ON THE IMAGE OF MILNOR’S K_{4} IN

QUILLEN’S K_{4} FOR NUMBER FIELDS

John Rognes

May 22nd–24th 1999

Introduction

Let F be a field, with multiplicative group F^{×}. Its Milnor K-theory [Mi1] is the graded ring K_{∗}^{M}(F) defined as the tensor algebra T (F^{×}) = ⨁_{n≥0}(F^{×})^{⊗n} divided by the two-sided homogeneous ideal generated by the terms x⊗(1−x) for x ≠ 0, 1. The image of x_{1} ⊗ ⋯ ⊗ x_{n} ∈ (F^{×})^{⊗n} in K_{n}^{M}(F) is written {x_{1}, …, x_{n}}.

Quillen’s higher algebraic K-theory K_{∗}(F) is also a graded ring [Qu], and the isomorphism F^{×} ≅ K_{1}(F) extends to a ring homomorphism T (F^{×}) → K_{∗}(F). The product in algebraic K-theory satisfies {x} · {1 − x} = 0 for x ≠ 0, 1, so this ring homomorphism factors through Milnor K-theory. We obtain a canonical homomorphism

κ_{n}: K_{n}^{M}(F) ⟶ K_{n}(F)

for all n ≥ 0. This is an isomorphism for n = 0, 1, more or less by definition, and for n = 2 by H. Matsumuto’s theorem [Mi2, §12].

Now suppose F is a number field, with r_{1} real embeddings. By the Bass–Tate theorem [BT], K_{n}^{M}(F) ≅ (ℤ/2)^{r_{1}} for all n ≥ 3. Multiplication by the symbol ϵ = {−1} induces a homomorphism ϵ·: K_{n}^{M}(F) → K_{n+1}^{M}(F) which is a surjection for n = 1, 2, and an isomorphism for n ≥ 3. Hence all classes in K_{n}^{M}(F) have the form ϵ^{n−1} · {x} = {−1, …, −1, x} for x ∈ F^{×}, when n ≥ 1. The class ϵ^{n} = {−1, …, −1} has nonzero image in K_{n}^{M}(ℝ), hence is nonzero in K_{n}^{M}(F) when F is formally real, i.e., when r_{1} > 0.

It was proved by J.M. Shapiro in [Sh] that κ_{3} injects K_{3}^{M}(F) into K_{3}(F). There is a ring map from the stable homotopy groups of spheres π_{∗}^{S} → K_{∗}(F) taking the Hopf map η ∈ π_{1}^{S} to ϵ ∈ K_{1}(F). Since η^{4} = 0 in π_{4}^{S} = 0, the symbol ϵ^{4} ∈ K_{4}^{M}(F) is in the kernel of κ_{4}. It follows that the image of K_{n}^{M}(F) in K_{n}(F) is zero for n ≥ 5. This leads us to ask what happens in the remaining case, n = 4. In many cases it is known that the image is zero. It is the purpose of this paper to give a sometimes nonzero lower bound on the image of K_{4}^{M}(F) ≅ (ℤ/2)^{r_{1}} in K_{4}(F).

We prove the following main theorem:

Theorem 0.1. Let F be a number field with r_{1} real embeddings, and R_{F} = O_{F} [1/2] its ring of 2-integers. Let s be the number of distinct primes in O_{F} dividing (2). Let t = rk_{2} Pic(R_{F}) and u = rk_{2} Pic_{+}(R_{F}) be the 2-rank of the Picard group of R_{F},

Typeset by AMS-TEX

and the narrow Picard group of R_{F}, respectively. The 2-rank ρ = rk_{2} im(κ_{4}) of the image of the canonical map

κ_{4} : K_{4}^{M}(F) ≅ (ℤ/2)^{r_{1}} → K_{4}(F)

satisfies u − t ≤ ρ ≤ s + u − 1. If F is formally real then ρ < r_{1}, otherwise ρ = 0.

As a simple example consider F = ℚ(\sqrt{7}), the real quadratic number field with discriminant 28. It has r_{1} = 2, t = 0 and u = 1, so the image of κ_{4} is precisely ℤ/2 in this case.

1. Number fields and cohomology

We review and fix some basic notation.

