The Étale Cohomology Ring of the Ring of Integers of a Number Field

NUMBER FIELD

ERIC AHLQVIST AND MAGNUS CARLSON

Abstract. We compute the cohomology ring H

(X, Z/nZ) 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.

Contents

1. Introduction 1

1.1. Organization 2

2. Background on the ´ etale cohomology of a number field 3

2.1. The Artin–Verdier site of a number field 3

2.2. Artin–Verdier duality for general number fields 8

2.3. Galois coverings in ˜ X ´ et 10

3. The cohomology ring of a number field 12

4. A non-vanishing criterion for Kim’s invariant 24

References 25

1. Introduction

Let K be a number field and let X = Spec O K . In this article we compute the ´ etale cohomology ring H ^∗ (X, Z/nZ) ¹ for arbitrary n ≥ 1. As an application of this computation, we give a non-vanishing criterion for an abelian “arithmetic” Chern–Simons invariant developed by Minhyong Kim. The ring H ^∗ (X, Z/nZ) holds non-trivial arithmetic information: for n = 2 it has been used by the second author and Tomer Schlank [CS16] in the study of embedding problems, while Maire [Mai17] used it to study the unramified Fontaine–Mazur conjecture at p = 2 and the 2-cohomological dimension of Galois groups with restricted ramification. It is our hope that our computation of H ^∗ (X, Z/nZ) will find interesting applications in the not too distant future.

We now move on to stating our results, and we assume for simplicity of exposition that K is a totally imaginary number field. However, the theorems in the main text are stated for arbitrary number fields, under the added proviso that one works with ´ etale cohomology on the Artin–Verdier site (see Definition 2.1). Recall that Artin–Verdier duality allows us to identify H ⁱ (X, Z/nZ) with Ext ³⁻ⁱ X (Z/nZ, G ^m,X ) ^∼ , where ^∼ denotes the Pontryagin dual. One can further identify Ext ¹ _X (Z/nZ, G ^m,X ) with the group Z 1 /B 1 , (see Corollary 2.15) where

Z 1 = {(a, a) ∈ K ^× ⊕ Div K| − div(a) = na} , B 1 = {(b ⁿ , − div(b)) ∈ K ^× ⊕ Div K|b ∈ K ^× } ,

1 If X is not totally imaginary, we use a “modified ´ etale cohomology” that takes the infinite primes into account

1

Div K is the group of fractional ideals, and div(a) it the fractional ideal associated to a. Hence we obtain the following list (see Corollary 2.15):

H ⁱ (X, Z/nZ) =



 

 

 



 

 

 



Z/nZ if i = 0

(Pic(X)/n) ^∼ if i = 1 (Z ₁ /B ₁ ) ^∼ if i = 2 µ n (X) ^∼ if i = 3

0 if i > 3.

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 represented by a cyclic unramified extension L/K of degree d|n and let σ ∈ Gal(L/K) be 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

hx ∪ y, (a, b)i = hy, N L|K (I) ^n/d + n ² 2d bi

where (a, b) ∈ Ext ¹ _X (Z/nZ, G ^m,X ) and I ∈ Div L is a 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, hx ∪ y, (a, b)i = 0 if and only if ⁿ _2d

b+ N _L|K (I) ^n/d is in the image of N _{M |K} .

It should be noted that the formula for this cup product is closely related to a pairing of McCallum–

Sharifi, as discussed in Remark 3.8. Our computation of the map H ² (X, Z/nZ) × H ¹ (X, Z/nZ) → H ³ (X, Z/nZ) is captured by:

Proposition 1.2. Let X = Spec O K be the ring of integers of a totally number field K and identify H ² (X, Z/nZ) and H ³ (X, Z/nZ) with Ext ¹ X (Z/nZ, G ^m,X ) ^∼ and µ n (K) ^∼ respectively, where ^∼ denotes the Pontryagin dual. Let x ∈ H ¹ (X, Z/nZ) be represented by the cyclic extension L/K of degree d|n, unramified at all places (including the infinite ones) and let σ ∈ Gal(L/K) be 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

hx ∪ y, ξi = hy, (a, a)i.

Recall that there is an isomorphism inv : H ³ (X, µ n ) → Z/nZ, as noted by Kim in [Kim15]. Given this remark, our formula for Kim’s invariant is as follows.

Proposition 1.3. Let K be a totally imaginary number field containing a primitive nth root of unity and let X = Spec O K be its ring of integers. Let C n be the cyclic group of order n, c 1 ∈ H ¹ (C n , Z/nZ) ∼ = Hom Ab (Z/nZ, Z/nZ) correspond to the identity and c ² = β n (c 1 ) ∈ H ² (C n , Z/nZ) be its image under the Bockstein homomorphism. Suppose that we have a continuous homomorphism f : π 1 (X, x) → C n

corresponding to the unramified Kummer extension L/K, where L = K(v ^1/n ) for some v ∈ K ^× such that there is an a ∈ Div K satisfying na = − div(v). Then Kim’s invariant, inv(f _X ^∗ (c)) ∈ Z/nZ vanishes if and only if a is in the image of the norm map N _L|K : Cl L → Cl K, i.e. if and only if Art _L|K (a) = 0.

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 computa- tion to arbitrary n.

1.1. Organization. In Section 2 we recall some material on the ´ etale 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.

2. Background on the ´ etale 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 the ´ etale cohomology of totally imaginary number fields. From the point-of-view of ´ etale cohomology, X behaves as a 3-manifold and satisfies an arithmetic version of Poincar´ e duality, namely Artin–Verdier duality. This duality states that for any constructible sheaf F, the ´ etale 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 modified ´ etale 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. 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 is ´etale 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 point ¯ y 6= y. A complex prime of Y is a pair of points y ₁ , y ₂ : Spec C → Y such that y 1 6= y ₂ and y ₁ = ¯ y 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, denoted ˜ X ´ et , is the site with objects pairs (Y, M ), where g : Y → X is a scheme that is separated and ´ etale 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

f : (Y 1 , M 1 ) → (Y 2 , M 2 )

in ˜ X ´ et is a morphism of X-schemes such that f (M 1 ) ⊂ M 2 . A family of morphisms {f i : (Y i , M i ) → (W, N )} i∈I is a covering if ∪ i f i (Y i ) = W and ∪ i f i (M i ) = N.

