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
114 B. Morin or (more directly) by the Pontryagin dual of the map R = H 1 T ( ¯X L , ˜R) −→ H 1 T ( ¯X L , ˜S 1 ) = π 1 ( ¯X L ,p) D . COROLLARY 4.6. Let ¯X L be a topos over T satisfying Hypotheses 4.1 and 4.2. Then there is a fundamental class θ ∈ H 1 (X L , ˜R). If the fundamental group π 1 ( ¯X L ,p)is representable by a locally compact group, then θ ∈ H 1 (X L , ˜R) = Hom cont (Pic( ¯X), R) is the canonical continuous morphism θ : Pic( ¯X) → R. Proof. The canonical map π : ¯X L → B Pic( ¯X) induces a map π ∗ : H 1 (B Pic( ¯X) , ˜R) −→ H 1 ( ¯X L , ˜R). The direct image of the unique map T → Sets is exact, hence we have H 1 (B Pic( ¯X) , ˜R) = H 0 (T ,H 1 T (B Pic( ¯X) , ˜R)) = Hom Top (Pic( ¯X), R). Therefore, the usual continuous morphism α : Pic( ¯X) → R is a distinguished element α ∈ H 1 (B Pic( ¯X) , ˜R). We define the fundamental class as θ := π ∗ (α) ∈ H 1 ( ¯X L , ˜R). Note that the fundamental class ϕ can also be defined by θ := f ∗ (Id R ) ∈ H 1 ( ¯X L , ˜R), where Id R is the distinguished non-zero element of H 1 (B R , ˜R) = Hom Top (R, R). Finally, if the fundamental group π 1 ( ¯X L ,p)is representable by a locally compact group, then the map π ∗ : H 1 (B Pic( ¯X) , ˜R) −→ H 1 ( ¯X L , ˜R) is an isomorphism, and θ can be identified with α. Indeed, Theorem 4.3 yields in this case that H 1 ( ¯X L , ˜R) = Hom cont (π 1 ( ¯X L ,p),R) 4.3. The fundamental group and unramified class field theory = Hom cont (π 1 ( ¯X L ,p) ab , R) = Hom cont (Pic( ¯X), R). ✷ There exist complexes R W (ϕ ! Z) and R W (Z) of sheaves on the Artin–Verdier étale topos whose hypercohomology is the conjectural Lichtenbaum cohomology with and without compact support respectively (see [13]). This suggests the existence of a canonical morphism of topoi γ : ¯X L −→ ¯X et such that Rγ ∗ Z = R W (Z),where ¯X et denotes the Artin–Verdier étale topos of X. On the one hand, the complex R W (Z) yields a canonical map H n ( ¯X et , Z) −→ H n L ( ¯X, Z) ✷
The Weil-étale fundamental group of a number field I 115 for any n ≥ 0. In degree n = 2, this map Pic(X) D = (π 1 ( ¯X et ) ab ) D = H 2 ( ¯X et , Z) −→ H 2 L ( ¯X, Z) := Pic 1 ( ¯X) D (13) is the dual map of the canonical morphism Pic 1 ( ¯X) → Pic(X) = Cl(F ). On the other hand, the morphism γ would induce a morphism of abelian fundamental groups π 1 ( ¯X L ,p) DD −→ π 1 ( ¯X et ,q) DD ≃ π 1 ( ¯X et ) ab , (14) where q is a geometric point of ¯X such that the following diagram commutes. ¯X L T p γ e T ¯X et q Sets Note that q is uniquely determined by p since the unique map e T : T → Sets has a canonical section s (see [7, IV. 4.10]). Indeed, we have e T ◦ s = Id hence q ≃ q ◦ e T ◦ s ≃ γ ◦ p ◦ s. (15) The map (14) needs to be compatible with the canonical map (13). In other words, the following morphism should be the reciprocity map of class field theory: More precisely, the diagram Pic( ¯X) ≃ π 1 ( ¯X L ,p) DD −→ π 1 ( ¯X et ) ab . (16) Pic( ¯X) Pic(X) = Cl(F ) π 1 ( ¯X L ,p) DD (14) π1 ( ¯X et ) ab should be commutative, where Pic( ¯X) → Pic(X) = Cl(F ) is the canonical map, Cl(F ) → π 1 ( ¯X et ) ab is the isomorphism of unramified class field theory and Pic( ¯X) → π 1 ( ¯X L ,p) DD is the isomorphism defined in Theorem 4.3. 