aa r X i v : . [ m a t h . N T ] M a y Weil-´etale Cohomology over p -adic Fields David A. Karpuk
Abstract
We establish duality results for the cohomology of the Weil group of a p -adic field, anal-ogous to, but more general than, results from Galois cohomology. We prove a dualitytheorem for discrete Weil modules, which implies Tate-Nakayama Duality. We defineWeil-smooth cohomology for varieties over local fields, and prove a duality theoremfor the cohomology of G m on a smooth, proper curve with a rational point. This lasttheorem is analogous to, and implies, a classical duality theorem for such curves.
1. Introduction1.1 Background and Motivation
Arithmetic applications of Weil groups have been a popular topic in recent years, starting withthe article [Lic05] of Lichtenbaum. In this article, Lichtenbaum defines a cohomology theory forvarieties over finite fields, Weil-´etale cohomology, wherein the Weil group plays the role that theGalois group plays in ´etale cohomology. The Weil-´etale cohomology groups of the sheaf Z ona smooth, projective variety are shown to be finitely generated abelian groups. The resultingsecondary Euler characteristics provide a cohomological interpretation of the order of vanishingand leading coefficient of the zeta function of the variety at t = 1.If k is a finite field, the ´etale and Weil-´etale cohomology groups of X = Spec k with Z coefficients are given, respectively, by H i ( ˆ Z , Z ) = Z if i = 00 if i = 1 Q / Z if i = 20 if i > H i ( Z , Z ) = Z if i = 0 Z if i = 10 if i > . In general, taking Weil-´etale cohomology of Spec k shifts the Q / Z ’s that appear in the ´etalecohomology groups down a degree and turning them into Z ’s. In this sense, Weil-´etale cohomologyof Spec k determines the ´etale cohomology, as is made precise by Lemma 1.2 of [Lic05].Let K be a p -adic local field with absolute Galois group G and Weil group W . Let L be thecompletion of the maximal unramified extension of K , and let ¯ L be an algebraic closure of L containing an algebraic closure ¯ K of K . If X = Spec K , the ´etale and Weil-´etale cohomologygroups of X with G m coefficients are given, respectively, by H i ( G, ¯ K × ) = K × if i = 00 if i = 1 Q / Z if i = 20 if i > H i ( W, ¯ L × ) = K × if i = 0 Z if i = 10 if i > Primary: 14F20, Secondary: 14G20, 11S25
Keywords: ´etale cohomology, Galois cohomology avid A. Karpuk
For any connected, commutative algebraic group A /K , the groups H i ( W, A ( ¯ L )) determine thegroups H i ( G, A ( ¯ K )) up to isomorphism, which is made precise by our Theorem (4.1.5).Lichtenbaum’s computation of the groups H i ( X W , Z ) for a curve X over a finite field, andtherefore his interpretation of special values of zeta functions, relies on a duality theorem stated interms of cup-product in Weil-´etale cohomology. The main theorems presented here are analogousduality theorems for the Weil-´etale cohomology of zero and one-dimensional schemes over p -adicfields. The former is the Weil analogue of Tate-Nakayama Duality, and the latter is the Weilanalogue of Lichtenbaum Duality for curves over p -adic fields.Our main theorem concerning the cohomology of W -modules is Theorem (3.3.1). Let Γ W : W -Mod → A b be the fixed-points functor, and let R Γ W : D ( W ) → D ( Z ) be its derived functor. Fora W -module M , let M D = Hom( M, ¯ L × ). There is a natural map in D ( Z ), ψ ( M ) : R Γ W ( M D ) → R Hom( R Γ W ( M ) , Z [ − , induced by a cup-product pairing. Our theorem is the following: Theorem 3.3.1.
Suppose that M is finitely generated as an abelian group. Then ψ ( M ) is anisomorphism in D ( Z ).This theorem implies Tate-Nakayama Duality, which in turn implies the main theorem ofLocal Class Field Theory. This theorem was originally proven by Jiang in [Jia06] using Tate-Nakayama Duality, under the assumption that M is a G -module. We have removed this assump-tion, and provided a proof which is independent of the main results of Galois cohomology.Our second main theorem is a duality theorem for the Weil-´etale cohomology of smooth,projective, geometrically connected curves X/K , which contain a rational point. Let Γ X be theglobal sections functor on the Weil-´etale site of X , and let R Γ X be the derived functor of Γ X .Our duality theorem for X is the following: Theorem 5.5.2.
Let
X/K be a smooth, projective, geometrically connected curve over K , suchthat X ( K ) = ∅ . There is a symmetric pairing R Γ X ( G m ) ⊗ L R Γ X ( G m ) → Z [ − , such that the induced map R Γ X ( G m ) → R Hom( R Γ X ( G m ) , Z [ − U ∗ K → Hom( U K , Q / Z ), which is not surjective. This self-duality of G m is the Weil analogue of the classical duality theorems of the article [Lic69]. In fact, for curveswith rational points, one can deduce the main result of [Lic69], namely that there is a naturalisomorphism Br( X ) → Pic( X ) ∗ .More recent work by Lichtenbaum [Lic09], Flach [Fla08], Morin [Mor11a] and [Mor11b], andFlach and Morin [FM10] has been done towards a definition of Weil-´etale cohomology for schemesof finite type over Spec Z . These approaches have been partially successful in giving a Weil-´etalecohomological interpretation of special values of zeta functions and L -functions of such schemes,2 eil-´etale Cohomology over p -adic Fields but a fully satisfying global theory is still lacking. Hopefully a better understanding of Weil-´etalecohomology over p -adic fields will help provide a link between the respective theories over finiteand global fields. If R is a commutative ring, then D ( R ) denotes the bounded derived category of R -modules. If G is a discrete group, then D ( G ) denotes the bounded derived category of G -modules; if G isprofinite, then we will always restrict our attention to discrete, continuous G -modules, unlessstated otherwise. If G is a discrete group and M is a G -module, then by H ∗ ( G, M ) we mean thetraditional group cohomology of G with coefficients in M . If G is profinite, or an extension of adiscrete group by a profinite group, and M is a discrete G -module, then by H ∗ ( G, M ) we meanGalois cohomology in the sense of [Ser02].We briefly recall the notion of cohomological dimension. Let G be a discrete group, or anextension of a discrete group by a profinite group. The cohomological dimension of G is definedto be the smallest integer n such that for all m > n and all torsion G -modules M , we have H m ( G, M ) = 0 (provided such an n exists). We write cd( G ) for the cohomological dimension of G . The strict cohomological dimension of G is defined to be the smallest integer n such that forall m > n and all G -modules M , we have H m ( G, M ) = 0 (provided such an n exists); we denotethe strict cohomological dimension of G by scd( G ).If f : X → Y is a map between two cochain complexes, we use f i to denote the inducedmap in i th cohomology. We will often write an exact triangle X → Y → Z → X [1] as simply X → Y → Z , with the X [1] being implied.If M is an abelian group and n is an integer, we use M [ n ] and M/n to denote the kernel andcokernel, respectively, of the multiplication-by- n maps on M . If M and N are abelian groups withsome possible extra structure (for example, M and N could be G -modules), then Hom( M, N ) willalways mean Hom Z ( M, N ), and similarly for Ext and ⊗ . When working in the derived categoryof abelian groups, we will write R Hom for R Hom D ( Z ) . If X and Y are objects in D ( Z ), we willwrite Ext i ( X, Y ) for their i th hyperext group, see Definition 10.7.1 of [Wei94].For a topological abelian group A , we define A ∗ = Hom cont ( A, Q / Z ), where Q / Z has thediscrete topology. If A is the group of rational points of some commutative algebraic groupscheme over a local field, it will be understood that A is endowed with the natural topologycoming from the local field.If X is any Grothendieck site, we denote by S ( X ) the category of sheaves of abelian groupson X . We let D ( X ) denote the bounded derived category of S ( X ).
2. Weil Groups of Finite Fields
We begin with a duality theorem for the cohomology of the Weil group w of a finite field k , andrestate the theorem as a duality theorem in Weil-´etale cohomology. Since w ≃ Z as abstractgroups, this is mostly an exercise in homological algebra. It will be useful to have at the readythe following explicit description of the Z -dual of a complex. Proposition . Let C be a bounded cochain complex of abelian groups, considered as anobject in D ( Z ) . Let Z [ − n ] be the cochain complex with Z in degree n and everywhere else.Then for all i we have short exact sequences → Ext ( H n − i +1 ( C ) , Z ) → Ext i ( C, Z [ − n ]) → Hom ( H n − i ( C ) , Z ) → . (1)3 avid A. Karpuk Proof.
This follows easily by applying R Hom( C, − ) to the exact sequence 0 → Z → Q → Q / Z → Proposition . Let R be a commutative ring with , and suppose that M is an R [ w ] -module. Let D be an injective R -module on which w acts trivially, so that H ( w , D ) = D . Thenthe cup-product pairing H i ( w , M ) ⊗ R H − i ( w , Hom R ( M, D )) → D induces isomorphisms H − i ( w , Hom R ( M, D )) ≃ Hom R ( H i ( w , M ) , D ) for i = 0 , . Proof.
Applying the functor Hom R ( − , D ) to the exact sequence 0 → M w → M σ − → M → M w → → Hom R ( M w , D ) → Hom R ( M, D ) σ − → Hom R ( M, D ) → Hom R ( M w , D ) → . The proposition now follows from the fact that H ( w , Hom R ( M, D )) and H ( w , Hom R ( M, D ))are, respectfully, the kernel and cokernel of the map σ − R ( M, D ) to itself.
Example . Let R = Z , D = Q / Z , and suppose that M is finite. The canonical isomorphism M = M ∗∗ yields the following perfect pairings of finite abelian groups: H i ( w , M ) ⊗ H − i ( w , M ∗ ) → Q / Z . Example . Let R be a field F of characteristic zero, and suppose that M = V is a finite-dimensional representation of w . Let D = F be the trivial representation, and consider the dualrepresentation Hom F ( V, D ). By (2.1.1) we have the following perfect pairings of vector spacesover F : H i ( w , V ) ⊗ F H − i ( w , Hom F ( V, F )) → F. Suppose that M is a bounded complex of w -modules, considered as an object in D ( w ). Let usdefine its dual complex by M D := R Hom( M, Z ) ∈ D ( w ), which comes equipped with a canonicalpairing M ⊗ L M D → Z in D ( w ). The projection map R Γ w ( Z ) → Z [ −
1] gives a cup-productpairing R Γ w ( M ) ⊗ L R Γ w ( M D ) → Z [ − ψ ( M ) : R Γ w ( M D ) → R Hom( R Γ w ( M ) , Z [ − . (2) Proposition . Suppose that M is finite. Then ψ ( M ) is an isomorphism. Proof.
