The étale cohomology ring of the ring of integers of a number field

We compute the cohomology ring H∗(X,Z/nZ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^*(X,{{\mathbb {Z}}}/n{{\mathbb {Z}}})$$\end{document} for X the spectrum of the ring of integers of a number field K. As an application, we give a non-vanishing formula for an invariant defined by Minhyong Kim.

Z 1 /B 1 , (see Corollary 2.15) where Z 1 = {(a, a) ∈ K × ⊕ Div K : div(a) + na = 0} , Div K is the group of fractional ideals, and div(a) is the fractional ideal associated to a. Hence we obtain the following list (see Corollary 2.15): An element x ∈ H 1 (X, Z/nZ) is given by a Z/nZ-torsor Y → X, which corresponds to a cyclic unramified extension L/K of degree d|n together with a choice of generator σ ∈ Gal(L/K). The structure of the cup product map H 1 (X, Z/nZ) × H 1 (X, Z/nZ) → H 2 (X, Z/nZ) is given by the following Proposition.
Proposition 1.1. Let X = Spec O K be the ring of integers of a totally imaginary number field K, and identify H 2 (X, Z/nZ) with Ext 1 X (Z/nZ, G m,X ) ∼ , where ∼ denotes the Pontryagin dual. Let x ∈ H 1 (X, Z/nZ) be an element corresponding to a cyclic unramified extension L/K of degree d|n and a choice of generator σ ∈ Gal(L/K). For an element y ∈ H 1 (X, Z/nZ) ∼ = (Cl K/n Cl K) ∼ represented by an unramified cyclic extension M/K, we have that where (a, b) ∈ Ext 1 X (Z/nZ, G m,X ) and I ∈ Div L is any fractional ideal such that b n/d O L = I − σ(I) + div(t) for some t ∈ L × such that N L|K (t) = a −1 . In particular, x ∪ y, (a, b) = 0 if and only if a ∈ Div K satisfying na = − div(v). Then Kim Lastly, the methods used to compute H * (X, Z/nZ) are similar to the methods used to compute H * (X, Z/2Z) in [CS16]. However, there are some new insights needed in order to generalize the computation to arbitrary n.
1.1. Organization. In Section 2 we recall some material on theétale cohomology of a number field and the Artin-Verdier site. What we cover in this section is classical and can be found in [Bie87] and [Maz73]. In Section 3 we then move on to determine the structure of H * (X, Z/nZ). In Section 4 we first recall the invariant defined by Minhyong Kim in [Kim15], whereafter we state the non-vanishing criteria.

Background on theétale cohomology of a number field
Let K be a number field and X = Spec O K its ring of integers. In the beautiful paper [Maz73], Mazur investigates theétale cohomology of totally imaginary number fields. From the point-of-view ofétale cohomology, X behaves as a 3-manifold and satisfies an arithmetic version of Poincaré duality, namely Artin-Verdier duality. This duality states that for any constructible sheaf F , theétale cohomology group H i (X, F ) is Pontryagin dual to Ext 3−i X (F, G m,X ), where G m,X is the sheaf of units. For fields K that are not totally imaginary, Artin-Verdier duality holds only modulo the 2-primary part. To remedy this, one must instead consider constructible sheaves on a modifiedétale site of X which takes the infinite primes into account. The purpose of this section is to recall some results from [Bie87] where the duality results we will need are proven. In an appendix to [Hab78], Zink removes the 2-primary restriction as well, but works with a modified cohomology which we will not use. For a very readable account of Zink's results, we recommend [CM16].
2.1. The Artin-Verdier site of a number field. In the following subsection we recall some of the main results that we need from [Bie87]. We emphasize that the results of this subsection (2.1) are not new, but due to the heavy amount of (possibly non-standard) notation, we include it to avoid confusion.
If X is the ring of integers of a number field, we let X ∞ = {x 1 , . . . , x n } be the infinite primes of X. An infinite prime is either a real embedding K → R or a pair of conjugate complex embeddings of K into C. If Y is a scheme that isétale over X, a real archimedean prime of Y is a point y : Spec C → Y which factors through Spec R. If y does not factor through Spec R, then by the conjugation action on C we obtain a pointȳ = y. A complex prime of Y is a pair of points y 1 , y 2 : Spec C → Y such that y 1 = y 2 and y 1 =ȳ 2 . Finally, we define Y ∞ to be the set whose elements are the real and complex primes of Y .
Definition 2.1. The Artin-Verdier site of X, denotedXé t , is the site with objects pairs (Y, M ), where g : Y → X is a scheme that is separated andétale over X, M ⊂ Y ∞ , and g(M ) ⊂ X ∞ is unramified, i.e, if p ∈ M is a complex prime, then its image is complex as well. A morphism Note that any morphism f : (Y 1 , M 1 ) → (Y 2 , M 2 ) inXé t has the property that f : Y 1 → Y 2 isétale and that f : M 1 → M 2 is unramified, i.e., if p ∈ M 1 is a complex prime, then f (p) is complex as well. The fact that Definition 2.1 gives a site is found in [Bie87, Prop. 1.2]. We define Sh(Xé t ) to be the category of abelian sheaves onXé t . It will be convenient to have a more concrete description of Sh(Xé t ). As above, we let X ∞ = {x 1 , . . . , x n } be the infinite primes of X. Fix a separable closureK of K. For each infinite prime x i , we fix an extension x i of x i toK. We then let I xi be the decomposition group of x i (note that I xi ∼ = Z/2Z if x i is real, and that I xi is trivial if x i is complex). Since I xi ⊂ Gal(K/K), if we let j : Spec K → X be the map induced from the inclusion, we see that for anyétale sheaf F on X, the pull-back j * F , viewed as a Galois module, has a natural action of I xi , and that we thus can take the fixed points with respect to this action and form (j * F ) I x i . One can also view x i as giving an absolute value on K; let K xi be the completion with respect to x i . If then i : Spec K xi → X is the natural map, then (j * F ) I x i is isomorphic to the global sections of i * F , and we will sometimes write F (K xi ) to denote (j * F ) I x i .
