A Canonical Representation of Tate Sequences

(b) The rows of (3.3) are induced by tensoring with R the exact sequences

0→ a

S<∞

Hⁱ(k(v)W,Z)→H_cⁱ⁺¹(YW,Z)→H_cⁱ⁺¹((SpecOF)W,Z)→0, i= 0,1.

Letχ₀ :=χ_c

(0, H⁰(k(v)_W,Z), H¹(k(v)_W,Z)),(`ψ)

. Applying Example 3.2 to (3.3), we see that

χ_c(Y) = ±χ_c(SpecO_F)χ₀.

Because Hⁱ(k(v)W,Z) is torsion free, and we has seen in (a) that`ψ sends 1v

tol_v :σ7→logN(v), so χ₀ =±Π_S<∞logN(v). Thus,

χ_c(Y) =±Rh w

Y

S<∞

logN(v) =±R_Sh_S w

(c.f. [15] Lemma 2.1 for the last equality).

Proof. Consider the spectral sequence

E₂^pq = Ext^p_Z(H^−q(C^•),Z)⇒H^p+q(RHom(C^•,Z)),

in which C^• =τ^≤3RΓ_c(Y_W,Z).

It is clear that E₂^pq = 0 for all p < 0 or q > 0, i.e. the E2 has only non-trivial terms on the 4th quadrant

0 0

X_S =X_S^∨∨ 0

U_S/tor =U_S^∨∨ Ext¹((PicY)^D,Z) 0 Ext¹(µ^D_F,Z) 0 We have exact sequences

0→PicY →H⁻¹ →X_S →0 and 0→µ_F →H⁻² →U_S/tor →0.

Thus, H⁻¹ = XS ⊕ Pic(Y) and H⁻² = (US/tor) ⊕ µF = US. This shows that RHom(τ^≤3RΓ_c(Y_W,Z),Z)[−2] has only two non-trivial cohomologies H⁰ = U_S and H¹ =XS⊕PicY.

From now on we assume that S is large enough so that PicY = 0 and is stable under the action of G. We will prove RHom(τ^≤3RΓ_c(Y_W,Z),Z)[−2] is indeed quasi- isomorphic to ΨS, the canonical Tate sequence. (see the introduction section).

Lemma 3.2. Let A and B ∈ D, the derived category of Z[G]-modules.

a) If Hⁱ(B) areZ[G]-injective for all i, then

HomD(A,B)∼=Y

i

Hom_G(Hⁱ(A), Hⁱ(B)).

b) If Hⁱ(A) areZ[G]-cohomologically trivial, then we have an exact sequence

0→Y

i

Ext¹_G(Hⁱ(A), Hⁱ⁻¹(B))→HomD(A,B)→Y

i

HomG(Hⁱ(A), Hⁱ(B))→0.

Proof. a) By [16], there is a spectral sequence

Y

i

Ext^p_G(Hⁱ(A), H^q+i(B))⇒H^p+q(RHom_Z[G](A,B)).

By condition, Ext^p(Hⁱ(A), H^q+i(B)) is non-trivial only if p = 0, so the above spectral sequence simply shows

HomD(A,B) =H⁰(RHom_Z[G](A,B)) = Y

i

Hom_G(Hⁱ(A), Hⁱ(B)).

b) In the same spectral sequence, since Hⁱ(A) are cohomologically trivial, Hⁱ(A) are of projective dimension 1, so Ext^p_G(Hⁱ(A), H^q+i(B)) = 0 for all p ≥ 2.

Therefore, on the position with p+q = 0, we have only 2 non-trivial groups E₂^0,0 andE₂^1,−1, and both the difference maps are 0. This gives the desired short exact sequence.

Lemma 3.3. The canonical exact triangle

RHom_Z(RΓ_c(Y_et´,Z),Z)→RHom_Z(RΓ_c(Y_´et,Z),Q)→RHom_Z(RΓ_c(Y_´et,Z),Q/Z)→

is isomorphic to the exact triangle

Ψ⁰_S[2]→X_S⊗_ZQ[1]→Ψf_S⁰[3]→

whereΨ⁰_S is the image ofΨS inExt²_G(XS,UbS)andΨfS

0 is the image ofΨS inExt³_G(XS⊗_Z Q/Z,Ub_S).