Note that any morphism f : (Y ₁ , M ₁ ) → (Y ₂ , M ₂ ) in ˜ X _{´ et} has the property that f : Y ₁ → Y ₂ is ´ etale 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( ˜ X _{´ et} ) to be the category of abelian sheaves on ˜ X _{´ et} . It will be convenient to have a more concrete description of Sh( ˜ X _{´ et} ). As above, we let X _∞ = {x ₁ , . . . , x _n } be the infinite primes of X. Fix a separable closure ¯ K of K. For each infinite prime x i , we fix an extension x e i of x i to ¯ K. We then let I

x e

be the decomposition group of x e i (note that I

x e

∼ = Z/2Z if x ⁱ is real, and that I

x e

is trivial if x i is complex). Since I

x e

⊂ Gal( ¯ K/K), if we let j : Spec K → X be the map induced from the inclusion, we see that for any ´ etale sheaf F on X, the pull-back j ^∗ F, viewed as a Galois module, has a natural action of I

x e

, and that we thus can take the fixed points with respect to this action and form (j ^∗ F ) ^I

. One can also view x _i as giving an absolute value on K; let K _x

be the completion with respect to x _i . If then i : Spec K _x

→ X is the natural map, then (j ^∗ F ) ^I

is isomorphic to the global sections of i ^∗ F, and we will sometimes write F (K x

) to denote (j ^∗ F ) ^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

S = (F _∞ , F, {σ _x } _x∈X

)

where F ∞ = {F ∞ (x)} x∈X

∈ Sh(X ∞ ), i.e. F ∞ is a product of abelian groups, one for each infinite prime, F is an abelian sheaf on X ´ et and for x ∈ X ∞ , σ x : F ∞ (x) → (j ^∗ F ) ^I

is a morphism of abelian groups. A morphism

f : S ₁ = (F _∞ , F, {σ _x } x∈X

) → S ₂ = (F _∞ ⁰ , F ⁰ , {σ _x ⁰ } x∈X

)

is a pair of maps f 1 : F ∞ → F _∞ ⁰ , f 2 : F → F ⁰ commuting with σ x , σ _x ⁰ , x ∈ X ∞ , upon pulling back f 2 by j : Spec K → X.

The following proposition shows precisely that the above definition gives us a concrete description of Sh( ˜ X ´ et ).

Proposition 2.3 ([Bie87, Prop. 1.2]). The category S _X defined above is equivalent to the category of abelian sheaves on ˜ X _{´ et} .

Proposition 2.3 is the archimedean analogue for how one for a scheme Y and a closed subscheme Z ⊂ Y with complement U, can reconstruct Sh(Y ) from Sh(Z) and Sh(U ) with gluing data (see [Mil80, Thm. II.3.10]).

We will from now on often identify the category Sh( ˜ X _{´ et} ) with S _X . The following proposition shows how Sh(X _{´ et} ), Sh( ˜ X _{´ et} ) and Sh(X _∞ ) relate to each other.

Proposition 2.4 ([Bie87, Prop. 1.1]). There are geometric morphisms φ _∗ : Sh(X ´ et )  Sh( ˜ X ´ et ) : φ ^∗ , κ _∗ : Sh(X _∞ )  Sh( ˜ X ´ et ) : κ ^∗ .

Let F ∈ Sh(X _{´ et} ), S = (G _∞ , G, {σ _x } _x∈X

) ∈ Sh( ˜ X _{´ et} ), and K ∈ Sh(X _∞ ). Then φ _∗ F = ({(j ^∗ F ) ^I

}, F, id),

φ ^∗ (S) = G , κ ∗ K = (K, 0, 0) ,

κ ^∗ S = G _∞ .

Further, φ ^∗ has a left adjoint, denoted by φ _! , while κ _∗ has a right adjoint κ ^! . These satisfy the following formulas:

φ _! (F ) = (0, F, 0) , κ ^! (S) = {ker σ x } x∈X

.

Remark 2.5. The forgetful functor ˜ X _{´ et} → X ´ et is a morphism of sites, and thus gives rise to a geometric morphism

φ ˜ _∗ : Sh(X ´ et )  Sh( ˜ X ´ et ) : ˜ φ ^∗ .

Under the identification of Sh( ˜ X ´ et ) with a category of triples as in Proposition 2.3, ˜ φ _∗ and ˜ φ ^∗ are identified with the functors φ _∗ and φ ^∗ respectively.

Note that if A ∈ Sh(X _{´ et} ) is the constant sheaf on X _{´ et} with value A, then φ _∗ (A) is isomorphic to the constant sheaf on Sh( ˜ X _{´ et} ) 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 I y ˜ such that I y ˜ ⊂ I x ˜ . We have a natural map π : Y → X and we will now define a push-forward functor π ∗ : Sh( ˜ Y ´ et ) → Sh( ˜ X ´ et ) and a pull-back functor

π ^∗ : Sh( ˜ X ´ et ) → Sh( ˜ Y ´ et ) .

This will be done by identifying Sh( ˜ Y ´ et ) and Sh( ˜ X ´ et ) with categories of triples as in Definition 2.2. Note that given a map π as above, we have geometric morphisms

π _∗ : Sh(Y _∞ )  Sh(X ∞ ) : π ^∗

and

π _∗ : Sh(Y ´ et )  Sh(X ^{´ et} ) : π ^∗ , and we want to combine these to get a geometric morphism

π ∗ : Sh( ˜ Y ´ et )  Sh( ˜ X ´ et ) : π ^∗ .

Given an extension L/K as above we have the following commutative diagram

Spec L Y

Spec K X.

π i

π j

If (F _∞ , F, {σ _y } _y∈Y ) ∈ Sh( ˜ Y _{´ et} ), we see that to construct π _∗ : Sh( ˜ Y _{´ et} ) → Sh( ˜ X _{´ et} ), we must, for each x ∈ X _∞ , give a natural map

σ _x ⁰ : π ∗ (F ∞ )(x) = M

y/x

F ∞ (y) → (j ^∗ π ∗ F ) ^I

and this is what is done in the discussion that follows. One sees that j ^∗ π ∗ F is naturally isomorphic to π ∗ i ^∗ F, and that for x ∈ X ∞ ,

(π _∗ i ^∗ F ) ^I

∼ = (i ^∗ F )(Spec ¯ K ^I

⊗ K L).

