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

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134 B. Morin hence κ 2 is the morphism κ 2 : Hom c (Pic 1 ( ¯X), S 1 ) −→ Hom((⊕ v|∞ W k(v) ) 1 , S 1 ) = Hom((⊕ v|∞ W k(v) ) 1 , R), where the first map is the Pontryagin dual of (⊕ v|∞ W k(v) ) 1 → Pic 1 ( ¯X). Recall that we have the exact sequence of topological groups 0 → (⊕ v|∞ W k(v) ) 1 /(O × F /μ F ) → Pic 1 ( ¯X) → Cl(F ) → 0. Then it is straightforward to check that there is a canonical identification and that the map R 2 is the morphism H 2 ( ¯X et ,τ ≤2 Rγ ∗ (ϕ ! Z)) ⊗ R = Hom(O × F , R) R 2 : Hom(O × F , R) −→ Hom c((⊕ v|∞ W k(v) ) 1 , R) which is the inverse of the isomorphism induced by the natural map O F × → (⊕ v|∞W k(v) ) 1 . In other words R2 −1 : Hom c ((⊕ v|∞ W k(v) ) 1 , R) −→ Hom(O F × , R) is induced by the natural map O F × → (⊕ v|∞W k(v) ) 1 . ✷ The morphisms ∪θ, R 1 and R 2 have been made explicit during the proof of Theorem 6.10 and in Theorem 6.11. The result of Corollary 6.12 follows. COROLLARY 6.12. We have a commutative diagram ( ∑ v|∞ R)/R D Hom(O F × , R) R 1 R 2 ( ∑ v|∞ R)/R ∪θ Homc ((⊕ v|∞ W k(v) ) 1 , R) where D is the transpose of the usual regulator map O × F ⊗ R −→ ( ∑ v|∞ R) + . We denote by ϕ : X L → ¯X L the natural open embedding, and by H n c (X L, A) := H n ( ¯X L ,ϕ ! A) the cohomology with compact support with coefficients in the abelian sheaf A. <strong>THE</strong>OREM 6.13. (Lichtenbaum’s formalism) Assume that F is totally imaginary. Let ¯X L be any topos satisfying Properties (1)–(9) above. We denote by τ ≤2 Rγ ∗ the truncated functor of the total derived functor Rγ ∗ ,whereγ is the morphism given by Property (1). Then the following are true. • H n ( ¯X et ,τ ≤2 Rγ ∗ (ϕ ! Z)) is finitely generated and zero for n ≥ 4. • The canonical map is an isomorphism for any n ≥ 0. H n ( ¯X et ,τ ≤2 Rγ ∗ (ϕ ! Z)) ⊗ R −→ H n c (X L, ˜R)
The Weil-étale fundamental group of a number field I 135 • There exists a fundamental class θ ∈ H 1 ( ¯X L , ˜R). The complex of finite-dimensional vector spaces ···→Hc n−1 (X L , ˜R) → Hc n (X L, ˜R) → Hc n+1 (X L , ˜R) →··· defined by cup product with θ, is acyclic. • The vanishing order of the Dedekind zeta function ζ F (s) at s = 0 is given by ord s=0 ζ F (s) = ∑ n≥0(−1) n n dim R H n c (X L, ˜R). • The leading term coefficient ζF ∗ (s) at s = 0 is given by the Lichtenbaum Euler characteristic ζF ∗ (s) =±∏ |H n ( ¯X et ,τ ≤2 Rγ ∗ (ϕ ! Z)) tors | (−1)n /det(Hc n (X L, ˜R), θ, B ∗ ), n≥0 where B n is a basis of H n ( ¯X et ,τ ≤2 Rγ ∗ (ϕ ! Z))/tors. In particular, those results hold for the Weil-étale topos ¯X W defined in Section 6. Proof. This follows from Theorems 6.5, 6.10, 6.11, Corollary 6.12 and from the analytic class number formula. ✷ 6.3. The sheaf R 2 γ ∗ Z The étale sheaf R 2 γ ∗ Z is the sheaf associated to the presheaf P 2 Z : Et ¯X −→ Ab Ū ↦−→ (C 1 Ū )D . Recall that if Ū is connected of function field K(Ū),thenCŪ is the S-idèle class group of K(Ū),whereS is the set of places of K(Ū)not corresponding to a point of Ū.Inotherwords, if we set K = K(U) then CŪ = C K,S is the S-idèle class group of K defined by the exact sequence ∏ O K × v → C K → C K,S → 0. v∈U The compact group C 1 Ū is then defined as the kernel of the canonical map C Ū → R× .Note that such a finite set S does not necessarily contain all the archimedean places. The restriction maps of the presheaf P 2 Z are induced by the canonical maps C ¯V → C Ū (well defined for any étale map ¯V → Ū of connected étale ¯X-schemes). By class field theory, we have a covariantly functorial exact sequence of compact topological groups 0 → D 1 Ū → C1 Ū → π 1(Ū et ) ab → 0, where π 1 (Ū et ) ab is the abelian étale fundamental group of Ū and D 1 is the connected Ū component of 1 in C 1 Ū .Hereπ 1(Ū) ab is defined as the abelianization of the profinite fundamental group of the Artin–Verdier étale topos ¯X et /yŪ ≃ Ū et . If we denote the function
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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