Extensions with restricted ramification and duality for arithmetic schemes

By Alexander Schmidt at Heidelberg

Let X be a (d + 1)-dimensional smooth arithmetic scheme lying smoothly and properly over an open subset U of Spec(O_{K}) for some algebraic number field K with geometrically connected fibres of dimension d. It is well-known, that there is a duality pairing for the sheaf cohomology of X, which connects the etale cohomology groups of a locally constant constructible sheaf F with the cohomology with compact support of the sheaf F^{D}(d), where (d) means the d-fold Tate-twist (see [6]). In this paper we will prove a duality theorem, which is exclusively formulated in terms of the ordinary sheaf cohomology, not using cohomology groups with compact support. One could also view the result as a computation of the cohomology with compact support in terms of ordinary sheaf cohomology using a dualizing sheaf. With the above notations let U = Spec(O_{K,S}) where S is a finite set of places of K including the archimedian places. We write S^{f} for the set of finite places in S and S_{∞} for the set of archimedian places of K, thus S = S^{f} ∪ S_{∞}. As usual we denote the maximal extension of K, unramified outside S by K_{S}. For an integer n and an abelian group A we denote the kernel of the n-multiplication map by _{n}A. Let C_{S^{f}} (K_{S}) be the S^{f}-idele class group of the field K_{S} (for a definition see below) and we put I_{n,U} := _{n}C_{S^{f}} (K_{S}). The Gal(K_{S}/K)-module I_{n,U} is the direct limit of its finite submodules and hence defines a sheaf of ℤ/nℤ-modules on U_{et}. Then the following holds.

Theorem 1 Let U be an open subscheme of Spec(O_{K}), where K is a finite extension of ℚ. Further let

π : X⟶U

be a smooth and proper morphisms of schemes with fibres of pure dimension d. Let n be an integer invertible on U and assume K to be totally imaginary if n is even. Then for every locally constant, constructible sheaf F of ℤ/nℤ-modules on X_{et} the cup-product

H_{et}^{i}(X, F) × H_{et}^{2d+2−i}(X, Hom(F, π^{∗}I_{n,U}(d)))⟶^{∪}H_{et}^{2d+2}(X, π^{∗}I_{n,U}(d))⟶^{∼}ℤ/nℤ

defines a perfect pairing of finite abelian groups for all i.

Using generalized Poincaré-duality the heart of the proof of theorem 1 is the case d = 0, which can be reformulated in terms of Galois groups with restricted ramification of algebraic number fields. The proof combines a result of Wingberg on pro-p-extensions with restricted ramification with a theorem of Washington on the behaviour of the prime-to-p part of ideal class groups in ℤ_{p}-extensions of algebraic number fields. For a profinite group G and a prime number p we denote the maximal pro-p factor group of G by G(p). Using the language of higher etale homotopy groups we prove the following theorem for the case of arithmetic surfaces.

Theorem 2 Assume that the fibres of π : X⟶U are smooth, projective curves of nonzero genus and let p be a prime number, which is invertible on U. Then one has:

π_{i}^{et}(X)(p) = 0 for all i ≥ 2.

Combining theorems 1 and 2 we conclude:

Theorem 3 Assume that the fibres of π : X⟶U are smooth, projective curves of genus g and let p be a prime number invertible on U. Assume that p is odd or that K is totally imaginary. Then the following holds for the etale fundamental group G := π_{1}^{et}(X):

(i) If g ≥ 1 then G is a profinite duality group at p of dimension 4.

(ii) If g = 0 then G is a profinite duality group at p of dimension 2.

Remark: a) If p = 2 and K is not totally imaginary the theorem remains true, if we replace the word duality group by virtual duality group, i.e. there is an open subgroup being a duality group.

b) Under certain restrictions to X and U in was shown in [8], that the maximal pro-p factor group of G has similar duality properties.