Proof. Suppose Ψ_S is the complex A → B and Ψe_S is the complex A → B →

X_S ⊗_Z Q. Recall that Ψ_S presents a class of Ext²_G(X_S,U_S) and Ψ⁰_S the image of Ψ_S under the map Ext²_G(X_S,U_S) −→^∼ Ext²_G(X_S,Ub_S). Indeed , Ψ⁰_S is the complex (A⊕Uˆ_S)/U_S =: A⁰ → B. Similarly, Ψe⁰_S is the complex A⁰ → B → X_S ⊗_Z Q. Re- call that RHom(RΓ_c(Y_et´,Z),Q/Z)−→^qis Ψe⁰_S by definition ([2] Prop. 3.1). Also, as Q is injective, Hⁱ(RHom_Z(RΓ_c(Y_´et,Z),Q)) = Hom_Z(H_c⁻ⁱ(Y_´et,Z),Q). Thus,

RHom_Z(RΓc(Y´et,Z),Q)−→^qis XS⊗_ZQ[1].

We claim that the following diagram commutes for a suitable choice of the left vertical quasi-isomorphism.

RHom_Z(RΓ_c(Y_et´,Z),Q)[−3] ^//RHom_Z(RΓ_c(Y_et´,Z),Q/Z)[−3]

XS⊗_ZQ[−2] ^//

qis

OO

Ψe⁰_S.

qis

OO (3.4)

Since the both horizontal maps induce the canonical projectionX_S⊗_ZQ→X_S⊗_ZQ/Z onH², the right vertical quasi-isomorphism induces the identity map on H² and the left vertical one can be induced by any G-automorphism of X_S ⊗_ZQ, so there is a suitable choice of the quasi-isomorphism so that the above diagram commutes forH² groups.

Let A be X_S⊗_ZQ[−2] and B be RHom_Z(RΓ_c(Y_et´,Z),Q/Z)[−3]. Clearly, Hⁱ(A) are Z[G]-cohomologically trivial. So we can apply lemma 3.2(b) to A and B, and it induces an exact sequence

0 = Ext¹_G(X_S⊗_ZQ, H¹(B)(= 0))−→Hom_D(A,B)−→Hom_G(H²(A), H²(B))→0.

Thus, HomD(A,B) ∼= Hom_G(H²(A), H²(B)). Therefore, the commutative on H² implies the diagram (3.4) indeed commutes. By taking exact triangles of each row, and note that cone(X_S⊗_ZQ[−2]→Ψe⁰_S)[−1]'Ψ⁰_S[2], we get the desired isomorphism of exact triangles.

Theorem 3.3. Let Y = SpecO_F,S, where S is large enough so that PicY = 0 and is

stable under the action of G. Then

RHom_Z(τ^≤3RΓc(YW,Z),Z)[−2]−→^qis ΨS,

where Ψ_S is the Tate sequence representing the canonical class of Ext²_G(X_S, U_S).

Proof. Here, we use the same notation with previous lemma. Clearly, we have a canonical exact triangle

Ψ_S →Ψ⁰_S →Ub_S/U_S[0]→.

On the other hand, by applying RHom_Z(−,Z) to exact triangle (3.1), we have another exact triangle

RHom_Z(τ^≤3RΓ_c(Y_W,Z),Z)[−2]→RHom_Z(RΓ_c(Y_et´,Z),Z)[−2]→RHom_Z(Hom_Z(U_S,Q),Z)[−1]

The previous lemma ensures that Ψ⁰_S−→^qis RHom_Z(RΓ_c(Y_et´,Z),Z)[−2].

Also, observe that RHom_Z(Hom_Z(U_S,Q),Z)[−1]−→^qis Ub_S/U_S[0] and we have seen that RHom_Z(τ^≤3RΓ_c(Y_W,Z),Z)[−2] has the same cohomologies as Ψ_S. We claim that there is an isomorphism between these two exact triangles,

RHom_Z(τ^≤3RΓ_c(Y_W,Z),Z)[−2] ^//Ψ⁰_S ^{β //}RHom_Z(Hom_Z(U_S,Q),Z)[−1] ^//

Ψ_S

δ

OO //Ψ⁰_S ^{γ //} bU_S/U_S[0]

α

OO //.

