_{p})^{d} ⊕ ℤ/w_{i}^{(p)}(E), & n = 2i − 1. \end{matrix} \}

Proof. It is classical that the groups H_{et}^{∗}(E; ℤ/p^{ν}) are finitely generated groups, and that the H_{et}^{∗}(E; ℤ/p) are finitely generated ℤ_{p}-modules. Since this implies that the groups K_{n}(E; ℤ/p) are finitely generated, this implies (by IV.2.9) that K_{n}(E; ℤ_{p}) is an extension of the Tate module of K_{n−1}(E) by the p-adic completion of K_{n}(E). Since the Tate module of K_{2i−1}(E) is trivial by 7.3, this implies that K_{2i}(E; ℤ_{p}) ≅ \underleftarrow{lim} K_{2i}(E)/p^{ν} ≅ ℤ/w_{i}^{(p)}(E).

By 7.3 and IV.2.9, K_{n}(V; ℤ_{p}) ≅ K_{n}(E; ℤ_{p}) for all n > 0. Hence it suffices to consider the p-adic group K_{2i−1}(V; ℤ_{p}). By 7.3 and IV.2.9 again, K_{2i−1}(V; ℤ_{p}) is the direct sum of the finite p-group ℤ/w_{i}^{(p)}(E) and two finitely generated torsionfree ℤ_{p}-modules: the Tate module of D_{i−1} and T_{i} ⊗_{ℤ} ℤ_{p}. All that is left is to calculate the rank of K_{2i−1}(V; ℤ_{p}).

Wagoner proved in [Wag] that the ℚ_{p}-vector space K_{n}(V; ℤ_{p})⊗ℚ has dimension [E : ℚ_{p}] when n is odd and n ≥ 3. (Wagoner’s continuous K-groups were identified with K_{∗}(E; ℤ_{p}) in [Pa].) Hence K_{2i−1}(V; ℤ_{p}) has rank d = [E : ℚ_{p}]. □

Corollary 7.4.1. For i > 1 and all large ν we have

K_{2i−1}(E; ℤ/p^{ν}) ≅ H_{et}^{1}(E, μ_{p^{ν}}^{⊗i}) ≅ (ℤ/p^{ν})^{[E:ℚ_{p}]} ⊕ ℤ/w_{i}^{(p)}(E) ⊕ ℤ/w_{i−1}^{(p)}(E).

Proof. This follows from Universal Coefficients and Theorem 7.4. □

Corollary 7.4.2. K_{3}(V) contains a torsionfree subgroup isomorphic to ℤ_{(p)}^{d}, whose p-adic completion is isomorphic to K_{3}(V; ℤ_{p}) ≅ (ℤ_{p})^{d}.

Proof. Combine 7.4 with Moore’s theorem III.6.2.4 that D_{1} is torsionfree. □

Remark 7.4.3. Surprisingly, the cohomology groups H_{et}^{1}(E; μ_{m}^{⊗i}) (for m = p^{ν}) were not known before the K-group K_{2i−1}(E; ℤ/m) was calculated, circa 2000.

Warning 7.5. Unfortunately, I do not know how to reconstruct the homotopy groups K_{n}(V) from the information in 7.4. Any of the ℤ_{p}’s in K_{2i−1}(V; ℤ_{p}) could come from either a ℤ_{(p)} in K_{2i−1}(V) or a ℤ/p^{∞} in K_{2i−2}(V). Another problem is illustrated by the case n = 1, V = ℤ_{p} and p ≠ 2. The information that \underleftarrow{lim} K_{1}(V)/p^{ν} ≅ (1 + πV)^{×} ≅ V is not enough to deduce that K_{1}(V) ⊗ ℤ_{(p)} ≅ V.

Even if we knew that \underleftarrow{lim} K_{n}(V)/p^{ν} = ℤ_{p}, we would still not be able to determine the underlying abelian group K_{n}(V) exactly. To see why, note that the extension 0 → ℤ_{(p)} → ℤ_{p} → ℤ_{p}/ℤ_{(p)} → 0 doesn’t split, because there are no p-divisible elements in ℤ_{p}, yet ℤ_{p}/ℤ_{(p)} ≅ ℚ_{p}/ℚ is a uniquely divisible abelian group. For example, I doubt that the extension 0 → ℤ_{(p)}^{d} → K_{3}(V) → U_{3} → 0 splits in Corollary 7.4.1.

Here are some more cases when I can show that the ℤ_{p}’s in K_{2i−1}(V; ℤ_{p}) come from torsionfree elements in K_{2i−1}(E); I do not know any example where a ℤ/p^{∞} appears in K_{2i}(E).

Example 7.6. If k > 0, then K_{4k+1}(ℤ_{2}) contains a subgroup T isomorphic to ℤ_{(2)}, and the quotient K_{4k+1}(ℤ_{2})/(T ⊕ℤ/w_{i}(ℚ_{2}) is uniquely divisible. (By Exercise 2.3, w_{i}(ℚ_{2}) = 2(2^{2k+1} − 1).) This follows from Rognes’ theorem [R1, 4.13] that the map from K_{4k+1}(ℤ) ⊗ ℤ_{2} ≅ ℤ_{2} ⊕ (ℤ/2) to K_{4k+1}(ℤ_{2}; ℤ_{2}) is an isomorphism for all k > 1. (The information about the torsion subgroups, missing in [R1], follows from 7.4 and 7.4.1.) Since this map factors through K_{4k+1}(ℤ_{2}), the assertion follows.

Example 7.7. Let F be a totally imaginary number field of degree d = 2r_{2} over ℚ, with s prime ideals over p, and let E_{1}, ..., E_{s} be the completions of F at these primes. By Borel’s Theorem IV.1.18, there is a subgroup of K_{2i−1}(F) isomorphic to ℤ^{r_{2}}; its image in ⊕K_{2i−1}(E_{j}) is a subgroup of rank at most r_{2}, while ⊕K_{2i−1}(E_{j}; ℤ_{p}) has rank 2r_{2} = ∑[E_{j} : ℚ_{p}]. So these subgroups of K_{2i−1}(E_{j}) can account for at most half of the torsionfree part of ⊕K_{2i−1}(E_{j}; ℤ_{p}).

Example 7.8. Suppose that F is a totally real number field, of degree d = r_{1} over ℚ, and let E_{1}, ..., E_{s} be the completions of F at the prime ideals over p. By Borel’s Theorem IV.1.18, there is a subgroup of K_{4k+1}(F) isomorphic to ℤ^{d} for all k > 0; its image in ⊕K_{4k+1}(E_{j}) is a subgroup of rank d. Although ⊕K_{4k+1}(E_{j}; ℤ_{p}) also has rank d = ∑[E_{j} : ℚ_{p}], this does not imply that the p-adic completion ℤ_{p}^{d} of the subgroup injects into ⊕K_{4k+1}(E_{j}; ℤ_{p}).

Implications like this are related to Leopoldt’s conjecture, which states that the torsionfree part ℤ_{p}^{d−1} of (O_{F})^{×}⊗ℤ_{p} injects into the torsionfree part ℤ_{p}^{d} of ∏_{j=1}^{s} O_{E_{j}}^{×}; see [Wash, 5.31]. This conjecture has been proven when F is an abelian extension of ℚ; see [Wash, 5.32].

When F is a totally real abelian extension of ℚ, and p is a regular prime, Soulé shows in [Sou81, 3.1, 3.7] that the torsion free part ℤ_{p}^{d} of K_{4k+1}(F) ⊗ ℤ_{p} injects into ⊕K_{4k+1}(E_{j}; ℤ_{p}) ≅ (ℤ_{p})^{d}, because the cokernel is determined by the Leopoldt p-adic L-function L_{p}(F, ω^{2k}, 2k+1), which is a p-adic unit in this favorable scenario. Therefore in this case we also have a summand ℤ_{(p)}^{d} in each of the groups K_{4k+1}(E_{j}).

We conclude this section with a description of the topological type of the p-adic completions (see IV.2.9) K̂(V)_{p} and K̂(E)_{p}, when p is odd, due to Hesselholt and Madsen [HM]. Recall that F Ψ^{k} denotes the homotopy fiber of Ψ^{k} − 1 : ℤ × BU → BU. Since Ψ^{k} = k^{i} on π_{2i}(BU) = ℤ when i > 0, and π_{2i−1}(BU) = 0, we see that π_{2i−1}F Ψ^{k} ≅ ℤ/(k^{i} − 1), and that all even homotopy groups of F Ψ^{k} vanish, except for π_{0}(F Ψ^{k}) = ℤ.

Theorem 7.9. ([HM, thm. D]) Let E be a local field, of degree d over ℚ_{p}, with p odd. Then after p-completion, there is a number k (given below) so that

K̂(V) ≃ SU × U^{d−1} × F Ψ^{k} × BF Ψ^{k},  K̂(E) ≃ U^{d} × F Ψ^{k} × BF Ψ^{k}.

The number k is defined as follows. As in Proposition 2.2, let p^{a} be the number of p-primary roots of unity in E(μ_{p}) and set r = [E(μ_{p}) : E]. If γ is a topological generator of ℤ_{p}^{×}, then k = γ^{n}, where n = p^{a−1}(p − 1)/r. (See Exercise 7.4.)

EXERCISES

7.1 Show that M = ∏_{i=1}^{∞} ℤ_{p} is not a free ℤ_{p}-module. Hint: Consider the submodule S of all (a_{1}, ...) in M where all but finitely many a_{i} are divisible by p^{ν} for all ν. If M were free, S would also be free. Show that S/p is countable, and that ∏ p^{i}ℤ_{p} is an uncountably generated submodule of S. Hence S cannot be free. 7.2 When E = 𝔽_{q}[[π]], show that the subgroup W (𝔽_{q}) = 1 + π𝔽_{q}[[π]] of units is a module over ℤ_{p}, by defining (1 + πf)^{a} for all power series f and all a ∈ ℤ_{p}. If {u_{i}} is a basis of 𝔽_{q} over 𝔽_{p}, show that every element of W (𝔽_{q}) is uniquely the product of terms (1 + u_{i}t^{n})^{a_{ni}}, where a_{ni} ∈ ℤ_{p}. This shows that W (𝔽_{q}) is a countably infinite product of copies of ℤ_{p}. Using Ex. 7.1, conclude that W (𝔽_{q}) is not a free ℤ_{p}-module.

7.3 Show that the first étale Chern classes K_{2i−1}(E; ℤ/p^{ν}) ≅ H^{1}(E, μ_{p^{ν}}^{⊗i}) are natural isomorphisms for all i and ν. 7.4 In Theorem 7.9, check that π_{2i−1}F Ψ^{k} ≅ ℤ_{p}/(k^{i} − 1) is ℤ/w_{i}(E) for all i.

§8. NUMBER FIELDS AT PRIMES WHERE cd = 2 47

§8. Number fields at primes where cd = 2

In this section we quickly obtain a cohomological description of the odd torsion in the K-groups of a number field, and also the 2-primary torsion in the K-groups of a totally imaginary number field. These are the cases where cd_{ℓ}(O_{S}) = 2; see [Milne2, 4.10]. This bound forces the motivic spectral sequence 4.2 to degenerate completely, leaving an easily-solved extension problem.

Classical Data 8.1. Let O_{S} be a ring of integers in a number field F. By Chapter IV, 1.18 and 6.9, the groups K_{n}(F) are finite when n is even and nonzero; if n is odd and n ≥ 3 the groups K_{n}(F) are the direct sum of a finite group and ℤ^{r}, where r is r_{2} when n ≡ 3 (mod 4) and r_{1} + r_{2} when n ≡ 1 (mod 4). Here r_{1} is the number of real embeddings of F, and r_{2} is the number of complex embeddings (up to conjugacy), so that [F : ℚ] = r_{1} + 2r_{2}. The formulas for K_{0}(O_{S}) = ℤ ⊕ Pic(O_{S}) and K_{1}(O_{S}) = O_{S}^{×} ≅ ℤ^{r_{2}+|S|−1} ⊕ μ(F) are different; see II.2.6.3 and III.1.3.6.

The Brauer group of O_{S} is determined by the sequence

(8.1.1)  0 → Br(O_{S}) → (ℤ/2)^{r_{1}} ⊕ ∐_{\begin{matrix}v∈S \\ finite \end{matrix}}(ℚ/ℤ) ⟶^{add} ℚ/ℤ → 0.

