On pro-p fundamental groups of marked arithmetic curves

0.35 MB·English

by  Alexander Schmidt

On pro-p fundamental groups of
marked arithmetic curves
9
0
Alexander Schmidt
0
2
January 16, 2009
n
a
J
6 Abstract
1
Letk beaglobal field,pan oddprimenumberdifferentfrom char(k) and

S, T disjoint, finite sets of primes of k. Let GT(k)(p) = G(kT(p)|k) be the
] S S
T Galois group of the maximal p-extension of k which is unramified outside S
N andcompletelysplitatT. WeprovetheexistenceofafinitesetofprimesS0,
which can be chosen disjoint from any given set M of Dirichlet density zero,
.
h suchthatthecohomologyofGTS∪S0(k)(p)coincideswiththeétalecohomology
at oftheassociatedmarkedarithmeticcurve. Inparticular,cd GTS∪S0(k)(p)=2.
m Furthermore,wecanchooseS0 insuchawaythatkST∪S0(p)realizes themax-
[ imalp-extensionkp(p)ofthelocal fieldkp forall p∈S∪S0,thecup-product
H1(GTS∪S0(k)(p),Fp)⊗H1(GTS∪S0(k)(p),Fp)→H2(GTS∪S0(k)(p),Fp)issurjec-
2 tiveandthedecomposition groupsoftheprimesinS establishafreeproduct
v inside GTS∪S0(k)(p). This generalizes previous work of the author where sim-
3 ilar results were shown in the case T = ∅ under the restrictive assumption
6 p∤#Cl(k) and ζp ∈/ k.
8
1
. 1 Introduction
6
0
8 Let k be a number field, S a finite set of places of k and p a prime number. We
0
denote the maximal p-extension of k which is unramified outside S by k (p) and
: S
v put
i G (k)(p)=G(k (p)k).
X S S
|
r The group G (k)(p) is rather well (but far from being completely) understood in
a S
the case that S contains all primes dividing p (see Chapter X of [NSW] for an
overviewontheresultsknown). Besidesthe factthatthe groupG (k)(p)isfinitely
S
presentable,untilrecentlynotmuchwasknowninthe casethatS doesnotcontain
all places above p.
In2005,J.Labute[La1]consideredthe casek =Qandfoundthe firstexamples
of pairs (S,p) such that G (Q)(p) has cohomological dimension 2 and p is not
S
in S. This progress was possible by using the newly developed theory of mild pro-
p-groups. The author then generalized Labute’s results to arbitrary number fields
[S1, S2], whereas the focus was on the K(π,1)-property. This property says that
the cohomology of the group G (k)(p) coincides with the étale cohomology of the
S
scheme Spec( )rS, in particular, it implies cohomological dimension 2. However,
k
for a given number field k, finitely many prime numbers had to be excluded: the
divisorsoftheclassnumberandsuchpwithζ k. Ithasturnedoutnowthatthis
p
∈
restriction can be removed if one considers a more general problem from the very
beginning,namelythequestionforthegroupGT(k)(p). HereS andT arefinitesets
S
of places, kT(p) is the maximal p-extension of k which is unramified outside S and
S
completely split at T, and GT(k)(p) = G(kT(p)k). This group is also interesting
S S |
in the function field case.
1
Inthis paper weprovewithoutassumptionsonS andT that, byadding finitely
many primes to S, we achieve a situation where GT(k)(p) has cohomological di-
S
mension 2 and further nice properties. Moreover,we can avoidan arbitrarily given
set of Dirichlet density zero (in particular, the places above p) when choosing the
additional primes. The precise result is the following:
Theorem 1.1. Let k be a global field and let p be an odd prime number different
from char(k). Let S, T and be pairwise disjoint sets of places of k, where S
M
and T are finite and has Dirichlet density δ( )=0. Then there exists a finite
M M
set of places S of k which is disjoint from S T and such that the following
0
∪ ∪M
assertions hold.