In the top exact triangle, we replaced RHom_Z(RΓ_c(Y_´et,Z),Z)[−2] by Ψ⁰_S. By taking the long exact sequence induced by the top exact triangle, we see thatH²(β) = i◦p where p : Ub_S → Ub_S/U_S is the canonical projection and i ∈ Aut(Ub_S/U_S). Clearly, H²(γ) = p. As RHom_Z(Hom_Z(U_S,Q),Z)[−1]−→^qis Ub_S/U_S[0], we may take α to be the morphism induced by i. Then the right square commutes on H². By lemma 3.2, becauseUb_S/U_Sis uniquely divisible and thus injective,H²(β) =H²(α)◦H²(γ) implies β =α◦γ. Asαis a quasi-isomorphism, so isδ by property of derived categories.

Remark. It is equivalent to say that τ^≤3RΓ_c(Y_W,Z)−→^qis RHom(Ψ_S,Z)[−2] and this

is an evidence that, at least for open subschemes of spectra of number rings, the Weil-

´

etale cohomology can be obtained from usual cohomology theories without using the Weil groups.

Chapter 4

The Construction of R G m

Throughout this chapter, we assume that F is a totally imaginary number field, X = SpecO_F, and U = SpecO_L,S is any connected ´etale neighborhood of X. We also assume that K is a subfield of L such that the extension L/K is Galois and G:= Gal(L/K).

When P ic(U) = 0 and S is a G-stable set containing all the archimedean places and those ramified in K(U)/Q, there exists a canonical Tate sequence:

Ψ_U : Ψ⁰_S →Ψ¹_S,

which represents a canonical class of Ext²_G(X_S, U_S). By the construction of Tate sequences, if U⁰ = SpecOL⁰,S⁰ is ´etale over U, then there is a morphism from ΨS to Ψ_S⁰ that fits into the following morphism of complexes

0 ^//U_S

//Ψ⁰_S

//Ψ¹_S

//X_S ^//

β

0

0 ^//U_S^{0 //}Ψ⁰_S0 //Ψ¹_S0 //X_S^{0 //}0, where β((a_v)v∈S) = (a_w := [L_w :F_v]·a_v)_w|v,w∈S⁰

.

In fact {X_S} and {β} define an X_et´-presheaf X, and we denote its associated sheaf by Xf. In section 4.1, we construct a complex RGm ∈ Sh(X_´et) which rep- resents a canonical class of Ext²_X

et(Xf,Gm) = Zb (see below). The cohomologies of RΓ(U_´et, RGm) and RHom_Z τ^≤3RΓ_c(U_W,Z), Z

[−2] are the same for arbitrary U.

Furthermore, in section 4.2, we prove that

RΓ(U´et, RG^m)−→^qis RHom τ^≤3RΓc(UW,Z),Z

[−2], (4.1)

inD(Z[G]), when S is stable under the action of G.

This implies RΓ(Uet´, RG^m) and RHom_Z τ^≤3RΓc(UW,Z), Z

[−2] are canonically quasi-isomorphic in D(Z) for any U. Also, whenU is small enough so that the Tate sequence exists, then by (4.1) and by the help of Theorem 3.3, one can see that

RΓ(U_et´, RGm)'Ψ_U

inD(Z[G]).

Thus, one can conclude that the complex RΓ(U_´et, RGm) generalizes the Tate sequences and itsZ-dual defines Weil-´etale cohomology ofS-integers without trunca- tions.

4.1 The Definition

LetM be the sheaf sending a connected ´etaleU →X to the group L

w∈UA_w, where A_w is Q (resp. Z) if w -∞ (resp. w | ∞), and the transition map M(U)→ M(V) for an X-morphism V →U is

M

v∈U

A_v −→ M

w∈V

A_w

a_v 7→ X

w|v

[K(V)_w :K(U)_v]a_v.

It is clear that the projectionsL

v∈UA_v L

v∈UA_v, for any ´etaleU →X, define an epimorphism of sheaves M →L

v∈Xiv,∗A_v.

