On products in algebraic K-theory

Dominique Arlettaz, Grzegorz Banaszak and Wojciech Gajda

Abstract. This paper investigates the product structure in algebraic K-theory. The first objective is to understand the relationships between products and the kernel of the Hurewicz homomorphism relating the algebraic K-theory of any ring to the integral homology of its linear groups. The second part of the paper is devoted to the ring of integers ℤ. Using recent results of V. Voevodsky and the 2-adic cyclotomic elements, we determine completely the products in K_{∗}(ℤ) tensored with the ring of 2-adic integers.

Introduction

The purpose of this paper is to study the product homomorphism

⋆ : K_{i}(R) ⊗ K_{k}(ℤ) ⟶ K_{i+k}(R)

in the algebraic K-theory of any ring R with identity, for positive integers i and k. Our first goal is to exhibit very strong connections between the image of that product and the kernel of the non-stable Hurewicz homomorphisms relating the K-groups of R to the integral homology groups of its linear groups

h_{i} : K_{i}(R) = π_{i}BGL(R)^{+} ⟶ H_{i}BGL(R)^{+} ≅ H_{i}GL(R) for i ≥ 1 ,

respectively h_{i} : K_{i}(R) → H_{i}E(R) for i ≥ 2 and h_{i} : K_{i}(R) ⟶ H_{i}St(R) for i ≥ 3 , where GL(R) is the infinite general linear group (considered as a discrete group) over R, E(R) its subroup generated by elementary matrices, and St(R) the infinite Steinberg group over R. A universal approximation of the exponent of the kernel and some information on the cokernel of these Hurewicz homomorphisms have been obtained in [2], [3] and Section 5 of [4]. Our argument is based on the understanding, from various viewpoints, of the stable Hurewicz homomorphism between the algebraic K-theory and the homology of the K-theory spectrum. We establish in particular the following result (see Theorem 3.2):

For any ring R and any integer i ≥ 2 , the image of ⋆ : K_{i}(R) ⊗ K_{1}(ℤ) → K_{i+1}(R) is
contained in the kernel of the Hurewicz homomorphisms h_{i+1} : K_{i+1}(R) → H_{i+1}GL(R) and
h_{i+1} : K_{i+1}(R) → H_{i+1}E(R) ; the same holds for h_{i+1} : K_{i+1}(R) → H_{i+1}St(R) if i ≥ 3 .

In low dimensions, we actually prove exactness results for any ring R (see Theorems 4.1 and 4.3).

The second and third authors were partially supported by the KBN grant 2 P03A 00511.

1991 Mathematics Subject Classification: Primary 19 D 55; Secondary 19 D 50, 19 F 27, 20 G 10, 55 N 15, 55 P 42.

(a) There is an exact sequence

K_{4}(R) ⟶^{h_{4}} H_{4}E(R) ⟶ Γ(K_{2}(R)) ⟶ K_{3}(R) ⟶^{h_{3}} H_{3}E(R) ⟶ 0 ,

where Γ(−) is the quadratic functor defined on abelian groups by J.H.C. Whitehead in Section 5 of [43]; moreover, ker h_{3} is isomorphic to K_{2}(R) ⋆ K_{1}(ℤ) .

(b) There is an exact sequence

K_{5}(R) ⟶^{h_{5}} H_{5}St(R) ⟶ K_{3}(R) ⊗ K_{1}(ℤ) ⟶^{⋆} K_{4}(R) ⟶^{h_{4}} H_{4}St(R) ⟶ 0

and the kernel of h_{5} fits into a short exact sequence

0 ⟶ K_{4}(R) ⋆ K_{1}(ℤ) ⟶ ker h_{5} ⟶ Q ⟶ 0 ,

where Q is a quotient of the subgroup of elements of order 2 in the group K_{3}(R) .

The second objective of the paper is to compute explicitely products in the algebraic K-theory of the ring of integers ℤ. First of all, we determine in low dimensions the products K_{i}(ℤ) ⋆ K_{k}(ℤ) , the homology groups of SL(ℤ) and St(ℤ) , and the Hurewicz homomorphism (see Proposition 5.1). Secondly, we consider maps

K_{i}(ℤ) ⊗ K_{k}(ℤ) ⟶^{⋆} K_{i+k}(ℤ) ⟶ K_{i+k}(ℤ) ⊗ ℤ_{ˆ2}

for all positive integers i and k, where the second arrow is the tensor product of K_{i+k}(ℤ) with the inclusion of ℤ into the ring of 2-adic integers ℤ_{ˆ2} . We call these maps 2-adic products for K_{∗}(ℤ) and continue to denote them by the symbol ⋆. We are able to calculate all such 2-adic products (see Theorems 5.6, 5.7, Corollary 5.8, Remark 5.9 and Theorem 8.2):

The 2-adic product ⋆ : K_{i}(ℤ) ⊗ K_{k}(ℤ) ⟶ K_{i+k}(ℤ) ⊗ ℤ_{ˆ2} is trivial except if i ≡ k ≡ 1 mod 8 or i ≡ 1 mod 8 and k ≡ 2 mod 8 (or i ≡ 2 mod 8 and k ≡ 1 mod 8). If i ≡ k ≡ 1 mod 8 or i ≡ 1 mod 8 and k ≡ 2 mod 8, then its image is cyclic of order 2.

Almost all these products are deduced from a topological argument based on results by M. Bökstedt [11], V. Voevodsky [37] and C. Weibel [41].

Observe that the 2-adic products in the K-theory of ℤ have a very small image. On the other hand, we observe in Theorem 6.4 that the image of the product

⋆ : K_{1}(E) ⊗ K_{2n−1}(E) ⟶ K_{2n}(E)

is huge when E is a cyclotomic field and n and odd integer.

The last part of the paper is devoted to the construction of a 2-adic version of the cyclotomic elements of C. Soulé (see [34], Sections 4.3 and 4.4, p. 239–240). For any odd integer n ≥ 3 , we define the group of 2-adic cyclotomic elements in the 2-adic K-theory of ℤ, C(n−1) ∈ K_{2n−1}(ℤ)⊗ℤ_{ˆ2} , and the group of 2-adic cyclotomic elements in étale cohomology, C^{ét}(n − 1) ∈ H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n)) , and establish the isomorphism (see Corollary 7.11)

H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n)) ≅ C^{ét}(n − 1) ⊕ ℤ/2 .

Since the case of the prime 2 is not often covered in the literature, we include explicit computations with the 2-adic cyclotomic elements. The use of the 2-adic cyclotomic elements enables us to complete our investigation of the 2-adic products in K_{∗}(ℤ) (see Theorem 8.2). The crucial point of our argument is the proof of the 2-adic Quillen-Lichtenbaum conjecture by V. Voevodsky [37]:

The Dwyer-Friedlander map at the prime 2

K_{i}(ℤ) ⊗ ℤ_{ˆ2} ⟶ K_{i}^{ét}(ℤ[\frac{1}{2}])

is an isomorphism for all integers i ≥ 2 .

We actually derive this result (see Theorem 8.1) from Weibel’s computation [41] of the 2-torsion of K_{∗}(ℤ) and from a result of W. Dwyer and E. Friedlander [18]. We also give another proof of some of the above vanishing results on the 2-adic products for K_{∗}(ℤ) (see Proposition 8.5).

We finally mention in Proposition 8.7 an interesting relationship between products in algebraic K-theory and the kernel of the Dwyer-Friedlander map.

For any odd prime l and any integer n ≥ 2 , the image of the product map

⋆ : K_{2n−1}(ℤ) ⊗ K_{2n−1}(ℤ) ⟶ K_{4n−2}(ℤ)

is contained in the kernel of the Dwyer-Friedlander map K_{4n−2}(ℤ) → K_{4n−2}^{ét}(ℤ[\frac{1}{l}]) .

The paper is organized as follows. In Section 1, we give a new construction of the Whitehead exact sequence for spectra. Section 2 presents another approach of the study of the Hurewicz homomorphism for spectra using the so-called Postnikov cofibrations. Section 3 is devoted to general results on the relations between products in algebraic K-theory and the kernel of the stable and of the non-stable Hurewicz homomorphism. Section 4 provides the above exact sequences involving the K-groups and the homology groups of the linear groups in dimensions ≤ 5 . Section 5 contains the results on the calculation of products in the algebraic Ktheory of the ring of integers ℤ which follow from the topological argument. Section 6 concerns the product in the K-theory of cyclotomic fields. In Section 7, we construct the various groups of 2-adic cyclotomic elements. We finally discuss in Section 8 the Dwyer-Friedlander map (at the prime 2) relating the algebraic K-theory of ℤ to its étale K-theory and compute the last product in K_{∗}(ℤ) .

Throughout the paper, all rings are supposed to have an identity. We write R^{×} for the group of units in a ring R. For a prime number l, ℤ_{l̂} denotes the ring of l-adic integers, and G_{l} the l-torsion subgroup of an abelian group G. We consider all ordinary homology groups with (trivial) coefficients in ℤ except if explicitly mentioned. The étale cohomology groups will be denoted by H_{ét}^{i} with étale sheaves as coefficients. For an abelian group G, K(G, s) is the Eilenberg-MacLane space having all homotopy groups trivial except for G in dimension s and H(G) the Eilenberg-MacLane spectrum having all homotopy groups trivial except for G in dimension 0 . If X is any CW-complex or any CW-spectrum and i any integer, we write α_{i} : X → X[i] for its i-th Postnikov section (i.e., π_{k}X[i] = 0 for k > i and (α_{i})_{∗} : π_{k}X →^{≅} π_{k}X[i] for k ≤ i) and γ_{i} : X(i) → X for the fiber of α_{i} ; in other words, X(i) is the i-connected cover of X . For j ≥ i + 1 , X(i, j] denotes X(i)[j] , whose homotopy groups are π_{k}X(i, j] = 0 if k ≤ i or k > j and π_{k}X(i, j] ≅ π_{k}X if i + 1 ≤ k ≤ j .

The second and third authors would like to thank the Swiss National Science Foundation for financial support and the Institut de Mathématiques of the University of Lausanne for its hospitality during January 1997 when the main part of this paper was completed. They also thank the Max-Planck-Institut für Mathematik in Bonn for support and hospitality during their stay in 1997.

1. The Whitehead exact sequence for spectra

Let S be the sphere spectrum and S ⟶^{α_{0}} S[0] = H(ℤ) its 0-th Postnikov section. By taking the smash product of any spectrum X with the cofibration S(0) ⟶^{γ_{0}} S ⟶^{α_{0}} S[0] , where S(0) is 0-connected, one obtains the cofibration of spectra

X ∧ S(0) ⟶^{id∧γ_{0}} X ∧ S ≃ X ⟶^{id∧α_{0}} X ∧ H(ℤ) ,

whose homotopy exact sequence is the long Whitehead exact sequence

⋯ ⟶ π_{i}(X ∧ S(0)) ⟶^{χ̄_{i}} π_{i}X ⟶^{h̄_{i}} H_{i}X ⟶^{ν̄_{i}} π_{i−1}(X ∧ S(0)) ⟶ ⋯

of X ; here i is any integer, ν̄_{i} is the connecting homomorphism, χ̄_{i} is induced by (id ∧ γ_{0}) and h̄_{i} by (id ∧ α_{0}) , i.e., h̄_{i} is the stable Hurewicz homomorphism. The groups π_{i}(X ∧ S(0)) are usually denoted by Γ_{i}(X) : that definition coincides actually with the homotopy groups of the fiber of the Dold-Thom map (see [16]) and it was recently proved in [31] that they are isomorphic to the groups introduced in the original paper [43] by J.H.C. Whitehead.

Now, let us assume that the spectrum X is (r − 1)-connected for some integer r. The advantage of the above approach is that one can compute the groups Γ_{i}(X) with the Atiyah-Hirzebruch spectral sequence for the S(0)-homology of X :

E_{s,t}^{2} ≅ H_{s}(X; π_{t}S(0)) ⟹ Γ_{s+t}(X) .

Notice that E_{s,t}^{2} = 0 if s ≤ r − 1 or t ≤ 0 . This implies in particular that Γ_{i}(X) = 0 for i ≤ r (Hurewicz theorem) and that (ρ_{1}ρ_{2} ⋯ ρ_{i−r})Γ_{i}(X) = 0 for i ≥ r + 1 , where ρ_{k} denotes the exponent of the homotopy group π_{k}S for k ≥ 1 (see also [31] for another proof and [4] for corresponding results for the generalized Hurewicz homomorphisms). The first interesting Gamma group of an (r − 1)-connected spectrum X is

Γ_{r+1}(X) ≅ E_{r,1}^{2} ≅ π_{r}X ⊗ π_{1}S

(this was in fact established a long time ago by J.H.C. Whitehead, see for instance Section 14 of [43]). Our first goal is to understand the homomorphism χ̄_{r+1} : Γ_{r+1}(X) ≅ π_{r}X ⊗ π_{1}S → π_{r+1}X . Let us start with the following general result on the external product π_{i}X ⊗ π_{k}S ⟶^{∧} π_{i+k}(X ∧ S) ≅ π_{i+k}X (see [36], p. 270 for the definition of the external product).

Lemma 1.1. Let X be any spectrum, i and k two integers with k ≥ 1 . Then the image of the external product ∧ : π_{i}X ⊗ π_{k}S → π_{i+k}X is contained in the kernel of the stable Hurewicz homomorphism h̄_{i+k} : π_{i+k}X → H_{i+k}X for all integers i and all positive integers k.

Proof. The commutative diagram

π_{i}X ⊗ π_{k}S(0) ⟶_{≅}^{(id)_{∗}⊗(γ_{0})_{∗}} π_{i}X ⊗ π_{k}S
↓ ∧  ↓ ∧
Γ_{i+k}(X)  ⟶^{χ̄_{i+k}} π_{i+k}X

shows that the image of ∧ : π_{i}X ⊗ π_{k}S → π_{i+k}X is contained in image χ̄_{i+k} ≅ ker h̄_{i+k} . Another proof of this fact is given by Lemma 1 of [5].

In the case where X is (r − 1)-connected and i = r, k = 1 , we have the following exactness result:

Proposition 1.2. For an (r − 1)-connected spectrum X , the homomorphism χ̄_{r+1} : Γ_{r+1}(X) → π_{r+1}X in the Whitehead exact sequence is exactly the external product ∧ : π_{r}X ⊗ π_{1}S → π_{r+1}X .

Proof. Consider the commutative diagram

H_{r}X ⊗ H_{1}S(0) ⟶_{≅}^{∧} H_{r+1}(X ∧ S(0))
↑ ≅  ↑ ≅
π_{r}X ⊗ π_{1}S(0) ⟶^{∧} π_{r+1}(X ∧ S(0)) = Γ_{r+1}(X)
≅ ↓ (id)_{∗}⊗(γ_{0})_{∗}  ↓ χ̄_{r+1}
π_{r}X ⊗ π_{1}S ⟶^{∧} π_{r+1}(X ∧ S) ≅ π_{r+1}X .

The top horizontal homomorphism is an isomorphism by Künneth formula and the two top vertical arrows, which are Hurewicz homomorphisms are isomorphisms since X is (r − 1)-connected, S(0) is 0-connected and X ∧ S(0) is r -connected. Consequently, the external product in the middle of the diagram is an isomorphism. The homomorphism (id)_{∗} ⊗ (γ_{0})_{∗} is an isomorphism because (γ_{0})_{∗} : π_{1}S(0) ⟶^{≅} π_{1}S . Therefore, χ̄_{r+1} is exactly the external product π_{r}X ⊗ π_{1}S ⟶^{∧} π_{r+1}X .

Corollary 1.3. For any (r − 1)-connected spectrum X , the following sequence is exact:

⋯ ⟶ Γ_{r+2}(X) ⟶^{χ̄_{r+2}} π_{r+2}X ⟶^{h̄_{r+2}} H_{r+2}X ⟶^{ν̄_{r+2}} π_{r}X ⊗ π_{1}S ⟶^{∧} π_{r+1}X ⟶^{h̄_{r+1}} H_{r+1}X ⟶ 0 .