Since Spec ¯ K ^I

⊗ K L = q _y/x Spec ¯ K ^I

, where y ranges over the primes lying over x, we see that (π _∗ i ^∗ F ) ^I

∼ = M

y/x

(i ^∗ F ) ^I

. If x ∈ X _∞ , we let

θ _∗ : M

y/x

(i ^∗ F ) ^I

→ (j ^∗ π _∗ F ) ^I

be the isomorphism that is the composite of the isomorphism

M

y/x

(i ^∗ F ) ^I

→ (π ∗ i ^∗ F ) ^I

followed by the natural isomorphism (π _∗ i ^∗ F ) ^I

→ (j ^∗ π _∗ F ) ^I

. We now construct σ ⁰ _x : π _∗ (F _∞ )(x) = L

y/x F _∞ (y) → (j ^∗ π _∗ F ) ^I

as the composite of M

y/x

F _∞ (y) −−−−−→ ^⊕

^σ

M

y/x

(i ^∗ F ) ^I

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 and j : Spec K → X, i : Spec L → Y 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

π _∗ : Sh( ˜ Y ´ et ) → Sh( ˜ X ´ et ) takes

(F _∞ , F, {σ _y } _y∈Y

) to

(π ∗ F ∞ , π ∗ F, {σ _x ⁰ } x∈X

) where π _∗ F is the push-forward in Sh(X ´ et ), π _∗ F _∞ (x) = L

y/x F _∞ (y), and σ _x ⁰ : π ∗ F ∞ (x) = M

y/x

F ∞ (y) → (j ^∗ π ∗ F ) ^I

is the map L

y/x F ∞ (y) −−−−−→ ^⊕

^σ

L

y/x (i ^∗ F ) ^I

followed by the map θ ∗ just defined.

Having defined the push-forward functor π _∗ : Sh( ˜ Y ´ et ) → Sh( ˜ X ´ et ) we now define the pull-back functor.

Just as above, we want to combine the two pull-back functors π ^∗ : Sh(X ´ et ) → Sh(Y ´ et ) and π ^∗ : Sh(X ∞ ) → Sh(Y ∞ ) to a functor

π ^∗ : Sh( ˜ X ´ et ) → Sh( ˜ Y ´ et ),

To do this, we follow the strategy of Bienenfeld [Bie87]. The map θ _∗ above gives a natural isomorphism θ _∗ : π _∗ τ Y ∼ = τ X π _∗ ,

where τ X : Sh(X ´ et ) → Sh(X ∞ ) is the functor which takes F to {(j ^∗ F ) ^I

: x ∈ X _∞ } and τ _Y : Sh(Y _{´ et} ) → Sh(Y _∞ ) is the functor which takes G to

{(i ^∗ G) ^I

: y ∈ Y ∞ }.

If F ∈ Sh(X _{´ et} ), we have, since π _∗ is right adjoint to π ^∗ , a natural unit morphism η F : F → π _∗ π ^∗ F.

We now let

θ ^∗ : π ^∗ τ X F → τ Y π ^∗ F be the map adjoint to the composite

τ _X F −−→ τ ^η

_X π _∗ π ^∗ F ^θ

−1

−−→ π

_∗ τ _Y π ^∗ F.

It is clear that this gives a natural transformation θ ^∗ : π ^∗ τ X ⇒ τ Y π ^∗ . Given F = (F ∞ , F, {σ x } x∈X

) ∈ Sh( ˜ X ´ et ), we define σ ⁰ as the composite

σ ⁰ : π ^∗ (F ∞ ) ^π

∗

(Π

σ

)

−−−−−−→ π ^∗ (τ X F ) ^θ

∗

−→ τ Y (π ^∗ F ).

Evaluating σ ⁰ at a point y ∈ Y _∞ , we get a map

σ ⁰ _y : π ^∗ (F ∞ )(y) → (i ^∗ π ^∗ F ) ^I

.

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

π ^∗ : Sh( ˜ X ´ et ) → Sh( ˜ Y ´ et ) takes (F _∞ , F, {σ _x } _x∈X

) to

(π ^∗ (F ∞ ), π ^∗ (F ), {σ ⁰ _y } y∈Y

) where σ _y ⁰ : (π ^∗ F _∞ )(y) → (i ^∗ π ^∗ F ) ^I

is defined as above.

Proposition 2.8. Let L/K be a finite extension of number fields and let π : Y = Spec O L → Spec O K = X be the natural projection. Then the functor

π _∗ : Sh( ˜ Y ´ et ) → Sh( ˜ X ´ et ) is right adjoint to

π ^∗ : Sh( ˜ X ´ et ) → Sh( ˜ Y ´ et ).

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 ∞ ) of ˜ X ´ et , and suppose further that (Y, Y ∞ ) ∼ = q ⁿ _i=1 (Y i , (Y i ) ∞ ) where Y i = Spec O L

for L i /K a finite extension of K. Let us set

Sh( ˜ Y ´ et ) := Π ⁿ _i=1 Sh( ˜ (Y i ) _{´ et} ).

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 (π i ) ^∗ : Sh( ˜ X ´ et )  Sh( ˜ (Y i ) _{´ et} ) : (π i ) ∗

define an adjunction

π ^∗ : Sh( ˜ X _{´ et} ) → Sh( ˜ Y _{´ et} ) : π _∗ .