We define F^• to be the complex

F⁰

P //F¹

Q,

where F⁰ =ker(M →L

v∈Xiv,∗Av) is the X-´etale sheaf U 7→ M

w∈U−U

A_w,

for any connected U →X and P

(U)((a_w)) =P

w 1

[K(U)w:Fv]a_w. Note that if U doesn’t contain all the finite places of F, then P

(U) is surjective.

This implies P

is an epimorphism in the category Sh(X_´et) as any U → X can be covered by some {Ui}, where Ui doesn’t contain all the finite places ofK(Ui), for all i. Consequently, F^• '(kerP

)[0] inD(Sh(X_´et)).

Remark. a. The way that we define the transition maps of the sheaf F⁰ makes X a sub-presheaf of kerP

. In fact, we have the following exact sequence of presheaves

0→X →kerX

→Brf →0,

where Brf is the presheaf U 7→H²(U,Gm). By taking associated sheaves, we see that Xf∼= kerP

as ^a(Hⁱ(−,G)) = 0 for any sheaf G wheni≥1.

b. Let M_v = lim−→LM_L,v where L runs over all finite extension of F and M_L,v = L

wAw where the sum runs over all placeswofLlying overv and the transition map is defined similarly to those of X. We denote Mv the sheaf associated to the Galois module Mv. Equivalently, in the level of sheaves, Mv = lim−→LπL,∗Av

where π_L : SpecL^v −→ SpecF is the canonical morphism and L^v is the fixed field of Dv ∩Gal(L/F). Clearly, M = L

v∈Xα∗Mv, where α : SpecF → X.

Moreover, consider the Cartesian diagram of schemes

SpecL^v

πL

αL //u(L^v)

π_L⁰

SpecF ^{α //}X,

whereu(L^v) = SpecO_L^v. Henceα∗lim−→LπL,∗A_v = lim−→Lα∗πL,∗A_v = lim−→Lπ_L,∗⁰ αL,∗A_v = lim−→Lπ⁰_L,∗A_v. (Note that α_L,∗A_v is the constant A_v on Spec(L^v).)

Lemma 4.1. The sheaf F⁰ is cohomologically trivial.

Proof. Since F⁰ fits into the short exact sequence

0→F⁰ →M

v∈X

α_∗Mv →M

v∈X

i_v,∗A_v →0, (4.2)

which is derived from the short exact sequence of sections:

0→ M

v∈U−U

A_v →M

v∈U

A_v →M

v∈U

A_v →0.

Since there is no real place and A_v = Q when v - ∞, the cohomology of iv,∗A_v concentrates at degree zero. Also, the above exact sequence remains exact when we pass to global sections. Thus, we only need to show that α∗Mv is cohomologically trivial for any v.

Recall that α∗Mv = lim−→Lπ_L,∗⁰ A_v. Whenv is finite,α∗M_v is acyclic asA_v =Qand inductive limit and π_L,∗⁰ both preserve acyclic sheaves (as π⁰_L is finite).

Whenv|∞,I_v is trivial, soM_v =Ind^G₁^FA_v and thenM_v is cohomologically trivial.

Therefore i^∗_w(R^qα∗M_v) =H^q(I_w, M_v) = 0 for q >0. The Leray spectral sequence for α,

H^p(U_´et, R^qα∗M_v)⇒H^p+q(F, M_v),

for any ´etale U over X, is degenerated, i.e. H^p(U_´et, α∗M_v) = H^p(F, M_v) = 0 for p >0. Thus the result follows.

Proposition 4.1. For any connected ´etale U →X,

H^p(U_et´,F^•) =











ker Σ(U) p= 0,

coker Σ(U) = Q/Im(Σ(U)) p= 1,

0 p≥2.

In particular, H⁰(X_´et,F^•) = 0, H¹(X_´et,F^•) = Q, H⁰((X − v)_´et,F^•) = 0 and H¹((X−v)_´et,F^•) = Q/A_v.

Corollary 4.1. The canonical morphism H_v¹(X_´et,F^•) → H¹(X_´et,F^•) is an inclu- sion of A_v into Q.

Now we define a complex RGm, up to quasi-isomorphism, ofSh(X_´et) by an exact triangle

Gm→RGm →F^•[−1]→, or equivalently an exact triangle

RGm →F^•[−1]−→^γ Gm[1]→.