2. Postnikov cofibrations

The purpose of this section is to present another approach of the study of the Hurewicz homomorphism. For an (r − 1)-connected spectrum X , consider for all integers i ≥ r + 1 the cofibrations of spectra

Σ^{i}H(π_{i}X) ⟶^{γ_{i−1}} X[i] ⟶^{α_{i−1}} X[i − 1] ,

where α_{i−1} is the (i − 1)-st Postnikov section of X[i] : let us call them the Postnikov cofibrations of X . The associated homology exact sequences are

⋯ ⟶ H_{i+1}X[i] ⟶^{(α_{i−1})_{∗}} H_{i+1}X[i − 1] →^{¯∂} \underbrace{H_{i}(Σ^{i}H(π_{i}X))}_{≅π_{i}X} ⟶^{(γ_{i−1})_{∗}} \underbrace{H_{i}X[i]}_{≅H_{i}X} ⟶^{(α_{i−1})_{∗}} H_{i}X[i − 1] ⟶ 0 ,

and it is easy to check that (γ_{i−1})_{∗} is the stable Hurewicz homomorphism h̄_{i} . Thus, we obtain the following

Proposition 2.1. Let X be an (r − 1)-connected spectrum and i an integer ≥ r + 1 . There is an exact sequence

⋯ ⟶ H_{i+1}X[i] ⟶^{(α_{i−1})_{∗}} H_{i+1}X[i − 1] ⟶^{¯∂} π_{i}X ⟶^{h̄_{i}} H_{i}X ⟶^{(α_{i−1})_{∗}} H_{i}X[i − 1] ⟶ 0 .

Now let us try to understand the homomorphism ¯∂ for the cases i = r + 1 and i = r + 2 .

Proposition 2.2. (a) For any (r − 1)-connected spectrum X , there is an exact sequence

0 ⟶ H_{r+2}X[r + 1] ⟶^{φ̄} π_{r}X ⊗ π_{1}S ⟶^{¯∂} π_{r+1}X ⟶^{h̄_{r+1}} H_{r+1}X ⟶ 0 .

(b) The homomorphism ¯∂ is again exactly the external product ∧ : π_{r}X ⊗ π_{1}S → π_{r+1}X .

Proof. Let us look at the Postnikov cofibration of X for i = r + 1 ,

Σ^{r+1}H(π_{r+1}X) ⟶^{γ_{r}} X[r + 1] ⟶^{α_{r}} X[r] ≃ Σ^{r}H(π_{r}X) ,

and take its homology exact sequence

\underbrace{H_{r+2}(Σ^{r+1}H(π_{r+1}X))}_{=0} ⟶ H_{r+2}X[r + 1] ⟶^{φ̄} H_{r+2}(Σ^{r}H(π_{r}X)) ⟶^{¯∂}

π_{r+1}X ⟶^{h̄_{r+1}} H_{r+1}X ⟶ \underbrace{H_{r+1}(Σ^{r}H(π_{r}X))}_{=0} ,

where φ̄ is written for (α_{r})_{∗} . Since Σ^{r}H(π_{r}X) is an Eilenberg-MacLane spectrum, it is clear that

H_{r+2}(Σ^{r}H(π_{r}X)) ≅ Γ_{r+1}(Σ^{r}H(π_{r}X)) ≅ π_{r}X ⊗ π_{1}S ,

and we get assertion (a). Then, consider the map α_{0} : S → H(ℤ) and denote by ζ the composition

(id ∧ α_{0}) γ_{r} : Σ^{r+1}H(π_{r+1}X) ⟶ X[r + 1] ≃ X[r + 1] ∧ S ⟶ X[r + 1] ∧ H(ℤ) ,

and by F its fiber. By smashing with H(ℤ) the cofibration obtained by looping the base spectrum of the cofibration Σ^{r+1}H(π_{r+1}X) ⟶^{γ_{r}} X[r + 1] ⟶^{α_{r}} Σ^{r}H(π_{r}X) , we get the commutative diagram

Σ^{r−1}H(π_{r}X) ∧ H(ℤ) ⟶ Σ^{r+1}H(π_{r+1}X) ∧ H(ℤ) ⟶^{γ_{r}∧id} X[r + 1] ∧ H(ℤ)
↑  ↑ id∧α_{0}  ↑ id
F  ⟶  Σ^{r+1}H(π_{r+1}X)  ⟶^{ζ} X[r + 1] ∧ H(ℤ)
↓  ↓ γ_{r}  ↓ id
X[r + 1] ∧ S(0)  ⟶^{id∧γ_{0}}  X[r + 1]  ⟶^{id∧α_{0}} X[r + 1] ∧ H(ℤ)

in which all rows are cofibrations. Then look at their homotopy exact sequences

H_{r+2}X[r + 1] ⟶ \overbrace{H_{r+1}(Σ^{r−1}H(π_{r}X))}^{≅H_{r+2}(Σ^{r}H(π_{r}X))} ⟶^{¯∂} \overbrace{H_{r+1}(Σ^{r+1}H(π_{r+1}X))}^{≅π_{r+1}X} ⟶_{=h̄_{r+1}}^{(γ_{r}∧id)_{∗}}  \overbrace{H_{r+1}X[r + 1]}^{≅H_{r+1}X}
↑ =  ↑  Hurewicz ↑ ≅  ↑ =
H_{r+2}X[r + 1] ⟶  π_{r+1}F  ⟶  π_{r+1}X  ⟶^{ζ_{∗}} H_{r+1}X[r + 1]
↓ =  ↓  (γ_{r})_{∗} ↓ ≅  ↓ =
H_{r+2}X[r + 1] ⟶  \underbrace{Γ_{r+1}(X[r + 1])}_{≅Γ_{r+1}(X)}  ⟶^{χ̄_{r+1}}  \underbrace{π_{r+1}X[r + 1]}_{≅π_{r+1}X}  ⟶^{h̄_{r+1}}  \underbrace{H_{r+1}X[r + 1]}_{≅H_{r+1}X} .

Observe that the three horizontal arrows on the left of the diagram are injective and conclude by the five lemma that the two vertical arrows starting from π_{r+1}F are isomorphisms: assertion (b) can then be deduced from Proposition 1.2.

Remark 2.3. It follows from Corollary 1.3 and Proposition 2.2 that the cokernel of h̄_{r+2} : π_{r+2}X → H_{r+2}X is isomorphic to image ν̄_{r+2} = ker (∧ : π_{r}X ⊗ π_{1}S → π_{r+1}X) ≅ H_{r+2}X[r + 1] for any (r − 1)-connected spectrum X .

Similarly, we can investigate the stable Hurewicz homomorphism in dimension r+2 . Consider the Postnikov cofibration of an (r − 1)-connected spectrum X for i = r + 2 and its homology exact sequence

⋯ ⟶ H_{r+3}X[r + 1] ⟶^{ψ̄} \underbrace{H_{r+2}(Σ^{r+2}H( pi_{r+2}X))}_{≅π_{r+2}X} ⟶^{h̄_{r+2}} \underbrace{H_{r+2}X[r + 2]}_{≅H_{r+2}X} ⟶ H_{r+2}X[r + 1] ⟶ 0 ,

where ψ̄ is written for ¯∂. The next two lemmas describe the group H_{r+3}X[r + 1] and the homomorphism ψ̄.

Lemma 2.4. There is an exact sequence

⋯ ⟶ π_{r+1}X ⊗ π_{1}S ⟶^{θ̄} H_{r+3}X[r + 1] ⟶^{η̄} _{2}(π_{r}X) ⟶ 0 ,

where _{2}(π_{r}X) denotes the subgroup of elements of order dividing 2 in the group π_{r}X .

Proof. Let us look again at the cofibration

Σ^{r+1}H(π_{r+1}X) ⟶^{γ_{r}} X[r + 1] ⟶^{α_{r}} Σ^{r}H(π_{r}X)

and at its homology exact sequence

⋯ ⟶ H_{r+3}(Σ^{r+1}H(π_{r+1}X)) ⟶^{θ̄} H_{r+3}X[r + 1] ⟶^{η̄} H_{r+3}(Σ^{r}H(π_{r}X)) ⟶ 0 ,

where θ̄ and η̄ are the homomorphisms induced by γ_{r} and α_{r} respectively. It turns out that H_{r+3}(Σ^{r+1}H(π_{r+1}X)) ≅ Γ_{r+2}(Σ^{r+1}H(π_{r+1}X)) ≅ π_{r+1}X ⊗ π_{1}S because of the results of Section 1 and that H_{r+3}(Σ^{r}H(π_{r}X)) ≅ _{2}(π_{r}X) , according to Théorème 2 of [14].

Lemma 2.5. The composition ψ̄ θ̄: π_{r+1}X ⊗ π_{1}S → π_{r+2}X is the external product ∧.

Proof. The obvious map X(r) → X provides the commutative diagram of cofibrations

Σ^{r+2}H(π_{r+2}X) ⟶ X(r, r + 2] ⟶ X(r, r + 1] ≃ Σ^{r+1}H(π_{r+1}X)
↓ id  ↓  ↓ γ_{r}
Σ^{r+2}H(π_{r+2}X) ⟶ X[r + 2] ⟶ X[r + 1]

which induces the commutative square

\underbrace{H_{r+3}(Σ^{r+1}H(π_{r+1}X))}_{≅π_{r+1}X⊗π_{1}S} ⟶ \underbrace{H_{r+2}(Σ^{r+2}H(π_{r+2}X))}_{≅π_{r+2}X}

↓ θ̄  ↓ =
H_{r+3}X[r + 1]  ⟶^{ψ̄} \underbrace{H_{r+2}(Σ^{r+2}H(π_{r+2}X))}_{≅π_{r+2}X} .

Then, the statement of Proposition 2.2 for the r-connected spectrum X(r) shows that the top horizontal arrow is the external product.

We may summarize our results on the stable Hurewicz homomorphism h̄_{r+2} as follows.

Proposition 2.6. Let X be an (r − 1)-connected spectrum.

(a) There is an exact sequence

⋯ ⟶ H_{r+3}X[r + 1] ⟶^{ψ̄} π_{r+2}X ⟶^{h̄_{r+2}} H_{r+2}X ⟶ H_{r+2}X[r + 1] ⟶ 0 .

(b) The kernel of h̄_{r+2} fits into the short exact sequence

0 ⟶ ∧(π_{r+1}X ⊗ π_{1}S) ⟶ ker h̄_{r+2} ⟶ Q ⟶ 0 ,

where Q is a quotient of _{2}(π_{r}X) .

Remark 2.7. Since π_{i}X[r + 1] = 0 for i ≥ r + 2 , H_{r+3}X[r + 1] ≅ Γ_{r+2}(X[r + 1]) . The Postnikov section X → X[r + 1] induces a map f between the Atiyah-Hirzebruch spectral sequences

H_{s}(X; π_{t}S(0)) ⟹ Γ_{s+t}(X) and H_{s}(X[r + 1]; π_{t}S(0)) ⟹ Γ_{s+t}(X[r + 1]) .

The lines s + t = r + 2 in these spectral sequences give the following picture:

H_{r+2}(X; π_{1}S) ⟶^{d^{2}} π_{r}X ⊗ π_{2}S ⟶ Γ_{r+2}(X) ⟶ H_{r+1}(X; π_{1}S) ⟶ 0
↓ f_{1}  ≅ ↓ f_{2}  ↓ f_{3}  ≅ ↓ f_{4}
H_{r+2}(X[r + 1]; π_{1}S) ⟶^{d^{2}} π_{r}X[r + 1] ⊗ π_{2}S ⟶ Γ_{r+2}(X[r + 1]) ⟶ H_{r+1}(X[r + 1]; π_{1}S) ⟶ 0 .

By the universal coefficient theorem, one has H_{r+2}(X; π_{1}S) ≅ (H_{r+2}X ⊗ π_{1}S) ⊕ Tor(H_{r+1}X, π_{1}S) and H_{r+2}(X[r + 1]; π_{1}S) ≅ (H_{r+2}(X[r + 1]) ⊗ π_{1}S) ⊕ Tor(H_{r+1}X, π_{1}S) ; thus, one can check that f_{1} is surjective because of Whitehead’s theorem and deduce from the five lemma that

Γ_{r+2}(X) ≅ Γ_{r+2}(X[r + 1]) ≅ H_{r+3}X[r + 1] .

Moreover, one can show with the argument of the proof of Proposition 2.2 (b) that the homomorphism ψ̄ of Proposition 2.6 (a) is actually χ̄_{r+2} : Γ_{r+2}(X) → π_{r+2}X of Corollary 1.3. Consequently, the part

H_{r+3}X[r + 1] ⟶^{ψ̄} π_{r+2}X ⟶^{h̄_{r+2}} H_{r+2}X

of the sequence given by Proposition 2.6 (a) is a piece of the Whitehead exact sequence.

Remark 2.8. It follows from Lemma 2.4 and the previous remark that the group Γ_{r+2}(X) is described by the exact sequence

⋯ ⟶ π_{r+1}X ⊗ π_{1}S ⟶^{θ̄} Γ_{r+2}(X) ⟶^{η̄} _{2}(π_{r}X) ⟶ 0 ,

and in particular that its exponent divides 4 (this was already known by [10], Section 4).

Remark 2.9. All exact sequences introduced in Sections 1 and 2 are obviously natural in X .

3. Products and Hurewicz homomorphisms in algebraic K-theory

If R is any ring, let us denote by X_{R} the connective K-theory spectrum of R, i.e., a (−1)-connected Ω-spectrum whose 0-th space is the infinite loop space BGL(R)^{+} × K_{0}(R) . We shall also consider the (r −1)-connected spectra X_{R}(r −1) for r ≥ 0 , i.e., the fiber of the Postnikov section X_{R} → X_{R}[r −1] , and call γ_{r−1} the obvious map X_{R}(r − 1) → X_{R} . Observe that K_{i}(R) ≅ π_{i}X_{R}(r − 1) for i ≥ r. Remember that the infinite loop spaces corresponding to X_{R}(0) , X_{R}(1) and X_{R}(2) are BGL(R)^{+} , BE(R)^{+} and BSt(R)^{+} respectively. If R and R′ are two rings, there is a pairing μ : X_{R} ∧ X_{R′} → X_{R⊗R′} and the product in algebraic K-theory is defined as follows

⋆ : K_{i}(R) ⊗ K_{k}(R′) ≅ π_{i}X_{R} ⊗ π_{k}X′_{R} ⟶^{∧} π_{i+k}(X_{R} ∧ X_{R′}) ⟶^{μ_{∗}} π_{i+k}X_{R⊗R′} ≅ K_{i+k}(R ⊗ R′)

for any two integers i ≥ 0 and k ≥ 0 (see for instance [22], Proposition 2.4.2). We shall actually concentrate our attention to the special case where R′ is the ring of integers ℤ: the goal of Sections 3 and 4 is to investigate the relationships between the image of the product

⋆ : K_{i}(R) ⊗ K_{k}(ℤ) ⟶ K_{i+k}(R ⊗ ℤ) ≅ K_{i+k}(R)

and the kernel of the stable and the non-stable Hurewicz homomorphism.

Remember that X_{ℤ} is a ring spectrum and let us call j : S → X_{ℤ} its identity. Notice that j corresponds to the map BΣ_{∞}^{+} → BGL(ℤ)^{+} given by the inclusion of the infinite symmetric group Σ_{∞} into GL(ℤ) . This map j induces an isomorphism j_{∗} : π_{1}S ⟶^{≅} π_{1}X_{ℤ} ≅ K_{1}(ℤ) and the image of j_{∗} : π_{k}S → K_{k}(ℤ) for k ≥ 2 is described in [25] and [29]. For any ring R, the above pairing μ provides then X_{R} with an X_{ℤ}-module structure. Let us first translate the results of Sections 1 and 2 in terms of algebraic K-theory.

Proposition 3.1. Let R be a ring, i and k two integers with i ≥ 0 , k ≥ 1 , and consider an element x ∈ K_{i}(R) and an element y ∈ K_{k}(ℤ) belonging to the image of j_{∗} : π_{k}S → K_{k}(ℤ) .

(a) For all r ≤ i, x⋆y is an element of the kernel of the stable Hurewicz homomorphism h̄_{i+k} : K_{i+k}(R) → H_{i+k}X_{R}(r − 1) .

(b) If k ≤ i − 1 , then x ⋆ y is an element of the kernel of the non-stable Hurewicz homomorphisms h_{i+k} : K_{i+k}(R) → H_{i+k}E(R) and h_{i+k} : K_{i+k}(R) → H_{i+k}GL(R) .

(c) If i ≥ 3 and k ≤ i − 1 , then x ⋆ y is an element of the kernel of h_{i+k} : K_{i+k}(R) → H_{i+k}St(R) .