Because M is finite, we have identifications M D = M ∗ [ −
1] and Ext i ( R Γ w ( M ) , Z [ − H − i ( w , M ) ∗ , the second of which follows from (2.1.2). The map ψ ( M ) i is the map H i − ( w , M ∗ ) → H − i ( w , M ) ∗ induced by cup-product, which is an isomorphism by (2.1.2). Proposition . Suppose that M is free and finitely generated as an abelian group. Then ψ ( M ) is an isomorphism. Proof. (The author thanks Thomas Geisser for pointing out this argument, which is presented inLemma 7.5 of [Gei10].) Let L ( − ) w be the derived functor of M M w . There is a natural equiv-alence L ( − ) w ≃ R Γ w ( − )[1]. Combined with the adjunction map Hom w ( M, Z ) ≃ Hom( M w , Z )induced by the evaluation pairing, taking derived functors yields the isomorphisms R Hom w ( M, Z ) ≃ R Hom( L ( M ) w , Z ) ≃ R Hom( R Γ w ( M )[1] , Z ) ≃ R Hom( R Γ w ( M ) , Z [ − eil-´etale Cohomology over p -adic Fields which gives the result. Theorem . Let M be a bounded complex of w -modules, whose cohomology groups arefinitely generated as abelian groups. Then the map ψ ( M ) of (2) is an isomorphism. Proof.
This follows easily from the above two propositions; one first proves it first for M con-centrated in degree 0 by considering the exact sequence 0 → M tors → M → M/M tors →
0. Thegeneral result follows by induction on the length of the complex.
Let us restate the duality theorem of the last section in terms of Weil-´etale cohomology. For ascheme X which is finite type over k , let X W denote X endowed with the Weil-´etale topology,as defined in [Lic05]. Let Γ X : S ( X W ) → A b be the global sections functor, and let R Γ X : D ( X W ) → D ( Z ) be its derived functor.Now let X = Spec k . By Proposition 2.2 of [Lic05], Weil-´etale sheaves on X W are simply w -modules. Theorem (2.1.6) has the following rephrasing in terms of Weil-´etale cohomology: Theorem . Let F be a bounded complex of Weil-´etale sheaves on X = Spec k , such that thecohomology sheaves H i ( F ) correspond to finitely generated w -modules. Let F D = R Hom X ( F, Z ) .Then the cup-product pairing R Γ X ( F ) ⊗ L R Γ X ( F D ) → Z [ − induces an isomorphism R Γ X ( F D ) → R Hom ( R Γ X ( F ) , Z [ − in the derived category of abelian groups. We remind the reader of Lichtenbaum’s duality theorem for the Weil-´etale cohomology ofcurves, proved in [Lic05]. Let
X/k be a smooth, geometrically connected curve. For simplic-ity we assume X is projective, though Lichtenbaum does not make this assumption. One has H ( X W , G m ) = Z and H i ( X W , G m ) = 0 for i > Theorem . ([Lic05], Theorem 5.1) Let F be a locally constant Weil-´etale sheaf on X ,representable by a finitely generated abelian group, and let F D = R Hom X ( F, G m ) . The cup-product pairing R Γ X ( F ) ⊗ L R Γ X ( F D ) → Z [ − induces an isomorphism R Γ X ( F D ) → R Hom ( R Γ X ( F ) , Z [ − in the derived category of abelian groups. The following two lemmas will prove useful in the next chapter, and are generalizations of theadditive and multiplicative Hilbert Theorem 90 for finite fields.
Lemma . Let M be a w -module which is finitely generated and free as an abelian group.Then:(i) H ( w , Hom ( M, ¯ k × )) is finite,(ii) H ( w , Hom ( M, ¯ k )) is finite, avid A. Karpuk (iii) and H ( w , Hom ( M, U L )) is profinite. Proof. (i) Let f : M → ¯ k × be w -equivariant. Then f ( σm ) = f ( qm ), so f factors through thefinite group M/ ( σ − q ) M . Since ¯ k × has only finitely many elements of any particular order, theimage of f lands in a finite set which is independent of f . Therefore there are only finitely manypossible f .(ii) Choose a Z -basis of M and a matrix A representing the action of σ on M ∼ = Z r . This Z -basis provides us with an isomorphism Hom( M, ¯ k ) ≃ ¯ k r . Under this isomorphism, elements of H ( w , Hom( M, ¯ k )) correspond to solutions of the equation Ax = x q , for x ∈ ¯ k r (here raising tothe q th power is done component-wise).Writing the coordinates of x as X , . . . , X r , we see that we must show that the affine variety V defined by the equations P ri =1 a ij X i − X qj = 0 for 1 j r is finite, where A = ( a ij ). Let V j ⊂ ¯ k r be the hypersurface defined by P ri =1 a ij X i − X qj , so that V = T j V j . Let v ∈ V , andlet T v be the tangent space to V at v . Then 0 = dim ker( A ) = dim T v > dim V, hence V hasdimension zero and is therefore finite.(iii) Since H ( w , − ) commutes with inverse limits, we have H ( w , Hom(
M, U L )) = lim ←− i H ( w , Hom(
M, U L /U ( i ) L )) . Inducting on the sequences 0 → U ( i ) L /U ( i +1) L → U L /U ( i +1) L → U L /U ( i ) L →
0, we see from parts (i)and (ii) that the terms appearing in the above inverse limit are all finite, hence the result.
Lemma . Let M be a w -module which is finitely generated and free as an abelian group.Then:(i) H ( w , Hom ( M, ¯ k × )) = 0 .(ii) H ( w , Hom ( M, ¯ k )) = 0 .(iii) H ( w , Hom ( M, U L )) = 0 . (See Chapter XIII, Proposition 15 of [Ser79] for the case M = Z .) Proof. (i) Writing the group law on ¯ k × additively, we wish to show that the map ( σ −
1) :Hom( M, ¯ k × ) → Hom( M, ¯ k × ) is surjective. Choosing a Z -basis of M determines an isomorphismHom( M, ¯ k × ) ≃ (¯ k × ) r and a matrix A ∈ GL r ( Z ) representing the action of σ on M . Directcalculation shows that under our isomorphism Hom( M, ¯ k × ) ≃ (¯ k × ) r , the map σ − qA − − k × ) r → (¯ k × ) r . Multiplying by the automorphism − A , we conclude that it sufficesto prove that A − q is surjective, as a map from (¯ k × ) r to itself.It is clear that q is not an eigenvalue of A , since det( A ) = ±
1, but the characteristic polynomialof A has integer coefficients. It follows that the map A − q : Z r → Z r is injective with finitecokernel C . Tensoring with ¯ k × , we obtain an exact sequence (¯ k × ) r A − q → (¯ k × ) r → C ⊗ ¯ k × → . But C ⊗ ¯ k × = 0, because ¯ k × is divisible and C is finite.(ii) As in part (i), we choose a Z -basis for M and a matrix A representing the action of σ on M . On the group Hom( M, ¯ k ), direct calculation shows that (( σ − f )( m ) = f ( A − m ) q − f ( m ),and replacing m with Am and multiplying by −
1, we are reduced to showing that the map f f ◦ A − f q is surjective. Under the isomorphism Hom( M, ¯ k ) ≃ ¯ k r , this is the map x Ax − x q ,where raising to the q th power is done component-wise. A proof of the surjectivity of this mapis contained in Corollary 5.1.2 of [GHKR10].(iii) Recall that U L comes equipped with a filtration · · · ⊂ U ( i ) L ⊂ · · · ⊂ U (1) L ⊂ U L of w -modules, with U L /U (1) L ≃ ¯ k × , and higher successive quotients all isomorphic to ¯ k . Let f : M → eil-´etale Cohomology over p -adic Fields U L , and reduce modulo U (1) L . We obtain a map ¯ f : M → ¯ k × , and by part (i) we can write¯ f = ( σ −
1) ¯ f for some ¯ f : M → ¯ k × . As M is free, we can lift ¯ f to a map f : M → U L , andwe have f = ( σ − f + g for some g : M → U (1) L . It is clear that by repeating this process, wecan write f = P i > ( σ − f i = ( σ − P i > f i ), where f i : M → U ( i ) L .
3. Weil Groups of p -adic Fields We now turn our attention to p -adic fields. The main theorem of this chapter is a duality theoremfor the cohomology of the Weil group of a p -adic field, which can be interpreted as a dualitytheorem for the Weil-´etale cohomology of the spectrum of the field.Let K be a local field of characteristic zero, with finite residue field k . Let K ur be its maximalunramified extension, and let L be the completion of K ur . Let ¯ K be an algebraic closure of K ,and let ¯ L be an algebraic closure of L containing ¯ K .Let g = Gal( K ur /K ) = Gal(¯ k/k ) ≃ ˆ Z , which has subgroup w ≃ Z consisting of integralpowers of the Frobenius. The action of w on K ur extends by continuity to L . We let I =Gal( ¯ K/K ur ) be the inertia subgroup of G = Gal( ¯ K/K ). It follows from Krasner’s Lemma (seeLemma 8.1.6 of [NSW08]) applied to the extension K ur ⊂ L that I = Gal( ¯ L/L ).We define the Weil group W of K to be the pullback of w under the surjection G → g . Thediagram 0 / / I / / W / / (cid:15) (cid:15) w / / (cid:15) (cid:15) / / I / / G / / g / / W is such that I is open, and translation by any preimage of σ is a homeomorphism.By Chapter XIII, Lemma 1 of [Ser79], the fixed points of w acting on L are exactly K ; it isimmediate that the fixed points of W acting on ¯ L are also K . Similar remarks apply to w and W acting on the L and ¯ L -points, respectively, of some commutative algebraic group scheme definedover K .The exact sequence 0 → I → W → w → H i ( w , H j ( I, M )) ⇒ H i + j ( W, M ) for any W -module M . Because scd( w ) = 1, thisspectral sequence degenerates to a collection of short exact sequences0 → H ( w , H i − ( I, M )) → H i ( W, M ) → H ( w , H i ( I, M )) → i >
0. Recall from Chapter II § I ) = 1 and scd( I ) = 2; it follows that H i ( W, M ) = 0 for all i > M ; we therefore suppress any mention of H i ( W, M ) for i > m and Finite W -Modules Proposition . The cohomology groups H i ( W, ¯ L × ) are given by H i ( W, ¯ L × ) = K × if i = 0 Z if i = 10 if i > . Proof. (See [Lic99].) The field L is a C , hence H i ( I, ¯ L × ) vanishes for i >
1. It follows immediately7 avid A. Karpuk from (5) that H i ( W, ¯ L × ) = 0 for i >
2. For i = 1, (5) yields an isomorphism H ( w , L × ) ≃ H ( W, ¯ L × ). The long exact sequence of w -cohomology of 0 → U L → L × → Z → H ( W, ¯ L × ) = H ( w , Z ) = Z , since H ( w , U L ) = 0 by (2.3.2). Corollary . Let µ n be the group of n th roots of unity in ¯ L × , and let µ = lim −→ n µ n . Then H i ( W, µ n ) = µ n ( K ) if i = 0 K × / ( K × ) n if i = 1 Z /n Z if i = 20 if i > ,and thus H ( W, µ ) = lim −→ n H ( W, µ n ) = Q / Z . Proof.