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( ˜ X ´ et ) is the category of sheaves on a site, it is a Grothendieck topos, so we have cohomology functors H ⁱ ( ˜ X, −) : Sh( ˜ X ´ et ) → Ab, defined as the derived functors of the global sections functor. With notation as in Proposition 2.4, if F ∈ Sh(X ´ et ), we want to relate H ⁱ ( ˜ X, F ) with H ⁱ (X, φ ^∗ F ). For any sheaf F on X ´ et we have a short exact sequence

0 → φ ! F → φ ∗ F → k ∗ k ^∗ φ ∗ F → 0

in Sh( ˜ X ´ et ). If we take F to be equal to Z, and let S ∈ Sh( ˜ X ´ et ), we get by applying Ext ⁱ _{X ˜} (−, S) to this short exact sequence a long exact sequence

· · · → H ⁱ (X ∞ , κ ^! S) → H ⁱ ( ˜ X, S) → H ⁱ (X, φ ^∗ S) → H ⁱ⁺¹ (X ∞ , κ ^! S) → · · · which is the local cohomology sequence for X _∞ .

Lemma 2.10. If F ∈ Sh(X _{´ et} ) is such that for each x ∈ X _∞ , j ^∗ F is a cohomologically trivial I _{x ˜} -module, then H ⁱ (X, F ) ∼ = H ⁱ ( ˜ X, φ _∗ F ).

Proof. This follows from the local cohomology sequence for X _∞ and [Bie87, Lemma 3.7]. Indeed, in the latter lemma, it is shown that if S = (G _∞ , G, {σ _x } x∈X

) ∈ Sh( ˜ X _{´ et} ), then

H ⁱ (X ∞ , κ ^! S) =



 

  L

x∈X

ker σ x if i = 0 L

x∈X

coker σ x if i = 1 L

x∈X

H ⁱ⁻¹ (I x ˜ , j _x ^∗ G) if i ≥ 2.

If F ∈ Sh(X _{´ et} ) then σ _x , x ∈ X _∞ associated to φ _∗ F is an isomorphism, so that H ⁱ (X _∞ , κ ^! φ _∗ F ) = 0 for i = 0, 1. 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 = 2 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, I _{x ˜} is trivial so that j ^∗ F is cohomologically

trivial. If we consider a real place x, then I _{x ˜} ∼ = 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 by ^∼ denote the functor

RHom Ab (−, Q/Z) : D(Ab) ^op → D(Ab).

Let F ∈ D( ˜ X _{´ et} ) and denote by G m,X the sheaf of units on X. Then the morphism A : RΓ( ˜ X, F ) → RHom X ˜ (F, φ _∗ G m,X )[3] ^∼ ,

is defined to be the adjoint to the composition map

RHom X ˜ (Z, F ) × RHom X ^˜ (F, φ _∗ G m,X ) → RHom X ˜ (Z, φ ∗ G m,X )

followed by the trace map RHom X ˜ (Z, φ ^∗ G m,X ) → Q/Z[−3] (see [Bie87, Prop. 2.7]). The following is the version of Artin–Verdier duality that we need.

Theorem 2.12 ([Bie87, Thm 5.1]). Let F be a constructible sheaf on X _{´ et} . Assume that for each x _i ∈ X _∞ , I x e

acts trivially on j ^∗ F. Then the map

A : RΓ( ˜ X, φ _∗ F ) → RHom X ˜ (φ _∗ F, φ _∗ G m,X )[3] ^∼ is an isomorphism in D(Ab).

Note that the hypothesis of the theorem is satisfied if F is a locally constant 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 [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 X _{´ et} such that for each x _i ∈ X ∞ , I

x e

acts trivially on each term in j ^∗ F and j ^∗ G. Then, the map

RΓ( ˜ X, φ _∗ F ) −−−−−−−→ RΓ( ˜ ^{RΓ( ˜ X,φ}

^{f )} X, φ _∗ G) corresponds under Artin–Verdier duality to the map

RHom X ˜ (φ _∗ F, φ _∗ G m,X )[3] ^{∼ RHom}

^(φ

^f,φ

^G

^)[3]

∼

−−−−−−−−−−−−−−−−−→ RHom X ˜ (φ _∗ G, φ _∗ G m,X )[3] ^∼ .

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⁻ⁱ _˜

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 ² _{X ˜} (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 the ´ etale sheaf

D iv ^{(X) = M}

p

Z /p ,

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

j _∗ G m,K

−−→ div D iv ^{X ,}

where j _∗ G m,K 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 − → Z ⁿ

of constant sheaves on ˜ X, 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. We now form the complex H om ^(E n , φ ∗ C), whose components are

φ _∗ j _∗ G ^m,K

(φ

div,n

)

−−−−−−−−→ φ _∗ D iv ^{X ⊕ φ} ∗ j _∗ G ^m,K

n+φ

div

−−−−−−→ φ _∗ D iv ^X.

Since E is a complex of locally free sheaves, we have

H om ^(E n , φ _∗ C) ∼ = RH om (Z/nZ, φ ∗ G m,X )

in D( ˜ X _{´ et} ). The plan for computing Ext ⁱ _{X ˜} (Z/nZ, φ ∗ G m,X ) is to use the hypercohomology spectral sequence, applied to H om ^(E n , φ ∗ C). To do this we need the cohomology of φ ∗ G m,K and φ ∗ D iv X as input.

Proposition 2.14 ([Bie87, Prop. 2.5, Prop. 2.6]). Let φ ∗ j ∗ G m,K be as above. Then

H ⁱ ( ˜ X, φ _∗ j _∗ G m,K ) =



 

 

 

 

K ^× if i = 0 0 if i = 1 Br 0 K if i = 2 0 if i > 2

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 ∗ D iv ^{X we have}

H ⁱ ( ˜ X, φ _∗ D iv ^{X) =}



 

 

 

  L

x∈X Z if i = 0

0 if i = 1

L

x∈X Br K x if i = 2

0 if i > 2

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, H om ^(E n , φ ∗ C)) → RHom(Z/nZ, φ ∗ G m,X ),

induced from the map Γ → RΓ. Since Γ( ˜ X, H om ^(E n , φ ∗ C)) is 2-truncated, this map will factor through τ ^≤2 (RHom(Z/nZ, φ ^∗ G m,X )). We denote by

ψ : Γ( ˜ X, H om ^(E n , φ _∗ C)) → τ ^≤2 (RHom(Z/nZ, φ ∗ G m,X ))

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 .

