Alg`ebre homologique/Homological Algebra The 2

Algèbre homologique/Homological Algebra
The 2-torsion in the K-theory of the Integers
Charles Weibel
Abstract — Using recent results of Voevodsky, Suslin-Voevodsky and BlochLichtenbaum, we completely determine the 2-torsion subgroups of the K-theory of
the integers Z. The result is periodic of order 8, and there are no 2-torsion elements
except those which have been known for over 20 years. There is no 2-torsion except
for the Z/2 summands in degrees 8n + 1 and 8n + 2, the Z/16 in degrees 8n + 3
and the image of the J-homomorphism in degrees 8n + 7. In particular, the 2-part
of ζ(1 − 2n) is twice the 2-part of the ratio |K4n−2 (Z)|/|K4n−1 (Z)| for all n > 0.
This corrects a conjecture of Lichtenbaum.
La torsion 2-primaire de la K-théorie de Z
Résumé — A partir de résultats récents de Voevodsky, Suslin-Voevodsky et
Bloch-Lichtenbaum, nous déterminons la torsion 2-primaire de la K-théorie de Z,
qui est périodique de période 8. Il n’existe pas d’autres éléments 2-primaires que
ceux connus depuis 20 ans. La torsion 2-primaire se réduit à Z/2 en degrés 8n + 1
et 8n + 2, à Z/16 en degré 8n + 3, et à l’image de l’homomorphisme J en degré
8n + 7. La partie 2-primaire de ζ(1 − 2n) est donc le double de la partie 2-primaire
de |K4n−2 (Z)|/|K4n−1 (Z)| pour tout n > 0. Ce résultat corrige un conjecture de
Lichtenbaum.
K0 (Z) = Z
K1 (Z) = Z/2
K2 (Z) = Z/2
K3 (Z) = Z/48
K4 (Z) = 0
K5 (Z) = Z ⊕ (3-torsion)
K6 (Z) = (odd)
K7 (Z) = Z/240 ⊕ (odd)
K8n (Z) = (odd) for n ≥ 1
K8n+1 (Z) = Z ⊕ Z/2 ⊕ (odd) for n ≥ 1
K8n+2 (Z) = Z/2 ⊕ (odd)
K8n+3 (Z) = Z/16 ⊕ (odd)
K8n+4 (Z) = (odd)
K8n+5 (Z) = Z ⊕ (odd)
K8n+6 (Z) = (odd)
K8n+7 (Z) = (Z/wi ) ⊕ (odd), i = 4(n + 1).
Table 1. The K-theory of Z. La K-théorie de Z.
Version française abrégée – Les résultats récents de Voevodsky [V] nous permettent de déterminer la torsion 2-primaire de la K-théorie de Z, qui est périodique
de période 8. Il n’existe pas d’autres éléments 2-primaires que ceux connus depuis
20 ans [Q] [Br]. Comme les groupes Kj (Z) sont de type fini [Q1], et que leur rang
est connu par [Bo], nous pouvons énoncer le résultat suivant.
Théorème 1. Les groupes K∗ (Z) sont données dans la Table 1 ci-dessus, où wi
dénote la plus grande puissance de 2 qui divise 4i, et le symbole “(odd)” dans la
table représente un groupe fini d’ordre impair.
Les premières assertions non déjà connues sont que K4 (Z) = 0 et que la seule
torsion dans K5 (Z) est 3-primaire. Cela découle des résultats de Lee-Szczarba et
Soulé [Sou], de Rognes [R], et de notre calcul qui montre qu’il n’existe pas de torsion
2-primaire.
Comme la partie 2-primaire de ζ(1 − 2n) = (−1)n Bn /2n est 12 wi , cela implique:
Typeset by AMS-TEX
1
2
Corollaire 2. La partie 2-primaire de la formule conjecturée par Lichtenbaum [L]
est correcte à un facteur 2 près:
1
1
|K4n−2 (Z)|
=
= ζ(1 − 2n)
|K4n−1 (Z)|
w2n
2
à un facteur impair près.
Pour obtenir ces résultats, nous utilisons la suite spectrale
(†)
p−q
E2pq = Het
(F ; Z/2) ⇒ K−p−q (F ; Z/2)
(p ≤ 0, p ≥ q).
C’est la suite spectrale de Bloch et Lichtenbaum [BL], le terme E2pq étant déduit
de l’isomorphisme
j
Het (F ; Z/2), j ≤ i
i
∼
CH (F, 2i − j; Z/2) =
0,
j > i,
voir les travaux récents de Voevodsky [V] et Suslin [S1] [SV].
Nous commençons par calculer la suite spectrale (†) pour R et pour les corps
i
i
locaux Qp . Comme Het
(Q; Z/2) → Het
(R; Z/2) est un isomorphisme pour i 6= 1, 2
(voir [T]), nous pouvons utiliser ces calculs pour déterminer les différentielles dans
la suite spectrale pour Q.
Théorème 5. Pour n ≥ 0, la K-théorie modulo 2 de Q est donnée par la Table 2
ci-dessous.
La comparaison des suites exactes de localisation en K-théorie et en cohomologie
étale, pour Q et pour les Qp , donne le résultat suivant.
Théorème 7. Pour n ≥ 0, la K-théorie modulo 2 de Z[ 12 ] est donnée par la Table 3
ci-dessous.
∼ Kj (Z[ 1 ]; Z/2) pour j ≥ 2.
Ce résultat implique le Théorème 1, car Kj (Z; Z/2) =
2
En effet, tous les termes de la Table 3 proviennent des facteurs connus de K∗ (Z).
Le Théorème 1 implique aussi que Kj (Z[ 12 ]) ⊗ Ẑ2 est la K-théorie étale dyadique
Kjet (Z[ 12 ]). Bökstedt, et plus tard Dwyer et Friedlander [DF1, Proposition 4.2], ont
remarqué que cela implique le résultat suivant.
Corollaire 8. Soit K(R)2̂ le complété dyadique de l’espace de la K-théorie K(R)
d’un anneau R. On fixe un plongement de Ẑ3 dans le corps C des nombres complexes. Alors le carré suivant est homotopiquement cartésien:
K(Z[ 12 ])2̂ −−−−→