One can choose carefully a morphismγ ∈Hom_D(Sh(X

´

et))(F^•[−1],G^m[1]) ∼= Ext²_X

´

et(F^•,G^m) so that RGm has the cohomologies that we expected. Before computing the group Ext²_X

´

et(F^•,G^m), we need to derive the following lemmas.

Lemma 4.2. Let I be an inductive system of index that can be refined by an in- dex which is isomorphic to Z, then the canonical morphism Ext^s_C(lim−→iAi, B) −→

lim←−iExt^s_C(A_i, B) is an isomorphism if any of the following conditions holds a. the projective system {Ext^s−1_C (Ai, B)} satisfies the Mittag-Leffler condition, b. Ext^s−1_C (A_i, B) is a finite group for all i.

Proof. Because I can be refined by Z, so lim−→IA_i = lim−→ZA_i and lim←−IExt^p_C(A_i, B) = lim←−ZExt^p_C(A_i, B). Thus, we only need to consider the case that I ∼=Z. When I ∼=Z,

by [13], there is a spectral sequence

lim←−^rIExt^s_C(Ai, B)⇒Ext^r+s_C (lim−→IAi, B).

This spectral sequence is certainly degenerated as we know that lim←−

r is vanishing for any inductive system of abelian groups when r ≥ 2 (cf. [17] 3.5). Therefore, we get exact sequences

0→lim←−¹IExt^s−1_C (A_i, B)→Ext^s_C(lim−→IA_i, B)→lim←−¹IExt^s_C(A_i, B)→0,

for all s≥0.

The Mittag-Leffler condition implies lim←−¹Ext^s−1_C (A_i, B) = 0 and condition (b) also implies the vanishing of lim←−

1Ext^s−1_C (A_i, B) by [7] 2.3. Thus, the isomorphism is established under both conditions.

Lemma 4.3. a.

Ext^p

Xet´(Q,Gm) =







0 p= 2,

ZbN

ZQ=Q0

qQq p= 3, where Q0

qQq = AQ/R is the restricted product of the Qp’s with respect to the Zp’s.

b.

Ext^p

Xet´(α∗Mv,Gm) =











0 p= 2,

H³(X_´et,Gm) = Q/Z p= 3and v | ∞, lim←−nH³(X,G^m) =ZbN

ZQ p= 3and v -∞.

Proof. a. Since Q= lim−→nZ andHⁱ(X´et,G^m) are finite groups fori= 1,2, we have

Ext^p

Xet´(Q,Gm) = lim←−nH^p(X_et´,Gm).

AsH²(X_´et,Gm) = 0, one sees that Ext²_X

´

et(Q,Gm) = 0. Also, lim←−nH³(X_et´,Gm) =

lim←−nQ/Z= Hom(Q,Q/Z) =ZbN

ZQ. b. As Mv = lim−→Lπ_L,∗⁰ A_v,

Ext^p

Xet´(α∗Mv,Gm) = Ext^p

X´et(lim−→Lπ⁰_L,∗A_v,Gm).

When v is an infinite place, A_v =Z, and by the norm theorem,

Ext^p

X´et(π⁰_L,∗Z,Gm) = H^p(u(L^v)_et´,Gm),

for all p. Note that F is totally imaginary and u(L^v) contains all the fi- nite places of L^v, so H¹(u(L^v)_´et,Gm) = P ic(u(L^v)), H²(u(L^v)_´et,Gm) = 0, and H³(u(L^v)_´et,Gm) = Q/Z. Since H¹(u(L^v)_et´,Gm) and H²(u(L^v)_´et,Gm) are both finite groups, by Lemma 4.2, we actually get

Ext^p

X´et(α∗Mv,Gm) = lim←−LH^p(u(L^v)_et´,Gm) for p = 2,3. Clearly, Ext²_X

´

et(α∗Mv,G^m) = H²(u(L^v)´et,G^m) = 0. Moreover, for any finite extension K over L, the transition map H³(u(K^v)_et´,Gm) → H³(u(L^v)´et,G^m) is an isomorphism. So, we conclude that Ext³_X

´

et(α∗Mv,G^m) = H³(X_´et,Gm) = Q/Z.

