ALGEBRAIC K-THEORY OF RINGS OF INTEGERS

IN LOCAL AND GLOBAL FIELDS

C. Weibel

August 21, 2004

Abstract. This survey describes the algebraic K-groups of local and global fields, and the K-groups of rings of integers in these fields. We have used the result of Rost and Voevodsky to determine the odd torsion in these groups.

Introduction

The problem of computing the higher K-theory of a number field F, and of its rings of integers O_{F}, has a rich history. Since 1972, we have known that the groups K_{n}(O_{F}) are finitely generated [Q3], and known their ranks [Bo], but have only had conjectural knowledge about their torsion subgroups [Li] [Li2] [Bei] until 1997 (starting with [We2]). The resolutions of many of these conjectures by Suslin, Voevodsky, Rost and others have finally made it possible to describe the groups K_{∗}(O_{F}). One of the goals of this survey is to give such a description; here is the odd half of the answer (the integers w_{i}(F) are even, and are defined in section 2):

Theorem 0.1. Let O_{S} be a ring of S-integers in a number field F. Then K_{n}(O_{S}) ≅ K_{n}(F) for each odd n ≥ 3, and 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):

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}(O_{S}) ≅ 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}

In particular, K_{n}(ℚ) ≅ ℤ for all n ≡ 5 (mod 8) (as w_{i} = 2; see 2.11). 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; see 2.8).

The proof of 0.1, will be given in 6.2, 6.5, and section 7 below.

Typeset by AMS-TEX

We also know the order of the groups K_{n}(ℤ) when n ≡ 2 (mod 4), and know that they are cyclic for n < 20, 000 (see 7.12 — conjecturally, they are cyclic for every n ≡ 2). If B_{k} denotes the k^{th} Bernoulli number (2.10), and c_{k} denotes the numerator of B_{k}/4k, then |K_{4k−2}(ℤ)| is: c_{k} for k even, and 2c_{k} for k odd; see 7.11.

Although the groups K_{4k}(ℤ) are conjectured to be zero, at present we only know that these groups have odd order, with no prime factors less than 10^{7}. This conjecture follows from, and implies, Vandiver’s conjecture in number theory (see 8.5 below). In Table 0.2, we have summarized what we know for n < 20, 000; conjecturally the same pattern holds for all n (see 8.6–8).

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

Table 0.2. 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^{7}.

For n ≤ 3, the groups K_{n}(ℤ) were known by the early 1970’s; see section 1. The right hand sides were also identified as subgroups of K_{n}(ℤ) by the late 1970’s; see sections 2 and 3. The 2-primary torsion was resolved in 1997 (section 7), but the rest of Table 0.2 only follows from the recent Voevodsky-Rost theorem (sections 6 and 8).

The K-theory of local fields, and global fields of finite characteristic, is richly interconnected with this topic. The other main goal of this article is to survey the state of knowledge here too.

In section 1, we describe the structure of K_{n}(O_{F}) for n ≤ 3; this material is relatively classical, since these groups have presentations by generators and relations.

The cyclic summands in theorem 0.1 are a special case of a more general construction, due to Harris and Segal. For all fields F, the odd-indexed groups K_{2i−1}(F) have a finite cyclic summand which up to a factor of 2 is detected by a variation of Adams’ e-invariant. These summands are discussed in section 2.

There are also canonical free summands related to units, discovered by Borel, and (almost periodic) summands related to the Picard group of R, and the Brauer group of R. These summands were first discovered by Soulé, and are detected by étale Chern classes. These summands are discussed in section 3.

The K-theory of a global field of finite characteristic is handled in section 4. In this case, there is a smooth projective curve X whose higher K-groups are finite, and are related to the action of the Frobenius on the Jacobian variety of X. The orders of these groups are related to the values of the zeta function ζ_{X}(s) at negative integers.

The K-theory of a local field E containing ℚ_{p} is handled in section 5. In this case, we understand the p-completion, but do not understand the actual groups K_{∗}(E).

K-THEORY OF INTEGERS 3

In section 6, we handle the odd torsion in the K-theory of a number field. This is a consequence of the Voevodsky-Rost theorem. These techniques also apply to the 2-primary torsion in totally imaginary number fields, which gives 0.1(a).

The 2-primary torsion in real number fields (those with an embedding in ℝ) is handled in section 7; this material is taken from [RW], and uses Voevodsky’s theorem in [V].

Finally, we consider the odd torsion in K_{2i}(ℤ) in section 8; the odd torsion in K_{2i−1}(ℤ) is given by 0.1. The torsion occurring in the groups K_{2i}(ℤ) only involves irregular primes, and is determined by Vandiver’s conjecture (8.5). The lack of torsion for regular primes was first guessed by Soulé in [So].

The key technical tool which makes calculations possible for local and global fields is the motivic spectral sequence, from motivic cohomology H_{M}^{∗} to algebraic K-theory. With coefficients ℤ/m, the spectral sequence for X is:

(0.3)  E_{2}^{p,q} = H_{M}^{p−q}(X; ℤ/m(−q)) ⇒ K_{−p−q}(X; ℤ/m).

This formulation assumes that X is defined over a field [SV]; a similar motivic spectral sequence was established by Levine in [Le, (8.8)] over a Dedekind domain, in which the group H_{M}^{n}(X, ℤ(i)) is defined to be the (2i − n)th hypercohomology on X of the complex of higher Chow group sheaves z^{i}.

When 1/m ∈ F, Voevodsky and Rost proved in [V] (m = 2^{ν}) and [V03] (m odd) that H_{M}^{n}(F, ℤ/m(i)) is isomorphic to H_{ét}^{n}(F, μ_{m}^{⊗i}) for n ≤ i and zero if n > i. That is, the E_{2}-terms in this spectral sequence are just étale cohomology groups.

If X = Spec(R), where R is a Dedekind domain with F = frac(R) and 1/m ∈ R, a comparison of the localization sequences for motivic and étale cohomology (see [Le] and [So, p.268]) shows that H_{M}^{n}(X, ℤ/m(i)) is: H_{ét}^{n}(X, μ_{m}^{⊗i}) for n ≤ i; the kernel of H_{ét}^{n}(X, μ_{m}^{⊗i}) → H_{ét}^{n}(F, μ_{m}^{⊗i}) for n = i + 1; and zero if n ≥ i + 2. That is, the E_{2}-terms in the fourth quadrant are étale cohomology groups, but there are also modified terms in the column p = +1. For example, we have E_{2}^{1,−1} = Pic(X)/m. This is the only nonzero term in the column p = +1 when X has étale cohomological dimension at most two for ℓ-primary sheaves (cd_{ℓ}(X) ≤ 2), as will often occur in this article.

Writing ℤ/ℓ^{∞}(i) for the union of the étale sheaves ℤ/ℓ^{∞}(i), we also obtain a spectral sequence for every field F:

(0.4)  E_{2}^{p,q} = \{ \begin{matrix}H_{ét}^{p−q}(F; ℤ/ℓ^{∞}(−q)) & for q ≤ p ≤ 0, \\ 0 & otherwise \end{matrix} ⇒ K_{−p−q}(F; ℤ/ℓ^{∞}),

and a similar spectral sequence for X which can have nonzero entries in the column p = +1. If cd_{ℓ}(X) ≤ 2 it is:

(0.5) E_{2}^{p,q} = \{ \begin{matrix}H_{ét}^{p−q}(X; ℤ/ℓ^{∞}(−q)) & for q ≤ p ≤ 0, \\ Pic(X) ⊗ ℤ/ℓ^{∞} & for (p,q)=(+1,-1), \\ 0 & otherwise \end{matrix} ⇒ K_{−p−q}(X; ℤ/ℓ^{∞}).

Periodicity for ℓ = 2 0.6. 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 (0.3), a canonical element of H_{ét}^{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 [FS] [Le]) 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 (0.3) to itself. For p ≤ 0 these maps are isomorphisms, induced by E_{2}^{p,q} ≅ H_{ét}^{p−q}(X, ℤ/2); we shall refer to these isomorphisms as periodicity isomorphisms.

Since the Voevodsky-Rost result has not been published yet (see [V03]), it is appropriate for us to indicate exactly where it has been invoked in this survey. In addition to 0.1, 0.2, (0.4) and (0.5) in this introduction, the Voevodsky-Rost theorem is used in theorem 4.7, section 6, 7.10–12 and in section 8.

Acknowledgements

Many people have graciously assisted me in assembling the information in this survey. The author specifically would like to thank Spencer Bloch, Bill Dwyer, Lars Hesselholt, Bruno Kahn, Steve Mitchell, Paul Østvær, John Rognes, Neil Sloane and Christophe Soulé.

The author would also like to thank the Newton Institute, for its hospitality during most of the writing process. The author was also partially supported by an NSA grant.

§1. Classical K-theory of Number Fields

Let F be a number field, i.e., a finite extension of ℚ, and let O_{F} denote the ring of integers in F, i.e., the integral closure of ℤ in F. The first few K-groups of F and O_{F} have been known since the dawn of K-theory. We quickly review these calculations in this section.

When Grothendieck invented K_{0} in the late 1950’s, it was already known that over a Dedekind domain R (such as O_{F} or the ring O_{S} of S-integers in F) every projective module is the sum of ideals, each of which is projective and satisfies I ⊕ J ≅ IJ ⊕ R. Therefore K_{0}(R) = ℤ ⊕ Pic(R). Of course, K_{0}(F) = ℤ.

In the case R = O_{F} the Picard group was already known as the Class group of F, and Dirichlet had proven that Pic(O_{F}) is finite. Although not completely understood to this day, computers can calculate the class group for millions of number fields. For cyclotomic fields, we know that Pic(ℤ[μ_{p}]) = 0 only for p ≤ 19, and that the size of Pic(ℤ[μ_{p}]) grows exponentially in p; see [Wash].

Example 1.1 (Regular primes). A prime p is called regular if Pic(ℤ[μ_{p}]) has no elements of exponent p, i.e., if p does not divide the order h_{p} of Pic(ℤ[μ_{p}]). Kummer proved that this is equivalent to the assertion that p does not divide the numerator of any Bernoulli number B_{k}, k ≤ (p − 3)/2 (see 2.10 and [Wash, 5.34]). 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 p = 37, 59, 67, 101, 103 and 131. About 39% of the primes less than 4 million are irregular.

The historical interest in regular primes is that Kummer proved Fermat’s Last Theorem for regular primes in 1847. For us, certain calculations of K-groups become easier at regular primes. (See section 8.)

We now turn to units. The valuations on F associated to the prime ideals ℘ of O_{F} show that the group F^{×} is the product of the finite cyclic group μ(F) of roots of unity and a free abelian group of infinite rank. Dirichlet showed that the group of units of O_{F} is the product of μ(F) and a free abelian group of rank r_{1} + r_{2} − 1, where r_{1} and r_{2} are the number of embeddings of F into the real numbers ℝ and complex numbers ℂ, respectively.

The relation of the units to the class group is given by the “divisor map” (of valuations) from F^{×} to the free abelian group on the set of prime ideals ℘ in O_{F}. The divisor map fits into the “Units-Pic” sequence:

0 → O_{F}^{×} → F^{×} ⟶^{div} ⊕_{℘}ℤ → Pic(O_{F}) → 0.

If R is any commutative ring, the group K_{1}(R) is the product of the group R^{×} of units and the group SK_{1}(R) = SL(R)/[SL(R), SL(R)]. Bass-Milnor-Serre proved in [BMS] that SK_{1}(R) = 0 for any ring of S-integers in any global field. Applying this to the number field F we obtain:

(1.2)  K_{1}(O_{F}) = O_{F}^{×} ≅ μ(F) × ℤ^{r_{1}+r_{2}−1}.

For the ring O_{S} of S-integers in F, the sequence 1 → O_{F}^{×} → O_{S}^{×} → ℤ[S] ⟶^{div} Pic(O_{F}) → Pic(O_{S}) → 1 yields:

(1.2.1)  K_{1}(O_{S}) = O_{S}^{×} ≅ μ(F) × ℤ^{|S|+r_{1}+r_{2}−1}.

The 1967 paper [BMS] was instrumental in discovering the group K_{2} and its role in number theory. Garland proved in [Gar] that K_{2}(O_{F}) is a finite group. By [Q2], we also know that it is related to K_{2}(F) by the localization sequence:

0 → K_{2}(O_{F}) → K_{2}(F) ⟶^{∂} ⊕_{℘}k(℘)^{×} → 0.

Since the map ∂ was called the tame symbol, the group K_{2}(O_{F}) was called the tame kernel in the early literature. Matsumoto’s theorem allowed Tate to calculate K_{2}(O_{F}) for the quadratic extensions ℚ(\sqrt{−d}) of discriminant < 35 in [BT]. In particular, we have K_{2}(ℤ) = K_{2}(ℤ[\frac{1+\sqrt{−7}}{2}]) = ℤ/2 on \{−1, −1\}, and K_{2}(ℤ[i]) = 0.

Tate’s key breakthrough, published in [Ta1], was the following result, which was generalized to all fields by Merkurjev and Suslin (in 1982).

Theorem 1.3. (Tate [Ta1]) If F is a number field and R is a ring of S-integers in F such that 1/ℓ ∈ R then K_{2}(R)/m ≅ H_{ét}^{2}(R, μ_{m}^{⊗2}) for every prime power m = ℓ^{ν}. The ℓ-primary subgroup of K_{2}(R) is H_{ét}^{2}(R, ℤ_{ℓ}(2)), which equals H_{ét}^{2}(R, μ_{m}^{⊗2}) for large ν.

If F contains a primitive m^{th} root of unity (m = ℓ^{ν}), there is a split exact sequence:

0 → Pic(R)/m → K_{2}(R)/m → _{m}Br(R) → 0.