Corollary 2.15. Let X = Spec O K for K a number field. Then

Ext ⁱ _{X ˜} (Z/nZ, φ ^∗ G m,X ) =



 

 

 



 

 

 



µ _n (K) if i = 0 Z 1 /B 1 if i = 1 Pic X/n if i = 2 Z/nZ if i = 3 0 if i > 3.

Here

Z ₁ = {(a, a) ∈ K ^× ⊕ Div X| − div(a) = na}

and

B ₁ = {(b ⁿ , − div(b)) ∈ K ^× ⊕ Div X|b ∈ K ^× }.

The above corollary 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 on X with value Z/nZ, Theorem 2.12 applies. Thus, with Z ˜ ¹ and B 1 as in Corollary 2.15:

H ⁱ ( ˜ X, Z/nZ) =



 

 

 



 

 

 



Z/nZ if i = 0

(Pic(X)/n) ^∼ if i = 1 (Z 1 /B 1 ) ^∼ if i = 2 µ _n (K) ^∼ if i = 3

0 if i > 3.

2.3. Galois coverings in ˜ X _{´ et} . Recall that if G is an abelian group, then any element x ∈ H ¹ (X, G) 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 ( ˜ X _{´ et} , G) can be represented by a Galois G-cover in ˜ X _{´ et} . To make sense of this, we must of course define what a Galois covering should be in the category ˜ X _{´ et} . This is done in the following definition. A similar definition was made by Zink in [Hab78, Appendix 2, 2.6.1].

Definition 2.16. Let X = Spec O _K be the ring of integers of a number field, and let f : (Y, M ) → (X, X _∞ ) be an object of ˜ X _{´ et} . 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 Galois covering with Galois group G in X ´ et .

(2) The action of G on M is free, and f induces an isomorphism f : M/G → X _∞ .

Remark 2.17. 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.

For any Galois G-covering ˜ Y = (Y, Y ∞ ) → (X, X ∞ ), the functor h Y ˜ = Hom X ˜

(−, ˜ Y ) gives rise to a (right) G-torsor in Sh( ˜ X ´ et ). We thus have a map

c : {Galois G-coverings of (X, X ∞ )}

∼ → {G-torsors on ˜ X ´ et }

∼

where ∼ means we are passing to isomorphism classes. The following lemma is a straightforward application of descent theory (see for example [Mil80, Theorem 4.3]).

Lemma 2.18. 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

F 1 ∨ G F 2 = (F 1 × F 2 )/G

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 2.18 and the fact that Sh( ˜ X _{´ et} ) is a Grothendieck topos gives the following result.

Lemma 2.19. Let X = Spec O K be the ring of integers of a number field. Then if G is the constant

sheaf associated to an abelian group, any element x ∈ H ¹ ( ˜ X, G) can be represented by a Galois cover

(Y, Y _∞ ) → (X, X _∞ ) in ˜ X _{´ et} with Galois group G.

In Remark 2.17, 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

p : Ind ^G _H ((Z, Z _∞ )) = q ⁿ _i=1 (Z, Z _∞ )g i → (X, X _∞ )

on the component (Z, Z _∞ )g i as just the map q : (Z, Z _∞ ) → (X, X _∞ ). We now define the action of G on Ind ^G _H (Z) = q ⁿ _i=1 (Z, Z _∞ )g i . If x ∈ (Z, Z _∞ )g i , g ∈ G, then there exists a unique coset representative g j

such that g i g = hg j for some h ∈ H. We then let

xg := xh ∈ (Z, Z _∞ )g _j .

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 2.20. Let G be a finite group and H ⊂ G a subgroup. Then for any Galois H-cover q : (Z, Z _∞ ) → (X, X _∞ ), the induced cover

Ind ^G _H ((Z, Z _∞ )) → (X, X _∞ )

is a Galois G-cover. Conversely, any Galois G-cover p : (Y, Y _∞ ) → (X, X _∞ ) is isomorphic to a Galois G-covering Ind ^G _H ((Z, Z _∞ )) for a subgroup H ⊂ G unique up to conjugation, and a connected Galois H-covering (Z, Z _∞ ) → (X, X _∞ ), unique up to isomorphism.

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 eachother. Further, (Y ₁ , (Y ₁ ) _∞ ) is a Galois H-cover since Y is a Galois G-cover, and it is clear that

(Y, Y _∞ ) ∼ = Ind ^G _H ((Y ₁ , (Y ₁ ) _∞ )).

The unicity claims are left to the reader. 

Note that Lemma 2.20 has the consequence that any

x ∈ H ¹ ( ˜ X, Z/nZ) ∼ = (Cl K/n Cl K) ^∼

can be represented by a unramified cyclic extension L/K of degree d dividing n, in the sense that x can be represented by a Galois Z/nZ-covering of the form

Ind ^Z/nZ

Z/dZ ((Y, Y _∞ )) → (X, X _∞ ), where Y = Spec O L .

We now move on to the last lemma of the section. Let p : ˜ Y → ˜ X be a Galois cover of ˜ X with Galois group G. If F is an abelian sheaf on ˜ X ´ et , we say that F is p-split if p ^∗ F is a constant sheaf on ˜ Y ´ et . 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.

Lemma 2.21. Let p : ˜ Y → ˜ X be a Galois cover of ˜ X with Galois group G. Then the category of p-split

abelian sheaves on ˜ X _{´ et} is equivalent to the category of left G-modules.

3. 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

x ∪ − : H ⁱ ( ˜ X, Z/nZ) → H ⁱ⁺¹ ( ˜ X, Z/nZ).

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 2.19 and Lemma 2.20, 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. By Lemma 2.13, the map c x is, under Artin–Verdier duality, dual to the map

c ^∼ _x : Ext ^3−(i+1) _˜

X (Z/nZ X ^˜ , φ ∗ G m,X ) → Ext ³⁻ⁱ _˜

X (Z/nZ X ^˜ , φ ∗ G m,X ).

We will compute the map c ^∼ _x under the identifications of Corollary 2.15, and under the choice of a generator of Gal(L/K). Different choices of generator will yield different maps, but of course they are all isomorphic.

So to summarize, to compute c _x will mean that we compute the map c ^∼ _x under the identifications of Corollary 2.15, and then dualize. This computation depends on the choice of a generator of Gal(L/K).

