On the K(π,1)-property for rings of integers in the mixed case

0.18 MB·English

by  Alexander Schmidt

On the K(π, 1)-property for rings of
8 integers in the mixed case
0
0
2 by Alexander Schmidt

n
a February 2, 2008
J

4
1
Abstract

] WeinvestigatetheGaloisgroupGS(p)ofthemaximalp-extensionun-
T
ramified outside a finite set S of primes of a numberfield in the (mixed)
N
case,whenthereareprimesdividingpinsideandoutsideS. Weshowthat
h. thecohomology ofGS(p)is‘often’isomorphic tothe´etale cohomology of
t the scheme Spec(ØkrS), in particular, GS(p) is of cohomological dimen-
a sion2then. Wededucethisfromtheresultsinourpreviouspaper[Sch2],
m
which mainly dealt with thetame case.
[

1 1 Introduction
v
3
Let Y be a connected locally noetherian scheme and let p be a prime number.
0
1 We denote the ´etale fundamental group of Y by π1(Y) and its maximal pro-
2 p factor group by π (Y)(p). The Hochschild-Serre spectral sequence induces
1
. natural homomorphisms
1
0
8 φi :Hi(π1et(Y)(p),Z/pZ)−→Heit(Y,Z/pZ), i≥0,
0
: and we call Y a ‘K(π,1) for p’ if all φ are isomorphisms; see [Sch2] Proposi-
v i
tion 2.1 for equivalent conditions. See [Wi2] for a purely Galois cohomological
i
X
approachto the K(π,1)-property. Our main result is the following
r
a Theorem 1.1. Let k be a number field and let p be a prime number. Assume
thatk doesnotcontainaprimitivep-throotofunityandthattheclassnumber
of k is prime to p. Then the following holds:
Let S be a finite set of primes of k and let T be a set of primes of k of
Dirichlet density δ(T) = 1. Then there exists a finite subset T ⊂ T such that
1
Spec(Ø )r(S∪T ) is a K(π,1) for p.
k 1
Remarks. 1. If S contains the set S of primes dividing p, then Theorem 1.1
p
holds with T = ∅ and even without the condition ζ ∈/ k and Cl(k)(p) = 0,
1 p
see [Sch2], Proposition 2.3. In the tame case S ∩S = ∅, the statement of
p
Theorem 1.1 is the main result of [Sch2]. Here we provide the extension to the
‘mixed’ case ∅ S∩S S .
p p
1
2. For agivennumber fieldk,allbut finitely manyprime numbersp satisfy the
condition of Theorem 1.1. We conjecture that Theorem 1.1 holds without the
restricting assumption on p.
LetS be a finite setofplacesofa numberfieldk. Letk (p)be the maximal
S
p-extension of k unramified outside S and put GS(p) = Gal(kS(p)|k). If SR
denotes the set of real places of k, then GS∪SR(p) ∼= π1(Spec(Øk)rS)(p) (we
have G (p) = G (p) if p is odd or k is totally imaginary). The following
S S∪SR
Theorem 1.2 sharpens Theorem 1.1.
Theorem 1.2. The set T ⊂T in Theorem 1.1 may be chosen such that
1
(i) T consists of primes p of degree 1 with N(p)≡1modp,
1
(ii) (k (p)) =k (p) for all primes p∈S∪T .
S∪T1 p p 1
Note that Theorem 1.2 provides nontrivial information even in the case S ⊃
S , where assertion (ii) was only known when k contains a primitive p-th root
p
of unity (Kuz’min’s theorem, see [Kuz] or [NSW], 10.6.4 or [NSW2], 10.8.4,
respectively) and for certain CM fields (by a result of Mukhamedov, see [Muk]
or [NSW], X §6 exercise or [NSW2], X §8 exercise, respectively).
ByTheorem3.3below,Theorem1.2providesmanyexamplesofG (p)being
S
adualitygroup. Ifζ ∈/ k,thisisinterestingeveninthecasethatS ⊃S ,where
p p
examples of G (p) being a duality group were previously known only for real
S
abelianfieldsandforcertainCM-fields(see[NSW],10.7.15and[NSW2],10.9.15,
respectively, and the remark following there).