In preparation of the following definition, consider the category of finite sets over X ∞ , i.e., the category whose objects are given by pairs (A, f ) where A is a finite set and f : A → X ∞ is a function, and where the morphisms between objects are given by commutative triangles. This category becomes a Grothendieck site if we define a covering to be given by surjective morphisms. We then let Sh(X ∞ ) be the category of sheaves on this site. Note that to give a sheaf F ∞ ∈ Sh(X ∞ ) is the same as giving a collection of abelian groups, F x , one for each x ∈ X ∞ .
Definition 2.2. The category S X is the category whose objects are given by triples is a product of abelian groups, one for each infinite prime, F is an abelian sheaf on Xé t and for x ∈ X ∞ , σ x : The following proposition shows precisely that the above definition gives us a concrete description of Sh(Xé t ).
Proposition 2.3 ([Bie87, Prop. 1.2]). The category S X defined above is equivalent to the category of abelian sheaves onXé t . Proposition 2.3 is the archimedean analogue ofétale recollement, i.e., how one reconstructs Sh(Y ) from Sh(Z) and Sh(U ) with gluing data, where Z is a closed subscheme of Y with complement U (see [Mil80,Thm. II.3.10]).
We will from now on often identify the category Sh(Xé t ) with S X . The following proposition shows how Sh(Xé t ), Sh(Xé t ) and Sh(X ∞ ) relate to each other.
Further, φ * has a left adjoint, denoted by φ ! , while κ * has a right adjoint κ ! . These satisfy the following formulas: Remark 2.5. The forgetful functorXé t → Xé t is a morphism of sites, and thus gives rise to a geometric morphismφ * : Sh(Xé t ) ⇄ Sh(Xé t ) :φ * . Under the identification of Sh(Xé t ) with a category of triples as in Proposition 2.3,φ * andφ * are identified with the functors φ * and φ * respectively.
Note that if A ∈ Sh(Xé t ) is the constant sheaf on Xé t with value A, then φ * (A) is isomorphic to the constant sheaf on Sh(Xé t ) with value A.
If L/K is an extension of number fields, let Y = Spec O L and X = Spec O K . For each infinite prime y ∈ Y ∞ lying over x ∈ X ∞ , choose the decomposition group Iỹ such that Iỹ ⊂ Ix. We have a natural map π : Y → X and we will now define a push-forward functor π * : Sh(Ỹé t ) → Sh(Xé t ) and a pull-back functor π * : Sh(Xé t ) → Sh(Ỹé t ) . This will be done by identifying Sh(Ỹé t ) and Sh(Xé t ) with categories of triples as in Definition 2.2. Note that given a map π as above, we have geometric morphisms π * : Sh(Yé t ) ⇄ Sh(Xé t ) : π * , and we want to combine these to get a geometric morphism Given an extension L/K as above we have the following commutative diagram If (F ∞ , F, {σ y } y∈Y∞ ) ∈ Sh(Ỹé t ), we see that to construct π * : Sh(Ỹé t ) → Sh(Xé t ), we must, for each x ∈ X ∞ , give a natural map and this is what is done in the discussion that follows. Since π is finiteétale, by looking at the stalk at a separable closure, we see that the natural map j * π * F → π * i * F is an isomorphism. For x ∈ X ∞ , and since SpecK Ix ⊗ K L = ∐ y/x SpecK Iỹ , where y ranges over the primes lying over x, we see that If x ∈ X ∞ , we let θ * : y/x (i * F ) Iỹ → (j * π * F ) Ix be the isomorphism that is the composite of the isomorphism with θ * . This allows us to construct the claimed push-forward, which we record in the following definition.
Definition 2.6. Let L/K be an extension of number fields, X = Spec O K , Y = Spec O L , and let π : Y → X be the natural projection. Further, let j : Spec K → X and i : Spec L → Y be the maps that are induced by inclusion. Denote by X ∞ and Y ∞ the infinite places of X and Y respectively. Then the push-forward functor Having defined the push-forward functor π * : Sh(Ỹé t ) → Sh(Xé t ) we now define the pull-back functor. Just as above, we want to combine the two pull-back functors π * : Sh(Xé t ) → Sh(Yé t ) and π * : Sh(X ∞ ) → Sh(Y ∞ ) to a functor π * : Sh(Xé t ) → Sh(Ỹé t ) , To do this, we follow the strategy of Bienenfeld [Bie87]. The map θ * above gives a natural isomorphism , since π * is right adjoint to π * , we have a natural unit morphism η F : F → π * π * F . We now let be the map adjoint to the composite It is clear that this gives a natural transformation θ * : π * τ X ⇒ τ Y π * . Given we define σ ′ as the composite Evaluating σ ′ at a point y ∈ Y ∞ , we get a map σ ′ y : π * (F ∞ )(y) → (i * π * F ) Iȳ . This allows us to define the pull-back, which we record in the following definition.