Here _{m}Br(R) denotes {x ∈ Br(R)|mx = 0}. If we compose with the inclusion of K_{2}(R)/m into K_{2}(R; ℤ/m), Tate’s proof shows that the left map Pic(R) → K_{2}(R; ℤ/m) is multiplication by the Bott element β ∈ K_{2}(R; ℤ/m) corresponding to a primitive mth root of unity. The quotient _{m}Br(R) of K_{2}(R) is easily calculated from the sequence:

(1.3.1)  0 → Br(R) → (ℤ/2)^{r_{1}} ⊕ ∐_{\begin{matrix}v∈S \\ finite \end{matrix}}(ℚ/ℤ) → ℚ/ℤ → 0.

Example 1.4. Let F = ℚ(ζ_{ℓ^{ν}}) and R = ℤ[ζ_{ℓ^{ν}} , 1/ℓ], where ℓ is an odd prime and ζ_{ℓ^{ν}} is a primitive ℓ^{ν}th root of unity. Then R has one finite place, and r_{1} = 0, so Br(R) = 0 via (1.3.1), and K_{2}(R)/ℓ ≅ Pic(R)/ℓ. Hence the finite groups K_{2}(ℤ[ζ_{ℓ^{ν}} , 1/ℓ]) and K_{2}(ℤ[ζ_{ℓ^{ν}}]) have ℓ-torsion if and only if ℓ is an irregular prime.

For the groups K_{n}(O_{F}), n > 2, different techniques come into play. Homological techniques were used by Quillen in [Q3] and Borel in [Bo] to prove the following result. Let r_{1} (resp., r_{2}) denote the number of real (resp., complex) embeddings of F; the resulting decomposition of F ⊗_{ℚ} ℝ shows that [F : ℚ] = r_{1} + 2r_{2}.

Theorem 1.5. (Quillen-Borel) Let F be a number field. Then the abelian groups K_{n}(O_{F}) are all finitely generated, and their ranks are given by the formula:

rank K_{n}(O_{F}) = \{ \begin{matrix}r_{1} + r_{2}, & if n ≡ 1 (mod 4); \\ r_{2}, & if n ≡ 3 (mod 4). \end{matrix}

In particular, if n > 0 is even then K_{n}(O_{F}) is a finite group. If n = 2i − 1, the rank of K_{n}(O_{F}) is the order of vanishing of the function ζ_{F} at 1 − i.

There is a localization sequence relating the K-theory of O_{F}, F and the finite fields O_{F} /℘; Soulé showed that the maps K_{n}(O_{F}) → K_{n}(F) are injections. This proves the following result.

Theorem 1.6. Let F be a number field.

a) If n > 1 is odd then K_{n}(O_{F}) ≅ K_{n}(F).

b) If n > 1 is even then K_{n}(O_{F}) is finite but K_{n}(F) is an infinite torsion group fitting into the exact sequence

0 → K_{n}(O_{F}) → K_{n}(F) → ⨁_{℘⊂O_{F}} K_{n−1}(O_{F} /℘) → 0.

For example, the groups K_{3}(O_{F}) and K_{3}(F) are isomorphic, and hence the direct sum of ℤ^{r_{2}} and a finite group. The Milnor K-group K_{3}^{M}(F) is isomorphic to (ℤ/2)^{r_{1}} by [BT], and injects into K_{3}(F) by [MS].

The following theorem was proven by Merkurjev and Suslin in [MS]. Recall that F is said to be totally imaginary if it cannot be embedded into ℝ, i.e., if r_{1} = 0 and r_{2} = [F : ℚ]/2. The positive integer w_{2}(F) is defined in section 2 below, and is always divisible by 24.

Theorem 1.7. (Structure of K_{3}F) Let F be a number field, and set w = w_{2}(F).

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

b) If F has a real embedding then K_{3}^{M}(F) ≅ (ℤ/2)^{r_{1}} is a subgroup of K_{3}(F) and:

K_{3}(F) ≅ ℤ^{r_{2}} ⊕ ℤ/(2w) ⊕ (ℤ/2)^{r_{1}−1}.

Examples 1.7.1. a) When F = ℚ we have K_{3}(ℤ) = K_{3}(ℚ) ≅ ℤ/48, because w_{2}(F) = 24. This group was first calculated by Lee and Szcarba.

b) When F = ℚ(i) we have w_{2}(F) = 24 and K_{3}(ℚ(i)) ≅ ℤ ⊕ ℤ/24.

c) When F = ℚ(\sqrt{±2}) we have w_{2}(F) = 48 because F (i) = ℚ(ζ_{8}). For these two fields, K_{3}(ℚ(\sqrt{2}) ≅ ℤ/96 ⊕ ℤ/2, while K_{3}(ℚ(\sqrt{−2}) ≅ ℤ ⊕ ℤ/48.

Classical techniques have not been able to proceed much beyond this. Although Bass and Tate showed that the Milnor K-groups K_{n}^{M}(F) are (ℤ/2)^{r_{1}} for all n ≥ 3, and hence nonzero for every real number field (one embeddable in ℝ, so that r_{1} ≠ 0), we have the following discouraging result.

Lemma 1.8. Let F be a real number field. The map K_{4}^{M}(F) → K_{4}(F) is not injective, and the map K_{n}^{M}(F) → K_{n}(F) is zero for n ≥ 5.

Proof. The map π_{1}^{s} → K_{1}(ℤ) sends η to [−1]. Since π_{∗}^{s} → K_{∗}(ℤ) is a ring homomorphism and η^{4} = 0, the Steinberg symbol {−1, −1, −1, −1} must be zero in K_{4}(ℤ). But the corresponding Milnor symbol is nonzero in K_{4}^{M}(F), because it is nonzero in K_{4}^{M}(ℝ). This proves the first assertion. Bass and Tate prove [BT] that K_{n}^{M}(F) is in the ideal generated by {−1, −1, −1, −1} for all n ≥ 5, which gives the second assertion. □

Remark 1.9. Around the turn of the century, homological calculations by Rognes [R4] and Elbaz-Vincent/Gangl/Soulé [EGS] proved that K_{4}(ℤ) = 0, K_{5}(ℤ) = ℤ, and that K_{6}(ℤ) has at most 3-torsion. These follow from a refinement of the calculations by Lee-Szczarba and Soulé in [So1] that there is no p-torsion in K_{4}(ℤ) or K_{5}(ℤ) for p > 3, together with the calculation in [RW] that there is no 2-torsion in K_{4}(ℤ), K_{5}(ℤ) or K_{6}(ℤ).

The results of Rost and Voevodsky imply that K_{7}(ℤ) ≅ ℤ/240 (see [We2]). It is still an open question whether or not K_{8}(ℤ) = 0.

§2. The e-invariant

The odd-indexed K-groups of a 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 F be a field, with separable closure F̄ and Galois group G = Gal(F̄/F). The abelian group μ of all roots of unity in F̄ is a G-module. For all i, we shall write μ(i) for the abelian group μ, made into a G-module by letting g ∈ G act as ζ ↦ g^{i}(ζ). (This modified G-module structure is called the ith Tate twist of the usual structure.) Note that the abelian group underlying μ(i) is isomorphic to ℚ/ℤ if char(F) = 0 and ℚ/ℤ[1/p] if char(F) = p ≠ 0. For each prime ℓ ≠ char(F), we write ℤ/ℓ^{∞}(i) for the ℓ-primary G-submodule of μ(i), so that μ(i) = ⊕ℤ/ℓ^{∞}(i).

For each odd n = 2i−1, Suslin proved [Su1, Su2] that the torsion subgroup of K_{2i−1}F̄ is naturally isomorphic to μ(i). It follows that there is a natural map

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

If μ(i)^{G} is a finite group, we write w_{i}(F) for its order, so that μ(i)^{G} ≅ ℤ/w_{i}(F). This is the case for all local and global fields (by 2.3.1 below). We shall call e the e-invariant, since the composition π_{2i−1}^{s} → K_{2i−1}(ℚ) ⟶^{e} ℤ/w_{i}(ℚ) is Adams’ complex e-invariant by [Q5].

The target group μ(i)^{G} is always the direct sum of its ℓ-primary Sylow subgroups ℤ/ℓ^{∞}(i)^{G}. The orders of these subgroups are determined by the roots of unity in the cyclotomic extensions F (μ_{ℓ^{ν}}). Here is the relevant definition.

Definition 2.2. Fix a prime ℓ. For any field F, we define integers w_{i}^{(ℓ)}(F) by

w_{i}^{(ℓ)}(F) = max{ℓ^{ν} | Gal(F (μ_{ℓ^{ν}})/F) has exponent dividing i}

for each integer i. If there is no maximum ν we set w_{i}^{(ℓ)}(F) = ℓ^{∞}.

Lemma 2.3. Let F be a field and set G = Gal(F̄/F). Then ℤ/ℓ^{∞}(i)^{G} is isomorphic to ℤ/w_{i}^{(ℓ)}(F). Thus the target of the e-invariant is ⨁_{ℓ} ℤ/w_{i}^{(ℓ)}(F).

Suppose in addition that w_{i}^{(ℓ)}(F) is 1 for almost all ℓ, and is finite otherwise. Then the target of the e-invariant is ℤ/w_{i}(F), where w_{i}(F) = ∏ w_{i}^{(ℓ)}(F).

Proof. Let ζ be a primitive ℓ^{ν}th root of unity. Then ζ^{⊗i} is invariant under g ∈ G (the absolute Galois group) precisely when g^{i}(ζ) = ζ, and ζ^{⊗i} is invariant under all of G precisely when the group Gal(F (μ_{ℓ^{ν}})/F) has exponent i. □

Corollary 2.3.1. Suppose that F (μ_{ℓ}) has only finitely many ℓ-primary roots of unity for all primes ℓ, and that [F (μ_{ℓ}) : F] → ∞ as ℓ → ∞. Then the w_{i}(F) are finite for all i ≠ 0.

This is the case for all local and global fields.

Proof. For fixed i ≠ 0, the formulas in 2.7 and 2.8 below show that each w_{i}^{(ℓ)} is finite, and equals one except when [F (μ_{ℓ}) : F] divides i. By assumption, this exception happens for only finitely many ℓ. Hence w_{i}(F) is finite. □

Example 2.4 (finite fields). Consider a finite field 𝔽_{q}. It is a pleasant exercise to show that w_{i}(𝔽_{q}) = q^{i} − 1 for all i. Quillen computed the K-theory of 𝔽_{q} in [Q1], showing that K_{2i}(𝔽_{q}) = 0 for i > 0 and that K_{2i−1}(𝔽_{q}) ≅ ℤ/w_{i}(𝔽_{q}). In this case, the e-invariant is an isomorphism.

The key part of the following theorem, i.e., the existence of a ℤ/w_{i} summand, was discovered in the 1975 paper [HS] by Harris and Segal; the splitting map was constructed in an ad hoc manner for number fields (see 2.5.2 below). The canonical nature of the splitting map was only established much later, in [DFM], [Ham] and [Ka2].

The summand does not always exist when ℓ = 2; for example K_{5}(ℤ) = ℤ but w_{3}(ℚ) = 2. The Harris-Segal construction fails when the Galois groups of cyclotomic field extensions are not cyclic. With this in mind, we call a field F non–exceptional if the Galois groups Gal(F (μ_{2^{ν}})/F) are cyclic for every ν, and exceptional otherwise. There are no exceptional fields of finite characteristic. 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}).

Theorem 2.5. Let R be an integrally closed domain containing 1/ℓ, and set w_{i} = w_{i}^{(ℓ)}(R). If ℓ = 2, we suppose that R is non-exceptional. Then each K_{2i−1}(R) has a canonical direct summand isomorphic to ℤ/w_{i}, 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 Harris-Segal summand of K_{2i−1}(R).

Example 2.5.1. If R contains a primitive ℓ^{ν}th root of unity ζ, we can give a simple description of the subgroup ℤ/ℓ^{ν} of the Harris-Segal summand. In this case, H_{ét}^{0}(R, μ_{ℓ^{ν}}^{⊗i}) ≅ μ_{ℓ^{ν}}^{⊗i} is isomorphic to ℤ/ℓ^{ν}, on generator ζ ⊗ ⋯ ⊗ ζ. If β ∈ K_{2}(R; ℤ/ℓ^{ν}) is the Bott element corresponding to ζ, the Bott map ℤ/ℓ^{ν} → K_{2i}(R; ℤ/ℓ^{ν}) sends 1 to β^{i}. (This multiplication is defined unless ℓ^{ν} = 2^{1}.) The Harris-Segal map, restricted to ℤ/ℓ^{ν} ⊆ ℤ/m, is just the composition

μ_{ℓ^{ν}}^{⊗i} ≅ ℤ/ℓ^{ν} ⟶^{Bott} K_{2i}(R; ℤ/ℓ^{ν}) → K_{2i−1}(R).

Remark 2.5.2. Harris and Segal [HS] originally constructed the Harris-Segal map by studying the homotopy groups of the space BN^{+}, where N is the union of the wreath products μ ⋉ Σ_{n}, μ = μ_{ℓ^{ν}}. Each wreath product embeds in GL_{n}(R[ζ_{ℓ^{ν}}]) as the group of matrices whose entries are either zero or ℓ^{ν}th roots of unity, each row and column having at most one nonzero entry. Composing with the transfer, this gives a group map N → GL(R[ζ_{ℓ^{ν}}]) → GL(R) and hence a topological map BN^{+} → GL(R)^{+}.

From a topological point of view, BN^{+} is the zeroth space of the spectrum Σ^{∞}(Bμ_{+}), and is also the K-theory space of the symmetric monoidal category of finite free μ-sets. The map of spectra underlying BN^{+} → GL(R)^{+} is obtained by taking the K-theory of the free R-module functor from finite free μ-sets to free R-modules.