1

In this section we prove a duality theorem for Galois groups with restricted ramification of algebraic number fields which will imply theorem 1 for the case d = 0. Let K be a number field and let S be a finite set of places of K which contains all archimedian places. By K_{S} we denote the maximal extension of K, which is unramified outside S and we denote Gal(K_{S}/K) by G_{S}. For an intermediate field K ⊂ L ⊂ K_{S} we use the following notations:

S(L) the set of places of L lying above S, E_{L,S} the group of S(L)-units of L , J_{L} the idele group of L, J_{L,S} the S-idele group of L, C_{S}(L) the S-idele class group of L, Cl_{S}(L) the S-ideal class group of L.

We further use similar notations for the set of finite primes S^{f} in S. The definitions are the same word-by-word, however we could not find them in the literature and therefore we give the definitions. Let L be a finite extension of K in K_{S}:

J_{L,S^{f}} {(a_{w}) ∈ J_{L} | a_{w} = 1 for w ∉ S^{f}} ≅ ∏_{v∈S^{f}} L_{v}^{×},

E_{L,S^{f}} =E_{L,S} -the group of S^{f}-units, C_{L,S^{f}} J_{S^{f}} (L)/E_{S^{f}} (L) .

Further we denote:

J_{S^{f}} = \underrightarrow{lim}J_{L,S^{f}},  E_{S^{f}} = \underrightarrow{lim}E_{L,S^{f}},  C_{S^{f}} = \underrightarrow{lim}C_{L,S^{f}},

where the limit runs through all finite subextensions L of K in K_{S}. Remark: If K is totally imaginary, every extension of K is automatically unramified at the archimedian places. In this situation the pair (G_{S}, C_{S^{f}}) is a P -class formation for the set P of prime numbers l with l^{∞}|#G_{S} having analogous properties like the usual P -class formation (G_{S}, C_{S}) (see proposition 10 below). Each element of C_{S}(K) which can be represented by an idele with support in the set of archimedian places is an universal norm for (G_{S}, C_{S}). The advantage of (G_{S}, C_{S^{f}}) is that it is free of this ”redundancy at infinity”, i.e. the group of universal norms in C_{S^{f}} (K) is much smaller.

Let p be a prime number and we assume K to be totally imaginary if p = 2. By S_{p}(K) we denote the set of places of K dividing p. For an abelian group A we will denote the subgroup of elements which are annihilated by a power of p by T or_{p}(A).

Theorem 4 If S ⊇ S_{p}(K) ∪ S_{∞}(K) then G_{S} is a duality group at p of dimension 2 with p-dualizing module I = T or_{p}(C_{S^{f}} (K_{S})), i.e. for every finite discrete p-primary G_{S}-module M and all i, the cup-product

H^{i}(G_{S}, M) × H^{2−i}(G_{S}, Hom(M, I)) ⟶^{∪} H^{2}(G_{S}, I)⟶^{∼}ℚ_{p}/ℤ_{p}

defines a perfect pairing of finite groups.

We will prove theorem 4 in several steps.

Proposition 5 Under the above assumptions one has cd_{p}(G_{S}) = 2. For a finite discrete p-primary G_{S}-module M the cohomology groups H^{i}(G_{S}, M) are finite for all i. The p-dualizing module of G_{S} is isomorphic to T or_{p}(C_{S^{f}} (K_{S})).

Proof: The statement about the cohomological dimension as well as the finiteness statement on the cohomology can be found at various places in the literature (see [6], [4]). Following the description of the dualizing module of an arbitrary profinite group in [10],[11] it holds for the p-dualizing module I of G_{S}:

I = lim ; ⟶_{n^{L,cor^{∗}}}H^{2}(Gal(K_{S}/L), ℤ/p^{n}ℤ)^{∗},

where L runs through the finite subextensions of K in K_{S}. Since μ_{2p} ∈ K_{S} we can take the limit over all L which are totally imaginary. Using Tate’s long exact sequence ([6] I§4 Thm.4.10) we obtain the following exact sequence:

0⟶H^{0}(Gal(K_{S}/L), μ_{p^{n}})⟶ ∏_{v∈S^{f} (L)} H^{0}(Gal(K̄_{v}/L_{v}), μ_{p^{n}}) →

→ H^{2}(Gal(K_{S}/L), ℤ/p^{n}ℤ)^{∗}⟶H^{1}(Gal(K_{S}/L), μ_{p^{n}}).