With that said, let us compute c _x . 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 on X ˜ _{´ et} . By Lemma 2.19, we can represent x ∈ H ¹ ( ˜ X, Z/nZ) by a Galois covering

p : ˜ Y → ˜ X.

Since p is finite ´ etale, we have by Proposition 2.8 that p _∗ is also left adjoint to p ^∗ , so that there is a trace map

T : p _∗ p ^∗ Z/nZ X ˜ → Z/nZ X ˜

which is adjoint to the identity p ^∗ Z/nZ X ˜ → p ^∗ Z/nZ X ˜ . This gives us a short exact sequence (1) 0 → ker T − → p ^u _∗ p ^∗ Z/nZ X ˜

− T → Z/nZ X ˜ → 0.

Denote by C n the cyclic group of order n. 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 2.21, p _∗ p ^∗ Z/nZ X ˜

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 trace map

T : p ∗ p ^∗ Z/nZ X ˜ → Z/nZ X ˜

corresponds, under the given equivalence of categories, to the augmentation map

 : 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 T corresponds to ker  under the stated equivalence of categories. Thus, exact sequence (1) corresponds, under the equivalence in Lemma 2.21 to the exact sequence

0 → ker  → Z/nZ[C n ] − → Z/nZ → 0. ^

It is easy to see that ker , as a Z/nZ-module, is free on the elements g−1, g ∈ C ⁿ . There is a C n -equivariant map

s : ker  → Z/nZ taking e − 1 to 1. If we let P be the pushout of

Z/nZ[C n ] ← ker  − → Z/nZ, ^s we have the commutative diagram

0 ker  Z/nZ[C n ] Z/nZ 0

0 Z/nZ P Z/nZ 0.

s

N

The short exact sequence

(2) 0 Z/nZ P Z/nZ 0

corresponds under the equivalence given by Lemma 2.21 to the short exact sequence

(3) 0 → Z/nZ X ^˜ → F → Z/nZ X ˜ → 0

of sheaves on ˜ X ´ et and we have a commutative diagram

(4)

0 ker T p _∗ p ^∗ Z/nZ X ^˜ Z/nZ X ^˜ 0

0 Z/nZ X ˜ F Z/nZ X ˜ 0.

f

u T r

Lemma 3.1. Let x ∈ H ¹ ( ˜ X, Z/nZ) be represented by the Z/nZ-torsor p : ˜ Y → ˜ X. Then the connecting homomorphism H ⁱ ( ˜ X, Z/nZ) → H ⁱ⁺¹ ( ˜ X, Z/nZ) arising from the short exact sequence (3) is given by cup product with x ∈ H ¹ ( ˜ X, Z/nZ).

Proof. The strategy is the same as in [CS16, Lemma 5.1]. By [MLM94, VIII, Theorem 7], x, viewed as a Z/nZ-torsor, corresponds to a geometric morphism

k _x : Sh( ˜ X _{´ et} ) → BC _n

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 C

(Z/nZ, Z/nZ) as follows.

We let Z/nZ[T ] be the C n -module whose elements are given by formal sums P

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  _T : Z/nZ[T ] → Z/nZ given by mapping P

i a i [t i ] → P

i a i . The kernel ker  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  → 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 _C

(Z/nZ, Z/nZ) to be the short exact sequence we get by pushout along f ^T of the exact sequence

0 → ker  T → Z/nZ[T ] → Z/nZ → 0.

It is well-known that this give an isomorphism Tors C

(Z/nZ) → Ext ¹ C

(Z/nZ, Z/nZ). We then see that µ(U _Z/nZ ) corresponds to the short exact sequence

(5) 0 → Z/nZ → P → Z/nZ → 0.

If we take cohomology, then the connecting homomorphism Ext ⁱ _C

n

(Z/nZ, Z/nZ) → Ext ⁱ⁺¹ C

(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 ¹ _{X ˜} (Z/nZ X ^˜ , Z/nZ X ^˜ )

classifying it. From what we just have shown, this element corresponds to the element x ∈ H ¹ ( ˜ X, Z/nZ).

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

H ⁱ ( ˜ X, Z/nZ) ^δ

//

 =

H ⁱ⁺¹ ( ˜ X, ker T )

f



H ⁱ ( ˜ X, Z/nZ) ^c

// H ⁱ⁺¹ ( ˜ X, Z/nZ)

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

δ x : Z/nZ X ^˜ → ker T [1], we see that the map

RΓ( ˜ X, δ x ) : RΓ( ˜ X, Z/nZ) → RΓ( ˜ X, ker T [1]) corresponds under Artin–Verdier duality to

RHom X ˜ (δ x , φ ∗ G m,X ) : RHom X ˜ (ker T [1], φ ∗ G m,X )[3] ^∼ → RHom X ˜ (Z/nZ X ^˜ , φ ∗ G m,X ).

This shows that δ x is Pontryagin dual to the map δ ^∼ _x : Ext ^3−(i+1) _˜

X (ker T, φ _∗ G m,X ) → Ext ³⁻ⁱ _˜

X (Z/nZ X ^˜ , φ _∗ G m,X )

which we get by applying H ³⁻ⁱ to RHom X ˜ (δ x , φ _∗ G m,X ). 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

Ext ^3−(i+1) _˜

X (Z/nZ X ^˜ , φ _∗ G ^m,X ) ^f

∗

−→ Ext ^3−(i+1) _˜

X (ker T, φ _∗ G ^m,X ) ^δ

∼

−−→ Ext

³⁻ⁱ _˜

X (Z/nZ, φ ∗ G ^m,X ).