y
K(Ẑ3 )2̂ −−−−→
K(R)2̂


y
K(C)2̂ .
The recent spectacular results of Voevodsky [V] allow us to completely determine
the 2-torsion in the K-theory groups Kj (Z) associated to the integers Z. The result
is that there are no new elements; all the 2-torsion has been known for over 20 years.
Since each group Kj (Z) is finitely generated by [Q1], and their ranks were computed
by Borel [Bo], our 2-primary calculation yields the following assertion.
3
Theorem 1. The groups K∗ (Z) are given in Table 1. Here wi = wi (Q) is defined
(for even i) to be the largest power of 2 dividing 4i. The symbol “(odd)” in the table
denotes a finite group of odd order.
Of course there is nothing new up to K3 (Z). The first new entry is the assertion
that K4 (Z) = 0 and that K5 (Z) has at most 3-torsion. This follows from our
calculation that there is no 2-torsion in K4 (Z) or K5 (Z), because Lee-Szczarba and
Soulé [Sou] showed that there is no p-torsion in K4 (Z) or K5 (Z) for p > 3, and
Rognes [R] has shown that there is no 3-torsion in K4 (Z) either.
Essentially, the 2-torsion summands of Kj (Z) all come from the stable homotopy
groups πjs of spheres via the natural map πjs → Kj (Z), and were found by Quillen
in [Q2]. When j is 8n + 1 or 8n + 2, the Z/2-summand is generated by the image
of Adams’ element µj . When j = 8n + 3 the 2-Sylow subgroup of J(πj O) is cyclic
of order 8, and is contained in a Z/16 summand of Kj (Z); this result is due to
Browder [Br], although Quillen proved in [Q2] that J(πj O) injects into Kj (Z).
When j = 8n + 7 there is a cyclic summand of Kj (Z) isomorphic to the subgroup
J(πj O) of πjs . The 2-Sylow subgroup of J(π8n+7 O) is isomorphic to Z/wi , where
i = 4(n + 1) and wi is defined in Theorem 1; see [C, p. 284].
The number wi also arises in the following ways; see [W, Theorem 6.7]. If i = 2n
0
∼
then: (1) for ν 0, Het
(Q, µ⊗i
2ν ) = Z/wi ; (2) the 2-part of the denominator of
1
n
ζ(1 − 2n) = (−1) Bn /2n is 2 w2n . The first of these is used to detect the image of
J, while the second lets us relate the orders of K-groups to the zeta function.
Corollary 2. The 2-part of the formula conjectured by Lichtenbaum in [L] holds
up to a factor of 2:
|K4n−2 (Z)|
1
1
≡
≡ ζ(1 − 2n)
|K4n−1 (Z)|
w2n
2
(up to odd torsion).
We now turn to the derivation of these results. Voevodsky proved [V] that the
j
Galois symbol KjM (F )/2KjM (F ) → Het
(F ; Z/2) is an isomorphism for every field
F of characteristic 6= 2 and every j. Suslin and Voevodsky have shown in [S1] and
[SV] that this implies that Bloch’s higher Chow groups (with coefficients mod 2)
[B] for a field F are:
i
CH (F, 2i − j; Z/2) =
j
Het
(F ; Z/2), j ≤ i
0,
j > i.
Bloch and Lichtenbaum have defined a third quadrant spectral sequence in [BL],
converging to groups K∗ (F ). Using K-theory with coefficients modulo 2 in their
construction yields a similar third quadrant spectral sequence; using the SuslinVoevodsky formula above, we may rewrite it as
(†)
p−q
E2pq = Het
(F ; Z/2) ⇒ K−p−q (F ; Z/2)
(p ≤ 0, p ≥ q).
Note that every column E2p∗ of (†) equals the mod 2 étale cohomology of F .
4
Example 3 (Local fields). Let Fv be a local field with residue field kv of chari
acteristic 6= 2. By [T] we know that Het
(Fv ; Z/2) is: Z/2 for i = 0, 2, (Z/2)2
for i = 1, and 0 for i > 2. Thus the spectral sequence (†) degenerates to yield
the well-known calculation that for j > 0 the group Kj (Fv ; Z/2) is Z/2 ⊕ Z/2 for
