Reinterpretations of the Brauer-Manin pairing

In this section we use exact sequence (3.3) and Lemma 3.2.1 to give other formu-lations of the Brauer-Manin pairing

X(A_k)×B(X)→Q/Z.

Our X is still an arbitrary smooth geometrically integral variety over a number fieldk, and we assumeX(A_k)6= /0.

We start with a couple of well-known observations. Since the elements of B(X)are locally constant by definition, the maps

b: B(X)→Q/Z (3.5)

given by evaluation on an adelic point(P_v)do not depend on the choice of(P_v), so defining the pairing is equivalent to defining the mapb. There is a commutative diagram with exact rows

injective. The first map in the top row is then injective because so is the left vertical map, by the Hasse principle for Brauer groups. Applying the snake lemma to the diagram we thus have a map

B(X) =ker(Br_aX →^M

v∈Ω

Br_aX_v)→coker(Brk→^M

v∈Ω

Brk_v)∼=Q/Z.

Lemma 3.3.1 The above map equals the mapb: B(X)→Q/Z.

Proof: Forα ∈B(X)the valueb(α)is defined by lifting firstα toα⁰∈Br₁X, then sending α⁰ to an element of ⊕_vBrk_v via a family of local sections (s_v : Br₁X →Brk_v) determined by an adelic point ofX, and finally taking the sum of local invariants. Since each s_v factors through Br₁(X×_kk_v), this yields the same element as the snake lemma construction.

Now observe that in view of Lemma 3.2.1 one may also obtain the diagram (3.6) by taking the long exact hypercohomology sequence coming from the dia-gram

0→[k¯^×→0]→ [k(X¯ )^×→DivX] → [k(X¯ )^×/k¯^×→DivX]→0

↓ ↓ ↓

0→^M

v∈Ω

[k¯^×_v →0]→^M

v∈Ω

[k¯v(X)^×→DivXv]→^M

v∈Ω

[k¯v(Xv)^×/k¯_v^×→DivXv]→0,

whereX_v:=X×_kk¯_v. The zeros on the right in (3.6) come from the fact that the groupsH³(k,G_m)andH³(k_v,G_m)all vanish.

Remark 3.3.2 Note in passing that the sectionss_vused in the above proof come from Galois-equivariant splittings

[k¯_v(X)^×→Div(X×_kk¯_v)]→[k¯^×_v →0] (3.7) of the base change of the extension (3.3) to ¯k_v. As maps of complexes, the latter are given in degree−1 by a natural splitting of the inclusion map ¯k^×_v →k¯_v(X)^× coming fromP_v as constructed e.g. in ([63], Theorem 2.3.4 (b)), and in degree 0 by the zero map. In particular, the extension (3.3) is locally split.

As in Remark 3.2.4 (2), we now pass to sheaves over the ´etale site of Sm/U, whereU ⊂SpecOk is a suitable open subset. We can then extend the upper row of the last diagram to an exact sequence

0→G_m[1]→K D(X)→K D⁰(X)→0 (3.8)

of complexes of ´etale sheaves onSm/U, where

K D(X):= [K_X^× →Div_X/U] and

K D⁰(X):= [K_X^×/G_m→Div_X/U].

By Lemma 3.2.2 we have Br_aX ∼=H¹(k,[k(X)¯ ^×/k¯^×→DivX]), so each element of Br_aX comes from an element in H¹(U,K D⁰(X)). Now assume moreover α ∈Br_aX is locally trivial, i.e. lies inB(X). For a finite place v of k we have H¹(k^h_v,j_v^h∗K D⁰(X))∼=H¹(k_v,j^∗_vK D⁰(X)), where k_v^h is the henselisation ofk atv, and j_v^h: Speck^h_v →Uas well as j_v: Speck_v→Uare the natural maps. This is shown using the quasi-isomorphism of Lemma 3.2.2, and then reasoning as in the proof of ([30], Lemma 2.7). Next recall that in the case whenkis totally imagi-nary the arithmetic compact support hypercohomologyH¹_c(U,F)of a complex of sheavesF is defined byHⁱ_c(SpecOk,j_!F), where j:U→SpecOk is the natural inclusion. It fits into a long exact sequence

· · · →H¹_c(U,F)→H¹(U,F)→ ^M

v∈SpecOk\U

H¹(k^h_v,j^h∗_v F)→. . .

