ÉTALE DESCENT FOR TWO-PRIMARY ALGEBRAIC

K-THEORY OF TOTALLY IMAGINARY NUMBER FIELDS

J. Rognes and C. Weibel

Abstract. We show that the natural map from 2-adic algebraic K-theory to 2-adic étale K-theory induces an isomorphism in positive degrees for rings of 2-integers in totally imaginary number fields. In this sense, 2-adic algebraic K-theory of totally imaginary number fields satisfies étale descent in positive degrees.

Introduction

We prove the following theorem.

Theorem 0.1. Let ℓ = 2, and let F be a totally imaginary number field with ring of ℓ-integers R_{F} = O_{F} [\frac{1}{ℓ}]. The Dwyer–Friedlander map

ϕ_{F} : K(R_{F})_{ℓ}^{∧} ⟶ K^{ét}(R_{F})_{ℓ}^{∧}

induces isomorphisms

ϕ_{F ∗}: K_{∗}(R_{F}; ℤ_{ℓ}) ⟶ K_{∗}^{ét}(R_{F}; ℤ_{ℓ})

for all ∗ ≥ 1. Hence ϕ_{F} induces a homotopy equivalence of connected components.

In this sense 2-adic algebraic K-theory for totally imaginary number fields satisfies étale descent, since étale K-theory does so, more or less by construction.

The Dwyer–Friedlander map in question was constructed in [DF], where it was shown that 2-adically ϕ_{F ∗} is surjective in positive degrees when F contains \sqrt{−1}. Keeping this hypothesis it was proven in [RW, 0.4] that ϕ_{F ∗} is an isomorphism in positive degrees. The present note uses a Tate spectral sequence to extend this result to arbitrary totally imaginary number fields.

The authors would like to thank B. Kahn and R. Thomason for discussions, dating from 1992, about the Tate spectral sequence.

1. Descent and codescent

Let F be a totally imaginary number field, and let ℓ be any prime, even or odd. We shall write R_{F} for the ring O_{F} [\frac{1}{ℓ}] of ℓ-integers in F. We are interested in the étale cohomology of the sheaves ℚ_{ℓ}/ℤ_{ℓ}(i) and ℤ_{ℓ}(i), always supposing i ≥ 2.

We shall use the following facts, which may be found in [So] and [RW]. First, H_{ét}^{n}(R_{F}; ℚ_{ℓ}/ℤ_{ℓ}(i)) = 0 unless n = 0, 1, and H_{ét}^{n}(R_{F}; ℤ_{ℓ}(i)) = 0 unless n = 1, 2. Next, the groups H_{ét}^{0}(R_{F}; ℚ_{ℓ}/ℤ_{ℓ}(i)) = ℤ/w_{i}^{(ℓ)}(F) and H_{ét}^{2}(R_{F}; ℤ_{ℓ}(i)) are finite.

Typeset by AMS-TEX

Also H_{ét}^{1}(R_{F}; ℤ_{ℓ}(i)) is a finitely generated ℤ_{ℓ} -module, and H_{ét}^{1}(R_{F}; ℚ_{ℓ}/ℤ_{ℓ}(i)) is Pontryagin dual to one such.

Now suppose E/F is a Galois extension of totally imaginary number fields, unramified outside the prime 2. This implies that Spec(R_{E}) is a Galois covering of Spec(R_{F}), with group G = Gal(E/F). For each étale coefficient sheaf M on R_{F}, there is a first quadrant Hochschild–Serre spectral sequence with E_{2}-term

E_{2}^{p,q} = H^{p}(G; H_{ét}^{q}(R_{E}; M))

converging to H_{ét}^{p+q}(R_{F}; M). See e.g. [Mi, III.2.20]. Writing A^{G} for the invariants H^{0}(G; A) of a G-module A, this yields natural isomorphisms

H_{ét}^{0}(R_{F}; ℚ_{ℓ}/ℤ_{ℓ}(i)) ⟶^{≅} H_{ét}^{0}(R_{E}; ℚ_{ℓ}/ℤ_{ℓ}(i))^{G}
H_{ét}^{1}(R_{F}; ℤ_{ℓ}(i)) ⟶^{≅} H_{ét}^{1}(R_{E}; ℤ_{ℓ}(i))^{G}

compatible with the restriction maps for E/F. The second isomorphism uses that H_{ét}^{0}(R_{E}; ℤ_{ℓ}(i)) = 0 for i ≠ 0.