For an abelian group A and an integer q let _{q}A = {a ∈ A | qa = 0} be the q-torsion subgroup. For a prime ℓ let A{ℓ} = ⋃_{ν≥0} _{ℓ^{ν}} A ⊆ A be the ℓ-power torsion subgroup. We define the ℓ-rank rk_{ℓ} A of A as the dimension of the ℤ/ℓ-vector space ℓA. When A is finite, this equals the ℤ/ℓ-dimension of A/ℓ. For abelian groups A and B we write A ⋊ B for an extension of B by A, such that there is a short exact sequence 0 → A → A ⋊ B → B → 0.

Let F be a number field, with r_{1} real embeddings and r_{2} pairs of complex embeddings. Let O_{F} be its ring of integers, and let R_{F} = O_{F} [1/2] be its ring of 2-integers. Let s be the number of distinct prime ideals in O_{F} over the rational prime ideal (2). By Dirichlet’s unit theorem, R_{F}^{×} ≅ ℤ^{r_{1}+r_{2}+s−1} × μ(R_{F}), where the group μ(R_{F}) of roots of unity in R_{F} is a finite cyclic group of even order.

Let Pic(R_{F}) be the Picard group of R_{F}, and let Pic_{+}(R_{F}) be the narrow Picard group. These can be defined as follows: Let

e: F^{×} → ⨁_{𝔓∤(2)} ℤ

be the divisor map, taking x = a/b ∈ F^{×} with a, b ∈ O_{F} to the sequence e(x) = (v_{𝔓}(a) − v_{𝔓}(b))_{𝔓} where v_{𝔓} is the valuation at 𝔓. Then the Picard group is Pic(R_{F}) = cok(e). Let F_{+}^{×} ⊆ F^{×} be the subgroup of totally positive units, i.e., the kernel of the sign map

σ: F^{×} → (ℤ/2)^{r_{1}}

defined as the sum over the r_{1} real embeddings of F of the corresponding maps F^{×} → ℝ^{×} → ℝ^{×}/2 ≅ ℤ/2. Let e_{+} = e|F_{+}^{×} be the restricted divisor map. Then the narrow Picard group is Pic_{+}(R_{F}) = cok(e_{+}). Let t = rk_{2} Pic(R_{F}) and u = rk_{2} Pic_{+}(R_{F}) be the 2-rank of the Picard group and the narrow Picard group of R_{F}, respectively.

The Brauer group Br(R_{F}) of R_{F} can be defined in terms of Azumaya algebras or Galois cohomology, and is determined by the Brauer–Hasse–Noether theorem, asserting that there is an exact sequence

0 → Br(R_{F}) → (ℤ/2)^{r_{1}} ⊕ (ℚ/ℤ)^{s} → ℚ/ℤ → 0 .

Thus Br(R_{F}) ≅ (ℤ/2)^{r_{1}} ⊕ (ℚ/ℤ)^{s−1}.

Let R be a commutative ring. For M an étale sheaf over Spec(R), let H_{ét}^{n}(R; M) denote the nth étale cohomology group of Spec(R) with coefficients in the sheaf M. With M = 𝔾_{m} (the multiplicative group) there are standard isomorphisms H_{ét}^{0}(R; 𝔾_{m}) ≅ R^{×}, H_{ét}^{1}(R; 𝔾_{m}) ≅ Pic(R) and H_{ét}^{2}(R; 𝔾_{m}) ≅ Br(R) [Ho]. With M = μ_{q} (the qth roots of unity) and 1/q ∈ R, the short exact Kummer sequence of étale sheaves

0 → μ_{q} → 𝔾_{m} ⟶^{(−)^{q}} 𝔾_{m} → 0

induces a long exact sequence in étale cohomology. There result natural isomorphisms H_{ét}^{0}(R; μ_{q}) ≅ μ_{q}(R), H_{ét}^{1}(R; μ_{q}) ≅ R^{×}/(R^{×})^{q} ⋊_{q} Pic(R) and H_{ét}^{2}(R; μ_{q}) ≅ Pic(R)/q ⋊_{q} Br(R). For integers i we write ℤ/q(i) = μ_{q}^{⊗i} for the ith tensor power of μ_{q}, as an étale sheaf. By a theorem of J. Tate, the natural map

α^{n}: H_{ét}^{n}(R_{F}; ℤ/q(i)) → ⨁^{r_{1}} H_{ét}^{n}(ℝ; ℤ/q(i))

is an isomorphism for n ≥ 3. Here H_{ét}^{n}(ℝ; ℤ/q(i)) ≅ H^{n}(Gal(ℂ/ℝ); ℤ/q(i)) is easily computed using the periodic resolution for the group cohomology on the right.