Consider the long exact sequence in cohomology of the Kummer sequence 0 → µ n → ¯ L × → ¯ L × →
0. Using (3.1.1), we see that the long exact sequence reads0 → µ n ( K ) → K × n → K × → H ( W, µ n ) → Z n → Z → H ( W, µ n ) → Theorem . Let M be a finite W -module.(i) The groups H i ( W, M ) are finite for all i , and vanish for i > .(ii) We have cd ( W ) = 2 .(iii) (Weil-Tate Local Duality) Let M ′ = Hom ( M, µ ) with its natural W -action. Then the cup-product pairing H i ( W, M ) ⊗ H − i ( W, M ′ ) → Q / Z is a perfect pairing of finite groups. Proof.
The proof of (i) is the same as for the Galois group, see Chapter II, § w ) = 1 and cd( N ) = 1. From the short exact sequences of 5, we seethat cd( W )
2. But H ( W, µ n ) = 0, hence the equality cd( W ) = 2. The proof of (iii) is thesame as for Tate Local Duality, as in Chapter II, § W when using Shapiro’s Lemma. H ( W, M ) for Finitely Generated M The vanishing theorem of this section is needed to prove the duality theorem of the next section,but it is interesting in its own right. As a consequence of this vanishing theorem, we deduce atheorem of Rajan (and offer a slight correction to his proof).
Theorem . Let M be a W -module which is finitely generated as an abelian group. Then H ( W, M ) = 0 . Proof.
The results of the previous section prove that H ( W, M ) = 0 for finite M , so we mayassume M is free. Let Q be the rationals with trivial W -action; then M Q = M ⊗ Q and P = M Q /M fit into the exact sequence 0 → M → M Q → P → W -modules. The Q -vector space M Q is uniquely divisible, therefore H ( I, M Q ) = H ( I, M Q ) = 0, which implies by (5) that H ( W, M Q ) = H ( W, M Q ) = 0. Therefore H ( W, P ) ≃ H ( W, M ), so it suffices to show thisformer group vanishes. 8 eil-´etale Cohomology over p -adic Fields Let P n denote the n -torsion of P . By Weil-Tate Local Duality (3.1.3) we have H ( W, P ) ≃ lim −→ n H ( W, P n ) ≃ lim −→ n Hom W ( P n , µ ) ∗ ≃ (lim ←− n Hom W ( P n , µ )) ∗ ≃ Hom W ( P, µ ) ∗ , where lim ←− n Hom W ( P n , µ ) has its profinite topology. From these isomorphisms, we see that itsuffices to show Hom W ( P, µ ) = 0.Let f : P → µ be W -equivariant, and let I ′ be an open, normal subgroup of I such that I ′ acts trivially on M . Then for all τ ∈ I ′ and e ∈ P , we have f ( e ) = f ( τ · e ) = f ( e ) τ . It followsthat f ( P ) ∩ µ p ∞ is finite, where p = char( k ). Since P has no finite quotients and we wish toprove that f ( P ) = 1, we may assume that f ( P ) ⊆ µ ′ , the group of roots of unity of order primeto p . On µ ′ , W acts via the natural action of σ , which is ζ ζ q , where q = k .Let a ∈ W be a preimage of σ . Applying the W -equivariance of f to a , we see that f ( a · e ) = f ( e ) q = f ( q · e ) for all e ∈ P . Therefore ( a − q )( P ) ⊆ ker( f ). We claim that ( a − q ) : P → P issurjective, which implies f = 1. By the Snake Lemma, it suffices to prove that a − q is surjectiveon M Q . Surjectivity of a − q on M Q is equivalent to injectivity, and injectivity will hold if andonly if q is not an eigenvalue of a . But q cannot be an eigenvalue of a , because det( a ) = ± a has integer coefficients. Remark . The above theorem does not immediately imply that H ( W, M ) = 0 for all W -modules M , since one cannot write an arbitrary W -module as a direct limit of modules whichare finitely generated as abelian groups. For example, let w act on M = L i ∈ Z Z by shifting theindices in the obvious way, and let W act on M through the quotient map W → w .Suppose now that A is a topological W -module; that is A is a topological abelian group with acontinuous action of W . For such A , we can define cohomology groups H icc ( W, A ) using complexesof continuous cochains. It follows from Corollary 2.4 of [Lic09] and Corollary 2 of [Fla08] thatthe groups H icc ( W, A ) agree with the topological group cohomology used in [Lic09] and [Fla08],which is defined using the classifying topos BW .The non-discrete topological W -modules in which we are interested are all complex manifolds,and hence Remark 2.2 of [Lic09] shows that the groups H icc ( W, A ) also agree with the cohomologygroups H iM ( W, A ) defined by Moore in [Moo76] and used by Rajan in [Raj04]. If A is discrete,then H icc ( W, A ) can be identified with the Galois cohomology groups H i ( W, A ); this follows fromthe remarks preceding Lemma 1 of [Raj04].
Theorem . Let T be an algebraic torus over C , equipped with an action of W via alge-braic automorphisms. Then H cc ( W, T ( C )) = 0 . In particular, for T = G m , we conclude that H ( W, C × ) = 0 . Proof.
From Corollary 8 of [Fla08], it follows that H icc ( I, V ) = 0 for any finite-dimensionalcomplex vector space V and any i >
1. Thus the spectral sequence of Moore (quoted by Rajanas Proposition 5 of [Raj04]) gives isomorphisms H i ( w , H cc ( I, V )) = H icc ( W, V ) , and thus thislatter group vanishes for i > X ∗ ( T ) denote the cocharacter group of T ; it is a finitely generated discrete W -module.There is a short exact sequence of topological W -modules,0 → X ∗ ( T ) → X ∗ ( T ) ⊗ C → T ( C ) → . The groups H icc ( W, X ∗ ( T ) ⊗ C ) vanish for i > H i ( W, X ∗ ( T )) vanish for i > avid A. Karpuk In [Raj04], Rajan proves that H M ( W, T ( C )) = 0, for those tori with an action of W thatcomes from an action of G . His proof seems to be slightly flawed, because his Proposition 6asserts that H ( G, A ) → H ( W, A ) is an isomorphism for any G -module A , when in fact thismap has a kernel for A = Z . However, Rajan’s proof of the vanishing of H M ( W, T ( C )) ultimatelyrelies only on the surjectivity of H ( G, A ) → H ( W, A ), which holds by by (4.1.3).Theorem (3.2.3) implies that the map H ( W, GL n ( C )) → H ( W, PGL n ( C )) is surjective,which says exactly that every projective complex representation of W lifts to an affine represen-tation. Let Γ W : W -Mod → A b be the fixed-points functor, which has derived functor R Γ W : D ( W ) →D ( Z ), so that H i ( R Γ W ( M )) = H i ( W, M ) for any W -module M . By (3.1.1), there is a naturalprojection map R Γ W ( ¯ L × ) → Z [ − M is any bounded complex of W -modules, we set M D := R Hom( M, ¯ L × ). The cup-product pairing R Γ W ( M ) ⊗ L R Γ W ( M D ) → Z [ − ψ ( M ) : R Γ W ( M D ) → R Hom( R Γ W ( M ) , Z [ − . (6) Theorem . Suppose that M is a bounded complex of W -modules, whose cohomology groupsare finitely generated as abelian groups. Then the map ψ ( M ) of (6) is an isomorphism in D ( Z ) . By induction on the length of the complex, it is clear that it suffices to prove the result for M concentrated in degree 0. Furthermore, for finite M , (3.3.1) is easily seen to be equivalent to(3.1.3). So it suffices to prove (3.3.1) for finitely generated, free W -modules M . For the rest ofthe section, we let M be such a module.The proof of this theorem relies on several lemmas. To begin, the discrete valuation L × → Z induces a valuation ¯ L × → Q . Let U ¯ L be the kernel of this map, so that we have a short exactsequence 0 → U ¯ L → ¯ L × → Q → W -modules. Taking W -cohomology, it follows easily that H ( W, U ¯ L ) = Q / Z , and H i ( W, U ¯ L ) =0 for i >
3. Cup-product therefore induces the map R Γ W (Hom( M, U ¯ L )) / / (cid:15) (cid:15) R Hom( R Γ W ( M ) , Q / Z [ − (cid:15) (cid:15) R Γ W ( M D ) (cid:15) (cid:15) ψ ( M ) / / R Hom( R Γ W ( M ) , Z [ − (cid:15) (cid:15) R Γ W (Hom( M, Q )) / / R Hom( R Γ W ( M ) , Q [ − D ( Z ). Lemma . To prove Theorem (3.3.1), it suffices to show that the maps H i ( W, Hom ( M, U ¯ L )) → H − i ( W, M ) ∗ and H j ( W, Hom ( M, Q )) → Hom ( H − j ( W, M ) , Q )10 eil-´etale Cohomology over p -adic Fields induced by cup-product are isomorphisms for all i > and all j > . Proof.
By the Five Lemma, it to prove Theorem (3.3.1), it suffices to prove that the top andbottom maps of (7) are isomorphisms. Because Q and Q / Z are injective abelian groups, this iseasily seen to be equivalent to the statement of the lemma. Proposition . The map H j ( W, Hom ( M, Q )) → Hom ( H − j ( W, M ) , Q ) induced by cup-product is an isomorphism for all j > . Proof.
Because the quotient M Q /M has only trivial maps to Q , our map can be identified with H j ( W, Hom Q ( M Q , Q )) → Hom Q ( H − j ( W, M Q ) , Q ) . Since H i ( I, V ) vanishes for i >
1, proving the proposition is equivalent, by (5), to showing thatthe map H j ( w , Hom Q ( M I Q , Q )) → Hom Q ( H − j ( w , M I Q ) , Q )is an isomorphism for all j >
0. This follows immediately from our results on w -duality for vectorspace coefficients.To prove that the maps H i ( W, Hom(
M, U ¯ L )) → H − i ( W, M ) ∗ are all isomorphisms, we willuse (5). In particular, for every i >
0, we have a map of short exact sequences0 (cid:15) (cid:15) (cid:15) (cid:15) H ( w , H i − ( I, Hom(
M, U ¯ L ))) / / (cid:15) (cid:15) H ( w , H − i ( I, M )) ∗ (cid:15) (cid:15) H i ( W, Hom(
M, U ¯ L )) / / (cid:15) (cid:15) H − i ( W, M ) ∗ (cid:15) (cid:15) H ( w , H i ( I, Hom(
M, U ¯ L )) / / (cid:15) (cid:15) H ( w , H − i ( I, M )) ∗ (cid:15) (cid:15) i >
1, that the top andbottom horizontal arrows are isomorphisms. It should be noted that the top and bottom arrowsare not isomorphisms for all i >
0: take M = Z and i = 0, then the bottom arrow is the zeromap U K → L ′ /L be a finite Galois extension with group H and degree e . Consider the long exactsequence in H -cohomology of 0 → U L ′ → L ′× → Z →
0. Since the groups H i ( H, L ′× ) and H i ( H, Q ) vanish for i >
1, the long exact sequence reads0 → U L → L × → Z → H ( H, U L ′ ) → H ( H, U L ′ ) = Z /e Z . Taking the limit over all L ′ /L givesan identification H ( I, U ¯ L ) = Q / Z . 11 avid A. Karpuk Throughout the proof of the next proposition, we will make use of the Tate cohomologygroups ˆ H ∗ ( H, − ), for a finite group H . For definitions and basic properties of Tate cohomologygroups, see Chapter VIII of [Ser79]. Proposition . The maps H i ( w , H ( I, Hom ( M, U ¯ L ))) → H − i ( w , H ( I, M )) ∗ are isomorphisms for all i > . Proof.