In the general case there are corrective terms coming from the real places; see the discussion at the beginning of§3 in [30] (but note the misprint in formula (8) there: the ˆk_vshould bek_v in that paper’s notation). It then follows from the above discussion that we may liftα ∈B(X)to an elementα_U∈H¹_c(U,K D⁰(X))for sufficiently smallU.

There is a cup-product pairing

∪: Ext¹_Sm/U(K D⁰(X),G_m[1])×H¹_c(U,K D⁰(X))→H_c³(U,G_m)∼=Q/Z, where the Ext-group is taken in the category of ´etale sheaves onSm/U, and the last isomorphism comes from global class field theory (see [47], p. 159). We shall be interested in the classEX∪αU, whereEX is the class of the sequence (3.8 )in Ext¹_Sm/U(K D⁰(X),G_m[1]). Note that there is a commutative diagram

Ext¹_Sm/U(K D⁰(X),G_m[1])×H¹_c(U,K D⁰(X))→H_c³(U,G_m)∼=Q/Z,

↓ ↓∼= ↓ id

Ext_U¹(g_∗K D⁰(X),G_m[1])×H¹_c(U,g_∗K D⁰(X))→H_c³(U,G_m)∼=Q/Z,

where g_∗ is the natural pushforward (or restriction) map from the ´etale site of Sm/U to the small ´etale site ofU, and the group in the bottom row is an Ext-group for ´etale sheaves onU. The left vertical map exists because the functorg_∗ is exact (and G_m as an ´etale sheaf on U is the pushforward by g of the G_m on Sm/U). The middle isomorphism comes from the fact that the hypercohomology of complexes of sheaves on the big ´etale site ofU equals the hypercohomology on the small ´etale site. So instead of EX we may work with its image g_∗EX in Ext_U¹(g_∗K D⁰(X),G_m[1]), and omit theg_∗from the notation when no confusion is possible. Note that the generic stalk ofg_∗EX is the class of the extension (3.3) in the group Ext¹_k([k(X)¯ ^×/k¯^×→DivX],k¯^×[1]).

Proposition 3.3.3 With notations as above, we have EX∪α_U =b(α).

Before starting the proof, note that though one has several choices forα_U, the cup-product depends only on α. Indeed two choices of α_U differ by an element of the direct sum of the groupsH⁰(k^h_v,K D⁰(X)), and the cup-product of each such group with Ext_U¹(K D⁰(X),G_m[1]) factors through the cup-product with Ext¹_kh

v(j_v^∗K D⁰(X),G_m[1]). But the image of the classg_∗EX in these groups is 0, because the extension is locally split (Remark 3.3.2).

Proof: In order to avoid complicated notation, we do the verification in the case whenU is totally imaginary and the simpler definition of compact support co-homology is available, and leave the general case to anxious readers. We may work over the small ´etale site ofU by the previous observations; in particular, we identify the complexesK D(X)andK D⁰(X)with their images underg_∗.

The cup-product EX∪α_U is none but the image of α by the boundary map H¹_c(U,K D⁰(X))→H_c³(U,G_m)coming from the long exact hypercohomology sequence associated with (3.8). Now consider the commutative exact diagram

0 → G_m[1] → K D(X) → K D⁰(X) → 0

↓ ↓ ↓

0→^M

v/∈U

j_v∗j^∗_vG_m[1]→^M

v/∈U

j_v∗j^∗_vK D(X)→^M

v/∈U

j_v∗j^∗_vK D⁰(X)→0

of complexes of ´etale sheaves onU, and denote the cones of the vertical maps by C,CK andC_K⁰ , respectively. The groupH¹(U,C_K⁰ [−1])may be identified with H¹_c(U,K D⁰(X))(apply, for instance, [47], Lemma II.2.4 and its proof with our U asV and our SpecOk asU), so we may view α_U as an element of the former group. The cup-productEX∪αU maps to a class in H²(U,C[−1]) =H¹(U,C).

But when one makes U smaller and smaller and passes to the limit, this class yields an element in the cokernel of the map Brk→ ⊕_vBrk_v^h which (noting the isomorphism Brk_v^h∼= Brk_v) is precisely the one obtained by the snake-lemma construction at the beginning of this section. It remains to apply Lemma 3.3.1.