Note that with q = 2, the étale sheaf ℤ/2(i) is independent of i, hence can be briefly written ℤ/2. The Kummer and Tate formulas above specialize to:

H_{ét}^{0}(R_{F}; ℤ/2) ≅ μ_{2}(R_{F}) ≅ ℤ/2
H_{ét}^{1}(R_{F}; ℤ/2) ≅ R_{F}^{×}/2 ⋊_{2} Pic(R_{F}) ≅ (ℤ/2)^{r_{1}+r_{2}+s+t}
H_{ét}^{2}(R_{F}; ℤ/2) ≅ Pic(R_{F})/2 ⋊_{2} Br(R_{F}) ≅ (ℤ/2)^{r_{1}+s+t−1}
H_{ét}^{n}(R_{F}; ℤ/2) ≅ H_{ét}^{n}(ℝ; ℤ/2)^{r_{1}} ≅ (ℤ/2)^{r_{1}}

with n ≥ 3 in the last row.

We define H̃_{ét}^{1}(R_{F}; ℤ/2) as the kernel of the natural map α^{1} : H_{ét}^{1}(R_{F}; ℤ/2) → ⨁^{r_{1}} H_{ét}^{1}(ℝ; ℤ/2) ≅ (ℤ/2)^{r_{1}}. Likewise for H̃_{ét}^{1}(F; ℤ/2) and H̃_{ét}^{2}(R_{F}; ℤ/2).

Proposition 1.1. There is a natural exact sequence

0 → H̃_{ét}^{1}(R_{F}; ℤ/2) → H_{ét}^{1}(R_{F}; ℤ/2) ⟶^{α^{1}} (ℤ/2)^{r_{1}} →

→ Pic_{+}(R_{F})/2 ⟶^{π} Pic(R_{F})/2 → 0 .

Hence rk_{2} cok(α^{1}) = u − t and rk_{2} H̃_{ét}^{1}(R_{F}; ℤ/2) = r_{2} + u + s.

Proof. Consider the following commutative diagram:

\xymatrix{ & H̃_{ét}^{1}(R_{F} ℤ/2) \ar[r] \ar[d] & H̃_{ét}^{1}(F; ℤ/2) \ar[d] \ar[dr]^{˜∂} \\ 0 \ar[r] & H_{ét}^{1}(R_{F}; ℤ/2) \ar[r] \ar[dr]_{α^{1}} & H_{ét}^{1}(F; ℤ/2) \ar[r]_{∂} \ar[d]^{σ/2} & ⨁_{𝔓∤(2)} H_{ét}^{0}(k_{𝔓}; ℤ/2) \\  &  & (ℤ/2)^{r_{1}} \ar[d] \\  &  & 0 .}

The middle row is part of the exact localization sequence in mod 2 étale cohomology, where we write k_{𝔓} for the residue field of F at the prime ideal 𝔓 ⊂ R_{F}. Since Pic(F) = 0 we can identify H_{ét}^{1}(F; ℤ/2) ≅ F^{×}/2, and the sign map σ descends to a homomorphism σ/2: H_{ét}^{1}(F; ℤ/2) → (ℤ/2)^{r_{1}}. It is surjective by the approximation theorem. Its restriction to H_{ét}^{1}(R_{F}; ℤ/2) ⊂ H_{ét}^{1}(F; ℤ/2) is α^{1}, and the kernel of σ/2 is denoted H̃_{ét}^{1}(F; ℤ/2). We let ˜∂ be the restriction of the boundary map ∂ to H̃_{ét}^{1}(F; ℤ/2).

Then we have an exact sequence

0 → H̃_{ét}^{1}(R; ℤ/2) → H_{ét}^{1}(R; ℤ/2) ⟶^{α^{1}} (ℤ/2)^{r_{1}} → cok(∂) → cok(˜∂) → 0 .

It remains to identify the two cokernels.