Harris and Segal split this map by choosing a prime p which is primitive mod ℓ, and is a topological generator of ℤ_{ℓ}^{×}. Their argument may be interpreted as saying that if 𝔽_{q} = 𝔽_{p}[ζ_{ℓ^{ν}}] then the composite map Σ^{∞}(Bμ_{+}) → K(R) → K(𝔽_{q}) is an equivalence after KU-localization.

If F is an exceptional field, a transfer argument using F (\sqrt{−1}) shows that there is a cyclic summand in K_{2i−1}(R) whose order is either w_{i}(F), 2w_{i}(F) or w_{i}(F)/2. If F is a totally imaginary number field, we will see in 6.5 that the Harris-Segal summand is always ℤ/w_{i}(F). The following theorem, proven in section 7.5 below (see [RW]), 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 summand in K_{2i−1}(O_{F}) is 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).

Here are the formulas for the numbers w_{i}^{(ℓ)}(F), taken from [HS, p. 28], and from [W1, 6.3] when ℓ = 2. Let log_{ℓ}(n) be the maximal power of ℓ dividing n, i.e., the ℓ-adic valuation of n. By convention let log_{ℓ}(0) = ∞.

Proposition 2.7. Fix a prime ℓ ≠ 2, and let F be a field of characteristic ≠ ℓ. Let a be maximal such that F (μ_{ℓ}) contains a primitive ℓ^{a}th root of unity. Then if r = [F (μ_{ℓ}) : F] and b = log_{ℓ}(i) the numbers w_{i}^{(ℓ)} = w_{i}^{(ℓ)}(F) are:

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

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

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

Proof. Since ℓ is odd, G = Gal(F (μ_{ℓ^{a+ν}})/F) is a cyclic group of order rℓ^{ν} for all ν ≥ 0. 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}}. □

Example 2.7.1. If F = ℚ(μ_{p^{ν}}) and 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). Of course, p|w_{i}(F) for all i.

Proposition 2.8. (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. Let i be any integer, and let b = log_{2}(i). 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}.

Example 2.9 (local fields). If E is a local field, finite over ℚ_{p}, then w_{i}(E) is finite by 2.3.1. Suppose that the residue field is 𝔽_{q}. Since (for ℓ ≠ p) the number of ℓ-primary roots of unity in E(μ_{ℓ}) is the same as in 𝔽_{q}(μ_{ℓ}), we see from 2.7 and 2.8 that w_{i}(E) is w_{i}(𝔽_{q}) = q^{i} − 1 times a power of p.

If p > 2 the p-adic rational numbers ℚ_{p} have w_{i}(ℚ_{p}) = q^{i} − 1 unless (p − 1)|i; if i = (p − 1)p^{b}m (p ∤ m) then w_{i}(ℚ_{p}) = (q^{i} − 1)p^{1+b}.

For p = 2 we have w_{i}(ℚ_{2}) = 2(2^{i} − 1) for i odd, because ℚ_{2} is exceptional, and w_{i}(ℚ_{2}) = (2^{i} − 1)2^{2+b} for i even, i = 2^{b}m with m odd.

Bernoulli numbers 2.10. 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 alway 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].

Although the numerator of B_{k} is difficult to describe, Kummer’s congruences show that if p is regular it does not divide the numerator of any B_{k}/k (see [Wash, 5.14]). Thus only irregular primes can divide the numerator of B_{k}/k (see 1.1).

Remark 2.10.1. We have already remarked in 1.1 that if a prime p divides the numerator of some B_{k}/k then p divides the order of Pic(ℤ[μ_{p}]). 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. Thus the denominator of ζ(1 − 2k) is \frac{1}{2}w_{2k}(ℚ).

Remark 2.10.2. 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 parallizable manifolds; see [MSt, App.B].

From 2.10, 2.7 and 2.8 it is easy to verify the following important result.

Lemma 2.11. If i is odd then w_{i}(ℚ) = 2 and w_{i}(ℚ(\sqrt{−1})) = 4. If i = 2k is even then w_{i}(ℚ) = w_{i}(ℚ(\sqrt{−1})), and this integer is the denominator of B_{k}/4k. The prime ℓ divides w_{i}(ℚ) exactly when (ℓ − 1) divides i.

Example 2.11.1. For F = ℚ or ℚ(\sqrt{−1}), w_{2} = 24, w_{4} = 240, w_{6} = 504 = 2^{3} · 3^{2} · 7, w_{8} = 480 = 2^{5} · 3 · 5, w_{10} = 264 = 2^{3} · 3 · 11, and w_{12} = 65520 = 2^{4} · 3^{2} · 5 · 7 · 13.

The w_{i} are the orders of the Harris-Segal summands of K_{3}(ℚ[\sqrt{−1}]), K_{7}(ℚ[\sqrt{−1}]), …, K_{23}(ℚ[\sqrt{−1}]) by 2.5. In fact, we will see in 6.5 that K_{2i−1}(ℚ[\sqrt{−1}]) ≅ ℤ ⊕ ℤ/w_{i} for all i ≥ 2.

By 2.6, the orders of the Harris-Segal summands of K_{7}(ℚ), K_{15}(ℚ), K_{23}(ℚ), … are w_{4}, w_{8}, w_{12}, etc., and the orders of the Harris-Segal summands of K_{3}(ℚ), K_{11}(ℚ), K_{19}(ℚ), … are 2w_{2} = 48, 2w_{6} = 1008, 2w_{10} = 2640, etc. In fact, these summands are exactly the torsion subgroups of the groups K_{2i−1}(ℚ).

Example 2.12. The image of the natural maps π_{n}^{s} → K_{n}(ℤ) capture most of the Harris-Segal summands, and were analyzed by Quillen in [Q5]. 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 [We2].) 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 2.6. 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 [Q5], and Browder showed that the order of the summand was 2w_{i}(ℚ).

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).

Example 2.13. 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 2.7 and 2.8. 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. If n ≡ 3 (mod 4), the groups K_{n}(ℤ[ζ +ζ^{−1}]) = K_{n}(F) are finite by 1.5; the order of their Harris-Segal summands are given by theorem 2.6, and have an extra p-primary factor not detected by the image of J when n ≢ −1 (mod 2p − 2).

Birch-Tate Conjecture 2.14. 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 [Ta] conjectured 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 theorem 1.3. The two-primary part 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]), which was proven by Wiles in loc. cit. for abelian extensions of ℚ. Thus 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.

§3. Étale Chern classes

We have seen in 1.2 and 1.3 that H_{ét}^{1} and H_{ét}^{2} are related to K_{1} and K_{2}. In order to relate them to higher K-theory, it is useful to have well-behaved maps. In one direction, we use the étale Chern classes introduced in [So], but in the form found in Dwyer-Friedlander [DF].

In this section, we construct the maps in the other direction. Our formulation is due to Kahn [Ka, Ka1, Ka2]; they were introduced in [Ka1], where they were called “anti-Chern classes.” Kahn’s maps are an efficient reorganization of the constructions of Soulé [So] and Dwyer-Friedlander [DF]. Of course, there are higher Kahn maps, but we do not need them for local or global fields so we omit them here.

If F is a field containing 1/ℓ, there is a canonical map from K_{2i−1}(F; ℤ/ℓ^{ν}) to H_{ét}^{1}(F, μ_{ℓ^{ν}}^{⊗i}), called the first étale Chern class. It is the composition of the map to the étale K-group K_{2i−1}^{ét}(F; ℤ/ℓ^{ν}) followed by the edge map in the Atiyah-Hirzebruch spectral sequence for étale K-theory [DF]. For i = 1 it is the Kummer isomorphism from K_{1}(F; ℤ/ℓ^{ν}) = F^{×}/F^{×ℓ^{ν}} to H_{ét}^{1}(F, μ_{ℓ^{ν}}).

For each i and ν, we can construct a splitting of the first étale Chern class, at least if ℓ is odd (or ℓ = 2 and F is non-exceptional). Let F_{ν} denote the smallest field extension of F over which the Galois module μ_{ℓ^{ν}}^{⊗i−1} is trivial, and let Γ_{ν} denote the Galois group of F_{ν} over F. Kahn proved in [Ka] that the transfer map induces an isomorphism

H_{ét}^{1}(F_{ν}, μ_{ℓ^{ν}}^{⊗i})_{Γ_{ν}} ⟶^{≅} H_{ét}^{1}(F, μ_{ℓ^{ν}}^{⊗i}). Note that because H_{ét}^{1}(F_{ν}, μ_{ℓ^{ν}}) ≅ F_{ν}^{×}/ℓ^{ν} we have an

isomorphism of Γ_{ν}-modules H_{ét}^{1}(F_{ν}, μ_{ℓ^{ν}}^{⊗i}) ≅ (F_{ν}^{×}) ⊗ μ_{ℓ^{ν}}^{⊗i−1}.

Definition 3.1. The Kahn map H_{ét}^{1}(F, μ_{ℓ^{ν}}^{⊗i}) → K_{2i−1}(F; ℤ/ℓ^{ν}) is the composition

H_{ét}^{1}(F, μ_{ℓ^{ν}}^{⊗i}) ⟵^{≅} H_{ét}^{1}(F_{ν}, μ_{ℓ^{ν}}^{⊗i})_{Γ_{ν}} =[F_{ν}^{×} ⊗ μ_{ℓ^{ν}}^{⊗i−1}]_{Γ_{ν}} ⟶^{Harris-Segal}
→ [(F_{ν}^{×}) ⊗ K_{2i−2}(F_{ν}; ℤ/ℓ^{ν})]_{Γ_{ν}} ⟶^{∪}K_{2i−1}(F_{ν}; ℤ/ℓ^{ν})_{Γ_{ν}} ⟶^{transfer} K_{2i−1}(F; ℤ/ℓ^{ν}).

Compatibility 3.1.1. Let F be the quotient field of a discrete valuation ring whose residue field k contains 1/ℓ. Then the Kahn map is compatible with the Harris-Segal map in the sense that for m = ℓ^{ν} the diagram commutes.

H_{ét}^{1}(F, μ_{m}^{⊗i}) ⟶^{∂} H_{ét}^{0}(k, μ_{m}^{⊗i−1})
Kahn↓  ↓^{Harris-Segal}
K_{2i−1}(F; ℤ/m) ⟶^{∂} K_{2i−2}(k; ℤ/m)

To see this, one immediately reduces to the case F = F_{ν}. In this case, the Kahn map is the Harris-Segal map, tensored with the identification H_{ét}^{1}(F, μ_{m}) ≅ F^{×}/m, and both maps ∂ amount to the reduction mod m of the valuation map F^{×} → ℤ.

Theorem 3.2. Let F be a field containing 1/ℓ. If ℓ = 2 we suppose that F is nonexceptional. Then for each i the Kahn map H_{ét}^{1}(F, μ_{ℓ^{ν}}^{⊗i}) → K_{2i−1}(F; ℤ/ℓ^{ν}) is an injection, split by the first étale Chern class.

The Kahn maps are compatible with change of coefficients. Hence it induces maps H_{ét}^{1}(F, ℤ_{ℓ}(i)) → K_{2i−1}(F; ℤ_{ℓ}) and H_{ét}^{1}(F, ℤ/ℓ^{∞}(i)) → K_{2i−1}(F; ℤ/ℓ^{∞}).

Proof. When ℓ is odd (or ℓ = 2 and \sqrt{−1} ∈ F), the proof that the Kahn map splits the étale Chern class is given in [Ka1], and is essentially a reorganization of Soulé’s proof in [So] that the first étale Chern class is a surjection. up to factorials. (Cf. [DF]). When ℓ = 2 and F is non-exceptional, Kahn proves in [Ka2] that this map is a split injection. □

Corollary 3.3. Let O_{S} be a ring of S-integers in a number field F, with 1/ℓ ∈ O_{S}. If ℓ = 2, assume that F is non-exceptional. Then the Kahn maps for F induce injections H_{ét}^{1}(O_{S}, μ_{ℓ^{ν}}^{⊗i}) → K_{2i−1}(O_{S}; ℤ/ℓ^{ν}), split by the first étale Chern class.

Proof. Since H_{ét}^{1}(O_{S}, μ_{ℓ^{ν}}^{⊗i}) is the kernel of H_{ét}^{1}(F, μ_{ℓ^{ν}}^{⊗i}) → ⊕_{℘}H_{ét}^{0}(k(℘), μ_{m}^{⊗i−1}), and K_{2i−1}(O_{S}; ℤ/ℓ^{ν}) is the kernel of K_{2i−1}(F; ℤ/ℓ^{ν}) → ⊕_{℘}K_{2i−2}(k(℘); ℤ/ℓ^{ν}) by 1.6, this follows formally from 3.1.1. □

Example 3.4. If F is a number field, the first étale Chern class detects the torsionfree part of K_{2i−1}(O_{F}) = K_{2i−1}(F) described in 1.5. In fact, it induces isomorphisms K_{2i−1}(O_{S}) ⊗ ℚ_{ℓ} ≅ K_{2i−1}^{ét}(O_{S}; ℚ_{ℓ}) ≅ H_{ét}^{1}(O_{S}, ℚ_{ℓ}(i)).

To see this, choose S to contain all places over some odd prime ℓ. Then 1/ℓ ∈ O_{S}, and K_{2i−1}(O_{S}) ≅ K_{2i−1}(F). A theorem of Tate states that

rank H_{ét}^{1}(O_{S}, ℚ_{ℓ}(i)) − rank H_{ét}^{2}(O_{S}, ℚ_{ℓ}(i)) = \{ \begin{matrix}r_{2}, & i even; \\ r_{1} + r_{2}, & i odd. \end{matrix}

We will see in 3.8 below that H_{ét}^{2}(O_{S}, ℚ_{ℓ}(i)) = 0. Comparing with 1.5, we see that the source and target of the first étale Chern class

K_{2i−1}(O_{S}) ⊗ ℤ_{ℓ} → K_{2i−1}^{ét}(O_{S}; ℤ_{ℓ}) ≅ H_{ét}^{1}(O_{S}, ℤ_{ℓ}(i))

have the same rank. By 3.2, this map is a split surjection (split by the Kahn map), whence the claim.