(i) The group GT (k)(p) has cohomologicaldimension 2 and the cup-product
S∪S0
H1(GT (k)(p),F ) H1(GT (k)(p),F ) H2(GT (k)(p),F )
S∪S0 p ⊗ S∪S0 p −→ S∪S0 p
is surjective.
(ii) We have
kT (p) =k (p)
S∪S0 p p
for all p S S , i.e. the extension of global fields kT (p)k realizes the
∈ ∪ 0 S∪S0 |
maximal p-extension k (p) of the local field k for all p S S .
p p 0
∈ ∪
(iii) The decomposition groups of the primes in S establish a free product in the
group GT (k)(p), i.e. the natural homomorphism
S∪S0
G(k (p)k ) G(kT (p)kS∪T(p))
p∈S(k∗S∪T(p)) p | p −→ S∪S0 | S0
S0
is an isomorphism of pro-p-groups.
(iv) For each discrete p-torsion GT (p)-module M, the edge morphisms of the
S∪S0
Hochschild-Serre spectral sequence for the universal pro-p-covering
Hi(GT (k)(p),M) Hi (Xr(S S ),T,M)
S∪S0 −→ et ∪ 0
areisomorphismsforalli 0. HereXistheuniquelydefinedone-dimensional,
≥
regular and connected scheme which is proper over Spec(Z) and has function
field k, and Hi (Xr(S S ),T,M) denotes the étale cohomology of the
et 0
∪
markedarithmeticcurve(Xr(S S ),T)withvaluesinthe sheafdefinedby
0
∪
M (see section 2).
Remarks 1.2. 1. The reason why we excluded the prime number p = 2 in The-
orem 1.1 is not the usual problem with the real places, but that there is no good
theory of mild pro-2-groups at hand at the moment.
2. Naturally the question arises whether properties (i)–(iv) then also hold for the
group GTS∪S0′(k)(p), where S0′ is an arbitrary finite set of places which contains S0
andisdisjointfromT. Wewillprovethisinsection8undertheassumptionthatno
prime in S′ rS splits completely in the extension kT (p)k (see Theorem 8.1).
0 0 S∪S0 |
3. Similar results for the full group GT(k), i.e. without passage to the maximal
S
pro-p factor group, seem to be out of reach at the moment.
In the number field case, the places above p do not play a special role in The-
orem 1.1, in particular, we do not assume that S contains the set S of primes
p
dividingp. ButeveninthenumberfieldcasewithS S andT =∅,Theorem1.1
p
⊃
provides new information: assertion (ii) was known up to now only if k contains a
primitivep-throotofunity(Kuz’min’sTheorem,see[Ku]or[NSW],10.8.4),aswell
as for certain CM-fields (see [Mu] or [NSW], X §8 Exercise). By (iii), after adding
2
finitely many primes to S S , the decomposition groups of the primes above p
p
⊃
establish a free product inside the group G (k)(p). So far this was only known for
S
sets of places S of Dirichlet density 1 (see [NSW], 9.4.4), but not for finite sets of
places.
In the “tame” case S S = ∅ with T = ∅, assertions (i), (ii) and (iv) were
p
∩
shown in [S2], however only if p ∤ #Cl(k) and ζ / k. Previous results in the
p
∈
“mixed”case∅ S S S hadbeenachievedbyK.Wingberg[Wi1],Ch.Maire
p p
∩
[Mai] and D. Vogel [Vo].
We will see in section 9 that Theorem 1.1 provides a large class of examples of
GT(k)(p) being a pro-p duality group (see Theorem 9.6). In the number field case
S
with S S and T = ∅, this was known if k contains a primitive p-th root of
p
⊃
unity byresultsofWingberg(see[Wi2]or[NSW], 10.9.8). Inthecaseζ / k,there
p
∈
existed merely results for real-abelian fields and for certain CM-fields (see [NSW],
10.9.15 and the Remark below).