PreviousresultsinthemixedcasehadbeenachievedbyK.Wingberg[Wi1],
Ch.Maire[Mai]andD.Vogel[Vog]. Thoughnotexplicitlyvisibleinthispaper,
the presentprogressin the subject was only possible due to the results on mild
pro-p groups obtained by J. Labute in [Lab].
I wouldlike to thank K.Wingbergfor pointing outthat the proofofPropo-
sition 8.1 in my paper [Sch2] did not use the assumption that the sets S and
S′ are disjoint from S . This was the key observation for the present paper.
p
The main part of this text was written while I was a guest at the Department
of Mathematical Sciences of TokyoUniversity and of the ResearchInstitute for
Mathematical Sciences in Kyoto. I want to thank these institutions for their
kind hospitality.
2 Proof of Theorems 1.1 and 1.2
We start with the observation that the proofs of Proposition 8.2 and Corol-
lary8.3 in[Sch2]did notuse the assumptionthatthe setsS andS′ aredisjoint
from S . Therefore, with the same proof (which we repeat for the convenience
p
of the reader) as in loc. cit., we obtain
2
Proposition2.1. Letkbeanumberfieldandletpbeaprimenumber. Assume
k to be totally imaginary if p = 2. Put X = Spec(Ø ) and let S ⊂ S′ be finite
k
sets of primes of k. Assume that XrS is a K(π,1) for p and that G (p) 6= 1.
S
Further assume that each p ∈ S′rS does not split completely in k (p). Then
S
the following hold.
(i) XrS′ is a K(π,1) for p.
(ii) kS′(p)p =kp(p) for all p∈S′rS.
Furthermore, the arithmetic form of Riemann’s existence theorem holds, i.e.,
setting K =k (p), the natural homomorphism
S
∗ T(Kp(p)|Kp)−→Gal(kS′(p)|K)
p∈S′\S(K)
isanisomorphism. HereT(K (p)|K )istheinertiagroupand∗denotesthefree
p p
pro-p-product of a bundle of pro-p-groups,cf. [NSW], Ch.IV,§3. In particular,
Gal(kS′(p)|kS(p)) is a free pro-p-group.
Proof. The K(π,1)-property implies
Hi(GS(p),Z/pZ)∼=Heit(XrS,Z/pZ)=0 for i≥4,
hence cd G (p) ≤ 3. Let p ∈ S′rS. Since p does not split completely in
S
k (p) and since cd G (p) < ∞, the decomposition group of p in k (p)|k is a
S S S
non-trivial and torsion-free quotient of Zp ∼= Gal(kpnr(p)|kp). Therefore kS(p)p
is the maximal unramified p-extension of k . We denote the normalization of
p
anintegralnormalscheme Y in analgebraicextensionL ofits function fieldby
Y . Then (XrS) is the universalpro-p coveringof XrS. We consider the
L kS(p)
´etaleexcisionsequenceforthepair((XrS) ,(XrS′) ). Byassumption,
kS(p) kS(p)
XrS is a K(π,1) for p, hence Hi ((XrS) ,Z/pZ) = 0 for i ≥ 1 by [Sch2],
et kS
Proposition2.1. Omitting the coefficients Z/pZ from the notation, this implies
isomorphisms
Hi (XrS′) →∼ ′ Hi+1 ((XrS) )
et(cid:0) kS(p)(cid:1) M p (cid:0) kS p(cid:1)
p∈S′rS(kS(p))
for i≥1. Here (and in variants also below) we use the notational convention
′
Hi+1 ((XrS) ) := lim Hi+1 ((XrS) ) ,
M p (cid:0) kS(p) p(cid:1) −→ M p (cid:0) K p(cid:1)
p∈S′rS(kS(p)) K⊂kS(p)p∈S′rS(K)
where K runs through the finite extensions of k inside k (p). As k (p) real-
S S
izes the maximal unramified p-extension of k for all p ∈ S′rS, the schemes
p
((XrS) ) , p∈S′rS(k (p)), have trivial cohomology with values in Z/pZ
kS(p) p S
and we obtain isomorphisms
Hi((k (p)) )→∼ Hi+1 ((XrS) )
S p p (cid:0) kS(p) p(cid:1)
for i≥1. These groups vanish for i≥2. This implies
Hi ((XrS′) )=0
et kS(p)
3
for i ≥ 2. Since the scheme (XrS′) is the universal pro-p covering of
kS′(p)
(XrS′) , the Hochschild-Serre spectral sequence yields an inclusion
kS(p)
H2(Gal(kS′(p)|kS(p)))֒→He2t((XrS′)kS(p))=0.