For the case v is a finite place of F, there is a slightly difference, as nowA_v = Q= lim−→ ¹ⁿZ and so α∗Mv = lim−→L,nπ⁰_L,∗Z. We claim that

Ext^p

X´et(lim−→L,nπ⁰_L,∗Z,Gm) = lim←−L,nH^p(u(L^v)_et´,Gm),

for p = 2,3. Indeed, because Ext¹_u(Lv)et´(π_L,∗⁰ Z,Gm) = H¹(u(L^v)_´et,Gm) = P ic(u(L^v)) is a finite groups, and Ext²_u(Lv)et´(π_L,∗⁰ Z,G^m) = H²(u(L^v)´et,G^m) = 0, by Lemma 4.2, one gets Ext^p

Xet´(α∗Mv,Gm) = lim←−L,nH^p(u(L^v)_et´,Gm) for p = 2,3. Again, by Lemma 4.2, one get Ext^p_X

´

et(Mv,G^m) = lim←−L,nExt^p_u(Lv)(Z,G^m)

for p= 2,3. Hence

Ext^p

Xet´ (M_v,Gm) =







0 p= 2,

lim←−nH³(X_et´,Gm) p= 3.

Proposition 4.2.

Ext^p_X

´

et(F⁰,Gm) =







0 p= 2,

Q

v|∞H³(X_´et,Gm) = Q

X∞Q/Z p= 3.

Proof. LetFv⁰ :=ker(α∗Mv →iv,∗A_v). Then it arises a long exact sequence

∂^p−1

−−−→Ext^p_X

´

et(α∗M_v,Gm)→Ext^p_X

´

et(F_v⁰,Gm)→Ext^p_X

´

et(iv,∗A_v,Gm)−→^∂p Ext^p+1_X

´

et(α∗M_v,Gm)→. Since Ext²_X

´

et(α∗Mv,Gm) = 0 by the previous lemma, and it is easy to see that Ext⁴_X

´

et(iv,∗Av,G^m) = 0, we have the following exact sequence, 0→Ext²_X

´

et(F_v⁰,Gm)→Ext³_X

´

et(iv,∗A_v,Gm)→Ext³_X

´

et(α∗Mv,Gm)→Ext³_X

´

et(F_v⁰,Gm)→0.

By the adjunction of i_v,∗ and i^!_v and Lemma 4.2 (asH_v²(X_et´,G^m) = 0),

Ext³_X

´

et(iv,∗A_v,Gm) =







H_v³(Xet´,G^m) = 0 v | ∞, lim←−nH_v³(X_et´,Gm) =Q0

qQq v -∞.

Therefore, Ext²_X

´

et(Fv⁰,G^m) = 0 and Ext³_X

´

et(Fv⁰,G^m) = Ext³_X

´

et(α_∗Mv,G^m) =Q/Z when v|∞.

We claim that the morphism Ext³_X

´

et(i_v,∗A_v,G^m) → Ext³_X

´

et(α_∗M_v,G^m) is an iso- morphism when v - ∞. For this, observe that there is a commutative diagram of

sheaves

α∗Mv = lim−→Lπ_L,∗⁰ A_v //iv,∗A_v

Av =π_F,∗⁰ Av

OO 66nnnnnnnnnnnnn

where A_v → i_v,∗A_v is the canonical morphism induced by adjunction of i^∗_v and i_v,∗. Applying Ext³(−,Gm) to the above diagram, we get another commutative diagram :

Ext³(α∗Mv,Gm)

lim←−nH_v³(X´et,G^m)

uukkkkkkkkⁱhkkkkkk

oo

lim←−nH³(X_´et,Gm).

All the groups in the above diagram are isomorphic to ZbN

ZQ = Q

qQ^q. Note the h is an isomorphism as the natural morphismH_v³(X_et´,Gm)→H³(X_et´,Gm) is an isomorphism ([4] Prop.3.2). Therefore, i is an injection from Q

qQ^q to itself. Since Hom_Z(Qp,Qq) = 0 isp6=qandQqare fields, one sees thatihas to be an isomorphism.

It follows that Ext²_X

´

et(Fv⁰,G^m) and Ext³_X

´

et(Fv⁰,G^m) are vanishing.