The notation A_{(ℓ)} will denote the localization of an abelian group A at the prime ℓ, and the notation _{ℓ}A will denote the subgroup {a ∈ A | ℓa = 0}.

Theorem 8.2. Let F be a number field, and let O_{S} be a ring of integers in F. Fix a prime ℓ; if ℓ = 2 we suppose F totally imaginary. Then for all n ≥ 2:

K_{n}(O_{S})_{(ℓ)} ≅ \{ \begin{matrix}H_{et}^{2}(O_{S}[1/ℓ]; ℤ_{ℓ}(i + 1)) & for n = 2i > 0; \\ ℤ_{(ℓ)}^{r_{2}} ⊕ ℤ/w_{i}^{(ℓ)}(F) & for n = 2i − 1, i even; \\ ℤ_{(ℓ)}^{r_{2}+r_{1}} ⊕ ℤ/w_{i}^{(ℓ)}(F) & for n = 2i − 1, i odd. \end{matrix}

Proof. Set R = O_{S}[1/ℓ]. For each prime ideal 𝔭 over ℓ, K_{n−1}(R/𝔭) has no ℓ-torsion by IV.1.13. By the localization sequence (V, (6.6) or 6.8), K_{n}(O_{S})_{(ℓ)} = K_{n}(R)_{(ℓ)}. Thus we may replace O_{S} by R = O_{S}[1/ℓ]. Since the rank of K_{n}(R) is classically known (see 8.1), it suffices by IV.2.9 and Ex. IV.2.6 to determine K_{2i−1}(R){ℓ} = K_{2i}(R; ℤ/ℓ^{∞}) and K_{2i}(R){ℓ} = K_{2i}(R; ℤ_{ℓ}).

If F is a number field, the étale ℓ-cohomological dimension of R (and of F) is 2, unless ℓ = 2 and r_{1} > 0 (F has a real embedding). Since H_{et}^{2}(R; ℤ/ℓ^{∞}(i)) = 0 by Ex. 8.1, the motivic spectral sequence 4.2 with coefficients ℤ/ℓ^{∞} has at most one nonzero entry in each total degree at the E_{2} page. Thus we may read off:

K_{n}(R; ℤ/ℓ^{∞}) ≅ \{ \begin{matrix}H^{0}(R; ℤ/ℓ^{∞}(i)) = ℤ/w_{i}^{(ℓ)}(F) & for n = 2i ≥ 2, \\ H^{1}(R; ℤ/ℓ^{∞}(i)) & for n = 2i − 1 ≥ 1. \end{matrix}

The description of K_{2i−1}(R){ℓ} = K_{2i}(R; ℤ/ℓ^{∞}) follows.

The same argument works for coefficients ℤ_{ℓ}; for i > 0 we have H_{et}^{n}(R, ℤ_{ℓ}(i)) = 0 for n ≠ 1, 2, so the spectral sequence (4.2) degenerates to yield K_{2i−1}(R; ℤ_{ℓ}) ≅ H_{et}^{1}(R, ℤ_{ℓ}(i)) and K_{2i}(R; ℤ_{ℓ}) ≅ H_{et}^{2}(R, ℤ_{ℓ}(i+1)) (which is a finite group by Ex. 8.1). The description of K_{2i}(R){ℓ} = K_{2i}(R; ℤ_{ℓ}) follows. □

Corollary 8.3. For all odd ℓ and i > 0, K_{2i}(O_{S})/ℓ ≅ H_{et}^{2}(O_{S}[1/ℓ], μ_{ℓ}^{⊗i+1}). The same formula holds for ℓ = 2 if F is totally imaginary.

Proof. Immediate from 8.2 since H_{et}^{2}(R, ℤ_{ℓ}(i + 1))/ℓ ≅ H_{et}^{2}(R, μ_{ℓ}^{⊗i+1}). □

Example 8.3.1. Let F be a number field containing a primitive ℓ^{th} root of unity, ℓ ≠ 2, and let S be the set of primes over ℓ in O_{F}. If t is the rank of Pic(O_{S})/ℓ, then H_{et}^{2}(O_{S}, μ_{ℓ}) has rank t+|S|−1 by (8.1.1). Since H_{et}^{2}(O_{S}, μ_{ℓ}^{⊗i+1}) ≅ H_{et}^{2}(O_{S}, μ_{ℓ})⊗μ_{ℓ}^{⊗i}, it follows from Corollary 8.3 that K_{2i}(O_{S})/ℓ has rank t+|S|−1. Hence the ℓ-primary subgroup of the finite group K_{2i}(O_{F}) has t + |S| − 1 nonzero summands for each i ≥ 1.

Example 8.3.2. If ℓ ≠ 2 is a regular prime (see 2.4.1), we claim that K_{2i}(ℤ[ζ_{ℓ}]) has no ℓ-torsion. The case K_{0} is tautological since Pic(O_{F})/ℓ = 0 by definition. Setting R = ℤ[ζ_{ℓ}, 1/ℓ], we have |S| = 1 and Br(R) = 0 by (8.1.1). The case K_{2} is known classically; see III.6.9.3. By Theorem 8.2, K_{2i}(ℤ[ζ_{ℓ}])_{(ℓ)} ≅ H_{et}^{2}(R, ℤ_{ℓ}(i + 1)). By Example 8.3.1, H_{et}^{2}(R, ℤ_{ℓ}(i + 1)) = 0 and the claim now follows.

Note that every odd-indexed group K_{2i−1}(ℤ[ζ_{ℓ}]) ≅ ℤ^{r_{2}} ⊕ ℤ/w_{i}(F) has nontrivial ℓ-torsion, because w_{i}^{(ℓ)}(F) ≥ ℓ for all i by 2.2.

Combining Theorems 8.1 and 8.2, we obtain a description of K_{∗}(O_{S}) when F is totally imaginary. This includes exceptional number fields such as ℚ(\sqrt{−7}).

Theorem 8.4. Let F be a totally imaginary number field, and let O_{S} be the ring of S-integers in F for some set S of finite places. Then:

K_{n}(O_{S}) ≅ \{ \begin{matrix}ℤ ⊕ Pic(O_{S}), & for n = 0; \\ ℤ^{r_{2}+|S|−1} ⊕ ℤ/w_{1}(F), & for n = 1; \\ ⊕_{ℓ} H_{et}^{2}(O_{S}[1/ℓ]; ℤ_{ℓ}(i + 1)) & for n = 2i ≥ 2; \\ ℤ^{r_{2}} ⊕ ℤ/w_{i}(F) & for n = 2i − 1 ≥ 3. \end{matrix}

Proof. The cases n = 0, 1 and the ranks of K_{n} are part of the Classical Data 8.1. Since F is totally imaginary, the torsion comes from Theorem 8.2. □

Similarly, the mod-ℓ spectral sequence (4.2) collapses when ℓ is odd to yield the K-theory of O_{S} with coefficients ℤ/ℓ, as our next example illustrates.

Example 8.5. If O_{S} contains a primitive ℓ^{th} root of unity and 1/ℓ ∈ O_{S} then H^{1}(O_{S}; μ_{ℓ}^{⊗i}) ≅ O_{S}^{×}/O_{S}^{×ℓ} ⊕ _{ℓ} Pic(O_{S}) and H^{2}(O_{S}; μ_{ℓ}^{⊗i}) ≅ Pic(O_{S})/ℓ ⊕ _{ℓ}Br(O_{S}) for all i, so K_{0}(O_{S}; ℤ/ℓ) ≅ ℤ/ℓ ⊕ Pic(O_{S})/ℓ and

K_{n}(O_{S}; ℤ/ℓ) ≅ \{ \begin{matrix}O_{S}^{×}/O_{S}^{×ℓ} ⊕ _{ℓ} Pic(O_{S}) & for n = 2i − 1 ≥ 1, \\ ℤ/ℓ ⊕ Pic(O_{S})/ℓ ⊕ _{ℓ}Br(O_{S}) & for n = 2i ≥ 2. \end{matrix}

The ℤ/ℓ summands in degrees 2i are generated by the powers β^{i} of the Bott element β ∈ K_{2}(O_{S}; ℤ/ℓ) (see IV.2.5.2). In fact, K_{∗}(O_{S}; ℤ/ℓ) is free as a graded ℤ[β]- module on Pic(O_{S})/ℓ, K_{1}(O_{S}; ℤ/ℓ) and _{ℓ}Br(O_{S}) ⊆ K_{2}(O_{S}; ℤ/ℓ); this is immediate from the multiplicative properties of 4.2 described in 4.2.1. Taking the direct limit over S, we also have K_{0}(F) = ℤ/ℓ and

K_{n}(F; ℤ/ℓ) ≅ \{ \begin{matrix}F^{×}/F^{×ℓ}, & for n = 2i − 1 ≥ 1, \\ ℤ/ℓ ⊕ _{ℓ}Br(F) & for n = 2i ≥ 2. \end{matrix}

We conclude this section with a comparison to the odd part of ζ_{F} (1 − 2k).

Birch-Tate Conjecture 8.6. If F is a number field, the zeta function ζ_{F} (s) has a pole of order r_{2} at s = −1. Birch and Tate conjectured in 1970 that for totally real number fields (r_{2} = 0) we have

ζ_{F} (−1) = (−1)^{r_{1}}|K_{2}(O_{F})|/w_{2}(F).

The odd part of this conjecture was proven by Wiles in [Wi], using Tate’s calculation that K_{2}(O_{S})/m ≅ H_{et}^{2}(O_{S}, μ_{m}^{⊗2}) when 1/m ∈ O_{S} (see [Tate] or III(6.10.4)).

The two-primary part of the Birch-Tate conjecture is still open, but it is known to be a consequence of the 2-adic Main Conjecture of Iwasawa Theory (see Kolster’s appendix to [RW]). This was proven by Wiles for abelian extensions of ℚ in loc. cit., so the full Birch-Tate Conjecture holds for all abelian extensions of ℚ. For example, when F = ℚ we have ζ_{ℚ}(−1) = −1/12, |K_{2}(ℤ)| = 2 and w_{2}(ℚ) = 24; see the Classical Data 8.1.

To generalize the Birch-Tate Conjecture 8.6, we invoke the following deep result of Wiles [Wi, Thm. 1.6], which is often called the “Main Conjecture” of Iwasawa Theory.

Theorem 8.7 (Wiles). Let F be a totally real number field. If ℓ is odd and O_{S} = O_{F} [1/ℓ], then for all even integers 2k > 0 there is a rational number u_{k}, prime to ℓ, such that:

ζ_{F} (1 − 2k) = u_{k} \frac{|H_{et}^{2}(O_{S}, ℤ_{ℓ}(2k)|}{|H_{et}^{1}(O_{S}, ℤ_{ℓ}(2k)|}.

The numerator and denominator on the right side are finite (Ex. 8.1–8.2). Note that if F is not totally real then ζ_{F} (s) has a pole of order r_{2} at s = 1 − 2k.

We can now verify a conjecture of Lichtenbaum, made in [Li2, 2.4–2.6], which was only stated up to powers of 2.

Theorem 8.8. If F is totally real, and Gal(F/ℚ) is abelian, then for all k ≥ 1:

ζ_{F} (1 − 2k) = (−1)^{kr_{1}} 2^{r_{1}} \frac{|K_{4k−2}(O_{F})|}{|K_{4k−1}(O_{F})|}.

Proof. We first show that the left and right sides of 8.8 have the same power of each odd prime ℓ. The group H_{et}^{2}(O_{F} [1/ℓ], ℤ_{ℓ}(2k)) is the ℓ-primary part of K_{4k−2}(O_{F}) by Theorem 8.2. The group H_{et}^{1}(O_{F} [1/ℓ], ℤ_{ℓ}(2k)) in the numerator of 8.7 is K_{4k−1}(O_{F})_{(ℓ)} ≅ ℤ/w_{2k}^{(ℓ)}(F) by the proof of 8.2; the details are left to Exercise 8.3.