First, recognize ∂ as e/2, the divisor map reduced mod 2, under the identifications H_{ét}^{1}(F; ℤ/2) ≅ F^{×}/2 and H_{ét}^{0}(k_{𝔓}; ℤ/2) ≅ ℤ/2. Thus cok(∂) ≅ cok(e/2) ≅ cok(e)/2 ≅ Pic(R)/2.

Second, the surjection F^{×} → F^{×}/2 ≅ H_{ét}^{1}(F; ℤ/2) restricts to a surjection F_{+}^{×} → H̃_{ét}^{1}(F; ℤ/2) of the kernels of σ and σ/2, respectively. The composite

F_{+}^{×} → H̃_{ét}^{1}(F; ℤ/2) ⟶^{˜∂} ⨁_{𝔓∤(2)} H_{ét}^{0}(k_{𝔓}; ℤ/2)

is identified with (e_{+})/2, the restricted divisor map reduced mod 2. Hence cok(˜∂) ≅ cok((e_{+})/2) ≅ cok(e_{+})/2 ≅ Pic_{+}(R)/2.

Clearly Pic(R)/2 ≅ (ℤ/2)^{t} and Pic_{+}(R)/2 ≅ (ℤ/2)^{u} because the groups Pic(R) and Pic_{+}(R) are finite, and the remaining claims follow by exactness. □

We shall also make use of ℓ-adic étale cohomology, with coefficients in the prosheaf ℤ_{ℓ}(i) = {ℤ/ℓ^{ν}}_{ν}. Since each group H_{ét}^{n}(R_{F}; ℤ/2^{ν}(i)) is finite, there is an isomorphism H_{ét}^{n}(R_{F}; ℤ_{2}(i)) ≅ lim_{ν} H_{ét}^{n}(R_{F}; ℤ/2^{ν}(i)).

Lemma 1.2. Let i ≥ 2. The finite group H_{ét}^{2}(R_{F}; ℤ_{2}(i)) has 2-rank r_{1} + s + t − 1 for i even, and s + t − 1 for i odd.

Proof. The short exact sequence

0 → H_{ét}^{2}(R_{F}; ℤ_{2}(i))/2 → H_{ét}^{2}(R_{F}; ℤ/2) → _{2}H_{ét}^{3}(ℝ_{F}; ℤ_{2}(i)) → 0

and the isomorphism H_{ét}^{3}(R_{F}; ℤ_{2}(i)) ≅ 0 for i even, ≅ (ℤ/2)^{r_{1}} for i odd, ensures that the 2-rank of the finite group H_{ét}^{2}(R_{F}; ℤ_{2}(i)) is r_{1} + s + t − 1 for i even and s + t − 1 for i odd. □

2. Milnor K-theory and cohomology

One difficulty in handling the image of Milnor’s K_{4}^{M}(F) in K_{4}(F) is that the target group is infinite and contains divisible elements. Let F be a number field, with ring of 2-integers R_{F}, and keep the notation of the previous section.

Lemma 2.1. The image of K_{4}^{M}(F) in K_{4}(F) maps injectively to K_{4}(F; ℤ_{2}) by the coefficient extension homomorphism K_{4}(F) → K_{4}(F; ℤ_{2}).

Proof. Consider the following commutative diagram:

\xymatrix{K_{3}^{M}(F) \ar[r] \ar[d]_{ϵ·}^{≅} & K_{3}(R_{F})\{2\} \ar[r] \ar[d]_{ϵ·} & K_{3}(F) \ar[d]_{ϵ·} \\ K_{4}^{M}(F) \ar[r] & K_{4}(R_{F})\{2\} \ar[r] \ar[d] & K_{4}(F) \ar[d] \\  & K_{4}(R_{F}; ℤ_{2}) \ar[r] & K_{4}(F; ℤ_{2})}

The canonical map κ_{3} : K_{3}^{M}(F) → K_{3}(F) factors (uniquely) through K_{3}(R_{F}){2} because K_{3}^{M}(F) ≅ (ℤ/2)^{r_{1}} is 2-torsion by the Bass–Tate theorem [BT], and the natural map K_{3}(R_{F}) → K_{3}(F) is an isomorphism by a theorem of Soulé [So]. Since multiplication by the symbol ϵ = {−1} is an isomorphism from K_{n}^{M}(F) to K_{n+1}^{M}(F) for n ≥ 3, also the canonical map κ_{4} : K_{4}^{M}(F) → K_{4}(F) factors (uniquely) through K_{4}(R_{F}){2}. This explains the maps in the upper two rows of the diagram. The maps in the lower part of the diagram are all natural.