Hence Gal(kS′(p)|kS(p)) is a free pro-p-groupand
H1(Gal(kS′(p)|kS(p)))→∼ He1t((XrS′)kS(p))∼= M′ H1(kS(p)p).
p∈S′rS(kS(p))
We set K =k (p) and consider the natural homomorphism
S
φ: ∗ T(Kp(p)|Kp)−→Gal(kS′(p)|K).
p∈S′\S(K)
Bythecalculationofthecohomologyofafreeproduct([NSW],4.3.10and4.1.4),
φ is ahomomorphismbetweenfree pro-p-groupswhichinduces anisomorphism
on mod p cohomology. Therefore φ is an isomorphism. In particular,kS′(p)p =
kp(p) for all p ∈ S′rS. Using that Gal(kS′(p)|kS(p)) is free, the Hochschild-
Serre spectral sequence
E2ij =Hi(cid:0)Gal(kS′(p)|kS(p)),Hejt((XrS′)kS′(p))(cid:1)⇒Hei+t j((XrS′)kS(p))
induces an isomorphism
0=He2t((XrS′)kS(p))−∼→He2t((XrS′)kS′(p))Gal(kS′|kS).
Hence He2t((XrS′)kS′(p)) = 0, since Gal(kS′(p)|kS(p)) is a pro-p-group. Now
[Sch2], Proposition 2.1 implies that XrS′ is a K(π,1) for p.
In order to prove Theorem 1.1, we first provide the following lemma. For
an extension field K|k and a set of primes T of k, we write T(K) for the set of
prolongations of primes in T to K and δ (T) for the Dirichlet density of the
K
set of primes T(K) of K.
Lemma 2.2. Let k be a number field, p a prime number and S a finite set of
nonarchimedean primes of k. Let T be a set of primes of k with δ (T)=1.
k(µp)
Then there exists a finite subset T ⊂T such that all primes p∈S do not split
0
completely in the extension k (p)|k.
T0
Proof. By [NSW], 9.2.2(ii) or [NSW2], 9.2.3(ii), respectively, the restriction
map
H1(G (p),Z/pZ)−→ H1(k ,Z/pZ)
T∪S∪Sp∪SR Y p
p∈S∪Sp∪SR
is surjective. A class in α ∈ H1(G (p),Z/pZ) which restricts to an
T∪S∪Sp∪SR
unramified class αp ∈ Hn1r(kp,Z/pZ) for all p ∈ S ∪Sp ∪SR is contained in
H1(G (p),Z/pZ). Therefore the image of the composite map
T
H1(G (p),Z/pZ)֒→H1(G (p),Z/pZ)→ H1(k ,Z/pZ)
T T∪S∪Sp∪SR Y p
p∈S
4
containsthesubgroup H1 (k ,Z/pZ). Asthisgroupisfinite,itisalready
Qp∈S nr p
contained in the image of H1(G (p),Z/pZ) for some finite subset T ⊂T. We
T0 0
concludethatnoprimeinS splitscompletelyinthemaximalelementaryabelian
p-extension of k unramified outside T .
0
Proof of Theorems 1.1 and 1.2. As p 6= 2, we may ignore archimedean primes.
Furthermore,wemayremovetheprimesinS∪S andallprimesofdegreegreater
p
than 1 from T. In addition, we remove all primes p with N(p)6≡1modp from
T. After these changes, we still have δ (T)=1.
k(µp)
By Lemma2.2,we findafinite subsetT ⊂T suchthatnoprime inS splits
0
completely in k (p)|k. Put X = Spec(Ø ). By [Sch2], Theorem 6.2, applied
T0 k
to T and TrT , we find a finite subset T ⊂ TrT such that Xr(T ∪T ) is
0 0 2 0 0 2
a K(π,1) for p. Then Proposition 2.1 applied to T ∪T ⊂ S ∪T ∪T , shows
0 2 0 2
that also Xr(S∪T ∪T ) is a K(π,1) for p. Now put T =T ∪T ⊂T.