Proof. The first assertion is a consequence of Lemma 1.1 and the commutativity of the diagram

π_{i}X_{R}(r − 1) ⊗ π_{k}S ⟶^{∧} π_{i+k}X_{R}(r − 1) ⟶^{h̄_{i+k}} H_{i+k}X_{R}(r − 1)
≅ ↓ (γ_{r−1})_{∗}⊗id  ≅ ↓ (γ_{r−1})_{∗}
π_{i}X_{R} ⊗ π_{k}S  ⟶^{∧}  π_{i+k}X_{R}
↓ ≅⊗j_{∗}  ↓ ≅
K_{i}(R) ⊗ K_{k}(ℤ) ⟶^{⋆}  K_{i+k}(R) ,

where the bottom square commutes because X_{R} is an X_{ℤ}-module. In order to prove the last two assertions, consider the (i − 1)-connected cover BGL(R)^{+}(i − 1) of the CW-complex BGL(R)^{+} , for i ≥ k + 1 ≥ 2 . The iterated homology suspension σ : H_{i+k}BGL(R)^{+}(i − 1) → H_{i+k}X_{R}(i − 1) , which is an isomorphism since k ≤ i − 1 , and the commutative diagram

K_{i+k}(R) ⟶^{h_{i+k}} H_{i+k}BGL(R)^{+}(i − 1)
↓ =  σ ↓ ≅
K_{i+k}(R) ⟶^{h̄_{i+k}}  H_{i+k}X_{R}(i − 1)

show that h_{i+k} : K_{i+k}(R) → H_{i+k}BGL(R)^{+}(i−1) fulfills h_{i+k}(x⋆y) = 0 according to (a) for r = i. Since i ≥ 2 , assertion (b) then follows from the composition with the obvious homomorphism

H_{i+k}BGL(R)^{+}(i − 1) ⟶ H_{i+k}BE(R)^{+} ⟶ H_{i+k}BGL(R)^{+} .

If i ≥ 3 , this homomorphism factors even through H_{i+k}BSt(R)^{+} and we get (c).

Now, let us consider the case k = 1 and i = r: the fact that j_{∗} : π_{1}S → K_{1}(ℤ) is an isomorphism implies the following result, where X_{R}(i − 1, i + 1] is written for X_{R}(i − 1)[i + 1] .

Theorem 3.2. Let R be any ring.

(a) For any integer i ≥ 0 , there is a natural exact sequence

K_{i+2}(R) ⟶^{h̄_{i+2}} H_{i+2}X_{R}(i − 1) ⟶^{ν̄_{i+2}} K_{i}(R) ⊗ K_{1}(ℤ) ⟶^{⋆} K_{i+1}(R) ⟶^{h̄_{i+1}} H_{i+1}X_{R}(i − 1) ⟶ 0 .

Moreover, ker(⋆) = image ν̄_{i+2} ≅ H_{i+2}X_{R}(i − 1, i + 1] .

(b) For any integer i ≥ 2 , the image of ⋆ : K_{i}(R)⊗K_{1}(ℤ) → K_{i+1}(R) is contained in the kernel of the nonstable Hurewicz homomorphisms h_{i+1} : K_{i+1}(R) → H_{i+1}E(R) and h_{i+1} : K_{i+1}(R) → H_{i+1}GL(R) .

(c) For any integer i ≥ 3 , the image of ⋆ : K_{i}(R) ⊗ K_{1}(ℤ) → K_{i+1}(R) is contained in the kernel of h_{i+1} : K_{i+1}(R) → H_{i+1}St(R) .

Proof. Assertion (a) follows from Corollary 1.3 and Remark 2.3 for the spectrum X_{R}(i−1) since the diagram

K_{i}(R) ⊗ π_{1}S ⟶^{∧} π_{i+1}X_{R}
≅ ↓ id⊗j_{∗}  ↓ ≅
K_{i}(R) ⊗ K_{1}(ℤ) ⟶^{⋆} K_{i+1}(R)

commutes again because of the X_{ℤ}-module structure of X_{R} . Assertions (b) and (c) are direct consequences of Proposition 3.1. (b) and (c).

It is possible to obtain a similar information on the stable and the non-stable Hurewicz homomorphism in any dimension i ≥ r + 1 . Proposition 2.1 provides the exact sequence

⋯ ⟶ H_{i+1}X_{R}(r − 1, i] ⟶^{(α_{i−1})_{∗}} H_{i+1}X_{R}(r − 1, i − 1] ⟶^{¯∂}

K_{i}(R) ⟶^{h̄_{i}} H_{i}X_{R}(r − 1) ⟶^{(α_{i−1})_{∗}} H_{i}X_{R}(r − 1, i − 1] ⟶ 0 .

Proposition 3.3. Let R be any ring, i and r positive integers such that r ≤ i ≤ 2r − 1 , then the kernel of the non-stable Hurewicz homomorphism

h_{i} : K_{i}(R) → H_{i}BGL(R)^{+}(r − 1)

is exactly the image of ¯∂.

Proof. Let us consider the homology exact sequence of the Postnikov cofibration

Σ^{i}(H(K_{i}(R)) ⟶^{γ_{i−1}} X_{R}(r − 1, i] ⟶^{α_{i−1}} X_{R}(r − 1, i − 1] ,

and the corresponding homology exact sequence obtained from the Serre spectral sequence of the fibration of CW-complexes

K(K_{i}(R), i) ⟶ BGL(R)^{+}(r − 1, i] ⟶ BGL(R)^{+}(r − 1, i − 1] .

We obtain the commutative diagram

H_{i+1}X_{R}(r − 1, i − 1]  ⟶^{¯∂} K_{i}(R) ⟶^{h̄_{i}}  H_{i}X_{R}(r − 1)
↑ σ  ↑ =  ↑
H_{i+1}BGL(R)^{+}(r − 1, i − 1] ⟶^{∂} K_{i}(R) ⟶^{h_{i}} H_{i}BGL(R)^{+}(r − 1) ,

where the horizontal sequences are exact and the three vertical arrows are iterated suspensions. The left iterated homology suspension σ is surjective if i + 1 ≤ 2r and even an isomorphism if i + 1 ≤ 2r − 1 (see [42], p. 382); consequently we may conclude that image ∂ = image ¯∂.

4. Products and the non-stable Hurewicz homomorphism in low dimensions

The purpose of this section is to study the relationships between the algebraic K-theory of a ring R and the integral homology of its linear groups in low dimensions. In dimension 2 , the following isomorphisms are known (see [3]):

K_{2}(R) ≅ H_{2}E(R) and H_{2}GL(R) ≅ K_{2}(R) ⊕ Λ^{2}(K_{1}(R)) .

Let us start by looking at dimensions 3 and 4 . Let Γ(−) be the quadratic functor defined on abelian groups by J.H.C. Whitehead in Section 5 of [43]: if Y is a simply connected CW-complex, then Γ_{3}(Y) turns out to be isomorphic to Γ(π_{2}Y) .

Theorem 4.1. For any ring R, there is a natural exact sequence

K_{4}(R) ⟶^{h_{4}} H_{4}E(R) ⟶^{ν_{4}} Γ(K_{2}(R)) ⟶^{χ_{3}} K_{3}(R) ⟶^{h_{3}} H_{3}E(R) ⟶ 0

and ker h_{3} is isomorphic to the image of the product homomorphism ⋆ : K_{2}(R) ⊗ K_{1}(ℤ) → K_{3}(R) . In particular, H_{3}E(R) ≅ K_{3}(R)/(K_{2}(R) ⋆ K_{1}(ℤ)) .

Proof. Consider the exact sequence given by Proposition 2.2 for X = X_{R}(1) and r = 2 , and also the corresponding exact sequence obtained from the Serre spectral sequence of the fibration of CW-complexes

K(K_{3}(R), 3) ⟶ BE(R)^{+}[3] ⟶ K(K_{2}(R), 2) .

We get the commutative diagram

0 ⟶ H_{4}X_{R}(1, 3] ⟶^{φ̄} \overbrace{H_{4}(Σ^{2}H(K_{2}(R)))}^{K_{2}(R)⊗K_{1}(ℤ)} ⟶^{⋆} K_{3}(R) ⟶^{h̄_{3}} H_{3}X_{R}(1) ⟶ 0
↑  ↑ σ  ↑ =  ↑
0 ⟶ H_{4}BE(R)^{+}[3] ⟶^{φ} H_{4}(K(K_{2}(R), 2)) ⟶^{∂} K_{3}(R) ⟶^{h_{3}} H_{3}BE(R)^{+} ⟶ 0 ,

where the vertical arrows are iterated suspensions. According to the proof of Proposition 3.3, the iterated homology suspension σ is surjective and thus image ∂ = K_{2}(R)⋆K_{1}(ℤ) . We may also look at the Whitehead exact sequence (see [43]) of the space BE(R)^{+} :

⋯ ⟶ Γ_{4}(BE(R)^{+}) ⟶^{χ_{4}} K_{4}(R) ⟶^{h_{4}} H_{4}E(R) ⟶^{ν_{4}} Γ_{3}(BE(R)^{+}) ⟶^{χ_{3}} K_{3}(R) ⟶^{h_{3}} H_{3}E(R) ⟶ 0 .

It turns out that H_{4}(K(K_{2}(R), 2)) ≅ Γ_{3}(K(K_{2}(R), 2)) ≅ Γ(K_{2}(R)) ≅ Γ_{3}(BE(R)^{+}) and the argument of the proof of Proposition 2.2 shows again that one can replace a piece of that Whitehead exact sequence by the bottom sequence of the above diagram. This proves the assertion.

Remark 4.2. This extends the result for fields given in [35], Corollary 5.2, to the case of any ring R.

In order to understand the 4-dimensional and 5-dimensional Hurewicz homomorphisms, let us use exactly the same idea (but now for r = 3 ) for the 2-connected CW-complex BSt(R)^{+} , respectively the 2-connected K-theory spectrum X_{R}(2) .

Theorem 4.3. Let R be any ring.

(a) There is a natural exact sequence

H_{6}X_{R}(2, 4] ⟶^{ψ̄} K_{5}(R) ⟶^{h_{5}} H_{5}St(R) ⟶^{ν_{5}} K_{3}(R) ⊗ K_{1}(ℤ) ⟶^{⋆} K_{4}(R) ⟶^{h_{4}} H_{4}St(R) ⟶ 0 .

In particular, H_{4}St(R) ≅ K_{4}(R)/(K_{3}(R) ⋆ K_{1}(ℤ)) .

(b) There is a natural exact sequence

K_{4}(R) ⊗ K_{1}(ℤ) ⟶^{θ̄} H_{6}X_{R}(2, 4] ⟶^{η̄} _{2}(K_{3}(R)) ⟶ 0 .

(c) The composition ψ̄ θ̄ is the product map ⋆ : K_{4}(R)⊗K_{1}(ℤ) → K_{5}(R) . Consequently, there is a natural

short exact sequence

0 ⟶ K_{4}(R) ⋆ K_{1}(ℤ) ⟶ ker h_{5} ⟶ Q ⟶ 0 ,

where Q is a quotient of _{2}(K_{3}(R)) .

Proof. As in the previous proof, we use Proposition 2.2, but consider in this case the following commutative diagram:

0 ⟶ H_{5}X_{R}(2, 4] ⟶^{φ̄} \overbrace{H_{5}(Σ^{3}H(K_{3}(R)))}^{K_{3}(R)⊗K_{1}(ℤ)} ⟶^{⋆} K_{4}(R) ⟶^{h̄_{4}} H_{4}X_{R}(2) ⟶ 0
↑  ↑ σ  ↑ =  ↑
0 ⟶ H_{5}BSt(R)^{+}[4] ⟶^{φ} H_{5}(K(K_{3}(R), 3)) ⟶^{∂} K_{4}(R) ⟶^{h_{4}} H_{4}BSt(R)^{+} ⟶ 0 .

However, this time, σ is even an isomorphism. Observe that H_{5}BSt(R)^{+}[4] is isomorphic to the kernel of ⋆ : K_{3}(R) ⊗ K_{1}(ℤ) → K_{4}(R) . The Whitehead exact sequence of BSt(R)^{+} is

⋯ ⟶ Γ_{5}(BE(R)^{+}) ⟶^{χ_{5}} K_{5}(R) ⟶^{h_{5}} H_{5}St(R) ⟶^{ν_{5}} Γ_{4}(BSt(R)^{+}) ⟶^{χ_{4}} K_{4}(R) ⟶^{h_{4}} H_{4}St(R) ⟶ 0

and it is easy to check that Γ_{4}(BSt(R)^{+}) ≅ K_{3}(R) ⊗ K_{1}(ℤ) ≅ H_{5}(K(K_{3}(R), 3)) . In order to understand the kernel of h_{5} , let us use the exact sequence given by Proposition 2.6 for r = 3 , i = 5 , and the exact sequence coming from the homology Serre spectral sequence of the fibration of CW-complexes

K(K_{5}(R), 5) ⟶ BSt(R)^{+}[5] ⟶ BSt(R)^{+}[4] .

H_{6}X_{R}(2, 4] ⟶^{ψ̄} K_{5}(R) ⟶^{h̄_{5}} H_{5}X_{R}(2) ⟶ H_{5}X_{R}(2, 4] ⟶ 0
↑ σ  ↑ =  ↑  ↑
H_{6}BSt(R)^{+}[4] ⟶^{ψ} K_{5}(R) ⟶^{h_{5}} H_{5}BSt(R)^{+} ⟶ H_{5}BSt(R)^{+}[4] ⟶ 0 ,

where the vertical arrows are iterated suspensions. According to the proof of Proposition 3.3, the iterated homology suspension σ is surjective and therefore image ψ = image ψ̄. On the other hand, the group H_{6}X_{R}(2, 4] and the image of ψ̄ may be described by Proposition 2.6 and Lemmas 2.4 and 2.5.

The following corollary follows from the five lemma and the argument of the proofs of Theorems 4.1 and 4.3.

Corollary 4.4. For any ring R, the iterated homology suspensions H_{3}E(R) ≅ H_{3}BE(R)^{+} → H_{3}X_{R}(1) and H_{4}St(R) ≅ H_{4}BSt(R)^{+} → H_{4}X_{R}(2) are isomorphisms.

Remark 4.5. Observe that h_{4} : K_{4}(R) → H_{4}St(R) is an isomorphism up to 2-torsion. This produces the following consequence of Proposition 9 of [8]. Let l be an odd prime, ξ_{l} a primitive l-root of unity of order l. Let R = ℤ[ξ_{l} + ξ_{l}^{−1}] be the ring of integers of the maximal real subfield of the cyclotomic field ℚ(ξ_{l}) . The vanishing of the group H_{4}St(R) in this case would imply the Kummer-Vandiver conjecture for the prime l.

5. Products in the algebraic K-theory of the ring of integers ℤ

This section is devoted to the study of products in the algebraic K-theory of the ring of integers ℤ:

⋆ : K_{i}(ℤ) ⊗ K_{k}(ℤ) ⟶ K_{i+k}(ℤ) .

Let us start by describing the results on low-dimensional products given by Section 4 in the case where R = ℤ.

Proposition 5.1.

(a) The product homomorphism ⋆ : K_{i}(ℤ) ⊗ K_{1}(ℤ) → K_{i+1}(ℤ) is an isomorphism if i = 1 , injective if i = 2 , and trivial if i ≢ 1 or 2 mod 8.

(b) The product homomorphism ⋆ : K_{i}(ℤ) ⊗ K_{2}(ℤ) → K_{i+2}(ℤ) is trivial if i ≢ 1 mod 8.

(c) H_{4}SL(ℤ) ≅ ℤ/2 and H_{4}St(ℤ) = 0 .

(d) There is a short exact sequence

0 ⟶ K_{5}(ℤ) ⟶^{h_{5}} H_{5}St(ℤ) ⟶^{ν_{5}} ℤ/2 ⟶ 0 .

Proof. The assertion (a) is well known for i = 1 . Theorem 4.1 produces the exact sequence

K_{4}(ℤ) ⟶^{h_{4}} H_{4}SL(ℤ) ⟶^{ν_{4}} \underbrace{Γ(ℤ/2)}_{ℤ/4} ⟶^{χ_{3}} \underbrace{K_{3}(ℤ)}_{ℤ/48} ⟶^{h_{3}} \underbrace{H_{3}SL(ℤ)}_{ℤ/24} ⟶ 0

(see [1], [20] and [43], Sections 5 and 13) and asserts that the product ⋆ : K_{2}(ℤ) ⊗ K_{1}(ℤ) → K_{3}(ℤ) is injective. Recently, C. Weibel deduced from the work of V. Voevodsky [37] the complete calculation of the 2-torsion of the algebraic K-theory of ℤ (see Table 1 of [41]). This, together with an argument of J. Rognes, shows that K_{4}(ℤ) = 0 and implies that H_{4}SL(ℤ) is cyclic of order 2 . Moreover, K_{i}(ℤ) is a finite odd torsion group if i is a positive integer ≡ 0 , 4 , or 6 mod 8. Therefore, K_{i}(ℤ)⋆K_{1}(ℤ) = 0 if i ≡ 0 , 4 , or 6 mod 8 or if i+1 ≡ 0 , 4 , or 6 mod 8. This gives (a), and (b) follows from (a) since K_{2}(ℤ) = K_{1}(ℤ)⋆K_{1}(ℤ) . Note that the first author proved the triviality of ⋆ : K_{3}(ℤ) ⊗ K_{1}(ℤ) → K_{4}(ℤ) in [5] before Weibel’s proof of the vanishing of K_{4}(ℤ) . Now, let us apply Theorem 4.3. The map ψ̄ is actually the connecting homomorphism of the homology exact sequence of the cofibration

Σ^{5}(H(K_{5}(ℤ)) ⟶ X_{ℤ}(2, 5] ⟶ X_{ℤ}(2, 4] .