The second étale Chern class is constructed in a similar fashion. Assuming that ℓ is odd, or that ℓ = 2 and F is non-exceptional, so that the e-invariant splits by 2.5, then for i ≥ 1 there is also a canonical map

K_{2i}(F; ℤ/ℓ^{ν}) → H_{ét}^{2}(F, μ_{ℓ^{ν}}^{⊗i+1}),

called the second étale Chern class. It is the composition of the map to the étale K-group K_{2i}^{ét}(F; ℤ/ℓ^{ν}), or rather to the kernel of the edge map K_{2i}^{ét}(F; ℤ/ℓ^{ν}) → H_{ét}^{0}(F, μ_{ℓ^{ν}}^{⊗i}), followed by the secondary edge map in the Atiyah-Hirzebruch spectral sequence for étale K-theory [DF].

Even if ℓ = 2 and F is exceptional, this composition will define a family of second étale Chern classes K_{2i}(F) → H_{ét}^{2}(F, μ_{ℓ^{ν}}^{⊗i+1}) and hence K_{2i}(F) → H_{ét}^{2}(F, ℤ_{ℓ}(i + 1)). This is because the e-invariant (2.1) factors through the map K_{2i}(F; ℤ/ℓ^{ν}) → K_{2i−1}(F).

For i = 1, the second étale Chern class K_{2}(F)/m → H_{ét}^{2}(F, μ_{m}^{⊗2}) is just Tate’s map, described in 1.3; it is an isomorphism for all F by the Merkurev-Suslin theorem.

Using this case, Kahn proved in [Ka] that the transfer always induces an isomorphism H_{ét}^{2}(F_{ν}, μ_{ℓ^{ν}}^{⊗i})_{Γ_{ν}} ⟶^{≅} H_{ét}^{2}(F, μ_{ℓ^{ν}}^{⊗i}). Here, F_{ν} and Γ_{ν} = Gal(F_{ν}/F) are as in 3.1 above, and if ℓ = 2 we assume that F is non-exceptional. As before, we have an isomorphism of Γ_{ν}-modules H_{ét}^{2}(F_{ν}, μ_{ℓ^{ν}}^{⊗i+1}) ≅ K_{2}(F_{ν}) ⊗ μ_{ℓ^{ν}}^{⊗i−1}.

Definition 3.5. The Kahn map H_{ét}^{2}(F, μ_{ℓ^{ν}}^{⊗i+1}) → K_{2i}(F; ℤ/ℓ^{ν}) is the composition

H_{ét}^{2}(F, μ_{ℓ^{ν}}^{⊗i+1}) ⟵^{≅} H_{ét}^{2}(F_{ν}, μ_{ℓ^{ν}}^{⊗i+1})_{Γ_{ν}} =[K_{2}(F_{ν}) ⊗ μ_{ℓ^{ν}}^{⊗i−1}]_{Γ_{ν}} ⟶^{Harris-Segal}
→ [K_{2}(F_{ν}) ⊗ K_{2i−2}(F_{ν}; ℤ/ℓ^{ν})]_{Γ_{ν}} ⟶^{∪}K_{2i}(F_{ν}; ℤ/ℓ^{ν})_{Γ_{ν}} ⟶^{transfer} K_{2i}(F; ℤ/ℓ^{ν}).

Compatibility 3.5.1. Let F be the quotient field of a discrete valuation ring whose residue field k contains 1/ℓ. Then the first and second Kahn maps are compatible with the maps ∂, from H_{ét}^{2}(F) to H_{ét}^{1}(k) and from K_{2i}(F; ℤ/m) to K_{2i−1}(k; ℤ/m). The argument here is the same as for 3.1.1.

As with 3.2, the following theorem was proven in [Ka1, Ka2].

Theorem 3.6. Let F be a field containing 1/ℓ. If ℓ = 2 we suppose that F is nonexceptional. Then for each i ≥ 1 the Kahn map H_{ét}^{2}(F, μ_{ℓ^{ν}}^{⊗i+1}) → K_{2i}(F; ℤ/ℓ^{ν}) is an injection, split by the second étale Chern class.

The Kahn map is compatible with change of coefficients. Hence it induces maps H_{ét}^{2}(F, ℤ_{ℓ}(i + 1)) → K_{2i}(F; ℤ_{ℓ}) and H_{ét}^{2}(F, ℤ/ℓ^{∞}(i + 1)) → K_{2i}(F; ℤ/ℓ^{∞}).

Corollary 3.7. Let O_{S} be a ring of S-integers in a number field F, with 1/ℓ ∈ O_{S}. If ℓ = 2, assume that F is non-exceptional. Then for each i > 0, the Kahn maps induce injections H_{ét}^{2}(O_{S}, ℤ_{ℓ}(i + 1)) → K_{2i}(O_{S}; ℤ_{ℓ}), split by the second étale Chern class.

Proof. Since H_{ét}^{2}(O_{S}, ℤ_{ℓ}(i + 1)) is the kernel of H_{ét}^{2}(F, ℤ_{ℓ}(i + 1)) → ⊕_{℘}H_{ét}^{1}(k(℘), ℤ_{ℓ}(i)), and K_{2i}(O_{S}; ℤ_{ℓ}) is the kernel of K_{2i}(F; ℤ_{ℓ}) → ⊕_{℘}K_{2i−1}(k(℘); ℤ_{ℓ}), this follows formally from 3.5.1. □

Remark 3.7.1. For each ν, H_{ét}^{2}(O_{S}, μ_{ℓ^{ν}}^{⊗i+1}) → K_{2i}(O_{S}; ℤ/ℓ^{ν}) is also a split surjection, essentially because the map H_{ét}^{2}(O_{S}, ℤ_{ℓ}(i+1)) → H_{ét}^{2}(O_{S}, μ_{ℓ^{ν}}^{⊗i+1}) is onto; see [Ka1, 5.2].

The summand H_{ét}^{2}(O_{S}, ℤ_{ℓ}(i)) is finite by the following calculation.

Proposition 3.8. Let O_{S} be a ring of S-integers in a number field F with 1/ℓ ∈ O_{S}. Then for all i ≥ 2, H_{ét}^{2}(O_{S}, ℤ_{ℓ}(i)) is a finite group, and H_{ét}^{2}(O_{S}, ℚ_{ℓ}(i)) = 0.

Finally, H_{ét}^{2}(O_{S}, ℤ/ℓ^{∞}(i)) = 0 if ℓ is odd, or if ℓ = 2 and F is totally imaginary.

Proof. If ℓ is odd or if ℓ = 2 and F is totally imaginary, then H_{ét}^{3}(O_{S}, ℤ_{ℓ}(i)) = 0 by [Se], so H_{ét}^{2}(O_{S}, ℤ/ℓ^{∞}(i)) is a quotient of H_{ét}^{2}(O_{S}, ℚ_{ℓ}(i)). Since H_{ét}^{2}(R, ℚ_{ℓ}(i)) = H_{ét}^{2}(R, ℤ_{ℓ}(i)) ⊗ ℚ, it suffices to prove the first assertion for i > 0. But H_{ét}^{2}(O_{S}, ℤ_{ℓ}(i)) is a summand of K_{2i−2}(O_{S}) ⊗ ℤ_{ℓ} for i ≥ 2 by 3.7, which is a finite group by theorem 1.5.

If ℓ = 2 and F is exceptional, the usual transfer argument for O_{S} ⊂ O_{S′} ⊂ F (\sqrt{−1}) shows that the kernel A of H_{ét}^{2}(O_{S}, ℤ_{2}(i)) → H_{ét}^{2}(O_{S′}, ℤ_{2}(i)) has exponent 2. Since A must inject into the finite group H_{ét}^{2}(O_{S}, μ_{2}), A must also be finite. Hence H_{ét}^{2}(O_{S}, ℤ_{2}(i)) is also finite, and H_{ét}^{2}(R, ℚ_{ℓ}(i)) = H_{ét}^{2}(R, ℤ_{ℓ}(i)) ⊗ ℚ = 0. □

Taking the direct limit over all finite S yields:

Corollary 3.8.1. Let F be a number field. Then H_{ét}^{2}(F, ℤ/ℓ^{∞}(i)) = 0 for all odd primes ℓ and all i ≥ 2.

Example 3.8.2. The Main Conjecture of Iwasawa Theory, proved by Mazur and Wiles [MW], implies that (for odd ℓ) the order of the finite group H_{ét}^{2}(ℤ[1/ℓ], ℤ_{ℓ}(2k)) is the ℓ-primary part of the numerator of ζ_{ℚ}(1 − 2k). See for example [RW, Appendix A] or [KNF, 4.2 and 6.3]. Note that by Euler’s formula 2.10.1 this is also the ℓ-primary part of the numerator of B_{k}/2k, where B_{k} is the Bernoulli number discussed in 2.10.

Real number fields 3.8.3. If ℓ = 2, the vanishing conclusion of 3.8.1 still holds when F is totally imaginary. However, it fails when F has r_{1} > 0 embeddings into ℝ:

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

One way to do this computation is to observe that, by 3.8, H^{2}(O_{S}; ℤ/2^{∞}(i)) has exponent 2. Hence the Kummer sequence is:

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

Now plug in the values of the right two groups, which are known by Tate-Poitou duality: H^{3}(O_{S}; ℤ/2) ≅ (ℤ/2)^{r_{1}}, while H^{3}(O_{S}; ℤ/2^{∞}(i)) is: (ℤ/2)^{r_{1}} for i even, and 0 for i odd.

3.8.4. Suppose that F is totally real (r_{2} = 0), and set w_{i} = w_{i}^{(ℓ)}(F). If i > 0 is even then H^{1}(O_{S}, ℤ_{ℓ}(i)) ≅ ℤ/w_{i}; this group is finite. If i is odd then H^{1}(O_{S}, ℤ_{ℓ}(i)) ≅ ℤ_{ℓ}^{r_{1}} ⊕ℤ/w_{i}; this is infinite. These facts may be obtained by combining the rank calculations of 3.4 and 3.8 with (2.1) and universal coefficients.

Theorem 3.9. For every number field F, and all i, the Adams operation ψ^{k} acts on K_{2i−1}(F) ⊗ ℚ as multiplication by k^{i}.

Proof. The case i = 1 is well known, so we assume that i ≥ 2. If S contains all places over some odd prime ℓ we saw in 3.4 that K_{2i−1}(O_{S}) ⊗ ℚ_{ℓ} ≅ K_{2i−1}^{ét}(O_{S}; ℚ_{ℓ}) ≅ H_{ét}^{1}(O_{S}, ℚ_{ℓ}(i)). Since this isomorphism commutes with the Adams operations, and Soulé has shown in [So3] the ψ^{k} = k^{i} on H_{ét}^{1}(O_{S}, ℚ_{ℓ}(i)), the same must be true on K_{2i−1}(O_{S}) ⊗ ℚ_{ℓ} = K_{2i−1}(F) ⊗ ℚ_{ℓ}. □

§4. Global fields of finite characteristic

A global field of finite characteristic p is a finitely generated field F 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 a 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.

The group K_{0}(X) = ℤ ⊕ Pic(X) is finitely generated of rank two by a theorem of Weil. In fact, there is a finite group J(X) such that Pic(X) ≅ ℤ ⊕ J(X). For K_{1} and K_{2}(X), the localization sequence of Quillen [Q2] implies that there is an exact sequence

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

By classical Weil reciprocity, the cokernel of ∂ is 𝔽_{q}^{×}, so K_{1}(X) ≅ 𝔽_{q}^{×} × 𝔽_{q}^{×}. Bass and Tate proved in [BT] that 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].

Theorem 4.1. Let X be a smooth projective curve over a finite field of characteristic p. For n ≥ 1, the groups K_{n}(X) are finite groups of order prime to p.

Proof. Tate proved that K_{n}^{M}(F) = 0 for all n ≥ 3. By Geisser and Levine’s theorem [GL], 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)

and a diagram chase, it follows that K_{n}(X) is uniquely p-divisible. Now Quillen proved in [GQ] that the groups K_{n}(X) are finitely generated abelian groups. A second diagram chase shows that the groups K_{n}(X) must be finite. □

Corollary 4.2. 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. Classically, K_{1}(R) = R^{×} ⊕ SK_{1}(R) and the units of R are well known. The computation that SK_{1}(R) = 0 is proven in [BMS]. The rest follows from the localization sequence K_{n}(X) → K_{n}(X′) → ⊕_{x∈S}K_{n−1}(x). □

The e-invariant 4.3. 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.

Classically, 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 or not 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 4.1. The following theorem determines the abelian group structure of the K_{n}(X̄) as well as the action of the Galois group on them. It depends upon Suslin’s theorem (see [Su4]) that for i ≥ 1 and ℓ ≠ p the groups H_{M}^{n}(X̄, ℤ/ℓ^{∞}(i)) equal the groups H_{ét}^{n}(X̄, ℤ/ℓ^{∞}(i)).

Theorem 4.4. Let X be a smooth projective curve over 𝔽_{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 shows that K_{n}(X̄; ℤ/ℓ^{∞}) is isomorphic to the ℓ-primary 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 (0.5); by Suslin’s theorem, the terms E_{2}^{p,q} vanish for q < 0 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_{ét}^{0}(X̄, ℤ/ℓ^{∞}(i)) = ℤ/ℓ^{∞}(i) of 4.3, and are therefore split surjections. Finally, we note that as Galois modules we have H_{ét}^{1}(X̄, ℤ/ℓ^{∞}(i)) ≅ J(X̄)_{ℓ}(i − 1), and (by Poincaré Duality [Mi, V.2]) H_{ét}^{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 4.4 and 2.4 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 algebraic Frobenius φ^{∗} (induced by φ^{−1}) on cohomology.

