THE WEIL-´ETALE FUNDAMENTAL GROUP OF A NUMBER FIELD I

The Weil-étale fundamental group of a number field I 133
THEOREM 6.10. The morphism ∪θ obtained by cup product with the fundamental class θ is
the canonical isomorphism
∪θ : H 1 ( ¯X L ,ϕ !˜R) = (
∏v|∞ R)/R −→ H 2 ( ¯X L ,ϕ !˜R) = Homc (( ∑ v|∞ W k(v)) 1 , R)
v ↦−→ θ v ◦ p v ,
where p v : ( ∑ w|∞ W k(w)) 1 → W k(v) is given by the projection.
Proof. By Property (8), the morphism
H 1 (B Pic( ¯X) , ˜R) = H 1 ( ¯X L , ˜R) −→ H 1 ( ∐ v∈X ∞
B Wk(v) , ˜R) = ∏ v|∞ H 1 (B Wk(v) , ˜R)
θ ↦−→ (θ v ) v|∞ ,
which is induced by ∐ v∈X ∞
B Wk(v) → ¯X L → B Pic( ¯X) , sends fundamental class to
fundamental class. Hence the cup-product morphism ∪θ is induced by
∏
∏
(∪θ v ) v|∞ : H 0 ( ¯X L ,i ∞∗˜R) = R −→ H 1 ( ¯X L ,i ∞∗˜R) = Hom c (B Wk(v) , ˜R)
v|∞
and the result follows. Note that ∪θ is well defined, since ∪θ( ∑ v|∞ v) = ∑ v|∞ θ v is the
canonical map ∑ v|∞ W k(v) → R, which vanishes on ( ∑ v|∞ W k(v)) 1 .
✷
THEOREM 6.11. For any n ≥ 1, the morphism
v|∞
R n : H n ( ¯X et ,τ ≤2 Rγ ∗ (ϕ ! Z)) ⊗ R −→ H n ( ¯X L ,ϕ !˜R),
induced by the morphism of sheaves ϕ ! Z → ϕ !˜R, is an isomorphism.
Proof. We denote by
κ n : H n ( ¯X et ,τ ≤2 Rγ ∗ (ϕ ! Z)) −→ H 1 ( ¯X L ,ϕ !˜R)
the morphism induced by ϕ ! Z → ϕ !˜R. The result is obvious for n ≠ 1, 2. The result is also
clear for n = 1, since κ 1 is canonically identified with
( ∐
)/
( ∐
)/
H 0 B Wk(v) , Z H 0 ( ¯X L , Z) → H 0 B Wk(v) , ˜R H 0 ( ¯X L , ˜R),
v∈X ∞ v∈X ∞
hence R 1 is the identity. Assume that n = 2. On the one hand, we have canonically
H 2 ( ¯X et ,τ ≤2 Rγ ∗ (ϕ ! Z)) = H 2 ( ¯X L , Z) = H 1 ( ¯X L , ˜S 1 )/H 1 ( ¯X L , ˜R).
On the other hand, we have
( ∐
)/
( ∐
H 2 ( ¯X L ,ϕ !˜R) = H 1 B Wk(v) , ˜R H 1 ( ¯X L , ˜R) = H 1 B Wk(v) , ˜S
)/H 1 1 ( ¯X L , ˜R)
v|∞ v|∞
and the map R 2 is induced by
( ∐
) ( ∑
)
H 1 ( ¯X L , ˜S 1 ) = Hom c (Pic( ¯X), S 1 ) −→ H 1 B Wk(v) , ˜S 1 = Hom c W k(v) , S 1
v|∞
v|∞
which is in turn induced by the canonical morphism ∑ v|∞ W k(v) → Pic( ¯X), as it follows
from Property (8). We have the exact sequence
Hom c (Pic( ¯X), R) → Hom c (Pic( ¯X), S 1 ) → Hom c (Pic 1 ( ¯X), S 1 ) → 0,