Consequently, Ext²_X

´

et(F⁰,Gm) = 0 and Ext³_X

´

et(F⁰,Gm) = M

v∈X

Ext³_X

´

et(Fv⁰,Gm) =M

v|∞

Q/Z

Proposition 4.3. There is a canonical isomorphism Ext²_X

´

et(F^•,Gm)∼=Zb. Proof. The exact sequence

0−→kerX

−→F⁰ −→Q−→0,

induces a long exact sequence

Ext²_X

´

et(F⁰,G^m)(= 0)→Ext²_X

´

et(kerX

,G^m)→Ext³_X

´

et(Q,G^m)−→^∆ M

v|∞

H³(X´et,G^m)→,

where ∆ is a copies of the canonical map π: Hom(Q,Q/Z)→Hom(Z,Q/Z).

Clearly, ker ∆ = kerπ = Hom_Z(Q/Z,Q/Z) =Zb. As a consequence Ext²_X

´

et(kerP

,G^m) = ker∆ = ker(π) = Zb.

Let γ be the class in Ext²_X

´

et(F^•,G^m) that corresponds to the generator 1 of Zb, and RGm is the complex decided by the exact triangle

RG^m →F^•[−1]−→^γ G^m[1]→.

To compute the ´etale cohomology of the complex RGm, we need to determine the coboundary morphisms ∂_Uⁱ :Hⁱ(U_´et,F^•)−−→^{^γ} Hⁱ⁺²(U_et´,Gm), for any connected ´etale U →X.

Lemma 4.4. For any connected ´etale U →X, let S :=U −U, we have a. ∂_U⁰ is the canonical projection

(M

v∈S

A_v)^Σ=0 →(M

v∈S

A_v/Z)^Σ=0,

b. ∂_U¹ is an isomorphism if S is non-empty, and it is the canonical projection Q→Q/Z otherwise.

c. ∂_Uⁱ is isomorphic for all i≥2.

In particular, all the ∂ⁱ are surjective for all i≥0.

Proof. We first deal with the second part of (b). Consider the following commutative

diagram

H¹(X_et´, kerP

) × Ext²_X

´

et(kerP ,G^m)

^ //H³(X_´et,Gm)

H⁰(X_et´,Q)

o

OO

× Ext³_X

´

et(Q,Gm) ^{^ //}H³(X_´et,Gm) lim−→H⁰(X_et´,_n¹Z) × lim←−Ext³_X

´

et(¹_nZ,G^m)^{^ //}H³(X_´et,Gm)

It is easy to see that the bottom cup product sends pair (a,1) to (a mod Z) and thus a ^ γ = a ^ 1 = a modZ. This shows ∂¹ : H¹(X´et,F^•) → H³(Xet´,G^m) is the canonical projection Q → Q/Z. The general case that U = U follows from the commutative diagram

H¹(X_´et,F^•)

=

^γ modZ

//H³(X_´et,Gm)

=

H¹(U_´et,F^•) ^{^γ //}H³(U_´et,Gm).

For a Zariski open subset j :U ,→X, set i: X−U ,→X, we have the following commutative diagram

0 ^//H⁰(U_´et,F^•) ^//

^γ

L

v /∈UH_v¹(X_´et,F^•)

^γ

P //H¹(X´et,F^•)

^γ

//H¹(U_´et,F^•)

^γ

//0

0 ^//H²(U_et´,G^m) ^//L

v /∈UH_v³(X_´et,G^m)

P //H³(X_et´,Gm) ^//H³(U_et´,G^m) ^//0,

where the rows are exact sequences induced by the short exact sequence

0−→j_!j^∗F −→F −→i∗i^∗F −→0, for any F ∈Sh(X).

The morphism H⁰(Uet´, kerP

) −−→^{^γ} H²(U´et,G^m) is completely determined by H¹(X_´et, kerP

)−−→^{^γ} H³(X_´et,Gm) because Σ restricts to an inclusion onH_v¹(X_et´,F^•) = Av. (This can be seen by putU to beX−v and use the factH⁰((X−v)et´, kerP

) =

0). Therefore, by the commutativity, ^ γ : H⁰(U_et´,F^•) → H²(U_´et,Gm) maps (L

v∈U−UA_v)^Σ=0 to (L

v∈U−UA_v/Z)^Σ=0 by taking modular by Z componentwise.

