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

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122 B. Morin Note that C K,S∞ has a finite filtration such that the quotients of the form Fil n /Fil n+1 are either finite or connected. Recall that, for w ultrametric, O K × w is given with the filtration so that O × K w is the profinite group O × K w = U 0 w ⊇ U 1 w ⊇ U 2 w ⊇··· O K × w = lim U 0 ←− w /Un w . Hence the exact sequence (24) provides CŪ with a structure of a topological pro-group. More precisely, we have C K,S = lim C ←− K,S /, where runs over the system of open subgroups of ∏ w∈S 0 O K × w . Definition 5.2. We define CŪ as the topological pro-group CŪ := {C K,S /, for open in ∏ w∈S 0 O × K w }. The pro-group CŪ can also be seen as the locally compact group C K,S endowed with the filtration C K,S ⊇ ∏ O K × w ⊇ ∏ Uw 1 ⊇ ∏ Uw 2 ⊇··· . (25) w∈S 0 w∈S 0 w∈S 0 Indeed the sequence { n := ∏ w∈S 0 Uw n , for n ≥ 0} is cofinal in the system of open ⊆ ∏ w∈S 0 O K × w . Hence the pro-group CŪ can be defined as follows: CŪ := {C K,S / n for n ≥ 0}. PROPOSITION 5.3. For any map ¯V → Ū of connected étale ¯X-schemes, the map N : C ¯V → CŪ , induced by the usual norm map, is compatible with the pro-group structures of C ¯V and CŪ . For any Galois étale cover ¯V → Ū, the usual Galois action of Gal( ¯V/Ū) on C ¯V is compatible with the pro-group structure of C ¯V . For any finite étale map ¯V → Ū, the natural morphism CŪ → C ¯V is compatible with the pro-group structures of C ¯V and C Ū . For any connected étale ¯X-schemes Ū, the reciprocity morphism is a morphism of topological pro-groups. rŪ : CŪ −→ π 1 (Ū et ) ab Proof. Concerning the first three statements, we just have to remark that those morphisms are all compatible with the filtration (25), which is clear. The reciprocity morphism rŪ is defined by the topological class formation (π 1 (Ū et ,qŪ), lim C −→ ¯V ),where ¯V runs over the filtered system of pointed étale covers of (Ū,qŪ) (see [15, Proposition 8.3.8 and Theorem 8.3.12]). Recall that the group Uv n is mapped, by class field theory, onto the nth-ramification subgroup (G n v )ab ⊂ G ab K v ⊂ G ab K,S = π 1(Ū et ,qŪ ) ab . Hence rŪ is a morphism of topological pro-groups. ✷
The Weil-étale fundamental group of a number field I 123 5.3.2. The morphism ϕŪ has dense image. By Property (1), the map γ : ¯X L → ¯X et is connected, i.e. γ ∗ is fully faithful. It follows immediately that the morphism γŪ : Ū L := ¯X L /γ ∗ Ū −→ ¯X et /U = Ū et is connected as well. Chose a T -point pŪ of Ū L and let qŪ be the geometric point of Ū defined by pŪ as in Section 4.3. We have a commutative square Ū L γŪ Ū et B π1 (Ū L ,pŪ ) B ϕŪ B sm π 1 (Ū et ,qŪ ) where the vertical maps are both connected. Indeed, the inverse image of the morphism Ū L → B π1 (Ū L ,pŪ ) (respectively of the morphism Ū et → B sm ) is the inclusion of π 1 (Ū et ,qŪ ) the full subcategory of sums of locally constant objects SLC T (Ū L )↩→ Ū L (respectively SLC(Ū et )↩→ Ū et ). Hence the previous diagram shows that B ϕŪ : B π1 (Ū L ,pŪ ) −→ Bsm π 1 (Ū et ,qŪ ) is connected as well. This morphism is induced by the morphism of strict topological pro-groups: ϕŪ : π 1 (Ū L ,pŪ ) −→ π 1 (Ū et ,qŪ). Consider π 1 (Ū L ,pŪ ) as a projective system of locally compact groups (W α ) α∈A and π 1 (Ū et ,qŪ ) as a projective system of finite groups (G β ) β∈B .ThenϕŪ is given by a family, indexed over B, of compatible morphisms W α → G β . More precisely, we have ϕŪ ∈ Hom((W α ) α∈A ,(G β ) β∈B ) := lim lim ←− β∈B −→ Hom c(W α∈A α ,G β ). Definition 5.4. We say that ϕŪ has dense image if all those maps W α → G β are surjective. The fact that the morphism B ϕŪ is connected implies that ϕŪ has dense image in that sense. Indeed, assume that one of the maps W α → G β is not surjective. Then the functor ϕ ∗ : BG sm β → B Wα , sending a G β -set E to the (sheaf represented by the) discrete W α -space E on which W α acts via W α → G β , is not fully faithful. But we have the commutative diagram of categories B π1 (Ū L ,pŪ ) B ∗ ϕŪ B sm π 1 (Ū et ,qŪ ) B Wα ϕ ∗ B sm G β where the vertical arrows are fully faithful functors. Hence the fact that ϕ ∗ is not fully faithful implies that B ∗ ϕŪ is not fully faithful. We have obtained the following result. PROPOSITION 5.5. Let ¯X L be a topos satisfying Properties (1)–(9). Then for any connected étale ¯X-scheme Ū the morphism of topological pro-groups ϕŪ has dense image.
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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