By the localization sequence in algebraic K-theory, the natural map K_{4}(R_{F}) → K_{4}(F) is injective, as is the inclusion K_{4}(R_{F}){2} ⊆ K_{4}(R_{F}). Since K_{4}(R_{F}) is finite, the coefficient extension homomorphism K_{4}(R_{F}){2} → K_{4}(R_{F}; ℤ_{2}) is also injective. Finally, by the localization sequence in algebraic K-theory with ℤ_{2}- coefficients, the natural map K_{4}(R_{F}; ℤ_{2}) → K_{4}(F; ℤ_{2}) is injective. This uses that the preceding term in the exact sequence is π_{4}(⋁_{𝔓∤(2)} K(k_{𝔓})_{2}^{∧}) = 0. Hence the image of K_{4}^{M}(F) in K_{4}(F) is isomorphic to the image of K_{4}^{M}(F) in each of the other groups K_{4}(R_{F}){2}, K_{4}(R_{F}; ℤ_{2}) and K_{4}(F; ℤ_{2}), respectively. □

The following observation shows why the image of κ_{4} cannot be detected in K_{4}(F; ℤ/2^{ν}) for ν ≤ 3.

Lemma 2.2. The image of K_{4}^{M}(F) in K_{4}(F) is divisible by 24.

Proof. Any class in K_{4}^{M}(F) has the form y = {−1, −1, −1, x} = ϵ^{3} · {x} for some x ∈ F^{×}. Here {−1, −1, −1} = ϵ^{3} lifts to K_{3}(ℚ) ≅ K_{3}(ℤ) ≅ ℤ/48, where it represents the class of order 2, hence is divisible by 24. Let λ ∈ K_{3}(ℚ) be a generator. Then y = 24 · λ · {x} in K_{4}(F), as claimed. □

3. Algebraic K-theory and cohomology

We next invoke the extended Bloch–Lichtenbaum spectral sequence of M. Levine, see [BL, 1.3.4] and [Le2, (1.8)]. In its integral form, it is a spectral sequence from motivic cohomology to algebraic K-theory.

Let A be a regular commutative ring of finite type over a regular base ring B of Krull dimension ≤ 1. For example, A can be a number field F, with B = ℚ, or A can be a ring of ℓ-integers R_{F}, with B = ℤ[1/ℓ]. Then Levine extends the definition of the motivic cohomology groups H^{n}(A; ℤ(i)) given by Bloch, Suslin and Voevodsky in the field case. He then constructs a homological, first quadrant, strongly convergent algebra spectral sequence:

(3.1)  E_{p,q}^{2} = H^{q−p}(A; ℤ(q)) ⟹ K_{p+q}(A) .

Likewise, there is a mod m version

(3.2)  E_{p,q}^{2} = H^{q−p}(A; ℤ/m(q)) ⟹ K_{p+q}(A; ℤ/m)

converging to algebraic K-theory with mod m coefficients, see [Le2, (1.9)].

We may tensor the integral spectral sequence (3.1) with ℤ_{ℓ}, to obtain an ℓ-adic spectral sequence. Its E^{2}-term is H^{p−q}(A; ℤ(q)) ⊗ ℤ_{ℓ}, which when each group H^{∗}(A; ℤ/ℓ^{ν}(q)) is finite can be identified with H^{p−q}(A; ℤ_{ℓ}(q)). Likewise its abutment is K_{p+q}(A)⊗ℤ_{ℓ}, which when each group K_{∗}(A; ℤ/ℓ^{ν}) is finite can be identified with K_{p+q}(A; ℤ_{ℓ}) = π_{p+q}(K(A)_{ℓ}^{∧}).