Definition 2.7. Let L/K be an extension of number fields, X = Spec O K , Y = Spec O L , and let π : Y → X be the natural projection and j : Spec K → X, i : Spec L → Y be the maps induced by inclusion. Then the pull-back functor is right adjoint to π * : Sh(Xé t ) → Sh(Ỹé t ) . If further L/K is unramified (at all places, including the infinite ones), then π * is left adjoint to π * as well.
Proof. The first part follows from [Bie87, Lemma 1.9 and Proposition 1.11]. To prove the second part, one can either use [Bie87, Proposition 1.11] or calculate it directly, which we leave to the reader.
Remark 2.9. Suppose that we have an object π : (Y, Y ∞ ) → (X, X ∞ ) ofXé t , and suppose further that . One easily sees that this is the category of sheaves on a natural Artin-Verdier site associated to (Y, Y ∞ ). It is clear that if we let π i : Y i → X be the restriction of π to Y i , that we can, using the adjunctions The functor π * will then be the pushforward and π * the pullback. It also clear, for formal reasons, that if for each i, (π i ) * is also left adjoint to (π i ) * , then the functor π * is left adjoint to π * . We will sometimes need to use π * and π * when Y is not connected in Section 3.
Since Sh(Xé t ) is the category of sheaves on a site, it is a Grothendieck topos, so we have cohomology functors H i (X, −) : Sh(Xé t ) → Ab, defined as the derived functors of the global sections functor. With notation as in in Sh(Xé t ). If we take F to be equal to Z, and let S ∈ Sh(Xé t ), we get by applying Ext iX (−, S) to this short exact sequence a long exact sequence which is the local cohomology sequence for X ∞ (see [Bie87, Proposition 1.4]).
Proof. This follows from the local cohomology sequence for X ∞ and [Bie87, Lemma 3.7]. Indeed, in the latter lemma, it is shown The assumption that j * F is cohomologically trivial shows that H i (X ∞ , κ ! φ * F ) = 0 for i ≥ 1 and by the local cohomology sequence we are done.
The following proposition shows that the cohomology ring H * (X, Z/nZ) will always be isomorphic to the cohomology ring H * (X, Z/nZ) unless n is even and X has real places.
Proposition 2.11. Let X = Spec O K be the ring of integers of a number field. Then the cohomology rings H * (X, Z/nZ) and H * (X, Z/nZ) are isomorphic if either K is totally imaginary or if n is odd.
Proof. We apply Lemma 2.10. For each complex place x of K, Ix is trivial so that j * F is cohomologically trivial. If we consider a real place x, then Ix ∼ = Z/2Z, so we must show that j * (Z/nZ) ∼ = Z/nZ (with trivial Galois action) is cohomologically trivial as a Z/2Z-module if n is odd, but this is obvious.

2.2.
Artin-Verdier duality for general number fields. We now move on to stating the duality result which will be needed for our later computation of the cup product. The notation in this subsection is the same as in Section 2.1. We will denote by ∼ the functor Let F ∈ D(Xé t ) and denote by G m,X the sheaf of units on X. Then the morphism is defined to be the adjoint to the composition map The following is the version of Artin-Verdier duality that we need.
Note that the hypothesis of the theorem is satisfied if F is a locally constant constructible sheaf on X that is split by a morphism p : Y = Spec O L → Spec O K = X, such that L/K is unramified at all places, including the infinite ones. The proof of the following lemma is just as in [CS16, Lemma 4.1].
Lemma 2.13. Let X = Spec O K be the ring of integers of a number field and let f : F → G be a morphism between bounded complexes of constructible sheaves on Xé t such that for each x i ∈ X ∞ , I xi acts trivially on each term in j * F and j * G. Then, the map

corresponds under Artin-Verdier duality to the map
We end this section by computing H i (X, Z/nZ). By Theorem 2.12, and the fact that φ * Z/nZ = Z/nZ, this is the same as computing Ext 3−ĩ X (Z/nZ, φ * G m,X ). For i = 0, 1, 3 and i > 3 this can be found in [Bie87, Prop. 2.13], but just as in [Maz73], Ext 1X (Z/nZ, φ * G m,X ) is not explicitly determined. In the paper [CS16], the second author and Tomer Schlank gave a concrete interpretation when X = Spec O K is the ring of integers of a totally imaginary number field, and we will now use the same method. The following presentation will be brief; for more details we refer to the aforementioned paper by the second author and Schlank. Consider on X theétale sheaf where we let p range over all closed points of X and Z /p means that we consider the skyscraper sheaf at that point. We now define the complex C, which is a resolution of G m,X , as where j * is the inclusion of the generic point, j * G m,K is in degree 0, and the map div is as in [Mil80,II 3.9]. By push-forward with φ * , we get a complex φ * C. It is easy to see that the complex φ * C is a resolution of φ * G m,X . We now define E n as the complex Z n − → Z of constant sheaves onX, where the non-zero terms are in degree −1 and 0. This is of course just a resolution of Z/nZ, concentrated in degree 0. Let Hom denote the internal hom in the derived category of abelian sheaves and consider the complex Hom(E n , φ * C), whose components are Since E n is a complex of free sheaves, we have in D(Xé t ). The plan for computing Ext iX (Z/nZ, φ * G m,X ) is to use the hypercohomology spectral sequence, applied to Hom(E n , φ * C). To do this we need the cohomology of φ * j * G m,K and φ * Div X as input.
Proposition 2.14 ([Bie87, Prop. 2.5, Prop. 2.6]). Let φ * j * G m,K be as above. Then where Br 0 K is the subgroup of the Brauer group of K which has zero local invariants at the real infinite primes. For φ * Div X we have where the direct sum ranges over the closed points in X and Br K x is the Brauer group of the completion of K at x.