When M is discrete, there is also a second quadrant Tate spectral sequence with E_{2}-term

E_{2}^{−p,q} = H_{p}(G; H_{ét}^{q}(R_{E}; M))

converging to H_{ét}^{−p+q}(R_{F}; M). See [Se, Ch.I, App.1] or [Ka, 3.1]. If we write A_{G} for the coinvariants H_{0}(G; A) of a G-module A, the edge maps are the corestriction maps H_{ét}^{q}(R_{E}; M)_{G} → H_{ét}^{q}(R_{F}; M). When M = ℚ_{ℓ}(i)/ℤ_{ℓ}(i) and i ≥ 2 then H_{ét}^{q}(R_{E}; M) = 0 for q ≥ 2, so the edge map

H_{ét}^{1}(R_{E}; ℚ_{ℓ}/ℤ_{ℓ}(i))_{G} ⟶^{≅} H_{ét}^{1}(R_{F}; ℚ_{ℓ}/ℤ_{ℓ}(i))

is a natural isomorphism for E/F. Since its construction assumes that the coefficient module M is discrete, the Tate spectral sequence does not apply directly with M = ℤ_{ℓ}(i). Nonetheless:

Lemma 1.1. Let E/F be a Galois extension of totally imaginary number fields. For i ≥ 2, the corestriction map induces an isomorphism

H_{ét}^{2}(R_{E}; ℤ_{ℓ}(i))_{G} ⟶^{≅} H_{ét}^{2}(R_{F}; ℤ_{ℓ}(i)).

Proof. The extension ℤ_{ℓ} → ℚ_{ℓ} → ℚ_{ℓ}/ℤ_{ℓ} induces a short exact sequence

0 → H_{ét}^{1}(R_{E}; ℤ_{ℓ}(i)) ⊗ ℚ_{ℓ}/ℤ_{ℓ} ⟶ H_{ét}^{1}(R_{E}; ℚ_{ℓ}/ℤ_{ℓ}(i)) ⟶ H_{ét}^{2}(R_{E}; ℤ_{ℓ}(i)) → 0 .

This uses H_{ét}^{2}(R_{E}; ℚ_{ℓ}(i)) = 0. Since this sequence is natural with respect to automorphisms of E over F, it is G-equivariant.

The corestriction map induces a map of exact sequences

\xymatrix{H_{ét}^{1}(R_{E}; ℤ_{ℓ}(i))_{G} ⊗ ℚ_{ℓ}/ℤ_{ℓ} \ar[r] \ar[d] & H_{ét}^{1}(R_{E}; ℚ_{ℓ}/ℤ_{ℓ}(i))_{G} \ar[r] \ar[d]^{≅} & H_{ét}^{2}(R_{E}; ℤ_{ℓ}(i))_{G} \ar[d] \\ H_{ét}^{1}(R_{F}; ℤ_{ℓ}(i)) ⊗ ℚ_{ℓ}/ℤ_{ℓ} \ar[r] & H_{ét}^{1}(R_{F}; ℚ_{ℓ}/ℤ_{ℓ}(i)) \ar[r] & H_{ét}^{2}(R_{F}; ℤ_{ℓ}(i))}

The upper and lower right horizontal maps are surjective, and the lower left horizontal map is injective. The middle vertical map is an isomorphism by the Tate spectral sequence. Hence the right hand vertical map is surjective. Its kernel is a finite group, and isomorphic to the cokernel of the left hand vertical map, which is divisible. Hence these groups are zero, and the right hand corestriction map must be an isomorphism. □

2. Proof of the Theorem

Let K_{∗}(R_{F}; ℤ_{ℓ}) and K_{∗}^{ét}(R_{F}; ℤ_{ℓ}) denote the algebraic and étale K-theory of R_{F} with ℓ-adic coefficients, respectively. We need to show that the universal maps

ϕ_{F} : K_{∗}(R_{F}; ℤ_{ℓ}) → K_{∗}^{ét}(R_{F}; ℤ_{ℓ})

are isomorphisms.

The edge maps in the Dwyer–Friedlander spectral sequence [DF] induce natural isomorphisms

DF_{i,1} : K_{2i−1}^{ét}(R_{F}; ℤ_{ℓ}) ⟶^{≅} H_{ét}^{1}(R_{F}; ℤ_{ℓ}(i))
DF_{i,2} : K_{2i−2}^{ét}(R_{F}; ℤ_{ℓ}) ⟶^{≅} H_{ét}^{2}(R_{F}; ℤ_{ℓ}(i))

in total degrees ∗ ≥ 1. When ℓ = 2, the Bloch–Lichtenbaum spectral sequence [BL] leads (via [RW, App. B]) to natural isomorphisms

BL_{i,1} : K_{2i−1}(R_{F}; ℤ_{ℓ}) ⟶^{≅} H_{ét}^{1}(R_{F}; ℤ_{ℓ}(i))
BL_{i,2} : K_{2i−2}(R_{F}; ℤ_{ℓ}) ⟶^{≅} H_{ét}^{2}(R_{F}; ℤ_{ℓ}(i))

in total degrees ∗ ≥ 1.

