An arithmetic site for the rings of integers of algebraic number fields

By Alexander Schmidt at Heidelberg

The study of extensions with prescribed ramification plays a central role in algebraic number theory. It is a general hope to benefit from the analogy between number fields and function fields of trancendency degree 1 over finite fields, because for the latter the situation is much better understood. In this paper we introduce a new kind of restriction for the ramification of primes in extensions of algebraic number fields and we study the corresponding covering type over the ring of integers. We will prove that the associated fundamental group has a description similar to that of the étale fundamental group of a smooth, projective curve over a finite field.

Let K be a number field, p be a prime number and let S be a finite set of places of K containing all archimedian places and all places dividing p. Extensions of K which are unramified outside S correspond to étale covers of the open subscheme Spec(O_{K,S}) of Spec(O_{K}). We denote the maximal pro-p Galois extension of K which is unramified outside S by K_{S}(p) and we write G_{S}(K)(p) for the Galois group Gal(K_{S}(p)/K). By the results of Wingberg [W1],[W2] there is (aside some exceptional cases) a strong analogy between the group G_{S}(K)(p) and the étale pro-p fundamental group π_{1}^{et}(C\{P_{1}, …, P_{n}})(p) of a smooth, projective curve over a finite field of characteristic ≠ p, which is punctured in finitely many points. Both are (in the generic case) pro-p duality groups of dimension 2 and their dualizing modules have an analogue description in terms of class field theory.

In the geometric case there is a canonical surjection

ϕ : π_{1}^{et}(C\\{P_{1}, …, P_{n}\})(p) {− \atop ↠} π_{1}^{et}(C)(p),

where π_{1}^{et}(C)(p) is the étale pro-p fundamental group of the complete curve. Assume that at least one nontrivial p-torsion point of the Jacobian of C is rational and that a primitive p^{th} root of unity is contained in the finite ground field. Then π_{1}^{et}(C)(p) is a Poincaré group of dimension 3 and the kernel of ϕ is isomorphic to the free product of étale inertia groups by a pro-p version of the Riemann existence theorem.

Following the analogy it is natural to ask, whether a corresponding surjection also exists in the number field case. However the étale fundamental group π_{1}^{et}(Spec(O_{K})) is much too small and the question about the relations between the inertia groups of different primes in the group G_{S}(K)(p) is in general a difficult unsolved problem.

Thus we are led to look for a larger quotient of G_{S}(K)(p) which should appear as the pro-p fundamental group for some Grothendieck topology on Spec(O_{K}). Precisely we will do the following:

For all number fields K we will construct a Grothendieck-topology (called positive topology) on Spec(O_{K}), which is finer than the étale topology and which allows a certain type of pure-wild ramification which we will call positive ramification (p.r.). The cyclotomic ℤ_{p}-extension of K is p.r. for all p. This agrees with the philosophy of Iwasawa-theory, which predicts that the cyclotomic ℤ_{p}-extension of a number field (although ramified itself) should be viewed as a good analog to the constant field extensions of function fields. Associated with the positive topology for every number field K there is an integer d(K) which is the least common multiple of all ”bad” primes of K. The number d(K) divides 2 · disc(K/ℚ) but in most cases it is much smaller. If K is a cyclotomic field then it holds d(K) = 2. We will prove the following Poincaré-duality theorem for the sheaf cohomology of the p.r. site.

Theorem 1 Let K be a number field and X = Spec(O_{K}). For every integer n with (n, d(K)) = 1 there is a canonical trace map tr : H_{pos}^{3}(X, μ_{n})⟶^{∼}ℤ/nℤ and for every locally constant constructible sheaf F of ℤ/nℤ-modules on X_{pos} the cup-product:

H_{pos}^{i}(X, F) × H_{pos}^{3−i}(X, Hom(F, μ_{n}))⟶H_{pos}^{3}(X, μ_{n})⟶^{∼}ℤ/nℤ

induces a perfect pairing of finite groups for all i.

If S is a finite set of places of K containing all archimedian places and all places dividing p then there is a natural isomorphism:

G_{S}(K)(p)⟶^{∼}π_{1}^{pos}(Spec(O_{K,S}))(p)

between the group G_{S}(K)(p) and the pro-p fundamental group of the open subscheme Spec(O_{K,S}) ⊂ Spec(O_{K}) furnished with the positive topology. Hence the pro-p fundamental group of Spec(O_{K})_{pos} is in a canonical way a quotient of the group G_{S}(K)(p). We conjecture that for (p, d(K)) = 1 the group π_{1}^{pos}(Spec(O_{K}))(p) has a similar structure as the étale pro-p fundamental group of a smooth, projective curve over a finite field of characteristic ≠ p. The following theorem gives an affirmative answer for abelian number fields. For the definition of the p-genus g_{p}(K) of a number field K we refer the reader to section 6.

Theorem 2 Let p ≠ 2 be a prime number and let K be an abelian number field containing a primitive p^{th} root of unity. Then the following holds for the positive pro-p fundamental group G = π_{1}^{pos}(Spec(O_{K}))(p)

(i) If g_{p}(K) > 0 then G is a Poincaré pro-p group of dimension 3 with dualizing module μ_{p^{∞}}.

(ii) If g_{p}(K) = 0 then G is a free pro-p group of rank r with 1 ≤ r ≤ 1 + \frac{1}{2}r_{2}(K), where r_{2}(K) is the number of complex places of K.

Remark: The theorem holds in fact not only for abelian number fields. It is enough that (p, d(K)) = 1 and that K is of ”CM-type at p” (see section 7). The rank of the free pro-p group in (ii) is equal to 1 + r_{2}(K^{+,p}), where K^{+,p} is the (+)-field of K with respect to p.

Furthermore a ”Riemann existence” theorem can be established, i.e. the kernel of the canonical surjection

G_{S}(K)(p) {− \atop ↠} π_{1}^{pos}(Spec(O_{K}))(p)

is the free pro-p product of inertia groups for the positive topology.

In the first step of the construction of the positive site on Spec(O_{K}) we deal with local fields. For a sufficiently rich class of local fields, called orientable local fields, k/ℚ_{p} (p odd) we construct an involution serving as a local analog of the complex conjugation. For a given global field K there are at most finitely many primes such that the completion of K is not orientable. For an orientable local field k we define a canonical extension k̃/k which is the maximal extension of k with at most pure-wild ramification that is positive with respect to the fixed local involution.

For the positive site the local fields k̃ will play the same role as the maximal unramified extension k^{nr} in étale topology: An extension L/K of number fields is defined to be positively ramified at a prime 𝔭 of L if the associated local extension L_{𝔭}/K_{𝔭} is a subextension of K̃_{𝔭}/K_{𝔭}.

In the first sections we define the positive topology and some preliminary calculations are made. The proof of the global duality theorem 1 combines a local duality theorem, a comparison result between positive and étale cohomology and the duality theorem of Artin-Verdier.

In section 6 we investigate a Picard group P̂ic for the positive topology and we discuss its connection with the first positive cohomology group of the multiplicative group as well as to the group of isometry classes of p-adically metrized line bundles on X = Spec(O_{K}). In particular we show Theorem 3 Let K be a number field and X = Spec(O_{K}). Let p be a prime number with (p, d(K)) = 1. Then there exists a canonical homomorphism:

Metr(p, X) {− \atop ↠}^{rec} π_{1}^{pos}(X)(p)^{ab}

from the group Metr(p, X) of isometry classes of p-adically metrized line bundles on X onto the abelianized pro-p fundamental group of X_{pos}. The homomorphism rec is surjective with p-divisible kernel. In particular rec induces an isomorphism on pro-p completions.

Eventually we investigate the associated non-abelian fundamental groups of the positive site and prove theorem 2 as well as a ”Riemann existence” theorem.

In a forthcoming paper we will investigate p-adic L-functions associated to the positive site. These satisfy functional equations and are strongly related to the p-adic L-functions of Kubota/ Leopoldt and Deligne/Ribet.

The idea of looking for ramification which is positive with respect to some involution is due to K.Wingberg. In his paper [W3] he applied this method to a pair (K, p) where the number field K is of CM-type with maximal totally real subfield K^{+} and p is an odd prime number with ζ_{p} ∈ K which has the additional property that no prime dividing p splits in the extension K/K^{+}. In this situation Wingberg constructed a canonical p-extension K̃ of K which contains the cyclotomic ℤ_{p}- extension K_{∞} of K. The field K̃ is the maximal p-extension of K which is unramified outside p and ”positively decomposed” at all primes dividing p. Using Iwasawa-theory and the classification of Demushkin groups he proved that the Galois group Gal(K̃/K_{∞}) has the same structure like the pro-p completion of the fundamental group of a smooth Riemann surface. In the proof the standard conjecture μ = 0 of Iwasawa theory (proved for abelian number fields by Ferrero and Washington) is always assumed to be true. However the situations which can be covered by this method are rather rare especially because of the ”non-split”-condition. There have been generalizations of this approach by Wingberg [W4] and the author [Sch] but always only special situations can be handled and more and more (unproven) Iwasawa-theoretic assumptions have to be made.

The advantage of the geometric approach made in this paper is that we can deal with all number fields and for every number field K there is only a finite set of exceptional primes, which can be thought as places where Spec(K) has ”bad reduction”. Also we get rid of Iwasawa theoretic assumptions.

The author is deeply indebted to K.Wingberg for numerous discussions and many valuable comments concerning this paper.

1 Positive ramification

Let p be a prime number. We denote the maximal Galois extension of odd degree (resp. p-power degree) of a field k by k(odd) (resp. k(p)). If k is a local field the maximal unramified extension of k is denoted by k^{nr}. Definition: A field k/ℚ_{p}, p odd, is preorientable if k ⊂ ℚ_{p}^{pre} := ℚ_{p}^{nr}(ζ_{p} + ζ_{p}^{−1})(odd)(ζ_{p}). Since < ρ >= Gal(ℚ_{p}^{pre}/ℚ_{p}^{nr}(ζ_{p} + ζ_{p}^{−1})(odd)) ≅ ℤ/2ℤ is a normal subgroup of Gal(ℚ_{p}^{pre}/ℚ_{p}) the automorphism ρ acts on every preorientable field. The involution ρ extends the automorphism: ℚ_{p}(ζ_{p}) → ℚ_{p}(ζ_{p}); ζ_{p} ↦ ζ_{p}^{−1} and we think of it as a local analog of the complex conjugation. The following definition is motivated by the intertwining between the local involution and the Tate-twist. Definition: Let k/ℚ_{p}, p odd, be preorientable. The field of ρ-invariant elements in k is denoted by k^{+}. We say that k is orientable if either k = k^{+} or if ζ_{p} ∈ k^{nr}. Example: Every abelian extension of ℚ_{p} (p odd) is preorientable. The author still has no reasonable idea for 2-adic positive ramification. We make the convention that no 2-adic number field is (pre)orientable.

Definition: An extension l/k is positively ramified (p.r.) if l ⊂ ℚ_{p}^{pre}k (in ℚ̄_{2} if p = 2) and if the Galois closure l̂/k of l over k has at most pure wild ramification. By k̃ we denote the maximal p.r. extension of k. By definition k̃ is a p-extension of k^{nr}. If p ≠ 2 then k contains the maximal preorientable subfield k_{1} := k ∩ ℚ_{p}^{pre} and the following holds:

k̃ = k̃_{1}k = k_{1}^{+^{nr}}(p)k.

Let K ⊂ L be (not necessary finite) global number fields and 𝔓 be a prime ideal of L. We denote the restriction of 𝔓 to K by 𝔭 and the residue characteristic by p. By L_{𝔓} (resp.K_{𝔭}) we denote the associated local field. Definition: The extension L/K is called positively ramified (p.r.) at the prime 𝔓 if L_{𝔓}/K_{𝔭} is a local p.r. extension. Remark: The following extensions are everywhere p.r.:

1. The cyclotomic ℤ̂-extension of a number field.

2. The maximal outside the places above p unramified p-extension of ℚ(ζ_{p} + ζ_{p}^{−1}).

3. Everywhere unramified extensions. Now we define a category ℭ which will contain all schemes of our interest.

Ob(ℭ) = finite disjoint unions of Spec(S^{−1}O_{K}) with K/ℚ algebraic and

S ⊂ O_{K}^{×} multiplicatively closed.

Mor(ℭ)= morphisms of schemes. If K is a number field and 𝔓 a prime of K then Spec(K) ,Spec(O_{K}) ,Spec((O_{K})_{𝔓}) are objects of ℭ. The local field K_{𝔓} and its ring of integers O_{K_{𝔓}} are not in ℭ but the henselization of (O_{K})_{𝔓} and its field of fractions are objects in the category which is sufficient for our purposes. The category ℭ has fibre products which are the normalizations of the fibre products in (Schemes).