We have a map Γ(X, Hom(E n , φ * C)) → RHom(Z/nZ, φ * G m,X ) , induced from the map Γ → RΓ. Since Γ(X, Hom(E n , φ * C)) is 2-truncated, this map will factor through τ ≤2 (RHom(Z/nZ, φ * G m,X )). We denote by the resulting map. As in [CS16, Lemma 4.2], ψ is quickly shown to be an isomorphism. This allows one to compute the Ext-groups we are after. If a ∈ K × we let div(a) be the divisor in Div(X) determined by the corresponding fractional ideal in O K . Here Remark 2.16. Note that we have Ext i X (Z/nZ, G m,X ) ∼ = H i (X fl , µ µ µ n ) for all i, where X fl denotes the big fppf site on X. This can be seen as follows: we have quasi-isomorphisms (Z n − → Z) ≃ Z/nZ and µ µ µ n ≃ (G m n − → G m ) of complexes of fppf sheaves and we get R Hom(Z/nZ, The corollary above gives us a concrete description of H i (X, Z/nZ) for all i. Indeed, to remind the reader, since φ * Z/nZ is the constant sheaf onX with value Z/nZ, Theorem 2.12 applies. Thus, with Z 1 and B 1 as in Corollary 2.15: 2.3. Galois coverings. Let G be a finite group. Any element x in the pointed set H 1 (X, G) of right G-torsors in Xé t can be represented by a Galois G-cover f : Y → X.
The goal of what remains in this section is to show that any element x ∈ H 1 (Xé t , G) can be represented by a Galois G-cover inXé t . To make sense of this, we must of course define what a Galois covering should be in the categoryXé t . This is done in the following definition. A similar definition was made by Zink in [Hab78, Appendix 2, 2.6.1].
Definition 2.17. Let X = Spec O K be the ring of integers of a number field, and let f : (Y, M ) → (X, X ∞ ) be an object ofXé t . Assume that the finite group G acts on f to the right. Then we say that f is a Galois covering with Galois group G if: (1) f : Y → X is a (not necessarily connected) Galois covering with Galois group G in Xé t .
(2) The action of G on M is free, and f induces an isomorphism f : M/G → X ∞ .
Remark 2.18. Note that if f : (Y, M ) → (X, X ∞ ) is a Galois covering with Galois group G, then M = Y ∞ . Further, it is clear that every connected Galois covering with Galois group G gives rise to a Galois extension L/K with Galois group G that is unramified at all places, including the infinite ones. Conversely, given a Galois extension L of K with Galois group G that is unramified at the finite as well as at the infinite places, one gets a connected Galois covering with Galois group G.
Lemma 2.19. Let G be a finite group. Then the map c defined above is a bijection.
If G is abelian, then the set of isomorphism classes of (right) G-torsors has the structure of an abelian group. Indeed, if F 1 , F 2 are G-torsors, define where g ∈ G acts by taking (x, y) to (xg −1 , yg). The operation ∧ G respects isomorphism classes and descends to an operation on the set of isomorphism classes of G-torsors. One can then verify that ∧ G gives the set of isomorphism classes of (right) G-torsors the structure of an abelian group with unit the trivial G-torsor. In any Grothendieck topos T , we have that if G is an abelian group, then the group H 1 (T , G) is isomorphic to the isomorphism classes of G-torsors on T . Now lemma 2.19 and the fact that Sh(Xé t ) is a Grothendieck topos gives the following result.
Lemma 2.20. Let X = Spec O K be the ring of integers of a number field. Then if G is the constant sheaf associated to a finite group, any element x ∈ H 1 (X, G) can be represented by a Galois cover (Y, Y ∞ ) → (X, X ∞ ) inXé t with Galois group G.
In Remark 2.18, we noted that any connected Galois covering p : (Y, Y ∞ ) → (X, X ∞ ) with finite Galois group G was represented by an unramified extension of K. If Y is not connected, we now show that Y is isomorphic to a covering that is induced from a subgroup of G. If H ⊂ G is a subgroup of G and q : (Z, Z ∞ ) → (X, X ∞ ) is a Galois H-covering, we can form the induced cover (Y, Y ∞ ) := Ind G H ((Z, Z ∞ )) → (X, X ∞ ) as follows. Let n = [G : H] and choose a set {g 1 , . . . , g n } of right coset representatives of H in G. For each g i , we denote by (Z, Z ∞ )g i a copy of (Z, Z ∞ ), which should be seen as marked by g i . We then define the map then there exists a unique coset representative g j such that g i g = hg j for some h ∈ H. We then let This is just the Galois-covering analogue of an induced representation. The induced coverings generate all Galois coverings in the sense of the following Lemma.
Lemma 2.21. Let G be a finite group and H ⊂ G a subgroup. Then for any Galois H-cover q :

up to conjugation, and a connected Galois
Proof. This is standard, so we will indicate the proof and leave some of the details to the reader. It is clear that Ind G H ((Z, Z ∞ )) is a Galois G-cover. To see that any Galois G-cover arises in this way for a unique subgroup H ⊂ G and a unique connected Galois H-cover q : (Z, Z ∞ ) → (X, X ∞ ), let p : (Y, Y ∞ ) → (X, X ∞ ) be a Galois G-cover. If Y is connected, the statement is trivial, since we then can take H = G and (Z, Z ∞ ) = (Y, Y ∞ ). Thus we assume that Y is not connected, say Y 1 , . . . , Y n are its components. We let H ⊂ G consist of those g ∈ G such that (Y 1 , (Y 1 ) ∞ )g ⊂ (Y 1 , (Y 1 ) ∞ ). Choose g 1 , . . . , g n such that ((Y 1 , (Y 1 ) ∞ ))g i = (Y i , (Y i ) ∞ ). We then see that the (Y i , (Y i ) ∞ ) are isomorphic to each other. Further, (Y 1 , (Y 1 ) ∞ ) is a Galois H-cover since Y is a Galois G-cover, and it is clear that

The unicity claims are left to the reader.