Example 4.5. φ^{∗} acts trivially on H_{ét}^{0}(X̄, ℚ_{ℓ}) = ℚ_{ℓ} and H_{ét}^{2}(X̄, ℚ_{ℓ}(1)) = ℚ_{ℓ}. It acts as q^{−i} on the twisted groups H_{ét}^{0}(X̄, ℚ_{ℓ}(i)) and H_{ét}^{2}(X̄, ℚ_{ℓ}(i + 1).

Weil’s proof in 1948 of the Riemann Hypothesis for Curves implies that the eigenvalues of φ^{∗} acting on H_{ét}^{1}(X̄, ℚ_{ℓ}(i)) have absolute value q^{1/2−i}.

Since H_{ét}^{n}(X, ℚ_{ℓ}(i)) ≅ H_{ét}^{n}(X̄, ℚ_{ℓ}(i))^{G}, a perusal of these cases shows that we have H_{ét}^{n}(X, ℚ_{ℓ}(i)) = 0 except when (n, i) is (0, 0) or (2, 1).

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

(4.5.1)  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 4.6. For a smooth projective curve X over 𝔽_{q}, ℓ ∤ q and i ≥ 2 we have: (1) H_{ét}^{n+1}(X, ℤ_{ℓ}(i)) ≅ H_{ét}^{n}(X, ℤ/ℓ^{∞}(i)) ≅ H_{ét}^{n}(X̄, ℤ/ℓ^{∞}(i))^{G} for all n;

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

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

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

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

Proof. Since i ≥ 2, we see from 4.5 that H_{ét}^{n}(X, ℚ_{ℓ}(i)) = 0. Since ℚ_{ℓ}/ℤ_{ℓ} = ℤ/ℓ^{∞}, this yields H_{ét}^{n}(X, ℤ/ℓ^{∞}(i)) ≅ H_{ét}^{n+1}(X, ℤ_{ℓ}(i)) for all n.

Since each H^{n} = H_{ét}^{n}(X̄, ℤ/ℓ^{∞}(i)) is a quotient of H_{ét}^{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 for X̄ → X collapses for i > 1 to yield exact sequences

(4.6.1)  0 → H_{ét}^{n}(X, ℤ/ℓ^{∞}(i)) → H_{ét}^{n}(X̄, ℤ/ℓ^{∞}(i)) ⟶^{φ^{∗}−1} H_{ét}^{n}(X̄, ℤ/ℓ^{∞}(i)) → 0.

In particular, H_{ét}^{n}(X, ℤ/ℓ^{∞}(i)) = 0 for n>2. Since H_{ét}^{2}(X̄, ℤ/ℓ^{∞}(i)) ≅ ℤ/ℓ^{∞}(i − 1) this yields H_{ét}^{2}(X, ℤ/ℓ^{∞}(i)) ≅ ℤ/ℓ^{∞}(i − 1)^{G} = ℤ/w_{i−1}. We also see that H_{ét}^{1}(X, ℤ/ℓ^{∞}(i)) is the group of invariants of the Frobenius, i.e., J(X)_{ℓ}(i − 1)^{φ^{∗}}. □

Given the calculation of K_{n}(X̄)^{G} in 4.4 and the calculation of H_{ét}^{n}(X, ℤ/ℓ^{∞}(i)) in 4.6, we see that the natural map K_{n}(X) → K_{n}(X̄)^{G} is a surjection, split by the Kahn maps 3.2 and 3.6. Thus the real content of the following theorem is that K_{n}(X) → K_{n}(X̄)^{G} is an isomorphism.

Theorem 4.7. 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 4.1. 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 (0.5), which degenerates by 4.6. □

The Zeta Function 4.8. 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 (4.5 above), given Grothendieck’s formula P (t) = det(1 − φ^{∗}t), where φ^{∗} is regarded as an endomorphism of H_{ét}^{1}(X̄; ℚ_{ℓ}). Note that by 4.5 the action of φ^{∗} on H_{ét}^{0}(X̄; ℚ_{ℓ}) has det(1 − φ^{∗}t) = (1 − t), and the action on H_{ét}^{2}(X̄; ℚ_{ℓ}) has det(1 − φ^{∗}t) = (1 − qt).

Here is application of theorem 4.7, which goes back to Thomason (see [Th, (4.7)] and [Li3]). Let #A denote the order of a finite abelian group A.

Corollary 4.9. 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_{ét}^{2}(X; ℤ_{ℓ}(i))}{#H_{ét}^{1}(X; ℤ_{ℓ}(i)) · #H_{ét}^{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 4.6 and 4.7. The second equality follows from the formula for ζ_{X}(1 − i) in 4.8. □

Iwasawa modules 4.10. It is worth noting that the group H_{ét}^{1}(X, ℤ/ℓ^{∞}(i)) is the (finite) group of invariants M^{#}(i)^{φ^{∗}} of the ith twist of the Pontrjagin dual M^{#} of the Iwasawa module M = M_{X}. By definition M_{X} is the Galois group of X̂ over X_{∞} = X ⊗_{𝔽_{q}} F_{q}(∞), where the field F_{q}(∞) is obtained from 𝔽_{q} by adding all ℓ-primary roots of unity, and X̂ is the maximal unramified pro-ℓ abelian cover of X_{∞}. It is known that the Iwasawa module M_{X} is a finitely generated free ℤ_{ℓ}-module, and that its dual M^{#} is a finite direct sum of copies of ℤ/ℓ^{∞} [DM, 3.22]. This viewpoint was developed in [DM], and the corresponding discussion of Iwasawa modules for number fields is in [MKH].

§5. Local Fields

Let E be a local field of residue characteristic p, with (discrete) valuation ring V and residue field 𝔽_{q}. It is well known 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 free ℤ_{p}-module.

In the equi-characteristic case, where char(E) = p, it is well known that V ≅ 𝔽_{q}[[π]] and E = 𝔽_{q}((π)) [Se], so μ(E) = 𝔽_{q}^{×}, and U_{1} = W (𝔽_{q}) has rank [𝔽_{q} : 𝔽_{p}] over ℤ_{p} = W (𝔽_{p}). The decomposition of K_{1}(V) = V^{×} is evident here. Here is a description of the abelian group structure on K_{n}(V) for n > 1.

Theorem 5.1. 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} so that

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

Proof. The map K_{n−1}(𝔽_{q}) → K_{n}(E) sending x to {x, π} splits the localization sequence, yielding the decomposition of K_{n}(E). If U_{n} denotes the kernel of the canonical map K_{n}(V) → K_{n}(𝔽_{q}), then naturality yields K_{n}(V) = U_{n} ⊕ K_{n}(𝔽_{q}). By Gabber’s rigidity theorem [Gab], U_{n} is uniquely ℓ-divisible for ℓ ≠ p and n > 0. It suffices to show that U_{n} is uncountable and uniquely p-divisible when n ≥ 2.

Tate showed that the Milnor groups K_{n}^{M}(E) are uncountable, uniquely divisible for n ≥ 3, and that the same is true for the kernel U_{2} of the norm residue map K_{2}(E) → μ(E); see [Ta2]. If n ≥ 2 then K_{n}^{M}(E) is a summand of the Quillen K-group K_{n}(E) by [Su3]. On the other hand, Geisser and Levine proved in [GL] that the complementary summand is uniquely p-divisible. □

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 by the convergent power series for x ↦ ln(x).

The group μ(E) of roots of unity in E (or V) is identified with (𝔽_{q}^{∗}) × μ_{p^{∞}}(E), where the first factor arises from Teichmüller’s theorem that V^{×} → 𝔽_{q}^{×} ≅ ℤ/(q − 1) has a unique splitting, and μ_{p^{∞}}(E) denotes the finite group of p-primary roots of unity in E. There seems to be no simple formula for the order of the cyclic p-group μ_{p^{∞}}(E)

For K_{2}, there is a norm residue symbol K_{2}(E) → μ(E) and we have the following result; see [WK, III.6.6].

Moore’s Theorem 5.2. The group K_{2}(E) is the product of a finite group, isomorphic to μ(E), and an uncountable, uniquely divisible abelian group U_{2}. In addition,

K_{2}(V) ≅ μ_{p^{∞}}(E) × U_{2}.

Proof. The fact that the kernel U_{2} of the norm residue map is divisible is due to C. Moore, and is given in the Appendix to [Mil]. The fact that U_{2} is torsionfree (hence uniquely divisible) was proven by Tate [Ta2] when char(F) = p, and by Merkurjev [Merk] when char(F) = 0. □

Since the transcendence degree of E over ℚ is uncountable, it follows from Moore’s theorem and the arguments in [Mil] that the Milnor K-groups K_{n}^{M}(E) are uncountable, uniquely divisible abelian groups for n ≥ 3. By [Su3], this is a summand of the Quillen K-group K_{n}(E). As in the equicharacteristic case, K_{n}(E) will contain an uncountable uniquely divisible summand about which we can say very little.

To understand the other factor, we typically proceed a prime at a time. This has the advantage of picking up the torsion subgroups of K_{n}(E), and detecting the groups K_{n}(V)/ℓ. For p-adic fields, the following calculation reduces the problem to the prime p.

Proposition 5.3. If i > 0 there is a summand of K_{2i−1}(V) ≅ K_{2i−1}(E) isomorphic to K_{2i−1}(𝔽_{q}) ≅ ℤ/(q^{i} − 1), detected by the e-invariant. The complementary summand is uniquely ℓ-divisible for every prime ℓ ≠ p, i.e., a ℤ_{(p)}-module.

There is also a decomposition K_{2i}(E) ≅ K_{2i}(V) ⊕ K_{2i−1}(𝔽_{q}), and the group K_{2i}(V) is uniquely ℓ-divisible for every prime ℓ ≠ p, i.e., a ℤ_{(p)}-module.

Proof. Pick a prime ℓ. By Gabber’s rigidity theorem [Gab], the groups K_{n}(V; ℤ/ℓ^{ν}) are isomorphic to K_{n}(𝔽_{q}; ℤ/ℓ^{ν}) for n > 0. Since the Bockstein spectral sequences are isomorphic, and detect all finite cyclic ℓ-primary summands of K_{n}(V) and K_{n}(𝔽_{q}) [WK, 5.9.12], it follows that K_{2i−1}(V) has a cyclic summand isomorphic to ℤ/w_{i}^{(ℓ)}(E), and that the complement is uniquely ℓ-divisible. Since K_{n}(V; ℤ/ℓ) ≅ ℤ/ℓ, we also see that K_{2i}(V) is uniquely ℓ-divisible. As ℓ varies, we get a cyclic summand of order w_{i}(E) in K_{2i−1}(V) whose complement is a ℤ_{(p)}-module.

If x ∈ K_{2i−1}(V), the product {x, π} ∈ K_{2i}(E) maps to the image of x in K_{2i−1}(𝔽_{q}) under the boundary map ∂ in the localization sequence. Hence the summand of K_{2i−1}(V) isomorphic to K_{2i−1}(𝔽_{q}) lifts to a summand of K_{2i}(E). This breaks the localization sequence up into split short exact sequences 0→K_{n}(V)→K_{n}(E)→K_{n−1}(𝔽_{q})→ 0. □

Completed K-theory 5.4. It will be convenient to fix a prime ℓ and pass to the ℓ-adic completion K̂(R) of the K-theory space K(R), where R is any ring. We also write K_{n}(R; ℤ_{ℓ}) for π_{n}K̂(R). Information about these groups tells us about the groups K_{n}(R, ℤ/ℓ^{ν}) = π_{n}(K(R); ℤ/ℓ^{ν}), because these groups are isomorphic to π_{n}(K̂(R); ℤ/ℓ^{ν}) for all ν.

If the groups K_{n}(R; ℤ/ℓ^{ν}) are finite, then K_{n}(R; ℤ_{ℓ}) is an extension of the Tate module of K_{n−1}(R) by the ℓ-adic completion of K_{n}(R). (The Tate module of an abelian group A is the inverse limit of the groups Hom(ℤ/ℓ^{ν}, A).) For example, K_{n}(ℂ; ℤ_{ℓ}) vanishes for odd n and for even n equals the Tate module ℤ_{ℓ} of K_{n−1}(ℂ). If in addition the abelian groups K_{n}(R) are finitely generated, there can be no Tate module and we have K_{n}(R; ℤ_{ℓ}) ≅ K_{n}(R) ⊗_{ℤ} ℤ_{ℓ} ≅ \underleftarrow{lim} K_{n}(R; ℤ/ℓ^{ν}).

Warning 5.4.1. Even if we know K_{n}(R; ℤ_{ℓ}) for all primes, we may not still be able to determine the underlying abelian group K_{n}(R) exactly from this information. For example, consider the case n = 1, R = ℤ_{p}. We know that K_{1}(R; ℤ_{p}) ≅ (1 + pR)^{×} ≅ ℤ_{p}, p ≠ 2, but this information does not even tell us that K_{1}(R) ⊗ ℤ_{(p)} ≅ ℤ_{p}. To see why, note that the extension 0 → ℤ_{(p)} → ℤ_{p} → ℤ_{p}/ℤ_{(p)} → 0 doesn’t split; there are no p-divisible elements in ℤ_{p}, yet ℤ_{p}/ℤ_{(p)} ≅ ℚ_{p}/ℚ is a uniquely divisible abelian group.

We now consider the p-adic completion of K(E). By 5.3, it suffices to consider the p-adic completion of K(V).

Write w_{i} for the numbers w_{i} = w_{i}^{(p)}(E), which were described in 2.9. For all i, and ℓ^{ν} > w_{i}, the étale cohomology group H^{1}(E, μ_{p^{ν}}^{⊗i}) is isomorphic to (ℤ/p^{ν})^{d} ⊕ ℤ/w_{i} ⊕ ℤ/w_{i−1}, d = [E : ℚ_{p}]. By duality, the group H^{2}(E, μ_{p^{ν}}^{⊗i+1}) is also isomorphic to ℤ/w_{i}.