Essential for the proof of Theorem 1.1 are arithmetic duality theorems, Hasse
principles for the cohomology, Labute’s theory of mild pro-p-groups and the tech-
nique of free products of bundles of profinite groups as developed in Chapter IV
of [NSW].
The author thanks Ph. Lebacque for his comments about an earlier version of
this paper.
2 The étale site of a marked curve
LetY beaone-dimensional,noetherianandregularschemeandletT beafiniteset
ofclosedpointsofY. Asusual,wedenotebyEt(Y)thecategoryofétalemorphisms
of finite type Y′ Y.
→
Definition 2.1. ThecategoryEt(Y,T)isthe fullsubcategoryofEt(Y)whichcon-
sists of all objects f : Y′ Y such that for each closed point y′ Y′ with
→ ∈
y =f(y′) T theresiduefieldextensionk(y′)k(y)istrivial. Theétalesite(Y,T)et
∈ |
ofthecurveY markedinT consistsofthecategoryEt(Y,T)withsurjectivefamilies
as coverings.
Obviously, we have (Y,∅)et = Yet. For T1 T2, we have a natural morphism
⊂
ι:(Y,T1)et (Y,T2)et and hence homomorphisms
→
Hi (Y,T ,F) Hi (Y,T ,ι∗F)
et 2 et 1
−→
foreachsheafF ofabeliangroupson(Y,T2)et andforalli 0. Foraproperclosed
≥
(hence finite) subset M Y, we define the local cohomology groups H∗ (Y,T, )
⊂ M −
in the usual manner as the right derivatives of the functor
F ker Γ(Y,T,F) Γ(Y rM,T rM,F) .
7−→ →
(cid:0) (cid:1)
Inthe samewayasforusualthe étalecohomology,oneprovesexcision,i.e.wehave
Hi (Y,T,F)= Hi(Yh,Th,F),
M ∼ x x x
x∈M
M
where Yh is the henselization of Y in x and Th is the preimage of T in Yh (i.e.
x x x
Th = x ifx T andTh =∅otherwise). Alsotheconstructionofthecup-product
x { } ∈ x
is completely analogous to the usual étale site: for sheaves F1,F2 on (Y,T)et and
3
i,j 0, we have a cup-product pairing
≥
Hi (Y,T,F ) Hj (Y,T,F ) ∪ Hi+j(Y,T,F F ),
et 1 et 2 et 1 2
× −→ ⊗
satisfying the using properties.
Also in an analogous way, one constructs the fundamental group. We consider
the full subcategory FEt(Y,T) of the finite morphisms Y′ Y in Et(Y,T). This
→
category satisfies the axioms of a Galois category ([SGA1], V, 4). Choosing a
geometric point x¯ in Y rT, we have the fibre functor
FEt(Y,T) (Sets), (Y′ Y) Mor (x¯,Y′),
Y
−→ → 7→
whose group of automorphisms is the étale fundamental group of (Y,T) by defini-
tion. We denote this groupby πet(Y,T,x¯). Assume that Y is connected. Then the
1
fundamental groupsto different base points are isomorphic,the isomorphismbeing
canonicaluptoinnerautomorphisms,andwewillfrequentlyexcludethebasepoint
fromthe notation. The fundamentalgroupis profinite andclassifiesétale coverings
of Y in which every point of T splits completely. The groupHe1t(Y,T,Fp) classifies
cyclic coverings of degree p of (Y,T). Therefore we have isomorphisms
H1(πet(Y,T)(p),F ) ∼ H1(πet(Y,T),F ) ∼ H1(Y,T,F ),
1 p −→ 1 p −→ et p
et et
where π (Y,T)(p) denotes the maximal pro-p factor group of π (Y,T). This
1 1
isomorphism is visible in the Hochschild-Serre spectral sequence as follows. We
consider the universal pro-p-covering (^Y,T)(p) of (Y,T). This is a pro-object in
FEt(Y,T) and the projection
(^Y,T)(p) (Y,T)
−→
is Galois with group πet(Y,T)(p). For a discrete p-torsion πet(Y,T)(p)-module M,
1 1
we obtain the spectral sequence
Eij =Hi πet(Y,T)(p),Hj (^Y,T)(p),M Hi+j(Y,T,M)
2 1 et ⇒ et
(cid:16) (cid:0) (cid:1)(cid:17)
and, in particular, edge morphisms
φ :Hi(πet(Y,T)(p),M) Hi (Y,T,M), i 0.