Going to the limit over L and n we obtain:

I ≅ coker(μ_{p^{∞}} ⟶^{diag} ∏_{v∈S^{f} (K)}(Ind_{G_{v}}^{G_{S}}μ_{p^{∞}})),

where G_{v} denotes the decomposition group in G_{S} of a fixed extension of v to K_{S}. Using the exact sequence

0⟶E_{S^{f}} (K_{S})⟶J_{S^{f}} (K_{S})⟶C_{S^{f}} (K_{S})⟶0

and the fact, that E_{S^{f}} (K_{S}) is p-divisible we obtain the exact sequence

0⟶T or_{p}(E_{S}(K_{S}))⟶T or_{p}(J_{S}(K_{S}))⟶T or_{p}(C_{S}(K_{S}))⟶0.

As μ_{p^{∞}} ⊂ E_{S^{f}} (K_{S}) and by the definition of J_{S^{f}} we get the exact sequence

0⟶μ_{p^{∞}} ⟶^{diag} ∏_{v∈S^{f} (K)}(Ind_{G_{v}}^{G_{S}}μ_{p^{∞}})⟶T or_{p}(C_{S^{f}} (K_{S}))⟶0,

which proves the proposition. □ In order to prove theorem 4 we have to verify the vanishing of the following limit for i = 0, 1 (cf. [10],[11]):

D_{i}(ℤ/pℤ) =^{def} lim ; ⟶_{{U⊂G_{S} \atop cor^{∗}}}H^{i}(U, ℤ/pℤ)^{∗},

where the limit runs through the open subgroups of U and the transition maps are the duals of the corestriction maps. Is is easy to see that D_{0}(ℤ/pℤ) = 0 since p^{∞}|#G_{S}. In order to prove the vanishing of D_{1}(ℤ/pℤ) the following theorem is crucial.

Theorem 6 If S ⊃ S_{p} ∪ S_{∞} then C_{S}(K_{S}) is p-divisible.

Remark: It is easily seen, that the p-divisibility of C_{S}(K_{S}) is equivalent to the fact, that the local field (K_{S})_{v} is a p-closed local field for all v ∈ S(K_{S}). If S omits only finitely many primes of K this is an easy consequence of the theorem of Grunwald-Hasse-Wang (see [7]). However if S is finite this is non-trivial. If S does not contain S_{p} one even does not know, whether the supernatural order of G_{S} is divisible by p^{∞}.

In [13] Wingberg investigates the similar pro-p situation for theorem 6. In order to refer to his result we introduce the following notations:

K_{S}(p) the maximal pro-p subextension of K in K_{S}, G_{S}(p) the Galois group of K_{S}(p)/K, K_{v} the completion of the number field K at the prime v, K_{v}(p) the maximal pro-p extension of K_{v}, G_{v} the Galois group of K_{v}(p)/K_{v}, T_{v} the inertia group of v in G_{v}, G_{v} the decomposition group of v in G_{S}(p), H_{1}⋆H_{2} the free pro-p product of the pro-p groups H_{1} and H_{2}.

Using a theorem of Kuz’min, Wingberg proved the following theorem see [13] :

Theorem 7 (Wingberg [13]) Assume μ_{2p} ⊂ K and S ⊃ S_{p}(K) ∪ S_{∞} and suppose that C_{S}(K_{S}(p)) is not p-divisible. Then there exists a prime v ∈ S_{p}(K) such that the decomposition group G_{v} of v in the extension K_{S}(p)/K is the full group G_{S}(p) and the following holds: There exists a finite set T of primes of K containing S such that the homomorphism

ϕ : ⋆_{v′≠v^{v′∈S\S∞}} G_{v′} ⋆ ⋆_{v′∈T \S}(G_{v′}/T_{v′}) ⟶^{∼} G_{S}(p)

induced by the maps: ϕ_{v′} : G_{v′} ⟶ Gal(K(p)/K) −_{↠}^{can} G_{S}(p) is an isomorphism.