Theorem 5.5. 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} \}

Moreover, 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.

Finally, each K_{2i}(V) is the direct sum of a uniquely divisible group, a divisible p-group and a subgroup isomorphic to ℤ/w_{i}^{(p)}(E).

Proof. If p > 2 the first part is proven in [HM, thm. A]. (It also follows from the spectral sequence (0.3) for E, using the Voevodsky-Rost theorem.) In this case, theorem 3.2 and a count shows that the étale Chern classes K_{2i−1}(E; ℤ/p^{ν}) → H_{ét}^{1}(E; μ_{p^{ν}}^{⊗i}) are isomorphisms. If p = 2 this is proven in [RW, 1.12]; surprisingly, this implies that the Harris-Segal maps and Kahn maps are even defined when E is an exceptional 2-adic field.

Now fix i and set w_{i} = w_{i}^{(p)}(E). Since the Tate module of any abelian group is torsionfree, and K_{2i}(E; ℤ_{p}) is finite, we see that the Tate module of K_{2i−1}(E) vanishes and the p-adic completion of K_{2i}(E) is ℤ/w_{i}. Since this is also the completion of the ℤ_{(p)}-module K_{2i}(V) by 5.3, the decomposition follows from the structure of ℤ_{(p)}- modules. (This decomposition was first observed in [Ka1, 6.2].) □

Remark 5.5.1. The fact that these groups were finitely generated ℤ_{p}-modules of rank d was first obtained by Wagoner in [Wg], modulo the identification in [Pa] of Wagoner’s continuous K-groups with K_{∗}(E; ℤ/p).

Unfortunately, I do not know how to reconstruct the “integral” homotopy groups K_{n}(V) from the information in 5.5. 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). Here are some cases when I can show that they come from torsionfree elements; I do not know any example where a ℤ/p^{∞} appears.

Corollary 5.6. 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} ⊕ ℤ/w_{2}^{(p)}.

Proof. Combine 5.5 with Moore’s theorem 5.2 and 5.3. □

I doubt that the extension 0 → ℤ_{(p)}^{d} → K_{3}(V) → U_{3} → 0 splits.

Example 5.7. If k > 0, K_{4k+1}(ℤ_{2}) contains a subgroup T_{k} isomorphic to ℤ_{(2)} × ℤ/w_{i}, and the quotient K_{4k+1}(ℤ_{2})/T_{k} is uniquely divisible. (By 2.9, w_{i} = 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 [RW].) Since this map factors through K_{4k+1}(ℤ_{2}), the assertion follows.

Example 5.8. Let F be a totally imaginary number field of degree d = 2r_{2} over ℚ, and let E_{1}, ..., E_{s} be the completions of F at the prime ideals over p. There is a subgroup of K_{2i−1}(F) isomorphic to ℤ^{r_{2}} by theorem 1.5; 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 d = ∑[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 5.9. 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. For k > 0, there is a subgroup of K_{4k+1}(F) isomorphic to ℤ^{d} by theorem 1.5; its image in ⊕K_{4k+1}(E_{j}) is a subgroup of rank d, while ⊕K_{4k+1}(E_{j}; ℤ_{p}) has rank d = ∑[E_{j} : ℚ_{p}]. However, 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.

Leopoldt’s conjecture 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 [So2, 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 Lfunction 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 with a description of the topological type of K̂(V) and K̂(E), when p is odd. 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 for i > 0, and the other homotopy group of BU vanish, 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 5.10. ([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. Set r = [E(μ_{p}) : E], and let p^{a} be the number of p-primary roots of unity in E(μ_{p}). If r is a topological generator of ℤ_{p}^{×}, then k = r^{n}, n = p^{a−1}(p − 1)/r. It is an easy exercise, left to the reader, to check that π_{2i−1}F Ψ^{k} ≅ ℤ_{p}/(k^{i} − 1) is ℤ/w_{i} for all i.

§6. 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, which forces the motivic spectral sequence (0.5) to degenerate completely.

The following trick allows us to describe the torsion subgroup of the groups K_{n}(R). Recall that the notation A{ℓ} denotes the ℓ-primary subgroup of an abelian group A.

Lemma 6.1. For a given prime ℓ, ring R and integer n, suppose that K_{n}(R) is a finite group, and that K_{n−1}(R) is a finitely generated group. Then K_{n}(R){ℓ} ≅ K_{n}(R; ℤ_{ℓ}) and K_{n−1}(R){ℓ} ≅ K_{n}(R; ℤ/ℓ^{∞}).

Proof. For large values of ν, the finite group K_{n}(R; ℤ/ℓ^{ν}) is the sum of K_{n}(R){ℓ} and K_{n−1}(R){ℓ}. The transition from coefficients ℤ/ℓ^{ν} to ℤ/ℓ^{ν−1} (resp., to ℤ/ℓ^{ν+1}) is multiplication by 1 and ℓ (resp., by ℓ and 1) on the two summands. Taking the inverse limit (resp., direct limit) yields the groups K_{n}(R; ℤ_{ℓ}) and K_{n}(R; ℤ/ℓ^{∞}), respectively. □

Example 6.1.1. By 1.6, the lemma applies to a ring O_{S} of integers in a number field F, with n even. For example, theorem 1.3 says that K_{2}(O_{S}){ℓ} = K_{2}(O_{S}; ℤ_{ℓ}) ≅ H_{ét}^{2}(O_{F} [1/ℓ], ℤ_{ℓ}(2)), and of course K_{1}(O_{S}){ℓ} = K_{2}(O_{S}; ℤ/ℓ^{∞}) is the group ℤ/w_{1}^{(ℓ)}(F) of ℓ-primary roots of unity in F.

We now turn to the odd torsion in the K-groups of a number field. The ℓ-primary torsion is described by the following result, which which is based on [RW] and uses the Voevodsky-Rost theorem. The notation A_{(ℓ)} will denote the localization of an abelian group A at the prime ℓ.

Theorem 6.2. Fix an odd prime ℓ. Let F be a number field, and let O_{S} be a ring of integers in F. If R = O_{S}[1/ℓ], then for all n ≥ 2:

K_{n}(O_{S})_{(ℓ)} ≅ \{ \begin{matrix}H_{ét}^{2}(R; 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. By 1.6 we may replace O_{S} by R without changing the ℓ-primary torsion. By 6.1 and 1.5, it suffices to show that K_{2i}(R; ℤ_{ℓ}) ≅ H_{ét}^{2}(R; ℤ_{ℓ}(i + 1)) and K_{2i}(R; ℤ/ℓ^{∞}) ≅ ℤ/w_{i}^{(ℓ)}(F). Note that the formulas for K_{0}(O_{S}) and K_{1}(O_{S}) are different; see (1.2.1).

If F is a number field and ℓ ≠ 2, the étale ℓ-cohomological dimension of F (and of R) is 2. Since H_{ét}^{2}(R; ℤ/ℓ^{∞}(i)) = 0 by 3.8.1, the Voevodsky-Rost theorem implies that the motivic spectral sequence (0.5) has only two nonzero diagonals, except in total degree zero, and collapses at E_{2}. This gives

(6.2.1)  K_{n}(O_{S}; 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}(O_{S}){ℓ} follows from 6.1 and 1.5.

The same argument works for coefficients ℤ_{ℓ}; for i > 0 we have H_{ét}^{n}(R, ℤ_{ℓ}(i)) = 0 for n ≠ 1, 2, so the spectral sequence degenerates to yield K_{2i}(R; ℤ_{ℓ}) ≅ H_{ét}^{2}(R, ℤ_{ℓ}(i)). (This is a finite group by 3.8.) The description of K_{2i}(R){ℓ} follows from 6.1 and 1.5. □

Because H_{ét}^{2}(R, ℤ_{ℓ}(i + 1))/ℓ ≅ H_{ét}^{2}(R, μ_{ℓ}^{⊗i+1}), we immediately deduce:

Corollary 6.3. For all odd ℓ and i > 0, K_{2i}(O_{S})/ℓ ≅ H_{ét}^{2}(O_{S}[1/ℓ], μ_{ℓ}^{⊗i+1}).

Remark 6.4. Similarly, the mod-ℓ spectral sequence (0.3) collapses to yield the Ktheory of O_{S} with coefficients ℤ/ℓ, ℓ odd. For example, if O_{S} contains a primitive ℓth root of unity and 1/ℓ 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_{n}(O_{S}; ℤ/ℓ) ≅ \{ \begin{matrix}ℤ/ℓ ⊕ Pic(O_{S})/ℓ, & n = 0 \\ 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 2.5.1). In fact, K_{∗}(O_{S}; ℤ/ℓ) is free as a graded ℤ[β]-module on K_{0}(O_{S}; ℤ/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 (0.3).

When F is totally imaginary, we have a complete description of K_{∗}(O_{S}). The 2primary torsion was first calculated in [RW]; the odd torsion comes from theorem 6.2.

Theorem 6.5. 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 for all n ≥ 2:

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

Proof. The case n = 1 comes from (1.2.1), and the odd torsion comes from 6.2, so it suffices to check the 2-primary torsion. This does not change if we replace O_{S} by R = O_{S}[1/2], by 1.6. By 6.1 and 1.5, it suffices to show that K_{2i}(R; ℤ_{2}) ≅ H_{ét}^{2}(R; ℤ_{2}(i + 1)) and K_{2i}(R; ℤ/2^{∞}) ≅ ℤ/w_{i}^{(2)}(F).

Consider the mod 2^{∞} motivic spectral sequence (0.5) for the ring R, converging to K_{∗}(R; ℤ/2^{∞}). It is well known that cd_{2}(R) = 2, and H_{ét}^{2}(R; ℤ/2^{∞}(i)) = 0 by 3.8.1. Hence the spectral sequence collapses; except in total degree zero, the E_{2}-terms are concentrated on the two diagonal lines where p = q, p = q + 1. This gives

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

The description of K_{2i−1}(R){2} follows from 6.1 and 1.5.

The same argument works for coefficients ℤ_{2}; for i > 0 we have H_{ét}^{n}(R, ℤ_{2}(i)) = 0 for n ≠ 1, 2, so (0.5) degenerates to yield K_{2i}(R; ℤ_{2}) ≅ H_{ét}^{2}(R, ℤ_{2}(i)). (This is a finite group by 3.8). The description of K_{2i}(R){2} follows from 6.1 and 1.5. □

Example 6.6. Let F be a number field containing a primitive ℓth root of unity, and let S be the set of primes over ℓ in O_{F}. If t is the rank of Pic(R)/ℓ, then H_{ét}^{2}(R, ℤ_{ℓ}(i))/ℓ ≅ H_{ét}^{2}(R, μ_{ℓ}^{⊗i}) ≅ H_{ét}^{2}(R, μ_{ℓ}) ⊗ μ_{ℓ}^{⊗i−1} has rank t + |S| − 1 by (1.3.1). By 6.5, the ℓ-primary subgroup of K_{2i}(O_{S}) has t + |S| − 1 nonzero summands for each i ≥ 2.

Example 6.7. If ℓ ≠ 2 is a regular prime, we claim that K_{2i}(ℤ[ζ_{ℓ}]) has no ℓ-torsion. (The case K_{0} is tautological by 1.1, and the classical case K_{2} is 1.4.) Note that the group K_{2i−1}(ℤ[ζ_{ℓ}]) ≅ ℤ^{r_{2}} ⊕ ℤ/w_{i}(F) always has ℓ-torsion, because w_{i}^{(ℓ)}(F) ≥ ℓ for all i by 2.7(a). Setting R = ℤ[ζ_{ℓ}, 1/ℓ], then by 6.5,

K_{2i}(ℤ[ζ_{ℓ}]) ≅ H_{ét}^{2}(R, ℤ_{ℓ}(i + 1)) ⊕ (finite group without ℓ-torsion).

Since ℓ is regular, we have Pic(R)/ℓ = 0, and we saw in 1.4 that Br(R) = 0 and |S| = 1. By 6.6, H_{ét}^{2}(R, ℤ_{ℓ}(i + 1)) = 0 and the claim now follows.

We conclude with a comparison to the odd part of ζ_{F} (1−2k), generalizing the Birch-Tate Conjecture 2.14. If F is not totally real, ζ_{F} (s) has a pole of order r_{2} at s = 1 − 2k. We need to invoke the following deep result of Wiles [Wi], which is often called the “Main Conjecture” of Iwasawa Theory.

Theorem 6.8 (Wiles). Let F be a totally real number field. If ℓ is odd and O_{S} = O_{F} [1/ℓ] then for all even i = 2k > 0:

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

where u_{i} is a rational number prime to ℓ.

The numerator and denominator on the right side are finite by 3.4. Lichtenbaum’s conjecture follows, up to a power of 2, by setting i = 2k:

Theorem 6.9. If F is totally real, then

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

Proof. By the functional equation, the sign of ζ_{F} (1 − 2k) is (−1)^{kr_{1}}. It suffices to show that the left and right sides of 6.9 have the same power of each odd prime ℓ. The group H_{ét}^{2}(O_{F} [1/ℓ], ℤ_{ℓ}(i)) is the ℓ-primary part of K_{2i−2}(O_{F}) by 6.2. The group H_{ét}^{1}(O_{F} [1/ℓ], ℤ_{ℓ}(i)) on the bottom of 6.8 is ℤ/w_{i}^{(ℓ)}(F) by 3.8.4, and this is isomorphic to the ℓ-primary subgroup of K_{2i−1}(O_{F}) by theorem 6.2. □

§7. 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 2.12). A final reason is that the groups K_{∗}(F; ℤ/2) do not have a natural multiplication, because of the structure of the mod 2 Moore space ℝℙ^{2}.