Definition: A morphism ϕ : X⟶Y in ℭ is positively ramified (p.r.) if -ϕ is flat and of finite type -for all connected components X_{0} ⊂ X, Y_{0} ⊂ Y, ϕ(X_{0}) ⊂ Y_{0} the field extension K(X_{0})/K(Y_{0}) is p.r. at every prime which corresponds to a point of X_{0}. For an object X of ℭ we define the small site X_{pos} to be the category of p.r. morphisms Y ⟶X with surjective families as coverings. Since a flat morphism of finite type is open ([M1] Thm. 2.12) and all schemes in ℭ are affine (hence quasicompact) we conclude that the site X_{pos} is Noetherian for all objects X ∈ ℭ. By Sh(X_{pos}) we denote the category of sheaves of abelian groups on X_{pos}. A morphism ϕ : X⟶Y in ℭ defines a morphism of sites: ϕ : X_{pos}⟶Y_{pos}. The associated functors ϕ_{∗}, ϕ^{∗} and ϕ_{!} (the latter being defined if ϕ is p.r.) satisfy the well known adjunctions. Since base change over an étale morphism preserves the normality of schemes, we have a morphism of sites X_{pos}⟶X_{et} , which is induced by the identity on X. For a sheaf F on X_{pos} we will denote the direct image in Sh(X_{et}) also by F, at least as long no confusion is possible. A sheaf on the flat site F ∈ Sh(X_{fl}) restricts to a presheaf on X_{pos} which is not necessarily a sheaf, because of the different fibre products in both categories. However we get a sheaf in every case, when F is representable since normalizations are injective on the ring level in our situation. As it is well known from the general theory the category Sh(X_{pos}) is abelian and has sufficiently many injective objects. We will denote the cohomology of a sheaf F ∈ Sh(X_{pos}) by H_{pos}^{∗}(X, F). Definition: An object X ∈ ℭ is called (pre)orientable at a point x if the local field associated with O_{X,x} is (pre)orientable. We say that X is (pre)orientable at an integer n if X is (pre)orientable at all points with residue characteristic dividing n. For a finite global field K the extension K/ℚ is unramified at almost all primes. Since the maximal unramified extension of ℚ_{p} (p odd) is orientable the following definition makes sense. Definition: For a finite number field K the number d(K) is defined to be the smallest positive integer such that K_{𝔭} is orientable for all primes 𝔭 not dividing d(K).

It easily follows that d(K)|2 disc(K/ℚ). Since ℚ_{p}(ζ_{n}) (p odd, n arbitrary) is orientable it holds d(K) = 2 for every cyclotomic field. Further (p, d(K)) = 1 if p is odd and K is an abelian number field containing ζ_{p}.

2 Preliminary Calculations

For technical reasons it is useful to add certain further objects to the category ℭ, which will play the role of points. These are however in general only locally ringed spaces rather than schemes. Definition: A point is a locally ringed space with a single point as underlying topological space, together with a henselian ring A, such that Spec(A) ∈ ℭ. We denote such a point by Specm(A) and notice that Specm(A) is a scheme (and already an object of ℭ) if and only if A is a field. We define the category ℭ_{0} by the following data:

Ob(ℭ_{0}): finite disjoint unions of objects of ℭ and of points.

Mor(ℭ_{0}): Morphisms of locally ringed spaces. ℭ is a full subcategory of ℭ_{0}. We say that a morphism X⟶P = Specm(A) is tale (p.r.) if X is a finite union of points and the associated scheme morphism Spec(O_{X})⟶Spec(A) is tale (p.r.). Definition: A ring R, Spec(R) ∈ ℭ is strictly positive, if there are no connected p.r. coverings of Spec(R). A geometric point x̃ of X_{pos} is a morphism x̃ : Specm(R)⟶X with R strictly positive. For a sheaf F ∈ Sh(X_{pos}) and a geometric point x̃ : Specm(R)⟶X we define the stalk

Fx̃ =^{df} Γ(Specm(R), x̃^{∗}F).

One easily verifies that a sequence of sheaves in Sh(X_{pos}) is exact if and only if it is stalkwise exact for every geometric point. Assume X to be connected and fix a geometric point x̃ → X. One observes that the category Fpr/X of schemes finite and p.r. over X together with the fibre functor F : Fpr/X⟶(Sets); F(Y) = Mor_{X}(x̃, Y) satisfies the axiomatic conditions for a Galois theory (see [SGA1] Exp.V,4.). Therefore we have the notion of Galois coverings and a positive fundamental group. The fundamental group of X at its generic point is the Galois group of the maximal extension of K(X), which is p.r. at all primes which are points of X. In the following we will always omit the base point writing simply π_{1}^{pos}(X). The same proof as in the case of tale cohomology (compare [M1] III Thm.2.20) shows that we have a Hochschild-Serre spectral sequence associated to a p.r. Galois covering:

Proposition 2.1 (Hochschild-Serre spectral sequence) Let Y/X be a finite Galois covering in ℭ with Galois group G and let F be sheaf for the positive site of X. Then there is a spectral sequence

E_{2}^{ij} = H^{i}(G, H_{pos}^{j}(Y, F_{|Y})) ⇒ H_{pos}^{i+j}(X, F).

Since the site X_{pos} is noetherian for every object X of ℭ_{0} positive cohomology commutes with inductive limits of sheaves and [Ar] III 3.5 implies the following lemma.

Lemma 2.2 Let X ∈ ℭ_{0} and let X_{i} be a projective system of objects in ℭ_{0}/X. Let X_{∞} = lim ; ⟵_{i}X_{i}

and for a sheaf F ∈ Sh(X_{pos}) we denote the base change to X_{i} and X_{∞} by F_{i} and F_{∞}. Then

lim ; ⟶_{i}H_{pos}^{n}(X_{i}, F_{i}) = H_{pos}^{n}(X_{∞}, F_{∞}) for all n.

For a field K, Spec(K) ∈ ℭ we have by definition: Spec(K)_{pos} = Spec(K)_{et} and hence positive cohomology calculates as Galois cohomology. Let ℭ ∋ X = Spec(R) where R is a one dimensional henselian local ring and let x = Specm(R) be the closed point. Denoting the quotient field of R by K the Hochschild-Serre spectral sequence implies the following Lemma 2.3 Under the above assumptions the following holds for every sheaf F ∈ Sh(X_{pos})

H_{pos}^{i}(X, F) = H_{pos}^{i}(x, F_{|x}) = H^{i}(Gal(K̃/K), M(F)) ,

where K̃ is the maximal field extension of K where the the normalization of R lies p.r. over R and M(F) denotes the discrete Gal(K̃/K) module attached to F via the global sections of the normalizations of R. If R is strictly henselian, this group is a pro-p group, where p is the residue characteristic of R.

Using the fact, that the extensions of a henselian field are in 1-1 correspondence to the extensions of its completion (Krasners lemma) one observes Gal(K̃/K) ≅ Gal(K̃_{𝔭}/K_{𝔭}), where K̃_{𝔭} is the field defined in section 1. Hence for the computation of positive cohomology we are reduced to the calculation of the cohomology of certain extensions of local fields. We freely use the results on Galois cohomology for local fields as the are to be found in [Ser].

Proposition 2.4 Let ℭ ∋ X = Spec(R) with R henselian of dimension 1. Then

(i) For every sheaf F we have H_{pos}^{i}(X, F) = 0 for i ≥ 3.

(ii) If F is a torsion sheaf and the residue characteristic is odd, then

H_{pos}^{i}(X, F) = 0 for i ≥ 2.

(iii) In the case of residue characteristic 2 part (ii) remains true for the prime-to-2 part of the cohomology.

Proof: Using lemma 2.2 we reduce to the case where the quotient field K̂ of the completion of R is a finite extension of ℚ_{p}. Assume p ≠ 2 for a moment. If K̂ is not preorientable we can change to the maximal preorientable subfield of K̂ without changing the Galois group G = Gal(K̃/K) of which we are interested in the cohomology. Then [K̂ : K̂^{+}] = 1 or 2 and ζ_{p} ∉ K̂^{+^{nr}}. Setting K̂^{+} = K̂ in the case p = 2 we conclude that in all cases G ≅ Gal(K̂^{+^{nr}}(p)/K̂^{+}). This group has cohomological dimension 1 for p ≠ 2 because ζ_{p} ∉ K̂^{+^{nr}} and by standard arguments of [Ser]. For p = 2 this group has strict cohomological dimension 2 and cohomological l-dimension 1 for l ≠ 2. □ The explicit calculations in [Ser] II§5 imply the following

Corollary 2.5 Let ℭ ∋ X = Spec(R) with R henselian of dimension 1, and assume that the quotient field K̂ of the completion of R is a finite, preorientable extension of ℚ_{p} (p odd). Then

dim_{ℤ/pℤ}H_{pos}^{1}(X, ℤ/pℤ) = [K̂^{+} : ℚ_{p}] + 1.

Returning to the general case we will see, that for cases, where all residue characteristics are prime to the torsion of the sheaves, étale and positive cohomology coincide. For an integer n we denote the localization of ℤ at the multiplicative closed subset of integers which are relatively prime to n by ℤ_{(n)}.

Proposition 2.6 Let X ∈ ℭ and n an integer invertible on X. Then for every F ∈ Sh(X_{pos}) the canonical homomorphism

H_{et}^{i}(X, F) ⊗ ℤ_{(n)}⟶^{∼}H_{pos}^{i}(X, F) ⊗ ℤ_{(n)}

is an isomorphism for all i ∈ ℤ.

Proof: Denoting the canonical morphism by f : X_{pos}⟶X_{et} it is obviously sufficient to show that for the higher direct image sheaves R^{i}f_{∗}(F) it holds R^{i}f_{∗}(F) ⊗ ℤ_{(n)} = 0 for i > 0. We verify this on stalks. For a geometric point x̄ of X_{et} we have

(R^{i}f_{∗}F)x̄ = H_{pos}^{i}(Spec(O_{X,x̄}), F)

and since (n, char(k(x̄))) = 1 by assumption, we are ready using lemma 2.3. □

Let X ∈ ℭ and Z ⊂ X be a closed subset (in the Zariski sense). For a sheaf F on X_{pos} we define

the group of sections with support in Z as usual by Γ_{Z}(X, F) =^{df} ker(Γ(X, F)⟶Γ(X\Z, F)), and H_{Z}^{i}(X, F) =^{df} R^{i}Γ_{Z}(X, −)(F). Similar arguments as in the tale case show that associated to every sheaf F we have natural long exact sequences for the local cohomology:

… → H_{Z}^{i}(X, F) → H_{pos}^{i}(X, F) → H_{pos}^{i}(X\Z, F) → ….

Proposition 2.7 (Excision) Let X, X′ ∈ ℭ , Z ⊂ X, Z′ ⊂ X′ be closed subschemes, and let f : X′⟶X be an étale morphism, such that the restriction of f to Z′ is an isomorphism f_{|Z′} : Z′⟶^{∼}Z and f(X′\Z′) ⊂ X\Z. Then for all sheaves F ∈ Sh(X_{pos}) and all i ≥ 0 the induced homomorphism H_{Z}^{i}(X, F)⟶H_{Z′}^{i}(X′, f^{∗}F) is an isomorphism.

Proof: Since f is étale, it is p.r. and by the adjunctions f_{!} ⊣ f^{∗} ⊣ f_{∗} we see that f^{∗} is exact and sends injective sheaves to injective sheaves. Hence it is enough to show the proposition for i = 0. But the remaining problem only involves the restriction of F to the étale site and by the well-known excision lemma of étale cohomology ([M1] III Prop.1.27) we are ready. □

Corollary 2.8 Let X ∈ ℭ and z be a closed point of X. Then

H_{z}^{i}(X, F)⟶^{∼}H_{z}^{i}(Spec(O_{X,z}^{h}), F)

for all i and all sheaves F on X_{pos}.

Let ℭ ∋ Spec(R), R henselian and R̂ = OK̂, K̂/ℚ_{p} (p odd) finite and preorientable. The local complex conjugation ρ acts on K and since R is henselian ρ decomposes the group of principal units of R into a direct sum of eigenspaces:

U_{K}^{1} = U_{K}^{+} ⊕ U_{K}^{−}.