Corollary 8 If C_{S}(K_{S}(p)) is not p-divisible then there exists a prime v ∈ S_{p}(K) such that the following inequality holds

[K_{v} : ℚ_{p}] ≥ ∑_{v′≠v^{v′∈Sp(K)}}[K_{v′} : ℚ_{p}].

Proof of the corollary: As μ_{p} ⊂ K the following (in)equality holds by [9] II§5 Thm.4 for a prime v′ ∈ S_{p}(K):

rank G_{v′} ≤ rank G_{v′} = [K_{v′} : ℚ_{p}] + 2.

Now assume that #S_{p}(K) > 1 (otherwise the statement of the corollary is trivial) and assume that C_{S}(K_{S}(p)) is not p-divisble. By theorem 7 there exists a prime v ∈ S_{p}(K) such that G_{S} = G_{v} and the free product decomposition of G_{S} yields the following inequalities:

[K_{v} : ℚ_{p}] = rank G_{v} − 2
≥ rank G_{v} − 2 = rank G_{S} − 2
≥ ( ∑_{v′≠v^{v′∈Sp(K)}} rank G_{v′}) − 2 = ( ∑_{v′≠v^{v′∈Sp(K)}}([K_{v′} : ℚ_{p}] + 2)) − 2
≥ ∑_{v′≠v^{v′∈Sp(K)}}[K_{v′} : ℚ_{p}],

which proves the corollary. □ In order to prove theorem 6 it is obviously sufficient to prove that the group C_{S}(L_{S}(p)) is p-divisible for a cofinal set of finite extensions L of K in K_{S}. Since μ_{2p} ⊂ K_{S} we assume without loss of generality that μ_{2p} ⊂ K, in particular K is totally imaginary, containing the imaginary abelian field ℚ(ζ_{2p}). Now what we need is a method of leaving the bad situations described in theorem 7. For this we use a result of Washington, where k_{n} denotes the unique subextension of degree p^{n} in the cyclotomic ℤ_{p}-extension of a number field k and h_{n} denotes the class number of k_{n}.

Theorem 9 (Washington [12]) Let k be an imaginary abelian number field and let

H = {l | l prime number, l divides h_{n} for some n}.

Then H is infinite.

Now returning to our situation assume that L is a finite subextension of K in K_{S} and that C_{S}(L_{S}(p)) is not p-divisible. Claim: There is a finite extension L′/L contained in K_{S} such that C_{S}(L′_{S}(p)) is p-divisible. Proof of the claim: Following theorem 9, choose a prime number l > p and n such that: (i) l|h(ℚ(ζ_{p^{n}})) and (ii) (l, [L : ℚ]) = 1. By class field theory there exists a finite cyclic unramified extension F/ℚ(ζ_{p^{n}}) of degree l. As the only prime of ℚ(ζ_{p^{n}}) which divides p is principal it completely splits in F, hence there are l different primes dividing p in F. By condition (ii) we see that F and L(ζ_{p^{n}}) are linearly disjoint over ℚ(ζ_{p^{n}}). Hence every prime v of L(ζ_{p^{n}}) dividing p splits into l different primes in LF. Therefore the field LF has the property that for every prime dividing p there are at least l − 1 other primes having the same absolute local degree. By corollary 8 we obtain that the group C_{S}((LF)_{S}(p)) is p-divisible which proves the claim. Thus we have proved theorem 6. □ Now we are able to prove the vanishing of D_{1} (see above).