When F is a number field, A = R_{F} is its ring of ℓ-integers, and ℓ = 2, Voevodsky’s proof of the Milnor conjecture [Vo2] together with [SV], implies that there are natural isomorphisms

H^{n}(R_{F}; ℤ/2^{ν}(i)) ≅ H_{ét}^{n}(R_{F}; ℤ/2^{ν}(i))
H^{n}(R_{F}; ℤ_{2}(i)) ≅ H_{ét}^{n}(R_{F}; ℤ_{2}(i))

for all n ≤ i, while H^{n}(R_{F}; ℤ/2^{ν}(i)) = 0 and H^{n}(R_{F}; ℤ_{2}(i)) = 0 for n > i. By finiteness of the groups H_{ét}^{n}(R_{F}; ℤ/2^{ν}(i)) and K_{n}(R_{F}; ℤ/2^{ν}) for all n, i, the discussion of the previous paragraph applies, and we have the following 2-adic extended Bloch–Lichtenbaum spectral sequence:

(3.3)  E_{p,q}^{2} = H_{ét}^{q−p}(R_{F}; ℤ_{2}(q)) ⟹ K_{p+q}(R_{F}) ⊗ ℤ_{2}

with p, q ≥ 0.

Consider the integral extended Bloch–Lichtenbaum spectral sequence in the case A = F. In total degree 1 the edge homomorphism E_{0,1}^{2} = H^{1}(F; ℤ(1)) → K_{1}(F) ≅ F^{×} is an isomorphism. There is a cup product in motivic cohomology which satisfies the relation {x} · {1 − x} = 0 for x ≠ 0, 1, thus providing a map K_{n}^{M}(F) → H^{n}(F; ℤ(n)) for all n ≥ 0. This is an isomorphism by the Suslin–Nesterenko theorem [NS, 4.9]. Also by multiplicativity of the spectral sequence, we can identify the edge homomorphism E_{0,n}^{2} = H^{n}(F; ℤ(n)) → K_{n}(F) of the extended Bloch– Lichtenbaum spectral sequence with the canonical map κ_{n}: K_{n}^{M}(F) → K_{n}(F). This reduces the question of determining the image of κ_{n} to that of determining the E^{∞}-term E_{0,n}^{∞}.

We now consider the spectral sequences (3.1) for A = F, (3.3) for A = F and (3.3) for A = R_{F}. The edge homomorphisms are compatible, yielding a commutative diagram

\xymatrix{H^{4}(F; ℤ(4)) \ar[r] \ar[d] & K_{4}(F) \ar[d] \\ H^{4}(F; ℤ_{2}(4)) \ar[r] & K_{4}(F; ℤ_{2}) \\ H^{4}(R_{F}; ℤ_{2}(4)) \ar[r] \ar[u] & K_{4}(R_{F}; ℤ_{2}) \ar[u]}

The left hand groups are all isomorphic to K_{4}^{M}(F) ≅ ⨁^{r_{1}} H^{4}(ℝ; ℤ_{2}(4)) ≅ (ℤ/2)^{r_{1}} by the Suslin–Nesterenko theorem and Voevodsky’s proof of the Milnor conjecture.

Thus both left hand vertical maps are isomorphisms. Also the lower right hand vertical map is injective, and the image of K_{4}^{M}(F) in K_{4}(F) is isomorphic to that in K_{4}(F; ℤ_{2}), by 2.1. Thus the image of κ_{4} is isomorphic to the image of the bottom edge homomorphism H^{4}(R_{F}; ℤ_{2}(4)) → K_{4}(R_{F}; ℤ_{2}), of the spectral sequence (3.3).

The spectral sequence (3.3) for A = ℝ has E^{2}-term

E_{p,q}^{2} = H_{ét}^{q−p}(ℝ; ℤ_{2}(q)) ≅ \{ \begin{matrix}ℤ/2 & for p even, \\ 0 & for p odd. \end{matrix}

It abuts to K_{∗}(ℝ; ℤ_{2}) ≅ π_{∗}(ko; ℤ_{2}), hence there are surjective differentials d_{p,q}^{2} for all p ≡ 2 mod 4, while the remaining differentials are all zero.

Similarly, in the spectral sequence (3.2) for A = ℝ with ℤ/2-coefficients the E^{2}- term is E_{p,q}^{2} = H_{ét}^{q−p}(ℝ; ℤ/2(q)) ≅ ℤ/2 for all p, q ≥ 0. The abutment is π_{∗}(ko; ℤ/2), and the pairing of (3.3) with (3.2) implies that there are surjective differentials d_{p,q}^{2} for all p ≡ 2, 3 mod 4, while the remaining differentials are all zero.