We carry out the analogous construction for the completion K̂ of K. Denoting the prime ideal of R by 𝔭 it holds:

Proposition 2.9 Let R be a one dimensional henselian ring with quotient field K. Assume that K̂ is a finite preorientable extension of ℚ_{p}. Then the following holds:

\begin{matrix}H_{𝔭}^{0}(Spec(R), ℚ_{p}/ℤ_{p}) = 0, & H_{𝔭}^{2}(Spec(R), ℚ_{p}/ℤ_{p}) = (U − ; K̂)^{∗}, \\ H_{𝔭}^{1}(Spec(R), ℚ_{p}/ℤ_{p}) = 0, & H_{𝔭}^{i}(Spec(R), ℚ_{p}/ℤ_{p}) = 0 for i ≥ 3. \end{matrix}

Proof: We have

H_{pos}^{i}(Spec(R)\𝔭, ℚ_{p}/ℤ_{p}) = 0 for i ≥ 2,

since the strict cohomological dimension of a local field and hence also of a henselian field equals 2. Local class field theory implies an isomorphism

H_{pos}^{1}(Spec(R)\𝔭, ℚ_{p}/ℤ_{p}) ≅ ℚ_{p}/ℤ_{p} ⊕ U ∗ ; K̂.

Lemma 2.3 and local class field theory imply

H_{pos}^{1}(Spec(R), ℚ_{p}/ℤ_{p}) = ℚ_{p}/ℤ_{p} ⊕ (U + ; K̂)^{∗}.

Using proposition 2.4 and the long exact sequence for the local cohomology we obtain that H_{𝔭}^{i}(Spec(R), ℚ_{p}/ℤ_{p}) = 0 for i = 0 and i ≥ 3 and the following sequence is exact:

0⟶H_{𝔭}^{1}(Spec(R), ℚ_{p}/ℤ_{p})⟶ℚ_{p}/ℤ_{p} ⊕ (U + ; K̂)^{∗} ⟶ ℚ_{p}/ℤ_{p} ⊕ (UK̂)^{∗}⟶H_{𝔭}^{2}(Spec(R), ℚ_{p}/ℤ_{p})⟶0.

This implies the remaining statements. □

3 Local duality

In this section we define a sheaf 𝔾̂_{m,X} for ℭ ∋ X = Spec(R) with R henselian, which will play the role of the multiplicative group for X_{pos}. It fits into a Kummer sequence also for the prime p, the residue characteristic of R. Let first R be of dimension 1 and let K be the quotient field of R. By μ_{prim} we denote the group of roots of unity in K with order prime to p. If p ≠ 2 we denote the maximal preorientable subfield in K by K′ and we define the group

U^{pos}(R) := μ_{prim} ⊕ U_{K′}^{−} ⊂ U(R) .

If p = 2 we set U^{pos}(R) := μ_{prim} and for a field L: U^{pos}(L) := L^{×}. A p.r. covering of a henselian ring in ℭ is a finite disjoint union of such objects. Therefore we can state the following Definition/Lemma Let Spec(R) ∈ ℭ with R henselian. Then

𝔾̂_{m}(Spec(R)) := U^{pos}(R)

defines a presheaf on X_{pos} with X = Spec(R). This presheaf is a sheaf. If R = K is a field it holds 𝔾̂_{m}(K) = 𝔾_{m}(K).

Proposition 3.1 Assume that R is henselian and that X = Spec(R) is orientable at n. Then the following sequence of sheaves in Sh(X_{pos}) is exact:

0⟶μ_{n}⟶𝔾̂_{m}⟶^{·n}𝔾̂_{m}⟶0.

Proof: Arguing stalkwise we can assume R to be strictly positive. The statement is trivial if R is a field and thus we assume R to be 1-dimensional. The kernel of the n-multiplication map on 𝔾̂_{m} is μ_{n} (if the residue characteristic of R is 2 then n is odd by assumption). Hence it remains to show that the group Γ(X, 𝔾̂_{m}) = U^{pos}(R) is n-divisible. If the local complex conjugation acts trivial on R then U^{pos}(R) = μ_{prim}. Otherwise U^{pos}(R) = μ_{prim} ⊕ U_{K}^{−} and a primitive n^{th} root of unity is contained in R = R^{nr}. We conclude that the adjunction of the n^{th} root of an element of U^{pos}(R) defines a p.r. extension. But R is strictly positive which concludes the proof. □ In the following let X = Spec(R) where R is the henselization of the ring of integers of a finite number field at a prime 𝔭|p. We denote the quotient field of R by K, the maximal preorientable subfield of K by K′ and K̃ is the field defined in lemma 2.3. R̃ is the normalization of R in K̃. We denote the imbedding of the generic point u = Spec(K) by j : u⟶X and the imbedding of the closed point x = Specm(R) by i : x⟶X.

For further use we will calculate the cohomology of the sheaves 𝔾_{m} and 𝔾̂_{m} for the positive topology. For an abelian torsion group A we denote the p-part (resp. prime-to-p part) by A(p) (resp. A^{(p)}).

Proposition 3.2 Under the above assumptions the following holds:

H_{pos}^{0}(X, 𝔾_{m}) = U_{K},  H_{x}^{0}(X, 𝔾_{m}) = 0,
H_{pos}^{1}(X, 𝔾_{m}) = K̃^{×}/K^{×}UK̃ ≅ ℚ_{p}/ℤ_{p},  H_{x}^{1}(X, 𝔾_{m}) = K̃^{×}/UK̃,
H_{pos}^{2}(X, 𝔾_{m}) = Br(K)(p) ≅ ℚ_{p}/ℤ_{p},  H_{x}^{2}(X, 𝔾_{m}) = 0,
H_{pos}^{i}(X, 𝔾_{m}) = 0 for i ≥ 3,  H_{x}^{3}(X, 𝔾_{m}) = Br(K)^{(p)} ≅ ⨁_{l≠p} ℚ_{l}/ℤ_{l},
H_{x}^{i}(X, 𝔾_{m}) = 0 for i ≥ 4.

Proof: We have the following exact sequence of Gal(K̃/K)-modules.

0⟶UK̃⟶K̃^{×}⟶K̃^{×}/UK̃⟶0.

Since K̃ is an infinite p-extension over the maximal unramified extension K^{nr} of K we have an isomorphism K̃^{×}/UK̃ ≅ ℤ[\frac{1}{p}] (with trivial Galois action) and hence

\begin{matrix}H^{0}(Gal(K̃/K), K̃^{×}/UK̃) = K̃^{×}/UK̃, & H^{0}(Gal(K̃/K), K̃^{×}) = K^{×}, \\ H^{1}(Gal(K̃/K), K̃^{×}/UK̃) = 0, & H^{1}(Gal(K̃/K), K̃^{×}) = 0, \\ H^{2}(Gal(K̃/K), K̃^{×}/UK̃) = ⨁_{l≠p} ℚ_{l}/ℤ_{l}, & H^{2}(Gal(K̃/K), K̃^{×}) = Br(K), \\ H^{i}(Gal(K̃/K), K̃^{×}/UK̃) = 0 for i ≥ 3, & H^{i}(Gal(K̃/K), K̃^{×}) = 0 for i ≥ 3. \end{matrix}

It is well known that the group of units is cohomologically trivial under unramified extensions. Hence (using the Hochschild-Serre spectral sequence for K̃/K^{nr}/K) the groups H^{i}(Gal(K̃/K), UK̃) are p-torsion groups for i ≥ 1 and they vanish for i ≥ 3 since Gal(K̃/K) has the strict cohomological dimension 2. Hence we get the following exact sequences:

0 → U_{K} → K^{×} → K̃^{×}/UK̃ → H^{1}(Gal(K̃/K), UK̃) → 0,

0 → H^{2}(Gal(K̃/K), UK̃) → Br(K) → ⨁_{l≠p} ℚ_{l}/ℤ_{l} → 0.

By lemma 2.3 there are isomorphisms H_{pos}^{i}(X, 𝔾_{m}) ≅ H^{i}(Gal(K̃/K), UK̃) for all i. From the excision sequence and from the long exact sequence for the Galois cohomology attached to 0 → UK̃ → K̃^{×} → K̃^{×}/UK̃ → 0 it follows H_{x}^{i}(X, 𝔾_{m}) ≅ H^{i−1}(Gal(K̃/K), K̃^{×}/UK̃). This proves the proposition. □

The sheaf 𝔾̂_{m} shows a better local behavior. As above let R be the henselization of the ring of integers of a finite number field at a prime 𝔭|p and X = Spec(R).

Proposition 3.3

H_{pos}^{i}(X, 𝔾̂_{m}) = 0 for i ≥ 1.

Proof: By lemma 2.3 the sheaf cohomology calculates as the group cohomology of the Galois group of the maximal extension K̃ of K = ℚ(R), where the normalization of R lies p.r. over R. By the definition of p.r. extensions we can assume that the completion of K is preorientable if the residue characteristic is odd, since the Galois group and the module in question are not changed when we substitute K by its maximal preorientable subfield. It is well known that a module M under a finite group G is cohomologically trivial if there is an integer i, such that Ĥ^{i}(H, M) = Ĥ^{i+1}(H, M) = 0 for every subgroup H ⊂ G. Hence the proposition is proved, if we show that for every finite Galois extension L/K, where the normalization R_{L} of R in L lies p.r. over R:

Ĥ^{0}(Gal(L/K), U^{pos}(R_{L})) = 0 = H^{1}(Gal(L/K), U^{pos}(R_{L})).

The statement for Ĥ^{0} follows from local class field theory and from the definition of p.r. extensions. The unit group is cohomologically trivial under unramified extensions and hence the same is true for U^{pos} as a direct summand. By the Hochschild-Serre spectral sequence we can restrict to a purely ramified p-extension L/K. By reasons of order the group μ_{prim} is cohomologically trivial, so it remains to show that H^{1}(L/K, U_{L}^{−}) vanishes in the case p ≠ 2 and K preorientable. If the local complex conjugation acts trivially on L then U_{L}^{−} = 0. Hence we can restrict to the case that the field L^{+} of ρ-invariant elements has index 2 in L. By the definition of p.r. extensions the extension L^{+}/K^{+} is Galois. Therefore the groups L_{−}^{×} = {x ∈ L^{×} | x^{ρ} = x^{−1}} ⊂ L^{×} and L_{+}^{×} = {x ∈ L^{×} | x^{ρ} = x} ⊂ L^{×} are modules under Gal(L/K). The kernel and the cokernel of the multiplication map

m : L_{+}^{×} ⊕ L_{−}^{×}⟶L^{×}.

are annihilated by 2, since ker(m) = {±1} and for every x ∈ L^{×} it holds: x^{2} = (x^{ρ}x)(x^{ρ}x^{−1}). Hence by Hilbert Satz 90:

0 = H^{1}(L/K, L^{×}) = H^{1}(L/K, L_{+}^{×}) ⊕ H^{1}(L/K, L_{−}^{×}).

Consequently we are ready if we show that the quotient L_{−}^{×}/U_{L}^{−} has order prime to p. For an x ∈ L_{−}^{×} it holds v_{L}(x) = v_{L}(ρ(x)) = v_{L}(x^{−1}) = −v_{L}(x). Hence L_{−}^{×} ⊂ U(L). The extension L/L^{+} is ramified and hence ρ acts trivial on μ_{prim}(L). Therefore L_{−}^{×} ⊂ {±1} × U_{L}^{−} and the proof is complete. □

Proposition 3.4 Under the above assumptions the following holds:

H_{x}^{0}(X, 𝔾̂_{m}) = 0,  H_{x}^{2}(X, 𝔾̂_{m}) = 0,
H_{x}^{1}(X, 𝔾̂_{m}) = K^{×}/U^{pos}(R), H_{x}^{3}(X, 𝔾̂_{m}) = Br(K) ≅ ℚ/ℤ,
H_{x}^{i}(X, 𝔾̂_{m}) = 0 for i ≥ 4.

Proof: By the long exact sequence for the local cohomology and proposition 3.3 we get: H_{x}^{1}(X, 𝔾̂_{m}) = coker(H_{pos}^{0}(X, 𝔾̂_{m}) → H_{pos}^{0}(u, 𝔾̂_{m})) and H_{x}^{i}(X, 𝔾̂_{m}) = H_{pos}^{i−1}(u, 𝔾̂_{m}) for i ≥ 2. Now the result follows from 𝔾̂_{m,u} = 𝔾_{m,u} and from the well-known calculation of the cohomology of the multiplicative group of local resp. henselian fields. □ Let G := π_{1}^{pos}(x) = Gal(K̃/K) and D := K̃^{×}/U^{pos}(R̃).

Proposition 3.5 The pair (G, D) is a class formation and for every open subgroup H ⊂ G it holds:

D^{H} = (K̃^{H})^{×}/U^{pos}(R̃^{H}).

Proof: We consider the exact sequence of G-modules

0⟶U^{pos}(R̃)⟶K̃^{×}⟶D⟶0.