It is of course possible to consider the analogous cofibration for the sphere spectrum S

Σ^{5}(H(π_{5}(S)) ⟶ S(2, 5] ⟶ S(2, 4] .

The identity j : S → X_{ℤ} of the ring spectrum X_{ℤ} induces the commutative diagram

H_{6}S(2, 4] ⟶ π_{5}(S) = 0
↓ j_{∗}  ↓
H_{6}X_{ℤ}(2, 4] ⟶^{ψ̄} K_{5}(ℤ)

which shows that ψ̄ j_{∗} = 0 . Now, look at the following commutative diagram where the bottom homomorphism is given by the second assertion of Theorem 4.3:

H_{6}S(2, 4] ⟶^{≅} H_{6}(Σ^{3}H(π_{3}(S))) ≅ _{2}(π_{3}(S))
↓ j_{∗}  ↓ ≅
H_{6}X_{ℤ}(2, 4] ⟶^{η̄} H_{6}(Σ^{3}H(K_{3}(ℤ))) ≅ _{2}(K_{3}(ℤ)) .

The top horizontal arrow is an isomorphism because the vanishing of π_{4}(S) exhibits an equivalence S(2, 4] ≃ Σ^{3}H(π_{3}(S)) . The right vertical arrow is an isomorphism since the homomorphism π_{3}(S) → K_{3}(ℤ) , induced by j , is injective (remember that π_{3}(S) ≅ ℤ/24 and K_{3}(ℤ) ≅ ℤ/48 ). Therefore, there exists a splitting τ : _{2}(K_{3}(ℤ)) → H_{6}X_{ℤ}(2, 4] of η̄ such that τ is the composition of an isomorphism _{2}(K_{3}(ℤ)) ⟶^{≅} H_{6}S(2, 4] with j_{∗} . It then follows from the vanishing of the composition ψ̄ j_{∗} that ψ̄ τ = 0 . Consequently, the group Q of Theorem 4.3 (c) is trivial if R = ℤ since Q = image (η̄τ) , and ker h_{5} ≅ image (ψ̄ θ̄) is the image of the product map ⋆ : K_{4}(ℤ) ⊗ K_{1}(ℤ) → K_{5}(ℤ) . Consequently, there is an exact sequence

0 ⟶ K_{4}(ℤ) ⋆ K_{1}(ℤ) ⟶^{ψ̄ θ̄} K_{5}(ℤ) ⟶^{h_{5}} H_{5}St(ℤ) ⟶^{ν_{5}} K_{3}(ℤ) ⊗ K_{1}(ℤ) ⟶^{⋆} K_{4}(ℤ) ⟶_{≅}^{h_{4}} H_{4}St(ℤ) ⟶ 0 .

The fact that K_{4}(ℤ) = 0 provides the vanishing of H_{4}St(ℤ) and the short exact sequence (d) (which was already given in Section 6 of [6])

0 ⟶ K_{5}(ℤ) ⟶^{h_{5}} H_{5}St(ℤ) ⟶^{ν_{5}} ℤ/2 ⟶ 0 .

According to Theorem 1 of [21], Theorem 1 of [32] and Table 1 of [41], K_{5}(ℤ) ≅ ℤ ⊕ T , where T is a finite abelian 3-group: thus, H_{5}St(ℤ) ≅ ℤ ⊕ T and h_{5} is multiplication by 2 on the infinite cyclic factor of K_{5}(ℤ) .

Corollary 5.2. Let R be a ring such that the homomorphism ℓ_{∗} : K_{3}(ℤ) → K_{3}(R) (induced by the obvious map ℓ : ℤ → R) induces an isomorphism ℓ_{∗} ⊗ id : K_{3}(ℤ) ⊗ ℤ/2 ⟶^{≅} K_{3}(R) ⊗ ℤ/2 (for instance, if R = ℚ or any localization of ℤ). Then the non-stable Hurewicz homomorphism h_{4} : K_{4}(R) → H_{4}St(R) is an isomorphism.

Proof. This follows from the commutativity of the diagram

K_{3}(ℤ) ⊗ K_{1}(ℤ) ⟶^{⋆} K_{4}(ℤ) = 0
≅ ↓ ℓ_{∗}⊗id  ↓ ℓ_{∗}
K_{3}(R) ⊗ K_{1}(ℤ) ⟶^{⋆} K_{4}(R) ⟶^{h_{4}} H_{4}St(R) ⟶ 0 ,

where the rows are the exact sequences given by Theorem 4.3 (a).

The argument of the proof of Proposition 5.1 (d) produces also the next two corollaries

Corollary 5.3. Let R be a ring such that the composition of j : S → X_{ℤ} with the obvious map X_{ℤ} → X_{R} induces an isomorphism _{2}(π_{3}(S)) ⟶^{≅} _{2}(K_{3}(R)) . Then K_{4}(R) ⋆ K_{1}(ℤ) ≅ ker (h_{5} : K_{5}(R) → H_{5}St(R)) .

Corollary 5.4. If R is a ring as in Corollary 5.3 (for instance, if R = ℤ or any localization of ℤ), then the iterated homology suspension H_{5}St(R) ≅ H_{5}BSt(R)^{+} → H_{5}X_{R}(2) is an isomorphism.

Proof. By hypothesis, one has actually the exact sequences

H_{6}(Σ^{4}(H(K_{4}(R))) ⟶^{ψ̄ θ̄} K_{5}(R) ⟶^{h̄_{5}} H_{5}X_{R}(2) ⟶ H_{5}X_{R}(2, 4] ⟶ 0
↑ σ  ↑ =  ↑  ↑ σ′
H_{6}K(K_{4}(R), 4) ⟶^{ψθ} K_{5}(R) ⟶^{h_{5}} H_{5}BSt(R)^{+} ⟶ H_{5}BSt(R)^{+}[4] ⟶ 0 ,

and the iterated homology suspension σ is clearly an isomorphism; the same is true for σ′ because of the commutativity of

H_{5}X_{R}(2, 4] ⟶^{≅} H_{5}(Σ^{3}H(K_{3}(R))
↑ σ′  ↑ ≅
H_{5}BSt(R)^{+}[4] ⟶^{≅} H_{5}(K(K_{3}(R), 3)) .

The assertion then follows from the five lemma.

In the remainder of this section, we consider maps

K_{i}(ℤ) ⊗ K_{k}(ℤ) ⟶^{⋆} K_{i+k}(ℤ) ⟶ K_{i+k}(ℤ) ⊗ ℤ_{ˆ2}

for all positive integers i and k, where the second arrow is the tensor product of K_{i+k}(ℤ) with the inclusion ℤ ↪ ℤ_{ˆ2} . We call these maps 2-adic products for K_{∗}(ℤ) and continue to denote them by the symbol ⋆. Again because of Table 1 of [41], K_{i}(ℤ) is a finite odd torsion group if i is a positive integer ≡ 0 , 4 , or 6 mod 8 and ≅ ℤ ⊕ (finite odd torsion group) if i ≡ 5 mod 8; thus, the only 2-adic products which can be non trivial are the following:

\begin{matrix}K_{8s+1}(ℤ) ⊗ K_{8t+1}(ℤ) & ⟶^{⋆} & K_{8(s+t)+2}(ℤ) ⊗ ℤ_{ˆ2} \\ K_{8s+1}(ℤ) ⊗ K_{8t+2}(ℤ) & ⟶^{⋆} & K_{8(s+t)+3}(ℤ) ⊗ ℤ_{ˆ2} \\ K_{8s+2}(ℤ) ⊗ K_{8t+5}(ℤ) & ⟶^{⋆} & K_{8(s+t)+7}(ℤ) ⊗ ℤ_{ˆ2} \\ K_{8s+2}(ℤ) ⊗ K_{8t+7}(ℤ) & ⟶^{⋆} & K_{8(s+t+1)+1}(ℤ) ⊗ ℤ_{ˆ2} \\ K_{8s+3}(ℤ) ⊗ K_{8t+7}(ℤ) & ⟶^{⋆} & K_{8(s+t+1)+2}(ℤ) ⊗ ℤ_{ˆ2} \\ K_{8s+5}(ℤ) ⊗ K_{8t+5}(ℤ) & ⟶^{⋆} & K_{8(s+t+1)+2}(ℤ) ⊗ ℤ_{ˆ2} \end{matrix}

for s and t ≥ 0 . We shall determine five of these products in the present section and the remaining one in Section 8.

The inclusion ℤ ↪ ℝ induces a map λ : BGL(ℤ)^{+} → BO and the induced homomorphism

λ_{∗} : K_{∗}(ℤ) ⟶ π_{∗}BO

is a ring homomorphism since it can be written as the composition λ_{∗} : K_{∗}(ℤ) → K_{∗}(ℂ) → π_{∗}BO, where both arrows are ring homomorphisms (see [13], p. 50 and p. 58). One can understand the kernel of λ_{∗} at the prime 2 by the following argument. If p is a prime ≡ 3 or 5 mod 8, M. Bökstedt introduced in [11] (see also [26] and Section 4 of [18]) a space J(p) which is defined by the pull-back diagram

J(p) ⟶^{λ′} BO
↓ f_{p}  ↓ c
F Ψ^{p} ⟶^{b} BU ,

where F Ψ^{p} is the fiber of (Ψ^{p} − 1) : BU → BU (recall that F Ψ^{p} ≃ BGL(𝔽_{p})^{+} by Theorem 7 of [27]), b the Brauer lifting and c the complexification. The fibers of the horizontal maps are homotopy equivalent to the unitary group U ≃ SU × S^{1} . More precisely, Bökstedt was interested in the covering space JK(ℤ, p) of J(p) corresponding to the cyclic subgroup of order 2 of π_{1}J(p) ≅ ℤ ⊕ ℤ/2 . After completion at the prime 2 , he constructed a map

φ̃ : (BGL(ℤ)^{+})_{ˆ2} ⟶ JK(ℤ, p)_{ˆ2}

which induces a split surjection on all homotopy groups. Let us write λ̂ and λ̂′ for the 2-completion of the maps λ and λ′ respectively: it turns out that the composition

(BGL(ℤ)^{+})_{ˆ2} ⟶^{φ̃} JK(ℤ, p)_{ˆ2} ⟶ J(p)_{ˆ2} \underrightarrow{_{λ′}} BO_{ˆ2}

is exactly λ̂. Recall that the localization exact sequence in K-theory implies that

(BGL(ℤ[\frac{1}{2}])^{+})_{ˆ2} ≃ (BGL(ℤ)^{+})_{ˆ2} × (S^{1})_{ˆ2} .

Therefore, φ̃ provides a map

φ : (BGL(ℤ[\frac{1}{2}])^{+})_{ˆ2} ⟶ J(p)_{ˆ2}

which also induces a split surjection on all homotopy groups. Since the 2-torsion of K_{∗}(ℤ) is known by Table 1 of [41], it is easy to check that φ̃ and φ are actually homotopy equivalences. Consequently, we obtain (see also Corollary 8 of [41]):

Proposition 5.5. For all primes p ≡ 3 or 5 mod 8, there is a pull-back diagram

(BGL(ℤ)^{+})_{ˆ2} × (S^{1})_{ˆ2} \underrightarrow{_{λ′}} BO_{ˆ2}
↓ _{f_{p}}  ↓ _{c}
(F Ψ^{p})_{ˆ2}  \underrightarrow{_{b}} BU_{ˆ2} .

Consequently, there is a fibration

SU_{ˆ2} ⟶^{η} (BGL(ℤ)^{+})_{ˆ2} \underrightarrow{_{λ}} BO_{ˆ2} .

This fibration induces the long exact sequence

⋯ ⟶ π_{i}SU ⊗ ℤ_{ˆ2} ⟶^{η_{∗}} K_{i}(ℤ) ⊗ ℤ_{ˆ2} \underrightarrow{_{λ_{∗}}} π_{i}BO ⊗ ℤ_{ˆ2} ⟶ π_{i−1}SU ⊗ ℤ_{ˆ2} ⟶ · · · .

Remember that π_{i}SU = 0 if i is even and π_{i}SU ≅ ℤ if i is odd ≥ 3 .

Theorem 5.6. The 2-adic products

K_{8s+3}(ℤ) ⊗ K_{8t+7}(ℤ) ⟶^{⋆} K_{8(s+t+1)+2}(ℤ) ⊗ ℤ_{ˆ2}

K_{8s+5}(ℤ) ⊗ K_{8t+5}(ℤ) ⟶^{⋆} K_{8(s+t+1)+2}(ℤ) ⊗ ℤ_{ˆ2}

are trivial for all integers s and t ≥ 0 .

Proof. For any product mentioned in the statement of the theorem, let us consider the commutative diagram

K_{i}(ℤ) ⊗ K_{k}(ℤ) ⟶^{⋆} K_{i+k}(ℤ) ⟶ K_{i+k}(ℤ) ⊗ ℤ_{ˆ2}
↓ λ_{∗}⊗λ_{∗}  ↓ λ_{∗}  ↓ _{λ_{∗}}
π_{i}BO ⊗ π_{k}BO ⟶ π_{i+k}BO ⟶ π_{i+k}BO ⊗ ℤ_{ˆ2} ,

where the bottom left horizontal arrow is the product map in π_{∗}BO and where the right horizontal arrows denote the tensor product with ℤ_{ˆ2} . Let x ∈ K_{i}(ℤ) and y ∈ K_{k}(ℤ) . One has clearly λ_{∗}(y) = 0 since π_{k}BO = 0 for k = 8t + 5 or k = 8t + 7 . Thus, λ_{∗}(x ⋆ y) = λ_{∗}(x)λ_{∗}(y) = 0 . This shows that the 2-adic product x ⋆ y ∈ K_{i+k}(ℤ) ⊗ ℤ_{ˆ2} belongs to the kernel of λ̂_{∗} , and consequently to the image of η_{∗} : π_{i+k}SU ⊗ ℤ_{ˆ2} → K_{i+k}(ℤ) ⊗ ℤ_{ˆ2} . Since i + k is even, the group π_{i+k}SU is trivial and x ⋆ y vanishes.

In the next theorem, we look at the groups K_{8s+1}(ℤ) for s ≥ 0 . Remember that K_{1}(ℤ) ≅ ℤ/2 and that K_{8s+1}(ℤ) ≅ ℤ ⊕ ℤ/2 ⊕ (finite odd torsion group) for s ≥ 1 by Table 1 of [41]. The fibration given by Proposition 5.5 provides the exact sequence

0 ⟶ π_{8s+1}SU ⊗ ℤ_{ˆ2} ≅ ℤ_{ˆ2} ⟶^{η_{∗}} K_{8s+1}(ℤ) ⊗ ℤ_{ˆ2} \underrightarrow{_{λ_{∗}}} π_{8s+1}BO ⊗ ℤ_{ˆ2} ≅ ℤ/2 ⟶ 0

if s ≥ 1 (if s = 0 , π_{1}SU = 0 ). Let us write x_{s} for the element of order 2 in K_{8s+1}(ℤ) . We denote by y_{s} , for s ≥ 1 , the generator of the infinite cyclic summand of K_{8s+1}(ℤ) whose image under the 2-adic completion K_{8s+1}(ℤ) → K_{8s+1}(ℤ) ⊗ ℤ_{ˆ2} is exactly the image of a generator of π_{8s+1}SU ⊗ ℤ_{ˆ2} under the

homomorphism η_{∗} : π_{8s+1}SU ⊗ ℤ_{ˆ2} ≅ ℤ_{ˆ2} → K_{8s+1}(ℤ) ⊗ ℤ_{ˆ2} .

Theorem 5.7. Consider the 2-adic product

K_{8s+1}(ℤ) ⊗ K_{8t+1}(ℤ) ⟶^{⋆} K_{8(s+t)+2}(ℤ) ⊗ ℤ_{ˆ2}

for any integers s, t ≥ 0 .

(a) If s and t are ≥ 1 , then y_{s} ⋆ y_{t} = 0 .

(b) If s ≥ 0 and t ≥ 1 , then x_{s} ⋆ y_{t} = 0 .

(c) For s and t ≥ 0 , x_{s} ⋆ x_{t} is the generator of K_{8(s+t)+2}(ℤ) ⊗ ℤ_{ˆ2} ≅ ℤ/2 .