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 the groups E_{2}^{p,q}, it is useful to label the nonzero elements of H_{ét}^{0}(ℝ, ℤ/2(i)) as β_{i}, writing 1 for β_{0}. Using the multiplicative pairing with (say) the spectral sequence ′E_{2}^{∗,∗} converging to K_{∗}(ℝ; ℤ/16), multiplication by the element η ∈ ′E_{2}^{0,−1} allows us to write the nonzero elements in the −ith column as η^{j}β_{i}. (See table 7.1.1 below)

From Suslin’s calculation of K_{n}(ℝ) in [Su2], 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).

Theorem 7.1. In the spectral sequence 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).

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}

1

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

The first 4 columns of E_{3}

Table 7.1.1. The mod 2 spectral sequence for ℝ.

Proof. Recall from 0.6 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.

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}}. □

7.1.2. The analysis with coefficients ℤ/2^{∞} is very similar, except that when p > q, E_{2}^{p,q} = H_{ét}^{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 7.1 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^{∞}.

7.1.3. The analysis 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_{ét}^{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.

In order to state the theorem 7.3 below for a ring O_{S} of integers in a number field F, we consider the natural maps (for n > 0) induced by the r_{1} real embeddings of F,

(7.2.0) 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; by 3.8.3, it is also an isomorphism for n = 2 and i ≥ 2. Write H̃^{1}(O_{S}; ℤ/2^{∞}(i)) for the kernel of α_{S}^{1}(i).

Lemma 7.2. The map H^{1}(F; ℤ/2^{∞}(i)) ⟶^{α^{1}(i)} (ℤ/2)^{r_{1}} is a split surjection for all even i. 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 map vertical map is a split surjection in the diagram:

F^{×}/F^{×2} ⟶^{≅} H^{1}(F, ℤ/2) ⟶ H^{1}(F, ℤ/2^{∞}(i))
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. □

We also write A ⋊ B for an abelian group extension of B by A.

Theorem 7.3. ( [RW, 6.9]) Let F be a real number field, and let R = O_{S} be a ring of S-integers in F containing O_{F} [\frac{1}{2}]. Then α_{S}^{1}(i) is onto when i = 4k > 0, and:

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

Proof. The morphism of spectral sequences (0.5), from that for O_{S} to the sum of r_{1} copies of that for ℝ, is an isomorphism on E_{2}^{p,q} except on the diagonal p = q (where it is an injection) and p = q + 1 (where we must show it is a surjection). When p ≡ +1 (mod 4), it follows from 7.1.2 that we may identify d_{2}^{p,q} with α_{S}^{p−q}. Hence d_{2}^{p,q} is an isomorphism if p ≥ 2 + q, and an injection if p = q. As in 7.1.2, the spectral sequence degenerates at E_{3}, yielding K_{n}(O_{S}; ℤ/2^{∞}) as proclaimed, except for two points: (a) the extension of ℤ/w_{4a+2} by ℤ/2^{r_{1}} when n = 8a + 4 is seen to be nontrivial by comparison with the extension for ℝ, and (b) when n = 8a + 6 it only shows that K_{n}(O_{S}; ℤ/2^{∞}) is the cokernel of α_{S}^{1}(4a + 4).

To resolve (b) we must show that α_{S}^{1}(4a + 4) is onto when a > 0. Set n = 8a + 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 1.6. But for sufficiently large S, the map α^{1}(4a + 4) is a surjection by 7.2, and hence K_{n}(O_{S}; ℤ/2^{∞}) = 0. □

Proof of Theorem 0.1. Let n > 0 be odd. By 1.5 and 1.6, it suffices to determine the torsion subgroup of K_{n}(O_{S}) = K_{n}(F). Since K_{n+1}(O_{S}) is finite, it follows that K_{n+1}(O_{S}; ℤ/ℓ^{∞}) is the ℓ-primary subgroup of K_{n}(O_{S}). By 6.5, we may assume F has a real embedding. By 6.2, we need only worry about the 2-primary torsion, which we can read off from 7.3, recalling from 2.8(b) that w_{i}^{(2)}(F) = 2 for odd i. □

To proceed further, we need to introduce the narrow Picard group and the signature defect of the ring O_{S}.

Narrow Picard group 7.4. 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 maps is the sign map σ: F^{×} ⟶ (ℤ/2)^{r_{1}}. The approximation theorem for F implies that σ is surjective. The group F_{+}^{×} of totally positive units in F is defined to be the kernel of σ.

Now let R = O_{S} be a ring of integers in F. The kernel of σ|_{R} : R^{×} → F^{×} → (ℤ/2)^{r_{1}} is the subgroup R_{+}^{×} of totally positive units in R. Since the sign map σ|_{R} factors through F^{×}/2 = H^{1}(F, ℤ/2), it also factors through α^{1} : H^{1}(R, ℤ/2) → (ℤ/2)^{r_{1}}. The signature defect j(R) of R is defined to be the dimension of the cokernel of α^{1}; 0 ≤ j(R) < r_{1} because σ(−1) ≠ 0. Note that j(F) = 0, and that j(R) ≤ j(O_{F}).

By definition, the narrow Picard group Pic_{+}(R) is the cokernel of the the restricted divisor map F_{+}^{×} → ⨁_{℘∉S} ℤ. (See [Co, 5.2.7]. This definition is due to Weber; Pic_{+}(O_{S}) is also called the ray class group Cl_{F}^{S}; see [Neu, VI.1].) 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.

A diagram chase (performed in [RW, 7.6]) shows that there is an exact sequence

(7.4.1) 0 → H̃^{1}(R; ℤ/2) ⟶ H^{1}(R; ℤ/2) ⟶^{α^{1}} (ℤ/2)^{r_{1}} ⟶ Pic_{+}(R)/2 ⟶ Pic(R)/2 → 0.

(H̃^{1}(R; ℤ/2) is defined as the kernel of α^{1}.) 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 R = O_{S}, then dim H^{1}(R; ℤ/2) = r_{1} + r_{2} + s + t and dim H^{2}(R; ℤ/2) = r_{1} + s + t − 1. This follows from (1.2.1) and (1.3.1), using Kummer theory. As in (7.2.0) and (7.4.1), define H̃^{n}(R; ℤ/2) to be the kernel of α^{n} : H^{n}(R; ℤ/2) → H^{n}(ℝ; ℤ/2)^{r_{1}} ≅ (ℤ/2)^{r_{1}}.

Lemma 7.4.2. Suppose that \frac{1}{2} ∈ R. Then dim H̃^{1}(R, ℤ/2) = r_{2} + s + u. Moreover, the map α^{2} : H^{2}(R, ℤ/2) → (ℤ/2)^{r_{1}} is onto, and dim H̃^{2}(R, ℤ/2) = t + s − 1.

Proof. The first assertion is immediate from (7.4.1). Since H^{2}(R; ℤ/2^{∞}(4)) ≅ (ℤ/2)^{r_{1}} by (3.8.3), the coefficient sequence for ℤ/2 ⊂ ℤ/2^{∞}(4) shows that H^{2}(R; ℤ/2) → H^{2}(R; ℤ/2^{∞}(4)) is onto. The final two assertions follow. □

Theorem 7.5. 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 = 8a, \\ H^{1}(O_{S}; ℤ/2) & for n = 8a + 1, \\ H^{2}(O_{S}; ℤ/2) ⋊ ℤ/2 & for n = 8a + 2, \\ (ℤ/2)^{r_{1}−1} ⋊ H^{1}(O_{S}; ℤ/2) & for n = 8a + 3, \\ (ℤ/2)^{j} ⋊ H^{2}(O_{S}; ℤ/2) & for n = 8a + 4, \\ (ℤ/2)^{r_{1}−1} ⋊ H̃^{1}(O_{S}; ℤ/2) & for n = 8a + 5, \\ (ℤ/2)^{j} ⊕ H̃^{2}(O_{S}; ℤ/2) & for n = 8a + 6, \\ H̃^{1}(O_{S}; ℤ/2) & for n = 8a + 7. \end{matrix}

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 (−3 ≤ p ≤ 0) of E_{3} = E_{∞}

Table 7.5.1. The mod 2 spectral sequence for O_{S}.

Proof. (Cf. [RW, 7.8].) As in the proof of Theorem 7.3, 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 7.4.2, we see from 0.6 that that the columns p ≤ 0 of E_{3} are 4-periodic, and all nonzero entries are described by Figure 7.5.1. (By (0.5), there is only one nonzero entry for p > 0, E_{3}^{+1,−1} = Pic(R)/2, and it is only important for n = 0.) By inspection, E_{3} = E_{∞}, yielding the desired description of the groups K_{n}(R, ℤ/2) in terms of extensions. We omit the proof that the extensions split if n ≡ 0, 6 (mod 8). □

The case F = ℚ has historical importance, because of its connection with the image of J (see 2.12 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_{8a−1}(ℤ) depends upon a.

Corollary 7.6. For n ≥ 0, the two-primary subgroups of K_{n}(ℤ) and K_{2}(ℤ[1/2]) are essentially periodic, of period eight, and are given by the following table. (When n ≡ 7 (mod 8), we set a = (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/16a & 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_{8a+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 0.1; w_{4a}^{(2)} is the 2-primary part of 16a by 2.8(c). Since s = 1 and t = u = 0, we see from 7.4.2 that dim H̃^{1}(ℤ[1/2]; ℤ/2) = 1 and that H̃^{2}(ℤ[1/2]; ℤ/2) = 0. By 7.5, 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 0.1, together with the ℤ/2 summand in K_{8a+2}(ℤ) provided by topology (see 2.12), 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 either theorem 0.1 or 7.3) 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 7.7. 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.

Proof. (Cf. [RW, 0.7].) 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 7.5. □

Example 7.7.1. 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_{8a+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 7.7. 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 7.7.2. (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 7.7 that K_{8a+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 [BS] and Rognes and Østvær [RØ] 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 [BS].)

A useful example is F = ℚ(\sqrt{2}). Note that the Steinberg symbols \{−1, −1, −1, −1\} and \{−1, −1, −1, 1 + \sqrt{2}\} generating K_{4}^{M}(F) ≅ (ℤ/2)^{2} must both vanish in K_{4}(ℤ[\sqrt{2}]), which we have seen has odd order. This is the case j = ρ = 0 of the following result.

Corollary 7.8. Let F be a real number field. Then the rank ρ of the image of K_{4}^{M}(F) ≅ (ℤ/2)^{r_{1}} in K_{4}(F) satisfies j(O_{F} [1/2]) ≤ ρ ≤ r_{1} − 1. The image (ℤ/2)^{ρ} lies in the subgroup K_{4}(O_{F}) of K_{4}(F), and its image in K_{4}(O_{S})/2 has rank j(O_{S}) for all O_{S} containing 1/2. In particular, the image (ℤ/2)^{ρ} lies in 2 · K_{4}(F).

Proof. By 1.8, we have ρ < r_{1} = rank K_{4}^{M}(F). The assertion that K_{4}^{M}(F) → K_{4}(F) factors through K_{4}(O_{F}) follows from 1.7, by multiplying K_{3}^{M}(F) and K_{3}(O_{F}) ≅ K_{3}(F) by [−1] ∈ K_{1}(ℤ). It is known [FS, 15.5] that the edge map H^{n}(F, ℤ(n)) → K_{n}(F) in the motivic spectral sequence agree with the usual map K_{n}^{M}(F) → K_{n}(F). By Voevodsky’s theorem, 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 7.5.1; 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 7.8.1. (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 7.7.

On the other hand, O_{S} = ℤ[\sqrt{7}, 1/7] still has ρ = 1, but now j = 0, and the 2-rank of K_{4}(O_{S}) is still one by 7.7. Hence the extension 0 → K_{4}(O_{F}) → K_{4}(O_{S}) → ℤ/48 → 0 of 1.6 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 1.6 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 7.8.2. Can ρ be less than the minimum of r_{1} − 1 and j + s + t − 1?

As in (7.2.0), 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 7.4.2, H̃^{2}(R; ℤ_{2}(i)) has 2-rank s + t − 1.

Theorem 7.9. ( [RW, 0.6]) 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_{ét}^{2}(R; ℤ_{2}(4a + 1)) & for n = 8a, \\ ℤ/2 & for n = 8a + 1, \\ H_{ét}^{2}(R; ℤ_{2}(4a + 2)) & for n = 8a + 2, \\ (ℤ/2)^{r_{1}−1} ⊕ ℤ/2w_{4a+2} & for n = 8a + 3, \\ (ℤ/2)^{ρ} ⋊ H_{ét}^{2}(R; ℤ_{2}(4a + 3)) & for n = 8a + 4, \\ 0 & for n = 8a + 5, \\ H̃_{ét}^{2}(R; ℤ_{2}(4a + 4)) & for n = 8a + 6, \\ ℤ/w_{4a+4} & for n = 8a + 7. \end{matrix}

Proof. When n = 2i − 1 is odd, this is theorem 0.1, since w_{i}^{(2)}(F) = 2 when n ≡ 1 (mod 4) by 2.8(b). When n = 2 it is 1.3. 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 1.5 (r = r_{1} + r_{2} or r_{2}). To compare this with theorem 7.3, 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_{ét}^{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 = 8a + 4. By mod 2 periodicity 0.6, the integer ρ of 7.8 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 7.9 with the odd torsion information in 6.2 and 6.9 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 7.10. 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_{ét}^{2}(O_{S}[1/ℓ]; ℤ_{ℓ}(i))|}{∏_{ℓ} |H_{ét}^{1}(O_{S}[1/ℓ]; ℤ_{ℓ}(i))|}.

