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

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130 B. Morin Proof. The hypercohomology spectral sequence H i ( ¯X et ,H j (τ ≤2 Rγ ∗ Z)) ⇒ H i+j ( ¯X et ,τ ≤2 Rγ ∗ Z) first gives H 0 ( ¯X 0 ,τ ≤2 Rγ ∗ Z) = Z and H 1 ( ¯X et ,τ ≤2 Rγ ∗ Z) = 0. On the other hand we have for any n ≤ 2. In particular, we have H n ( ¯X et ,τ ≤2 Rγ ∗ Z) = H n ( ¯X et ,Rγ ∗ Z) = H n ( ¯X L , Z) H 2 ( ¯X et ,τ ≤2 Rγ ∗ Z) = H 2 ( ¯X L , Z) = Pic 1 ( ¯X) D . Therefore, the spectral sequence above yields an exact sequence 0 → Cl(F ) D → Pic 1 ( ¯X) D → Hom(O ∗ F , Q) → Hom(O∗ F , Q/Z) → H3 ( ¯X L ,τ ≤2 Rγ ∗ Z) → 0. Here the maps Cl(F ) D → Pic 1 ( ¯X) D → Hom(OF ∗ , Q) are explicitly given. The first is induced by the morphism from the Weil-étale fundamental group to the étale fundamental group π 1 ( ¯X L ) ab −→ π 1 ( ¯X et ) ab given by Property (3), and the second is induced by the canonical morphism P 2 γ ∗ Z → R 2 (γ ∗ )Z (the map from a presheaf to its associated sheaf). It follows that the cokernel of the map Pic 1 ( ¯X) D → Hom(O ∗ F , Q) is We obtain an exact sequence Hom(O ∗ F , Q)/Hom(O∗ F , Z) ≃ Hom(O∗ F /μ F , Q/Z). 0 → Hom(O ∗ F /μ F , Q/Z) → Hom(O ∗ F , Q/Z) → H3 ( ¯X et ,τ ≤2 Rγ ∗ Z) → 0. (31) Let us denote by α : Hom(OF ∗ /μ F , Q/Z) → Hom(OF ∗ , Q/Z) the first map of the exact sequence (31). We need to show that α is the canonical map. There is a decomposition Hom(O ∗ F , Q/Z) = Hom(O∗ F /μ F , Q/Z) × μ D F and the composition (where p is the projection) p ◦ α : Hom(O ∗ F /μ F , Q/Z) → Hom(O ∗ F , Q/Z) → μD F must be 0 since Hom(OF ∗ /μ F , Q/Z) is divisible and μ D F is contained in the subgroup finite. It follows that the image of α hence α induces an injective morphism Hom(O ∗ F /μ F , Q/Z) ⊂ Hom(O ∗ F , Q/Z) ˜α : Hom(O ∗ F /μ F , Q/Z)↩→ Hom(O ∗ F /μ F , Q/Z). Since those two groups are both finite sums of Q/Z,thismap˜α needstobeanisomorphism. Indeed, the n-torsion subgroups are both finite of the same cardinality for any n, hence an
The Weil-étale fundamental group of a number field I 131 injective map must be bijective. Hence α has the same image as the canonical map, i.e. the image of the map induced by the quotient map O ∗ F → O∗ F /μ F . Hence the exact sequence (31) yields a canonical identification H 3 ( ¯X et ,τ ≤2 Rγ ∗ Z) = μ D F . Finally, we have H n ( ¯X et ,τ ≤2 Rγ ∗ Z) = 0 forn ≥ 4 since the diagonals of the hypercohomology spectral sequence are all trivial for n ≥ 4. ✷ Let ϕ : X L → ¯X L be the open embedding. By Property (7), we have an open/closed decomposition ϕ : X L → ¯X L ← ∐ B Wk(v) : i ∞ . v∈X ∞ In particular, we have adjoint functors ϕ ! ,ϕ ∗ ,ϕ ∗ . If we consider the induced functors on abelian sheaves, we have adjoint functors i∞ ∗ ,i ∞∗,i ∞ ! , showing that i ∞∗ is exact (on abelian objects). We obtain an exact sequence of sheaves 0 → ϕ ! ϕ ∗ A → A → ∏ i v∗ iv ∗ A → 0 (32) v|∞ for any abelian object A of ¯X L . THEOREM 6.5. We have canonically ⎧ 0 for n = 0 ( ∏ Z)/ Z for n = 1 ⎪⎨ H n X ∞ ( ¯X et ,τ ≤2 Rγ ∗ (ϕ ! Z)) = Pic 1 ( ¯X) D for n = 2 μ D F for n = 3 ⎪⎩ 0 for n ≥ 4. Proof. Let v ∈ X ∞ . By Property (7), we have γ ◦ i v = u v ◦ α v ,whereα v : B Wk(v) → Sets is the unique map. The morphisms i v and u v are both closed embeddings so that i v∗ and u v∗ are both exact, hence we have R(γ ∗ )i v∗ Z = u v∗ R(α v∗ )Z. This complex is concentrated in degree 0 since R n (α v∗ )Z = H n (B Wk(v) , Z) = 0forany n ≥ 1, and we have R 0 (γ ∗ )i v∗ Z = u v∗ Z. Hence the theorem follows from Theorem 6.4, exact sequence (32), and H ∗ ( ¯X et ,u v∗ Z) = H ∗ (Sets, Z). ✷ Remark 6.6. A complex quasi-isomorphic to τ ≤2 Rγ ∗ (ϕ ! Z) was constructed in [13] usinga more complicated method. 6.2. Dedekind zeta functions at s = 0 We denote by ( ⊕ v|∞ W k(v)) 1 the kernel of the canonical morphism ⊕ v|∞ W k(v) → R >0 .
Page 1 and 2: Kyushu J. Math. 65 (2011), 101-140
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THE WEIL-´ETALE FUNDAMENTAL GROUP OF A NUMBER FIELD I

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