Let M be an I -module which is finitely generated and free as an abelian group. We willshow that α : H ( I, Hom(
M, U ¯ L ))) → H ( I, M ) ∗ is an isomorphism. The proposition then follows easily from w -duality. If M = Z with trivial I -action, then α is the canonical map Q / Z → Z ∗ . Since cohomology commutes with direct sums,this proves the result for M with trivial action.Now let M be any finitely generated I -module, and choose a finite Galois extension L ′ /L with group H , such that the open subgroup I ′ = Gal( ¯ L/L ′ ) of N acts trivially on M . We mimicthe argument on page 128 of [Lic69]. Consider the diagrams0 / / H ( H, Hom(
M, U L ′ )) inf / / γ (cid:15) (cid:15) H ( I, Hom(
M, U ¯ L )) res / / α (cid:15) (cid:15) H ( I ′ , Hom(
M, U ¯ L )) α ′ (cid:15) (cid:15) / / ˆ H ( H, M ) ∗ / / H ( I, M ) ∗ tr ∗ / / M ∗ and H ( I ′ , Hom(
M, U ¯ L )) tr / / α ′ (cid:15) (cid:15) H ( I, Hom(
M, U ¯ L )) α (cid:15) (cid:15) M ∗ / / H ( I, M ) ∗ / / α ′ is the corresponding map for L ′ . By the previous paragraph, α ′ is an isomorphism.From the second of the above two diagrams, we see that α is surjective. To show that α is anisomorphism, we only need to show it is injective, and for this, it suffices to prove that γ is anisomorphism.By Chapter X, §
7, Proposition 11 and Chapter IX, §
5, Theorem 9 of [Ser79], Hom(
M, L ′× )is cohomologically trivial for H . Applying the exact functor Hom( M, − ) to 0 → U L ′ → L ′× → Z → H -cohomology gives us an isomorphism δ : ˆ H ( H, Hom( M, Z )) → H ( H, Hom(
M, U L ′ ))which commutes with cup-product in the sense that the diagramˆ H ( H, Hom( M, Z )) / / δ (cid:15) (cid:15) ˆ H ( H, M ) ∗ H ( H, Hom(
M, U L ′ )) γ / / ˆ H ( H, M ) ∗ commutes. By ([NSW08], Chapter III, Proposition 3.1.2) the top horizontal arrow is an isomor-phism, and we conclude that γ is an isomorphism.12 eil-´etale Cohomology over p -adic FieldsProposition . The map H ( w , H ( I, Hom ( M, U ¯ L ))) → H ( w , H ( I, M )) ∗ induced by cup-product is an isomorphism. Proof.
Let L ′ /L be a finite Galois extension with group H , such that I ′ = Gal( ¯ L/L ′ ) actstrivially on M . Using the same argument as in the previous proposition and ([NSW08], ChapterIII, Proposition 3.1.2), we see that the mapˆ H ( H, Hom(
M, U L ′ )) → H ( H, M ) ∗ is an isomorphism.We claim that there is an isomorphism H ( w , H ( N, Hom(
M, U ¯ L ))) = H ( w , ˆ H ( H, Hom(
M, U L ′ ))) . Together with the fact that the inflation map H ( H, M ) → H ( I, M ) is an isomorphism (since H ( I ′ , M ) = 0), this will suffice to prove the proposition. It is clear that H ( I, Hom(
M, U ¯ L )) = H ( H, Hom(
M, U L ′ )), which reduces us to showing that the natural map H ( w , H ( H, Hom(
M, U L ′ ))) → H ( w , ˆ H ( H, Hom(
M, U L ′ )))is an isomorphism.For any H -module P , let Nm H : P → H ( H, P ) be the norm map, defined by m P s ∈ H s · m .Consider the reduction mapNm H (Hom( M, U L ′ )) → Nm H (Hom( M, U L ′ /U (1) L ′ )) = Nm H (Hom( M, ¯ k × )) . The extension L ′ /L is totally ramified, hence H acts trivially on ¯ k × . Thus for any f : M → ¯ k × ,we have Nm H ( f )( m ) = X s ∈ H sf ( s − m ) = X s ∈ H f ( s − m ) = f (Nm H ( m ))which proves that Nm H (Hom( M, ¯ k × )) = Hom(Nm H ( M ) , ¯ k × ). As Nm H ( M ) ⊂ M is a finitelygenerated and free abelian group, we see from (2.3.2) that H ( w , Hom(Nm H ( M ) , ¯ k × )) = 0, andthe same argument shows that H ( w , Hom(Nm H ( M ) , ¯ k )) = 0. One now proceeds in the samefashion as in the proof of (2.3.2) to show that H ( w , Nm H (Hom( M, U L ′ ))) = 0. Proposition . The maps H i ( W, Hom ( M, U ¯ L )) → H − i ( W, M ) ∗ (9) are isomorphisms for all i > . For i = 0 , this is a topological isomorphism of profinite groups. Proof.
For i >
1, this follows immediately from (8), (3.3.4), and (3.3.5). For i = 0, multiplication-by- n Kummer sequences for M and Hom( M, U ¯ L ) give rise to a map of long exact cohomologysequences, the relevant part of which reads0 / / H (Hom( M, U ¯ L )) /n δ / / (cid:15) (cid:15) H (Hom( M, U ¯ L )[ n ]) / / ≀ (cid:15) (cid:15) H (Hom( M, U ¯ L ))[ n ] ≀ (cid:15) (cid:15) / / H ( M )[ n ] ∗ δ ∗ / / H ( M/n ) ∗ / / ( H ( M ) /n ) ∗ where we have written H i ( − ) for H i ( W, − ) for convenience. The middle arrow is Weil-Tate LocalDuality applied to the finite modules Hom( M, U ¯ L )[ n ] = Hom( M/n, µ n ) and M/n . The right-most13 avid A. Karpuk arrow is the map of (9) when i = 1, restricted to n -torsion; it is therefore an isomorphism. Weconclude by the Five Lemma that the left-most vertical arrow is an isomorphism for all n . Takingthe inverse limit, we conclude thatlim ←− n H ( W, Hom(
M, U ¯ L )) /n → H ( W, M ) ∗ is an isomorphism (recall that H ( W, M ) is torsion).It remains to show that the natural map H ( W, Hom(
M, U ¯ L )) → lim ←− n H ( W, Hom(
M, U ¯ L )) /n is an isomorphism; in other words, to show that H ( W, Hom(
M, U ¯ L )) is profinite. If I acts triviallyon M , then this is contained in the statement of (2.3.1). Otherwise, pick an open normal subgroup I ′ of I acting trivially on M , corresponding to a finite Galois extension L ′ /L with group H . Wehave H ( W, Hom(
M, U ¯ L )) = H ( w , H ( H, Hom(
M, U L ′ )))and the latter group is clearly profinite, hence we are done.Theorem (3.3.1) was proven by Jiang in his thesis (see Proposition 4.15 and Theorem 5.3 of[Jia06]). However, Jiang assumes that M is a G -module, and uses Tate-Nakayama Duality in hisproof. We have removed the condition that M be a G -module, and presented a proof which isindependent of the main results of Local Class Field Theory.A natural question to ask is whether the other map induced by the cup-product pairing,namely η ( M ) : R Γ W ( M ) → R Hom( R Γ W ( M D ) , Z [ − , (10)is an isomorphism. The next proposition shows that this map fails to be an isomorphism when M = Z with trivial action, due to the non-trivial natural topology on the cohomology groups ofthe complex R Γ W ( M D ). However, elucidating this map will prove useful in later sections, so wenow describe it explicitly. Proposition . The map η ( Z ) : R Γ W ( Z ) → R Hom ( R Γ W ( ¯ L × ) , Z [ − has the following properties:(i) η ( Z ) i is an isomorphism for i = 2 .(ii) η ( Z ) induces an isomorphism of H ( W, Z ) with the torsion subgroup U ∗ K of Ext ( R Γ W ( ¯ L × ) , Z [ − .The cohomology of both complexes vanishes outside of degrees through . Proof.
For i = 0 ,
1, explicit calculation using (2.0.1) shows that the cohomology of both sides is Z , and the map between them is the identity map. For i >
3, using (2.0.1) one shows easily thatthe cohomology of both complexes vanishes.The only assertion left to prove is that of (ii). Coming from the sequences 0 → Z → Z → Z /n Z → → µ n → U ¯ L → U ¯ L → / / H ( W, Z ) /n / / ≀ (cid:15) (cid:15) H ( W, Z /n Z ) δ / / ≀ (cid:15) (cid:15) H ( W, Z )[ n ] / / (cid:15) (cid:15) / / ( H ( W, U ¯ L ) ∗ ) /n / / H ( W, µ n ) ∗ δ ∗ / / U ∗ K [ n ] / / . eil-´etale Cohomology over p -adic Fields Here the left vertical arrow is the natural isomorphism Z /n Z → ˆ Z /n ˆ Z , and the middle arrow isthe isomorphism of Weil-Tate Local Duality. Therefore the right vertical arrow is an isomorphism,and by passing to the limit over all n , we see that H ( W, Z ) → U ∗ K is an isomorphism. Asimple calculation using (2.0.1) shows that Ext ( R Γ W ( ¯ L × ) , Z [ − U K , Z ), whose torsionsubgroup is U ∗ K . Corollary . The cohomology groups H i ( W, Z ) are given by H i ( W, Z ) = Z i = 0 , U ∗ K i = 20 i > . Proof.
This is contained in the proof of the previous proposition.
4. Local Class Field Theory via the Weil Group
In this section we show how to deduce the main theorems of Local Class Field Theory from thetheorems of the previous chapter. We prove, in particular, that Br( K ) = Q / Z and that there isa canonical isomorphism K × ⊗ ˆ Z → G ab . Any G -module can be given a W -module structure via the map W → G , and this process inducesrestriction maps H i ( G, M ) → H i ( W, M ) on cohomology. The following comparison theoremsdescribe these restriction maps.
Proposition . Let M be a torsion G -module. Then there are functorial isomorphisms H i ( G, M ) = H i ( W, M ) for all i > . Proof. (This proof appears in the unpublished note [Lic99] of Lichtenbaum.) Since Galois co-homology is always torsion, and g and w cohomology agree for torsion coefficients by ChapterXIII, Proposition 1 of [Ser79], the restriction maps H i ( g , H j ( N, M )) → H i ( w , H j ( N, M )) areisomorphisms for all i, j >
0. Thus the map of spectral sequences H i ( g , H j ( N, M )) + (cid:15) (cid:15) H i + j ( G, M ) (cid:15) (cid:15) H i ( w , H j ( N, M )) + H i + j ( W, M )is an isomorphism on the second page and therefore in the limit.