0 2 1 0 2
It remains to show Theorem 1.2. Assertion (i) holds by construction of T .
1
By[Sch2],Lemma4.1,alsoXr(S′∪T )isaK(π,1)forp. By[Sch2],Theorem3,
1
the field k (p) realizes k (p) for p∈T , showing (ii) for these primes. Finally,
T1 p 1
assertion (ii) for p∈S follows from Proposition 2.1.
3 Duality
We start by investigating the relation between the K(π,1)-property and the
universal norms of global units.
LetusfirstremoveredundantprimesfromS: Ifp∤pisaprimewithζ ∈/ k ,
p p
then every p-extension of the local field k is unramified (see [NSW], 7.5.1 or
p
[NSW2], 7.5.9, respectively). Therefore primes p ∈/ S with N(p) 6≡ 1modp
p
cannot ramify in a p-extension. Removing all these redundant primes from S,
we obtain a subset S ⊂ S, which has the property that G (p) = G (p).
min S Smin
Furthermore, by [Sch2], Lemma 4.1, XrS is a K(π,1) for p if and only if
XrS is a K(π,1) for p.
min
Theorem 3.1. Let k be a number field and let p be a prime number. Assume
that k is totally imaginary if p = 2. Let S be a finite set of nonarchimedean
primesof k. Thenanytwoofthe followingconditions(a)–(c)imply the third.
(a) Spec(O )rS is a K(π,1) for p.
k
(b) lim Ø× ⊗Z =0.
←−K⊂kS(p) K p
(c) (k (p)) =k (p) for all primes p∈S .
S p p min
Thelimitin(b)runsthroughallfiniteextensionsK ofk insidek (p). If(a)–(c)
S
hold, then also
lim Ø× ⊗Z =0.
←− K,Smin p
K⊂kS(p)
Remarks: 1. Assume that ζ ∈ k and S ⊃ S . Then (a) holds and condition
p p
(b) holds for p>2 if #S >r +2 (see [NSW2], Remark 2 after 10.9.3). In the
2
5
casek =Q(ζ ),S =S ,condition(b)holdsifandonlyifpisanirregularprime
p p
number.
2. Assume that S∩S = ∅ and S 6= ∅. If condition (a) holds, then either
p min
G (p)=1(whichonlyhappensinveryspecialsituations,see[Sch2],Proposition
S
7.4) or (b) holds by [Sch2], Theorem 3 (or by Proposition 3.2 below).
Proof of Theorem 3.1. We may assume S = S in the proof. Let K run
min
through the finite extensions of k in k (p) and put X =Spec(Ø ). Applying
S K K
the topological Nakayama-Lemma ([NSW], 5.2.18) to the compact Z -module
p
limØ× ⊗Z , we see that condition (b) is equivalent to
←− K p
(b)’ lim Ø×/p=0.
←−K⊂kS(p) K
Furthermore, by [Sch2], Proposition 2.1, condition (a) is equivalent to
(a)’ lim Hi ((XrS) ,Z/pZ)=0 for i≥1.
−→K⊂kS(p) et K
Condition(a)’alwaysholdsfori=1,i≥4,anditholdsfori=3providedthat
G (p) is infinite or S is nonempty or ζ ∈/ k (see [Sch2], Lemma 3.7). The flat
S p
·p
Kummer sequence 0→µ →G →G →0 induces exact sequences
p m m
0−→Ø×/p−→H1(X ,µ )−→ Pic(X)→0
K fl K p p
for all K. As the field k (p) does not have nontrivial unramified p-extensions,
S
class field theory implies
lim Pic(X )⊂ lim Pic(X )⊗Z =0.
←− p K ←− K p
K⊂kS(p) K⊂kS(p)
As we assumed k to be totally imaginary if p = 2, the flat duality theorem of
Artin-Mazur ([Mil], III Corollary 3.2) induces natural isomorphisms
He2t(XK,Z/pZ)=Hfl2(XK,Z/pZ)∼=Hfl1(XK,µp)∨.