all j > 0. Comparing with an unramified extension of Fv , we see that the tame
symbol K2i (Fv ; Z/2) → K2i−1 (kv ; Z/2) = Z/2 is the projection onto the summand
2
Het
(Fv ; Z/2), while the tame symbol K2i+1 (Fv ; Z/2) → K2i (kv ; Z/2) = Z/2 may
1
0
be identified with the canonical map Het
(Fv ; Z/2) → Het
(kv ; Z/2).
When F = R, every column of the spectral sequence (†) is the polynomial ring
∗
1
Het
(R; Z/2) = Z/2[η], where η is the nonzero element in Het
(R; Z/2). From [S] we
know that Kj (R; Z/2) is: Z/2 for j ≡ 0, 1, 3, 4 (mod 8); Z/4 for j ≡ 2 (mod 8); 0
for j ≡ 5, 6, 7 (mod 8). Thus the d2 differential out of E2−2,−2 must be nonzero.
Now multiplication by η commutes with the differentials, and induction yields the
following result.
Proposition 4. When F = R, the spectral sequence (†) degenerates at E 3 = E∞
to the mod 2 K-theory of R. Indeed, the differentials E2−p,q → E22−p,q−1 are isomorphisms for all p ≡ 2, 3 (mod 4) and all q ≤ −p.
e 1 (Q; Z/2)
We are now ready to attack the rational numbers Q. Let us write H
1
1
2
e (Q; Z/2) for the
for the kernel of the surjection Het (Q; Z/2) → Het (R; Z/2) and H
2
2
kernel of the surjection Het
(Q; Z/2) → Het
(R; Z/2).
Theorem 5. For n ≥ 0, the mod 2 K-theory of Q is given by Table 2. Here the
notation H o Z/2 denotes the nontrivial extension of Z/2 by the group H.
1
K8n+1 (Q; Z/2) = Het
(Q; Z/2)
1
K8n+3 (Q; Z/2) = Het
(Q; Z/2)
1
e
K8n+5 (Q; Z/2) = H (Q; Z/2)
e 1 (Q; Z/2)
K8n+7 (Q; Z/2) = H
2
K8n+2 (Q; Z/2) = Het
(Q; Z/2) o Z/2
2
K8n+4 (Q; Z/2) = Het
(Q; Z/2)
2
e
K8n+6 (Q; Z/2) = H (Q; Z/2)
e 2 (Q; Z/2) ⊕ Z/2
K8n+8 (Q; Z/2) = H
Table 2. The mod 2 K-theory of Q. La K-théorie modulo 2 de Q.
i
i
Proof. We know from [T] that the natural map Het
(Q; Z/2) → Het
(R; Z/2) is an
isomorphism for i 6= 1, 2. The differentials in (†) for F = Q are thus determined by
the differentials when F = R, and we can read off the result from E3 = E∞ . The
two extension problems are resolved by comparison with F = R and F = Qp ; cf.
[Br].
In order to pass from Q to Z[ 21 ] we need to break up the localization sequence:
Lemma 6. For each j > 0 there is a short exact sequence
1
∂
0 → Kj (Z[ ]; Z/2) → Kj (Q; Z/2) −
→ ⊕p6=2 Kj−1 (Z/p; Z/2) → 0.
2
Proof. Fix an even number j. Then the Universal Coefficient Theorem implies that
Kj (Q; Z/2) maps onto ⊕Kj−1 (Z/p; Z/2). In addition, since Kj (Z/p; Z/2) = Z/2
for all p 6= 2, the natural map from ⊕p6=2 Kj (Z/p; Z/2) = ⊕p6=2 Z/2 to Kj (Z[ 12 ]; Z/2)
factors through Pic(Z[ 12 ])/2 = 0.
5
Theorem 7. For n ≥ 0, the mod 2 K-theory of Z is given by Table 3.
K8n+1 (Z[ 12 ]; Z/2) = (Z/2)2
K8n+3 (Z[ 12 ]; Z/2) = (Z/2)2
K8n+5 (Z[ 12 ]; Z/2) = Z/2
K8n+7 (Z[ 12 ]; Z/2) = Z/2
K8n+2 (Z[ 12 ]; Z/2) = Z/4
K8n+4 (Z[ 12 ]; Z/2) = Z/2
K8n+6 (Z[ 12 ]; Z/2) = 0
K8n+8 (Z[ 12 ]; Z/2) = (Z/2)
Table 3. The mod 2 K-theory of Z[ 21 ]. La K-théorie modulo 2 de Z[ 21 ].
1
1
Proof. Since Het
(Q; Z/2) = Q∗ /Q∗2 and Het
(Z[ 12 ]; Z/2) = Z[ 12 ]∗ /Z[ 21 ]∗2 ∼
= (Z/2)2 ,
we have a short exact sequence
1
1
1
0 → Het
(Z[ ]; Z/2) → Het
(Q; Z/2) → ⊕p6=2 Z/2 → 0.
2
Comparing with the direct sum of the corresponding sequences for the local fields
Qp allows us to deduce the calculation of Kj (Z[ 12 ]; Z/2) for odd j. The calculation
for j even follows similarly from the commutative diagram (in which the left vertical
isomorphism follows from class field theory).
2
e et
H
(Q; Z/2)