By the functional equation, the sign of ζ_{F} (1−2k) is (−1)^{kr_{1}}. Therefore it remains to check the power of 2 in Theorem 8.8. By Theorem 9.12 in the next Section, the power of 2 on the right side equals |H_{et}^{2}(O_{F} [1/ℓ], ℤ_{2}(2k))|/|H_{et}^{1}(O_{F} [1/ℓ], ℤ_{2}(2k))|. By the 2-adic Main Conjecture of Iwasawa Theory, which is known for abelian F, this equals the 2-part of ζ_{F} (1 − 2k). □

EXERCISES

8.1 Suppose that ℓ is odd, or that F is totally imaginary. If R is any ring of integers in F containing 1/ℓ, and i ≥ 2, show that H_{et}^{2}(R, ℤ_{ℓ}(i)) is a finite group, and conclude that H_{et}^{2}(R, ℤ/ℓ^{∞}(i)) = 0. Hint: Use 4.2 to compare it to K_{2i−2}(R), which is finite by IV, 1.18, 2.9 and 8.8. Then apply Ex. IV.2.6. 8.2 Supose that F is any number field, and let R be any ring of integers in F containing 1/ℓ. Show that H_{et}^{2}(R, ℤ_{ℓ}(i)) is a finite group for all i ≥ 2, Hint: Let R′ be the integral closure of R in F (\sqrt{−1}). Use a transfer argument to show that the kernel A of H_{et}^{2}(R, ℤ_{ℓ}(i)) → H_{et}^{2}(R′, ℤ_{ℓ}(i)) has exponent 2; A is finite because it injects into H_{et}^{2}(R, μ_{2}). Now apply Exercise 8.1. 8.3 Let ℓ be an odd prime and F a number field. If i > 1, show that for every ring O_{S} of integers in F containing 1/ℓ,

H_{et}^{1}(O_{S}, ℤ_{ℓ}(i)) ≅ H_{et}^{1}(F, ℤ_{ℓ}(i)) ≅ ℤ_{ℓ}^{r} ⊕ ℤ/w_{i}^{(ℓ)}(F),

where r is r_{2} for even i and r_{1} + r_{2} for odd i. Hint: Compare to K_{2i−1}(F; ℤ_{ℓ}), as in the proof of 8.2. 8.4 It is well known that ℤ[i] is a principal ideal domain. Show that the finite group K_{n}(ℤ[i]) has odd order for all even n > 0. Hint: Show that H_{et}^{2}(ℤ[\frac{1}{2}, i], μ_{4}) = 0. 8.5 Show that K_{3}(ℤ[i]) ≅ ℤ ⊕ ℤ/24, K_{7}(ℤ[i]) ≅ ℤ ⊕ ℤ/240 and K_{4k+1}(ℤ[i]) ≅ ℤ ⊕ ℤ/4 for all k > 0. Note that the groups w_{i}(ℚ(\sqrt{−1})) are given in 2.1.2. 8.6 Let F be a number field. Recall (Ex. IV.7.10) that there is a canonical involution on K_{∗}(F), and that it is multiplication by −1 on K_{1}(F) = F^{×}. Show that it is multiplication by (−1)^{i} on K_{2i−1}(O_{F}) and K_{2i−2}(O_{F}) for i > 1. Hint: Pick an odd prime ℓ and consider the canonical involution on K_{2i−1}(F (ζ_{ℓ}); ℤ/ℓ).

§9. Real number fields at the prime 2

Let F be a real number field, i.e., F has r_{1} > 0 embeddings into ℝ. The calculation of the algebraic K-theory of F at the prime 2 is somewhat different from the calculation at odd primes, for two reasons. One reason is that a real number field has infinite cohomological dimension, which complicates descent methods. A second reason is that the Galois group of a cyclotomic extension need not be cyclic, so that the e-invariant may not split (see 2.1.2). A final reason, explained in IV.2.8, is that, while K_{∗}(F; ℤ/2^{ν}) is a graded ring for 2^{ν} = 8 and a graded-commutative ring for 2^{ν} ≥ 16, its graded product may be non-associative and non-commutative for 2^{ν} = 4, the groups K_{∗}(F; ℤ/2) do not have a natural multiplication.

For the real numbers ℝ, the mod 2 motivic spectral sequence has E_{2}^{p,q} = ℤ/2 for all p, q in the octant q ≤ p ≤ 0. In order to distinguish between these terms, it is useful to label the nonzero elements of H_{et}^{0}(ℝ, ℤ/2(i)) as β_{i}, writing 1 for β_{0}. Using the multiplicative pairing with the spectral sequence ′E_{2}^{∗,∗} converging to K_{∗}(ℝ), multiplication by the element η of ′E_{2}^{0,−1} = H^{1}(ℝ, ℤ(1)) allows us to write the nonzero elements in the −ith column as η^{j}β_{i}. (See Table 9.1.1 below)

From Suslin’s calculation of K_{n}(ℝ) in Theorem 3.1, we know that the groups K_{n}(ℝ; ℤ/2) are cyclic and 8-periodic (for n ≥ 0) with orders 2, 2, 4, 2, 2, 0, 0, 0 for n = 0, 1, …, 7. The unexpected case K_{2}(ℝ; ℤ/2) ≅ ℤ/4 is described in IV.2.5.1.

§9. REAL NUMBER FIELDS AT THE PRIME 2 51

Theorem 9.1. In the spectral sequence 4.2 converging to K_{∗}(ℝ; ℤ/2), all the d_{2} differentials with nonzero source on the lines p ≡ 1, 2 (mod 4) are isomorphisms. Hence the spectral sequence degenerates at E_{3}. The only extensions are the nontrivial extensions ℤ/4 in K_{8a+2}(ℝ; ℤ/2).

Proof. Recall from 4.8.1 that the mod 2 spectral sequence has periodicity isomorphisms E_{r}^{p,q} ⟶^{≃} E_{r}^{p−4,q−4}, p ≤ 0. Therefore it suffices to work with the columns −3 ≤ p ≤ 0. These columns are shown in Table 9.1.1.

Because K_{3}(ℝ; ℤ/2) ≅ ℤ/2, the differential closest to the origin, from β_{2} to η^{3}, must be nonzero. Since the pairing with ′E_{2} is multiplicative and d_{2}(η) = 0, we must have d_{2}(η^{j}β_{2}) = η^{j+3} for all j ≥ 0. Thus the column p = −2 of E_{3} is zero, and every term in the column p = 0 of E_{3} is zero except for {1, η, η^{2}}.

Similarly, we must have d_{2}(β_{3}) = η^{3}β_{1} because K_{5}(ℝ; ℤ/2) = 0. By multiplicativity, this yields d_{2}(η^{j}β_{3}) = η^{j+3}β_{1} for all j ≥ 0. Thus the column p = −3 of E_{3} is zero, and every term in the column p = −1 of E_{3} is zero except for {β_{1}, ηβ_{1}, η^{2}β_{1}}. □

p = −3  −2  −1  0

1

β_{1}  η
β_{2}  ηβ_{1}  η^{2}
β_{3}  ηβ_{2}  η^{2}β_{1}  η^{3}
ηβ_{3}  η^{2}β_{2}  η^{3}β_{1}  η^{4}

The first 4 columns of E_{2}

−3  −2  −1  0

1

β_{1}  η
0  ηβ_{1}  η^{2}
0  0  η^{2}β_{1}  0
0  0  0  0

The same columns of E_{3}

Table 9.1.1. The mod 2 spectral sequence for ℝ.

Variant 9.1.2. The spectral sequence 4.2 with coefficients ℤ/2^{∞} is very similar, except that when p > q, E_{2}^{p,q} = H_{et}^{p−q}(ℝ; ℤ/2^{∞}(−q)) is: 0 for p even; ℤ/2 for p odd. If p is odd, the coefficient map ℤ/2 → ℤ/2^{∞} induces isomorphisms on the E_{2}^{p,q} terms, so by Theorem 9.1 all the d_{2} differentials with nonzero source in the columns p ≡ 1 (mod 4) are isomorphisms. Again, the spectral sequence converging to K_{∗}(ℝ; ℤ/2^{∞}) degenerates at E_{3} = E_{∞}. The only extensions are the nontrivial extensions of ℤ/2^{∞} by ℤ/2 in K_{8a+4}(ℝ; ℤ/2^{∞}) ≅ ℤ/2^{∞}.

Variant 9.1.3. The analysis of the spectral sequence with 2-adic coefficients is very similar, except that (a) H^{0}(ℝ; ℤ_{2}(i)) is: ℤ_{2} for i even; 0 for i odd and (b) (for p > q) E_{2}^{p,q} = H_{et}^{p−q}(ℝ; ℤ/2^{∞}(−q)) is: ℤ/2 for p even; 0 for p odd. All differentials with nonzero source in the column p ≡ 2 (mod 4) are onto. Since there are no extensions to worry about, we omit the details.

We now consider the K-theory of the ring O_{S} of integers in a number field F with coefficients ℤ/2^{∞}. The E_{2} terms in the spectral sequence 4.2 are the (étale) cohomology groups H^{n}(O_{S}; ℤ/2^{∞}(i)). Following Tate, the r_{1} real embeddings of F define natural maps α_{S}^{n}(i):

(9.2) α_{S}^{n}(i): H^{n}(O_{S}; ℤ/2^{∞}(i)) ⟶ ⨁^{r_{1}} H^{n}(ℝ; ℤ/2^{∞}(i)) ≅ \{ \begin{matrix}(ℤ/2)^{r_{1}}, & i − n odd \\ 0, & i − n even. \end{matrix}

This map is an isomorphism for all n ≥ 3 by Tate-Poitou duality [Milne2, I(4.20)]. It is also an isomorphism for n = 2 and i ≥ 2, as shown in Exercise 9.1.

Write H̃^{1}(O_{S}; ℤ/2^{∞}(i)) for the kernel of α_{S}^{1}(i).

Lemma 9.3. For even i, H^{1}(F; ℤ/2^{∞}(i)) ⟶^{α^{1}(i)} (ℤ/2)^{r_{1}} is a split surjection. Hence H^{1}(O_{S}; ℤ/2^{∞}(i)) ≅ (ℤ/2)^{r_{1}} ⊕ H̃^{1}(O_{S}; ℤ/2^{∞}(i)) for sufficiently large S.

Proof. By the strong approximation theorem for units of F, the left vertical map is a split surjection in the diagram:

F^{×}/F^{×2} ⟶^{≅} H^{1}(F, ℤ/2) ⟶ H^{1}(F, ℤ/2^{∞}(i))
split onto↓^{⊕σ}  ↓  ↓^{α^{1}(i)}
(ℤ/2)^{r_{1}} = ⊕ ℝ^{×}/ℝ^{×2} ⟶^{≅} ⊕H^{1}(ℝ, ℤ/2) ⟶^{≅} ⊕H^{1}(ℝ, ℤ/2^{∞}(i)).

Since F^{×}/F^{×2} is the direct limit (over S) of the groups O_{S}^{×}/O_{S}^{×2}, we may replace F by O_{S} for sufficiently large S. □

The next result is taken from [RW, 6.9]. When n ≡ 5 (mod 8), we have an unknown group extension; to express it, we write A ⋊ B for an abelian group extension of B by A.

Theorem 9.4. Let F be a real number field, and let O_{S} be a ring of S-integers in F containing O_{F} [\frac{1}{2}]. Then α_{S}^{1}(4k) is onto (k > 0), and for all n ≥ 0:

K_{n}(O_{S}; ℤ/2^{∞}) ≅ \{ \begin{matrix}ℤ/w_{4k}(F) & for n = 8k, \\ H^{1}(O_{S}; ℤ/2^{∞}(4k + 1)) & for n = 8k + 1, \\ ℤ/2 & for n = 8k + 2, \\ H^{1}(O_{S}; ℤ/2^{∞}(4k + 2)) & for n = 8k + 3, \\ ℤ/2w_{4k+2} ⊕ (ℤ/2)^{r_{1}−1} & for n = 8k + 4, \\ (ℤ/2)^{r_{1}−1} ⋊ H^{1}(O_{S}; ℤ/2^{∞}(4k + 3)) & for n = 8k + 5, \\ 0 & for n = 8k + 6, \\ H̃^{1}(O_{S}; ℤ/2^{∞}(4k + 4)) & for n = 8k + 7. \end{matrix}

Proof. Consider the morphism α_{S} of spectral sequences (4.2) with coefficients ℤ/2^{∞}, from that for O_{S} to the direct sum of r_{1} copies of that for ℝ. By naturality, the morphism in the E_{2}^{p,q} spot is the map α_{S}^{p−q}(−q) of (9.2). By Tate-Poitou duality and Ex. 9.1, this is an isomorphism except on the two diagonals p = q, where it is the injection of ℤ/w_{−q}^{(2)}(F) into ℤ/2^{∞}, and on the critical diagonal p = q + 1.