We conclude that
(∗) −li→m He2t(XK,Z/pZ)∼=(cid:0) l←im− Ø×K/p (cid:1)∨.
K⊂kS(p) K⊂kS(p)
We first show the equivalence of (a) and (b) in the case S = ∅. If (a)’ holds,
then (∗) shows (b)’. If (b) holds, then ζ ∈/ k or G (p) is infinite. Hence we
p S
obtain(a)’fori=3. Furthemore,(b)’implies(a)’fori=2by(∗). Thisfinishes
the proof of the case S =∅.
Now we assume that S 6= ∅. For p ∈ S(K), a standard calculation of local
cohomology shows that
0 for i≤1,

H1(K ,Z/pZ)/H1 (K ,Z/pZ) for i=2,
Hpi(XK,Z/pZ)∼= p H2(K ,Zn/rpZ)p for i=3.
p
0 for i≥4.

6
Forp∈S =S ,everyproperGaloissubextensionofk (p)|k admits ramified
min p p
p-extensions. Hence condition (c) is equivalent to
(c)’ lim Hi(X ,Z/pZ)=0 for all i,
−→K⊂kS(p)Lp∈S(K) p K
and to
(c)” lim H2(X ,Z/pZ)=0.
−→K⊂kS(p)Lp∈S(K) p K
Consider the direct limit over all K of the excision sequences
···→ Hi(X ,Z/pZ)→Hi (X ,Z/pZ)→Hi ((XrS) ,Z/pZ)→··· .
M p K et K et K
p∈S(K)
Assume that (a)’ holds, i.e. the right hand terms vanish in the limit for i ≥ 1.
Then (∗) shows that (b)’ is equivalent to (c)”.
Now assume that (b) and (c) hold. As above, (b) implies the vanishing of
the middle term for i=2,3 in the limit. Condition (c)’ then shows (a)’.
We have proven that any two of the conditions (a)–(c) imply the third.
Finally,assumethat(a)–(c)hold. Tensoringtheexactsequences(cf.[NSW],
10.3.11 or [NSW2], 10.3.12,respectively)
0→Ø× →Ø× → (K×/U )→Pic(X )→Pic((XrS) )→0
K K,S M p p K K
p∈S(K)
by(theflatZ-algebra)Z ,weobtainexactsequencesoffinitelygenerated,hence
p
compact, Z -modules. Passing to the projective limit over the finite extensions
p
K of k inside k (p) and using limPic(X ) ⊗ Z = 0, we obtain the exact
S ←− K p
sequence
0→ lim Ø×⊗Z → lim Ø× ⊗Z → lim (K×/U )⊗Z →0.
←− K p ←− K,S p ←− M p p p
K⊂kS(p) K⊂kS(p) K⊂kS(p)p∈S(K)
Condition (c) and local class field theory imply the vanishing of the right hand
limit. Therefore (b) implies the vanishing of the projective limit in the middle.
If G (p) 6= 1 and condition (a) of Theorem 1.1 holds, then the failure in
S
condition(c) canonly come from primes dividing p. This followsfrom the next
Proposition3.2. Letkbeanumberfieldandletpbeaprimenumber. Assume
that k is totally imaginary if p = 2. Let S be a finite set of nonarchimedean
primes of k. If Spec(Ø )rS is a K(π,1)for p and G (p)6=1, then everyprime
k S
p∈S with ζ ∈k has an infinite inertia group in G (p). Moreover,we have
p p S
k (p) =k (p)
S p p
for all p∈S rS .
min p
7
Proof. We may assume S =S . Suppose p∈S with ζ ∈k does not ramify
min p p
in kS(p)|k. Setting S′ =Sr{p}, we have kS′(p)=kS(p), in particular,
H1(XrS′,Z/pZ)−∼→H1(XrS,Z/pZ).
et et
In the following, we omit the coefficients Z/pZ from the notation. Using the
vanishing ofH3(XrS), the´etaleexcisionsequenceyields a commutative exact
et
diagram
H2(GS(cid:127)_′(p)) ∼ // H2(GS(p))
≀
(cid:15)(cid:15) (cid:15)(cid:15)
H2(X)(cid:31)(cid:127) // H2(XrS′) α // H2(XrS) //H3(X) //// H3(XrS′).