Corollary . (Tate Local Duality) Let M be a finite G -module. The cup-product pairing H i ( G, M ) ⊗ H − i ( G, M ′ ) → Q / Z is a perfect pairing of finite groups. Proof.
This follows from (3.1.3), and (4.1.1) applied to the finite module M . Theorem . Let M be a discrete G -module. Then there are functorial isomorphisms(i) H ( G, M ) = H ( W, M ) and(ii) H ( G, M ) = H ( W, M ) tors , avid A. Karpuk (iii) there is a short exact sequence → H ( W, M ) ⊗ Q / Z → H ( G, M ) → H ( W, M ) → , (iv) and there are isomorphisms H i ( W, M ) = H i ( G, M ) for all i > . Proof.
By considering the exact sequence 0 → M tors → M → M/M tors → M is torsion-free. Since W is dense in G it is clear that H ( G, M ) = H ( W, M ). Kummer sequences give rise to a diagram0 / / H ( G, M ) /n / / ≀ (cid:15) (cid:15) H ( G, M/nM ) δ / / ≀ (cid:15) (cid:15) H ( G, M )[ n ] / / (cid:15) (cid:15) / / H ( W, M ) /n / / H ( W, M/nM ) δ / / H ( W, M )[ n ] / / . Since H ( G, M ) is all torsion, passing to the limit proves that H ( G, M ) = H ( W, M ) tors .For part (iii), consider the diagram0 / / H ( G, M ) /n / / β (cid:15) (cid:15) H ( G, M/n ) δ / / ≀ (cid:15) (cid:15) H ( G, M )[ n ] κ (cid:15) (cid:15) / / / / H ( W, M ) /n / / H ( W, M/n ) δ / / H ( W, M )[ n ] / / . From the Snake Lemma it is clear that κ is surjective and that ker( κ ) = coker( β ). Passing to thelimit over all n , we havelim −→ n coker( H ( G, M ) /n → H ( W, M ) /n ) = coker( H ( G, M ) ⊗ Q / Z → H ( W, M ) ⊗ Q / Z )= H ( W, M ) ⊗ Q / Z because H ( G, M ) ⊗ Q / Z = 0. The existence of our exact sequence is now clear.For i = 3, we have the diagram0 / / H ( G, M ) /n / / (cid:15) (cid:15) H ( G, M/n ) δ / / ≀ (cid:15) (cid:15) H ( G, M )[ n ] (cid:15) (cid:15) / / / / H ( W, M ) /n / / H ( W, M/n ) δ / / H ( W, M )[ n ] / / . and it follows from the exact sequence 0 → H ( W, M ) ⊗ Q / Z → H ( G, M ) → H ( W, M ) → H ( G, M )[ n ] = H ( W, M )[ n ], and passing tothe limit over all n gives H ( G, M ) = H ( W, M ). The result for i > i and considering Kummer sequences.The above theorem implies that the groups H i ( W, M ) determine the groups H i ( G, M ) up toisomorphism, since H ( W, M ) ⊗ Q / Z is an injective abelian group. The converse fails: take forexample M = Q with trivial action. Then H i ( G, Q ) = 0 for i >
1, but H ( W, Q ) = Hom( w , Q ) = Q . Thus the groups H i ( W, M ) contain more information than their Galois counterparts.
Corollary . The strict cohomological dimension of G is . Proof.
Let M be a G -module. Since the orbit under G of any m ∈ M is finite, we can write M as a direct limit of G -modules M n which are finitely generated as abelian groups. Then by theabove comparison theorem and (3.2.1), we have H ( G, M ) = lim −→ n H ( W, M n ) = 016 eil-´etale Cohomology over p -adic Fields Let A /K be a commutative algebraic group scheme defined over K . Then A ( ¯ K ) is naturallya G -module, and A ( ¯ L ) is naturally a W -module. The inclusion map A ( ¯ K ) → A ( ¯ L ) inducesrestriction maps H i ( G, A ( ¯ K )) → H i ( W, A ( ¯ L )). We have the following comparison theorem: Theorem . Let A be a connected commutative algebraic group scheme over K . There arefunctorial isomorphisms:(i) H ( G, A ( ¯ K )) = H ( W, A ( ¯ L )) and(ii) H ( G, A ( ¯ K )) = H ( W, A ( ¯ L )) tors , (iii) there is a short exact sequence → H ( W, A ( ¯ L )) ⊗ Q / Z → H ( G, A ( ¯ K )) → H ( W, A ( ¯ L )) → , (iv) and the higher cohomology groups H i ( G, A ( ¯ K )) and H i ( W, A ( ¯ L )) vanish for i > . Proof.
First we will show that the multiplication-by- n maps A → A are all surjective. Since A is connected, we have a short exact sequence0 → H → A → B → H is linear and B is an abelian variety (see Theorem 1.1 of [Con02]).As H is commutative, it is the product of a torus and a commutative unipotent group. It followsthat the multiplication-by- n maps on H and B are surjective, hence the same is true of A bythe Five Lemma.The n -torsion of A ( ¯ L ) is contained in A ( ¯ K ), hence there is a map of Kummer sequences0 / / A [ n ] / / A ( ¯ K ) n / / (cid:15) (cid:15) A ( ¯ K ) / / (cid:15) (cid:15) / / A [ n ] / / A ( ¯ L ) n / / A ( ¯ L ) / / . The rest of the proof is exactly as in (4.1.3).
Corollary . There is a natural isomorphism H ( W, ¯ L × ) ⊗ Q / Z = Br ( K ) , and thusBr ( K ) = Q / Z . Proof.
The isomorphism is the map in the short exact sequence of (4.1.5), and the second state-ment follows from (3.1.1).From now on, we denote the groups H i ( G, A ( ¯ K )) and H i ( W, A ( ¯ L )) by H i ( G, A ) and H i ( W, A ),respectively. Recall the main theorem (3.3.1) of the previous section, which states that for a W -module M which is finitely generated as an abelian group, cup-product gives a natural isomorphism R Γ W ( M D ) ∼ → R Hom( R Γ W ( M ) , Z [ − T /K is a torus with character group M = Hom( T, G m ). Then T ( ¯ L ) can be identified with M D as a W -module, and the duality theorem reads R Γ W ( T ) ∼ → R Hom( R Γ W ( M ) , Z [ − . We can use our duality theorem to prove the following:
Proposition . Let
T /K be a torus. Then H ( W, T ) = 0 . avid A. Karpuk Proof.
By the duality theorem, this group is isomorphic to Ext ( R Γ W ( M ) , Z [ − H ( W, M ) , Z ) ∼ → Ext ( R Γ W ( M ) , Z [ − , but the former groupvanishes because H ( W, M ) ⊆ M is finitely generated and free. Corollary . Let
T /K be a torus. Then there are natural isomorphisms(i) H ( G, T ) = H ( W, T ) (ii) H ( G, T ) = H ( W, T ) tors (iii) H ( G, T ) = H ( W, T ) ⊗ Q / Z Proof.
This follows immediately from (4.1.5) and (4.2.1).
Corollary . (Tate-Nakayama Duality) Let T /K be a torus with character group M . Thenthe map H i ( G, T ) → H − i ( G, M ) ∗ induced by cup-product is an isomorphism for i = 1 , , and an isomorphism for i = 0 uponpassing to the profinite completion of the left-hand side. Proof.
Using (5) one sees that H ( W, M ) is finitely generated, hence Ext( H ( W, M ) , Z ) ⊗ Q / Z vanishes. Our duality theorem therefore gives an isomorphism H ( W, T ) ⊗ Q / Z ∼ → Hom( H ( W, M ) , Z ) ⊗ Q / Z = H ( W, M ) ∗ which, by the previous corollary, can be identified with the map H ( G, T ) → H ( G, M ) ∗ .Now consider the case where i = 1. Our duality theorem gives an isomorphism H ( W, T ) tors ∼ → Ext( H ( W, M ) , Z ) = ( H ( W, M ) tors ) ∗ which, again by the previous corollary, can be identified with the map H ( G, T ) → H ( G, M ) ∗ .Finally we treat the case i = 0. Consider the commutative diagram0 / / Ext( H ( W, M ) , Z ) / / = (cid:15) (cid:15) H ( G, T ) / / (cid:15) (cid:15) Hom( H ( W, M ) , Z ) / / (cid:15) (cid:15) / / H ( W, M ) ∗ / / H ( G, M ) ∗ / / ( H ( W, M ) ⊗ Q / Z ) ∗ / / H ( G, T ) = H ( W, T ), our duality theorem, and(2.0.1). The bottom row is the dual of the exact sequence of (4.1.3). The left vertical arrow is theidentity since H ( W, M ) is all torsion, as can be seen easily from (5). The right vertical arrow isan isomorphism upon passing to the profinite completion of Hom( H ( W, M ) , Z ), hence the sameis true of the middle vertical arrow.Of course, for T = G m and i = 0, one recovers from Tate-Nakayama Duality the reciprocityisomorphism K × ⊗ ˆ Z ∼ → G ab of Local Class Field Theory, where − ⊗ ˆ Z denotes profinite com-pletion. Hence one can recover the main statements of Local Class Field Theory by studying thecohomology of the Weil group.
5. The Weil-smooth Topology on Schemes over K For any arbitrary scheme Y , let us recall the definition of the smooth site Y sm . The underlyingcategory is the category of schemes which are smooth and locally of finite type over Y , and the18 eil-´etale Cohomology over p -adic Fields coverings are the surjective families. In [vH04], van Hamel illustrates the utility of the smoothsite in the study of duality theorems; the cohomology groups coincide with those familiar fromthe ´etale site, but the internal hom functor is better suited to proving duality results.This chapter is devoted to introducing a variant of the Weil-´etale topology, the Weil-smooth topology. This definition is motivated by the definition of the Weil-´etale topology given by Jiangin [Jia06], and is related to the smooth topology in the same way that the Weil-´etale topologyis related to the ´etale topology. As with the smooth site, the internal hom functor on the Weil-smooth site is more appropriate for a functorial approach to duality results.
Throughout this section we fix a scheme X which is smooth and finite type over K . Definition . Let π : X L → X and π : X L → Spec L be the projections. We define the Weil-smooth topology W ( X ) to be the following Grothendieck topology:(i) The objects of W ( X ) are the schemes which are smooth and locally of finite type over X L .That is, they are the objects of the smooth site of X L .(ii) A morphism ( V f → X L ) → ( Z g → X L ) of objects in W ( X ), for connected V , is a map φ : V → Z of schemes, such that (a) π ◦ g ◦ φ = π ◦ f , and (b) there exists n ∈ Z such that σ n ◦ π ◦ f = π ◦ g ◦ φ , where σ is the Frobenius automorphism of L . If V is not connected,we impose these conditions component-wise.(iii) The coverings in W ( X ) are the surjective families.We let X W denote X endowed with the Weil-smooth topology.If Y is a scheme, G is a discrete group of automorphisms of Y , and F ∈ S ( Y sm ), we say that G acts on F if there are morphisms F → τ ∗ F of sheaves for all τ ∈ G , compatible in the obvioussense with the multiplication in G . We denote the category of sheaves on Y sm which carry a G -action by S ( Y sm ) G . Proposition . The category S ( X W ) is equivalent to the category S ( X L,sm ) w Proof.