i,M 1 −→ et ≥
Lemma2.2. LetY beaconnected,one-dimensional,noetherianandregularscheme
and let T be a finite set of closed points of Y. Then, for every discrete p-torsion
πet(Y,T)(p)-module M, the homomorphismsφ and φ are isomorphismsand
1 0,M 1,M
φ is injective. Furthermore, the following assertions are equivalent.
2,M
(i) φ is an isomorphism for all i and all M.
i,M
(ii) φi,Fp is an isomorphism for all i.
(iii) Heit((^Y,T)(p),Fp)=0 for all i 1.
≥
Proof. Byconstruction,wehaveHe1t((^Y,T)(p),Fp)=0,showingthe firstassertion.
Asπet(Y,T)(p)isapro-p-group,F istheonlysimplep-torsionπet(Y,T)(p)-module.
1 p 1
Hence (ii) implies (i) for finite M. Since cohomology of profinite groups as well as
étale cohomology commute with filtered direct limits, (ii) implies (i) for an arbi-
trary p-torsion πet(Y,T)(p)-module M. We can read off the implication (iii) (ii)
1 ⇒
directly from the spectral sequence. Finally, assume that (i) holds. Each class
in Heit((^Y,T)(p),Fp) is already in Heit(Y′,T′,Fp) for a finite intermediate covering
4
(Y′,T′) of (^Y,T)(p) (Y,T). The covering (Y′,T′) corresponds to an open sub-
group U ⊂ π1et(Y,T)→(p). Applying (i) to the π1et(Y,T)(p)-module IndUπ1et(Y,T)(p)Fp,
we see that each α Heit(Y′,T′,Fp) vanishes in Heit(Y′′,T′′,Fp) for a suitable cho-
∈
sen finite intermediate covering (Y′′,T′′) of (^Y,T)(p) (Y′,T′). This shows (iii)
→
and finishes the proof.
Definition2.3. IftheequivalentassertionsofLemma2.2aresatisfied,wesaythat
(Y,T) has the K(π,1)-property for p.
Remark 2.4. Lemma 2.2 and Definition 2.3 naturally extend to pro-objects. A
markedcurve(Y,T)hasthe K(π,1)-propertyforp if andonly ifthis is the casefor
its universal pro-p-covering(^Y,T)(p).
3 Calculations of cohomology groups
In the following, we want to calculate the étale cohomology of marked arithmetic
curves. Letk be alocalorglobalfieldandletp=char(k)beafixedprimenumber.
6
We denote the groupofp-throotsofunity byµ andputδ =1ifµ k andδ =0
p p
⊂
otherwise. All cohomology groups take values in the constant sheaf F , which we
p
exclude from the notation. Furthermore, we put
hi(−)=dimFpHeit(−) (=dimFpHeit(−,Fp)).
We start with a local computation. For a local field k which is not an extension
of Q , we use the convention [k : Q ]= 0. Furthermore, we denote the unramified
p p
cohomology by Hnir(k) and set
Hi (k)=Hi(k)/Hi (k).
/nr nr
By A∨ we denote the Pontryagindual of A.