D_{1}(ℤ/pℤ) = lim ; ⟶_{L}H^{1}(Gal(K_{S}/L), ℤ/pℤ)^{∗}
= lim ; ⟶_{L}Gal(K_{S}/L)^{ab}/p
= lim ; ⟶_{L}C_{S}(L)/p
= C_{S}(K_{S})/p = 0.

Thus the proof of theorem 4 is complete. □ In order to get a better understanding of theorem 4 we give the following proposition for the case of totally imaginary K. (Compare [6] I§4 4.2,4.5,4.6.)

Proposition 10 Let K be a totally imaginary number field and let S ⊃ S_{∞}(K) be a finite set of primes. Then the following holds.

(i) The pair (G_{S}, C_{S^{f}}) is a P-class formation for the set P of primes l with l^{∞}|#G_{S}.

(ii) The reciprocity map: rec : C_{S^{f}} (K)⟶^{rec}G_{S}^{ab} is surjective with divisible kernel.

(iii) If p is a prime number such that all primes dividing p are in S and if M is a finite p-primary G_{S}-module then the cup-product defines isomorphisms:

α^{r} : Ext_{G_{S}}^{r}(M, C_{S^{f}})⟶^{∼}H^{2−r}(G_{S}, M)^{∗}

for all r.

Proof: Using the definition of C_{S} and C_{S^{f}} we get the following commutative diagram with exact rows and columns:

0  0
↓  ↓
Ind^{G_{S}} ℂ^{×} = Ind^{G_{S}} ℂ^{×}
↓  ↓
0 ⟶ E_{S} ⟶  J_{S}  ⟶  C_{S}  ⟶ 0
∥  ↓  ↓
0 ⟶ E_{S^{f}} ⟶  J_{S^{f}}  ⟶ C_{S^{f}}  ⟶ 0
↓  ↓
0  0  .

Hence we obtain for all i ≥ 1 isomorphisms: H^{i}(G_{S}, C_{S})⟶^{∼}H^{i}(G_{S}, C_{S^{f}}). Since (G_{S}, C_{S}) is a P-class formation the same follows for (G_{S}, C_{S^{f}}). We call the reciprocity maps associated to these class formations by rec and rec_{f} and we denote the kernel of rec (rec_{f}) by D_{S}(K) (D_{S^{f}} (K)). It is well-known ([6] I§4 4.5), that rec is surjective and that D_{S}(K) is divisible. Hence we get the following commutative diagram with exact rows and columns:

0  0
↓  ↓
0 ⟶ ∏_{v∈S_{∞}(K)} ℂ^{×} ⟶ D_{S}(K) ⟶ D_{S^{f}} (K) ⟶ 0
∥  ↓  ↓
0 ⟶ ∏_{v∈S_{∞}(K)} ℂ^{×} ⟶ C_{S}(K) ⟶ C_{S^{f}} (K) ⟶ 0
↓^{rec}  ↓^{rec_{f}}
G_{S}^{ab} = G_{S}^{ab}
↓  ↓
0  0

This proves (ii). By Theorem 1.13 of [6] I §1 and by (ii) we obtain that α^{r} is an isomorphism for r ≥ 1 (cf. [6] I §4 Thm.4.6.(a) ). The statement for r = 0 follows from proposition 5. For r < 0 both groups are zero. □ Remark: Now we can conclude theorem 4 from proposition 10 (iii) and from theorem 6: C_{S^{f}} is a quotient of C_{S}, hence p-divisible. Therefore the spectral sequence

E_{2}^{pq} = H^{r}(G_{S}, Extq_{ℤ}(M, C_{S^{f}})) ⟹ Ext_{G_{S}}^{p+q}(M, C_{S^{f}})

degenerates to a sequence of isomorphisms: H^{p}(G_{S}, Hom(M, C_{S^{f}}))⟶^{∼}Ext_{G_{S}}^{p}(M, C_{S^{f}}), which by proposition 10 (iii) implies theorem 4.