When p ≡ +1 (mod 4), we saw in 9.1.2 that d_{2}^{p,q}(ℝ) is an isomorphism whenever q ≤ p < 0. It follows that we may identify d_{2}^{p,q}(O_{S}) with α_{S}^{p−q}. Therefore d_{2}^{p,q}(O_{S}) is an isomorphism if p ≥ 2 + q, and an injection if p = q. As in 9.1.2, the spectral sequence degenerates at E_{3}, yielding K_{n}(O_{S}; ℤ/2^{∞}) as proclaimed, except for two points: (a) when n = 8k + 4, the extension of ℤ/w_{4k+2} by (ℤ/2)^{r_{1}} is seen to be nontrivial by comparison with the extension for ℝ, and (b) when n = 8k + 6, it only shows that K_{n}(O_{S}; ℤ/2^{∞}) is the cokernel of α_{S}^{1}(4k + 4).

To resolve (b) we must show that the map α_{S}^{1}(4k+4) is onto when k ≥ 0. Set n = 8k + 6. Since K_{n}(O_{S}) is finite, K_{n}(O_{S}; ℤ/2^{∞}) must equal the 2-primary subgroup of K_{n−1}(O_{S}), which is independent of S by V.6.8. But for sufficiently large S, the map α^{1}(4k + 4) is a surjection by Lemma 9.3, and hence K_{n}(O_{S}; ℤ/2^{∞}) = 0. □

Theorem 9.5. Let O_{S} be a ring of S-integers in a number field F. Then for each odd n ≥ 3, the group K_{n}(O_{S}) ≅ K_{n}(F) is given by:

a) If F is totally imaginary, K_{n}(F) ≅ ℤ^{r_{2}} ⊕ ℤ/w_{i}(F);

b) If F has r_{1} > 0 real embeddings then, setting i = (n + 1)/2,

K_{n}(F) ≅ \{ \begin{matrix}ℤ^{r_{1}+r_{2}} ⊕ ℤ/w_{i}(F), & n ≡ 1 (mod 8) \\ ℤ^{r_{2}} ⊕ ℤ/2w_{i}(F) ⊕ (ℤ/2)^{r_{1}−1}, & n ≡ 3 (mod 8) \\ ℤ^{r_{1}+r_{2}} ⊕ ℤ/\frac{1}{2}w_{i}(F), & n ≡ 5 (mod 8) \\ ℤ^{r_{2}} ⊕ ℤ/w_{i}(F), & n ≡ 7 (mod 8) \end{matrix}

Note that these groups are determined only by the number r_{1}, r_{2} of real and complex places of F and the integers w_{i}(F).

Proof. Part (a), when F is totally imaginary, is given by Theorem 8.4. In case (b), since the rank is classically known (see 8.1), and K_{n}(O_{S}) ≅ K_{n}(F) by V.6.8, it suffices to determine the torsion subgroup of K_{n}(O_{S}). The odd torsion is given by Theorem 8.2, so we need only worry about the 2-primary torsion. Since K_{n+1}(O_{S}) is finite, it follows from Ex. IV.2.6 that the 2-primary subgroup of K_{n}(O_{S}) is K_{n+1}(O_{S}; ℤ/2^{∞}), which we can read off from Theorem 9.4, recalling from 2.3(b) that w_{i}^{(2)}(F) = 2 for odd i. □

Example 9.5.1. K_{n}(ℚ) ≅ ℤ for all n ≡ 5 (mod 8) as w_{i}(ℚ) = 2; see 2.1.2. More generally, if F has a real embedding and n ≡ 5 (mod 8), then K_{n}(F) has no 2-primary torsion, because \frac{1}{2}w_{i}(F) is an odd integer when i is odd; see 2.3(b).

The narrow Picard group

To determine the 2-primary torsion in K_{n}(O_{S}) when n is even, we need to introduce the narrow Picard group and the signature defect of the ring O_{S}. We begin with some terminology.

Each real embedding σ_{i} : F → ℝ determines a map F^{×} → ℝ^{×} → ℤ/2, detecting the sign of units of F under that embedding. The sum of these is the sign map σ: F^{×} ⟶ (ℤ/2)^{r_{1}}; it is surjective by the strong approximation theorem for F. The kernel F_{+}^{×} of σ is called the group of totally positive units in F, since it consists of units which are positive under every real embedding.

If R = O_{S} is a ring of integers in F, we write R_{+}^{×} for R^{×} ∩ F_{+}^{×}, the subgroup of totally positive units in R. Since the sign map σ factors through F^{×}/F^{×2} = H_{et}^{1}(F, ℤ/2), the restriction to R^{×} also factors through α^{1} : H_{et}^{1}(R, ℤ/2) → (ℤ/2)^{r_{1}}. This map is part of a family of maps

(9.6)  α^{n} : H_{et}^{n}(R, ℤ/2) → ⊕_{r_{1}}H_{et}^{n}(ℝ, ℤ/2) = (ℤ/2)^{r_{1}}

related to the maps α^{n}(i) in (9.2). By Tate-Poitou duality, α^{n} is an isomorphism for all n ≥ 3; it is a surjection for n = 2 (see Ex. 9.2). We will be interested in α^{1}. The following classical definitions are due to Weber; see [Co, 5.2.7] or [Neu, VI.1].

Definition 9.6.1. The signature defect j(R) of R is defined to be the dimension of the cokernel of α^{1}. Since the sign of −1 nontrivial, we have 0 ≤ j(R) < r_{1}. Note that j(F) = 0, and that j(O_{S}) ≤ j(O_{F}) for all S.

The narrow Picard group Pic_{+}(R) is defined to be the cokernel of the restricted divisor map F_{+}^{×} → ⨁_{℘∉S} ℤ of I.3.5; it is a finite group. Some authors call Pic_{+}(O_{S}) the ray class group and write it as Cl_{F}^{S}.

The kernel of the restricted divisor map is clearly R_{+}^{×}, and it is easy to see from this that there is an exact sequence

0 → R_{+}^{×} → R^{×} ⟶^{σ} (ℤ/2)^{r_{1}} → Pic_{+}(R) ⟶ Pic(R) → 0.

For simplicity we write H^{n}(R, ℤ/2) for H_{et}^{n}(R, ℤ/2) and, as in (9.2), we define H̃^{n}(R; ℤ/2) to be the kernel of α^{n}. A diagram chase (left to Ex. 9.3) shows that there is an exact sequence

(9.6.2) 0→H̃^{1}(R; ℤ/2) → H^{1}(R; ℤ/2) ⟶^{α^{1}} (ℤ/2)^{r_{1}} →Pic_{+}(R)/2 → Pic(R)/2→0.

Thus the signature defect j(R) is also the dimension of the kernel of Pic_{+}(R)/2 → Pic(R)/2. If we let t and u denote the dimensions of Pic(R)/2 and Pic_{+}(R)/2, respectively, then this means that u = t + j(R). If s denotes the number of finite places of O_{S}, then dim H^{1}(O_{S}; ℤ/2) = r_{1} + r_{2} + s + t and dim H^{2}(O_{S}; ℤ/2) = r_{1} + s + t − 1. This follows from 8.1 and (8.1.1), using Kummer theory.

Lemma 9.6.3. Suppose that \frac{1}{2} ∈ O_{S}. Then dim H̃^{1}(O_{S}, ℤ/2) = r_{2} + s + u, and dim H̃^{2}(O_{S}, ℤ/2) = t + s − 1.

Proof. The first assertion is immediate from (9.6.2). Since α^{2} is onto, the second assertion follows. □

Theorem 9.7. Let F be a real number field, and O_{S} a ring of integers containing \frac{1}{2}. If j = j(O_{S}) is the signature defect, then the mod 2 algebraic K-groups of O_{S} are given (up to extensions) for n > 0 as follows:

K_{n}(O_{S}; ℤ/2) ≅ \{ \begin{matrix}H̃^{2}(O_{S}; ℤ/2) ⊕ ℤ/2 & for n = 8k, \\ H^{1}(O_{S}; ℤ/2) & for n = 8k + 1, \\ H^{2}(O_{S}; ℤ/2) ⋊ ℤ/2 & for n = 8k + 2, \\ (ℤ/2)^{r_{1}−1} ⋊ H^{1}(O_{S}; ℤ/2) & for n = 8k + 3, \\ (ℤ/2)^{j} ⋊ H^{2}(O_{S}; ℤ/2) & for n = 8k + 4, \\ (ℤ/2)^{r_{1}−1} ⋊ H̃^{1}(O_{S}; ℤ/2) & for n = 8k + 5, \\ (ℤ/2)^{j} ⊕ H̃^{2}(O_{S}; ℤ/2) & for n = 8k + 6, \\ H̃^{1}(O_{S}; ℤ/2) & for n = 8k + 7. \end{matrix}

p = −3  −2  −1  0

1

β_{1}  H^{1}
0  H^{1}  H^{2}
0  H̃^{1}  H^{2}  (ℤ/2)^{r_{1}−1}
H̃^{1}  H̃^{2}  (ℤ/2)^{r_{1}−1}  (ℤ/2)^{j}
H̃^{2}  0  (ℤ/2)^{j}  0
0  0  0  0
The first 4 columns of E_{3} = E_{∞}

Table 9.7.1. The mod 2 spectral sequence for O_{S}.

Proof. (Cf. [RW, 7.8].) As in the proof of Theorem 9.4, we compare the spectral sequence for R = O_{S} with the sum of r_{1} copies of the spectral sequence for ℝ. For n ≥ 3 we have H^{n}(R; ℤ/2) ≅ (ℤ/2)^{r_{1}}. It is not hard to see that we may identify the differentials d_{2} : H^{n}(R, ℤ/2) → H^{n+3}(R, ℤ/2) with the maps α^{n}. Since these maps are described in 9.6, we see from periodicity 4.8.1 that the columns p ≤ 0 of E_{3} are 4-periodic, and all nonzero entries are described by Figure 9.7.1.

As in Example 4.6, the E_{2} page of the spectral sequence 4.2 has only one nonzero entry for p > 0, namely E_{3}^{+1,−1} = Pic(R)/2, and it only affects K_{0}(R; ℤ/2). By inspection, E_{3} = E_{∞}, yielding the desired description of the groups K_{n}(R, ℤ/2) in terms of extensions. The proof that the extensions split for n ≡ 0, 6 (mod 8) is left to Exercises 9.4 and 9.5. □

The case F = ℚ has historical importance, because of its connection with the image of J (see 2.1.3 or [Q5]) and classical number theory. The following result was first established in [We2]; the groups are not truly periodic only because the order of K_{8k−1}(ℤ) depends upon k.

Corollary 9.8. For n ≥ 0, the 2-primary subgroups of K_{n}(ℤ) and K_{n}(ℤ[1/2]) are essentially periodic, of period eight, and are given by the following table for n ≥ 2. (When n ≡ 7 (mod 8), we set k = (n + 1)/8.)

\begin{matrix}n (mod 8) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ K_{n}(ℤ)\{2\} & ℤ/2 & ℤ/2 & ℤ/16 & 0 & 0 & 0 & ℤ/16k & 0 \end{matrix}

In particular, K_{n}(ℤ) and K_{n}(ℤ[1/2]) have odd order for all n ≡ 4, 6, 8 (mod 8), and the finite group K_{8k+2}(ℤ) is the sum of ℤ/2 and a finite group of odd order. We will say more about the odd torsion in the next section.

Proof. When n is odd, this is Theorem 9.5; w_{4k}^{(2)} is the 2-primary part of 16k by 2.3(c). For R = ℤ[1/2] we have s = 1 and t = u = j = 0. By Lemma 9.6.3 we have dim H̃^{1}(R; ℤ/2) = 1 and H̃^{2}(R; ℤ/2) = 0. By 9.7, the groups K_{n}(ℤ[1/2]; ℤ/2) are periodic of orders 2, 4, 4, 4, 2, 2, 1, 2 for n ≡ 0, 1, ..., 7 respectively. The groups K_{n}(ℤ[1/2]) for n odd, given in 9.5, together with the ℤ/2 summand in K_{8k+2}(ℤ) provided by topology (see 2.1.3), account for all of K_{n}(ℤ[1/2]; ℤ/2), and hence must contain all of the 2-primary torsion in K_{n}(ℤ[1/2]). □

Recall that the 2-rank of an abelian group A is the dimension of Hom(ℤ/2, A). We have already seen (in Theorem 9.5) that for n ≡ 1, 3, 5, 7 (mod 8) the 2-ranks of K_{n}(O_{S}) are: 1, r_{1}, 0 and 1, respectively.