∼
y=
−−−−→
∂
K2i (Q; Z/2)


y
−−−−→ ⊕p6=2 K2i−1 (Z/p; Z/2)

=
y
∂
2
⊕p6=2 Het
(Qp ; Z/2) −−−−→ ⊕p6=2 K2i (Qp ; Z/2) −−−−→ ⊕p6=2 K2i−1 (Z/p; Z/2)
Theorem 7 implies Theorem 1. Indeed, Kj (Z; Z/2) ∼
= Kj (Z[ 12 ]; Z/2) for j ≥ 2,
and the known summands of Kj (Z) account for all of Kj (Z[ 12 ]; Z/2) in this range.
Theorem 1 also shows that Kj (Z[ 12 ])⊗ Ẑ2 is the 2-adic étale K-theory Kjet (Z[ 12 ])2̂ .
As observed by Bökstedt, and again by Dwyer and Friedlander in [DF1, Proposition 4.2], it also implies the following topological result. Let K(R)2̂ denote the
2-completion of the algebraic K-theory space K(R) of a ring R.
Corollary 8. Choose an embedding of Ẑ3 in the complex numbers C representing
the Brauer lifting of K(Z/3) → BU . Then the square
Z[ 12 ] −−−−→


y
R


y
Ẑ3 −−−−→ C
induces a homotopy cartesian square upon 2-adic completion.
K(Z[ 12 ])2̂ −−−−→


y
K(Ẑ3 )2̂ −−−−→
K(R)2̂


y
K(C)2̂
6
Example 9. Another case when the spectral sequence (†) degenerates is when F
2
is a totally imaginary number field. In this case Kj (F ; Z/2) is Z/2 ⊕ Het
(F ; Z/2)
1
for j > 0 even, and Het (F ; Z/2) for j odd. That is, Kj (F ; Z/2) equals the étale
K-theory Kjet (F ; Z/2) of Dwyer and Friedlander [DF]. However, the results of [DF]
√
do not apply here unless F contains −1.
When the primes over 2 generate the class group of F , the methods used for
Z allow us to calculate
√ the K-theory of the ring of integers in F . For example,
when F = Q(i), i = −1, we obtain the following result. Recall that the groups
K0 (Z[i]) = Z, K1 (Z[i]) = Z/4, K2 (Z[i]) = 0 and K3 (Z[i]) = Z ⊕ Z/24 are known.
Theorem 10. For all n ≥ 2, K2n (Z[i]) is a finite group of odd order, while
K2n−1 (Z[i]) = Z ⊕ Z/wn ⊕ (odd).
As in Theorem 1, wn = wn (Q(i)) denotes the largest power of 2 dividing 4n. (For
example, if n is odd then wn = 4.)
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[BL]
[Bo]
[Br]
[DF]
[DF1]
[L]
[Q]
[R]
[Sou]
[S]
[S1]
[SV]
[T]
[V]
[W]
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Department of Mathematics, Rutgers University, New Brunswick, N.J. 08903 U.S.A.
E-mail address: [email protected]