2

In this section we prove theorem 1. Let K be a number field, U ⊂ Spec(O_{K}) an open subscheme and let S be the finite set of places of K, containing the archimedian places and such that U = Spec(O_{K,S}). As it is well known every finite discrete G_{S}-module M defines a locally constant constructible sheaf on U_{et}. If M is a discrete G_{S} module, annihilated by some integer n, then M is the direct limit of its finite submodules and therefore defines a sheaf of ℤ/nℤ-modules on U_{et}, which is the direct limit of locally constant constructible sheaves. The following proposition is well-known.

Proposition 11 Assume that n is invertible on U. Then for all i:

H_{et}^{i}(U, M) ≅ H^{i}(G_{S}, M).

Proof: If M is finite this is [6]II§2 2.9. The general case follows since etale cohomology as well as Galois cohomology commute with direct limits. □ Therefore theorem 1 for d = 0 is equivalent to theorem 4 which is already proved. Further one knows that the p-dualizing module of a duality group is divisible (or else T or_{p}(C_{S^{f}} (K_{S})) is p-divisible as C_{S}(K_{S}) is p-divisible). For further use we note:

Corollary 12 The stalks of the sheaf I_{n,U} are injective ℤ/nℤ-modules.

Now let X,π,U,n be as in the introduction. By D(X, ℤ/nℤ) resp. D(U, ℤ/nℤ) we denote the derived category of ℤ/nℤ-module sheaves on X_{et} resp. U_{et}. As π and n fulfill the conditions of generalized Poincaré-duality we get the following isomorphism of objects in D(U, ℤ/nℤ) (see theorem 3.2.5. in [1] XVIII) :

Rπ_{∗}Hom_{D(X,ℤ/nℤ)}(F, π^{∗}I_{n,U}(d)[2d]) ≅ Hom_{D(U,ℤ/nℤ)}(Rπ_{∗}F, I_{n,U}). (∗)

Here (d) is the d-fold Tate-twist and [2d] is the shift by 2d. By corollary 12 the stalks of I_{n,U} (resp.π^{∗}I_{n,U}) are injective ℤ/nℤ-modules. Therefore we have an isomorphism

Hom_{D(X,ℤ/nℤ)}(F, π^{∗}I_{n,U}(d)[2d]) ≅ Hom_{D(X,ℤ/nℤ)}(F, π^{∗}I_{n,U}(d))[2d].

Applying RΓ(U, −) on the left hand side of (∗) we get a complex of abelian groups whose r^{th} cohomology group is isomorphic to

H_{et}^{r+2d}(X, Hom(F, π^{∗}I_{n,U}(d))).

By the proper-smooth base change theorem ([5]VI§4 4.2) the sheaves R^{i}π_{∗}F are locally constant and constructible on U. Applying RΓ(U, −) on the right hand side of (∗) we get get a complex of abelian groups whose r^{th} cohomology group by theorem 1 for d = 0 is isomorphic to

Hom(H_{et}^{2−r}(X, F), ℤ/nℤ).

This proves theorem 1. □

3

In this section we prove theorems 2 and 3. For a scheme X we denote by X_{et} its etale homotopy type, i.e. a pro-simplicial set. The etale homotopy groups of X are by definition the homotopy groups of X_{et} and it is well-known that these pro-groups are pro-finite, whenever the scheme X is noetherian, connected and geometrically unibranch ([2] Theorem 11.1). By X̃_{et} we denote the universal covering of X_{et}. If p is a prime number and Y_{·} is a pro-simplicial set, we denote the pro-p completion of Y_{·} by Y_{·}^{∧p}. The maximal pro-p factor group of a pro-group G is denoted by G(p). For the following we need

Proposition 13 Assume that Y_{·} is simply connected (i.e. π_{1}(Y_{·}) = 0) and that π_{i}(Y_{·}) is pro-finite for all i ≥ 2. Then we have isomorphisms for all i:

π_{i}(Y_{·})(p)⟶π_{i}(Y_{·}^{∧p}).