We will now compute δ _x ^∼ and c ^∼ _x by taking resolutions of ker T, φ _∗ G m,X and Z/nZ. Since, under the equivalence between locally constant sheaves split by p and C n -sets, ker T corresponds to ker , to resolve ker T, it is enough to find a resolution of ker , and that is what we will do. Let us denote the element P

g∈C

g by D C

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

K = (Z −−−−−−→ Z ⊕ Z[C ^(n,−D

⁾ n ] −−−−−→ Z[C ^D

⁺ⁿ n ])

that is, the first map is multiplication by n on the first factor and multiplication by −D C

on the second factor. The second map is multiplication by D C

on the first factor and by n on the second factor. The map K → ker  taking 1 ∈ Z[C ⁿ ] to e − 1 then exhibits K as a resolution of ker . By the equivalence of categories between C n -sets and locally constant sheaves split by p, we get the resolution

Z X ˜ d

−→ Z X ˜ ⊕ p _∗ p ^∗ Z X ˜ d

−→ p _∗ p ^∗ Z X ˜

of ker T. We will by abuse of notation also call the complex resolving ker T for K. Let us now resolve φ _∗ G m,X as in Section 2.2, by the complex C, defined as

φ ∗ j ∗ G m,K φ

div

−−−−→ φ ∗ D iv ^{X → 0.}

Let E n be the complex

Z X ^˜

− → Z n X ^˜

resolving Z/nZ X ^˜ . Then the maps u : ker T → p _∗ p ^∗ Z/nZ X ^˜ , T : p _∗ p ^∗ Z/nZ X ^˜ → Z/nZ X ˜ , and f : ker T → Z/nZ X ˜ from commutative diagram (4) lifts to morphisms of complexes ˆ u : K → p _∗ p ^∗ E n , b T : p _∗ p ^∗ E n → E n

and ˆ f : 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 map ˆ u is defined as:

Z Z ⊕ Z[C n ] Z[C n ]

0 Z[C n ] Z[C n ],

(n,−D

)

(0,e−1)

D

+n

e−1 n

while the map b T is given by

Z[C n ] Z[C n ]

Z Z.

n

 

n

Lastly, the map ˆ f is given in components as

Z Z ⊕ Z[C n ] Z[C n ]

0 Z Z,

(n,−D

)

(0,)

D

+n

 n

Let C(ˆ u) = p ∗ p ^∗ E n ⊕ K[1] denote the cone of ˆ u. Since b T ◦ ˆ u = 0, we have a map q(ˆ u) : C(ˆ u) −−−→ E ^{( b T ,0)} n . Lemma 3.2. The map q(ˆ u) : C(ˆ u) → E _n is a quasi-isomorphism.

Proof. Using the equivalence between locally constant sheaves split by p and C n -sets, this is a tedious

but straightforward computation which we leave to the reader. 

Summarizing our situation, we have the zig-zag

E _n ← ^{q(u) ˆ} −− − C(ˆ u) −−→ K[1] ^pr

and a map ˆ f : K[1] → E n [1]. The zig-zag represents δ x : Z/nZ X ^˜ → ker T [1]. We now apply H om ^{(−, C) to}

this zig-zag and ˆ f to get the zig-zag

(6) H om ^(E n , C) ^{q(u) ˆ}

∗

−−−→ H om ^{(C(ˆ u), C) pr}

∗

←−−

H om ^{(K[1], C)}

and the map

f ˆ ^∗ : H om ^(E n [1], C) → H om ^{(K[1], C).}

The map q(ˆ u) ^∗ is a quasi-isomorphism since q(ˆ u) 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

Hom(E n , C) Hom(C(ˆ u), C) Hom(K[1], C)

RHom(E n , C) RHom(C(ˆ u), C) RHom(K[1], C).

ˆ q(u)

s t

pr

RHom(ˆ q(u),C) RHom(pr

,C)

We want to prove the following lemma.

Lemma 3.3. The maps s, t and ˆ q(u) ^∗ induces isomorphism on H ⁱ for i = 0, 1, 2.

Before proving this lemma, we write out the complexes and the maps appearing in the zig-zag Hom(E _n , C) ^{q(u) ˆ}

∗

−−−→ Hom(C(ˆ u), C) ^pr

∗

←−− Hom(K[1], C)

^{f ˆ}

∗

←− Hom(E _n [1], C)

explicitly. To do this, we first need some notation. Since p : ˜ Y → ˜ X is a Galois covering, it can be written as Ind ^C _C

d

( ˜ Z) where

Z = (Z, Z ˜ _∞ ) = (Spec O L , (Spec O L ) _∞ )

for L an unramified field extension of K of degree d|n with Galois group C d . Note that ˜ Y has n/d components. We let σ be a fixed generator of Gal(L/K) and denote by σ ⁰ a generator of Gal(O _Y /O _K ).

We let the inclusion Gal(L/K) ⊂ Gal(O Y /O K ) take σ to σ ^0n/d . We then choose the set of right coset representatives {e, σ ⁰ , . . . , σ ^0n/d−1 } of C d ⊂ C n . Using this set of coset representatives, we fix an isomorphism ˜ Y ∼ = q ^n/d _i=1 Z. We write ˜

σ ⁰ − 1 : (L ^× ) ^n/d → (L ^× ) ^n/d , σ ⁰ − 1 : (Div Y ) ^n/d → (Div Y ) ^n/d to denote the maps taking a = (a ₁ , a ₂ , . . . , a _n/d ) ∈ (L ^× ) ^n/d to

σ ⁰ (a)/a := (σ(a _n/d )/a 1 , a 1 /a 2 . . . , a _n/d−1 /a _n/d ), I = (I 1 , . . . , I _n/d ) ∈ Div(Y )

to

σ ⁰ (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 ^× ,

N _{Y |X} : (Div L) ^n/d → Div K taking a = (a 1 , . . . , a n/d ) ∈ (L ^× ) ^n/d , I = (I 1 , . . . , I n/d ) ∈ (Div L) ^n/d to

N _{Y |X} (a) = Π ^n/d _j=1 (Π ^d−1 _i=0 σ ⁱ (a j )) and

N _{Y |X} (I) =

n/d

X

j=1

(

n−1

X

i=0

σ ⁱ (I _j ))

respectively. We also have the obvious inclusion maps i : K ^× → (L ^× ) ^n/d ,

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 H om (G, C). All the maps and differentials we have are then collected in

the diagram

K ^× (L ^× ) ^n/d

K ^× ⊕ Div K (Div L) ^n/d ⊕ (L ^× ) ^n/d ⊕ (L ^× ) ^n/d (L ^× ) ^n/d K ^×

Div K (Div L) ^n/d ⊕ (Div L) ^n/d ⊕ K ^× ⊕ (L ^× ) ^n/d (Div L) ^n/d ⊕ K ^× ⊕ (L ^× ) ^n/d K ^× ⊕ Div K