Proposition 3.1. Let k be a non-archimedean local field with char(k) = p. Let
6
X =Spec( ), x the closed point of X and T =∅ or T = x . Then the local étale
k
{ }
cohomology groups Hi(X,T) vanish for i 1 and i 4. Moreover,we have
x ≤ ≥
H1 (k) if T =∅,
H2(X,T)= /nr
x ∼( H1(k) if T = x ,
{ }
hence
h2(X,T)=δ+[k :Q ]+#T .
x p
Furthermore, we have H3(X,T) = H2(k) = µ (k)∨, hence h3(X,T) = δ, and the
x ∼ ∼ p x
following Euler-Poincarécharacteristic formula holds:
3
( 1)ihi(X,T)=[k:Q ]+#T .
− x p
i=0
X
Proof. Since X is henselian, we have isomorphisms Heit(X) ∼= Heit(x) for all i.
Furthermore, there are no nontrivial étale covers of (X,T) if T = x . Therefore
{ }
we obtain
1 für i=0,
hi(X,T)= 1 #T für i=1,
 −
0 für i 2.

≥

5
Furthermore, Xr{x} = Spec(k), hence Heit(Xr{x}) ∼= Hi(k). The local duality
theorem(see[NSW],Theorem7.2.6)showsHe2t(Xr{x})∼=µp(k)∨ and,by[NSW],
Corollary 7.3.9, we obtain
h1(Xr x )=1+δ+[k :Q ].
p
{ }
Finally,thenaturalhomomorphismHe1t(X) He1t(Xr x )isinjective. Therefore
→ { }
the statement of the proposition follows from the exact excision sequence
Hi(X,T) Hi (X,T) Hi (Xr x ) Hi+1(X,T) .
···→ x → et → et { } → x →···
Now let k be a global field and let X be the uniquely defined one-dimensional,
regularandconnectedschemewhichisproperoverSpec(Z)andhasfunctionfieldk
(i.e. X = Spec( ) if k is a number field and X is a smooth projective curve over
k
a finite field if k is a function field). Let S and T be disjoint finite sets of non-
archimedean places of k, i.e. disjoint finite sets of closed points of X.
We denote the set of archimedean places of k by S (hence S = ∅ if k is a
∞ ∞
function field). In Galois terminology (and without mention the base point), we
have
πet(XrS,T)=GT (k):=G(kT k),
1 S∪S∞ S∪S∞|
wherekT isthe maximalextensionofk whichisunramifiedoutsideS S and
S∪S∞ ∪ ∞
completely split at T. If K k is an intermediate extension of kT k, we denote
| S∪S∞|
by
(XrS,T)
K
the normalization (XrS) of the curve XrS in K, marked in the set T(K) of
K
prolongationsof placesof T to K. IfK k is finite, then (XrS,T) is an objectin
K
|
FEt(XrS,T), and a pro-object otherwise.
Now let p=char(k) be a fixed prime number. We denote the maximal elemen-
6
tary abelian p-extension of k inside kT by
S∪S∞
kT,el
S∪S∞
and note that
G(kST∪,eSl∞|k)∼=He1t(XrS,T)∨.
We denote the set of primes dividing p by S (hence S = ∅ in the function field
p p
case). Assuming p=2 or k totally imaginary if k is a number field, we may ignore
6
the archimedean primes, i.e. we have
GT(k)(p)=GT (k)(p) =πet(XrS,T)(p) .
S S∪S∞ 1
Part (iv) of our main result Theorem 1.1 say(cid:0)s that, for p =2, p =(cid:1) char(k), we can
6 6
achieve the K(π,1)-property for (XrS,T) by adding finitely many primes to S.
This is well known, even without adding primes, in the special case S S and
p
⊃
T =∅:
Proposition 3.2. Letk be a globalfield, p=char(k) a prime number andS S
p
6 ⊃
a finite, nonempty set of places of k. Then, for every discrete p-torsionG (k)-
S∪S∞
module M, the natural maps
Hi(G (k),M) Hi (XrS,M)
S∪S∞ −→ et
areisomorphismsforall i 0. If,moreover,M isadiscretep-torsionG (k)(p)-
≥ S∪S∞
module, then also the natural maps
Hi(G (k)(p),M) Hi(G (k),M)
S∪S∞ −→ S∪S∞
are isomorphisms for all i 0.
