K-Theory 16: 101–104, 1999.
c 1999 Kluwer Academic Publishers. Printed in the Netherlands.
101
Étale Descent for Two-Primary Algebraic K-Theory of
Totally Imaginary Number Fields
J. ROGNES1 and C. WEIBEL2
1 Department
2 Department
of Mathematics, University of Oslo, Norway. e-mail: rognes@math.uio.no
of Mathematics, Rutgers University, U.S.A. e-mail: weibel@math.rutgers.edu
(Received: March 1998)
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.
Mathematics Subject classifications (1991): 11R70, 19D50, 19F27.
Key words: Algebraic K-theory, étale K-theory, Bloch–Lichtenbaum spectral sequence, Dwyer–
Friedlander spectral sequence, étale descent.
0. Introduction
We prove the following theorem:
THEOREM 0.1. Let ` = 2, and let F be a totally imaginary number field with ring
of `-integers RF = OF [1/`]. The Dwyer–Friedlander map
φF : K(RF )∧` −→ K ét (RF )∧`
induces isomorphisms
φF ∗ : K∗ (RF ; Z` ) −→ K∗ét (RF ; Z` )
for all ∗ > 1. Hence, φF 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 [2], where it√was
shown that 2-adically φF ∗ is surjective in positive degrees when F contains −1.
Keeping this hypothesis it was proven in [5, 0.4] that φF ∗ is an isomorphism in
positive degrees. The present note uses a Tate spectral sequence to extend this result
to arbitrary totally imaginary number fields.
102
J. ROGNES AND C. WEIBEL
1. Descent and Codescent
Let F be a totally imaginary number field, and let ` be any prime, even or odd. We
shall write RF for the ring OF [1/`] of `-integers in F . We are interested in the étale
cohomology of the sheaves Q` /Z` (i) and Z` (i), always supposing i > 2.
We shall use the following facts, which may be found in [5, 7]. First, Hétn (RF ;
Q` /Z` (i)) = 0 unless n = 0, 1, and Hétn (RF ; Z` (i)) = 0 unless n = 1, 2. Next,
the groups Hét0 (RF ; Q` /Z` (i)) = Z/wi(`) (F ) and Hét2 (RF ; Z` (i)) are finite. Also
Hét1 (RF ; Z` (i)) is a finitely generated Z` -module, and Hét1 (RF ; Q` /Z` (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(RE ) is a Galois covering of
Spec(RF ), with group G = Gal(E/F ). For each étale coefficient sheaf M on RF ,
p,q
there is a first quadrant Hochschild–Serre spectral sequence with E2 -term E2 =
q
p+q
H p (G; Hét (RE ; M)) converging to Hét (RF ; M). See, e.g., [4, III. 2.20]. Writing
AG for the invariants H 0 (G; A) of a G-module A, this yields natural isomorphisms
∼
=
Hét0 (RF ; Q` /Z` (i)) −→ Hét0 (RE ; Q` /Z` (i))G ,
∼
=
Hét1 (RF ; Z` (i)) −→ Hét1 (RE ; Z` (i))G ,
compatible with the restriction maps for E/F . The second isomorphism uses that
Hét0 (RE ; Z` (i)) = 0 for i 6= 0.
When M is discrete, there is also a second quadrant Tate spectral sequence with
−p,q
q
−p+q
= Hp (G; Hét (RE ; M)) converging to Hét
(RF ; M). See [6, Ch. I,
E2 -term E2
App. 1] or [3, 3.1]. If we write AG for the co-invariants H0 (G; A) of a G-module
q
q
A, the edge maps are the corestriction maps Hét (RE ; M)G → Hét (RF ; M). When
q
M = Q` /Z` (i) and i > 2 then Hét (RE ; M) = 0 for q > 2, so the edge map
∼
=
Hét1 (RE ; Q` /Z` (i))G −→ Hét1 (RF ; Q` /Z` (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` (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
∼
=
Hét2 (RE ; Z` (i))G −→ Hét2 (RF ; Z` (i)).
Proof. The extension Z` → Q` → Q` /Z` induces a short exact sequence
0 → Hét1 (RE ; Z` (i)) ⊗ Q` /Z` → Hét1 (RE ; Q` /Z` (i)) → Hét2 (RE ; Z` (i)) → 0.
This uses Hét2 (RE ; Q` (i)) = 0. Since this sequence is natural with respect to automorphisms of E over F , it is G-equivariant.
ÉTALE DESCENT FOR TWO-PRIMARY ALGEBRAIC K-THEORY
103
The corestriction map induces a map of exact sequences
Hét1 (RE ; Z` (i))G ⊗ Q` /Z` −→ Hét1 (RE ; Q` /Z` (i))G −→ Hét2 (RE ; Z` (i))G




