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

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 .