≥
6
Proof. See [Mi2], II, Proposition2.9, for a proof of the first statement. The second
statement is a result of O. Neumann, see [NSW], Corollary 10.4.8.
Fromnowonweusethe followingnotation,wherep isaprimenumberdifferent
from char(k).
r the number of real places of k
1
r the number of complex places of k
2
r =r +r , the number of archimedean places of k
1 2
δ equals 1 if µ k and 0 otherwise
p
⊂
δ equals 1 if µ k and 0 otherwise
p p p
⊂
Cl (k) =Pic(XrS), the S-ideal class group of k
S
E =H0(XrS,G ), the group of S-units of k
k,S m
·n
A =ker(A A), where A is an abelian group and n N
n
→·n ∈
A/n =coker(A A), where A is an abelian group and n N.
→ ∈
Ifk haspositivecharacteristic,thenr1 =r2 =r =0,andEk,∅ isthe multiplicative
groupofthefinitefieldofconstants. Asbefore,weexcludetheconstantcoefficients
F from the notation in cohomology groups. We make the assumption p = 2 or k
p
6
totally imaginary in the number field case everywhere in the paper. Therefore we
make the following notational convention:
‘Set of places’ always means ‘set of non-archimedean places’.
Proposition3.3. LetS and T bedisjoint,finitesetsofplacesoftheglobalfieldk.
We assume p = 2 or k totally imaginary if k is a number field. Then the groups
6
Hi (XrS,T)=0 vanish for i 4 and we have
et
≥
3
χ(XrS,T):= ( 1)ihi(XrS,T)=r+#T [k :Q ].
p p
− −
Xi=0 p∈XS∩Sp
Proof. The assertion is well known in the case T = ∅ and S S : by [Mi2], II,
p
⊃
Theorem 2.13(a), we have χ(XrS)= r , and furthermore
2
−
r [k :Q ]=r +r [k :Q]= r ,
p p 1 2 2
− − −
pX∈Sp
where we set [k : Q] = 0, if k is a function field. We consider the exact excision
sequence
Hi(X ,T ) Hi (X,T) Hi (Xr(S T)) ,
···→ p p p → et → et ∪ →···
p∈S∪T
M
where X denotes the completionof X atp (the cohomologygroupsof the comple-
p
tion and the henselization coincide). We obtain the result for S =∅ and arbitrary
T by using this excision sequence for S = S and by applying Proposition 3.1. Fi-
p
nally,weobtaintheresultforarbitraryS fromthecaseS =∅byusingtheexcision
sequence
Hi(X ) Hi (X,T) Hi (XrS,T) ,
···→ p p → et → et →···
p∈S
M
and another application of Proposition 3.1.
7
Corollary 3.4. Let S and T be disjoint, finite sets of places of the global field k.
Then we have an exact sequence
H1(XrS,T)֒ H1(XrS) H1 (k ) H2(XrS,T)։H2(XrS)
et et nr p et et
→ → →
p∈T
M
and isomorphisms Hi(XrS,T) ∼ Hi(XrS) for i 3.
−→ ≥
Proof. This follows from Proposition 3.1 by comparing the excision sequences for
XrS and Xr(S T) as well as for (XrS,T) and Xr(S T).
∪ ∪
To obtain formulae for the individual cohomology groups, we introduce the
Kummer group VT(k). Let S and T be disjoint, finite sets of places of k and let
S
p=char(k) be a fixed prime number, which we exclude fromthe notation. We put
6
VT(k):= a k× a k×p for v S and a U k×p for v / T /k×p,
S { ∈ | ∈ v ∈ ∈ v v ∈ }
where U denote the unit group of the local field k (convention: U = k×, if v is
v v v v
archimedean). In terms of flat cohomology,we have
VT(k)=ker H1(Xr(S T),µ ) H1(k ,µ ) .