Proof: If G is an abelian pro-finite group, the canonical surjection: G⟶G(p) has a kernel with trivial p-Sylow subgroup, i.e. the supernatural order of the kernel is prime to p. Therefore G is a p-good pro-group in the sense of [2], i.e. for every finite p-primary G(p)-module M the canonical homomorphism: H^{i}(G(p), M)⟶^{infl}H^{i}(G, M) is an isomorphism for all i. Hence proposition 13 follows by induction on i from theorem 6.7. of [2]. □ Now let K be a number field, let U ⊂ Spec(O_{K}) be an open subscheme and let p be a prime number invertible on U. Then the following holds:

Proposition 14 The higher etale homotopy groups of U have no p-part, i.e.

π_{i}^{et}(U)(p) = 0 for i ≥ 2.

Further the canonical morphism:

U_{et}^{∧p}⟶K(π_{1}^{et}(U)(p), 1)

with π_{1}^{et}(U)(p) the maximal pro-p factor group of the etale fundamental group of U is a weak homotopy equivalence.

Proof: Let S be the finite subset of places of K, containing the archimedian places and such that U = Spec(O_{K,S}). Then G_{S}(K) ≅ π_{1}^{et}(U) and proposition 11 implies that the universal covering Ũ_{et} has no cohomology with values in p-primary coefficient groups. Hence the pro-p completion of Ũ_{et} is contractible and therefore the first statement of the proposition follows from proposition 13. By a theorem of O.Neumann (see [4] Prop.22) for every finite p-primary G_{S}(p)-torsion module M the canonical homomorphism H^{i}(G_{S}(p), M)⟶^{infl}H^{i}(G_{S}, M) is an isomorphism for all i. The same arguments as for the first statement then also show the second. □ Similar arguments apply in the geometric situation. Here we denote the genus of a curve C by g(C).

Proposition 15 Let k be a field and let C be a connected, smooth curve over k. Assume either that C is incomplete or g(C) > 0. Then

π_{i}^{et}(C) = 0 for i ≥ 2,

i.e. the canonical morphism

C_{et}⟶K(π_{1}^{et}(C), 1)

is a weak homotopy equivalence. If k is separably closed and p is an arbitrary prime number then also the canonical morphism

C_{et}^{∧p}⟶K(π_{1}^{et}(C)(p), 1)

is a weak homotopy equivalence.

Proof: First, if necessary, we replace k by a suitable extension, such that C is geometrically connected. We denote the base change of C to an separable closure k^{s} of k by C_{k^{s}}. Since the morphism C_{k^{s}}⟶C is pro-etale the canonical morphism:

C̃_{k^{s},et}⟶^{can}C̃_{et}

from the universal covering of C_{k^{s},et} to the universal covering of C_{et} is a weak homotopy equivalence. Therefore we can assume k to be separably closed also in the first statement. In order to show the first statement we show that the higher homotopy groups of C_{et} (which are abelian profinite groups) have no p-part for an arbitrary prime number p. If either C is incomplete or p = char(k) then H_{et}^{i}(C, ℤ/pℤ) = 0 for i ≥ 2. Therefore H^{i}(C̃_{et}, ℤ/pℤ) = 0 for i ≥ 1, which implies π_{i}^{et}(C)(p) = 0 for i ≥ 2. If C is complete and p ≠ char(k) then H_{et}^{i}(C, ℤ/pℤ) = 0 for i > 2 and Poincaré-duality ([5] V§2 Thm.2.1) implies an isomorphism: H_{et}^{2}(C, ℤ/pℤ) ≅ H_{et}^{0}(C, ℤ/pℤ)^{∗} = ℤ/pℤ. If C′ is a finite etale cover of C of degree n then the following diagram is commutative:

H_{et}^{2}(C, ℤ/pℤ) ⟶^{∼} ℤ/pℤ
↓^{can}  ↓^{·n}
H_{et}^{2}(C′, ℤ/pℤ) ⟶^{∼} ℤ/pℤ .

