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

132 B. Morin
THEOREM 6.7. We have canonical isomorphisms
⎧( ∏
)/
R R for n = 1
⎪⎨
H n v|∞
( ¯X L ,ϕ !˜R) =
Hom c (( ⊕ v|∞
⎪⎩
W k(v)) 1 , R) for n = 2
0 for n ≠ 1, 2.
Proof. The direct image i v∗ is exact hence the group H n ( ¯X L , ∏ v|∞ i v∗˜R) is canonically
isomorphic to
∏
R n = 0
⎧⎪
∏
⎨
v|∞ ( ∑
)
H n (B Wk(v)˜R) = Hom c W k(v) , R n = 1
v|∞
⎪ ⎩
v|∞
0 n ≥ 2.
Using the exact sequence (32), the result for n ≥ 3 follows from Property (9). By (32) we
have the exact sequence
0 → H 0 ( ¯X L ,ϕ !˜R) → H 0 ( ¯X W , ˜R) = R → ∏ v|∞
H n (B Wk(v)˜R)
= ∏ v|∞
R → H 1 ( ¯X L ,ϕ !˜R) → 0,
where the central map is the diagonal embedding. The result follows for n = 0, 1. For n = 2,
we have the exact sequence
( ∑
)
Hom c (Pic( ¯X), R) → Hom c W k(v) , R → H 2 ( ¯X L ,ϕ !˜R) → 1,
v|∞
where, by Property (8), the first map is induced by the canonical morphism ∑ v|∞ W k(v) →
Pic( ¯X). We obtain a canonical isomorphism
(( ∑
) 1 )
H 1 ( ¯X L ,ϕ !˜R) = Homc W k(v) , R . ✷
v|∞
Definition 6.8. We define the fundamental class θ ∈ H 1 ( ¯X L , ˜R) as the canonical morphism
θ ∈ H 1 ( ¯X L , ˜R) = Hom c (Pic( ¯X), R).
Recall that, for any closed point v ∈ ¯X, wehaveH n (B Wk(v) , ˜R) = R, Hom c (W k(v) , R)
and0forn = 0, 1andn ≥ 2 respectively.
Definition 6.9. For any closed point v ∈ ¯X, the v-fundamental class is the canonical
morphism θ v : W k(v) → R:
θ v ∈ H 1 (B Wk(v) , ˜R) = Hom c (W k(v) , R).
The morphism obtained by cup product with θ v is the canonical isomorphism
∪θ v : H 0 (B Wk(v) , ˜R) = R −→ H 1 (B Wk(v) , ˜R) = Hom c (W k(v) , R)
1 ↦−→ θ v .