This is the same proof as the analogous result for Weil-´etale sheaves on schemes overfinite fields; see Proposition 2.2 of [Lic05].Let ¯ X = X × K ¯ L and let φ : ¯ X → X L . In a slight abuse of notation, for any F ∈ S ( X W ) = S ( X L,sm ) w , we denote the pullback to ¯ X also by F . Proposition . The group F ( ¯ X ) is a W -module, and H ( W, F ( ¯ X )) = H ( w , F ( X L )) . Proof.
It is clear that F ( ¯ X ) is an I -module; we must show that this action extends to all of W .Let a ∈ W be a preimage of the Frobenius element σ of w . We have F ( ¯ X ) = lim −→ L ′ /L F ( X L ′ )where the limit ranges over all of the finite extensions L ′ of L .For a fixed finite extension L ′ /L , let I ′ ⊆ I be the open subgroup corresponding to L ′ , andlet a ′ ∈ W/I ′ be the image of a . There is a short exact sequence1 → Gal( L ′ /L ) → W/I ′ → w → a ′ determines a splitting W/I ′ ≃ Gal( L ′ /L ) ⋊ w . Because F isendowed with an action of w , the groups Gal( L ′ /L ) and w both act on F ( X L ′ ), and it is easyto check these actions are compatible with the decomposition of W/I ′ as a semi-direct product.19 avid A. Karpuk Therefore
W/I ′ acts on F ( X L ′ ), and passing to the limit over all L ′ shows that F ( ¯ X ) is a W -module. One can verify easily that a different choice of a gives an isomorphic W -module.The second statement follows from the description of the W -module structure, and the factthat H (Gal( L ′ /L ) , F ( X L ′ )) = F ( X L ) for any finite extension L ′ /L . Definition . Let F ∈ S ( X W ). Define the i th Weil-smooth cohomology group of X withcoefficients in F by setting Γ X ( F ) = H ( X W , F ) = H ( w , F ( X L )), and letting H i ( X W , F ) bethe i th right derived functor of H ( X W , − ) applied to F . If F is a complex of sheaves in D ( X W ),we let R Γ X ( F ) denote the derived functor of Γ X applied to F . Theorem . There are spectral sequences(i) H p ( w , H q ( X L,sm , F )) ⇒ H p + q ( X W , F ) , and(ii) H p ( W, H q ( ¯ X sm , F )) ⇒ H p + q ( X W , F ) for any F ∈ S ( X W ) . Proof.
To establish the first spectral sequence, note that we can factor Γ X as F F ( X L ) H ( w , F ( X L )). The functor F F ( X L ) preserves injectives, since it has as exact left adjointthe functor w -Mod → S ( X L,sm ) w which takes a w -module to the corresponding locally constantsheaf. Part (i) is now just the spectral sequence of composite functors.To see that the second spectral sequence holds, note that for any sheaf F ∈ S ( X L,sm ) w , wehave a spectral sequence H p ( I, H q ( ¯ X sm , F )) ⇒ H p + q ( X L,sm , F )of w -modules; see Theorem III.2.20 and Remark III.2.21(a) of [Mil80]. In derived category lan-guage, we have an isomorphism R Γ I ◦ R Γ ¯ X ( F ) ≃ R Γ X L ( F ). Applying R Γ w to both sides of thisisomorphism, we obtain R Γ W ◦ R Γ ¯ X ( F ) ≃ R Γ w ◦ R Γ I ◦ R Γ ¯ X ( F ) ≃ R Γ w ◦ R Γ X L ( F ) ≃ R Γ X ( F ) . The first isomorphism is simply the spectral sequence coming from the group extension 1 → I → W → w →
1, and the third isomorphism is from part (i). The isomorphism R Γ X ( F ) ≃ R Γ W ◦ R Γ ¯ X ( F ) defines the desired spectral sequence, as in Corollary 10.8.3 of [Wei94]. Corollary . Suppose that F ∈ S ( K W ) is defined by a smooth commutative group schemedefined over K . Then H p ( K W , F ) = H p ( W, F ( ¯ L )) . Proof.
This is immediate from the second spectral sequence of the previous theorem, and thefact that H q ( ¯ L sm , F ) = 0 for all q > F . This last statement follows, for example,from the fact that smooth and ´etale cohomology agree for sheaves given by smooth commutativegroup schemes; see § Adjusting notation slightly, let ρ : X L → X be the natural map. Following Proposition 2.4 of[Lic05], we use ρ to define a pair of adjoint functors. For any F ∈ S ( X sm ), the sheaf ρ ∗ F ∈ S ( X L )carries a natural w -action, and hence ρ ∗ defines a pullback functor ρ ∗ : S ( X sm ) → S ( X W ). Wedefine ρ w ∗ : S ( X W ) → S ( X ) by the rule ( ρ w ∗ G )( U ) = H ( w , G ( U L )) for any G ∈ S ( X W ).20 eil-´etale Cohomology over p -adic FieldsProposition . Let ρ ∗ and ρ w ∗ be as above. We have the following:(i) ρ ∗ is left adjoint to ρ w ∗ .(ii) ρ ∗ is exact, and therefore ρ w ∗ preserves injectives.(iii) If G/X is a smooth commutative group scheme which is locally of finite type, then there isnatural isomorphism ρ ∗ G ∼ → G L in S ( X W ) .(iv) For any G ∈ S ( X sm ) , there is a canonical map G → ρ w ∗ ρ ∗ G , which is an isomorphism when G is representable by a smooth commutative group scheme which is locally of finite type.(v) For any smooth sheaf F ∈ S ( X sm ) , there is a map of spectral sequences from H i ( G, H j ( X ¯ K,sm , F )) ⇒ H i + j ( X sm , F ) to H i ( W, H j ( ¯ X, φ ∗ ρ ∗ F )) ⇒ H i + j ( X W , ρ ∗ F ) . Proof.
Part (i) is proved in the usual manner, and part (ii) holds because pullback is always exact.To see part (iii), we imitate the proof of Chapter II, Remark 3.1(d) of [Mil80]. Let
G/X be asmooth commutative group scheme, which we identify with the sheaf it defines on X sm , and let F ∈ S ( X W ) be any Weil-smooth sheaf. By definition of ρ w ∗ we have H ( w , F ( G L )) ∼ → ( ρ w ∗ F )( G ) . By basic properties of representable sheaves, this implies thatHom X W ( G L , F ) ∼ → Hom X sm ( G, ρ w ∗ F )and the result follows by uniqueness of adjoints.The map described in part (iv) is the map induced by the adjunction map G → ρ ∗ ρ ∗ G , theimage of which lands in ρ w ∗ ρ ∗ G . When G is given by a smooth commutative group scheme, wehave by part (iii) that ρ ∗ G = G L , and thus the map is the natural map G ( U ) → H ( w , G L ( U L )).It is easy to see that this induces an isomorphism of sheaves. The map of spectral sequences inpart (v) is simply the map induced by the inclusion W → G and the projection ¯ X → X ¯ K .The exact functor ρ ∗ extends naturally to a functor ρ ∗ : D ( X sm ) → D ( X W ) between thecorresponding derived categories. If F ∈ D ( X sm ) is a complex such that H i ( F ) is representableby some smooth commutative group scheme G i /X for all i , then by the exactness of ρ ∗ wehave H i ( ρ ∗ F ) = ρ ∗ G i = G iL . If F ∈ S ( X sm ) is representable by a smooth commutative groupscheme which is locally of finite type, then we will often simply write F instead of ρ ∗ F for thecorresponding Weil-smooth sheaf it defines. Let
X/K be a smooth scheme of finite type over K , and let F, F ′ ∈ S ( X W ). We define the Weil-smooth sheaf hom by Hom( F, F ′ ) = Hom X W ( F, F ′ ) = Hom X L,sm ( F, F ′ ) , which carries a natural w -action. The functor Hom( F, − ) is left exact, and we denote by R Hom( F, − ) its derived functor. Lemma . For any
F, G ∈ D ( K sm ) , there is a canonical map Φ( F, G ) : ρ ∗ R Hom K sm ( F, G ) → R Hom ( ρ ∗ F, ρ ∗ G ) in D ( K W ) . If G = G m and F is a torus, an abelian variety, or a free finitely generated groupscheme, then Φ( F, G ) is an isomorphism. avid A. Karpuk Proof.
By standard adjointness properties (see [Wei94], Chapter 10.7.1), we have identificationsHom D ( K W ) ( ρ ∗ R Hom K sm ( F, G ) , R Hom( ρ ∗ F, ρ ∗ G ))= Hom D ( K sm ) ( R Hom K sm ( F, G ) , ρ w ∗ R Hom( ρ ∗ F, ρ ∗ G ))= Hom D ( K sm ) ( R Hom K sm ( F, G ) , R Hom K sm ( F, ρ w ∗ ρ ∗ G )) . We define Φ(
F, G ) to be the map induced by the canonical map G → ρ w ∗ ρ ∗ G of (5.2.1 (iv)),which is the identity map when G = G m .Now set G = G m , and suppose that F = M is a free finitely generated commutative groupscheme. In this case R Hom K sm ( M, G m ) = Hom K sm ( M, G m ) = T is a torus (see [vH04], Corollary1.4). Thus Φ( M, G m ) is the natural mapΦ( M, G m ) : T L → Hom K W ( M L , G m,L )which is clearly an isomorphism. For F = T a torus, the same argument, with the roles of T and M reversed, shows that Φ( F, G m ) is an isomorphism.For F = A an abelian variety we have R Hom K sm ( A, G m ) = Ext K sm ( A, G m ) = A t [ −
1] (see[vH04], Corollary 1.4), where A t is the dual abelian variety of A . The map Φ( A, G m ) now readsΦ( A, G m ) : A tL [ − → Ext K W ( A L , G m,L )which is an isomorphism by the compatibility of the Barsotti-Weil formula with base change.If F ∈ D ( K W ) is any bounded complex of sheaves, we define its Cartier Dual by F D := R Hom( F, G m ). If G ∈ D ( K sm ) we define G D sm := R Hom K sm ( G, G m ). The previous propositionessentially says that if we restrict ourselves to tori and their cocharacter groups, and abelianvarieties, we have ρ ∗ (( − ) D sm ) = ( ρ ∗ ( − )) D . Proposition . Let T be a torus over K with cocharacter group M , and A an abelianvariety over K with dual abelian variety A t . Then we have the following natural isomorphismsin D ( K W ) : M D ≃ T, T D ≃ M, A D ≃ A t [ − . (12) Proof.