Proof. The commutativity of the square

K_{8s+1}(ℤ) ⟶ K_{8s+1}(ℤ) ⊗ ℤ_{ˆ2}
↓ λ_{∗}  ↓ _{λ_{∗}}
π_{8s+1}BO ⟶^{≅} π_{8s+1}BO ⊗ ℤ_{ˆ2} ,

where the horizontal arrows denote the tensor product with ℤ_{ˆ2} , and the definition of y_{s} show that λ_{∗}(y_{s}) = 0 . We deduce similarily that λ_{∗}(y_{t}) = 0 . This implies the vanishing of λ̂_{∗}(y_{s} ⋆ y_{t}) and λ̂_{∗}(x_{s} ⋆ y_{t}) . The fact that π_{8(s+t)+2}SU = 0 then enables us to deduce the triviality of the products y_{s} ⋆ y_{t} and x_{s} ⋆ y_{t} in K_{8(s+t)+2}(ℤ)⊗ℤ_{ˆ2} . The image of x_{s} under λ_{∗} is the generator c_{s} of π_{8s+1}BO ≅ ℤ/2 and it is known that c_{s}c_{t} is non trivial in π_{8(s+t)+2}BO (see [36], p. 304). Therefore, it follows from the equality λ_{∗}(x_{s} ⋆x_{t}) = c_{s}c_{t} that the product x_{s} ⋆ x_{t} does not vanish.

Corollary 5.8. The products

K_{8s+2}(ℤ) ⊗ K_{8t+5}(ℤ) ⟶^{⋆} K_{8(s+t)+7}(ℤ) ⊗ ℤ_{ˆ2}

K_{8s+2}(ℤ) ⊗ K_{8t+7}(ℤ) ⟶^{⋆} K_{8(s+t+1)+1}(ℤ) ⊗ ℤ_{ˆ2}

are trivial for all s and t ≥ 0 .

Proof. According to Theorem 5.7,

K_{8s+2}(ℤ) ⊗ ℤ_{ˆ2} ≅ (K_{1}(ℤ) ⋆ K_{8s+1}(ℤ)) ⊗ ℤ_{ˆ2} .

This implies the assertion because the products K_{8s+1}(ℤ) ⊗ K_{8t+5}(ℤ) →^{⋆} K_{8(s+t)+6}(ℤ) ⊗ ℤ_{ˆ2} = 0 and K_{8s+1}(ℤ) ⊗ K_{8t+7}(ℤ) →^{⋆} K_{8(s+t+1)}(ℤ) ⊗ ℤ_{ˆ2} = 0 are obviously trivial (see Table 1 of [41]).

Remark 5.9. The last interesting product is

K_{8s+1}(ℤ) ⊗ K_{8t+2}(ℤ) ⟶^{⋆} K_{8(s+t)+3}(ℤ) ⊗ ℤ_{ˆ2} .

As in the previous corollary, we have

(K_{8s+1}(ℤ) ⋆ K_{8t+2}(ℤ)) ⊗ ℤ_{ˆ2} ≅ (K_{1}(ℤ) ⋆ K_{8s+1}(ℤ) ⋆ K_{8t+1}(ℤ)) ⊗ ℤ_{ˆ2}

and Theorem 5.7 asserts that the image of this product is at most a cyclic group of order 2. We shall determine it exactly in Theorem 8.2.

6. Products in the algebraic K-theory of cyclotomic fields

The results of Section 5 indicate that the 2-adic products in K_{∗}(ℤ) are trivial or have a very small image. In this section, we show that in the case of products in the K-theory of number fields, the image of product maps can be quite big. In the proof, we use the methods of [8] and [9]. Let us consider an odd prime number l, a positive integer m, and the cyclotomic field E = ℚ(ξ_{l^{m}}) obtained from ℚ by adding a primitive root of unity ξ_{l^{m}} of order l^{m} . Our goal is to show that for n odd, the product homomorphism

⋆ : K_{1}(E) ⊗ K_{2n−1}(E)_{l} ⟶ K_{2n}(E)_{l}

has a big image.

If R is a commutative ring, X_{R} is a ring spectrum with respect to μ : X_{R} ∧ X_{R} → X_{R⊗R} → X_{R} , where the first map is the pairing which was also called μ at the beginning of Section 3 (see [22], Proposition 2.4.2) and the second is induced by the multiplication R ⊗ R → R. The product structure of K_{∗}(R) , also denoted by ⋆, is given by the composition

⋆ : K_{i}(R) ⊗ K_{k}(R) ≅ π_{i}X_{R} ⊗ π_{k}X_{R} ⟶^{∧} π_{i+k}(X_{R} ∧ X_{R}) ⟶^{μ_{∗}} π_{i+k}X_{R} ≅ K_{i+k}(R) .

Recall that the K-theory with ℤ/l^{m}-coefficients (for a prime l and a positive integer m, with m ≥ 2 if l = 2 ) may be defined by K_{k}(R; ℤ/l^{m}) = π_{k}(M ∧ X_{R}) , where M is the mod l^{m} Moore spectrum (i.e., such that H_{0}M ≅ ℤ/l^{m} , H_{k}M = 0 for k ≠ 0 ). Notice that M is a ring spectrum with identity i_{M} and product μ_{M} (see [30], p. 22). We also consider the following products (see also [13]):

⋆ : K_{i}(R)⊗K_{k}(R; ℤ/l^{m}) ≅ π_{i}X_{R}⊗π_{k}(M ∧X_{R}) ⟶^{∧} π_{i+k}(M ∧X_{R}∧X_{R}) ⟶^{μ_{∗}} π_{i+k}(M ∧X_{R}) ≅ K_{i+k}(R; ℤ/l^{m})

and

⋆ : K_{i}(R; ℤ/l^{m}) ⊗ K_{k}(R; ℤ/l^{m}) ≅ π_{i}(M ∧ X_{R}) ⊗ π_{k}(M ∧ X_{R}) ⟶^{∧}

π_{i+k}(M ∧ M ∧ X_{R} ∧ X_{R}) ⟶^{(μ_{M} ∧μ)_{∗}} π_{i+k}(M ∧ X_{R}) ≅ K_{i+k}(R; ℤ/l^{m}) .

Remark 6.1. Since M is a ring spectrum, the diagram

S ∧ S ⟶^{id∧i_{M}} S ∧ M ⟶^{i_{M} ∧id} M ∧ M
↓ ≃  ↓ ≃  ↓ μ_{M}
S ⟶^{i_{M}} M ⟶^{id}  M

commutes and implies the compatibility of the three products, i.e., the commutativity of the diagram

K_{i}(R) ⊗ K_{k}(R) ⟶^{id⊗red} K_{i}(R) ⊗ K_{k}(R; ℤ/l^{m}) ⟶^{red⊗id} K_{i}(R; ℤ/l^{m}) ⊗ K_{k}(R; ℤ/l^{m})
↓ ⋆  ↓ ⋆  ↓ ⋆
K_{i+k}(R)  ⟶^{red}  K_{i+k}(R; ℤ/l^{m})  ⟶^{id}  K_{i+k}(R; ℤ/l^{m}) ,

where red is the map which is induced on K-theory by the reduction of coefficients mod l^{m} .

For any ring R, there is the following Bockstein long exact sequence

⋯ ⟶ K_{k}(R) ⟶^{l^{m}} K_{k}(R) ⟶ K_{k}(R; ℤ/l^{m}) \underrightarrow{𝔟} K_{k−1}(R) ⟶ ⋯,

where 𝔟 is the Bockstein homomorphism.

Lemma 6.2. For any x ∈ K_{i}(R) and any y ∈ K_{k}(R; ℤ/l^{m}) , one has 𝔟(x ⋆ y) = x ⋆ 𝔟(y) ∈ K_{i+k−1}(R) .

Proof. The Bockstein homomorphism 𝔟 is induced by the obvious map ε : M → Σ^{−1}S which fits into the commutative diagram

S ∧ M ⟶^{≃} M
↓ id∧ε  ↓ ε
S ∧ Σ^{−1}S ⟶^{≃} Σ^{−1}S

and provides the commutativity of

K_{i}(R) ⊗ K_{k}(R; ℤ/l^{m}) ⟶^{⋆} K_{i+k}(R; ℤ/l^{m})
↓ id⊗𝔟  ↓ 𝔟
K_{i}(R) ⊗ K_{k−1}(R) ⟶^{⋆} K_{i+k−1}(R)

and the statement of the lemma.

This lemma implies the formula

𝔟(T r_{E/ℚ}(u ⋆ β_{m}^{⋆n})) = T r_{E/ℚ}(u ⋆ 𝔟(β_{m}^{⋆n})) ,

where u is any element ∈ K_{1}(E) = E^{×} , β_{m} = β(ξ_{l^{m}}) ∈ K_{2}(E; ℤ/l^{m}) is the Bott element (see Definition 2.7.2 of [40]) and T r_{E/ℚ} is the transfer map (see [28], Section 4). Using this equality, we can rewrite the definition of the Stickelberg pseudosplitting homomorphism Λ from [7], Section IV.1, or [9], Definition 3.2, as follows.

Definition 6.3. There is a homomorphism

Λ : ⨁_{p} K_{2n−1}(𝔽_{p})_{l} ⟶ K_{2n}(ℚ)_{l} ,

where Λ = ∏_{p} Λ_{p} and Λ_{p} : K_{2n−1}(𝔽_{p})_{l} → K_{2n}(ℚ)_{l} is given by the formula

Λ_{p}(κ_{p}) = \{ \begin{matrix}T r_{E/ℚ}(λ_{b}(𝔭) ⋆ 𝔟(β_{k}^{⋆n})^{b^{n}γ_{l}}) & if l does not divide n, \\ T r_{E/ℚ}(λ_{b}(𝔭) ⋆ 𝔟(β_{k}^{⋆n})^{nb^{n}γ_{l}}) & if l divides n, \end{matrix}

where b is a natural number such that (b, w_{n+1}(ℚ)) = 1 and κ_{p} is a generator of the group K_{2n−1}(𝔽_{p})_{l} . In addition, λ_{b}(𝔭) ∈ E^{×} are the twisted Gauss sums (see [8], Definition 3) and γ_{l} = 1 + l^{n} + l^{2n} + l^{3n} … =

\frac{1}{1−l^{n}} ∈ ℤ_{l̂}.

Theorem 6.4. Let I be the image of the map

K_{1}(E) ⊗ K_{2n−1}(E)_{l} ⟶^{⋆} K_{2n}(E)_{l} ⟶^{T r_{E/ℚ}} K_{2n}(ℚ)_{l} ,

where n is an odd integer. Then the exponent of the group K_{2n}(ℚ)_{l}/I divides the number (#K_{2n}(ℤ)_{l})^{2} .

Proof. Consider the localization sequence

0 ⟶ K_{2n}(ℤ)_{l} ⟶ K_{2n}(ℚ)_{l} ⟶^{∂} ⨁_{p} K_{2n−1}(𝔽_{p})_{l} ⟶ 0 .

By Definition 6.3, we see that image Λ ⊆ I . On the other hand, by Proposition 2 of [8], the composition ∂ Λ acts by raising into the power with exponent an integer |(b^{n+1} − 1)ζ_{ℚ}(−n)|_{l}^{−1} (recall that the Gauss sums used in the construction of Λ depend on b), where ζ_{ℚ}(s) is the Riemann zeta function. Consider now x ∈ K_{2n}(ℚ)_{l} and z = ∂(x) ∈ ⨁_{p} K_{2n−1}(𝔽_{p})_{l} . The computations above show that for every b,

\frac{Λ(z)}{x^{|(b^{n+1}−1)ζ_{ℚ}(−n)|_{l}^{−1}}} ∈ K_{2n}(ℤ)_{l} .

The greatest common divisor of all |(b^{n+1} −1)|_{l}^{−1} over the integers b which are relatively prime to w_{n+1}(ℚ) equals the number |w_{n+1}(ℚ)|_{l}^{−1} , by Lemma 2.3 of [15]. Since the integer |w_{n+1}(ℚ)ζ_{ℚ}(−n)|_{l}^{−1} divides the number t of elements in the group K_{2n}(ℤ)_{l} , it follows that x^{t^{2}} ∈ image Λ ⊆ I .

Remark 6.5. Observe that in an abelian group the product of a torsion element and a nontorsion element is again nontorsion. Hence, every torsion element in K_{2n−1}(E) can be written as a quotient of two nontorsion elements. Consequently, the subgroup of K_{1}(E)⋆K_{2n−1}(E) generated by the subset {x⋆y | x ∈ K_{1}(E) , y ∈ K_{2n−1}(E) , y nontorsion } contains the subgroup K_{1}(E) ⋆ K_{2n−1}(E)_{l} of K_{2n}(E) .

Remark 6.6. It follows from Theorem 6.4 and Theorem 3.4 of [9] that the image of K_{1}(E) ⋆ K_{2n−1}(E)_{l} under the transfer T r_{E/ℚ} : K_{2n}(E) → K_{2n}(ℚ) contains the group of divisible elements D(n)_{l} ≅ K_{2n}^{ét}(ℤ[\frac{1}{l}]) from Section 5.2 of [8] (see also [7], Section IV.3).

7. 2-adic cyclotomic elements in K_{2n−1}(ℤ) ⊗ ℤ_{ˆ2}

In this section, we discuss a 2-adic version of cyclotomic elements (see [34], Sections 4.3 and 4.4, p. 239–240). We construct the subgroup of cyclotomic elements in the 2-adic K-group of ℤ

K_{2n−1}(ℤ) ⊗ ℤ_{ˆ2} ≅ K_{2n−1}(ℤ[\frac{1}{2}]) ⊗ ℤ_{ˆ2} ≅ \underleftarrow{lim}K_{2n−1}(ℤ[\frac{1}{2}]; ℤ/2^{m}) ,

where the inverse limit is taken over reduction maps

r_{m} : K_{2n−1}(ℤ[\frac{1}{2}]; ℤ/2^{m+1}) ⟶ K_{2n−1}(ℤ[\frac{1}{2}]; ℤ/2^{m}) ,

and the subgroup of étale 2-adic cyclotomic elements in

H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n)) ≅ \underleftarrow{lim}H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/2^{m}(n)) .

Since the case of the prime 2 is not often discussed in the literature, we shall make explicit computations. We assume that n is an odd integer and use the following notation:

ξ_{2^{m}} = exp(\frac{2πi}{2^{m}}) is a primitive root of unity of order 2^{m} , where m is a positive integer, E_{m} = ℚ(ξ_{2^{m}}) is the corresponding cyclotomic field, R_{m} = ℤ[ξ_{2^{m}}] is the ring of integers of E_{m} , R′_{m} = ℤ[\frac{1}{2}, ξ_{2^{m}}] ,

ℤ/2^{m}(n) is the n-th Tate twist of the sheaf of roots of unity on Spec R′_{k} for k ≥ 1 , ⟨g_{1}, g_{2}, ...g_{r}⟩ denotes the subgroup generated by elements g_{1} , g_{2}, …, g_{r} of a given abelian group G,

C_{m}^{+} = ⟨−1, , ξ2^{m}^{\frac{1−a}{2}} \frac{1 − ξ_{2^{m}}^{a}}{1 − ξ_{2^{m}}} , 1 < a < 2^{m−1} , a odd⟩ is the group of real cyclotomic units in E_{m}^{×}

(see [38], p. 145), C_{m} denotes the subgroup of R_{m}^{×} generated by C_{m}^{+} and ξ_{2^{m}} , C′_{m} is the subgroup of R_{m}^{′×} generated by C_{m} and 1 − ξ_{2^{m}}^{a} for 1 ≤ a < 2^{m} and a odd.

Remark 7.1. It is easy to show that C′_{m} is generated by ξ_{2^{m}} and 1 − ξ_{2^{m}}^{a} for 1 ≤ a < 2^{m} and a odd. Observe that the group C′_{m} is also generated by 1 − ξ_{2^{m}}^{a} and ξ_{2^{m}}(1 − ξ_{2^{m}}^{a}) for 1 ≤ a < 2^{m} and a odd.

Example 7.2. The group C′_{2} is generated by 1−i and 1+i because i(1−i) = 1+i and i(1+i) = −(1−i) .

Remark 7.3. We leave to the reader to check that

N_{E_{m}/ℚ}(ξ_{2^{m}}^{b}(1 − ξ_{2^{m}}^{a})) = 2

for all a, b ∈ ℤ with 1 ≤ a < 2^{m} and a odd, where N_{E_{m}/ℚ} denotes the norm map down to ℚ.

Let us assume for a moment that m ≥ 2 and consider the map

α_{m} : C′_{m} ⊗ ℤ/2^{m}(n − 1) ⟶ K_{2n−1}(R′_{m}; ℤ/2^{m})

given by the formula

α_{m}(c ⊗ ξ_{2^{m}}^{⊗ (n−1)}) = c ⋆ β_{m}^{⋆(n−1)} ,

where c ∈ C′_{m} ⊆ K_{1}(R′_{m}) and β_{m} = β(ξ_{2^{m}}) ∈ K_{2}(R′_{m}; ℤ/2^{m}) is the Bott element (see Definition 2.7.2 of [40]). The product ⋆ used in that formula is explained at the beginning of Section 6. We shall also use the transfer map in K-theory (see [28], Section 4)

T r_{E_{m}/ℚ} : K_{2n−1}(R′_{m}; ℤ/2^{m}) ⟶ K_{2n−1}(ℤ[\frac{1}{2}]; ℤ/2^{m}) .