p et et p et
Hence α is split-surjective and Z/pZ ∼= H3(X) →∼ H3(XrS′). This implies
p et
S′ =∅, hence S ={p}, and ζ ∈k. The same applies to every finite extension
p
of k in kS(p), hence p is inert in kS(p) = k∅(p). This implies that the natural
homomorphism
Gal(kpnr(p)|kp)−→G∅(k)(p)
is surjective. Therefore GS(p) = G∅(p) is abelian, hence finite by class field
theory. Since this group has finite cohomological dimension by the K(π,1)-
property, it is trivial, in contradiction to our assumptions.
This shows that all p ∈ S with ζ ∈ k ramify in k (p). As this applies to
p p S
every finite extension of k inside k (p), the inertia groups must be infinite. For
S
p∈S rS this implies k (p) =k (p).
min p S p p
Theorem 3.3. Let k be a number field and let p be a prime number. As-
sume that k is totally imaginary if p = 2. Let S be a finite nonempty set of
nonarchimedean primes of k. Assume that conditions (a)–(c) of Theorem 3.1
hold and that ζ ∈ k for all p ∈ S. Then G (p) is a pro-p duality group of
p p S
dimension 2.
Proof. Condition (a) implies H3(G (p),Z/pZ)→∼ H3(XrS,Z/pZ)=0. Hence
S et
cdG (p)≤2. Ontheotherhand,by(c),thegroupG (p)containsGal(k (p)|k )
S S p p
as a subgroup for all p∈S. As ζ ∈k for p∈S, these local groups have coho-
p p
mological dimension 2, hence so does G (p).
S
In order to show that G (p) is a duality group, we have to show that
S
D (G (p),Z/pZ):= lim Hi(U,Z/pZ)∨
i S −→
U⊂GS(p)
cor∨
vanish for i=0,1,where U runs through the open subgroupsof G (p) and the
S
transition maps are the duals of the corestriction homomorphisms; see [NSW],
3.4.6. The vanishing of D is obvious, as G (p) is infinite. Using (a), we
0 S
therefore have to show that
lim H1((XrS) ,Z/pZ)∨ =0.
−→ K
K⊂kS(p)
8
We put X = Spec(O ) and denote the embedding by j : (XrS) → X . By
k K K
the flat duality theorem of Artin-Mazur, we have natural isomorphisms
H1((XrS)K,Z/pZ)∨ ∼=Hfl2,c((XrS)K,µp)=Hfl2(XK,j!µp).
The excision sequence together with a straightforward calculation of local co-
homology groups shows an exact sequence
(∗) K×/K×p →H2(X ,jµ )→H2((XrS) ,µ ).
M p p fl K ! p fl K p
p∈S(K)
As ζ ∈ k and k (p) = k (p) for p ∈ S by assumption, the left hand term of
p p S p p
(∗)vanisheswhenpassingtothelimitoverallK. WeusetheKummersequence
to obtain an exact sequence
(∗∗) Pic((XrS) )/p−→H2((XrS) ,µ )−→ Br((XrS) ).
K fl K p p K
The left hand term of (∗∗) vanishes in the limit by the principal ideal theorem.
The Hasse principle for the Brauer group induces an injection
Br((XrS) )֒→ Br(K ).
p K M p p
p∈S(K)
As k (p) realizes the maximal unramified p-extension of k for p∈S, the limit
S p
of the middle term in (∗∗), and hence also the limit of then middle term in (∗)
vanishes. This shows that G (p) is a duality group of dimension 2.
S
Remark: The dualizing module can be calculated to
D ∼=torp CS(kS(p) ,
(cid:0) (cid:1)
i.e.Disisomorphictothep-torsionsubgroupintheS-id`eleclassgroupofk (p).
S
The proof is the same as in ([Sch1], Proof of Thm. 5.2), where we dealt with
the tame case.
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Alexander Schmidt, NWF I - Mathematik, Universita¨t Regensburg, D-93040 Regensburg,
Deutschland. email: [email protected]
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