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

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118 B. Morin Note that we have W k(w) ≃ Z for w ultrametric and W k(w) ≃ R × + for w archimedean. We denote by G k(w) := D w /I w the Galois group of the residue field k(w),whereD w and I w are, respectively, the decomposition and the inertia subgroups of G K at w. Hence G k(w) is the trivial group for w archimedean. There is a canonical morphism W k(w) −→ G k(w) (23) for any closed point w ∈ Ū. We consider the big classifying topos B Wk(w) and the small classifying topos BG sm k(w) , i.e. the category of continuous G k(w) -sets. In particular, BG sm k(w) is just the final topos Sets for w archimedean. The map (23) induces a morphism of toposes: α v : B Wk(w) −→ BG sm k(w) . We denote by T the topos of sheaves on the site (Top, J op ),whereTop is the category of Hausdorff locally compact spaces endowed with the open cover topology. If we need to use constant sheaves represented by non-locally compact spaces, then we can define T ′ := (Top h , J op ),whereTop h is the category of Hausdorff spaces, and consider the base change ¯X L × T T ′ to obtain a connected and locally connected topos over T ′ . Finally, if G is a strict pro-group object of T given by a covariant functor G : I → Gr(T ) where Gr(T ) denotes the category of groups in T and I is a small filtered category. We consider the pro-abelian group object G DD of T defined as the composite functor (−) DD ◦ G : I −→ Gr(T ) −→ Ab(T ). Let t : E → T be a connected and locally connected topos over T ,i.e.t is a connected and locally connected morphism. In particular, t ∗ has a left adjoint t ! . An object X of E is said to be connected over T if t ! X is the final object of T .AT -point of E is a section s : T → E of the structure map t, i.e.t ◦ s is isomorphic to Id T . 5.2. Expected properties (1) The conjectural Lichtenbaum topos ¯X L should be naturally associated to ¯X. There should be a canonical connected morphism from ¯X L to the Artin–Verdier étale topos: γ : ¯X L −→ ¯X et . (2) The conjectural Lichtenbaum topos ¯X L should be defined over T . The structure map t : ¯X L −→ T should be connected and locally connected, and ¯X L should have a T -point p. For any connected étale ¯X-scheme Ū, the object γ ∗ Ū of ¯X L should be connected over T . It follows that the slice topos Ū L := ¯X L /γ ∗ Ū −→ ¯X L −→ T is connected and locally connected over T , for any connected étale ¯X-scheme Ū, and has a T -point pŪ : T −→ Ū L .
The Weil-étale fundamental group of a number field I 119 Then the fundamental group π 1 (Ū L ,pŪ ) is well defined as a prodiscrete localic group in T . Moreover, π 1 (Ū L ,pŪ ) should be pro-representable by a locally compact strict pro-group, and we consider this fundamental group as a locally compact pro-group. By Corollary 3.3, we have π 1 (Ū L ,pŪ ) DD = π 1 (Ū L ,pŪ ) ab = π 1 (Ū L ) ab . We have a canonical connected morphism inducing a morphism Ū L := ¯X L /γ ∗ Ū −→ ¯X et /Ū = Ū et ϕŪ : π 1 (Ū L ,pŪ ) −→ π 1 (Ū et ,qŪ ), where qŪ is defined by pŪ as in (15). We obtain a morphism ϕ DD Ū : π 1(Ū L ) ab = π 1 (Ū L ,pŪ ) DD −→ π 1 (Ū et ,pŪ ) DD = π 1 (Ū et ) ab . (3) We should have a canonical isomorphism such that the composition ϕ DD Ū rŪ : CŪ ≃ π 1 (Ū L ) ab ◦ r Ū : C Ū ≃ π 1(Ū L ) ab −→ π 1 (Ū et ) ab is the reciprocity law of class field theory. This reciprocity morphism 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 cover of (Ū,qŪ ) (see [15, Proposition 8.3.8] and [15, Theorem 8.3.12]). (4) The isomorphism rŪ should be covariantly functorial for any map f : ¯V → Ū of connected étale ¯X-schemes. More precisely, such a map induces a morphism of toposes: f L : ¯V L := ¯X L / ¯V −→ Ū L := ¯X L /Ū hence a morphism of abelian pro-groups in T , ˜f L : π 1 ( ¯V L ) ab −→ π 1 (Ū L ) ab . Then the following diagram should be commutative: π 1 ( ¯V L ) ab ˜f L π 1 (Ū L ) ab r ¯V C ¯V rŪ CŪ N where N is induced by the norm map.
Page 1 and 2: Kyushu J. Math. 65 (2011), 101-140
Page 3 and 4: The Weil-étale fundamental group o
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THE WEIL-´ETALE FUNDAMENTAL GROUP OF A NUMBER FIELD I

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