In total degree 4, the spectral sequence (3.3) yields the exact sequence

(3.4)  H_{ét}^{1}(R_{F}; ℤ_{2}(3)) ⟶^{d_{2,3}^{2}} (ℤ/2)^{r_{1}} → K_{4}(R_{F}; ℤ_{2}) → H_{ét}^{2}(R_{F}; ℤ_{2}(3)) → 0 .

The image of κ_{4} is thus identified with the cokernel of the differential d_{2,3}^{2}. By naturality of (3.3) with respect to the r_{1} real embeddings R_{F} → F → ℝ, there is a commutative square:

\xymatrix{H_{ét}^{1}(R_{F}; ℤ_{2}(3)) \ar[r]^{d_{2,3}^{2}} \ar[d]^{α^{1}} & H_{ét}^{4}(R_{F}; ℤ_{2}(4)) \ar[d]^{α^{4}}_{≅} \\ ⨁^{r_{1}} H_{ét}^{1}(ℝ; ℤ_{2}(3)) \ar[r]^{d_{2,3}^{2}}_{≅} & ⨁^{r_{1}} H_{ét}^{4}(ℝ; ℤ_{2}(4))}

Since the right hand and lower maps are isomorphisms, the cokernel of d_{2,3}^{2} is identified with the cokernel of the natural homomorphism

α^{1} : H_{ét}^{1}(R_{F}; ℤ_{2}(3)) → ⨁^{r_{1}} H_{ét}^{1}(ℝ; ℤ_{2}(3)) .

This reduces the question about the image of κ_{4} to a question in étale cohomology, which however does not seem to be so easy to answer directly.

4. Two-rank formulas

We rewrite the exact sequence (3.4) as follows:

(4.1)  0 → im(κ_{4}) → K_{4}(R_{F}) → H_{ét}^{2}(R_{F}; ℤ_{2}(3)) → 0 .

We can compute the 2-rank of both the middle and right hand term in this exact sequence, which will provide a lower bound on the 2-rank, thus dimension, of the image group on the left.

Proposition 4.2. Let F be a formally real number field, i.e., with r_{1} > 0. For n ≥ 1, the 2-rank of K_{n}(R_{F}; ℤ_{2}) is given by the table:

n mod 8  0  1  2  3
rk_{2} K_{n}(R_{F}; ℤ_{2})  s + t − 1  1  r_{1} + s + t − 1  r_{1}
n mod 8  4  5  6  7
rk_{2} K_{n}(R_{F}; ℤ_{2})  s + u − 1  0  s + u − 1  1

Proof. Consider the mod 2 extended Bloch–Lichtenbaum spectral sequence (3.2) for R_{F}. Its E^{2}-term has

E_{p,q}^{2} = H_{ét}^{q−p}(R_{F}; ℤ/2) ≅ \{ \begin{matrix}0 & for q < p, \\ ℤ/2 & for q = p, \\ R^{×}/2 ⋊_{2} Pic(R) & for q = p + 1, \\ Pic(R)/2 ⋊_{2} Br(R) & for q = p + 2, \\ (ℤ/2)^{r_{1}} & for q ≥ p + 3. \end{matrix}

By naturality with respect to the r_{1} real embeddings, there are commutative squares:

\xymatrix{H_{ét}^{q−p}(R_{F}; ℤ/2) \ar[r]^{d_{p,q}^{2}} \ar[d]^{α^{q−p}} & H_{ét}^{q−p+3}(R_{F}; ℤ/2) \ar[d]^{α^{q−p+3}} \\ ⨁^{r_{1}} H_{ét}^{q−p}(ℝ; ℤ/2) \ar[r]^{d_{p,q}^{2}} & ⨁^{r_{1}} H_{ét}^{q−p+3}(ℝ; ℤ/2)}

For all q ≥ p the right hand vertical maps are isomorphisms, so d_{p,q}^{2} is identified with the natural homomorphism

α^{q−p}: H_{ét}^{q−p}(R_{F}; ℤ/2) → ⨁^{r_{1}} H_{ét}^{q−p}(ℝ; ℤ/2) ≅ (ℤ/2)^{r_{1}}

for p ≡ 2, 3 mod 4, and is zero otherwise.