By proposition 3.3 we know that U^{pos} is cohomologically trivial and it holds H^{1}(G, K̃^{×}) = 0 (Hilbert Satz 90) and H^{2}(G, K̃^{×}) = Br(K) = ℚ/ℤ, since K̃ contains K^{nr}. By the long exact cohomology sequence associated to the above sequence we get H^{1}(G, D) = 0 and H^{2}(G, D) = Br(K). For an open subgroup H ⊂ G and L = K̃^{H} we get the following commutative diagram:

H^{2}(G, D) ⟶^{Res_{G,H}} H^{2}(H, D)
↓^{≀}  ↓^{≀}
Br(K)  ⟶^{kan}  Br(L)
↓^{≀}  ↓^{≀}
ℚ/ℤ  ⟶^{[L:K]}  ℚ/ℤ  .

This proves the statement of the proposition. □ We will call a sheaf F on X_{pos} locally constant if there is a p.r. covering {U_{i}⟶X}, such that F_{|U_{i}} = \underline{A_{i}}_{|U_{i}}, with abelian groups A_{i}. F is called constructible if it has finite stalks and there is a Zariski-open subset U ⊂ Y such that F_{|U} is locally constant. The Galois module D from proposition 3.5 is the limit of its finitely generated submodules and hence gives rise to a sheaf over the closed point x_{pos}, which is the direct limit of locally constant sheaves. The defining equation D = K̃^{×}/U^{pos}(R̃) then reads as the exact sequence:

0⟶𝔾̂_{m}⟶j_{∗}𝔾_{m}⟶i_{∗}D⟶0. (∗)

Proposition 3.6 Let R be a 1-dimensional henselian ring such that ℭ ∋ X = Spec(R) is preorientable. Let x be the closed point of X. Then for every constructible sheaf F on x_{pos} the following pairing is non degenerate for all r

Ext_{x}^{r}(F, D) × H_{pos}^{2−r}(x, F)⟶H_{pos}^{2}(x, D) ≅ ℚ/ℤ.

If X is not preorientable the above pairing is non degenerate up to p-torsion, where p is the residue characteristic of R.

Proof: By [M2] I Th 1.8. it suffices to show, that the duality maps α^{0}(H, ℤ/nℤ) and α^{1}(H, ℤ/nℤ) (loc.cit.) are isomorphisms for every open subgroup H ⊂ G and every n (resp. every n prime to p). Since the situation is the same for an open H we restrict to the case H = G. By [M2] I Lemma 1.7. these maps have the following explicit meaning

α^{0}(G, ℤ/nℤ) : _{n}D^{G}⟶^{rec}_{n}G^{ab},

α^{1}(G, ℤ/nℤ) : D_{n}^{G}⟶^{rec}G_{n}^{ab},

where _{n}A (resp. A_{n}) denotes the kernel (resp. cokernel) of the n-multiplication map on an abelian group A and rec is the map, which is induced by the reciprocity morphism rec : D → G^{ab}. (Indeed the explicit description of α^{0} in [M2] is only correct if scd(G) ≤ 2, however this is true in our situation.) We first look for α^{0}. Since D^{G} = K^{×}/U^{pos}(R) is torsion free it holds _{n}D^{G} = 0. Further G has cohomological dimension 1 (resp. cohomological l-dimension 1 for every odd l if p = 2) and hence G^{ab} is torsion free (resp. free of odd torsion if p = 2), and therefore α^{0} is an isomorphism (resp. an isomorphism up to 2-torsion if p = 2). Now look for α^{1}. We denote the absolute Galois group of of K by G. By proposition 3.5 it holds D_{n}^{G} = K^{×}/U^{pos}(R)K^{×}^{n} and the obvious compatibility of class formations implies the following commutative diagram:

K^{×}/K^{×}^{n}  ⟶^{∼} G_{n}^{ab}
[figure]
K^{×}/U^{pos}(R)K^{×}^{n} ⟶^{α^{1}} G_{n}^{ab}.

Now assume that K is preorientable. By the definition of p.r. extensions an abelian extension L of K is p.r. if and only if U^{pos}(R) is contained in the norm group N_{L/K}(L^{×}) ⊂ K^{×}. This shows that α^{1} is an isomorphism. If K is not preorientable this remains true for all n prime to p. □

Theorem 3.7 Let X be preorientable and let F be a constructible sheaf of ℤ/nℤ-modules on X. Then for every i the cup product

Ext_{X}^{i}(F; 𝔾̂_{m}) × H_{x}^{3−i}(X, F)⟶H_{x}^{3}(X, 𝔾̂_{m}) ≅ ℚ/ℤ

induces a perfect pairing of finite abelian groups. If X is not preorientable the above pairing is perfect up to p-torsion, where p is the residue characteristic of X.

Proof: In view of the exact sequence

0⟶j_{!}j^{∗}F ⟶F ⟶i_{∗}i^{∗}F ⟶0

and the five lemma it suffices to proof the theorem for sheaves of the form j_{!}F and i_{∗}F. Using proposition 3.6 the proof proceeds word-by-word in the same way as the proof of the tale local duality theorem (see [M2] II Thm. 1.8.(b)). We omit the remaining reduction steps. □

Corollary 3.8 Assume that X is orientable. Let F be a locally constant constructible sheaf on X_{pos} and let F^{D} := Hom(F, 𝔾̂_{m}). Then the canonical pairing

H_{pos}^{r}(X, F^{D}) × H_{x}^{3−r}(X, F)⟶H_{x}^{3}(X, 𝔾̂_{m}) ≅ ℚ/ℤ

is non degenerate for every r.

Proof: We use the spectral sequence

H_{pos}^{r}(X, Ext_{X}^{s}((F, 𝔾̂_{m})) ⇒ Ext_{X}^{r+s}(F, 𝔾̂_{m}).

Since F is locally constant the stalk of Ext_{X}^{s}(F, 𝔾̂_{m}) at x̃ is equal to Ext^{s}(Fx̃, U^{pos}(R̃)) (Ext as abelian groups). But U^{pos}(R̃) is divisible (see prop. 3.1). The same argument shows, that also Ext_{X}^{s}(F, 𝔾̂_{m})_{u} vanishes. Hence the above spectral sequence degenerates to a sequence of isomorphisms

H_{pos}^{r}(X, F^{D})⟶^{∼}Ext_{X}^{r}(F, 𝔾̂_{m}).

□

Remark: Let k be a local field and let M be a finite G(k̃/k)-module. Defining

H_{pr}^{i}(k, M) := ker(H^{i}(k, M)⟶H^{i}(k̃, M))

it follows from corollary 3.8 that the groups H_{pr}^{i}(k, M) and H_{pr}^{2−i}(k, Hom(M, k̄^{×})) are their exact annihilators in the Tate pairing if the local field k is orientable. Thus the groups H_{pr}^{i} play a similar role like the groups H_{f}^{i} defined by Bloch and Kato (see [BK]).

4 The multiplicative group

In the local duality theorem of the last section we made extensive use of the sheaves 𝔾̂_{m} over the spectrum of a henselian ring. These where certain subsheaves of 𝔾_{m}. However it seems to be impossible to define a subsheaf in 𝔾_{m}_{X}, having restriction 𝔾̂_{m} to every point, if infinitely many different residue characteristics occur in X. However we can define partial multiplicative groups 𝔾̂_{m,n} for every integer n > 1. For n|N there is a canonical sheaf homomorphism 𝔾̂_{m,N}⟶𝔾̂_{m,n}, which leaves invariant essential cohomological information.

Let K be a number field, X be an open subscheme of Spec(O_{K}) and n > 1 an integer. Definition: For a X ∈ ℭ 𝔾̂_{m,n}(X) = {s ∈ 𝔾_{m}(X)|s ∈ U^{pos}(R)

for every geometric point x̃ : Specm(R)⟶X, the residue characteristic of which divides n. }.

For a henselian ring R, Spec(R) ∈ ℭ and a morphism f : Spec(R)⟶X we have f^{∗}(𝔾̂_{m,n,X}) = 𝔾̂_{m,Spec(R)}, if the residue characteristic of R divides n and f^{∗}(𝔾̂_{m,n,X}) = 𝔾_{m,Spec(R)} otherwise.

Proposition 4.1 (Kummer sequence) Assume that X ∈ ℭ is orientable at n. Then for every k|n the following sequence of sheaves in Sh(X_{pos}) is exact:

0⟶μ_{k}⟶𝔾̂_{m,n}⟶^{·k}𝔾̂_{m,n}⟶0.

Proof: We verify this on stalks. For a geometric point x̃ of residue characteristic dividing n this follows from proposition 3.1. For the other points the exactness follows from the corresponding exact sequence for étale cohomology. □ Given integers n|N there is a canonical map 𝔾̂_{m,N} ⟶^{can}𝔾̂_{m,n} inducing homomorphisms for all i:

α_{i} : H_{pos}^{i}(X, 𝔾̂_{m,N}) ⊗ ℤ_{(n)}⟶H_{pos}^{i}(X, 𝔾̂_{m,n}) ⊗ ℤ_{(n)}.

Proposition 4.2 (i) α_{i} is an isomorphism for i ≥ 2.

(ii) α_{1} is surjective with divisible kernel.

(iii) If n sufficiently large, then α_{0} is an isomorphism and the kernel of α_{1} uniquely divisible.

Proof: Denoting the residue characteristic of a point x of X (i.e. x = Specm(O_{X,𝔭}^{h}) for a closed Zariski point 𝔭 of X) by char(x) we set

S_{N,n} = {x ∈ X|char(x)|N, char(x)∤n }.

Denoting the imbedding of a closed point x by i_{x} : x → X we have the exact sequence of sheaves over X_{pos}

0⟶𝔾̂_{m,N}⟶𝔾̂_{m,n}⟶ ⨁_{x∈S_{N,n}} i_{x∗}𝔾_{m}_{x}/𝔾̂_{m,x}⟶0 .

We have H_{pos}^{i}(X, i_{x∗}𝔾_{m}/𝔾̂_{m}) = H_{pos}^{i}(x, 𝔾_{m}/𝔾̂_{m}) = H_{pos}^{i}(Spec(O_{X,𝔭}^{h}), 𝔾_{m}/𝔾̂_{m}) and for (i) it suffices to show that these groups only have torsion prime to n for i ≥ 1. This however follows from the long exact sequence for the positive cohomology attached to the sequence 0 → 𝔾̂_{m} → 𝔾_{m} → 𝔾_{m}/𝔾̂_{m} → 0 together with propositions 3.2 and 3.3. For the low-dimensional cohomology groups we get the exact sequence

0 → Γ(X, 𝔾̂_{m,N}) → Γ(X, 𝔾̂_{m,n}) → ⨁_{x∈S_{N,n}} U_{x}/U_{x}^{pos} → H_{pos}^{1}(X, 𝔾̂_{m,N}) → H_{pos}^{1}(X, 𝔾̂_{m,n}) → 0.

Since the residue characteristic of an x ∈ S_{N,n} is prime to n, and U_{x}^{pos} is a direct summand in U_{x}, which contains all torsion elements, the group U_{x}/U_{x}^{pos} is uniquely n-divisible. Hence the kernel of α_{1} is divisible. By the definition of 𝔾̂_{m,n} we have Γ(X, 𝔾̂_{m,n}) = μ_{K} for sufficiently large n if either n is odd or if X does not contain a point of residue characteristic 2 (there are points, where U^{pos} = μ_{prim}). Otherwise Γ(X, 𝔾̂_{m,n}) is the group of roots of unity of odd order for sufficiently large n. Hence if n is sufficiently large and n|N then the canonical map Γ(X, 𝔾̂_{m,N}) ⊗ ℤ_{(n)} → Γ(X, 𝔾̂_{m,n}) ⊗ ℤ_{(n)} is an isomorphism, which concludes the proof. □ Thus the cohomology of 𝔾̂_{m,n} is interesting for us after tensoring with ℤ_{(n)}.

Proposition 4.3 (Cohomology of 𝔾̂_{m,n})

\begin{aligned}H_{pos}^{1}(X, 𝔾̂_{m,n}) ⊗ ℤ_{(n)} & = & ((⨁_{𝔭∈X,𝔭∤n} K_{𝔭}^{h}^{×}/U(K_{𝔭}^{h}) ⊕ ⨁_{𝔭∈X,𝔭|n} K_{𝔭}^{h}^{×}/U_{𝔭}^{pos})/K^{×}) ⊗ ℤ_{(n)} \\ H_{pos}^{2}(X, 𝔾̂_{m,n}) ⊗ ℤ_{(n)} & = & ker(Br(K) → ⨁_{𝔭∈X} Br(K_{𝔭}^{h})) ⊗ ℤ_{(n)} \\ H_{pos}^{3}(X, 𝔾̂_{m,n}) ⊗ ℤ_{(n)} & = & coker(Br(K) → ⨁_{𝔭∈X} Br(K_{𝔭}^{h})) ⊗ ℤ_{(n)} \\ H_{pos}^{i}(X, 𝔾̂_{m,n}) & = & 0 for i ≥ 4. \end{aligned}

The sums run over all closed points of X.