∼

y
y=
y
1
1
2
Hét (RF ; Z` (i)) ⊗ Q` /Z` −−−→ Hét (RF ; Q` /Z` (i)) −−−→ Hét (RF ; Z` (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∗ (RF ; Z` ) and K∗ét (RF ; Z` ) denote the algebraic and étale K-theory of RF with
`-adic coefficients, respectively. We need to show that the universal maps
φF : K∗ (RF ; Z` ) −→ K∗ét (RF ; Z` )
are isomorphisms.
The edge maps in the Dwyer–Friedlander spectral sequence [2] induce natural
isomorphisms
∼
=
ét
(RF ; Z` ) −→ Hét1 (RF ; Z` (i)),
DFi,1 : K2i−1
∼
=
ét
(RF ; Z` ) −→ Hét2 (RF ; Z` (i))
DFi,2 : K2i−2
in total degrees ∗ > 1. When ` = 2, the Bloch–Lichtenbaum spectral sequence [1]
leads (via [5, App. B]) to natural isomorphisms
∼
=
BLi,1 : K2i−1 (RF ; Z` ) −→ Hét1 (RF ; Z` (i)),
∼
=
BLi,2 : K2i−2 (RF ; Z` ) −→ Hét2 (RF ; Z` (i))
in total degrees ∗ > 1.
√
Now √
let F be a totally imaginary number field, not
√ containing −1, and let
E = F ( −1) be the quadratic extension containing −1. Then G = Gal(E/F )
is cyclic of order 2. It is known that φE is surjective for all ∗ > 1, and that it is an
isomorphism for ` = 2, by [2] and [5, 0.4].
For ∗ = 2i − 1, we use the following commutative diagram:
BLi,1
φF
DFi,1
BLG
i,1
G
φE
G
DFi,1
=
=
=
ét (R ; Z ) −
Hét1 (RF ; Z` (i)) ←−−
−−− K2i−1 (RF ; Z` ) −−−→ K2i−1
−→ Hét1 (RF ; Z` (i))
F
` −∼
∼
=
=






∼
∼
y
y
y=
y=
ét (R ; Z )G −
Hét1 (RE ; Z` (i))G ←−−
−−−K2i−1 (RE ; Z` )G −−∼
−→K2i−1
−∼
−→Hét1 (RE ; Z` (i))G .
E
`
∼
104
J. ROGNES AND C. WEIBEL
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 φF is a
natural isomorphism in degree (2i − 1).
For ∗ = 2i − 2, we use the following commutative diagram:
(BLi,2 )G
(φE )G
(DFi,2 )G
BLi,2
φF
DFi,2
ét (R ; Z (i)) −−−−−−→ H 2 (R ; Z (i))
Hét2 (RE ; Z` (i))G ←−−−−−− K2i−2 (RE ; Z` (i))G −−−−−→ K2i−2
E
G
G
`
`
ét E
∼
∼
∼
=
=
=








∼


∼
y=
y
y
y=
ét (R ; Z (i)) −−−−−−−→ H 2 (R ; Z (i)).
Hét2 (RF ; Z` (i)) ←−−−−−−− K2i−2 (RF ; Z` (i)) −−−−−→ K2i−2
F
`
`
ét F
∼
=
∼
=
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 φF is also a natural isomorphism
in degree (2i − 2). This completes the proof of the theorem.
3. Acknowledgement
The authors would like to thank B. Kahn and R. Thomason for discussions, dating
from 1992, about the Tate spectral sequence.
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