By Corollary 1.4 of [vH04], the isomorphisms we are trying to demonstrate hold in D ( K sm ). Applying ρ ∗ to van Hamel’s isomorphisms and using (5.3.1), we arrive at the cor-responding isomorphisms in D ( K W ).Let us return now to an arbitrary smooth scheme X/K of finite type over K , and let F ∈D ( X W ). There is a Yoneda pairing F ′ ⊗ L R Hom( F ′ , F ) → F for any F ′ ∈ D ( X W ). Suppose that H n ( X W , F ) = 0, but H m ( X W , F ) vanishes for all m > n .Then by applying R Γ X ( − ) and projecting, we arrive at a pairing R Γ X ( F ′ ) ⊗ L R Γ X ( R Hom( F ′ , F )) → H n ( X W , F )[ − n ] (13)in D ( Z ), which we will also call the Yoneda pairing. If X = Spec K , F ′ = T is a torus withcocharacter group M , and F = G m , then by the above proposition we arrive at the pairing R Γ K ( T ) ⊗ L R Γ K ( M ) → Z [ −
1] of (3.3.1).
Proposition . Let π : X → K be a smooth, projective curve over a p -adic field K . Thenin D ( K W ) , there is a canonical isomorphism R Hom ( Rπ ∗ G m , G m [ − ∼ → Rπ ∗ G m eil-´etale Cohomology over p -adic Fields which induces a pairing Rπ ∗ G m ⊗ L Rπ ∗ G m → G m [ − . Proof.
Quite generally, Let S be a scheme, and let π : X → S be a smooth, proper curve over S . Deligne, in [Del73], has constructed an isomorphism τ R Hom( τ Rπ ∗ G m [1] , G m ) → τ Rπ ∗ G m (14)of sheaves on S fppf . Let us make this isomorphism explicit when S is the spectrum of a field F of characteristic zero. As noted in [vH04], in this case (14) holds even on the smooth site of S .Let F ( X ) denote the function field of X . Then in D ( X sm ), the complex G m [1] is isomorphicto the complex F ( X ) × [1] → Div X where the map takes a function to its divisor. Applying Rπ ∗ to this complex, we see that π ∗ G m [1] = G m [1] and R π ∗ G m [1] = Pic X , where by Pic X we meanthe sheaf on F sm defined by the Picard scheme of X/F .For U smooth over F , let Z [ X ( U )] be the free abelian group on the set of morphisms from U to X over F , and let Z X be the sheaf on F sm associated to U Z [ X ( U )]. There is a map Z X → Rπ ∗ G m [1] in D ( F sm ), given by taking a morphism U → X to its divisor in Div( X × F U ).Applying R Hom F sm ( − , G m ) to this map, we arrive at Deligne’s isomorphism R Hom F sm ( Rπ ∗ G m [1] , G m ) ∼ → R Hom F sm ( Z X , G m ) = Rπ ∗ G m , which encodes the auto-duality of the Jacobian of X , and the duality between the sheaves Z and G m . The identity R Hom F sm ( Z X , G m ) = Rπ ∗ G m follows from Yoneda’s Lemma (this is wherewe use the fact that X is smooth over S ).When F = K Deligne’s isomorphism reads ( Rπ ∗ G m [1]) D sm ∼ → Rπ ∗ G m . The cohomologysheaves of the complex Rπ ∗ G m are all free finitely generated group schemes, tori, or abelianvarieties, or extensions of such sheaves. Thus (5.3.1) and (5.3.2) imply that( ρ ∗ Rπ ∗ G m ) D ≃ ρ ∗ ( Rπ ∗ G m [1]) D sm ≃ ρ ∗ Rπ ∗ G m which is the desired canonical isomorphism. The pairing Rπ ∗ G m ⊗ L Rπ ∗ G m → G m [ −
1] is inducedby standard adjoint properties and a degree shift.On applying R Γ K ( − ) to each term in the pairing (5.3.3) and composing with the map R Γ K ( G m [ − → Z [ − R Γ X ( G m ) ⊗ L R Γ X ( G m ) → Z [ −
2] (15)in the derived category of abelian groups. We let λ ( X ) : R Γ X ( G m ) → R Hom( R Γ X ( G m ) , Z [ − λ ( X ) is an isomorphism. K with Abelian Variety Coefficients Studying the map λ ( X ) will require us to understand the cohomology of W acting on the ¯ L -points of the Jacobian of X . Therefore, it will be useful to establish a Weil group analogue ofTate’s duality theorem for abelian varieties over local fields, found in [Tat57].To that end, we devote this section to the pairing (13) when F = G m and F ′ = A is anabelian variety over K . Using (12) and shifting, we see that the Yoneda pairing induces a pairing R Γ K ( A ) ⊗ L R Γ K ( A t ) → Z avid A. Karpuk which is equal to the pairing induced by the biextension map A ⊗ L A t → G m [1]. We let τ ( A )denote the induced map, τ ( A ) : R Γ K ( A t ) → R Hom( R Γ K ( A ) , Z ) . (17)Our duality theorem for abelian varieties will describe to what extent τ ( A ) is an isomorphism. Lemma . H i ( W, A ) = 0 for i = 0 , . Proof.
The only non-trivial assertion is that H ( W, A ) = 0. For this, one can use the same proofas for the vanishing of H ( G, A ( ¯ K )); see Chapter II, § Lemma . Let Y be any of the following groups: Z p , O K , U K , A ( K ) . Then Hom ( Y, Z ) is zero. Proof.
First consider the case of Y = Z p . Let f : Z p → Z be a non-zero homomorphism; since theonly non-trivial subgroups of Z are isomorphic to Z , we may assume f is surjective. Composingwith the surjection Z → Z /p n Z , we see that f induces a surjection Z p → Z /p n Z . This lattermap must factor through Z p /p n Z p , and therefore f induces a surjection Z p /p n Z p → Z /p n Z . Asthese two finite groups have the same order, this is an isomorphism. It follows that ker( f ) ⊆ T n p n Z p = 0, and hence f is injective. Thus f is an isomorphism, which is a contradiction.Now suppose that Y = O K . The vanishing of Hom( O K , Z ) follows immediately from the factthat O K is a free Z p -module of rank equal to [ K : Q p ]. The result for Y = U K follows from thefact that U K contains a subgroup of finite index isomorphic to O K as abstract abelian groups.Similarly, A ( K ) contains a finite index subgroup isomorphic to dim A copies of O K . Lemma . The restriction map R Γ G ( A ) → R Γ W ( A ) is an isomorphism in D ( Z ) . Proof.
We must show that the maps H i ( G, A ) → H i ( W, A ) are isomorphisms for all i >
0. Inlight of (4.1.5), we only need to show that H ( W, A ) is torsion. Recall from (5) that the group H ( W, A ) fits into the exact sequence0 → H ( w , A ( L )) → H ( W, A ) → H ( w , H ( N, A )) → . The group on the right is torsion, because it is a subgroup of a Galois cohomology group. Thuswe are reducing to showing that H ( w , A ( L )) is torsion.Let A / O L be the N´eron model for A over the ring of integers of L ( A is the base change to O L of the N´eron model for A/K ; see [BLR90], Theorem 7.2.1 and Corollary 2.). Let A ⊆ A bethe subscheme whose special fiber is the identity component of the special fiber of A , and whosegeneric fiber is A L . We have an exact sequence of w -modules,0 → A ( O L ) → A ( O L ) → π (¯ k ) → , where π (¯ k ) is the group of connected components of the special fiber of A . This sequence isexact by Hensel’s Lemma; see Proposition I.3.8 of [Mil80]. Taking cohomology gives a short exactsequence H ( w , A ( O L )) → H ( w , A ( O L )) → H ( w , π (¯ k )) → , and it follows from Proposition 3 of [Gre63] that H ( w , A ( O L )) = 0. Since A ( O L ) = A ( L ), wehave that H ( w , A ( L )) = H ( w , π (¯ k )), which is finite. Theorem . The map τ ( A ) of (17) has the following properties:(i) τ ( A ) : A t ( K ) → Ext ( H ( W, A ) , Z ) = H ( W, A ) ∗ is an isomorphism of profinite groups.(ii) τ ( A ) : H ( W, A t ) → Ext ( A ( K ) , Z ) induces an isomorphism of H ( W, A t ) with the torsionsubgroup A ( K ) ∗ of Ext ( A ( K ) , Z ) . eil-´etale Cohomology over p -adic Fields The cohomology of both complexes vanishes outside of degrees 0 and 1. In particular, τ ( A ) i isinjective for all i . Proof.
By (2.0.1) and (5.4.2), the maps τ ( A ) i reduce to maps τ ( A ) i : H i ( W, A t ) → Ext( H − i ( W, A ) , Z ) . The group H ( W, A ) is torsion, hence has no non-zero maps to Q . Part (i) of the theorem nowfollows from (5.4.3) and Tate’s duality theorem on abelian varieties over local fields (see themain theorem of [Tat57]). The profinite group A ( K ) admits no continuous maps to Q , so A ( K ) ∗ injects into Ext( A ( K ) , Z ). Part (ii) of the theorem now follows again from (5.4.3) and Tate’stheorem. Before stating our Weil-smooth duality theorem for curves, we would like to remind the readerof Lichtenbaum’s duality theorem for curves over p -adic fields, and van Hamel’s approach to itsconstruction and proof of non-degeneracy. As always, let π : X → K be a smooth, projective,geometrically connected curve. Theorem . (Lichtenbaum, [Lic69]) There are natural pairings H i ( X sm , G m ) ⊗ H − i ( X sm , G m ) → Q / Z which induce isomorphisms H i ( X sm , G m ) ⊗ ˆ Z → H − i ( X sm , G m ) ∗ for all i , where H i ( X sm , G m ) has the natural topology coming from that on K . Lichtenbaum defines his pairing by explicitly evaluating representatives of the Brauer groupon divisor classes. Since our objects live in the derived category where the notion of “element”does not make sense, van Hamel’s functorial approach adapts better to the Weil-smooth situation.Let F = Rπ ∗ G m ∈ D ( X sm ), so that R Γ K sm ( F ) = R Γ X sm ( G m ). In [vH04], van Hamel’sapproach to Lichtenbaum’s duality theorem is to put an “ascending filtration” on F . That is, hedefines complexes F i and constructs a series of morphisms 0 → F → F → F = F in D ( K sm ).For each i >
0, van Hamel defines the i th graded piece G i to be the mapping cone of F i − → F i ,yielding an exact triangle F i − → F i → G i → F i − [1] in D ( K sm ).The sheaf F is defined by F := H ( F ) = G m , and F is defined to be the mapping cone ofthe composite F → Pic X [ − deg → Z [ − F → F → Z [ − → F [1] and G m → F → Pic X [ − → G m [1]in D ( K sm ). The graded pieces G i are the given by G i = G m i = 0Pic X [ − i = 1 Z [ − i = 20 i > . The filtration on F induces a “descending filtration” F D sm = F D sm → F D sm → F D sm → F D sm , and also induces triangles G D sm i → F D sm i → F D sm i − → G D sm i [1] for all i . The sheaves G D sm i are given by G D sm i = Z i = 0Alb X i = 1 G m [1] i = 20 i > . avid A. Karpuk To prove the non-degeneracy and perfectness results of the pairing, van Hamel then uses dualitytheorems for finitely generated group schemes, tori, and abelian varieties to analyze the pairings R Γ K sm ( G i ) ⊗ L R Γ K sm ( G D sm i ) → Q / Z [ − , and pieces together a duality theorem for R Γ K sm ( F ) = R Γ X sm ( G m ) using the Five Lemma. Wewill essentially copy this approach, by applying ρ ∗ to van Hamel’s filtration and exact triangles.We can now state and prove our duality theorem for the Weil-smooth cohomology of curves. Theorem . Let
X/K be a smooth, projective, geometrically connected curve over K , suchthat X ( K ) = ∅ . The map λ ( X ) : R Γ X ( G m ) → R Hom ( R Γ X ( G m ) , Z [ − induced by the pairing (15) has the following properties:(i) λ ( X ) i is an isomorphism for i = 0 , .(ii) λ ( X ) i is injective for i = 2 , .The cohomology of both complexes vanishes outside of degrees through . Proof.