Let n = q − p. The homomorphism α^{0} : ℤ/2 → (ℤ/2)^{r_{1}} is injective since F is assumed formally real. The homomorphism α^{1} : H_{ét}^{1}(R; ℤ/2) → (ℤ/2)^{r_{1}} was discussed in 1.1. The homomorphism α^{2} : H_{ét}^{2}(R_{F}; ℤ/2) → (ℤ/2)^{r_{1}} is surjective, with kernel denoted H̃_{ét}^{2}(R_{F}; ℤ/2). ((Proof ?)) The homomorphisms α^{n} for n ≥ 3 are all isomorphisms. This leaves an E_{3}-term concentrated in bidegrees (p, q) with 0 ≤ q − p ≤ 4, so located that there is no room for further differentials.

Thus, for n ≥ 1 the abutment of the spectral sequence is given by the formulas:

K_{n}(R_{F}; ℤ/2) ≅ \{ \begin{matrix}H̃_{ét}^{2}(R_{F}; ℤ/2) ⋊ ℤ/2 & for n ≡ 0 mod 8, \\ H_{ét}^{1}(R_{F}; ℤ/2) & for n ≡ 1 mod 8, \\ H_{ét}^{2}(R_{F}; ℤ/2) ⋊ ℤ/2 & for n ≡ 2 mod 8, \\ (ℤ/2)^{r_{1}−1} ⋊ H_{ét}^{1}(R_{F}; ℤ/2) & for n ≡ 3 mod 8, \\ (ℤ/2)^{u−t} ⋊ H_{ét}^{2}(R_{F}; ℤ/2) & for n ≡ 4 mod 8, \\ (ℤ/2)^{r_{1}−1} ⋊ H̃_{ét}^{1}(R_{F}; ℤ/2) & for n ≡ 5 mod 8, \\ (ℤ/2)^{u−t} ⋊ H̃_{ét}^{2}(R_{F}; ℤ/2) & for n ≡ 6 mod 8, \\ H̃_{ét}^{1}(R_{F}; ℤ/2) & for n ≡ 7 mod 8. \end{matrix}

By Borel’s theorem, the rational rank of K_{n}(R_{F}) is known, and the asserted 2-rank formulas follow by inspection. □

We can now prove the main theorem, stated in the introduction.

Proof of Theorem 0.1. By (4.1) we have the inequality

rk_{2} K_{4}(R_{F}; ℤ_{2}) ≤ rk_{2} im(κ_{4}) + rk_{2} H_{ét}^{2}(R_{F}; ℤ_{2}(3)) .

(This is an equality when the short exact sequence (4.1) is split, as will be the case when K_{4}(R_{F}) has exponent 2.) Substituting the formulas from 1.2 and 4.2 we obtain (s + u − 1) ≤ ρ + (s + t − 1) which we rewrite as ρ ≥ u − t. Clearly ρ ≤ rk_{2} K_{4}(R_{F}; ℤ_{2}) = s + u − 1. □

5. Quadratic number fields

The 2-rank of the Picard group and of the narrow Picard group of a quadratic number field was determined using genus theory by P.A. Østvær in his Master’s thesis [Ø].

Let F = ℚ(\sqrt{d}) with d > 0 square free.

Then u − t = 1 if (a) there exists a prime p | d with p ≡ 3 mod 4, and (b): d ≡ 5 mod 8, or p ≡ 1, 7 mod 8 for every odd prime p | d, or there exists an odd prime p | d with p ≡ 5, 7 mod 8.

The first few examples are d = 7, 14, 15, 21, 23, 30, 31, 35, 39, 42, 46, 47, 53, 55, 58, 61, 62, 65, 69 and 70.

References

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[SV] A.A. Suslin and V. Voevodsky, Bloch–Kato conjecture and motivic cohomology with finite coefficients, Cycles, Transfers and Motivic Homology Theories (V. Voevodsky, A. Suslin and E. Friedlander, eds.), Annals of Mathematics Studies, vol. 143, Princeton University Press, Princeton, N.J., 1998.

[Vo1] V. Voevodsky, Triangulated categories of motives over a field, Cycles, Transfers and Motivic Homology Theories (V. Voevodsky, A. Suslin and E. Friedlander, eds.), Annals of Mathematics Studies, vol. 143, Princeton University Press, Princeton, N.J., 1998.

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((Reference list incomplete.))