For general i : U → X, set U^a to be the largest open subscheme of U such that for any v ∈ U^a, all the valuations of K(U) above i(v) is contained in U^a. We have the commutative diagram

H⁰(U_{´et _},F^•) ^{^γ //}

H²(U_{´et _},Gm)

H⁰(U_´et^a,F^•) ^{^γ //}H²(U_´et^a,G^m)

H⁰(i(U^a)_et´,F^•) ^{^γ //}

OO

H²(i(U^a)_et´,Gm)

OO

More precisely,

L

w∈U−U

A_wΣ=0

^γ //

 _

L

w∈U−U

A_w/Z Σ=0

 _

L

w∈U−U^a

A_w Σ=0

^γ //

L

w∈U−U^a

A_w/Z Σ=0

L

v∈X−i(U^a)

A_vΣ=0

^γ//

OO

L

v∈X−i(U^a)

A_v/Z Σ=0

OO

Note that the cup products are canonical on v and w, and we have seen above that the bottom cup product is canonical projection. By chasing diagram, one sees that the middle one is also canonical projection, and so is the top one as the injection is canonical. It follows that ∂_U⁰ is the canonical projection.

To see the first part of part (b), we assume S is non-empty. When U_f * U, H³(U_´et,Gm) = 0 and H¹(U_et´,F^•) = 0 (Prop. 4.1). Therefore, we have ∂_U¹ = 0. For the case U_f ⊆ U, note that U is then ´etale over X as all the infinite places are

complex. Therefore, we have the following commutation diagram

Q=H¹(U_et´,F^•) ^{π //}

∂¹

U

H¹(U_´et,F^•) =Q/Z

∂_U¹

Q/Z=H³(U_´et,Gm) H³(U_´et,Gm) = Q/Z.

As we have seen, in the very beginning of the proof, that ∂_U¹ and π are the canonical projections, ∂_U¹ has to be the identity map on Q/Z by the commutative.

Part (c) is trivial because Hⁱ(U´et,F^•) =Hⁱ⁺²(Uet´,G^m) = 0 for all i≥2.

Now, we are ready to compute the ´etale cohomology of the complex RG^m. Proposition 4.4. Let U be ´etale over X, S =U −U and L=K(U), then

H^p(U´et, RG^m) =























O^×_L,S p= 0, Pic(O_L,S)⊕X_S p= 1,

0 p= 2,

Z(resp.0) p= 3 and S =∅(resp. S 6=∅),

0 p≥4.

Hence, these cohomologies coincide with those of theZ-dual ofτ^≤3RΓ_c(U_W,Z)[2] (c.f.

Prop. 3.5).

Proof. Consider the usual long exact sequence on cohomology induced by the exact triangle RGm → F^•[−1] −→^γ Gm[1] → and use the result that ∂ⁱ are surjective (Lemma 4.4), we get

H⁰(U_et´,Gm)−→^∼ H⁰(U_´et, RGm),

0→H¹(U_et´,Gm)→H¹(U_et´, RGm)→ker(∂⁰)→0 is exact, Hⁱ(U_et´, RGm) = ker(∂ⁱ⁻¹).

Thus,H⁰(U_et´, RGm) = O^×_L,S. Because ker∂⁰ = (⊕v∈SZ)^Σ=0 =X_S isZ-free, the mid- dle exact sequence splits and soH¹(U_et´, RG^m) = P ic(O_L,S)⊕X_S. Also,H³(U_et´, RG^m) =

0 (resp. Z) when S 6= ∅ (resp. S = ∅) and Hⁱ(U_´et, RGm) = 0 for all i ≥ 4, by Lemma 4.4 (b) and (c).

The computations of Prop.4.4 and 3.5 suggest that

Theorem 4.1. RΓ(U_´et, RGm)−→^qis RHom τ^≤3RΓ_c(U_W,Z),Z

[−2] in D(Z).

In the next section, we shall prove the following more general quasi-isomorphism

RΓ(U_´et, RGm)−→^qis RHom τ^≤3RΓ_c(U_W,Z),Z [−2],

in D(Z[G]), when S is stable under the action of G. Note that when G is the trivial group, this is just Theorem 4.1