As above, let F be the complex Rπ ∗ G m considered on the smooth site of K , so that R Γ X ( G m ) = R Γ K W ( ρ ∗ Rπ ∗ G m )Applying ρ ∗ to van Hamel’s filtration provides us with a filtration 0 → ρ ∗ F → ρ ∗ F → ρ ∗ F = ρ ∗ F on F which comes equipped with exact triangles ρ ∗ F i − → ρ ∗ F i → ρ ∗ G i → ρ ∗ F i − [1] . Now we apply ρ ∗ to the dual filtration, and note that by (5.3.2), ρ ∗ commutes with thedualizing functors in the sense that ρ ∗ ( G D sm i ) = ( ρ ∗ G i ) D for all i . Repeatedly applying the FiveLemma and (5.3.1) to van Hamel’s exact triangles shows that ρ ∗ ( F Di ) = ( ρ ∗ F i ) D for all i . Nowthe Yoneda pairing induces pairings R Γ K ( ρ ∗ G i ) ⊗ L R Γ K (( ρ ∗ G i ) D ) → Z [ − R Γ K ( ρ ∗ F i ) ⊗ L R Γ K (( ρ ∗ F i ) D ) → Z [ − D ( Z ) for all i . In a slight abuse of notation, we suppress ρ ∗ from now on.Let γ i : R Γ K ( G Di ) → R Hom( R Γ K ( G i ) , Z [ − D ( Z ). We will describe the maps γ i in terms of duality theorems we havealready proven.From (3.3.7) we see that γ = η ( Z ) has the following properties: γ i is an isomorphism for i = 2, and γ maps H ( W, Z ) isomorphically onto the torsion subgroup U ∗ K of Ext( K × , Z ).Let J X be the Jacobian variety of X . Any rational point of X determines an embedding X ֒ → J X defined over K , and thus a Weil-equivariant isomorphism Pic X ( ¯ L ) → J X ( ¯ L ). Hence wecan identify Pic X ( ¯ L ) with the ¯ L -points of an abelian variety defined over K , and apply (5.4.4)to Pic X and its dual abelian variety Alb X .From (5.4.4), we see that γ = τ (Pic X ) has the following properties: γ is an isomorphismwhich maps Alb X ( K ) isomorphically onto H ( W, Pic X ) ∗ , and γ is an injective map which maps H ( W, Alb X ) isomorphically onto the torsion subgroup Pic X ( K ) ∗ of Ext(Pic X ( K ) , Z ). From(3.3.1) we see that γ = ψ ( Z )[1] is an isomorphism.Now consider the maps φ i : R Γ K ( F Di ) → R Hom( R Γ K ( F i ) , Z [ − eil-´etale Cohomology over p -adic Fields induced by the Yoneda pairing. We can determine to what extent the maps φ i are isomorphisms,by using the triangles which defined G i and G Di . When i = 0 one has G = F = G m , hence φ = γ = η ( Z ) is the map of (3.3.7). When i = 1 we have a diagram R Γ K (Alb X ) γ / / (cid:15) (cid:15) R Hom( R Γ K (Pic X [ − , Z [ − (cid:15) (cid:15) R Γ K ( F D ) φ / / (cid:15) (cid:15) R Hom( R Γ K ( F ) , Z [ − (cid:15) (cid:15) R Γ K ( F D ) φ / / R Hom( R Γ K ( F D ) , Z [ − . It follows that φ is an isomorphism, φ is injective, and φ maps H ( K W , F D ) isomorphicallyonto the torsion subgroup U ∗ K of Ext ( R Γ K ( F ) , Z [ − i = 2 we have a diagram R Γ K ( G m [1]) ψ ( Z )[1] ∼ / / (cid:15) (cid:15) R Hom( R Γ K ( Z [ − , Z [ − (cid:15) (cid:15) R Γ K ( F D ) φ / / (cid:15) (cid:15) R Hom( R Γ K ( F ) , Z [ − (cid:15) (cid:15) R Γ K ( F D ) φ / / R Hom( R Γ K ( F ) , Z [ − . It follows that φ − = ψ ( Z ) is an isomorphism from K × to Hom( R Γ K ( F ) , Z [ − φ isan isomorphism, that φ is injective, and that φ is an isomorphism from H ( K W , F D ) to thetorsion subgroup U ∗ K of Ext ( R Γ K ( F ) , Z [ − λ ( X ) is the map induced by the isomorphism F [1] D → F of Weil-smooth sheaves, the map φ , and shifting degrees by one. In this section we compare the duality theorem of the previous section with the main theoremof [Lic69]. That the our pairing is compatible with the original pairing defined by Lichtenbaumfollows from the results of § Proposition . Suppose that
X/K is a smooth, proper variety over K , and let F be atorsion sheaf in S ( X sm ) . Then the restriction map R Γ X sm ( F ) → R Γ X ( F ) is an isomorphism in D ( Z ) . Proof.
By ([Mil80], Chapter VI, Corollary 2.6), the map R Γ X ¯ K,sm ( F ) → R Γ ¯ X sm ( F ) is an iso-morphism. The result now follows from (4.1.1), since R Γ X sm = R Γ G ◦ R Γ X ¯ K,sm and R Γ X = R Γ W ◦ R Γ ¯ X sm .By the previous proposition, smooth and Weil-smooth cohomology agree for the sheaf µ n ,hence we can use Kummer sequences to study the restriction maps H i ( X sm , G m ) → H i ( X W , G m ).27 avid A. Karpuk In particular there is a diagram0 / / H i ( X sm , G m ) /n δ / / res i /n (cid:15) (cid:15) H i +1 ( X sm , µ n ) / / ≀ (cid:15) (cid:15) H i +1 ( X sm , G m )[ n ] / / res i +1 [ n ] (cid:15) (cid:15) / / H i ( X W , G m ) /n δ / / H i +1 ( X W , µ n ) / / H i +1 ( X W , G m )[ n ] / / δ in : ker(res i +1 [ n ]) → coker(res i /n ) for any pair of integers i, n . Passing to the limit over all n , we obtain a canonical isomorphism δ i : ker(res i +1 | tors ) → coker(res i ⊗
1) (18)where res i ⊗ H i ( X sm , G m ) ⊗ Q / Z → H i ( X W , G m ) ⊗ Q / Z . Proposition . Let
X/K be a smooth, projective, geometrically connected curve over K such that X ( K ) = ∅ . The restriction maps res i : H i ( X sm , G m ) → H i ( X W , G m ) are described bythe following exact sequences: → H ( X sm , G m ) res → H ( X W , G m ) → H ( W, G m ) → → Br ( K ) → H ( X sm , G m ) res → H ( W, Pic X ) → → H ( W, Z ) ⊗ Q / Z → H ( X sm , G m ) res → H ( X W , G m ) → . (21) Proof.
The map of Hochschild-Serre spectral sequences computing smooth and Weil-smoothcohomology gives us a map of short exact sequences0 / / / / (cid:15) (cid:15) H ( X sm , G m ) ∼ / / res (cid:15) (cid:15) H ( G, Pic X ) / / ≀ (cid:15) (cid:15) / / H ( W, G m ) / / H ( X W , G m ) / / H ( W, Pic X ) / / X ( K ) = ∅ . The existence of (19) follows by applying the SnakeLemma.To prove the second existence of the second exact sequence, note that H ( X sm , G m ) is atorsion group, so ker(res ) = ker(res ). We will show that there is a natural identificationker(res ) = Br( K ). The long exact sequences of low degree from the Hochschild-Serre spectralsequences give us a map of short exact sequences0 / / Br( K ) / / (cid:15) (cid:15) H ( X sm , G m ) / / res (cid:15) (cid:15) H ( G, Pic X ) / / (cid:15) (cid:15) / / / / H ( X W , G m ) ∼ / / H ( W, Pic X ) / / . It follows from (4.1.3) and (4.1.5) that H ( G, Pic X ) can be identified with the torsion subgroupof H ( W, Pic X ) via the restriction map.On the other hand, consider the long exact sequence in W -cohomology of 0 → Pic X → Pic X → Z →
0. Since X ( K ) = ∅ , any rational point determines a Weil-equivariant degree 1divisor class on ¯ X , hence the map deg : H ( W, Pic X ) → Z is surjective. The relevant part of thelong exact sequence now reads H ( W, Pic X ) deg → Z → H ( W, Pic X ) → H ( W, Pic X ) → Z → , eil-´etale Cohomology over p -adic Fields and we have an identification H ( W, Pic X ) = H ( W, Pic X ) tors . That (19) and (20) are exact isnow clear.The only remaining task is to identify the kernel of res . By (18) we have an identificationker(res ) = coker(res ⊗ H ( X sm , G m ) is torsion, this last cokernel can be identifiedwith H ( X W , G m ) ⊗ Q / Z = H ( W, Pic X ) ⊗ Q / Z . The map on cohomology induced by the degreemap gives an isomorphism of this last group with H ( W, Z ) ⊗ Q / Z , since H ( W, Pic X ) ⊗ Q / Z =0. With the above comparison theorem, we can reprove the main result of [Lic69] for curves X/K containing a rational point.
Theorem . Suppose that
X/K is a smooth, projective, geometrically connected curve, suchthat X ( K ) = ∅ . Then the Lichtenbaum pairing H ( X sm , G m ) ⊗ H ( X sm , G m ) → Q / Z inducesan isomorphism H ( X sm , G m ) → H ( X sm , G m ) ∗ . Proof.
The map induced by the Lichtenbaum pairing fits into the diagram0 / / Br( K ) / / ≀ (cid:15) (cid:15) H ( X sm , G m ) / / (cid:15) (cid:15) H ( W, Pic X ) / / ≀ (cid:15) (cid:15) / / Z ∗ / / H ( X sm , G m ) ∗ / / Pic X ( K ) ∗ / / Acknowledgements
The research presented here constitutes the bulk of the author’s Ph.D. thesis, completed at theUniversity of Maryland, College Park. As such, the author would like to extend his sincerestgratitude towards his thesis adviser, Niranjan Ramachandran, for suggesting this topic, and forconsistently providing helpful comments, suggestions, and guidance.Thomas Geisser, Baptiste Morin, and Mathias Flach also deserve a great amount of theauthor’s gratitutde, for many helpful comments and questions regarding the main results of thisarticle. Lastly, the author would like to thank Thomas Haines and Larry Washington for helpfuldiscussions concerning his thesis.
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David A. Karpuk [email protected]@math.umd.edu