THE WEIL-´ETALE FUNDAMENTAL GROUP OF A NUMBER FIELD I
THE WEIL-´ETALE FUNDAMENTAL GROUP OF A NUMBER FIELD I THE WEIL-´ETALE FUNDAMENTAL GROUP OF A NUMBER FIELD I
128 B. Morin Proof. The result for n = 0 follows from the connectedness of Ū L → T given by Property (2). Indeed, we have H 0 (Ū W ,t ∗˜R) := (eT ∗ ◦ t ∗ )t ∗˜R = eT ∗˜R = R, where e T denotes the unique map e T : T → Sets. By Property (3), the result for n = 1 follows from H 1 (Ū L , ˜R) = Hom c (π 1 (Ū L ), R) := lim Hom −→ c (π 1 (Ū L ), R) = lim Hom −→ c (π 1 (Ū L ) ab , R) = Hom c (CŪ , R). The result for n ≥ 2 is given by Property (9). ✷ The maximal compact subgroup of CŪ , i.e. the kernel of the absolute value map CŪ → R >0 , is denoted by C 1 . Thus we have an exact sequence of topological groups Ū 1 → C 1 Ū → C Ū → R >0 → 1. The Pontraygin dual (C 1 Ū )D is discrete since C 1 is compact. Ū PROPOSITION 6.2. For any connected étale ¯X-scheme Ū, we have canonically ⎧ ⎪⎨ Z for n = 0 H n (Ū L , Z) = 0 for n = 1 ⎪⎩ (C 1 Ū )D for n = 2. Proof. As above, the result for n = 0 follows from the connectedness of Ū L → T given by Property (2). By Property (3), the result for n = 1 follows from H 1 (Ū L , Z) = Hom c (π 1 (Ū L ) ab , Z) = 0. By Property (3) we have canonical isomorphisms The exact sequence of topological groups H 1 (Ū L , ˜S 1 ) = Hom c (π 1 (Ū L ) ab , S 1 ) := lim Hom −→ c (π 1 (Ū L ) ab , S 1 ) = Hom c (lim π ←− 1 (Ū L ) ab , S 1 ) = Hom c (CŪ , S 1 ) = C D Ū . 0 → Z → R → S 1 → 0 induces an exact sequence 0 → Z → ˜R → ˜S 1 → 0
The Weil-étale fundamental group of a number field I 129 of abelian sheaves on Ū L . The induced long exact sequence 0 = H 1 (Ū L , Z) → H 1 (Ū L , ˜R) → H 1 (Ū L , ˜S 1 ) → H 2 (Ū L , Z) → H 2 (Ū L , ˜R) = 0 is canonically identified with 0 → Hom c (CŪ , R) → Hom c (CŪ , S 1 ) → H 2 (Ū L , Z) → 0 and we obtain H 2 (Ū L , Z) = (C 1 Ū )D . Recall that we have a canonical morphism γ : ¯X L → ¯X et . We consider the truncated functor τ ≤2 Rγ ∗ of the total derived functor Rγ ∗ . COROLLARY 6.3. We have γ ∗ Z = Z, R 1 (γ ∗ )Z = 0 and R 2 (γ ∗ )Z is the étale sheaf associated to the abelian presheaf P 2 γ ∗ Z : Et ¯X −→ Ab Ū ↦−→ (C 1 Ū )D . Proof. The sheaf R n (γ ∗ )Z is the sheaf associated to the presheaf Ū ↦→ H n ( ¯X L /γ ∗ Ū,Z). Hence the corollary follows immediately from Proposition 6.2. Note that it follows from Property (4) that the restriction map P 2 γ ∗ Z(Ū)= (C 1 Ū )D → P 2 γ ∗ Z( ¯V)= (C )D 1¯V is the Pontryagin dual of the canonical morphism C 1¯V → C1 (induced by the norm map), for Ū any ¯V → Ū in Et ¯X . ✷ The cohomology of sheaf R 2 γ ∗ Z associated to P 2 γ ∗ Z is computed in Section 6.3. The étale sheaf R 2 γ ∗ Z is acyclic for the global sections functor on ¯X et .Moreprecisely,wehave { Hom(O H n ( ¯X et ; R 2 × γ ∗ Z) = F , Q) for n = 0, 0 forn ≥ 1. Recall that Pic( ¯X) = C ¯X is the Arakelov–Picard group of F ,andthatμ F is the group of roots of unity in F . We compute below the hypercohomology of the complex of abelian étale sheaves τ ≤2 Rγ ∗ Z. THEOREM 6.4. We have ⎧ Z for n = 0 ⎪⎨ 0 for n = 1 H n ( ¯X et ,τ ≤2 Rγ ∗ Z) = Pic 1 ( ¯X) D for n = 2 μ D F for n = 3 ⎪⎩ 0 for n ≥ 4. Recall that the Artin–Verdier étale cohomology of Z is given by ⎧ Z for i = 0 ⎪⎨ 0 fori = 1 H i ( ¯X et , Z) = Cl(F ) D for i = 2 Hom(OF ∗ , Q/Z) for i = 3 ⎪⎩ 0 fori ≥ 4. ✷
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128 B. Morin<br />
Proof. The result for n = 0 follows from the connectedness of Ū L → T given by<br />
Property (2). Indeed, we have<br />
H 0 (Ū W ,t ∗˜R) := (eT ∗ ◦ t ∗ )t ∗˜R = eT ∗˜R = R,<br />
where e T denotes the unique map e T : T → Sets. By Property (3), the result for n = 1 follows<br />
from<br />
H 1 (Ū L , ˜R) = Hom c (π 1 (Ū L ), R)<br />
:= lim Hom −→ c (π 1 (Ū L ), R)<br />
= lim Hom −→ c (π 1 (Ū L ) ab , R)<br />
= Hom c (CŪ , R).<br />
The result for n ≥ 2 is given by Property (9).<br />
✷<br />
The maximal compact subgroup of CŪ , i.e. the kernel of the absolute value map<br />
CŪ → R >0 , is denoted by C 1 . Thus we have an exact sequence of topological groups<br />
Ū<br />
1 → C 1 Ū → C Ū → R >0 → 1.<br />
The Pontraygin dual (C 1 Ū )D is discrete since C 1 is compact.<br />
Ū<br />
PROPOSITION 6.2. For any connected étale ¯X-scheme Ū, we have canonically<br />
⎧<br />
⎪⎨ Z for n = 0<br />
H n (Ū L , Z) = 0 for n = 1<br />
⎪⎩ (C 1 Ū )D for n = 2.<br />
Proof. As above, the result for n = 0 follows from the connectedness of Ū L → T given by<br />
Property (2). By Property (3), the result for n = 1 follows from<br />
H 1 (Ū L , Z) = Hom c (π 1 (Ū L ) ab , Z) = 0.<br />
By Property (3) we have canonical isomorphisms<br />
The exact sequence of topological groups<br />
H 1 (Ū L , ˜S 1 ) = Hom c (π 1 (Ū L ) ab , S 1 )<br />
:= lim Hom −→ c (π 1 (Ū L ) ab , S 1 )<br />
= Hom c (lim π ←− 1 (Ū L ) ab , S 1 )<br />
= Hom c (CŪ , S 1 ) = C D Ū .<br />
0 → Z → R → S 1 → 0<br />
induces an exact sequence<br />
0 → Z → ˜R → ˜S 1 → 0
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