Note that Lemma 2.21 has the consequence that any x ∈ H 1 (X, Z/nZ) ∼ = (Cl K/n Cl K) ∼ can be represented by an unramified cyclic extension L/K of degree d dividing n, together with a choice of generator σ ∈ Gal(L/K), in the sense that x can be represented by a Galois Z/nZ-covering of the form Ind We now move on to the last lemma of the section. Let p :Ỹ →X be a Galois cover ofX with Galois group G. If F is an abelian sheaf onXé t , we say that F is p-split if p * F is a constant sheaf onỸé t . There is a natural action of G on p * F and in this manner we get a functor from the category of sheaves split by p to the category of G-modules. The proof of the following lemma follows once again from standard descent theory.

The cohomology ring of a number field
The aim of this section is to compute the cohomology ring H * (X, Z/nZ) for X = Spec O K the ring of integers of a number field. In [CS16], this was done when K is totally imaginary and n = 2; if K is a number field that is not totally imaginary, or if n > 2, the methods utilized in that paper have to be altered. To compute the cohomology ring, we note that the fact that H i (X, Z/nZ) = 0 for i > 3, together with graded commutativity of the cup product, shows that it is enough to calculate x ∪ y, where y ∈ H i (X, Z/nZ), i = 0, 1, 2 and x ∈ H 1 (X, Z/nZ). For i = 0 the result is obvious, so we are reduced to when i = 1, 2. We denote by c x the map First of all, we should be precise with what we mean when we say that we compute c x . Let us note that by Lemma 2.20 and Lemma 2.21, x can be represented by a Galois covering Ind Z/nZ Z/dZ ((Y, Y ∞ )) for Y = Spec O L the ring of integers of a cyclic extension L/K of degree d|n which is unramified at the finite as well as the infinite places. In other words, x is represented by L, together with a choice of generator σ ∈ Gal(L/K). By Lemma 2.13, the map c x is, under Artin-Verdier duality, dual to a map . We will compute the map c ∼ x under the identifications of Corollary 2.15. We start by showing that the map c x : H i (X, Z/nZ) → H i+1 (X, Z/nZ) can be identified with a connecting homomorphism coming from a certain exact sequence of sheaves onXé t . By Lemma 2.20, we can represent x ∈ H 1 (X, Z/nZ) by a Galois covering p :Ỹ →X . Since p is finiteétale, we have by Proposition 2.8 that p * is also left adjoint to p * , and the counit N : p * p * Z/nZX → Z/nZX will be called the norm. This terminology is bad from the point of view of algebraic geometry where it is usually called trace but it agrees well with the number theoretic terminology and hence we will stick with norm. The norm N is an epimorphism, which can be seen by pulling back sections along the cover p : Y → X.
This gives us a short exact sequence Denote by C n = Z/nZ. Let us note that under the equivalence between the category of sheaves split by the morphism p and the category of C n -sets given by Lemma 2.22, p * p * Z/nZX corresponds to the left C n -module which is the group ring Z/nZ[C n ] ∼ = Z/nZ[e]/(e n − 1) .
One also easily shows that the norm map N : p * p * Z/nZX → Z/nZX corresponds, under the given equivalence of categories, to the augmentation map ǫ : Z/nZ[C n ] → Z/nZ .
Here Z/nZ has the trivial C n -action, and the augmentation map is the map that takes g ∈ C n to 1. This gives us that ker N corresponds to ker ǫ under the stated equivalence of categories. Thus, exact sequence (1) corresponds, under the equivalence in Lemma 2.22 to the exact sequence It is easy to see that ker ǫ, as a Z/nZ-module, is free on the elements g − 1, for 0 = g ∈ C n . There is a C n -equivariant map s : ker ǫ → Z/nZ taking e − 1 to 1. If we let P be the pushout of ], x, viewed as a Z/nZ-torsor, corresponds to a geometric morphism k x : Sh(Xé t ) → BC n (i.e., an adjunction where the left adjoint preserves all finite limits) where BC n is the topos of C n -sets. In this topos, there is a universal Z/nZ-torsor, which we denote U Z/nZ . The underlying C n -set of U Z/nZ is just Z/nZ with C n acting by left translation, and Z/nZ acts on U Z/nZ by right translation. Given a Z/nZ-torsor T , we define µ(T ) ∈ Ext 1 Cn (Z/nZ, Z/nZ) as follows. We let Z/nZ[T ] be the C n -module whose elements are given by formal sums i a i [t i ], a i ∈ Z/nZ, t i ∈ T and where C n acts in the obvious way. There is a map ǫ T : Z/nZ[T ] → Z/nZ given by mapping i a i [t i ] → i a i . The kernel ker ǫ T is generated by elements of the form [t 1 ]−[t 2 ], t 1 , t 2 ∈ T and we define a map f T : ker ǫ → Z/nZ by mapping [t 1 ] − [t 2 ] to the unique g ∈ Z/nZ ∼ = C n such that gt 2 = t 1 . We then define µ(T ) ∈ Ext 1 Cn (Z/nZ, Z/nZ) to be the short exact sequence we get by pushout along f T of the exact sequence It is well-known that this gives an isomorphism Tors Cn (Z/nZ) → Ext 1 Cn (Z/nZ, Z/nZ). We then see that µ(U Z/nZ ) corresponds to a short exact sequence If we take cohomology, then the connecting homomorphism Ext i Cn (Z/nZ, Z/nZ) → Ext i+1 Cn (Z/nZ, Z/nZ) is given by the Yoneda product with µ(U Z/nZ ). If we pull-back the short exact sequence (5) by k * x we get the short exact sequence (3), and our claim now follows. Indeed, the connecting homomorphism from short exact sequence (3) is given by Yoneda product with the element in Ext 1X (Z/nZX , Z/nZX ) classifying it. Thus the connecting homomorphism is given by cup product with x.