S fl ∪ p −→ p p
p∈S
(cid:0) Y (cid:1)
Lemma 3.5. For S =∅, there is a natural exact sequence
0 E /p VT(k) Cl (k) 0.
k,T ∅ p T
−→ −→ −→ −→
In particular, we have dimFpV∅T(k)= dimF∅ppClT(k)+r−1+#T +δ, unless k is
a function field and T =∅, where dimFpV∅(k)=dimFppCl(k)+δ.
For arbitraryS and any additional place p / S T, we have an exact sequence
∈ ∪
0 VT (k) φ VT(k) U k×p/k×p.
−→ S∪{p} −→ S −→ p p p
In particular, VST(k) is finite. For p∈/ Sp, we have dimFpcoker(φ)≤δp.
Proof. Sending a V∅T(k) to the class [a] of the uniquely defined fractional T-
∈
Ideal a with (a)=ap, defines a surjective homomorphism V∅T(k) pClT(k) whose
→
kernel is E /p. This, together with Dirichlet’s unity theorem shows the first
k,T
statement. The second exact sequence follows directly from the definitions of the
objects occurring. For p ∈/ Sp, we have dimFpUpkp×p/kp×p = δp, showing the last
statement.
Theorem 3.6. Let S and T be disjoint, finite sets of places of the global field k.
Assumep=2orktotallyimaginaryifk isanumberfield. ThenHi (XrS,T)=0
et
6
for i 4 and
≥
h0(XrS,T) = 1,
h1(XrS,T) = 1+ δp−δ+dimFpVST(k)+ [kp :Qp]−r−#T,
pX∈S p∈XS∩Sp
h2(XrS,T) = δp−δ+dimFpVST(k)+θ,
p∈S
X
h3(XrS,T) = θ.
Here θ =1 if δ =1 and S =∅, and θ =0 otherwise.
8
Proof. The statement about h0 is trivial and the vanishing of the cohomology in
dimensiongreaterthanorequalto4isalreadycontainedinProposition3.3. Artin-
Verdier duality (see [Maz], 2.4 or [Mi2], Theorem 3.1) or étale Poincaré duality
([Mi1], V, Corollary 2.3), respectively, show that
He3t(X)∨ ∼=HomX(Fp,Gm)=µp(k).
Furthermore, the vanishing of Heit(Xp,Tp) for i 2 and the local duality theorem
≥
([NSW], Theorem 7.2.6) show isomorphisms Hp3(Xp,Tp)∨ ∼= H2(kp,Fp)∨ ∼= µp(kp)
for every point p of X. By Corollary 3.4, we have an isomorphism He3t(X,T) ∼
→
He3t(X)∼=µp(k)∨. We consider the excision sequence
H3(X) α H3(X,T) β H3(XrS,T) H4(X ).
p → et → et → p p
p∈S p∈S
M M
The right hand term vanishes by Proposition 3.1, hence β is surjective. The dual
map to α is the map
µ (k) µ (k ),
p p p
→
p∈S
M
which is injective, unless δ = 1 and S = ∅. We obtain h3(XrS,T) = 1 if δ = 1
and S =∅, and 0 otherwise. Using the isomorphismH1(GTS(k))→∼ He1t(XrS,T),
we obtain the assertion about h1 from the calculation of the first cohomology of
the groupGT(k), see [NSW], Theorem10.7.10. Finally, the result for h2 follows by
S
using the Euler-Poincarécharacteristic formula in Proposition 3.3.
The vanishing of the cohomology in dimension greater than 2 and Lemma 2.2
imply the following
Corollary 3.7. For S = ∅ and each intermediate extension K k of kT(p)k, the
6 | S |
following assertions are equivalent:
(i) (XrS,T) has the K(π,1)-property for p.