Proof: Going to the limit over the exact sequences for the local cohomology for smaller and smaller open subschemes of X we get the following exact sequence:

0 → Γ(X, 𝔾̂_{m,n}) → K^{×} → ⨁_{char(x)∤n} H_{x}^{1}(X_{x}^{h}, 𝔾_{m}) ⊕ ⨁_{char(x)|n} H_{x}^{1}(X_{x}^{h}, 𝔾̂_{m}) → H_{pos}^{1}(X, 𝔾̂_{m,n}) → 0.

The zero on the right hand side comes from H^{1}(K, 𝔾_{m}) = 0. The local cohomology groups have already been calculated (3.4, 3.2). Denoting the quotient field of O_{x}^{h} by K_{x} for char(x)∤n the canonical homomorphism K_{x}^{×}/U(K_{x})⟶K̃_{x}^{×}/U(K̃_{x}) becomes an isomorphism after tensoring with ℤ_{(n)}. This shows the statement for H^{1}. Since the groups H_{x}^{2}(X_{x}^{h}, 𝔾_{m}) and H_{x}^{2}(X_{x}^{h}, 𝔾̂_{m}) vanish the following sequence is exact:

0 → H_{pos}^{2}(X, 𝔾̂_{m,n}) → H^{2}(K, 𝔾_{m}) →

→ ⨁_{char(x)∤n} H_{x}^{3}(X_{x}^{h}, 𝔾_{m}) ⊕ ⨁_{char(x)|n} H_{x}^{3}(X_{x}^{h}, 𝔾̂_{m}) → H_{pos}^{3}(X, 𝔾̂_{m,n}) → 0.

This implies the statement about H^{2} and H^{3}. The higher cohomology groups vanish, standing between zeroes in the long exact sequence. □

Corollary 4.4 Assume that n is odd or that K is totally imaginary. Then

\begin{aligned}H_{pos}^{2}(Spec(O_{K}), 𝔾̂_{m,n}) ⊗ ℤ_{(n)} & = & 0 \\ H_{pos}^{3}(Spec(O_{K}), 𝔾̂_{m,n}) ⊗ ℤ_{(n)} & = & ℚ/ℤ ⊗ ℤ_{(n)}. \end{aligned}

If n is even and if K is not totally imaginary this holds up to 2-torsion.

Proof: This follows from proposition 4.3 and from the exact sequence

0⟶Br(K)⟶ ⨁_{𝔭} Br(K_{𝔭}^{h}) ⊕ ⨁_{varch.} Br(K_{v})⟶^{Σ}ℚ/ℤ⟶0.

□

Using the Kummer sequence we observe:

Corollary 4.5 If (n, d(K)) = 1 then there is a canonical isomorphism

H_{pos}^{3}(Spec(O_{K}), μ_{n})⟶^{∼}ℤ/nℤ.

5 Global duality

The aim of this section is to prove the following global duality theorem

Theorem 5.1 Let X = Spec(O_{K}), where K is a finite number field and let F be a locally constant constructible sheaf of ℤ/nℤ-modules on X_{pos}. If (n, d(K)) = 1 then the cup product

H_{pos}^{i}(X, F) × H_{pos}^{3−i}(X, Hom(F; 𝔾̂_{m,n}))⟶H_{pos}^{3}(X, 𝔾̂_{m,n}) ⊗ ℤ_{(n)}⟶^{∼}ℚ/ℤ ⊗ ℤ_{(n)}

defines a perfect pairing of finite abelian groups.

By corollary 4.5 and the Kummer sequence this theorem is equivalent to theorem 1 of the introduction. First we need a lemma, concerning the first local cohomology group. Let R be henselian with prime ideal 𝔭|p and let F be a sheaf of ℤ/nℤ-modules on Spec(R), which has the property, that F or F^{D} is locally constant. (This is equivalent only if (n, p) = 1, or if ζ_{p} ∈ R^{nr}, since in the remaining case, the sheaf μ_{n} is not locally constant.)

Lemma 5.2

H_{𝔭}^{1}(Spec(R), F) = 0.

Proof: We denote the quotient field of R by K and the field of lemma 2.3 by K̃. We first show, that H_{pos}^{0}(R, F) ≅ H^{0}(K, F). If F is locally constant let K̃/L/K be a Galois extension of K such that

F_{|R_{L}} = \underline{A}_{|R_{L}}

with a finite abelian group A, where R_{L} denotes the normalization of R in L. If F^{D} is locally constant, we choose L trivializing F^{D}. Then it holds with G = Gal(L/K):

H_{pos}^{0}(R, F) = H^{0}(G, H_{pos}^{0}(R_{L}, F)) = H^{0}(G, H^{0}(L, F)) = H^{0}(K, F).

In view of the long exact sequence for the local cohomology it remains to show, that the canonical map

H_{pos}^{1}(R, F)⟶H^{1}(K, F)

is injective. But using the result about the H^{0} for all p.r. extensions of R and lemma 2.3 this follows from the Hochschild-Serre sequence for the group extension

0⟶Gal(K̄/K̃)⟶Gal(K̄/K)⟶Gal(K̃/K)⟶0.
□

Proof of the theorem: Let S be a finite set of places of K, containing all archimedian places, all places dividing n, and which is big enough, such that the restriction of F on U_{et}, with U = X\S is locally constant in the étale topology. As usual, we denote the Galois group of the maximal extension of K, unramified outside S by G_{S}. By assumption (n, d(K)) = 1, in particular n is odd. Therefore all cohomology groups arising from archimedian places vanish and we will ignore them in the following. For a prime 𝔭 of K, we denote the henselization of O_{K} at 𝔭 by R_{𝔭} with quotient field K_{𝔭}. By proposition 2.4 we have H^{2}(K_{𝔭}, F) ≅ H_{𝔭}^{3}(R_{𝔭}, F). Now we consider the long exact sequence for the local cohomology, where the 0 on the left follows from lemma 5.2:

0 → H_{pos}^{1}(X, F) → H_{pos}^{1}(U, F) → ⨁_{𝔭∈S} H_{𝔭}^{2}(R_{𝔭}, F) → H_{pos}^{2}(X, F) → H_{pos}^{2}(U, F) → ⨁_{𝔭∈S} H_{𝔭}^{3}(R_{𝔭}, F).

By proposition 2.6 H_{pos}^{i}(U, F) = H_{et}^{i}(U, F) = H^{i}(G_{S}, F), and defining as usual

ker^{i}(F) = ker(H^{i}(G_{S}, F) → ⨁_{𝔭∈S} H^{i}(K_{𝔭}, F)),

we transform the above sequence in the following form:

0 → H_{pos}^{1}(X, F) → H^{1}(G_{S}, F) → ⨁_{𝔭∈S} H_{𝔭}^{2}(R_{𝔭}, F) → H_{pos}^{2}(X, F) → ker^{2}(F) → 0  (1)

Denoting Hom(F, 𝔾̂_{m,n}) by F^{D}, the duality theorem of Tate-Poitou implies the following exact sequence:

0 → ker^{1}(F) → H^{1}(G_{S}, F) → ⨁_{𝔭∈S} H^{1}(K_{𝔭}, F) → H^{1}(G_{S}, F^{D})^{∗} → ker^{2}(F) → 0.  (2)

We know that

H_{𝔭}^{2}(R_{𝔭}, F) = coker(H_{pos}^{1}(R_{𝔭}, F) ⟶ H^{1}(K_{𝔭}, F)),  (3)

and we use this to connect the sequences 1 and 2 in the following commutative diagram

ker^{1}(F) ↪ H^{1}(G_{S}, F) → ⨁_{𝔭∈S} H^{1}(K_{𝔭}, F) → H^{1}(G_{S}, F^{D})^{∗} ↠ ker^{2}(F)
[figure: ∩; β; ∥; α; ↓^{γ}; ∥]
H_{pos}^{1}(X, F) ↪ H^{1}(G_{S}, F) → ⨁_{𝔭∈S} H_{𝔭}^{2}(R_{𝔭}, F) → H_{pos}^{2}(X, F) ↠ ker^{2}(F) .

Explanations: The map α comes from (3) and is clearly surjective. Hence β exists and is injective. Using the cup product

H_{pos}^{1}(X, F^{D}) × H_{pos}^{2}(X, F)⟶H_{pos}^{3}(X, 𝔾̂_{m,n})⟶ℚ/ℤ ⊗ ℤ_{(n)}

γ is the composite map

H^{1}(G_{S}, F^{D})^{∗}⟶^{∼}H_{pos}^{1}(U, F^{D})^{∗}⟶H_{pos}^{1}(X, F^{D})^{∗}⟶H_{pos}^{2}(X, F).

The snake lemma implies the exact sequence

0 → ker^{1}(F) → H_{pos}^{1}(X, F) → ⨁_{𝔭∈S} H_{pos}^{1}(R_{𝔭}, F) → H^{1}(G_{S}, F^{D})^{∗} → H_{pos}^{2}(X, F) → 0. (4)

Comparing (4) with the dualized sequence (2) with F^{D} instead of F, we get the following commutative diagram with exact rows:

ker^{1}(F) ↪ H_{pos}^{1}(X, F) → ⨁_{𝔭∈S} H_{pos}^{1}(R_{𝔭}, F) → H^{1}(G_{S}, F^{D})^{∗} ↠ H_{pos}^{2}(X, F)
↓^{a}  ↓^{b}  ↓^{c}  ∥  ↓^{d}
ker^{2}(F^{D})^{∗} ↪ H_{pos}^{2}(X, F^{D})^{∗} → ⨁_{𝔭∈S} H_{𝔭}^{2}(R_{𝔭}, F^{D})^{∗} → H^{1}(G_{S}, F^{D})^{∗} ↠ H_{pos}^{1}(X, F^{D})^{∗}.

The maps b,c and d are induced by the cup product, c is an isomorphism by the local duality theorem. The map a is the well known duality map between the ker^{1} and ker^{2}. The commutativity of the left hand square follows, from the fact, that also a comes from a cup product pairing: Denoting the S-idele class group of K_{S} by C_{S} we have the following commutative diagram, where the commutativity of the lower part follows from [Ha] 5.1,5.2.

H_{pos}^{1}(X, F)  × H_{pos}^{2}(X, F^{D}) ⟶^{∪} H_{pos}^{3}(X, 𝔾̂_{m,n}) ⊗ ℤ_{(n)}
[figure: ∪; ∥]
ker^{1}(F)  × ker^{2}(F^{D}) ⟶  ℚ/ℤ ⊗ ℤ_{(n)}
[figure: ∩; ∥]
H^{0}(G_{S}, Hom(F^{D}, C_{S})) × H^{2}(G_{S}, F^{D}) ⟶^{∪}  H^{2}(G_{S}, C_{S})(n) .

We could also conclude the commutativity of the left hand square from the well-known comparison between the duality theorems of Tate-Poitou and Artin-Verdier. Therefore the five lemma implies the global duality theorem for i = 1, 2. It holds H_{pos}^{i}(U, F) = H_{et}^{i}(U, F) = 0 for i > 2, because cd_{l}(G_{S}) ≤ 2 for all l|n. By proposition 2.4 we get H_{𝔭}^{i}(R_{𝔭}, F) = 0 for i > 3. Hence there are no cohomology groups in dimension greater than three and it remains to proof the theorem for i = 0, 3. The exact sequence for the local cohomology

H_{pos}^{2}(U, F)⟶ ⨁_{𝔭∈S} H_{𝔭}^{3}(R_{𝔭}, F)⟶H_{pos}^{3}(X, F)⟶0

together with the Tate-Poitou theorem implies:

H_{pos}^{0}(U, F^{D})^{∗}⟶^{∼}H_{pos}^{3}(X, F).

But using lemma 5.2 for F^{D} it follows

H_{pos}^{0}(X, F^{D})⟶^{∼}H_{pos}^{0}(U, F^{D})

which implies the result for i = 0. The same arguments applied on F^{D} eventually show the case i = 3 and the proof of the theorem is complete. □

6 P̂ic and p-adically metrized line bundles

Throughout this section K will be a finite number field and X = Spec(O_{K}). We make the convention that an extension of number fields is p.r. at a place 𝔭|∞ iff it is unramified at 𝔭. For such a 𝔭|∞ we write

K_{𝔭}^{h} =^{df} K_{𝔭} ∈ {ℝ, ℂ} and U_{𝔭}^{pos} =^{df} {x ∈ K_{𝔭}^{×} | x̄ = x^{−1}} = {x ∈ K_{𝔭}^{×} | ||x|| = 1}.

By K^{pr} we denote the maximal (everywhere) positively ramified extension of K. If K is totally imaginary then: Gal(K^{pr}/K) ≅ π_{1}^{pos}(X).