4.4. The fundamental group and the closed embedding i v For any closed point v of ¯X, i.e. any non-trivial valuation of the number field F , we denote by W k(v) := F × v /O× F v the Weil group of the residue field k(v) at v, whereO × F v is the kernel of the valuation F × v → R× . Let U ⊆ X be an open sub-scheme. The conjectural Lichtenbaum cohomology with compact support is defined as (see the Introduction of [10]): H ∗ c (U, A) := H ∗ ( ¯X L ,ϕ ! A), where ϕ : U L := ¯X L /γ ∗ U −→ ¯X L
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114 B. Morin<br />
or (more directly) by the Pontryagin dual of the map<br />
R = H 1 T ( ¯X L , ˜R) −→ H 1 T ( ¯X L , ˜S 1 ) = π 1 ( ¯X L ,p) D .<br />
COROLLARY 4.6. Let ¯X L be a topos over T satisfying Hypotheses 4.1 and 4.2. Then there<br />
is a fundamental class θ ∈ H 1 (X L , ˜R). If the fundamental group π 1 ( ¯X L ,p)is representable<br />
by a locally compact group, then<br />
θ ∈ H 1 (X L , ˜R) = Hom cont (Pic( ¯X), R)<br />
is the canonical continuous morphism θ : Pic( ¯X) → R.<br />
Proof. The canonical map π : ¯X L → B Pic( ¯X) induces a map<br />
π ∗ : H 1 (B Pic( ¯X) , ˜R) −→ H 1 ( ¯X L , ˜R).<br />
The direct image of the unique map T → Sets is exact, hence we have<br />
H 1 (B Pic( ¯X) , ˜R) = H 0 (T ,H 1 T (B Pic( ¯X) , ˜R)) = Hom Top (Pic( ¯X), R).<br />
Therefore, the usual continuous morphism α : Pic( ¯X) → R is a distinguished element<br />
α ∈ H 1 (B Pic( ¯X) , ˜R). We define the fundamental class as<br />
θ := π ∗ (α) ∈ H 1 ( ¯X L , ˜R).<br />
Note that the fundamental class ϕ can also be defined by<br />
θ := f ∗ (Id R ) ∈ H 1 ( ¯X L , ˜R),<br />
where Id R is the distinguished non-zero element of H 1 (B R , ˜R) = Hom Top (R, R).<br />
Finally, if the fundamental group π 1 ( ¯X L ,p)is representable by a locally compact group,<br />
then the map<br />
π ∗ : H 1 (B Pic( ¯X) , ˜R) −→ H 1 ( ¯X L , ˜R)<br />
is an isomorphism, and θ can be identified with α. Indeed, Theorem 4.3 yields in this case<br />
that<br />
H 1 ( ¯X L , ˜R) = Hom cont (π 1 ( ¯X L ,p),R)<br />
4.3. The fundamental group and unramified class field theory<br />
= Hom cont (π 1 ( ¯X L ,p) ab , R)<br />
= Hom cont (Pic( ¯X), R). ✷<br />
There exist complexes R W (ϕ ! Z) and R W (Z) of sheaves on the Artin–Verdier étale topos<br />
whose hypercohomology is the conjectural Lichtenbaum cohomology with and without<br />
compact support respectively (see [13]). This suggests the existence of a canonical morphism<br />
of topoi<br />
γ : ¯X L −→ ¯X et<br />
such that Rγ ∗ Z = R W (Z),where ¯X et denotes the Artin–Verdier étale topos of X. On the one<br />
hand, the complex R W (Z) yields a canonical map<br />
H n ( ¯X et , Z) −→ H n L ( ¯X, Z)<br />
✷
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