The commutative diagram (4) shows that if δ x is the connecting homomorphism coming from the upper short exact sequence, then the diagram commutes. Our plan to compute the cup product is now to first compute the map δ x , and then to compose with f * . By the above commutative diagram, this agrees with the cup product map. By applying Lemma 2.13 to the map we see that the map RΓ(X, δ x ) : corresponds under Artin-Verdier duality to This shows that δ x is Pontryagin dual to the map In the same way we see that the map c x : H i (X, Z/nZ) → H i+1 (X, Z/nZ) is, under Artin-Verdier duality, Pontryagin dual to the map c ∼ x , which is the composite . We will now compute δ ∼ x and c ∼ x by taking resolutions of ker N, φ * G m,X and Z/nZ. Since, under the equivalence between locally constant sheaves split by p and C n -sets, ker N corresponds to ker ǫ, to resolve ker N , it is enough to find a resolution of ker ǫ, and that is what we will do. Let us denote the element g∈Cn g by ∆ and choose a generator e of C n , thus establishing an isomorphism Z[C n ] ∼ = Z[e]/(e n − 1). The resolution we will use is the following that is, the first map is multiplication by n on the first factor and multiplication by −∆ on the second factor. The second map is multiplication by ∆ on the first factor and by n on the second factor. The map K → ker ǫ taking 1 ∈ Z[C n ] to e − 1 then exhibits K as a resolution of ker ǫ. By the equivalence of categories between C n -sets and locally constant sheaves split by p, we get the resolution of ker N . We will by abuse of notation also denote by K, the complex resolving ker N . Let us now resolve φ * G m,X as in Section 2.2, by the complex C, defined as Note that φ * div is an epimorphism since div : j * G m,K → Div X is an epimorphism and j * Div X = 0. Let E n be the complex ZX n − → ZX resolving Z/nZX . We will now show that the maps u : ker N → p * p * Z/nZX , N : p * p * Z/nZX → Z/nZX , and f : ker N → Z/nZX from commutative diagram (4) lift to morphisms of complexesû : K → p * p * E n , N : p * p * E n → E n andf : K → E n . We will explain how these morphisms are defined for the corresponding C n -sets, using once again the equivalence between locally constant sheaves split by p and C n -sets. The mapû is defined as: ∆ n e−1 n while the map N is given by Lastly, the mapf is given in components as The map q(û) * is a quasi-isomorphism since q(û) is a quasi-isomorphism between complexes of locally free sheaves. Applying the global sections functor to the zig-zag (6) and using the natural transformation Γ → RΓ, we have the commutative diagram RHom(q(û),C) RHom(pr 2 ,C) We want to prove the following lemma.
Before proving this lemma, we write out the complexes and the maps appearing in the zig-zag explicitly. To do this, we first need some notation. Since p :Ỹ →X is a Galois covering, it can be written as Ind Cn for L an unramified field extension of K of degree d|n with Galois group C d . Note thatỸ has n/d components. We let σ be the fixed generator of Gal(L/K) corresponding to the element x ∈ H 1 (X, Z/nZ) that we started with and denote by σ ′ a choice of generator of Gal(Y /X) such that the inclusion Gal(L/K) ⊂ Gal(Y /X) takes σ to σ ′n/d . We then get a set of right coset representatives {e, σ ′ , . . . , σ ′n/d−1 } of Gal(L/K) ⊂ Gal(Y /X). Using this set of coset representatives, we fix an isomorphismỸ ∼ = ∐ n/d i=1Z . We write σ ′ − 1 : (L × ) n/d → (L × ) n/d and σ ′ − 1 : (Div L) n/d → (Div L) n/d to denote the maps taking a = (a 1 , a 2 , . . . , a n/d ) ∈ (L × ) n/d to σ ′ (a)/a := (σ(a n/d )/a 1 , a 1 /a 2 . . . , a n/d−1 /a n/d ) , and I = (I 1 , . . . , I n/d ) ∈ Div(Y ) to σ ′ (I) − I := (σ(I n/d ) − I 1 , I 1 − I 2 , . . . , I n/d−1 − I n/d ) respectively. There are also norm maps N Y |X : (L × ) n/d → K × and N Y |X : (Div L) n/d → Div K taking a = (a 1 , . . . , a n/d ) ∈ (L × ) n/d and I = (I 1 , . . . , I n/d ) ∈ (Div L) n/d to respectively. We also have the obvious diagonal inclusion maps i : K × → (L × ) n/d and i : Div K → (Div L) n/d . Lastly, we have the maps div : (L × ) n/d → (Div L) n/d taking a tuple a = (a 1 , . . . , a n ) ∈ (L × ) n/d to div(a) = (div(a 1 ), . . . , div(a n )) ∈ (Div L) n/d , where div(a i ) is the fractional ideal of L generated by a i . If we have a complex G, we will in the diagram that follows write G * to denote Hom(G, C). All the maps and differentials we have are then collected in the diagram Where, if we write the maps in matrix form, the differentials are as follows Proof of Lemma 3.2. We start by running the hypercohomology spectral sequence on Hom(C(û), C). Then we have E p,q 1 = H q (X, Hom p (C(û), C)) and the complex Hom(C(û), C) has the form Since π is finiteétale, the pushforward π * is exact and, using the Leray spectral sequence, we see that H n (X, π * F ) ∼ = H n (Y, F ) for allétale sheaves F on Y and all n ≥ 0. Furthermore, since π isétale, we have an isomorphism π * Div X ∼ = Div Y and π * j * G m,K is isomorphic a direct sum of sheaves (j L ) * G m,L , one for each component of Y , where j L is the inclusion of the generic point on Spec O L . It follows that for every p, the sheaf Hom p (C(û), C) is a direct sum of sheaves which by Proposition 2.14 only has cohomology in degree 0 and 2. Hence the E 1 -page can be visualized as This restriction is injective since the subgroup Br 0 L has no invariants at the complex points. On the E 2 -page, we see that no differential can hit E p,0 2 for p = 0, 1, 2. This shows that E p,0 2 = E p,0 ∞ = H p (RHom(C(q(û)), C)) for p = 0, 1, 2. But E p,0 2 = H p (Hom(C(û), C)) and hence H i (t) is an isomorphism for i = 0, 1, 2. If one uses the hypercohomology spectral sequence on Hom(E n , C), one sees that H i (s) is an isomorphism for i = 0, 1, 2 as well. The claim that q(û) * induces an isomorphism now follows. Indeed, we know that RHom(q(û), C) is an isomorphism, so that H i (RHom(q(û), C) • s) = H i (t • q(û) * ) is an isomorphism for i = 0, 1, 2. Since H i (t) is an isomorphism in these degrees, the statement follows.
We will now utilize this corollary to compute the map c ∼ x . We will use the maps and the notation of the diagram on page 16 freely in the proofs that follow. If x ∈ H 1 (X, Z/nZ) is represented by a Z/nZ-torsor of the form Ind the ring of integers of an unramified cyclic extension L/K of degree d, we will say that we identify x with the cyclic extension L/K of degree d|n together with a choice of generator σ ∈ Gal(L/K).
Lemma 3.4. Let x ∈ H 1 (X, Z/n) and identify x with a pair (L, σ), where L/K is a cyclic extension of degree d|n that is unramified at all places, including the infinite ones, and σ ∈ Gal(L/K) is a generator. Then the morphism where a ∈ K × and I ∈ Div K are elements such that a = b −n and IO L = div(b) respectively, where b ∈ L × is an element satisfying ξ n/d = σ(b)/b in L × .
Proof. We use Corollary 3.3. We see that (pr * 2 ) 1 • f * 1 takes the element ξ ∈ µ n (K) to (0, i(ξ), 1). We want to reduce this element, modulo the image of d 0 C(û) * so that it is in the image of q(û) * 1 . Note that the norm of ξ n/d ∈ L × is ξ n = 1, thus Hilbert's theorem 90 gives us an element b ∈ L × such that to get the element (div(b), 1, b −n ). Since ξ is a unit, we have div(b) = (div(b), div(b), . . . , div(b)) ∈ (Div L) n/d , and since ξ n = 1, we have b −n = (b −n , . . . , b −n ) ∈ (L × ) n/d . To show that this is in the image of the q(û) * 1 , we note that div(b) is invariant under the Galois action. Indeed, taking div of the equality (7), we see that div(b) = σ(div(b)). This implies that there is a fractional ideal I of Div K such that i(I) = div(b). Similarly, we see that b −n is invariant under the Galois action, so that there is some element a ∈ K × such that i(a) = b −n . It is then clear that q(û) * (a, I) = (div(b), 1, b −n ), so that our lemma follows.
Corollary 3.5. Suppose that in Lemma 3.4 the field K contains a primitive nth root of unity. Identify L with a Kummer extension L = K(v 1/n ) with v ∈ K × such that div(v) = na for some a ∈ Div(K). Choose a primitive root of unity ξ such that ξ n/d = σ(v 1/n )/v 1/n . Then Proof. In the statement of 3.4 we can in this situation choose b to be v 1/n . Since clearly v −1 lies under v −1/n and a lies under div(v 1/n ), our Corollary follows.