Therefore H^{2}(C̃_{et}, ℤ/pℤ) = 0 if and only if there are etale covers of C of degree divisible by arbitrary high powers of p. This is obviously the case if and only if g(C) > 0. This proves the first statement. The same arguments also prove the second statement. □

Now let π : X⟶U be smooth, proper with fibres which are curves of nonzero genus. We denote by U(i) = Ũ the pro-scheme representing the universal covering of U. By X(i) we denote the pro-scheme X ×_{U} U(i). Fixing a (geometric) base point u of U we denote X ×_{U} u by X_{u}. Further let p be a prime number invertible on U. Then by theorem 11.5. of [3] and propositions 14, 13 we get isomorphisms for all r ≥ 1:

π_{r}((X_{u})_{et}^{∧p})⟶^{∼}π_{r}(X(i)_{et}^{∧p}).

By proposition 15 we obtain π_{r}(X(i)_{et}^{∧p}) = 0 for r ≥ 2, in particular we have H_{et}^{r}(X(i), ℤ/pℤ) = 0 for r ≥ 2. If Y is a finite etale cover of X, the same argument applies for Y (i) (possibly Y is defined over an etale cover of U). Going to the limit we obtain for the universal cover X̃_{et} of X_{et}: H^{r}(X̃_{et}, ℤ/pℤ) = 0 for r ≥ 1. Since the groups π_{r}^{et}(X) are profinite they are p-good for r ≥ 2 and we obtain 0 = π_{r}(X̃_{et})(p) = π_{r}^{et}(X)(p) for r ≥ 2. This proves theorem 2. □ Now theorem 3 part (i) is an easy consequence of theorem 2 and theorem 1: For every finite p-primary π_{1}^{et}(X)-module M the canonical homomorphism

H^{r}(π_{1}^{et}(X), M)⟶H_{et}^{r}(X, M)

is an isomorphism for all r by theorem 2. Hence theorem 1 implies the duality statement (i). Part (ii) is even easier. As the fibres are simply connected we have an isomorphism π_{1}^{et}(X)⟶^{∼}π_{1}^{et}(U) which proves the assertion in view of theorem 1 for U and proposition 14. □

References

[1] M.Artin,A.Grothendieck,J.L.Verdier Theorie des Topos et Cohomolgie Etale des Schemas(SGA4) LNM 269,270,305

[2] M.Artin,B.Mazur Etale Homotopy LNM 100 Springer 1969

[3] E.M.Friedlander Etale homotopy of simplicial schemes Ann. of Math. Studies 104 Princeton Univ. Press 1982

[4] K.Haberland Galois cohomology of algebraic number fields Dt. Verlag der Wissenschaften, Berlin 1978

[5] J.S. Milne Etale Cohomology Princeton University Press 1980 [6] J.S.Milne Arithmetic Duality Theorems

Persp. in Math. 1 Academic Press Boston 1986

[7] J.Neukirch Einbettungsprobleme mit lokaler Vorgabe und freie Produkte lokaler Galoisgruppen J. f. reine u. ang. Math. 259(1973) 1-47

[8] A.Schmidt,K.Wingberg On the fundamental group of a smooth arithmetic surface Math. Nachr.159 (1992) 19-36

[9] J.P.Serre Cohomologie Galoisienne Lecture Notes in Math. 5 Springer 1964

[10] J.Tate Letter to Serre in [9]

[11] J.L.Verdier Dualité dans la cohomologie des groupes profinies in [9]

[12] L.C.Washington Class Numbers and ℤ_{p}-Extensions Math.Ann. 214 (1975) 177-193

[13] K.Wingberg On Galois groups of p-closed algebraic number fields with restricted ramification J. reine u. ang. Math. 400(1989) 185-202

Alexander Schmidt Universität Heidelberg Mathematisches Institut Im Neuenheimer Feld 288 D-69120 Heidelberg Germany