Definition 7.4. Let n be any odd integer ≥ 3 .

(a) Let C_{m}(n − 1) ⊆ K_{2n−1}(ℤ[\frac{1}{2}]; ℤ/2^{m}) denote the image of the composition T r_{E_{m}/ℚ} α_{m} for m ≥ 2 .

(b) We define C_{1}(n − 1) ⊆ K_{2n−1}(ℤ[\frac{1}{2}]; ℤ/2) to be the group r_{1}C_{2}(n − 1), where

r_{1} : K_{2n−1}(ℤ[\frac{1}{2}]; ℤ/4) ⟶ K_{2n−1}(ℤ[\frac{1}{2}]; ℤ/2)

is the reduction of coefficients on K-groups.

(c) Let C(n − 1) = \underleftarrow{lim}C_{m}(n − 1) which is a subgroup of

\underleftarrow{lim}K_{2n−1}(ℤ[\frac{1}{2}]; ℤ/2^{m}) ≅ K_{2n−1}(ℤ[\frac{1}{2}]) ⊗ ℤ_{ˆ2} ≅ K_{2n−1}(ℤ) ⊗ ℤ_{ˆ2} .

We call C(n − 1) the group of 2 -adic cyclotomic elements in K_{2n−1}(ℤ) ⊗ ℤ_{ˆ2} .

Similarly we can define 2-adic cyclotomic elements inside the étale cohomology group H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n)) . Since Pic(R′_{m})_{2} is a trivial group by [39], Satz C, p. 244, it follows from the Kummer exact sequence that we have the isomorphism

H_{ét}^{1}(R′_{m}; ℤ/2^{m}(n)) ≅ H_{ét}^{1}(R′_{m}; ℤ/2^{m}(1)) ⊗ ℤ/2^{m}(n − 1) ≅ R_{m}^{′×} ⊗ ℤ/2^{m}(n − 1) .

Hence we get the composition of maps

α_{m}^{ét} : C′_{m} ⊗ ℤ/2^{m}(n − 1) ⟶ R_{m}^{′×} ⊗ ℤ/2^{m}(n − 1) ⟶^{≅} H_{ét}^{1}(R′_{m}; ℤ/2^{m}(n)) ,

where the first map is induced by the natural injection. Note that the maps α_{m} and α_{m}^{ét} are only defined for m ≥ 2 .

Lemma 7.5. The map α_{m}^{ét} is an isomorphism for all m ≥ 2 .

Proof. It is sufficient to prove that R_{m}^{′×}/2^{m} ≅ C′_{m}/2^{m} . It follows from the commutative diagram of embeddings of groups

W U_{m} ⟶ R_{m}^{×}
↑  ↑
W C_{m}^{+} ⟶ C_{m} ,

where W = ⟨ξ_{2^{m}}⟩ and U_{m} is the group of units of the maximal real subfield of E_{m} , that the index [R_{m}^{×} : C_{m}] divides the product of indices [R_{m}^{×} : W U_{m}][W U_{m} : W C_{m}^{+}] . By [38], Corollary 4.13, p. 39 and Theorem 8.2, p. 145, we know that [R_{m}^{×} : W U_{m}] = 1 and that [W U_{m} : W C_{m}^{+}] = [U_{m} : C_{m}^{+}] is the class number which is odd (see [39], Satz C, p. 244). Thus, we obtain R_{m}^{×}/2^{m} ≅ C_{m}/2^{m} and, by the definition given at the beginning of this section, R_{m}^{′×}/2^{m} ≅ C′_{m}/2^{m} .

Definition 7.6. Let n be any odd integer ≥ 3 .

(a) If m ≥ 2 , we define C_{m}^{ét}(n − 1) to be the image of the composition

C′_{m} ⊗ ℤ/2^{m}(n − 1) ⟶_{≅}^{αét_{m}} H_{ét}^{1}(R′_{m}; ℤ/2^{m}(n)) ⟶^{T r_{Em/ℚ}} H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/2^{m}(n)) ,

where T r_{E_{m}/ℚ} denotes the transfer map in étale cohomology.

(b) For m = 1 we put C_{1}^{ét}(n − 1) to be the image of C_{2}^{ét}(n − 1) via the reduction of coefficients

r_{1} : H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/4(n)) ⟶ H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/2(n)) .

(c) Taking inverse limits with respect to m above we obtain a subgroup C^{ét}(n − 1) of

H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n)) ≅ \underleftarrow{lim}H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/2^{m}(n))

which we call the subgroup of étale 2-adic cyclotomic elements.

Observe that for m ≥ k ≥ 2 the norm map N_{E_{m}/E_{k}} : C′_{m} → C′_{k} is surjective by Remark 7.1. For m ≥ k > 2 , we have the following commutative diagram with exact rows which comes from the naturality of the Bockstein long exact sequence in étale cohomology:

(7.1)

H_{ét}^{1}(R′_{m}; ℤ/2^{k−1}(n)) ⟶ H_{ét}^{1}(R′_{m}; ℤ/2^{k}(n)) ⟶^{r′} H_{ét}^{1}(R′_{m}; ℤ/2(n))
↓ T r_{Em/E2}  ↓ T r_{Em/E2}  ↓ T r_{Em/E2}
H_{ét}^{1}(R′_{2}; ℤ/2^{k−1}(n)) ⟶ H_{ét}^{1}(R′_{2}; ℤ/2^{k}(n)) ⟶ H_{ét}^{1}(R′_{2}; ℤ/2(n)) .

For m ≥ k ≥ 2 , in a similar way to the proof of Lemma 7.5 we obtain the isomorphism

(7.2)  H_{ét}^{1}(R′_{m}; ℤ/2^{k}(n)) ≅ C′_{m} ⊗ ℤ/2^{k}(n − 1) .

By the projection formula in étale cohomology (see [33], Lemma 6) the right vertical map in Diagram (7.1) is surjective. In addition, the reduction of coefficients map r′ in the top exact sequence is surjective. By induction over k we can assume that the left vertical map is surjective. Hence, it follows that the middle vertical map in (7.1) is surjective (see Diagram 2.6 of [9]). Let us put

(7.3)  B_{m} = \{ \begin{matrix}r_{1}(α_{2}^{ét}(C′_{2} ⊗ ℤ/4(n − 1))) & if m = 1, \\ T r_{E_{m}/E_{2}}(α_{m}^{ét}(C′_{m} ⊗ ℤ/2^{m}(n − 1))) & if m ≥ 2, \end{matrix}

where r_{1} : H_{ét}^{1}(ℤ[\frac{1}{2}, i]; ℤ/4(n)) → H_{ét}^{1}(ℤ[\frac{1}{2}, i]; ℤ/2(n)) is the reduction of coefficients and i = ξ_{4} . From the surjectivity of the transfer maps in Diagram (7.1) we obtain the isomorphism

(7.4)  H_{ét}^{1}(ℤ[\frac{1}{2}, i]; ℤ_{ˆ2}(n)) ≅ \underleftarrow{lim}H_{ét}^{1}(ℤ[\frac{1}{2}, i]; ℤ/2^{m}(n)) ≅ \underleftarrow{lim}B_{m} ,

where the inverse limit is taken with respect to the reduction of coefficients in étale cohomology.

Lemma 7.7. For any k ≥ 1 , m ≥ 1 and n odd, H_{ét}^{1}(R′_{k}; ℤ/2^{m}(n)) ≅ H_{ét}^{1}(R′_{k}; ℤ_{ˆ2}(n))/2^{m} .

Proof. This is an exercise on long exact sequences in cohomology which are associated with the sequence of Galois modules

0 ⟶ ℤ_{ˆ2}(n) ⟶^{2^{m}} ℤ_{ˆ2}(n) ⟶ ℤ/2^{m}(n) ⟶ 0

and the sequence of sheaves

0 ⟶ ℤ/2(1) ⟶ 𝔾_{m} ⟶^{2} 𝔾_{m} ⟶ 0 ,

where 𝔾_{m} is the mulplicative group scheme on SpecR′_{k} . The sequence of Galois modules induces a long exact sequence in étale cohomology, which gives the short exact sequence

0 ⟶ H_{ét}^{1}(R′_{k}; ℤ_{ˆ2}(n))/2^{m} ⟶ H_{ét}^{1}(R′_{k}; ℤ/2^{m}(n)) ⟶ H_{ét}^{2}(R′_{k}; ℤ_{ˆ2}(n))[2^{m}] ⟶ 0 .

Here, H_{ét}^{2}(R′_{k}; ℤ_{ˆ2}(n))[2^{m}] denotes the subgroup {x ∈ H_{ét}^{2}(R′_{k}; ℤ_{ˆ2}(n)) | 2^{m}x = 0}. We want to prove that the group H_{ét}^{2}(R′_{k}; ℤ_{ˆ2}(n))[2^{m}] vanishes. Since the ℤ_{ˆ2} -module H_{ét}^{2}(R′_{k}; ℤ_{ˆ2}(n)) is finitely generated, we see that H_{ét}^{2}(R′_{k}; ℤ_{ˆ2}(n)) = 0 if and only if H_{ét}^{2}(R′_{k}; ℤ_{ˆ2}(n))/2 = 0 . The same long exact sequence in étale cohomology with m = 1 gives the short exact sequence

0 ⟶ H_{ét}^{2}(R′_{k}; ℤ_{ˆ2}(n))/2 ⟶ H_{ét}^{2}(R′_{k}; ℤ/2(n)) ⟶ H_{ét}^{3}(R′_{k}; ℤ_{ˆ2}(n))[2] ⟶ 0 .

In addition, by triviality of the Galois action on ℤ/2(1) , we get the isomorphism

H_{ét}^{2}(R′_{k}; ℤ/2(n)) ≅ H_{ét}^{2}(R′_{k}; ℤ/2(1)) ⊗ ℤ(n − 1) .

Now consider the long exact sequence in cohomology which is induced by the above exact sequence of sheaves: it gives the short exact sequence

0 ⟶ Pic(R′_{k})/2 ⟶ H_{ét}^{2}(R′_{k}; ℤ/2(1)) ⟶ Br(R′_{k})[2] ⟶ 0 ,

since H_{ét}^{1}(R′_{k}; 𝔾_{m}) = Pic(R′_{k}) and H_{ét}^{2}(R′_{k}; 𝔾_{m}) = Br(R′_{k}) by [23], p. 107 and p. 109. We know from [39], Satz C, p. 244, that Pic(R′_{k})/2 = 0 . On the other hand, it follows from class field theory (see Example 2.22(f), p. 109 of [23]) that:

Br(R′_{k}) is ℤ/2 if k = 1 and vanishes for k ≥ 2 ,
H_{ét}^{3}(R′_{k}; ℤ/2(n)) is ℤ/2 if k = 1 and vanishes for k ≥ 2 .

This implies that H_{ét}^{2}(R′_{k}; ℤ/2(n)) ≅ H_{ét}^{3}(R′_{k}; ℤ_{ˆ2}(n))[2] . Consequently, we get H_{ét}^{2}(R′_{k}; ℤ_{ˆ2}(n))/2 = 0 and may deduce the lemma.

By Theorem 10 of [41] and Theorem 1 (i) of [19], we know that

H_{ét}^{1}(ℤ[\frac{1}{2}, i]; ℤ_{ˆ2}(n)) ≅ ℤ_{ˆ2} ⊕ ℤ/4 .

Hence, by Lemma 7.7 applied for k = m = 2 we obtain:

(7.5)  H_{ét}^{1}(ℤ[\frac{1}{2}, i]; ℤ/4(n)) ≅ ℤ/4 ⊕ ℤ/4 .

On the other hand by (7.2) the map α_{2}^{ét} gives an isomorphism

(7.6)  H_{ét}^{1}(ℤ[\frac{1}{2}, i]; ℤ/4(n)) ≅ C′_{2} ⊗ ℤ/4(n − 1) .

According to Example 7.2, the group C′_{2} has two generators: 1 − i and 1 + i. The module ℤ/4(n − 1) has a canonical generator ξ_{4}^{⊗(n−1)} . Hence, using the isomorphism (7.6) we see that the elements

(1 − i) ⊗ ξ_{4}^{⊗(n−1)} and (1 + i) ⊗ ξ_{4}^{⊗(n−1)}

generate H_{ét}^{1}(ℤ[\frac{1}{2}, i]; ℤ/4(n)) . We will use these elements below to contruct explicitly a generator of the ℤ_{ˆ2} -module H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n)) . For m ≥ 2 we check directly that

(7.7)  \begin{aligned}N_{E_{m}/E_{2}}(1 − ξ_{2^{m}}) & = 1 − i \\ N_{E_{m}/E_{2}}(1 − ξ_{2^{m}}^{2^{m}−1}) & = 1 + i . \end{aligned}

Hence, by the definition of the inverse system (7.3) we obtain the two elements

e_{1} = (b_{m}^{(1)}), e_{2} = (b_{m}^{(2)}) ∈ \underleftarrow{lim} B_{m} ≅ H_{ét}^{1}(ℤ[\frac{1}{2}, i]; ℤ_{ˆ2}(n)) ,

where b_{m}^{(1)} and b_{m}^{(2)} are the elements of B_{m} defined as follows:

(7.8)

b_{m}^{(1)} = \{ \begin{matrix}r_{1}(α_{2}^{ét}((1 − i) ⊗ ξ_{4}^{⊗(n−1)})) & if m = 1, \\ T r_{E_{m}/E_{2}}(α_{m}^{ét}((1 − ξ_{2^{m}}) ⊗ ξ_{2^{m}}^{⊗(n−1)})) & if m ≥ 2, \end{matrix}

b_{m}^{(2)} = \{ \begin{matrix}r_{1}(α_{2}^{ét}((1 + i) ⊗ ξ_{4}^{⊗(n−1)})) & if m = 1, \\ T r_{E_{m}/E_{2}}(α_{m}^{ét}((1 − ξ_{2^{m}}^{2^{m}−1}) ⊗ ξ_{2^{m}}^{⊗(n−1)})) & if m ≥ 2. \end{matrix}

Lemma 7.8. The elements e_{1} and e_{2} are nontorsion elements in H_{ét}^{1}(ℤ[\frac{1}{2}, i]; ℤ_{ˆ2}(n)) .

Proof. If both e_{1} and e_{2} were torsion elements, then their images in the group (see Lemma 7.7)

H_{ét}^{1}(ℤ[\frac{1}{2}, i]; ℤ/4(n)) ≅ ℤ/4 ⊕ ℤ/4

under the reduction of coefficients mod 4 would be contained in the same ℤ/4 summand of that group. This is impossible because the images of e_{1} and e_{2} generate the group H_{ét}^{1}(ℤ[\frac{1}{2}, i], ℤ/4(n)) by construction. It follows that at least one of them is nontorsion. On the other hand the complex conjugation τ acts on the group H_{ét}^{1}(ℤ[\frac{1}{2}, i]; ℤ_{ˆ2}(n)). The automorphism τ sends e_{1} to e_{2} and e_{2} to e_{1}. Indeed,

b_{1}^{(1)τ} = r_{1}(α_{2}^{ét}((1 − i)^{τ(−1)^{n−1}} ⊗ ξ_{4}^{⊗(n−1)})) = r_{1}(α_{2}^{ét}((1 + i) ⊗ ξ_{4}^{⊗(n−1)})) = b_{1}^{(2)}

because (1 − i)^{τ} = 1 + i and n is odd by assumption. For m ≥ 2 we have

b_{m}^{(1)τ} = T r_{E_{m}/E_{2}}(α_{m}^{ét}((1 − ξ_{2^{m}})^{τ(−1)^{n−1}} ⊗ ξ_{2^{m}}^{⊗(n−1)})) = T r_{E_{m}/E_{2}}(α_{m}^{ét}((1 − ξ_{2^{m}}^{2^{m}−1}) ⊗ ξ_{2^{m}}^{⊗(n−1)})) = b_{m}^{(2)}

because (1 − ξ_{2^{m}})^{τ} = 1 − ξ_{2^{m}}^{2^{m}−1} and n is odd by assumption. Hence, we obtain that e_{1}^{τ} = e_{2} . In a similar way we prove that e_{2}^{τ} = e_{1} . It follows that both elements must be nontorsion.

Observe that by the isomorphism (7.2) and Definition 7.6 we have:

C^{ét}(n − 1) ≅ image (H_{ét}^{1}(ℤ[\frac{1}{2}, i]; ℤ_{ˆ2}(n)) ⟶^{T r_{E_{2}/E_{1}}} H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n))) .

