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

124 B. Morin
Let ¯V → Ū be a finite Galois étale cover of étale ¯X-schemes with Gal( ¯V/Ū)= G,and
consider the injective morphism of the topological pro-group
π 1 ( ¯V L ,p¯V )↩→ π 1(Ū L ,pŪ ).
In other words, if we see the fundamental groups of ¯V L and of Ū L as projective systems of
topological groups (W ′ α ) α∈A and (W α ) α∈A (indexed over the same filtered category A), the
previous map is given by a family of compatible injective morphisms of topological groups
W ′ α → W α. We can consider the quotient pro-object of T :
π 1 (Ū L ,pŪ )/π 1 ( ¯V L ,p¯V ) := (yW α/yW ′ α ) α∈A.
Then this projective system is in fact an essentially constant pro-group and we have an
isomorphism in T :
π 1 (Ū L ,pŪ )/π 1 ( ¯V L ,p¯V ) ≃ y(G).
More generally, for any finite étale map ¯V → Ū of étale ¯X-schemes the pro-object of T ,
π 1 (Ū L ,pŪ )/π 1 ( ¯V L ,p¯V ),
is essentially constant, endowed with an action of the pro-group object π 1 (Ū L ,pŪ ), andwe
have an isomorphism of finite π 1 (Ū L ,pŪ )-sets:
π 1 (Ū L ,pŪ )/π 1 ( ¯V L ,p¯V ) ≃ π 1(Ū et ,qŪ)/π 1 ( ¯V et ,q¯V ).
Therefore, for any finite étale map ¯V → Ū, the induced morphism
π 1 ( ¯V L ,p¯V ) −→ π 1(Ū L ,pŪ )
isgivenbyacompatible family of closed topological subgroups of finite index W ′ α ↩→ W α.
Moreover, we can choose an index α 0 ∈ A such that for any map α → α 0 in A, themap
W α /W ′ α → W α 0
/W ′ α 0
is a bijective map of finite sets. It follows that the usual transfer maps
tr α : W ab
α
′ab
−→ W α
are well defined and that they make the following square commutative.
W ′ab
α
tr α
W ab
α
W ′ab
α 0
tr α0
W ab
α 0
We obtain a morphism of locally compact topological pro-groups
tr : π 1 (Ū L ,pŪ ) ab −→ π 1 ( ¯V L ,p¯V )ab .
If ¯V → Ū is a Galois étale cover, then W α ′ is normal in W α for any α ∈ A, hence W α acts
on W α
′ab by conjugation. This action is certainly functorial in α hence π 1 (Ū L ,pŪ ) acts on
π 1 ( ¯V L ,p¯V )ab by conjugation. More precisely, we consider the topological pro-group
π 1 (Ū L ,pŪ ) × π 1 ( ¯V L ,p¯V )ab : A −→ Gr(T )
α ↦−→ W α × W α ′ab.