The local definition of 𝔾̂_{m} suggests the following :

Definition:

D̂iv(X) = ⨁_{all 𝔭} K_{𝔭}^{h}^{×}/U_{𝔭}^{pos}

P̂ic(X) = D̂iv(X)/(principal divisors),

where a principal divisor is given by the image of an element of K^{×} in D̂iv(X). As usual we denote the idele class group of K by C_{K}. The ramification theory of local and global class field theory implies that the composition of the canonical homomorphisms

⨁_{all 𝔭} K_{𝔭}^{h}^{×}⟶C_{K} {− \atop ↠} Gal(K̄/K)^{ab} {− \atop ↠} Gal(K^{pr}/K)^{ab}

gives rise to a canonical reciprocity homomorphism

rec : P̂ic(X)⟶Gal(K^{pr}/K)^{ab}.

Proposition 6.1 For every n the group π_{1}^{pos}(X)^{ab}/n is finite. If n is odd or K is totally imaginary then the induced homomorphism

rec_{n} : P̂ic(X)/n⟶π_{1}^{pos}(X)^{ab}/n

is surjective.

Proof: By definition a p.r. extension of degree n is unramified outside the divisors of n. Hence there are only finitely many of them, which proves the first statement. If n is odd or if K is totally imaginary then Gal(K^{pr}/K)^{ab}/n ≅ π_{1}^{pos}(X)^{ab}/n and the kernel of the reciprocity map rec : C_{K} {− \atop ↠} π_{1}^{pos}(X)^{ab}/n contains the subgroup

(∏_{𝔭∤n} U_{𝔭} × ∏_{𝔭|n} U_{𝔭}^{pos} × ∏_{𝔭|∞} K_{𝔭}^{×})/K^{×} ⊂ C_{K}.

This shows the statement of the proposition. □ The following proposition follows from proposition 4.3 by a straightforward calculation.

Proposition 6.2 Assume that n is odd. For every N with n|N there is an exact sequence

0 → A_{n,N} → P̂ic(X) ⊗ ℤ_{(n)} → H_{pos}^{1}(X, 𝔾̂_{m,N}) ⊗ ℤ_{(n)} → 0,

where the abelian group A_{n,N} is divisible. For sufficiently large N the group A_{n,N} is uniquely divisible.

Corollary 6.3 For (n, d(K)) = 1 the canonical homomorphism

rec_{n} : P̂ic(X)/n⟶π_{1}^{pos}(X)^{ab}/n

is an isomorphism.

Proof: Using the Kummer sequence for μ_{n} and 𝔾̂_{m,N} for sufficiently large N by corollary 4.4 we get an isomorphism

P̂ic(X)/n ≅ H_{pos}^{2}(X, μ_{n}).

The statement of the corollary then follows from the commutative diagram

P̂ic(X)/n ⟶^{rec_{n}} π_{1}^{pos}(X)^{ab}/n
↓^{≀}  ↓^{≀}
H_{pos}^{2}(X, μ_{n}) ⟶^{∼} H_{pos}^{1}(X, ℤ/nℤ)^{∗}.
□

For a prime number p we denote the p-torsion subgroup of an abelian group A by A(p).

Corollary 6.4 If p is a prime number with (p, d(K)) = 1 then there is a canonical isomorphism

H_{pos}^{1}(X, μ_{p^{∞}}) ≅ P̂ic(X)(p).

Proof: Using proposition 6.2 this follows from the Kummer sequence associated to 𝔾̂_{m,N} and p^{n} for N sufficiently large going to the limit over n. □ Remarks: 1. Analogously to the function field case (see [Taka]) one can show that if ζ_{p} ∉ K and (p, d(K)) = 1 then the Picard group gives rise to a class formation for p.r. p-extensions of K.

2. The compositum K_{∞} of the cyclotomic ℤ_{p}-extensions (all p) of K is p.r. over K. One can define a degree map:

deg : P̂ic(X)⟶Gal(K_{∞}/K) × ℝ_{>0}^{×} ≅ ℤ̂ × ℝ^{+}

in the following way: The first component of deg is induced by the reciprocity map: P̂ic(X) → Gal(K^{pr}/K)^{ab}↠Gal(K_{∞}/K) and the second is induced by the usual idele norm. It is easily verified that deg has dense image and we denote its kernel by P̂ic^{0}(X). Assume that μ_{n} ⊂ K and (n, d(K)) = 1. Then the global duality theorem together with proposition 6.2 implies the existence of a perfect pairing: _{n}P̂ic^{0}(X) × P̂ic^{0}(X)/n ⟶μ_{n}.

Now we fix an odd prime number p and assume that k/ℚ_{p} is an orientable local field. We will write the action of the local complex conjugation ρ on k by ρ(x) = x̄. Under a p-adic Hermitian metric on a finite dimensional k-vector space V we understand a non-degenerate Hermitian form < , >: V × V ⟶k, i.e. the usual axioms hold with ℂ replaced by k. (However there is no notion of positive definiteness in the p-adic context.) Let K be a number field with (p, d(K)) = 1. We denote the ring of p-integers of K by O_{S_{p}}. Definition: A p-adically metrized O_{S_{p}}-module is a finitely generated O_{S_{p}}-module M equipped with a p-adic Hermitian metric on

M_{𝔭} := M ⊗_{O_{Sp}} K_{𝔭}

for every place 𝔭|p of K. As in the case of archimedian metrized modules (cf. [N1] III §4) direct sums, tensor products etc. of metrized modules are metrized in a canonical way. A fractional O_{S_{p}}-ideal 𝔞 ⊂ K is always equipped with the trivial metric which is given by K_{𝔭} × K_{𝔭}⟶K_{𝔭}; < α, β >= αβ̄ for every 𝔭|p of K and every other metric on 𝔞 is given by a family (ϵ_{𝔭} ∈ (K_{𝔭}^{+})^{×})_{𝔭|p} and the rule < α, β >_{𝔭}= ϵ_{𝔭}αβ̄ ∈ K_{𝔭}. An element a ∈ K^{×} defines the metrized principal ideal [a] which is given by the principal O_{S_{p}}- fractional ideal (a) ⊂ K together with the metric given by the family (ϵ_{𝔭} ∈ (K_{𝔭}^{+})^{×})_{𝔭|p} with ϵ_{𝔭} = aā ∈ (K_{𝔭}^{+})^{×} for all 𝔭|p. We define the group Metr(p, X) of isomorphy classes of p-adically metrized line bundles on X to be the group of isometry classes of p-adically metrized invertible O_{S_{p}}-modules under the operation tensor product. Using the same arguments like in the case of archimedian metrics (cf. [N1]III§4) one observes that every p-adically metrized line bundle on X is isometric to a metrized O_{S_{p}}-fractional ideal and two of those are isometric if and only if they differ by a metrized principal ideal [a] for some a ∈ K^{×}.

Corollary 6.5 The homomorphism

ϕ : K^{×}⟶ ⨁_{𝔭finite^{𝔭∤p}} K_{𝔭}^{×}/U_{𝔭} ⊕ ⨁_{𝔭|p}(K_{𝔭}^{+})^{×}; a ↦ \{ \begin{matrix}a mod U_{𝔭} ∈ K_{𝔭}^{×}/U_{𝔭} & for & 𝔭∤p \\ aā ∈ (K_{𝔭}^{+})^{×} & for & 𝔭|p \end{matrix}

gives rise to an isomorphism

( ⨁_{𝔭finite^{𝔭∤p}} K_{𝔭}^{×}/U_{𝔭} ⊕ ⨁_{𝔭|p}(K_{𝔭}^{+})^{×})/im(ϕ)⟶^{∼}Metr(p, X).

Theorem 6.6 (i) There is a canonical injective homomorphism

H_{pos}^{1}(X, 𝔾̂_{m,p}) ⊗ ℤ_{(p)} ⟶ Metr(p, X) ⊗ ℤ_{(p)}

with uniquely divisible cokernel.

(ii) There is a canonical homomorphism

Metr(p, X) {− \atop ↠}^{rec} π_{1}^{pos}(X)(p)^{ab},

which is surjective with p-divisible kernel. Thus rec induces an isomorphism on the pro-p completion.

Proof: Part (i) easily follows from proposition 4.3 and corollary 6.5. Using H_{pos}^{2}(X, 𝔾̂_{m,p}) ⊗ ℤ_{(p)} = 0 (cor. 4.4) and the Kummer sequence for p^{r} (r = 1, 2….) we get the following isomorphisms:

Metr(p, X)/p^{r} ≅ H_{pos}^{1}(X, 𝔾̂_{m,p})/p^{r}
≅ H_{pos}^{2}(X, μ_{p^{r}})
≅ H_{pos}^{1}(X, ℤ/p^{r}ℤ)^{∗} (global duality)
≅ π_{1}^{pos}(X)(p)^{ab}/p^{r} .

This implies the following isomorphism in the projective limit

lim ; ⟵_{r}Metr(p, X)/p^{r} ≅ π_{1}^{pos}(X)(p)^{ab}.

Eventually by corollary 6.5 and by the finiteness of the class number one verifies that the canonical homomorphism

Metr(p, X)⟶lim ; ⟵_{r}Metr(p, X)/p^{r}

is surjective. In order to show that the kernel of rec is p-divisible by the snake lemma it suffices to show, that the induced homomorphism: rec : _{p}Metr(p, X)⟶^{rec}_{p}π_{1}^{pos}(X)^{ab} is surjective. The Kummer sequence together with (i) implies a surjection: H_{pos}^{1}(X, μ_{p}) {− \atop ↠} _{p}Metr(p, X) and we obviously have the injection: H_{pos}^{1}(X, ℚ_{p}/ℤ_{p})/p ⟶ H_{pos}^{2}(X, ℤ/pℤ). These fit together in the commutative diagram

H_{pos}^{1}(X, μ_{p}) {− \atop ↠} _{p}Metr(p, X)
↓^{≀}  ↓^{rec}
H_{pos}^{2}(X, ℤ/pℤ)^{∗} {− \atop ↠} _{p}π_{1}^{pos}(X)^{ab},

where the left hand vertical arrow is induced by the global duality theorem. This concludes the proof. □ Remark: If K is an abelian number field then the Leopoldt conjecture is true for K and p by the theorem of Brumer/Baker and a straightforward calculation using corollary 6.5 implies that the p-torsion subgroup Metr(p, X)(p) of Metr(p, X) is finite and hence the kernel of rec is uniquely p-divisible. Definition: Assume (p, d(K)) = 1. Then the p-genus g_{p}(K) (or g_{p}(X)) is defined by

g_{p}(K) = dim_{𝔽_{p}}Metr(p, X)(p)/p.

Proposition 6.7 The p-genus g_{p}(K) is finite and it holds:

g_{p}(K) = dim_{𝔽_{p}} P̂ic(X)(p)/p.

Proof: The first statement follows easily from part (ii) of theorem 6.6 and from the fact that π_{1}^{pos}(X)(p) is a finitely generated pro-p group. The second statement follows from proposition 6.2 for n = N = p and from part (i) of theorem 6.6. □

7 Fundamental groups

Let p be an odd prime number and let K be a number field with (p, d(K)) = 1. We write X = Spec(O_{K})_{pos}. The set of primes of K dividing p will be denoted by S_{p}. By the definition of p.r. covers a Galois cover of p-power degree of X\S_{p} is étale. Therefore there is a canonical surjection

G_{S_{p}}(K)(p)⟶^{∼}π_{1}^{pos}(X\S_{p})(p) {− \atop ↠} π_{1}^{pos}(X)(p).

We will investigate the structure of π_{1}^{pos}(X)(p). All results in this section are in direct analogy to the corresponding results in the function field case (see [Taka]). First we are interested in the relation between the cohomology of the fundamental group and the sheaf cohomology. Every discrete π_{1}^{pos}(X)(p)-module M defines a sheaf on X_{pos} which we will also denote by M. From the Hochschild-Serre spectral sequence we get canonical homomorphisms:

ϕ_{M,i} : H^{i}(π_{1}^{pos}(X)(p), M)⟶H_{pos}^{i}(X, M)

for all i. If M is a finite p-primary abelian group with trivial action then this map is an isomorphism for i = 1. We will say that

X is a K(π, 1) for p

if ϕ_{M,i} is an isomorphism for all i and every discrete finite p-primary π_{1}^{pos}(X)(p)-module M. This is equivalent to the fact that the higher homotopy groups π_{i} of the p-completion of the positive homotopy type of X vanish for i ≥ 2. First we establish a sufficient condition for X being a K(π, 1) for p. By definition to every closed normal subgroup N of G = π_{1}^{pos}(X)(p) it corresponds a pro-scheme X(N)/X which is a p.r. Galois pro-covering of X with Galois group G/N. It holds: π_{1}^{pos}(X(N))(p) = N. The universal pro-p covering X({1}) of X will also be denoted by X̃^{p}.