Recall that for n odd,

H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n)) ≅ ℤ_{ˆ2} ⊕ ℤ/2

by Theorem 1 (i) of [19] and Table 1 of [41], and consider the surjective map

ϕ : H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n)) ⟶ H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/2(n)) ≅ ⟨−1 , 2⟩ ⊗ ℤ/2(n − 1) ,

where ⟨−1 , 2⟩ = ℤ[\frac{1}{2}]^{×}/2 .

Lemma 7.9. The generator of the 2-torsion subgroup of H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n)) is sent by the map ϕ to (−1) ⊗ ξ_{2}^{⊗(n−1)} .

Proof. Let S_{0} = {2} be the set containing only the prime number 2 , ℚ_{S_{0}} the maximal extension of ℚ unramified outside 2 and G_{S_{0}} = G(ℚ_{S_{0}}/ℚ) . We know that

H_{ét}^{∗}(ℤ[\frac{1}{2}]; F) ≅ H^{∗}(G_{S_{0}}; F)

for F = ℤ/2^{m}(n) or F = ℤ_{ˆ2}(n) (see [24], Proposition 2.9, p. 209). We have the commutative diagram

(7.9)  \begin{matrix}H^{0}(G_{S_{0}}; ℤ/2(n)) & ⟶^{∂} & H^{1}(G_{S_{0}}; ℤ/2(n)) \\ ↓ = & & ↓ = \\ ℤ/2(n) & ⟶ & Hom(G(ℚ(i, \sqrt{2})/ℚ), ℤ/2(n)) , \end{matrix}

where the upper horizontal arrow is the connecting homomorphism in the long cohomology exact sequence associated with the exact sequence of G_{S_{0}} -modules

0 ⟶ ℤ/2(n) ⟶ ℤ/4(n) ⟶ ℤ/2(n) ⟶ 0 .

It is easy to observe that ∂ is an embedding. The Galois group G(ℚ(i, \sqrt{2})/ℚ) ≅ ℤ/2 ⊕ ℤ/2 is generated by the complex conjugation τ and the automorphism σ such that σ(i) = i and σ(\sqrt{2}) = −\sqrt{2} . Observe that the lower horizontal arrow in Diagram (7.9) sends the generator ξ_{2}^{⊗n} of ℤ/2(n) to a homomorphism f : G(ℚ(i, \sqrt{2})/ℚ) → ℤ/2(n) such that f(τ) = ξ_{2}^{⊗n} and f(σ) = 1^{⊗n} . Let O_{S_{0}} denote the ring of integers in ℚ_{S_{0}} and consider the following commutative diagram:

(7.10)  \begin{matrix}H^{0}(G_{S_{0}}; O_{S_{0}}^{×})/2 & ⟶_{≅}^{∂′} & H^{1}(G_{S_{0}}; ℤ/2(1)) \\ ↓ = & & ↓ = \\ ⟨−1 , 2⟩ & ⟶^{≅} & Hom(G(ℚ(i, \sqrt{2})/ℚ), ℤ/2(1)) . \end{matrix}

The upper horizontal arrow ∂′ in Diagram (7.10) is the connecting homomorphism in the long cohomology exact sequence associated with the exact sequence

0 ⟶ ℤ/2(1) ⟶ O_{S_{0}}^{×} ⟶^{2} O_{S_{0}}^{×} ⟶ 0 .

Tensoring the inverse map ∂′^{−1} with ℤ/2(n − 1) we get the isomorphism

(7.11)  H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/2(n)) ≅ ⟨−1 , 2⟩ ⊗ ℤ/2(n − 1)

which we constantly use. Hence, the isomorphism (7.11) is just the inverse of the lower horizontal arrow in Diagram (7.10) tensored with ℤ/2(n − 1) which we can compute explicitly. We are actually only interested in the image of (−1) tensored with ξ_{2}^{⊗(n−1)} . We observe that (−1) ⊗ ξ_{2}^{⊗(n−1)} is sent to a homomorphism f such that f(τ) = ξ_{2}^{⊗n} and f(σ) = 1 ⊗ ξ_{2}^{⊗(n−1)} = 1^{⊗n} . Consider the commutative diagram

(7.12)

H_{ét}^{0}(ℤ[\frac{1}{2}]; ℤ/2(n)) ⟶^{∂} H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/2^{m}(n))
↓ =  ↓
H_{ét}^{0}(ℤ[\frac{1}{2}]; ℤ/2(n)) ⟶^{∂} H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/2^{m−1}(n)) ,

where the horizontal arrows are the connecting homomorphisms in the long exact sequences associated with the exact sequences

0 ⟶ ℤ/2^{j−1}(n) ⟶ ℤ/2^{j}(n) ⟶ ℤ/2(n) ⟶ 0

for j = m, m+1 . The horizontal arrows in Diagram (7.12) are injective. In this way we get a commutative diagram

(7.13)

H_{ét}^{0}(ℤ[\frac{1}{2}]; ℤ/2(n)) ⟶ H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n))
↓ =  ↓
H_{ét}^{0}(ℤ[\frac{1}{2}]; ℤ/2(n)) ⟶^{∂} H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/2(n)) ≅ ⟨−1 , 2⟩ ⊗ ℤ/2(n − 1)

which implies that H_{ét}^{0}(ℤ[\frac{1}{2}]; ℤ/2(n)) maps isomorphically to the torsion subgroup of H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n)) . On the other hand, the diagrams (7.9), (7.10), (7.12), (7.13) show that the generator of H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n))_{2} maps to the element (−1) ⊗ ξ_{2}^{⊗(n−1)} in ⟨−1 , 2⟩ ⊗ ℤ/2(n − 1) .

Lemma 7.10. The elements e_{1} and e_{2} are sent by the transfer map T r_{E_{2}/E_{1}} to the same nontorsion element e whose image ẽ in the ℤ_{ˆ2}-module H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n))/torsion ≅ ℤ_{ˆ2} is a generator. In addition, the element e is sent to 2 ⊗ ξ_{2}^{⊗(n−1)} via the map ϕ of Lemma 7.9.

Proof. Since the transfer map T r_{E_{2}/E_{1}} commutes with the complex conjugation, we have:

T r_{E_{2}/E_{1}}(e_{2}) = T r_{E_{2}/E_{1}}(e_{1}^{τ}) = (T r_{E_{2}/E_{1}}(e_{1}))^{τ} = T r_{E_{2}/E_{1}}(e_{1}) .

Let us put e = T r_{E_{2}/E_{1}}(e_{1}) ∈ C^{ét}(n−1) . First we prove that e is nontorsion in the group H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n)) . The composition

H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n))/torsion ⟶^{χ} H_{ét}^{1}(ℤ[\frac{1}{2}, i]; ℤ_{ˆ2}(n))/torsion ⟶^{T r_{E_{2}/E_{1}}} H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n))/torsion ,

where χ is induced by the inclusion ℤ[\frac{1}{2}] ↪ ℤ[\frac{1}{2}, i] , is multiplication by 2 because of the basic property of the transfer. Let g be a generator of the group H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n))/torsion . Then χ(g) = a ẽ_{1} for some a ∈ ℤ_{ˆ2} , where ẽ_{1} denotes the image of e_{1} in the group H_{ét}^{1}(ℤ[\frac{1}{2}, i]; ℤ_{ˆ2}(n))/torsion . Hence, 2g = a T r_{E_{2}/E_{1}}(ẽ_{1}) and we see that a = 2^{i}u, where i = 0 or 1 and u ∈ (ℤ_{ˆ2})^{×} . It follows that the element T r_{E_{2}/E_{1}}(e_{1}) = e is nontorsion. In order to check that ẽ generates H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n))/torsion we consider the image of e in the group

H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n))/2 ≅ H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/2(n)) ≅ ⟨−1 , 2⟩ ⊗ ℤ/2(n − 1)

(see Lemma 7.7). It then follows by the projection formula in étale cohomology that the image of e equals T r_{E_{2}/E_{1}}(α_{2}^{ét}((1 − i) ⊗ ξ_{2}^{⊗(n−1)})) = 2 ⊗ ξ_{2}^{⊗(n−1)} ∈ ⟨−1 , 2⟩ ⊗ ℤ/2(n − 1),

where ξ_{2} = −1 . By Lemma 7.9, the image of the generator of the module H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n))_{2} is equal to (−1) ⊗ ξ_{2}^{⊗(n−1)} . Hence, the element ẽ must be a generator of the ℤ_{ˆ2}-module H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n))/torsion .

Corollary 7.11. For any odd integer n, there is a natural isomorphism of ℤ_{ˆ2}-modules

H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n)) ≅ C^{ét}(n − 1) ⊕ ℤ/2 .

We keep writing ẽ for the generator of C^{ét}(n − 1) which is the image via the last isomorphism of the generator of H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n))/torsion introduced in Lemma 7.10.

Proof. This follows from Lemmas 7.9 and 7.10.

8. 2-adic cyclotomic elements and products in K_{∗}(ℤ)

We start this section with a proof of the Quillen-Lichtenbaum conjecture for ℤ at the prime 2 wich uses Weibel’s computation [41] and the topological approach of Bökstedt [11] and Dwyer-Friedlander [18]. Since the decisive step in the proof of the conjecture was made by V. Voevodsky in [37], we attribute the theorem to him.

Theorem 8.1 (Voevodsky). The Dwyer-Friedlander map at the prime 2

K_{i}(ℤ) ⊗ ℤ_{ˆ2} ⟶ K_{i}^{ét}(ℤ[\frac{1}{2}])

is an isomorphism for all integers i ≥ 2 .

Proof. Proposition 5.5 above and Proposition 4.2 of [18] show immediately that the map

(BGL(ℤ[\frac{1}{2}])^{+})_{ˆ2} ≃ (BGL(ℤ)^{+})_{ˆ2} × (S^{1})_{ˆ2} ⟶ K̂^{ét}(ℤ[\frac{1}{2}])

which induces the Dwyer-Friedlander map is a homotopy equivalence (here K̂^{ét}(ℤ[\frac{1}{2}]) denotes the étale K-theory space of the ring ℤ[\frac{1}{2}] , see [17], Definition 4.3, p. 256). This provides the assertion.

With the next result we complete the investigation of the 2-adic products in K_{∗}(ℤ) (see Section 5, in particular Remark 5.9).

Theorem 8.2. The image of the 2-adic product map

⋆ : K_{8s+1}(ℤ) ⊗ K_{8t+2}(ℤ) ⟶ K_{8(s+t)+3}(ℤ) ⊗ ℤ_{ˆ2}

is cyclic of order 2 for every s and t ≥ 0 .

Proof. If s = t = 0 , observe that it is known, and also given by Proposition 5.1 (a), that the product map ⋆ : K_{1}(ℤ) ⊗ K_{2}(ℤ) → K_{3}(ℤ) is injective. In the general case, according to Remark 5.9, the image of the product map is at most a cyclic group of order 2 . To finish the proof we must show that the product map

⋆ : K_{8s+1}(ℤ) ⊗ K_{8t+2}(ℤ) ⟶ K_{8(s+t)+3}(ℤ) ⊗ ℤ_{ˆ2}

is nontrivial. The étale K-theory has a well-defined product which is compatible with the Dwyer-Friedlander map. We denote the product in algebraic and in étale K -theory by the same symbol ⋆. Hence, we have the following commutative diagram (see [17], Proposition 4.4, p. 256)

(8.1)  \begin{matrix}K_{8s+1}(ℤ) ⊗ K_{8t+2}(ℤ) & ⟶^{⋆} & K_{8(s+t)+3}(ℤ) \\ ↓ & & ↓ \\ K_{8s+1}^{ét}(ℤ[\frac{1}{2}]) ⊗ K_{8t+2}^{ét}(ℤ[\frac{1}{2}]) & ⟶^{⋆} & K_{8(s+t)+3}^{ét}(ℤ[\frac{1}{2}]) . \end{matrix}

The Dwyer-Friedlander spectral sequence ([17], Proposition 5.1, p. 260) provides the isomorphisms

(8.2)  K_{8s+1}^{ét}(ℤ[\frac{1}{2}]) ≅ H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(4s + 1))
(8.3)  K_{8t+2}^{ét}(ℤ[\frac{1}{2}]) ≅ H_{ét}^{2}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(4t + 2))

and the short exact sequence

(8.4)  0 ⟶ H_{ét}^{3}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(4(s + t) + 3)) ⟶ K_{8(s+t)+3}^{ét}(ℤ[\frac{1}{2}]) ⟶ H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(4(s + t) + 2)) ⟶ 0 .

The spectral sequence is compatible with the products in étale K-theory and the cup product in étale cohomology by [17], Proposition 5.4, p.262. Hence, we get the commutative diagram

(8.5)

H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(4s + 1)) ⊗ H_{ét}^{2}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(4t + 2)) ⟶^{∪} H_{ét}^{3}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(4(s + t) + 3))
↓  ↓
K_{8s+1}^{ét}(ℤ[\frac{1}{2}]) ⊗ K_{8t+2}^{ét}(ℤ[\frac{1}{2}])  ⟶^{⋆}  K_{8(s+t)+3}^{ét}(ℤ[\frac{1}{2}]) .

The isomorphisms (8.2) and (8.3), the exact sequence (8.4) and Diagram (8.5) show that the composition of the upper horizontal arrow with the right vertical arrow in Diagram (8.1) has image contained in H_{ét}^{3}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(4(s+t)+3)) considered as subgroup of K_{8(s+t)+3}^{ét}(ℤ[\frac{1}{2}]) via the exact sequence (8.4). We know by Table 1 of [41] that the 2-torsion subgroups of K_{8s+1}(ℤ) and of K_{8t+2}(ℤ) are cyclic of order 2 for all s, t ≥ 0 . Let x_{s} denote the generator of the summand ℤ/2 in the group K_{8s+1}(ℤ) and z_{t} the generator of ℤ/2 in K_{8t+2}(ℤ) . We want to show that the element x_{s} ⋆ z_{t} does not vanish. In order to simplify the notation, if M is a Galois module and k an integer, let us write M(k) for M ⊗ ℤ/16(k) . Then consider the commutative diagram

(8.6)

H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(4s + 1)) ⊗ H_{ét}^{2}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(4t + 2)) ⟶^{∪} H_{ét}^{3}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(4(s + t) + 3))
↓  ↓
H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/16(1))(4s) ⊗ H_{ét}^{2}(ℤ[\frac{1}{2}]; ℤ/16(2))(4t) ⟶^{∪} H_{ét}^{3}(ℤ[\frac{1}{2}]; ℤ/16(3))(4(s + t)) ,

where the vertical arrows are the reduction of the coefficients mod 16 . We shall show that the image of x_{s} ⋆ z_{t} in the group H_{ét}^{3}(ℤ[\frac{1}{2}]; ℤ/16(4(s + t) + 3)) via Diagrams (8.1), (8.5) and (8.6) is nontrivial. Because the Dwyer-Friedlander map is an isomorphism according to Theorem 8.1, we can check that the maps coming from the Dwyer-Friedlander spectral sequence

(8.7)  \begin{matrix}K_{8s+1}(ℤ) ⊗ ℤ_{ˆ2} & ⟶ & H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(4s + 1)) \\ K_{8t+2}(ℤ) ⊗ ℤ_{ˆ2} & ⟶ & H_{ét}^{2}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(4t + 2)) \end{matrix}

are isomorphisms. It follows that the elements x_{s} and z_{t} map to the 2-torsion elements in the cohomology groups via the above maps. Let us call their images x′_{s} and z′_{t} respectively. We also have an isomorphism K_{2}(ℤ) ≅ H_{ét}^{2}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(2)) . Since H_{ét}^{3}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(k)) = 0 for k even (see [19], proof of Corollary 1), this implies that the map

(8.8)  K_{2}(ℤ)/16 ⟶ H_{ét}^{2}(ℤ[\frac{1}{2}]; ℤ/16(2))

is an isomorphism. On the other hand, K_{2}(ℤ) is a cyclic group of order 2 generated by the Steinberg symbol {−1, −1}. Obviously the map (8.8) commutes with products, so {−1, −1} maps by (8.8) to (−1) ∪ (−1) , which is nontrivial. Both target and source of the map (8.8) are isomorphic to ℤ/2 . So we observe that z′_{t} goes to an element of the form [(−1) ∪ (−1)] ⊗ w_{t} in H_{ét}^{2}(ℤ[\frac{1}{2}]; ℤ/16(2))(4t) , where w_{t} stands for a generator of the module ℤ/16(4t) . Now let us have a look at the image of x′_{s} in H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/16(1))(4s) . We know by (7.11) that

(8.9)  H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/16(1)) = ⟨−1, 2⟩/16 ≅ ℤ/2 ⊕ ℤ/16 .