Corollary 9.9. For n ≡ 2, 4, 6, 8 (mod 8), n > 0, the respective 2-ranks of the finite groups K_{n}(O_{S}) are: r_{1} + s + t − 1, j + s + t − 1, j + s + t − 1 and s + t − 1.

Here j is the signature defect of O_{S} (9.6.1), s is the number of finite places of O_{S} and t is the rank of Pic(O_{S})/2.

Proof. Since K_{n}(R; ℤ/2) is an extension of Hom(ℤ/2, K_{n−1}R) by K_{n}(R)/2, and the dimensions of the odd groups are known, we can read this off from the list given in Theorem 9.7, using Lemma 9.6.3. □

Example 9.9.1. Consider F = ℚ(\sqrt{p}), where p is prime. When p ≡ 1 (mod 8), it is well known that t = j = 0 but s = 2. It follows that K_{8k+2}(O_{F}) has 2-rank 3, while the two-primary summand of K_{n}(O_{F}) is nonzero and cyclic when n ≡ 4, 6, 8 (mod 8).

When p ≡ 7 (mod 8), we have j = 1 for both O_{F} and R = O_{F} [1/2]. Since r_{1} = 2 and s = 1, the 2-ranks of the finite groups K_{n}(R) are: t + 2, t + 1, t + 1 and t for n ≡ 2, 4, 6, 8 (mod 8) by 9.9. For example, if t = 0 (Pic(R)/2 = 0) then K_{n}(R) has odd order for n ≡ 8 (mod 8), but the 2-primary summand of K_{n}(R) is (ℤ/2)^{2} when n ≡ 2 and is cyclic when n ≡ 4, 6.

Example 9.9.2. (2–regular fields) A number field F is said to be 2–regular if there is only one prime over 2 and the narrow Picard group Pic_{+}(O_{F} [\frac{1}{2}]) is odd (i.e., t = u = 0 and s = 1). In this case, we see from 9.9 that K_{8k+2}(O_{F}) is the sum of (ℤ/2)^{r_{1}} and a finite odd group, while K_{n}(O_{F}) has odd order for all n ≡ 4, 6, 8 (mod 8) (n > 0). In particular, the map K_{4}^{M}(F) → K_{4}(F) must be zero, since it factors through the odd order group K_{4}(O_{F}), and K_{4}^{M}(F) ≅ (ℤ/2)^{r_{1}}.

Browkin and Schinzel [BrwS] and Rognes and Østvær [ROst] have studied this case. For example, when F = ℚ(\sqrt{m}) and m > 0 (r_{1} = 2), the field F is 2-regular exactly when m = 2, or m = p or m = 2p with p ≡ 3, 5 (mod 8) prime. (See [BrwS].)

A useful example is F = ℚ(\sqrt{2}). Note that K_{4}^{M}(F) ≅ (ℤ/2)^{2} is generated by the Steinberg symbols \{−1, −1, −1, −1\} and \{−1, −1, −1, 1 + \sqrt{2}\}. Both symbols must vanish in K_{4}(ℤ[\sqrt{2}]), since this group has odd order. This is the case j = 0, r_{1} = 2 of the following result.

Let ρ denote the rank of the image of the group K_{4}^{M}(F) ≅ (ℤ/2)^{r_{1}} in K_{4}(F).

Corollary 9.10. Let F be a real number field. Then j(O_{F} [1/2]) ≤ ρ ≤ r_{1} − 1. The image (ℤ/2)^{ρ} of K_{4}^{M}(F) → K_{4}(F) lies in the subgroup K_{4}(O_{F}), and its image in K_{4}(O_{S})/2 has rank j(O_{S}) whenever S contains all primes over 2.

In particular, the image (ℤ/2)^{ρ} of K_{4}^{M}(F) lies in 2 · K_{4}(F).

Proof. By Ex. IV.1.12(d), {−1, −1, −1, −1} is nonzero in K_{4}^{M}(F) but zero in K_{4}(F). Since K_{4}^{M}(F) ≅ (ℤ/2)^{r_{1}} by III.7.2(d), we have ρ < r_{1}. The assertion that K_{4}^{M}(F) → K_{4}(F) factors through K_{4}(O_{F}) follows from the fact that K_{3}(O_{F}) = K_{3}(F) (see IV.6.8), by multiplying K_{3}^{M}(F) and K_{3}(O_{F}) ≅ K_{3}(F) by [−1] ∈ K_{1}(ℤ). We saw in 4.3 that the edge map H^{n}(F, ℤ(n)) → K_{n}(F) in the motivic spectral sequence agrees with the usual map K_{n}^{M}(F) → K_{n}(F). By Theorem 4.1 (due to Voevodsky), K_{n}^{M}(F)/2^{ν} ≅ H^{n}(F, ℤ(n))/2^{ν} ≅ H^{n}(F, ℤ/2^{ν}(n)). For n = 4, the image of the edge map from H^{4}(O_{S}, ℤ/2^{ν}(4)) ≅ H^{4}(F, ℤ/2^{ν}(4)) → K_{4}(O_{S}; ℤ/2) has rank j by Table 9.7.1; this implies the assertion that the image in K_{4}(O_{S})/2 ⊂ K_{4}(O_{S}; ℤ/2) has rank j(O_{S}). Finally, taking O_{S} = O_{F} [1/2] yields the inequality j(O_{S}) ≤ ρ. □

Example 9.10.1. (ρ = 1) Consider F = ℚ(\sqrt{7}), O_{F} = ℤ[\sqrt{7}] and R = O_{F} [1/2]; here s = 1, t = 0 and j(R) = ρ = 1 (the fundamental unit u = 8 + 3\sqrt{7} is totally positive). Hence the image of K_{4}^{M}(F) ≅ (ℤ/2)^{2} in K_{4}(ℤ[\sqrt{7}]) is ℤ/2 on the symbol σ = \{−1, −1, −1, \sqrt{7}\}, and this is all of the 2-primary torsion in K_{4}(ℤ[\sqrt{7}]) by 9.9.

On the other hand, O_{S} = ℤ[\sqrt{7}, 1/7] still has ρ = 1, but now j = 0, and the 2rank of K_{4}(O_{S}) is still one by 9.9. Hence the extension 0 → K_{4}(O_{F}) → K_{4}(O_{S}) → ℤ/48 → 0 of V.6.8 cannot be split, implying that the 2-primary subgroup of K_{4}(O_{S}) must then be ℤ/32.

In fact, the nonzero element σ is divisible in K_{4}(F). This follows from the fact that if p ≡ 3 (mod 28) then there is an irreducible q = a + b\sqrt{7} whose norm is −p = qq̄. Hence R′ = ℤ[\sqrt{7}, 1/2q] has j(R′) = 0 but ρ = 1, and the extension 0 → K_{4}(O_{F}) → K_{4}(O_{S}) → ℤ/(p^{2} − 1) → 0 of V.6.8 is not split. If in addition p ≡ −1 (mod 2^{ν}) — there are infinitely many such p for each ν — then there is an element v of K_{4}(R′) such that 2^{ν+1}v = σ See [We3] for details.

Question 9.10.2. Can ρ be less than the minimum of r_{1} − 1 and j + s + t − 1? As in (9.2), when i is even we define H̃^{2}(R; ℤ_{2}(i)) to be the kernel of α^{2}(i) :

H^{2}(R; ℤ_{2}(i)) → H^{2}(ℝ; ℤ_{2}(i))^{r_{1}} ≅ (ℤ/2)^{r_{1}}. By Lemma 9.6.3, H̃^{2}(R; ℤ_{2}(i)) has 2-rank s + t − 1. The following result is taken from [RW, 0.6].

Theorem 9.11. Let F be a number field with at least one real embedding, and let R = O_{S} denote a ring of integers in F containing 1/2. Let j be the signature defect of R, and write w_{i} for w_{i}^{(2)}(F).

Then there is an integer ρ, j ≤ ρ < r_{1}, such that, for all n ≥ 2, the two-primary subgroup K_{n}(O_{S}){2} of K_{n}(O_{S}) is isomorphic to:

K_{n}(O_{S})\{2\} ≅ \{ \begin{matrix}H_{et}^{2}(R; ℤ_{2}(4k + 1)) & for n = 8k, \\ ℤ/2 & for n = 8k + 1, \\ H_{et}^{2}(R; ℤ_{2}(4k + 2)) & for n = 8k + 2, \\ (ℤ/2)^{r_{1}−1} ⊕ ℤ/2w_{4k+2} & for n = 8k + 3, \\ (ℤ/2)^{ρ} ⋊ H_{et}^{2}(R; ℤ_{2}(4k + 3)) & for n = 8k + 4, \\ 0 & for n = 8k + 5, \\ H̃_{et}^{2}(R; ℤ_{2}(4k + 4)) & for n = 8k + 6, \\ ℤ/w_{4k+4} & for n = 8k + 7. \end{matrix}

Proof. When n = 2i − 1 is odd, this is Theorem 9.5, since w_{i}^{(2)}(F) = 2 when n ≡ 1 (mod 4) by 2.3(b). When n = 2 it is III.6.9.3. To determine the two-primary subgroup K_{n}(O_{S}){2} of the finite group K_{2i+2}(O_{S}) when n = 2i + 2, we use the universal coefficient sequence

0 → (ℤ/2^{∞})^{r} → K_{2i+3}(O_{S}; ℤ/2^{∞}) → K_{2i+2}(O_{S}){2} → 0,

where r is the rank of K_{2i+3}(O_{S}) and is given by 8.1 (r = r_{1}+r_{2} or r_{2}). To compare this with Theorem 9.4, we note that H^{1}(O_{S}, ℤ/2^{∞}(i)) is the direct sum of (ℤ/2^{∞})^{r} and a finite group, which must be H^{2}(O_{S}, ℤ_{2}(i)) by universal coefficients; see [RW, 2.4(b)]. Since α_{S}^{1}(i) : H^{1}(R; ℤ_{2}(i)) → (ℤ/2)^{r_{1}} must vanish on the divisible group (ℤ/2^{∞})^{r}, it induces the natural map α_{S}^{2}(i): H_{et}^{2}(O_{S}; ℤ_{2}(i)) → (ℤ/2)^{r_{1}} and

H̃^{1}(O_{S}, ℤ/2^{∞}(i)) ≅ (ℤ/2^{∞})^{r} ⊕ H̃^{2}(O_{S}, ℤ_{2}(i)).

This proves all of the theorem, except for the description of K_{n}(O_{S}), n = 8k + 4. By mod 2 periodicity 4.8.1, the integer ρ of 9.10 equals the rank of the image of H^{4}(O_{S}, ℤ/2(4)) ≅ H^{4}(O_{S}, ℤ/2(4k + 4)) ≅ (ℤ/2)^{r_{1}} in Hom(ℤ/2, K_{n}(O_{S})), considered as a quotient of K_{n+1}(O_{S}; ℤ/2). □

We can combine the 2-primary information in 9.11 with the odd torsion information in 8.2 and 8.8 to relate the orders of K-groups to the orders of étale cohomology groups. Up to a factor of 2^{r_{1}}, they were conjectured by Lichtenbaum in [Li2]. Let |A| denote the order of a finite abelian group A.

Theorem 9.12. Let F be a totally real number field, with r_{1} real embeddings, and let O_{S} be a ring of integers in F. Then for all even i > 0

2^{r_{1}} · \frac{|K_{2i−2}(O_{S})|}{|K_{2i−1}(O_{S})|} = \frac{∏_{ℓ} |H_{et}^{2}(O_{S}[1/ℓ]; ℤ_{ℓ}(i))|}{∏_{ℓ} |H_{et}^{1}(O_{S}[1/ℓ]; ℤ_{ℓ}(i))|}.