Now let F be a totally imaginary number field, not containing \sqrt{−1}, and let E = F (\sqrt{−1}) be the quadratic extension containing \sqrt{−1}. Then G = Gal(E/F) is cyclic of order 2. It is known that ϕ_{E} is surjective for all ∗ ≥ 1, and that it is an isomorphism for ℓ = 2, by [DF] and [RW, 0.4].

For ∗ = 2i − 1, we use the following commutative diagram:

\xymatrix{H_{ét}^{1}(R_{F}; ℤ_{ℓ}(i)) \ar[d]^{≅} & K_{2i−1}(R_{F}; ℤ_{ℓ}) \ar[l]_{BL_{i,1}}^{≅} \ar[d] \ar[r]^{ϕ_{F}} & K_{2i−1}^{ét}(R_{F}; ℤ_{ℓ}) \ar[r]^{DF_{i,1}}_{≅} \ar[d] & H_{ét}^{1}(R_{F}; ℤ_{ℓ}(i)) \ar[d]^{≅} \\ H_{ét}^{1}(R_{E}; ℤ_{ℓ}(i))^{G} & K_{2i−1}(R_{E}; ℤ_{ℓ})^{G} \ar[l]_{BL_{i,1}^{G}}^{≅} \ar[r]^{ϕ_{E}^{G}}_{≅} & K_{2i−1}^{ét}(R_{E}; ℤ_{ℓ})^{G} \ar[r]^{DF_{i,1}^{G}}_{≅} & H_{ét}^{1}(R_{E}; ℤ_{ℓ}(i))^{G}}

Before taking G-invariants, the bottom row consists of isomorphisms, and is Gequivariant by naturality with respect to the Galois-automorphisms of E/F. Hence the row of G-invariant submodules also consists of isomorphisms.

Using the descent isomorphism at both ends and convergence of the Dwyer– Friedlander and Bloch–Lichtenbaum spectral sequences we deduce that ϕ_{F} is a natural isomorphism in degree (2i − 1).

For ∗ = 2i − 2, we use the following commutative diagram:

\xymatrix{H_{ét}^{2}(R_{E}; ℤ_{ℓ}(i))_{G} \ar[d]^{≅} & K_{2i−2}(R_{E}; ℤ_{ℓ}(i))_{G} \ar[l]_{(BL_{i,2})_{G}}^{≅} \ar[d] \ar[r]^{(ϕ_{E})_{G}}_{≅} & K_{2i−2}^{ét}(R_{E}; ℤ_{ℓ}(i))_{G} \ar[r]^{(DF_{i,2})_{G}}_{≅} \ar[d] & H_{ét}^{2}(R_{E}; ℤ_{ℓ}(i))_{G} \ar[d]^{≅} \\ H_{ét}^{2}(R_{F}; ℤ_{ℓ}(i)) & K_{2i−2}(R_{F}; ℤ_{ℓ}(i)) \ar[l]_{BL_{i,2}}^{≅} \ar[r]^{ϕ_{F}} & K_{2i−2}^{ét}(R_{F}; ℤ_{ℓ}(i)) \ar[r]^{DF_{i,2}}_{≅} & H_{ét}^{2}(R_{F}; ℤ_{ℓ}(i))}

Before taking G-coinvariants, the top row consists of isomorphisms and is Gequivariant as before. Hence also the row of G-coinvariant quotient modules consists of isomorphisms. Using Lemma 1.1 we deduce that ϕ_{F} is also a natural isomorphism in degree (2i − 2). This completes the proof of the theorem.

References

[BL] S. Bloch and S. Lichtenbaum, A spectral sequence for motivic cohomology, K-theory preprint http://www.math.uiuc.edu/K-theory/0062 (1994).

[DF] W. Dwyer and E. Friedlander, Algebraic and étale K-theory, Trans. AMS 292 (1985), 247–280.

[Ka] B. Kahn, Deux théorèmes de comparaison en cohomologie étale; applications, Duke Math. J. 69 (1993), 137–165.

[Mi] J. S. Milne, Étale Cohomology, Princeton University Press, 1980.

[RW] J. Rognes and C. Weibel, Two-primary algebraic K-theory of rings of integers in number fields, K-theory preprint http://www.math.uiuc.edu/K-theory/0220 (1997).

[Se] J.-P. Serre, Cohomologie Galoisienne. 5eme ed., rev. et complete, Lecture Notes in Mathematics, vol. 5, Springer Verlag, 1994.

[So] C. Soulé, K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale, Invent. Math. 55 (1979), 251–295.

Dept. of Math., University of Oslo, Norway

   E-mail address: rognes@math.uio.no

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

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