Proposition 7.1 Assume that there is a closed normal subgroup N in π_{1}^{pos}(X)(p) such that

H_{pos}^{2}(X(N), ℤ/pℤ) = 0.

Then X is a K(π, 1) for p.

Proof: Writing G for π_{1}^{pos}(X)(p) we use the Hochschild-Serre spectral sequence

E_{pq}^{2} = H^{p}(G, H_{pos}^{q}(X̃^{p}, M_{|X̃^{p}}) ⇒ H_{pos}^{p+q}(X, M).

Evidently it suffices to show, that

H_{pos}^{q}(X̃^{p}, M_{|X̃^{p}}) = lim ; ⟶_{Y |X}H_{pos}^{q}(Y, M_{|Y}) = 0 for q > 0,

where Y runs over all finite connected Galois p.r. p-coverings of X. Considering sufficiently large coverings we can assume M_{|Y} to be constant and by an easy reduction it suffices to deal the case M = ℤ/pℤ. The result is trivial for q = 1 because there are no p-power Galois coverings of X̃^{p} by definition. For q = 3 the statement follows from the compatibility

H_{pos}^{3}(Y, ℤ/pℤ) ≅ H_{pos}^{0}(Y, μ_{p})^{∗}
↓^{can}  ↓^{[K(Y′):K(Y)]}
H_{pos}^{3}(Y′, ℤ/pℤ) ≅ H_{pos}^{0}(Y′, μ_{p})^{∗}

and from the fact, that the order of G is divisible by p^{∞}. Therefore the Hochschild-Serre spectral sequence yields an injection:

H^{2}(G, ℤ/pℤ) ⟶ H_{pos}^{2}(X, ℤ/pℤ)

and by the same argument for every closed normal subgroup N in G:

H^{2}(N, ℤ/pℤ) ⟶ H_{pos}^{2}(X(N), ℤ/pℤ).

For a N as in the proposition we get H^{2}(N, ℤ/pℤ) = 0 hence N is a free pro-p group. Therefore in the Hochschild-Serre spectral sequence

H^{i}(N, H_{pos}^{j}(X̃^{p}, ℤ/pℤ)) ⇒ H_{pos}^{i+j}(X(N), ℤ/pℤ)

the spectral term E_{2}^{ij} is trivial unless i ∈ {0, 1} and j ∈ {0, 2}. Hence the spectral sequence degenerates in E_{2} and we get the isomorphism:

H_{pos}^{2}(X̃^{p}, ℤ/pℤ)^{N} ≅ H_{pos}^{2}(X(N), ℤ/pℤ) = 0.

But N is a pro-p group and therefore H_{pos}^{2}(X̃^{p}, ℤ/pℤ) = 0. □

Proposition 7.2 Assume that (p, d(K)) = 1 and that X = Spec(O_{K}) is a K(π, 1) for p. Then the following holds.

(i) If ζ_{p} ∈ K the group π_{1}^{pos}(X)(p) is a Poincaré-group of dimension 3 with dualizing module μ_{p^{∞}}.

(ii) If ζ_{p} ∉ K the group π_{1}^{pos}(X)(p) is a duality group of dimension 2 with dualizing module P̂ic(X̃^{p})(p) if and only if P̂ic(X̃^{p}) is p-divisible.

Proof: Part (i) follows from the global duality theorem together with the property K(π, 1) for p. Assume that ζ_{p} ∉ K. Then it holds (writing G for π_{1}^{pos}(X)(p)):

H^{3}(G, ℤ/pℤ) ≅ H_{pos}^{3}(X, ℤ/pℤ) ≅ H_{pos}^{0}(X, μ_{p}) = 0

Hence cd_{p}(G) ≤ 2. Further we know that the cohomology of G with values in finite p-primary discrete modules is finite as it equals the positive cohomology. For the dualizing module I_{2}(G) of G it holds.

I_{2}(G) = lim ; ⟶_{U,n}H^{2}(U, ℤ/p^{n}ℤ)^{∗} = H_{pos}^{1}(X̃^{p}, μ_{p^{∞}}) = P̂ic(X̃^{p})(p) (corollary 6.4).

G is a duality group if and only if the following holds for i = 0, 1. (see [Tate],[Ver]):

D_{i}(G, ℤ/pℤ) = lim ; ⟶_{U}H^{i}(U, ℤ/pℤ)^{∗} = 0,

where the limit runs over all open subgroups and the maps are the duals of the corestriction maps. The order of G is infinite as the cyclotomic ℤ_{p}-extension is p.r. in all cases and hence D_{0}(G) = 0. Further it holds:

D_{1}(G, ℤ/pℤ) = lim ; ⟶_{Y/X}H_{pos}^{2}(Y, μ_{p}),

where the limit runs over all finite p-power Galois coverings of X. Hence D_{1}(G, ℤ/pℤ) = H_{pos}^{2}(X̃^{p}, μ_{p}) = P̂ic(X̃^{p})/p. This shows (ii) □ In the geometric case a complete curve is a K(π, 1) for a prime number p if there is a non trivial rational p-division point on the Jacobian or if ζ_{p} is not contained in the ground field (cf.[Taka]). We conjecture that the same is true for Spec(O_{K})_{pos} and we can prove this if K is totally p-real or p-CM: Definition: Let K be a number field an let p be a prime number with (p, d(K)) = 1. We say that

(i) K is totally p-real if for every prime 𝔭 dividing p of K it holds: K_{𝔭} = K_{𝔭}^{+}, (ii) K is totally p-imaginary if [K_{𝔭} : K_{𝔭}^{+}] = 2 for every prime 𝔭 of K dividing p,

(iii) K is a p-CM field if it is totally p-imaginary and if it contains a totally p-real subfield K^{+,p} of index 2. By r_{p}(K) we denote the number of complex places of K^{+,p}.

Remark: If K is an abelian number field and (p, d(K)) = 1 then K is either totally p-real or p-CM. If K contains a primitive p^{th} root of unity then K is p-CM. Examples: a) K = ℚ(ζ_{p}) is p-CM with maximal totally p-real subfield K^{+,p} = ℚ(ζ_{p} + ζ_{p}^{−1}). It holds r_{p}(K) = 0. K is totally q-real for every odd prime number q ≠ p.

b) If p, q are odd prime numbers then K = ℚ(ζ_{pq}) is p-CM with maximal totally p-real subfield field K^{+,p} = ℚ(ζ_{p} + ζ_{p}^{−1}, ζ_{q}). It holds r_{p}(K) = \frac{1}{2}(p − 1)(q − 1).

c) A number field K which is of CM-type in the usual sense (which should be called ∞-CM in our context) may or may not be p-CM and vice versa. If K is both CM and p-CM the (+)-fields may or may not be the same.

Theorem 7.3 Let K be p-CM and assume that K contains a primitive p^{th} root of unity. Then the following holds.

(i) If g_{p}(K) > 0 then X = Spec(O_{K}) is a K(π, 1) for p, hence π_{1}^{pos}(X)(p) is a Poincaré-group of dimension 3 with dualizing module μ_{p^{∞}}.

(ii) If g_{p}(K) = 0 then π_{1}^{pos}(X)(p) is a free pro-p group of rank r_{p}(K) + 1.

Proof: We denote π_{1}^{pos}(X)(p) by G and we write T or_{p}(H) for the subgroup of p-torsion elements in an abelian profinite group H. By theorem 6.6(ii) T or_{p}(G^{ab}) = 0 if and only if g_{p}(X) = 0. Decomposing G^{ab} into eigenspaces with respect to the involution ρ with < ρ >= ℤ/2ℤ = Gal(K/K^{+,p}) we get by the definition of p.r. extensions:

G^{ab} = (G^{ab})^{+} ⊕ (G^{ab})^{−} = G_{S_{p}}(K^{+,p})(p)^{ab} ⊕ Cl_{S_{p}}(K)(p)^{−},

where Cl_{S_{p}}(K) denotes the S_{p}-ideal class group of K. Hence if g_{p}(K) = 0 then Cl_{S_{p}}(K)(p)^{−} = 0, which implies by the results of [W2] that G_{S_{p}}(K^{+,p})(p) is a free pro-p group of rank r_{2}(K^{+,p}) + 1 = r_{p}(K) + 1. We will show that this quotient of G is indeed the full group. Let L be a finite extension of K in K_{S_{p}}^{+,p}(p)K. Then also L is p-CM and G_{S_{p}}(L^{+,p})(p) is a free pro-p group of rank r_{p}(L) + 1 as a closed subgroup of index [L : K] in the free pro-p group G_{S_{p}}(K^{+,p})(p). Hence by [W2] it holds Cl_{S_{p}}(L)(p)^{−} = 0. Denoting the pro-scheme corresponding to the field K_{S_{p}}^{+,p}(p)K by Y we see that Y is a pro-p Galois covering of X and

π_{1}^{pos}(Y)(p)^{ab} = lim ; ⟵_{L⊂K_{Sp}^{+,p}(p)K}Cl_{S_{p}}(L)(p) = 0.

Hence Y = X̃^{p} which proves (ii). Now assume g_{p}(K) ≠ 0. Then Cl_{S_{p}}(K)(p)^{−} ≠ 0 because otherwise G_{S_{p}}(K^{+,p})(p) would be a free pro-p group by [W2] and hence T or_{p}(G^{ab}) = 0 contradicting g_{p}(K) ≠ 0. Let S be the set of primes of K dividing p or ∞. Since Cl_{S_{p}}(K)(p)^{−} ≠ 0 the S-idele class group C_{S}(K_{S_{p}}^{+,p})(p) is p-divisible by [W1],[W2]. Let as above Y denote the pro-scheme corresponding to the field K_{S_{p}}^{+,p}(p)K. We will show

H_{pos}^{2}(Y, ℤ/pℤ) = 0,

which proves the theorem by propositions 7.1 and 7.2. The pro-scheme Y corresponds to a closed normal subgroup N in G. Global duality yields:

H_{pos}^{2}(Y, ℤ/pℤ)(1) = H_{pos}^{2}(Y, μ_{p}) ≅ lim ; ⟶_{U}H_{pos}^{1}(X(U), ℤ/pℤ)^{∗},

where U runs through all open normal subgroups in G containing N. Decomposing H_{pos}^{2}(Y, ℤ/pℤ) with respect to ρ into eigenspaces and making a twist by (1) we get:

H_{pos}^{2}(Y, ℤ/pℤ)^{+}(1) = H_{pos}^{2}(Y, μ_{p})^{−} ≅ lim ; ⟶_{L⊂K_{Sp}^{+,p}(p)K}Cl_{S_{p}}(L)^{−}/p = Cl_{S_{p}}(K_{S_{p}}^{+,p}(p)K)^{−}/p.

By a result of Neumann ([Neu] §4): Cl_{S_{p}}(K_{S_{p}}^{+,p}(p)K)(p) = 0 which proves H_{pos}^{2}(Y, ℤ/pℤ)^{+} = 0. Further we have for the (−)-part:

H_{pos}^{2}(Y, ℤ/pℤ)^{−}(1) = H_{pos}^{2}(Y, μ_{p})^{+} = lim ; ⟶_{L⊂K_{Sp}^{+,p}(p)K}G_{S_{p}}(L^{+,p})(p)^{ab}/p ≅ C_{S}(K_{S_{p}}^{+,p})/p = 0.

This proves the theorem. □

Theorem 7.4 Assume that K is totally p-real. Then X is a K(π, 1) for p and the following conditions are equivalent.

(i) P̂ic(X̃^{p}) is not p-divisible.

(ii) π_{1}^{pos}(X)(p) is a free pro-p group of rank r_{2}(K) + 1.

Furthermore if (i) resp. (ii) is false then π_{1}^{pos}(X)(p) is a duality group of dimension 2 with dualizing module P̂ic(X̃^{p})(p). If (i) resp. (ii) is true then g_{p}(K) = 0. The converse is true if the Leopoldt conjecture is true for K and p.

Proof: If K is totally p-real and 𝔭|p then all p-extensions of K_{𝔭} are p.r. Therefore it holds π_{1}^{pos}(X)(p) = G_{S_{p}}(K)(p) and H_{𝔭}^{i}(X, ℤ/pℤ) = 0 for all 𝔭|p and for all i. Using excision and proposition 2.6 we conclude that

H_{pos}^{i}(X, ℤ/pℤ) = H_{pos}^{i}(X\S_{p}, ℤ/pℤ)
= H_{et}^{i}(X\S_{p}, ℤ/pℤ)  ( by prop. 2.6)
= H^{i}(G_{S_{p}}(K), ℤ/pℤ)
= H^{i}(G_{S_{p}}(K)(p), ℤ/pℤ) (by [Neu] Cor.1)
= H^{i}(π_{1}^{pos}(X)(p), ℤ/pℤ).