The ℤ/2 summand above is generated by −1 and the ℤ/16 summand by 2 . The group H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/16(1)) has only three nontrivial elements of order 2 , namely −1 , 8 and −8 . Via the natural reduction map H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/16(1)) ⟶ H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/2(1)) the element −1 goes to −1 , 8 goes to 2 and −8 goes to −2 . Hence we deduce from Lemma 7.9 that x′_{s} maps to (−1) ⊗ w_{s} ∈ H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ/16(1))(4s) , where w_{s} is a generator of ℤ/16(4s) . By Diagrams (8.1), (8.5) and (8.6), the product x_{s} ⋆ z_{t} maps to the element [(−1) ∪ (−1) ∪ (−1)] ⊗ w_{s+t} in H_{ét}^{3}(ℤ[\frac{1}{2}]; ℤ/16(3))(4(s + t)) , where w_{s+t} = w_{s} ⊗ w_{t} is a generator of ℤ/16(4(s + t)) . But the element (−1) ∪ (−1) ∪ (−1) is the image of the symbol {−1, −1, −1} which comes from (−1) ⊗ {−1, −1} via the product map K_{1}(ℤ) ⊗ K_{2}(ℤ) ⟶ K_{3}(ℤ) . The injectivity of that map (see Proposition 5.1 (a)) implies that {−1, −1, −1} ≠ 0 . Consider finally the commutative diagram

(8.10)  \begin{matrix}K_{3}^{M}(ℤ[\frac{1}{2}])/16 & ⟶ & H_{ét}^{3}(ℤ[\frac{1}{2}]; ℤ/16(3)) \\ ↓ & & ↓ \\ K_{3}^{M}(ℚ)/16 & ⟶^{≅} & H_{ét}^{3}(ℚ; ℤ/16(3)) , \end{matrix}

where the horizontal arrows are the Galois symbol maps. The lower horizontal map is an isomorphism by Theorem 5.1 of [37]. The element \{−1, −1, −1\} considered in K_{3}^{M}(ℤ[\frac{1}{2}])/16 maps nontrivially to K_{3}^{M}(ℚ)/16 . Its image in H_{ét}^{3}(ℤ[\frac{1}{2}]; ℤ/16(3)) is equal to (−1) ∪ (−1) ∪ (−1) and must be nontrivial by Diagram (8.10). This proves that x_{s} ⋆ z_{t} does not vanish.

Our next goal in this section is to use the 2-adic cyclotomic elements constructed in Section 7 to reprove parts of Theorem 8.1 and of our results on the other 2-adic products in K_{∗}(ℤ) given in Section 5.

Proposition 8.3. The natural homomorphism

K_{2n−1}(ℤ) ⊗ ℤ_{ˆ2} ⟶ H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n))

of Dwyer and Friedlander provides a surjective homomorphism (respectively an isomorphism)

C(n − 1) ⟶ C^{ét}(n − 1) .

when n ≥ 3 and n ≡ 1 mod 4 (respectively n ≡ 3 mod 4).

Proof. Let F be a number field containing ℚ(i) and S a finite set of prime ideals containing prime ideals over 2 in the ring of integers O_{F} of F . The ring O_{S} denotes as usual the ring of S-integers in F . By [17], Proposition 4.4, p. 256, there is a surjective homomorphism

(8.11)  K_{2n−1}(O_{S}) ⊗ ℤ_{ˆ2} ⟶ K_{2n−1}^{ét}(O_{S}) ⟶^{≅} H_{ét}^{1}(O_{S}; ℤ_{ˆ2}(n))

obtained by taking the inverse limit of the homomorphisms

(8.12)  K_{2n−1}(O_{S}; ℤ/2^{m}) ⟶ K_{2n−1}^{ét}(O_{S}; ℤ/2^{m}) ⟶^{≅} H_{ét}^{1}(O_{S}; ℤ/2^{m}(n)) .

The maps (8.12) commute with products for m ≥ 2 , and with transfers and reductions of coefficients for m ≥ 1 . For m ≥ 2 we have the following commutative diagram:

(8.13)  \begin{matrix}C′_{m} ⊗ ℤ/2^{m}(n − 1) & ⟶^{α_{m}} & K_{2n−1}(R′_{m}; ℤ/2^{m}) \\ ↓ = & & ↓ \\ C′_{m} ⊗ ℤ/2^{m}(n − 1) & ⟶_{≅}^{αét_{m}} & K_{2n−1}^{ét}(R′_{m}; ℤ/2^{m}) . \end{matrix}

Consequently, by Definitions 7.4 and 7.6 we obtain a surjective map

(8.14)  C(n − 1) ⟶ C^{ét}(n − 1) .

If n ≡ 3 mod 4, then K_{2n−1}(ℤ) ⊗ ℤ_{ˆ2} ≅ ℤ_{ˆ2} by Table 1 of [41], and the map (8.14) is also injective.

Now, let us formulate the following corollary the proof of which does not use Theorem 8.1.

Corollary 8.4. For any integer i ≥ 2 , the homomorphism

K_{i}(ℤ) ⊗ ℤ_{ˆ2}/torsion ⟶ K_{i}^{ét}(ℤ[\frac{1}{2}])/torsion

induced by the Dwyer-Friedlander map is an isomorphism.

Proof. For i = 2n − 1 with n odd ≥ 3 , Proposition 8.3. implies that the map is surjective since

K_{2n−1}^{ét}(ℤ[\frac{1}{2}])/torsion ≅ C^{ét}(n − 1) ≅ ℤ_{ˆ2}

by Corollary 7.11. The map is injective because

K_{2n−1}(ℤ) ⊗ ℤ_{ˆ2}/torsion ≅ ℤ_{ˆ2}

by [12], and a surjective homomorphism of ℤ_{ˆ2} -modules ℤ_{ˆ2} → ℤ_{ˆ2} must be injective. This holds also in dimensions i = 2n − 1 , with n even, and i even because the corresponding algebraic and étale K-groups are finite according to [12] and [19], Theorem 1.

Now let us deduce parts of Theorems 5.6 and 5.7 from a simple calculation of Steinberg symbols and Weibel’s computation of the 2-torsion of K_{∗}(ℤ) (Table 1 of [41]). First of all observe that by Corollary 8.4, Corollary 7.11 and Proposition 8.3, we have the following sequence of isomorphisms for any odd n ≥ 3 :

(8.15)  K_{2n−1}(ℤ) ⊗ ℤ_{ˆ2}/torsion ≅ K_{2n−1}^{ét}(ℤ[\frac{1}{2}])/torsion ≅ H_{ét}^{1}(ℤ[\frac{1}{2}]; ℤ_{ˆ2}(n))/torsion ≅ C^{ét}(n − 1) .

Consider the generator ẽ of C^{ét}(n−1) given by Corollary 7.11. It is clear that there is a nontorsion element b_{n} ∈ K_{2n−1}(ℤ) which is sent by (8.15) onto ẽ. We observe that for n odd ≥ 3 , the element b_{n} can be chosen in such a way that the image of b_{n} in K_{2n−1}(ℤ)/torsion ≅ ℤ is an odd multiple of a generator. Hence, we can assume for our computations that the image of b_{n} in K_{2n−1}(ℤ)/torsion is a generator.

Proposition 8.5. Assume that n and m are odd integers ≥ 3 .

(a) The product map

⋆ : K_{1}(ℤ) ⊗ K_{2m−1}(ℤ) ⟶ K_{2m}(ℤ)

sends the subgroup ⟨−1⟩ ⊗ ⟨b_{m}⟩ to zero.

(b) The 2-adic product map

⋆ : K_{2n−1}(ℤ) ⊗ K_{2m−1}(ℤ) ⟶ K_{2(n+m−1)}(ℤ) ⊗ ℤ_{ˆ2}

sends the subgroup ⟨b_{n}⟩ ⊗ ⟨b_{m}⟩ to zero.

Proof. Since we are only interested in the image of the product (a) in the 2-torsion part of K_{2m}(ℤ) , it is sufficient to consider the following commutative diagram

(8.16)  \begin{matrix}K_{1}(ℤ)/4 ⊗ K_{2m−1}(ℤ)/4 & ⟶^{⋆} & K_{2m}(ℤ)_{2} \\ ↓ & & ↓ \\ K_{1}(ℤ)/4 ⊗ K_{2m−1}(ℤ; ℤ/4) & ⟶^{⋆} & K_{2m}(ℤ; ℤ/4) . \end{matrix}

According to Table 1 of [41], K_{2m}(ℤ) ⊗ ℤ_{ˆ2} ≅ ℤ/2 if m ≡ 1 mod 4 and the right vertical arrow is a well defined embedding. Hence, to prove claim (a) we need to show that ⟨−1⟩ ⊗ ⟨b_{m}⟩ maps to zero in K_{2m}(ℤ; ℤ/4) via the homomorphisms in Diagram (8.16). The left vertical arrow in Diagram (8.16) maps the element (−1) ⊗ b_{m} to

(−1) ⊗ T r_{E_{2}/E_{1}}(c ⋆ β_{2}^{⋆(m−1)})

for some c ∈ C′_{2} . Sending it further to K_{2m}(ℤ; ℤ/4) along the lower horizontal arrow and using the projection formula and associativity of the product for K -groups with ℤ/4 coefficients we find out that its image equals

T r_{E_{2}/E_{1}}(((−1) ⋆ c) ⋆ β_{2}^{⋆(m−1)}) .

The product ⋆ : K_{1}(R) ⊗ K_{1}(R) → K_{2}(R) with integral coefficients is just the Steinberg symbol { , } for any commutative ring with identity R (see [22], Proposition 2.2.3). Hence, we get:

\begin{aligned}(−1) ⋆ (1 − i) & = & \{−1, 1 − i\} & = & \{i, 1 − i\}^{2} = 1 \\ (−1) ⋆ (1 + i) & = & \{−1, 1 + i\} & = & \{−i, 1 + i\}^{2} = 1. \end{aligned}

But the group C′_{2} is generated by 1 − i and 1 + i, hence (−1) ⋆ c = 1 . This proves (a) for m ≡ 1 mod 4 and there is nothing to prove if m ≡ 3 mod 4 since K_{2m}(ℤ)_{2} = 0 by [41]. In order to prove (b) consider the commutative diagram

(8.17)  \begin{matrix}K_{2n−1}(ℤ)/4 ⊗ K_{2m−1}(ℤ)/4 & ⟶^{⋆} & K_{2(n+m−1)}(ℤ)_{2} \\ ↓ & & ↓ \\ K_{2n−1}(ℤ; ℤ/4) ⊗ K_{2m−1}(ℤ; ℤ/4) & ⟶^{⋆} & K_{2(n+m−1)}(ℤ; ℤ/4) . \end{matrix}

It is enough to show that the image of b_{n} ⊗b_{m} in the upper left corner of Diagram (8.17) maps to the trivial element in the lower right corner on its journey through the diagram. Indeed, b_{n} ⊗ b_{m} is sent to

T r_{E_{2}/E_{1}}(c_{1} ⋆ β_{2}^{⋆(n−1)}) ⊗ T r_{E_{2}/E_{1}}(c_{2} ⋆ β_{2}^{⋆(m−1)})

via the left vertical arrow in Diagram (8.17) for some c_{1}, c_{2} ∈ C′_{2} . Sending the last element along the bottom horizontal arrow and using projection formula we get the element

T r_{E_{2}/E_{1}}(c_{1} ⋆ β_{2}^{⋆(n−1)} ⋆ c_{2}c_{2}^{τ} ⋆ β_{2}^{⋆(m−1)}) ,

where τ denotes the complex conjugation. Since β_{2} commutes with elements c ∈ K_{1}(R) , we obtain the following equality:

c_{1} ⋆ β_{2}^{⋆(n−1)} ⋆ c_{2}c_{2}^{τ} ⋆ β_{2}^{⋆(m−1)} = ±c_{1} ⋆ c_{2}c_{2}^{τ} ⋆ β_{2}^{⋆(n+m−2)} .

On the other hand, we have the following equalities involving Steinberg symbols:

(8.18)  \begin{aligned}(1 − i) ⋆ (1 − i) & = \{1 − i, 1 − i\} = \{1 − i, −1\} = \{1 − i, i\}^{2} = 1 \\ (1 − i) ⋆ (1 + i) & = \{1 − i, 1 + i\} = \{1 − i, i(1 − i)\} = \{1 − i, i\}\{1 − i, 1 − i\} = 1 \\ (1 + i) ⋆ (1 − i) & = \{1 − i, 1 + i\}^{−1} = 1 \\ (1 + i) ⋆ (1 + i) & = \{1 + i, 1 + i\} = \{1 + i, i(1 − i)\} \\ & = \{1 + i, i\}\{1 + i, 1 − i\} = \{1 + i, i\} = \{1 + i, −1\} = 1 \end{aligned}

The group C′_{2} is generated by 1 − i and 1 + i (see Example 7.2). Hence, by bilinearity of the Steinberg symbol and by (8.18) we get c_{1} ⋆ c_{2}c_{2}^{τ} = {c_{1}, c_{2}c_{2}^{τ}} = 1 and it follows that

T r_{E_{2}/E_{1}}(c_{1} ⋆ β_{2}^{⋆(n−1)} ⋆ c_{2}c_{2}^{τ} ⋆ β_{2}^{⋆(m−1)}) = 1 .

So b_{n} ⊗ b_{m} maps to zero in K_{2(n+m−1)}(ℤ)_{2} via the horizontal arrow of Diagram (8.17) because the right vertical arrow is injective.

Remark 8.6. We can prove claim (a) in Proposition 8.5 without even knowing the 2 -torsion of the group K_{2m}(ℤ) . In order to see this, fix s > 1 such that 2^{s} annihilates the group K_{2m}(ℤ)_{2} . Consider the commutative diagram

K_{1}(ℤ) ⊗ K_{2m−1}(ℤ)  ⟶^{⋆}  K_{2m}(ℤ)_{2}
↓  ↓
K_{1}(ℤ)/2^{s} ⊗ K_{2m−1}(ℤ; ℤ/2^{s}) ⟶^{⋆} K_{2m}(ℤ; ℤ/2^{s}) ,

where the right vertical arrow is injective by assumptions. Now we argue in the same way as in the proof of (a) in Proposition 8.5. The crucial point is to prove that the element T r_{E/ℚ}((−1)⋆u_{s} ⋆β_{s}^{⋆(m−1)}) ∈ K_{2m}(ℤ; ℤ/2^{s}) is trivial for any u_{s} ∈ C′_{s} . But we know the generators of the group C′_{s} by Remark 7.1, so it is enough to observe the following equalities on Steinberg symbols:

{−1, ξ_{2^{s}}} = {−ξ_{2^{s}}, ξ_{2^{s}}}^{2^{s−1}} = 1 and {−1, 1 − ξ_{2^{s}}^{a}} = {ξ_{2^{s}}^{a}, 1 − ξ_{2^{s}}^{a}}^{2^{s−1}} = 1 .

Let us conclude this section by the following observation about the relationships between products in algebraic K-theory and the Dwyer-Friedlander map.

Proposition 8.7. For any odd prime l and any integer n ≥ 2 , the image of the product map

⋆ : K_{2n−1}(ℤ) ⊗ K_{2n−1}(ℤ) ⟶ K_{4n−2}(ℤ)

is contained in the kernel of the Dwyer-Friedlander map K_{4n−2}(ℤ) → K_{4n−2}^{ét}(ℤ[\frac{1}{l}]) .

Proof. The products in algebraic K-theory and étale K-theory commute with the Dwyer-Friedlander map. Observe that

K_{2n−1}^{ét}(ℤ[\frac{1}{l}]) = \{ \begin{matrix}ℤ_{l̂} & if n is odd, \\ ℤ/|w_{n}(ℚ)|_{l}^{−1} & if n is even. \end{matrix}

Hence K_{2n−1}^{ét}(ℤ[\frac{1}{l}]) ≅ H_{ét}^{1}(ℤ[\frac{1}{l}]; ℤ_{l}(n)) is cyclic. But the product in étale K-theory is just the cup product in étale cohomology. This shows that the product

K_{2n−1}^{ét}(ℤ[\frac{1}{l}]) ⊗ K_{2n−1}^{ét}(ℤ[\frac{1}{l}]) ⟶ K_{4n−2}^{ét}(ℤ[\frac{1}{l}])

is zero because the cup product H_{ét}^{1} ⊗ H_{ét}^{1} → H_{ét}^{2} is anticommutative.

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Dominique Arlettaz Institut de mathématiques, Université de Lausanne, CH-1015 Lausanne, Switzerland e-mail: dominique.arlettaz@ima.unil.ch

Grzegorz Banaszak Department of Mathematics, Adam Mickiewicz University, 60769 Poznań, Poland e-mail: BANASZAK@math.amu.edu.pl

Wojciech Gajda current address: Max-Planck-Institut für Mathematik, 53235 Bonn, Germany permanent address: Department of Mathematics, Adam Mickiewicz University, 60769 Poznań, Poland e-mail: GAJDA@math.amu.edu.pl