Proof. Since 2i − 1 ≡ 3 (mod 4), all groups involved are finite (see 8.1, Ex. 8.2 and Ex. 8.3.) Write h^{n,i}(ℓ) for the order of H_{et}^{n}(O_{S}[1/ℓ]; ℤ_{ℓ}(i)). By Ex. 8.3, h^{1,i}(ℓ) = w_{i}^{(ℓ)}(F). By 9.5, the ℓ-primary subgroup of K_{2i−1}(O_{S}) has order h^{1,i}(ℓ) for all odd ℓ and all even i > 0, and also for ℓ = 2 with the exception that when 2i − 1 ≡ 3 (mod 8) then the order is 2^{r_{1}}h^{1,i}(2).

By Theorems 8.2 and 9.11, the ℓ-primary subgroup of K_{2i−2}(O_{S}) has order h^{2,i}(ℓ) for all ℓ, except when ℓ = 2 and 2i − 2 ≡ 6 (mod 8) when it is h^{1,i}(2)/2^{r_{1}}. Combining these cases yields the formula asserted by the theorem. □

Theorem 9.12 was used in the previous section (Theorem 8.8) to equate the ratio of orders of the finite groups K_{4k−2}(O_{F}) and K_{4k−1}(O_{F}) with |ζ_{F} (1 − 2k)|/2^{r_{1}}.

EXERCISES

9.1 Suppose that F has r_{1} > 0 embeddings into ℝ. Show that

H_{et}^{2}(O_{S}; ℤ/2^{∞}(i)) ≅ H_{et}^{2}(F; ℤ/2^{∞}(i)) ≅ \{ \begin{matrix}(ℤ/2)^{r_{1}}, & i ≥ 3 odd \\ 0, & i ≥ 2 even. \end{matrix}

Using (8.1.1), determine H_{et}^{2}(O_{S}; ℤ/2^{∞}(1)). Hint: Compare F with F (\sqrt{−1}), and use Exercise 8.1 to see that H_{et}^{2}(O_{S}; ℤ/2^{∞}(i)) has exponent 2. Hence the Kummer sequence is:

0 → H_{et}^{2}(O_{S}; ℤ/2^{∞}(i)) → H_{et}^{3}(O_{S}; ℤ/2) → H_{et}^{3}(O_{S}; ℤ/2^{∞}(i)) → 0.

Now plug in the values of the H^{3} groups, which are known by (9.2).

9.2 Show that α^{2} is onto. Hint: Use Ex. 9.1 and the coefficient sequence for ℤ/2 ⊂ ℤ/2^{∞}(4) to show that the map H_{et}^{2}(R; ℤ/2) → H_{et}^{2}(R; ℤ/2^{∞}(4)) is onto. 9.3 Establish the exact sequence (9.6.2). (This is taken from [RW, 9.6].) 9.4 The stable homotopy group π_{8k}(QS^{0}; ℤ/2) contains an element β_{8k} of exponent 2 which maps onto the generator of K_{8k}(ℝ; ℤ/2) ≅ ℤ/2; see [RW, 5.1]. Use it to show the extension K_{8k}(O_{S}; ℤ/2) of ℤ/2 by H̃^{2}(O_{S}, ℤ/2) splits in Theorem 9.7. 9.5 Show that the extension K_{8k+6}(O_{S}; ℤ/2) splits in Thm. 9.7. Conclude that K_{8k+6}(O_{S})/2 ≅ H̃^{2}(O_{S}, ℤ/2) ⊕ (ℤ/2)^{j}. Hint: use Example 9.5.1. 9.6 Let R = O_{F} [1/2], where F is a real number field. Show that K_{8k+4}(R; ℤ/2) is an extension of _{2}Br(R) by Pic_{+}(R)/2.

Let Br_{+}(R) denote the kernel of the canonical map Br(R) → (ℤ/2)^{r_{1}} induced by (8.1.1). Show that K_{8k+7}(R; ℤ/2) ≅ Pic_{+}(R)/2 ⊕_{2}Br_{+}(R). (See [RW, 7.8].)

§10. The K-theory of ℤ

The determination of the groups K_{n}(ℤ) has been a driving force in the development of K-theory. We saw in Chapters II and III that the groups K_{0}(ℤ), K_{1}(ℤ) and K_{2}(ℤ) are related to very classical mathematics. In the 1970’s, homological methods led to the calculation of the rank of K_{n}(ℤ) by Borel (see 8.1) and K_{3}(ℤ) ≅ ℤ/48 by Lee and Szczarba (see Example 2.1.2 or [LSz]).

In order to describe the groups K_{n}(ℤ), we use the Bernoulli numbers B_{k}. We let c_{k} denote the numerator of B_{k}/4k; c_{k} is a product of irregular primes (see 2.4.1). We saw in Lemma 2.4 that the denominator of B_{k}/4k is w_{2k}, so B_{k}/4k = c_{k}/w_{2k}.

Theorem 10.1. For n ≢ 0 (mod 4) and n > 1, we have:

(1) If n = 8k + 1, K_{n}(ℤ) ≅ K_{n}(ℚ) ≅ ℤ ⊕ ℤ/2;

(2) If n = 8k + 2, |K_{n}(ℤ)| = 2c_{2k+1};

(3) If n = 8k + 3, K_{n}(ℤ) ≅ K_{n}(ℚ) ≅ ℤ/2w_{4k+2};

(4) If n = 8k + 5, K_{n}(ℤ) ≅ K_{n}(ℚ) ≅ ℤ;

(5) If n = 8k + 6, |K_{n}(ℤ)| = c_{2k+1};

(6) If n = 8k + 7, K_{n}(ℤ) ≅ K_{n}(ℚ) ≅ ℤ/c_{4k+4}.

As a consequence, for k ≥ 1 we have: \frac{|K_{4k−2}(ℤ)|}{|K_{4k−1}(ℤ)|} = \frac{B_{k}}{4k} = \frac{(−1)^{k}}{2} ζ(1 − 2k).

Proof. The equality B_{k}/4k = (−1)^{k}ζ(1−2k)/2 comes from 2.4.2. The equality of this with |K_{4k−2}(ℤ)|/|K_{4k−1}(ℤ)| comes from Theorem 8.8 (using 9.12). This gives the displayed formula.

When n is odd, the groups K_{n}(ℤ) were determined in Theorem 9.5, and K_{n}(ℤ) ≅ K_{n}(ℚ) by IV.6.8. Thus we may suppose that n = 4k − 2. Since the 2-primary torsion in K_{n}(ℤ) was determined in Corollary 9.8, we can ignore factors of 2. But up to a factor of 2, |K_{4k−1}(ℤ)| = w_{2k}(ℚ) so the displayed formula yields |K_{4k−2}(ℤ)|/w_{2k} = B_{k}/4k and hence |K_{4k−2}(ℤ)| = c_{k}. □

The groups K_{n}(ℤ) are much harder to determine when n ≡ 0 (mod 4). The group K_{4}(ℤ) was proven to be zero in the late 1990’s (see Remark 10.1.3 or [Rognes]). If n = 4i ≥ 8, the orders of the groups K_{4i}(ℤ) are known to be products of irregular primes ℓ, with ℓ > 10^{8}, and are conjectured to be zero; this conjecture follows from, and implies, Vandiver’s conjecture (stated in 10.8 below).

In Table 10.1.1, we have summarized what we know for n < 20, 000; conjecturally the same pattern holds for all n (see Theorem 10.2).

\begin{matrix}K_{0}(ℤ) = ℤ & K_{8}(ℤ) = (0?) & K_{16}(ℤ) = (0?) & K_{8k}(ℤ) = (0?), k ≥ 1 \\ K_{1}(ℤ) = ℤ/2 & K_{9}(ℤ) = ℤ ⊕ ℤ/2 & K_{17}(ℤ) = ℤ ⊕ ℤ/2 & K_{8k+1}(ℤ) = ℤ ⊕ ℤ/2 \\ K_{2}(ℤ) = ℤ/2 & K_{10}(ℤ) = ℤ/2 & K_{18}(ℤ) = ℤ/2 & K_{8k+2}(ℤ) = ℤ/2c_{2k+1} \\ K_{3}(ℤ) = ℤ/48 & K_{11}(ℤ) = ℤ/1008 & K_{19}(ℤ) = ℤ/528 & K_{8k+3}(ℤ) = ℤ/2w_{4k+2} \\ K_{4}(ℤ) = 0 & K_{12}(ℤ) = (0?) & K_{20}(ℤ) = (0?) & K_{8k+4}(ℤ) = (0?) \\ K_{5}(ℤ) = ℤ & K_{13}(ℤ) = ℤ & K_{21}(ℤ) = ℤ & K_{8k+5}(ℤ) = ℤ \\ K_{6}(ℤ) = 0 & K_{14}(ℤ) = 0 & K_{22}(ℤ) = ℤ/691 & K_{8k+6}(ℤ) = ℤ/c_{2k+2} \\ K_{7}(ℤ) = ℤ/240 & K_{15}(ℤ) = ℤ/480 & K_{23}(ℤ) = ℤ/65520 & K_{8k+7}(ℤ) = ℤ/w_{4k+4}. \end{matrix}

Table 10.1.1. The groups K_{n}(ℤ), n < 20, 000. The notation ‘(0?)’ refers to a finite group, conjecturally zero, whose order is a product of irregular primes > 10^{8}.

Relation to π_{n}^{s} 10.1.2. Using homotopy-theoretic techniques, the torsion subgroups of K_{n}(ℤ) had been detected by the late 1970’s, due to the work of Quillen [Qlet], Harris-Segal [HS], Soulé [Sou] and others.

As pointed out in Remark 2.1.3, the image of the natural maps π_{n}^{s} → K_{n}(ℤ) capture most of the Harris-Segal summands 2.5. When n is 8k + 1 or 8k + 2, there is a ℤ/2-summand in K_{n}(ℤ), generated by the image of Adams’ element μ_{n}. (It is the 2-torsion subgroup by 9.8.) Since w_{4k+1}(ℚ) = 2, we may view it as the Harris-Segal summand when n = 8k + 1. When n = 8k + 5, the Harris-Segal summand is zero by Example 9.5.1. When n = 8k + 7 the Harris-Segal summand of K_{n}(ℤ) is isomorphic to the subgroup J(π_{n}O) ≅ ℤ/w_{4k+4}(ℚ) of π_{n}^{s}.

When n = 8k + 3, the subgroup J(π_{n}O) ≅ ℤ/w_{4k+2}(ℚ) of π_{n}^{s} is contained in the Harris-Segal summand ℤ/(2w_{i}) of K_{n}(ℤ); the injectivity was proven by Quillen in [Qlet], and Browder showed that the order of the summand was 2w_{i}(ℚ).

The remaining calculations of K_{∗}(ℤ) depend upon the development of motivic cohomology, via the tools described in Section 4, and date to the period 1997–2007. The 2-primary torsion was resolved in 1997 using [V-MC] (see Section 9), while the order of the odd torsion (conjectured by Lichtenbaum) was only determined using the Norm Residue Theorem 4.1 of Rost and Voevodsky.

Homological methods 10.1.3. Lee-Szczarba [LSz] and Soulé [So78] used homological methods in the 1970s to show that K_{3}(ℤ) ≅ ℤ/48 and that there is no p-torsion in K_{4}(ℤ) or K_{5}(ℤ) for p > 3. Much later, Rognes [Rognes] and Elbaz-Vincent–Gangl–Soulé [EGS] refined this to show that K_{4}(ℤ) = 0, K_{5}(ℤ) = ℤ, and that K_{6}(ℤ) has at most 3-torsion. This used the calculation in [RW] (using [V-MC]) that there is no 2-torsion in K_{4}(ℤ), K_{5}(ℤ) or K_{6}(ℤ).

Our general description of K_{∗}(ℤ) is completed by the following assertion, which follows immediately from Theorems 10.1,10.9 and 10.10 below, It was observed independently by Kurihara [Kur] and Mitchell [Mit].

Theorem 10.2. If Vandiver’s conjecture holds, then the groups K_{n}(ℤ) are given by Table 10.2.1, for all n ≥ 2. Here k is the integer part of 1 + \frac{n}{4}.

\begin{matrix}n (mod 8) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ K_{n}(ℤ) & ℤ ⊕ ℤ/2 & ℤ/2c_{k} & ℤ/2w_{2k} & 0 & ℤ & ℤ/c_{k} & ℤ/w_{2k} & 0 \end{matrix}

Table 10.2.1. The K-theory of ℤ, assuming Vandiver’s Conjecture. When n is at most 20,000 and n ≡ 2 (mod 4), we show that the finite groups

K_{n}(ℤ) are cyclic in Examples 10.3 and 10.3.2. (The order is c_{k} or 2c_{k}, where k = (n + 2)/4, by Theorem 10.1.)