K
(ii) The homomorphism φ2,Fp : H2(GTS(K)(p)) → He2t((XrS,T)K) is surjective
and cd GT(K)(p) 2.
S ≤
4 Duality for marked arithmetic curves
Next we prove a duality theorem for marked arithmetic curves. For a sheaf F on
(XrS,T)et, we consider the Shafarevich-Tate groups
Xi(k,S,T,F)=ker Hi (XrS,T,F) Hi(k ,F) .
et p
−→
p∈S
(cid:0) M (cid:1)
We omit the constant coefficients F = F from the notation and we also omit T if
p
T =∅. Furthermore, we put
BT(k)=VT(k)∨.
S S
Theorem 4.1. Let k be a global field and let p be a prime number different from
char(k). Assume p = 2 or k totally imaginary if k is a number field. Let S and T
6
be disjoint, finite sets of places of k. Then there is a natural isomorphism
X2(k,S,T) ∼ BT(k).
−→ S
In particular, X2(k,S,T) is finite.
9
Remark4.2. Bypassingtothelimitoverallfinitesubsets,Theorem4.1generalizes
to the case of an arbitrary, i.e. not necessarily finite set of places S.
Proof of Theorem 4.1. We choose a sufficiently large finite set of places Σ such
that S T S Σ and Pic (XrΣ) is p-torsion free. Then the groups
∪ ∪ p ⊂ k(µp)
X1(k,Σ) and X1(k,Σ,µ ) vanish. From this follows the vanishing of X2(k,Σ)=
p (cid:0) (cid:1) ∼
X1(k,Σ,µ )∨ (byProposition3.2andPoitou-Tateduality,[NSW],Theorem8.6.7)
p
and the exactness of the sequence
0 VT(k) H1(XrΣ,µ ) H1(k ,µ ) H1 (k ,µ ).
−→ S −→ et p −→ p p × /nr p p
pY∈S p∈ΣY\(S∪T)
We dualize this sequence and consider the commutative diagram
H1(k ) H1 (k ) // H1(XrΣ,µ )∨ ////BT(k)
p × nr p et p S
pY∈S p∈(cid:127)Σ_Y\(S∪T)
(cid:15)(cid:15)
H1(XrΣ)(cid:31)(cid:127) // H1(k ) ////H1(XrΣ,µ )∨
et p et p
p∈Σ
Y
(cid:15)(cid:15)(cid:15)(cid:15)
H1(k ) H1 (k ).
p × /nr p
pY∈T p∈ΣY\(S∪T)
The middle row is part of the long exact sequence of Poitou-Tate (see [NSW],
Theorem 8.6.10), hence exact. The snake lemma yields the exactness of
(I) H1(XrΣ) H1(k ) H1 (k ) BT(k) 0.
et −→ p × /nr p −→ S −→
pY∈S p∈ΣY\(S∪T)
Using the commutative diagram
H2(XrS,T) //H2(XrΣ)
et et (cid:127)_
(cid:15)(cid:15) (cid:15)(cid:15)
H2(k )(cid:31)(cid:127) // H2(k ),
p p
p∈S p∈Σ
Y Y
we obtain
X2(k,S,T)=ker H2(XrS,T) H2(XrΣ) .
et et
−→
Therefore excision yields the exac(cid:0)t sequence (cid:1)
(II) H1(XrΣ) H1(k ) H1 (k ) X2(k,S,T) 0,
et −→ p × /nr p −→ −→
pY∈S p∈ΣY\(S∪T)
and a comparisonof (I) and (II) shows the statement of the theorem.
Corollary 4.3. Let k be a global field and let p be a prime number different from
char(k). Assume p = 2 or k totally imaginary if k is a number field. Let S and T
6
be disjoint, finite sets of places of k with VT(k)=0. Then the natural map
S
H2(XrS,T) H2(k )
et p
−→
p∈S
Y
10