Proof. (Cf. proof of 6.9.) Since 2i − 1 ≡ 3 (mod 4), all groups involved are finite (see 1.5, 3.8 and 3.8.4.) Write h^{n,i}(ℓ) for the order of H_{ét}^{n}(O_{S}[1/ℓ]; ℤ_{ℓ}(i)). By 3.8.4, h^{1,i}(ℓ) = w_{i}^{(ℓ)}(F). By 0.1, 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 6.2 and 7.9, 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. □

Corollary 7.11. For R = ℤ, the formula conjectured by Lichtenbaum in [Li2] holds up to exactly one factor of 2. That is, for k ≥ 1,

\frac{|K_{4k−2}(ℤ)|}{|K_{4k−1}(ℤ)|} = \frac{B_{k}}{4k} = \frac{(−1)^{k}}{2} ζ(1 − 2k).

Moreover, if c_{k} denotes the numerator of \frac{B_{k}}{4k}, then

|K_{4k−2}(ℤ)| = \{ \begin{matrix}c_{k}, & k even \\ 2 c_{k}, & k odd. \end{matrix}

Proof. The equality B_{k}/4k = (−1)^{k}ζ(1 − 2k)/2 comes from 2.10.1. By 6.9, the formula holds up to a factor of 2. By 2.11, the two-primary part of B_{k}/4k is 1/w_{2k}^{(2)}. By 2.8(c), this is also the two-primary part of 1/8k. By 7.6, the two-primary part of the left-hand side of 7.11 is 2/16 when k is odd, and the two-primary part of 1/8k when k = 2a is even. □

Examples 7.12. (K_{4k−2}(ℤ)) The group K_{4k−2}(ℤ) is cyclic of order c_{k} or 2c_{k} for all k ≤ 5000. For small k we need only consult 2.10 to see that the groups K_{2}(ℤ), K_{10}(ℤ), K_{18}(ℤ) and K_{26}(ℤ) are isomorphic to ℤ/2. We also have K_{6}(ℤ) = K_{14}(ℤ) = 0. (The calculation of K_{6}(ℤ) up to 3-torsion was given in [EGS].) However, c_{6} = 691, c_{8} = 3617, c_{9} = 43867 and c_{13} = 657931 are all prime, so 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 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 37torsion in every group K_{72a+62}(ℤ) (see 8.6 below).

For k < 5000, 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) and (2884, 101). However, K_{4k−2}(ℤ) is still cyclic with one ℤ/ℓ^{2} summand in these cases. To see this, we note that Pic(R)/ℓ ≅ ℤ/ℓ for these ℓ, where R = ℤ[ζ_{ℓ}]. Hence K_{4k−2}(R)/ℓ ≅ H^{2}(R, ℤ_{ℓ}(2k))/ℓ ≅ H^{2}(R, ℤ/ℓ(2k)) ≅ Pic(R) ≅ ℤ/ℓ. The usual transfer argument now shows that K_{4k−2}(ℤ)/ℓ is either zero or ℤ/ℓ for all k.

§8. The odd torsion in K_{∗}(ℤ)

We now turn to the ℓ-primary torsion in the K-theory of ℤ, where ℓ is an odd prime. By 2.11 and 6.2, the odd-indexed groups K_{2i−1}(ℤ) have ℓ-torsion exactly when i ≡ 0 (mod ℓ − 1). Thus we may restrict attention to the groups K_{2i}(ℤ), whose ℓ-primary subgroups are H_{ét}^{2}(ℤ[1/ℓ]; ℤ_{ℓ}(i + 1)) by 6.2.

Our method is to consider the cyclotomic extension ℤ[ζ] of ℤ, ζ = e^{2πi/ℓ}. Because the Galois group G = Gal(ℚ(ζ)/ℚ) is cyclic of order ℓ−1, prime to ℓ, the usual transfer argument shows that K_{∗}(ℤ) → K_{∗}(ℤ[ζ]) identifies K_{n}(ℤ) ⊗ ℤ_{ℓ} with K_{n}(ℤ[ζ]))^{G} ⊗ ℤ_{ℓ} for all n. Because K_{n}(ℤ) and K_{n}(ℤ[1/ℓ]) have the same ℓ-torsion (by the localization sequence), it suffices to work with ℤ[1/ℓ].

Proposition 8.1. When ℓ is an odd regular prime there is no ℓ-torsion in K_{2i}(ℤ).

Proof. Since ℓ is regular, we saw in example 6.7 that the finite group K_{2i}(ℤ[ζ]) has no ℓ-torsion. Hence the same is true for its G-invariant subgroup, K_{2i}(ℤ). □

It follows from this and 2.11 that K_{2i}(ℤ; ℤ/ℓ) contains only the Bockstein representatives of the Harris-Segal summands in K_{2i−1}(ℤ), and this only when 2i ≡ 0 (mod 2ℓ−2).

We can also describe the algebra structure of K_{∗}(ℤ; ℤ/ℓ) using the action of the cyclic group G = Gal(ℚ(ζ)/ℚ) on the ring K_{∗}(ℤ[ζ]; ℤ/ℓ). For simplicity, let us assume that ℓ is a regular prime. It is useful to set R = ℤ[ζ, 1/ℓ] and recall from 6.4 that K_{∗} = K_{∗}(R; ℤ/ℓ) is a free graded ℤ/ℓ[β]-module on the \frac{ℓ+1}{2} generators of R^{×}/ℓ ∈ K_{1}(R; ℤ/ℓ), together with 1 ∈ K_{0}(R; ℤ/ℓ).

By Maschke’s theorem, ℤ/ℓ[G] ≅ ∏_{i=0}^{ℓ−2} ℤ/ℓ is a simple ring; every ℤ/ℓ[G]-module has a unique decomposition as a sum of 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}^{⊗ℓ−1−i}.

For example, the G-module ⟨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 isomorphic to μ_{ℓ}^{⊗i}.

Example 8.2. 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,

H_{ét}^{1}(R, μ_{ℓ}) = R^{×}/R^{×ℓ} ≅ μ_{ℓ} ⊕ (ℤ/ℓ) ⊕ μ_{ℓ}^{⊗2} ⊕ ⋯ ⊕ μ_{ℓ}^{⊗ℓ−3}.

The root of unity ζ generates the G-submodule μ_{ℓ}, and the class of the unit ℓ of R generates the trivial submodule of R^{×}/R^{×ℓ}.

Tensoring with μ_{ℓ}^{⊗i−1} yields the G-module decomposition of R^{×} ⊗μ_{ℓ}^{⊗i}. If ℓ is regular this is K_{2i−1}(R; ℤ/ℓ) ≅ H_{ét}^{1}(R, μ_{ℓ}^{⊗i}) by 6.4. If i is even, exactly one term is ℤ/ℓ; if i is odd, ℤ/ℓ occurs only when i ≡ 0 (mod ℓ − 1).

Notation 8.2.1. Set R = ℤ[ζ_{ℓ}, 1/ℓ], For i = 0, ..., \frac{ℓ−3}{2}, pick a generator x_{i} of the Gsubmodule of R^{×}/R^{×ℓ} isomorphic to μ_{ℓ}^{⊗−2i}. The indexing is set up so that y_{i} = β^{2i}x_{i} is a G-invariant element of K_{4i+1}(R; ℤ/ℓ) ≅ H_{ét}^{1}(R, μ_{ℓ}^{⊗2i+1}). We may arrange that x_{0} = y_{0} is the unit [ℓ] in K_{1}(R; ℤ/ℓ).

The elements β^{ℓ−1} of H_{ét}^{0}(R, μ_{ℓ}^{⊗ℓ−1}) and v = β^{ℓ−2}[ζ] of H_{ét}^{1}(R, μ_{ℓ}^{⊗ℓ−1}) are also Ginvariant. By abuse of notation, we shall also write β^{ℓ−1} and v, respectively, for the corresponding elements of K_{2ℓ−2}(ℤ[1/ℓ]; ℤ/ℓ) and K_{2ℓ−3}(ℤ[1/ℓ]; ℤ/ℓ).

Theorem 8.3. 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 y_{i} ∈ K_{4i+1} (i = 0, ..., \frac{ℓ−3}{2}).

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 submodule generated by v and β^{ℓ−1} comes from the Harris-Segal summands of K_{4i−1}(ℤ). The submodule generated by the y’s comes from the ℤ summands in K_{4i+1}(ℤ).

Proof. K_{∗}(ℤ[1/ℓ]; ℤ/ℓ) is the G-invariant subalgebra of K_{∗}(R; ℤ/ℓ). Given 8.2, it is not very hard to check that this is just the subalgebra described in the theorem. □

Examples 8.3.1. When ℓ = 3, the groups K_{∗} = K_{∗}(ℤ[1/3]; ℤ/3) are 4-periodic of ranks 1, 1, 0, 1, generated by an appropriate power of β^{2} times one of {1, [3], v}.

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

Now suppose that ℓ is an irregular prime, so that Pic(R) has ℓ-torsion for R = ℤ[ζ, 1/ℓ]. Then H_{ét}^{1}(R, μ_{ℓ}) is R^{×}/ℓ ⊕_{ℓ} Pic(R) and H_{ét}^{2}(R, μ_{ℓ}) ≅ Pic(R)/ℓ by Kummer theory. This yields K_{∗}(R; ℤ/ℓ) by 6.4.

Example 8.4. Set R = ℤ[ζ_{ℓ}, 1/ℓ] and P = Pic(R)/ℓ. If ℓ is regular then P = 0 by definition 1.1. When ℓ is irregular, the G-module structure of P is not fully understood; see Vandiver’s conjecture 8.5 below. However, the following arguments show that P^{[i]} = 0, i.e., P contains no summands isomorphic to μ_{ℓ}^{⊗i}, for i = 0, −1, −2, −3.

The usual transfer argument shows that P^{G} ≅ Pic(ℤ[1/ℓ])/ℓ = 0. Hence P contains no summands isomorphic to ℤ/ℓ. By 1.4, we have a G-module 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 6.5. Again, the transfer argument shows that K_{n}(R)/ℓ^{G} ≅ K_{n}(ℤ[1/ℓ])/ℓ for n = 4, 6. These groups are known to be zero by [R4] and [EGS]; see 1.9. It follows that P contains no summands isomorphic to μ_{ℓ}^{⊗−2} or μ_{ℓ}^{⊗−3}.

Vandiver’s conjecture 8.5. 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-primary subgroup of Pic(ℤ[ζ_{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 12 million; see [Bu12]. We also known from 8.4 that μ_{ℓ}^{⊗i} does not occur as a summand of Pic(R)/ℓ for i = 0, −2.

Remark 8.5.1. 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 7.12), so Pic(R)/ℓ^{[5]} = ℤ/37 and Pic(R)/ℓ^{[k]} = 0 for k ≠ 5.

Historical Remark 8.5.2. 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 made his conjecture circa 1920. 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 called the assertion [Vandiver’s conjecture] “a 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 [Vandiver’s conjecture] was still unproven.

For the rest of this paper, we set R = ℤ[ζ_{ℓ}, 1/ℓ], where ζ^{ℓ} = 1.

Theorem 8.6. (Kurihara [Kur]) Let ℓ be an irregular prime number. Then the following are equivalent for every 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^{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. Set P = Pic(R)/ℓ. By Kummer theory (see 1.4), P ≅ H^{2}(R, μ_{ℓ}) and hence P ⊗ μ_{ℓ}^{⊗2k} ≅ H^{2}(R, μ_{ℓ}^{⊗2k+1}) as G-modules. Taking G-invariant subgroups shows that H^{2}(ℤ[1/ℓ], μ_{ℓ}^{⊗2k+1}) ≅ (P ⊗ μ_{ℓ}^{⊗2k})^{G} ≅ P^{[−2k]}. Hence (1) and (4) are equivalent.

By 6.3, K_{4k}(ℤ)/ℓ ≅ H^{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 8.7. 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 < 500, by 7.12.)

Proof. Set P = Pic(R)/ℓ. Vandiver’s conjecture also implies that each of the “odd” summands P^{[1−2k]} = P^{[ℓ−2k]} of P is cyclic, and isomorphic to ℤ_{ℓ}/c_{k}; see [Wash, 10.15] and 3.8.2 above. Since Pic(R) ⊗ μ_{ℓ}^{⊗2k−1} ≅ H^{2}(R, μ_{ℓ}^{⊗2k}), taking G-invariant subgroups shows that P^{[1−2k]} ≅ H^{2}(ℤ[1/ℓ], μ_{ℓ}^{⊗2k}). By theorem 6.2, this group is the ℓ-primary torsion in K_{4k−2}(ℤ[1/ℓ]). □

Using 2.10 and 2.11 we may write the Bernoulli number B_{k}/4k as c_{k}/w_{2k} in reduced terms, with c_{k} odd. The following result, which follows from theorems 0.1, 8.6 and 8.7, was observed independently by Kurihara [Kur] and Mitchell [Mit].

Corollary 8.8. If Vandiver’s conjecture holds, then K_{n}(ℤ) is given by Table 8.8.1, 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 8.8.1. The K-theory of ℤ, assuming Vandiver’s Conjecture.

Remark 8.9. 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 [BG, Proof of Prop.8]. It is an open question whether there are any divisible elements of even order.

For example, recall from 7.12 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} and s_{j} be respective generators of the summand of Pic(R)/ℓ and K_{1}(R; ℤ/ℓ) isomorphic to μ_{ℓ}^{⊗−j}. The following result follows easily from 6.4 and 8.2, using the proof of 8.3, 8.6 and 8.7. It was originally proven in [Mit]; another proof is given in the article [MKH] in this Handbook. (The generators s_{j}β^{j} were left out in [Mit2, 6.13].)

Theorem 8.10. 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 the (ℓ−3)/2 generators y_{i} described in 8.3, together with the generators t_{j}β^{j} ∈ K_{2j} and s_{j}β^{j} ∈ K_{2j+1}.

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Department of Mathematics, Rutgers University, U.S.A.

   E-mail address: weibel@math.rutgers.edu