Div K ⊕ (Div L) ^n/d ⊕ K ^× Div K ⊕ (Div L) ^n/d ⊕ K ^× Div K

Div K Div K

ˆ q(u)

d

n

d

ˆ q(u)

d

n

d

(pr

)

d

f

d

ˆ q(u)

d

(pr

)

d

f

d

d

(pr

)

d

f

(pr

)

Where, if we write the maps in matrix form, the differentials are as follows d ⁰ _E

n

= −n div



d ¹ _E

n

= div n
, d ⁰ _{C(ˆ u)}

=



 div 1 − σ ⁰

−n





d ¹ _{C(ˆ u)}

=









σ ⁰ − 1 div 0

n 0 div

0 −N Y |X 0

0 −n σ ⁰ − 1









d ² _{C(ˆ u)}

=





N _{Y |X} 0 div 0

n 1 − σ ⁰ 0 div

0 0 n −N Y |X



 d ³ _{C(ˆ u)}

= −n N _{Y |X} div

d ¹ _K[1]

=



 div

−N _{Y |X}

−n





d ² _K[1]

=





N _{Y |X} div 0

n 0 div

0 n −N _{Y |X}



 d ³ _K[1]

= d ³ _{C(ˆ u)}

and the differentials for E n [1] ^∗ are given by d ⁱ _E

n

[1]

= d ⁱ⁻¹ _E

n

. The maps are given as follows: ˆ q(u) ^∗ ₀ (a) = i(a), ˆ

q(u) ^∗ ₁ (a, b) = (i(b), 0, i(a)), ˆ

q(u) ^∗ ₂ (a) = (0, i(a), 0, 0), (pr ^∗ ₂ ) ₁ (a) = (0, a, 0), (pr ^∗ ₂ ) ₂ (a, b, c) = (a, 0, b, c), (pr ^∗ ₂ ) 3 and (pr ^∗ ₂ ) 4 are equal to the identity map, while f ₁ ^∗ = i,

f ₂ ^∗ (a, b) = (i(b), a, i(a)),

f ₃ ^∗ (a) = (a, i(a), 0).

We now proceed with the proof of Lemma 3.3.

Proof of Lemma 3.3. We start by running the hypercohomology spectral sequence on H om ^{(C(ˆ q(u)), C).}

Then we have

E ₁ ^p,q = H ^q ( ˜ X, Hom ^p (C(ˆ u), C)) and using Proposition 2.14 the E ₁ -page can be visualized as

0 2 4 6

0 2 4 6

Recall that ˜ Y ∼ = Ind ^C _C

d

( ˜ Z) for Z = Spec O L the ring of integers of an unramified extension L of K. The differential

d ^0,2 _I : E ₁ ^0,2 = (Br ₀ L) ^n/d → ( M

z∈Z

Br L _z ) ^n/d ⊕ (Br ₀ L) ^n/d ⊕ (Br ₀ L) ^n/d is injective, since the map (Br 0 L) ^n/d → ( L

z∈Z Br L z ) ^n/d coincides with the map (inv) ^n/d , where inv is the restriction to Br 0 L of the invariant map Br L → L

z∈ ˜ Z Br L z , where z ranges over all points of Z = (Spec O ˜ L , Z ∞ ) (including the complex points). 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 ₂ ^p,0 for p = 0, 1, 2. This shows that

E ₂ ^p,0 = E _∞ ^p,0 = H ^p (RHom(C(ˆ q(u)), C)) for p = 0, 1, 2. But

E ₂ ^p,0 = H ^p (Hom(C(ˆ u), C))

and hence H ⁱ (t) is an isomorphism for i = 0, 1, 2. If one uses the hypercohomology spectral sequence on H om (K, C), one sees that H ⁱ (s) is an isomorphism for i = 0, 1, 2 as well. The claim that ˆ q(u) ^∗ induces an isomorphism now follows. Indeed, we know that RHom(ˆ q(u), C) is an isomorphism, so that H ⁱ (RHom(ˆ q(u), C) ◦ s) = H ⁱ (t ◦ ˆ q(u) ^∗ ) is an isomorphism for i = 0, 1, 2. Since H ⁱ (t) is an isomorphism in

these degrees, the statement follows. 

Corollary 3.4. The map c ^∼ _x : Ext ⁱ _{X ˜} (Z/nZ X ^˜ , φ _∗ G m,X ) → Ext ⁱ⁺¹ _˜

X (Z/nZ X ^˜ , φ _∗ G m,X ) for i = 0, 1 is iso- morphic to H ⁱ (ˆ q(u) ^∗ ) ⁻¹ ◦ H ⁱ (pr ^∗ ₂ ) ◦ H ⁱ (f ^∗ ).

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 17 freely in the proofs that follow. If x ∈ H ¹ ( ˜ X, Z/nZ) is represented by a Z/nZ-torsor of the form Ind ^Z/nZ

Z/dZ ((Y, Y ∞ )) → X, for Y = Spec O L 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.

Lemma 3.5. Let x ∈ H ¹ ( ˜ X, Z/n) and identify x with a cyclic extension L/K of degree d|n that is unramified at all places, including the infinite ones, and choose a generator σ of Gal(L/K). Then the morphism

c ^∼ _x : Ext ⁰ _{X ˜} (Z/nZ X ^˜ , φ _∗ G m,X ) → Ext ¹ _{X ˜} (Z/nZ X ^˜ , φ _∗ G m,X ) sends ξ ∈ µ n (K) ∼ = Ext ⁰ _{X ˜} (Z/nZ X ^˜ , φ ∗ G m,X ) to

(a, I) ∈ H ¹ (Hom(E n , C)) ∼ = Ext ¹ X ˜ (Z/nZ X ^˜ , φ _∗ G m,X )

where a ∈ K ^× and I ∈ Div K are elements lying under b ⁻ⁿ and div(b) respectively, where b ∈ L ^× is an

element satisfying ξ ^n/d = σ(b)/b in L ^× .