Examples 10.3. For n at most 450, the group K_{n}(ℤ) is cyclic because its order is squarefree. For n ≤ 30 we need only consult 2.1.2 to see that the groups K_{2}(ℤ), K_{10}(ℤ), K_{18}(ℤ) and K_{26}(ℤ) are isomorphic to ℤ/2, while K_{6}(ℤ) = K_{14}(ℤ) = 0. Since c_{6} = 691, c_{8} = 3617, c_{9} = 43867 and c_{13} = 657931 are all prime, we have K_{22}(ℤ) ≅ ℤ/691, K_{30}(ℤ) ≅ ℤ/3617, K_{34}(ℤ) ≅ ℤ/2 ⊕ ℤ/43867 and K_{50} ≅ ℤ/2 ⊕ ℤ/657931.

The next hundred values of c_{k} are also squarefree: c_{10} = 283·617, c_{11} = 131·593, c_{12} = 103·2294797, c_{14} = 9349·362903 and c_{15} = 1721·1001259881 are all products of two primes, while c_{16} = 37 · 683 · 305065927 is a product of 3 primes. Hence K_{38}(ℤ) = ℤ/c_{10}, K_{42}(ℤ) = ℤ/2c_{11}, K_{46} = ℤ/c_{12}, K_{54}(ℤ) = ℤ/c_{14}, K_{58}(ℤ) = ℤ/2c_{15} and K_{62}(ℤ) = ℤ/c_{16} = ℤ/37 ⊕ ℤ/683 ⊕ ℤ/305065927.

Thus the first occurrence of the smallest irregular prime (37) is in K_{62}(ℤ); it also appears as a ℤ/37 summand in K_{134}(ℤ), K_{206}(ℤ), …, K_{494}(ℤ). In fact, there is 37-torsion in every group K_{72a+62}(ℤ) (see Ex. 10.2). This direct method fails for K_{454}(ℤ), because its order 2c_{114} is divisible by 103^{2}.

To go further, we need to consider the torsion in the groups K_{4k−2}(ℤ) on a primeby-prime basis. Since the 2-torsion has order at most 2 by 9.8, we may suppose that ℓ is an odd prime. Our method is to consider the cyclotomic extension ℤ[ζ_{ℓ}] of ℤ, ζ_{ℓ} = e^{2πi/ℓ}. Because K_{n}(ℤ) → K_{n}(ℤ[1/ℓ]) is an isomorphism on ℓ-torsion (by the Localization Sequence V.6.6), and similarly for K_{n}(ℤ[ζ_{ℓ}]) → K_{n}(ℤ[ζ_{ℓ}, 1/ℓ]), it suffices to work with ℤ[1/ℓ] and R = ℤ[ζ_{ℓ}, 1/ℓ].

The usual transfer argument 10.3.1. The ring extension ℤ[1/ℓ] ⊂ R is Galois and its Galois group G = Gal(ℚ(ζ)/ℚ) is cyclic of order ℓ − 1. The map i^{∗} : K_{∗}(ℤ) → K_{∗}(R) identifies K_{n}(ℤ[1/ℓ])_{(ℓ)} with K_{n}(R)_{(ℓ)}^{G} for all n, because i_{∗}i^{∗} is multiplication by |G| on K_{∗}(ℤ) and i^{∗}i_{∗} is multiplication by |G| on K_{n}(R)^{G} (see Ex. IV.6.13). This style of argument is called the usual transfer argument.

Example 10.3.2. The group K_{4k−2}(ℤ) is cyclic (of order c_{k} or 2c_{k}) for all k ≤ 5000. To see this, we observe that K_{4k−2}(ℤ)_{(ℓ)} is cyclic if ℓ^{2} does not divide c_{k}, and in this range only seven of the c_{k} are not square-free; see [OEIS, A090943]. The numerator c_{k} is divisible by ℓ^{2} only for the following pairs (k, ℓ): (114, 103), (142, 37), (457, 59), (717, 271), (1646, 67), (2884, 101) and (3151, 157). In each of these cases, we note that Pic(ℤ[ζ_{ℓ}]) = Pic(R)/ℓ ≅ ℤ/ℓ. By Example 8.3.1, K_{4k−2}(R)/ℓ ≅ Pic(R) ≅ ℤ/ℓ. The usual transfer argument (10.3.1) now shows that K_{4k−2}(ℤ)/ℓ is either 0 or ℤ/ℓ for all k. Since c_{k} is divisible by ℓ^{2} but not ℓ^{3}, K_{4k−2}(ℤ)_{(ℓ)} ≅ ℤ/ℓ^{2}.

Representations of G over ℤ/ℓ 10.4. When G is the cyclic group of order ℓ − 1, a ℤ/ℓ[G]-module is just a ℤ/ℓ-vector space on which G acts linearly. By Maschke’s theorem, ℤ/ℓ[G] ≅ ∏_{i=0}^{ℓ−2} ℤ/ℓ is a simple ring, so every ℤ/ℓ[G]-module has a unique decomposition as a sum of its ℓ − 1 irreducible modules. Since μ_{ℓ} is an irreducible G-module, it is easy to see that the irreducible G-modules are μ_{ℓ}^{⊗i}, i = 0, 1, ..., ℓ − 2. The “trivial” G-module is μ_{ℓ}^{⊗ℓ−1} = μ_{ℓ}^{⊗0} = ℤ/ℓ. By convention, μ_{ℓ}^{⊗i} = μ_{ℓ}^{⊗i+a(ℓ−1)} for all integers a.

For example, the G-submodule ⟨β^{i}⟩ of K_{2i}(ℤ[ζ]; ℤ/ℓ) generated by β^{i} is isomorphic to μ_{ℓ}^{⊗i}. It is a trivial G-module only when (ℓ − 1)|i.

If A is any ℤ/ℓ[G]-module, it is traditional to decompose A = ⊕A^{[i]}, where A^{[i]} denotes the sum of all G-submodules of A isomorphic to μ_{ℓ}^{⊗i}.

Example 10.4.1. Set R = ℤ[ζ_{ℓ}, 1/ℓ]. It is known that the torsionfree part R^{×}/μ_{ℓ} ≅ ℤ^{\frac{ℓ−1}{2}} of the units of R is isomorphic as a G-module to ℤ[G] ⊗_{ℤ[c]} ℤ, where c is complex conjugation. (This is sometimes included as part of Dirichlet’s theorem on units.) It follows that as a G-module,

R^{×}/R^{×ℓ} ≅ μ_{ℓ} ⊕ (ℤ/ℓ) ⊕ μ_{ℓ}^{⊗2} ⊕ ⋯ ⊕ μ_{ℓ}^{⊗ℓ−3}.

The first two terms μ_{ℓ} and ℤ/ℓ are generated by the root of unity ζ_{ℓ} and the class of the unit ℓ of R. It will be convenient to choose units x_{0} = ℓ, x_{1}, …, x_{(ℓ−3)/2} of R such that x_{i} generates the summand μ_{ℓ}^{−2i} of R^{×}/R^{×ℓ}; the notation is set up so that x_{i} ⊗ ζ_{ℓ}^{⊗2i} is a G-invariant element of R^{×} ⊗ μ_{ℓ}^{⊗2i}.

Example 10.4.2. The G-module decomposition of M = R^{×}⊗μ_{ℓ}^{⊗i−1} is obtained from Example 10.4.1 by tensoring with μ_{ℓ}^{⊗i−1}. If i is even, ℤ/ℓ occurs only when i ≡ 0 (mod ℓ − 1), corresponding to ζ^{⊗i}. If i is odd, exactly one term of M is ℤ/ℓ; M^{G} is ℤ/ℓ on the generator x_{j} ⊗ ζ_{ℓ}^{i−1}, where i ≡ 1 + 2j (mod ℓ − 1).

Torsion for odd regular primes

Suppose that ℓ is an odd regular prime. By definition, Pic(ℤ[ζ]) has no ℓ-torsion, and K_{1}(R)/ℓ ≅ R^{×}/R^{×ℓ} by III.1.3.6. Kummer showed that ℓ cannot divide the order of any numerator c_{k} of B_{k}/k (see 2.4.1). Therefore the case 2i = 4k − 2 of the following result follows from Theorem 10.1.

Proposition 10.5. When ℓ is an odd regular prime, the group K_{2i}(ℤ) has no ℓtorsion. Thus the only ℓ-torsion subgroups of K_{∗}(ℤ) are the Harris-Segal subgroups ℤ/w_{i}^{(ℓ)}(ℚ) of K_{2i−1}(ℤ) when i ≡ 0 (mod ℓ − 1).

Proof. Since ℓ is regular, we saw in Example 8.3.2 that the group K_{2i}(ℤ[ζ]) has no ℓ-torsion. Hence the same is true for its G-invariant subgroup, K_{2i}(ℤ). The restriction on i comes from Example 2.1.2. □

We can also describe the algebra structure of K_{∗}(ℤ; ℤ/ℓ). For this we set R = ℤ[ζ_{ℓ}, 1/ℓ] and G = Gal(ℚ(ζ)/ℚ), noting that K_{∗}(ℤ[1/ℓ]; ℤ/ℓ) ≅ K_{∗}(R; ℤ/ℓ)^{G} by the usual transfer argument (10.3.1). Recall from Example 8.5 that K_{∗}(R; ℤ/ℓ) is a free graded ℤ/ℓ[β]-module on \frac{ℓ+1}{2} generators: the x_{i} of R^{×}/R^{×ℓ} = K_{1}(R; ℤ/ℓ), together with 1 ∈ K_{0}(R; ℤ/ℓ).

Thus K_{2i}(R; ℤ/ℓ) ≅ ℤ/ℓ is generated by β^{i}, and is isomorphic to μ_{ℓ}^{⊗i} as a Gmodule. It follows that K_{2i}(R; ℤ/ℓ)^{G} is zero unless i = a(ℓ − 1), when it is ℤ/ℓ on the generator β^{i}. By abuse of notation, we shall write β^{ℓ−1} for the element of K_{2(ℓ−1)}(ℤ; ℤ/ℓ) corresponding to β^{ℓ−1}; if i = a(ℓ − 1) we shall write β^{i} for the element (β^{ℓ−1})^{a} of K_{2i}(ℤ; ℤ/ℓ) corresponding to β^{i} ∈ K_{2i}(R; ℤ/ℓ)^{G}.

By Example 8.5, K_{2i−1}(R; ℤ/ℓ) is just R^{×} ⊗ μ_{ℓ}^{⊗i−1} when ℓ is regular. The Gmodule structure was determined in Example 10.4.2: if i is even, exactly one term is ℤ/ℓ; if i is odd, ℤ/ℓ occurs only when i ≡ 1 (mod ℓ − 1).

Multiplying [ζ] ∈ K_{1}(R; ℤ/ℓ) by β^{ℓ−2} yields the G-invariant element v = [ζ] β^{ℓ−2} of K_{2ℓ−3}(R; ℤ/ℓ). Again by abuse of notation, we write v for the corresponding element of K_{2ℓ−3}(ℤ; ℤ/ℓ).

Similarly, multiplying x_{k} ∈ R^{×} = K_{1}(R) by β^{2k} ∈ K_{4k}(R; ℤ/ℓ) gives a Ginvariant element y_{k} = x_{k}β^{2k} of K_{4k+1}(R; ℤ/ℓ) with y_{0} = [ℓ] in K_{1}(R; ℤ/ℓ). Again by abuse of notation, we write y_{k} for the corresponding element of K_{4k+1}(ℤ; ℤ/ℓ).

Theorem 10.6. If ℓ is an odd regular prime then K_{∗} = K_{∗}(ℤ[1/ℓ]; ℤ/ℓ) is a free graded module over the polynomial ring ℤ/ℓ[β^{ℓ−1}]. It has (ℓ + 3)/2 generators: 1 ∈ K_{0}, v ∈ K_{2ℓ−3}, and the elements y_{k} ∈ K_{4k+1} (k = 0, ..., \frac{ℓ−3}{2}) described above.

Similarly, K_{∗}(ℤ; ℤ/ℓ) is a free graded module over ℤ/ℓ[β^{ℓ−1}]; a generating set is obtained from the generators of K_{∗} by replacing y_{0} by y_{0}β^{ℓ−1}.