All Galois p-coverings of X are also totally p-real hence going to the limit we obtain:

H_{pos}^{i}(X̃^{p}, ℤ/pℤ) = 0 for i ≥ 1.

Hence X is a K(π, 1) for p. Further it holds:

P̂ic(X̃^{p})/p = H_{pos}^{2}(X̃^{p}, μ_{p}) = lim ; ⟶_{K⊂L⊂K_{Sp} (p)}C_{S}(L)/p = C_{S}(K_{S_{p}}(p))/p.

The equivalence of (i)-(ii) then follows from the results of [W1],[W2]. Further (ii) implies g_{p}(K) = 0 by theorem 6.6. If we assume the Leopoldt conjecture for K and p then g_{p}(K) = 0 implies G_{S_{p}}(K)(p)^{ab} ≅ ℤ_{p}^{r_{2}(K)+1}. Again by the Leopoldt-conjecture it holds H^{2}(G_{S_{p}}(p), ℚ_{p}/ℤ_{p}) = 0 which implies (ii). □ Remark: In the remaining case when K is p-CM but ζ_{p} ∉ K we can so far only deal the case when r_{p}(K) = 0. Assuming the vanishing of the Iwasawa μ-invariant for the cyclotomic ℤ_{p}-extension (proved for abelian number fields) the result is the same as in theorem 7.4.

8 Riemann existence theorem

Returning to our starting problem we are going to determine the kernel of the canonical surjection

G_{S}(K)(p) {− \atop ↠} π_{1}^{pos}(X)(p),

where S is a finite set of places of K containing all primes dividing p. It will turn out that it is isomorphic to the free pro-p product of pro-p inertia groups of the points over S in X̃^{p}. First we need some properties of pro-p groups with an action of a finite group of order prime to p. Let G be a pro-p group with an action of a finite group A of order prime to p. (Usually such an action arises from an exact sequence 1 → G → G → A → 1 using a fixed section A → G.) For a pro-p group P we denote the maximal elementary p-abelian quotient by P̄. The 𝔽_{p}-vector space Ḡ carries the structure of a compact 𝔽_{p}[A]-module.

Proposition 8.1 Let V ⊂ Ḡ be a closed 𝔽_{p}[A] submodule. Then there is an A-invariant closed subgroup H ⊂ G such that the canonical homomorphism H̄ → Ḡ induces a 𝔽_{p}[A] isomorphism

H̄ →^{∼} V.

Proof: If G is finitely generated then this was proved by Boston, [Bo] prop. 2.3. In order to extend the result to non finitely generated groups one uses a decomposition

V ≅ ∏_{i∈I} W_{i},

into finite simple 𝔽_{p}[A]-modules W_{i} (cf. [CR] 3.12-3.15). We omit the details. □ Now assume that p ≠ 2 and A ≅ ℤ/2ℤ. There is an exact sequence:

1⟶G^{−}⟶G⟶G^{+}⟶1,

where G^{+} is the largest quotient of G on which A acts trivial and G^{−} is defined by this sequence.

Corollary 8.2 If G is a free pro-p group with an action of an involution then there are closed subgroups G(+), G(−) in G inducing a decomposition:

G ≅ G(+) ⋆ G(−).

The subgroup G(−) generates G^{−} as a normal subgroup and G(+) maps isomorphically onto G^{+} via the canonical map G(+) ⟶ G {− \atop ↠} G^{+}.

Proof: We decompose Ḡ via the action of A as Ḡ = Ḡ^{+} × Ḡ^{−}. By the last proposition there are closed subgroups G(+) and G(−) inducing isomorphisms \overline{G(+)}⟶^{∼}Ḡ^{+} and \overline{G(−)}⟶^{∼}Ḡ^{−}. Therefore we get a surjection:

G(+) ⋆ G(−) {− \atop ↠}^{π} G

and denoting the kernel of π by H the Hochschild-Serre sequence induces the exact sequence:

0⟶H̄_{G}⟶\overline{G(+)} × \overline{G(−)}⟶^{∼}Ḡ⟶0.

Hence H = 0 since G is a pro-p group. Therefore G is the free product of G(+) and G(−) which immediately also implies the remaining statements. □ Let as before K̃ be the quotient field of X̃^{p}. Assume that 𝔭 is a prime of K̃ and we denote its restriction to K by the same letter. We denote the maximal p-extension of the local field K_{𝔭} by K_{𝔭}(p). The group Gal(K_{𝔭}(p)/(K̃)_{𝔭}) is a free pro-p group because the degree of (K̃)_{𝔭} over K_{𝔭} is infinite. If 𝔭 is not a divisor of p then Gal(K_{𝔭}(p)/(K̃)_{𝔭}) is the classical inertia group of 𝔭 in the extension K_{𝔭}(p)/K_{𝔭} and we denote it by T_{𝔭}. It is isomorphic to ℤ_{p} if ζ_{p} ∈ K_{𝔭} and trivial otherwise.

Assume 𝔭|p. Then by corollary 8.2 the local complex conjugation induces a free product decomposition:

Gal(K_{𝔭}(p)/(K̃)_{𝔭}) ≅ Gal(K_{𝔭}(p)/(K̃)_{𝔭})(+) ⋆ Gal(K_{𝔭}(p)/(K̃)_{𝔭})(−).

Definition: We denote Gal(K_{𝔭}(p)/(K̃)_{𝔭})(−) by T_{𝔭}(−) and we call it the positive inertia group of the prime 𝔭 ∈ S_{p}(K̃), depending on the above decomposition. For the following we remind the reader of the definition of the constructible topology (see [EGA1] I§7.2.11)). For a (commutative) ring A the subsets f^{∗}(Spec(B)) ⊂ Spec(A), where f : A → B is a ring homomorphism, satisfy the axioms for closed sets in a topological space. The associated topology is the constructible topology. Spec(A) with the constructible topology is a Hausdorff, compact and totally disconnected topological space. Assume that K is a finite number field, S ⊂ Spec(O_{K}) is a set of (finite) primes and assume that L/K is a not necessary finite extension. Then the set S(L) ⊂ Spec(O_{L}) of primes of L lying over S is closed in the constructible topology. Whenever we will write a free product of groups indexed over S(L) then this will mean the free pro-p product of a locally constant family of pro-p groups over the Hausdorff, compact, totally disconnected and pointed space (S(L) ∪ {η}, η), where η denotes the generic point of Spec(O_{L}). For the definition of these free products see Gildenhuys and Ribes [GR1]. If the constructible topology on S(L) is discrete (which is always the case if L/ℚ is finite or if L is contained in a cyclotomic extension of a finite number field) then the free product of [GR1] coincides with the free product defined by Neukirch (see [N2] and [GR1] prop. 1.5).

Theorem 8.3 Let (p, d(K)) = 1 and assume that X = Spec(O_{K}) is a K(π, 1) for p. Let S be a finite set of primes of K containing S_{p}. Then there is a canonical exact sequence

1⟶ ⋆_{𝔭∈S\S_{p}(X̃^{p})} T_{𝔭} ⋆ ⋆_{𝔭∈S_{p}(X̃^{p})} T_{𝔭}(−)⟶G_{S}(K)(p)⟶π_{1}^{pos}(X)(p)⟶1.

Proof: Denote the kernel of the canonical surjection G_{S}(K)(p) {− \atop ↠} π_{1}^{pos}(X)(p) by H. The canonical homomorphisms

T_{𝔭}(−) ⟶ Gal(K_{𝔭}(p)/K̃_{𝔭})⟶H

for 𝔭 ∈ S_{p} (resp. with T_{𝔭} for 𝔭 ∈ S\S_{p}) induce a homomorphism

F := ⋆_{𝔭∈S\S_{p}(K̃)} T_{𝔭} ⋆ ⋆_{𝔭∈S_{p}(K̃)} T_{𝔭}(−)⟶^{α}H.

Since there are no everywhere p.r. p-extensions of K̃ by the definition of K̃ the image of α generates H as a normal subgroup. We conclude that α is surjective because H is a pro-p group. In order to show that α is an isomorphism it obviously suffices to show that

(i) H is a free pro-p group and

(ii) α induces an isomorphism F̄⟶^{∼}H̄ on the maximal elementary abelian quotients.

By definition H = G_{S}(K̃)(p) and hence:

H^{i}(H, ℤ/pℤ) = H_{et}^{i}(X̃^{p}\S, ℤ/pℤ) = H_{pos}^{i}(X̃^{p}\S, ℤ/pℤ).

Using the excision sequence for X̃^{p} and S, the fact that X is a K(π, 1) for p and proposition 2.9 we deduce H^{2}(H, ℤ/pℤ) = 0 (hence condition (i) is fulfilled) and we get the following isomorphism:

H^{1}(H, ℤ/pℤ)^{∗} ≅ lim ; ⟵_{K⊂L⊂K̃} ⨁_{𝔭∈S(L)}(U_{L_{𝔭}}^{−})/p,

where the limit runs over all finite subextensions L of K in K̃ and the transition maps are the norm maps. For every 𝔭 ∈ S_{p}(K̃) we have by definition:

H^{1}(T_{𝔭}(−), ℤ/pℤ)^{∗} = \overline{T_{𝔭}(−)}⟶^{∼}(H^{1}(Gal(K_{𝔭}(p)/(K̃)_{𝔭}), ℤ/pℤ)^{−})^{∗} = lim ; ⟵_{K_{𝔭}⊂L⊂(K̃)_{𝔭}}U_{L}^{−}/p,

and the analogous formula for 𝔭 ∈ S\S_{p} also holds true. Calculating the cohomology of the free pro-p product in terms of the cohomology of its factors T_{𝔭}(−) (resp. T_{𝔭}) by [GR2] Prop.3.1. it follows that condition (ii) is fulfilled, too. □

In order to benefit of theorem 8.3 one wants to understand the action of π_{1}^{pos}(X) on the free product of the inertia groups T_{𝔭}(−). However a priory these depend on the choice in making the free product decomposition

Gal(K_{𝔭}(p)/(K̃)_{𝔭}) ≅ Gal(K_{𝔭}(p)/(K̃)_{𝔭})(+) ⋆ T_{𝔭}(−).

The ambiguity disappears if the (+)-group is zero, i.e. if the maximal positively ramified p-extension of the local fields K_{𝔭} is realized by the maximal p.r. p-extension K̃ of the global field K. However we cannot expect this in general as the following example shows: Example: Let K = ℚ(ζ_{p}) and assume that the Iwasawa-invariant λ_{p}^{−} equals 1 (e.g. p = 37.) Then we have an exact sequence

1⟶ℤ_{p} × ℤ_{p}⟶Gal(K̃/K)⟶ℤ_{p}⟶1,

which is induced by the surjection: Gal(K̃/K) {− \atop ↠} Gal(K_{∞}/K). We conclude that every closed subgroup of Gal(K̃/K) has a generator rank less or equal to 3. Hence the local groups (having rank \frac{p−1}{2} + 1) cannot be subgroups. However we hope that the case λ = 1 is exceptional similar to the genus 1 case in geometry. The next proposition solves the question in a special case for a p-CM field K with r_{p}(K) ≠ 0. We do not know a proof for the same statement for any p-CM field with r_{p}(K) = 0.

Proposition 8.4 Let ζ_{p} ∈ K and assume that K is p-CM and CM in the usual sense (e.g. K/ℚ abelian). If all primes dividing p split in the extension K/K^{+} then the maximal local p.r. p-extension is realized by K̃ for every 𝔭 ∈ S_{p}(K) and it holds:

T_{𝔭}(−) = T_{𝔭}(K_{S_{p}}(p)/K̃) = Gal(K_{𝔭}(p)/(K̃)_{𝔭})

for all primes 𝔭 ∈ S_{p}(K).

Proof: Since K/K^{+,p} is ramified at p we conclude that K^{+,p} is a CM-field not containing ζ_{p}. Further the condition on the extension K/K^{+} implies that also in the extension K^{+,p}/(K^{+,p})^{+} all primes dividing p split. By the result of [Mu] already the extension (K^{+,p})_{S_{p}}(p)K/K realizes the maximal p.r. p-extensions of K_{𝔭} for all 𝔭 dividing p. Further since for every prime in S_{p}(K) there is another prime in S_{p}(K) having the same local degree the results of [W1] (esp. Thm.6) imply that the maximal local p-extensions of K_{𝔭} is realized by K_{S_{p}}(p). This proves the statement of the proposition. Further it follows that in this situation the extension K̃/K_{S_{p}}^{+,p}(p)K is unramified and totally decomposed at all primes dividing p. □

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Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany, e-mail: schmidt@harmless.mathi.uni-heidelberg.de