Note that since ξ is a primitive nth root of unity, Corollary 3.5 determines the map c ∼ x in full. We now determine the map Lemma 3.6. Let x ∈ H 1 (X, Z/n) and identify x with a pair (L, σ), where L/K is a cyclic extension of degree d|n that is unramified at all places, including the infinite ones, and σ is a generator of Gal(L/K). Then the morphism c ∼ x : Ext 1X (Z/nZX , φ * G m,X ) → Ext 2X (Z/nZX , φ * G m,X ) sends the element (b, b) ∈ H 1 (Hom(E n , C)) ∼ = Ext 1X (Z/nZX , φ * G m,X ) to In this formula, I is any fractional ideal of L satisfying the equality Proof. Once again, we use Corollary 3.3. We see that (pr * 2 ) • f * 2 takes the element We want to reduce this element modulo the image of d 1 C(û) to get an element of the form (0, i(J), 1, 1) for some ideal J ∈ Div K, since im(q(û) 2 ) = (0, i(Div(K)), 1, 1). Let σ be the generator of Gal(L/K) corresponding to x. Since b n/d O L is in the kernel of the map N L|K : Cl L → Cl K, Furtwängler's theorem [Lem07, IV, Theorem 1] gives us a fractional ideal b ′ ∈ Div L and an element a ∈ L × such that Note that div(N L|K (a)) = nb = div(b −1 ), so that N L|K (a) = b −1 u for some unit u ∈ K × . Since units are always norms in unramified extensions of local fields, Hasse's norm theorem [Has31] implies that there is a v ∈ L × such that N L|K (v) = u −1 . Now, since N L|K (div(v)) is the unit ideal, Hilbert's theorem 90 for ideals (see e.g. [Bem12, Proposition 2.1.1]) implies that there is an ideal J ∈ Div L such that div(v) = J − σ(J). Set I = b ′ − J and t = av. Then we see that Set I = (I, b + I, 2b + I, . . . , ( n d − 1)b + I) ∈ (Div L) n/d and t = (t, 1, . . . , 1). Denote by div(t) and t n respectively the element obtained by applying the operators div and (−) n component-wise and recall the definition of σ ′ − 1 after Lemma 3.2. Then If we reduce (i(b), 0, b, i(b)) modulo the element ) . Now, t n b n/d is in the kernel of N L|K : L × → K × , so by Hilbert's theorem 90 there is some element w ∈ L × such that w/σ(w) = t n b n/d . We claim that we can choose the element w as t n σ(t n− n d )σ 2 (t n− 2n d ) · · · σ d−1 (t n− n(d−1) d ) .
Proposition 3.11. Let X = Spec O K be the ring of integers of a number field K and identify H 2 (X, Z/nZ) and H 3 (X, Z/nZ) with Ext 1X (Z/nZX , φ * G m,X ) ∼ and µ n (K) ∼ respectively, where ∼ denotes the Pontryagin dual. Let x ∈ H 1 (X, Z/nZ) be represented by a pair (L, σ), where L/K is a cyclic extension of degree d|n, unramified at all places (including the infinite ones), and σ ∈ Gal(L/K) is a generator. Let ξ ∈ µ n (K) and choose b ∈ L × such that ξ n/d = σ(b)/b, and let a ∈ K × and a ∈ Div K be such that aO L = div(b) and a = b −n in L × . If y ∈ H 2 (X, Z/nZ), then x ∪ y, ξ = y, (a, a) .
Proof. This follows at once from Lemma 3.4.
Corollary 3.12. Let X = Spec O K be the ring of integers of a number field K containing all nth roots of unity, and identify H 2 (X, Z/nZ) and H 3 (X, Z/nZ) with Ext 1X (Z/nZX , φ * G m,X ) ∼ and µ n (K) ∼ respectively, where ∼ denotes the Pontryagin dual. Let x ∈ H 1 (X, Z/nZ) be represented by a pair (L, σ), where L = K(v 1/n ) Kummer extension of degree d = [L : K], with v ∈ K × such that div(v) = na for some a in Div(K), and σ ∈ Gal(L/K) is a generator. If y ∈ H 2 (X, Z/nZ), then x ∪ y = 0 if and only if y, (v −1 , a) = 0.
Proof. This is a direct consequence of Corollary 3.5. To see this, let ξ be a primitive nth root of unity such that σ(v 1/n )/v 1/n = ξ n/d . We then see that c x (y)(ξ) = y, (v −1 , a) . Since ξ generates µ n (K), the element c x (y) ∈ µ n (K) ∼ is zero if and only if c x (y), ξ = 0, so we have our Corollary.
Proposition 3.13. Let X = Spec O K be the ring of integers of a number field K and identify H 2 (X, Z/nZ) with Ext 1X (Z/nZX , φ * G m,X ) ∼ , where ∼ denotes the Pontryagin dual. Let x ∈ H 1 (X, Z/nZ) be represented by a pair (L, σ), where L/K is a cyclic extension of degree d|n, unramified at all places (including the infinite ones), and σ ∈ Gal(L/K) is a generator. For an element y ∈ H 1 (X, Z/nZ) ∼ = (Cl K/n Cl K) ∼ represented by an unramified cyclic extension M/K, we have that x ∪ y ∈ Ext 1X (Z/nZX , φ * G m,X ) ∼ satisfies the formula x ∪ y, (a, b) = y, N L|K (I) n/d + n 2 2d b where (a, b) ∈ Ext 1X (Z/nZX , φ * G m,X ) and I ∈ Div L is any fractional ideal such that b n/d O L = I − σ(I) + div(t) for some t ∈ L × such that N L|K (t) = a −1 . In particular, x ∪ y, (a, b) = 0 if and only if n 2 2d b + N L|K (I) n/d is in the image of N M|K .
Proof. By Artin reciprocity, y ∈ H 1 (X, Z/nZ) corresponds to a map Cl K/n Cl K → Z/nZ with kernel N M|K (Cl M ). The formula for x ∪ y is given by Lemma 3.6. Indeed, x ∪ y, (a, b) = y, c ∼ x (a, b) = y, where I is as in the proposition. The fact that x ∪ y, (a, b) = 0 if and only if n 2 2d b + N L|K (I) n/d is in the image of N M|K readily follows from this formula and the observation that the kernel of y : Cl K/n Cl K → Z/nZ is N M|K (Cl M ).