The ℤ/ℓ[β^{ℓ−1}]-submodule generated by v and β^{ℓ−1} comes from the Harris-Segal summands of K_{2i−1}(ℤ). The submodule generated by the y’s comes from the ℤ summands in K_{n}(ℤ), n ≡ 1 (mod 4).

Proof. K_{∗}(ℤ[1/ℓ]; ℤ/ℓ) is the G-invariant subalgebra of K_{∗}(R; ℤ/ℓ). Given 10.5, it is not very hard to check that this is just the subalgebra described in the theorem. Since ℓ − 1 is even, the elements y_{k}β^{a(ℓ−1)} are in K_{n}(ℤ; ℤ/ℓ) for some n ≡ 1 (mod 4). Since K_{n−1}(ℤ) has no ℓ-torsion by Proposition 10.5, K_{n}(ℤ; ℤ/ℓ) = K_{n}(ℤ)/ℓ. Since 1 ≤ 4k + 1 ≤ 2ℓ − 4, we have n ≡ 4k + 1 ≢ 0 (mod 2ℓ − 2) and hence K_{n}(ℤ) has no ℓ-torsion (combine 10.1 with 2.1.2). Hence the element y_{k}β^{a(ℓ−1)} must come from the ℤ-summand of K_{n}(ℤ[1/ℓ]). □

Examples 10.6.1. When ℓ = 3, the groups K_{n} = K_{n}(ℤ[1/3]; ℤ/3) are 4periodic of ranks 1, 1, 0, 1, generated by an appropriate power of β^{2} ∈ K_{4} times one of {1, [3], v}. Here v ∈ K_{3}.

When ℓ = 5, the groups K_{n} = K_{n}(ℤ[1/5]; ℤ/5) are 8-periodic, with respective ranks 1, 1, 0, 0, 0, 1, 0, 1 (n = 0, ..., 7), generated by an appropriate power of β^{4} ∈ K_{8} times one of {1, [5], y_{1}, v}. Here y_{1} ∈ K_{5} (x_{1} is the golden mean) and v ∈ K_{7}.

Torsion for irregular primes

Now suppose that ℓ is an irregular prime, so that Pic(R) has ℓ-torsion for R = ℤ[ζ, 1/ℓ]. Then H_{et}^{1}(R, μ_{ℓ}) is R^{×}/ℓ ⊕ _{ℓ} Pic(R) and H_{et}^{2}(R, μ_{ℓ}) ≅ Pic(R)/ℓ by Kummer theory and (8.1.1). This yields K_{∗}(R; ℤ/ℓ) by Example 8.5.

Set P = Pic(R)/ℓ. When ℓ is irregular, the G-module structure of P is not fully understood; see Vandiver’s conjecture 10.8 below.

Lemma 10.7. For i = 0, −1, −2, −3, P contains no summands isomorphic to μ_{ℓ}^{⊗i}, i.e., P^{[i]} = 0.

Proof. The usual transfer argument shows that P^{G} ≅ Pic(ℤ[1/ℓ])/ℓ = 0. Hence P contains no summands isomorphic to ℤ/ℓ. By III.6.9.3, there is a Gmodule isomorphism (P ⊗ μ_{ℓ}) ≅ K_{2}(R)/ℓ. Since K_{2}(R)/ℓ^{G} ≅ K_{2}(ℤ[1/ℓ])/ℓ = 0, (P ⊗ μ_{ℓ}) has no ℤ/ℓ summands — and hence P contains no summands isomorphic to μ_{ℓ}^{⊗−1}.

Finally, we have (P ⊗ μ_{ℓ}^{⊗2}) ≅ K_{4}(R)/ℓ and (P ⊗ μ_{ℓ}^{⊗3}) ≅ K_{6}(R)/ℓ by 8.3. Again, the transfer argument shows that K_{n}(R)/ℓ^{G} ≅ K_{n}(ℤ[1/ℓ])/ℓ for n = 4, 6. The groups K_{4}(ℤ) and K_{6}(ℤ) are known to be zero by [Rognes] and [EGS]; see 10.1.3. It follows that P contains no summands isomorphic to μ_{ℓ}^{⊗−2} or μ_{ℓ}^{⊗−3}. □

Vandiver’s conjecture 10.8. If ℓ is an irregular prime then Pic(ℤ[ζ_{ℓ} + ζ_{ℓ}^{−1}]) has no ℓ-torsion. Equivalently, the natural representation of G = Gal(ℚ(ζ_{ℓ})/ℚ) on Pic(ℤ[ζ_{ℓ}])/ℓ is a sum of G-modules μ_{ℓ}^{⊗i} with i odd.

This means that complex conjugation c acts as multiplication by −1 on the ℓ-torsion subgroup of Pic(ℤ[ζ_{ℓ}])/ℓ, because c is the unique element of G of order 2.

As partial evidence for this conjecture, we mention that Vandiver’s conjecture has been verified for all primes up to 163 million; see [BH]. We also known from Lemma 10.7 that μ_{ℓ}^{⊗i} does not occur as a summand of Pic(R)/ℓ for i = 0, −2.

Remark 10.8.1. The Herbrand-Ribet theorem [Wash, 6.17–18] states that ℓ|B_{k} if and only if (Pic R/ℓ)^{[ℓ−2k]} ≠ 0. Among irregular primes < 4000, this happens for at most 3 values of k. For example, 37|c_{16} (see 10.3), so (Pic R/ℓ)^{[5]} = ℤ/37 and (Pic R/ℓ)^{[k]} = 0 for k ≠ 5.

Historical Remark 10.8.2. What we now call “Vandiver’s conjecture” was actually discussed by Kummer and Kronecker in 1849–1853; Harry Vandiver was not born until 1882 and only started using this assumption circa 1920 (e.g., in [Van29] and [Van34]), but only retroactively claimed to have conjectured it “about 25 years ago” in the 1946 paper [Van46, p. 576].

In 1849, Kronecker asked if Kummer conjectured that a certain lemma ([Wash, 5.36]) held for all p, and that therefore p never divided h^{+} (i.e., Vandiver’s conjecture holds). Kummer’s reply [Kum, pp.114–115] pointed out that the Lemma could not hold for irregular p, and then referred to the assertion [Vandiver’s conjecture] as the noch zu beweisenden Satz (theorem still to be proven). Kummer also pointed out some of its consequences. In an 1853 letter (see [Kum, p.123]), Kummer wrote to Kronecker that in spite of months of effort, the assertion [now called Vandiver’s conjecture] was still unproven.

For the rest of this paper, we set R = ℤ[ζ_{ℓ}, 1/ℓ], and P = Pic(R)/ℓ.

Theorem 10.9. (Kurihara [Kur]) Let ℓ be an irregular prime number. Then the following are equivalent for every integer k between 1 and \frac{ℓ−1}{2}:

(1) Pic(ℤ[ζ])/ℓ^{[−2k]} = 0.

(2) K_{4k}(ℤ) has no ℓ-torsion;

(3) K_{2a(ℓ−1)+4k}(ℤ) has no ℓ-torsion for all a ≥ 0;

(4) H_{et}^{2}(ℤ[1/ℓ], μ_{ℓ}^{⊗2k+1}) = 0. In particular, Vandiver’s conjecture for ℓ is equivalent to the assertion that K_{4k}(ℤ) has no ℓ-torsion for all k < \frac{ℓ−1}{2}, and implies that K_{4k}(ℤ) has no ℓ-torsion for all k.

Proof. By Kummer theory and (8.1.1), P ≅ H_{et}^{2}(R, μ_{ℓ}). Hence P ⊗ μ_{ℓ}^{⊗2k} ≅ H_{et}^{2}(R, μ_{ℓ}^{⊗2k+1}) as G-modules. Taking G-invariant subgroups shows that

H_{et}^{2}(ℤ[1/ℓ], μ_{ℓ}^{⊗2k+1}) ≅ (P ⊗ μ_{ℓ}^{⊗2k})^{G} ≅ P^{[−2k]}.

Hence (1) and (4) are equivalent.

By 8.3, K_{4k}(ℤ)/ℓ ≅ H_{et}^{2}(ℤ[1/ℓ], μ_{ℓ}^{⊗2k+1}) for all k > 0. Since μ_{ℓ}^{⊗b} = μ_{ℓ}^{⊗a(ℓ−1)+b} for all a and b, this shows that (2) and (3) are separately equivalent to (4). □

Theorem 10.10. If Vandiver’s conjecture holds for ℓ then the ℓ-primary torsion subgroup of K_{4k−2}(ℤ) is cyclic for all k.

If Vandiver’s conjecture holds for all ℓ, the groups K_{4k−2}(ℤ) are cyclic for all k.

(We know that the groups K_{4k−2}(ℤ) are cyclic for all k < 5000, by 10.3.2.)

Proof. Vandiver’s conjecture also implies that each of the “odd” summands P^{[1−2k]} = P^{[ℓ−2k]} of P is cyclic; see [Wash, 10.15]. Taking the G-invariant subgroups of Pic(R) ⊗ μ_{ℓ}^{⊗2k−1} ≅ H_{et}^{2}(R, μ_{ℓ}^{⊗2k}), yields P^{[1−2k]} ≅ H_{et}^{2}(ℤ[1/ℓ], μ_{ℓ}^{⊗2k}). By Corollary 8.3, this group is the ℓ-torsion in K_{4k−2}(ℤ[1/ℓ])/ℓ. □

Remark 10.11. The elements of K_{2i}(ℤ) of odd order become divisible in the larger group K_{2i}(ℚ). (The assertion that an element a is divisible in A means that for every m there is an element b so that a = mb.) This was proven by Banaszak and Kolster for i odd (see [Ban, Thm. 2]), and for i even by Banaszak and Gajda [BaGj, Proof of Prop. 8]. It is an open question whether there are any divisible elements of even order.

For example, recall from 10.3 that K_{22}(ℤ) = ℤ/691 and K_{30}(ℤ) ≅ ℤ/3617. Banaszak observed [Ban] that these groups are divisible in K_{22}(ℚ) and K_{30}(ℚ), i.e., that the inclusions K_{22}(ℤ) ⊂ K_{22}(ℚ) and K_{30}(ℤ) ⊂ K_{30}(ℚ) do not split.

Let t_{j} (resp., s_{j}) be generators of the summand of Pic(R)/ℓ (resp. K_{1}(R; ℤ/ℓ)) isomorphic to μ_{ℓ}^{⊗−j}. The following result follows easily from Examples 8.5 and 10.4.1, using the proof of 10.6, 10.9 and 10.10. It was originally proven in [Mit].

Theorem 10.12. If ℓ is an irregular prime for which Vandiver’s conjecture holds, then K_{∗} = K_{∗}(ℤ; ℤ/ℓ) is a free module over ℤ/ℓ[β^{ℓ−1}] on 1, v ∈ K_{2ℓ−3}, the (ℓ − 3)/2 generators y_{k} ∈ K_{4k+1} described in Theorem 10.6, together with the generators t_{j}β^{j} ∈ K_{2j} and s_{j}β^{j} ∈ K_{2j+1} (j = 3, 5, ..., (ℓ − 8)).

EXERCISES

10.1 Let ℓ be an irregular prime and suppose that K_{n}(ℤ) has no ℓ-torsion for some positive n ≡ 0 (mod 4). Show that K_{4k}(ℤ) has no ℓ-torsion for every k satisfying n ≡ 4k (mod 2ℓ − 2). 10.2 Show that K_{n}(ℤ) has nonzero 37-torsion for all positive n ≡ 62 (mod 72), and that K_{n}(ℤ) has nonzero 103-torsion for all positive n ≡ 46 (mod 204). 10.3 Give a careful proof of Theorem 10.12, by using Examples 8.5 and 10.4.1 for ℤ[ζ_{ℓ}, 1/ℓ] to modify the proof of Theorem 10.6. 10.4 The Bockstein operation b : K_{n}(R; ℤ/ℓ) → K_{n+1}(R; ℤ/ℓ) is the boundary map in the long exact sequence associated to the coefficient sequence 0 → ℤ/ℓ → ℤ/ℓ^{2} → ℤ/ℓ → 0. Show that when R = ℤ the Bockstein sends v to β^{ℓ−1}, t_{j} to s_{j} and t_{j}β^{j} to s_{j}β^{j} in Theorems 10.6 and 10.12.