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
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112 B. Morin<br />
4.1. The abelian arithmetic fundamental group<br />
<strong>THE</strong>OREM 4.3. Let ¯X L be a topos over T satisfying Hypotheses 4.1 and 4.2. Then we have<br />
an isomorphism of topological groups<br />
π 1 ( ¯X L ,p) DD ≃ Pic( ¯X),<br />
where Pic( ¯X) denotes the Arakelov–Picard group of the number field F . In particular,<br />
if π 1 ( ¯X L ,p) is represented by a locally compact topological group, then we have an<br />
isomorphism of topological groups<br />
π 1 ( ¯X L ,p) ab ≃ Pic( ¯X).<br />
Proof. By Hypothesis 4.1 and Section 3.2, the fundamental group π 1 ( ¯X L ,p)is well defined<br />
as a group object of T . The basic idea is to use Corollary 3.6 to recover the abelian<br />
fundamental group. We have<br />
H 1 T ( ¯X L , A) = Hom T (π 1 ( ¯X L ,p),A)<br />
for any abelian object A of T . The exact sequence of topological groups<br />
induces an exact sequence<br />
0 → Z → R → S 1 → 0<br />
0 → Z → ˜R → ˜S 1 → 0<br />
of abelian sheaves in ¯X L ,where˜S 1 denotes t ∗ (y(S 1 )). Consider the induced long exact<br />
sequence of T -cohomology<br />
0 = H 1 T ( ¯X L , Z) → H 1 T ( ¯X L , ˜R) → H 1 T ( ¯X L , ˜S 1 ) → H 2 T ( ¯X L , Z) → H 2 T ( ¯X L , ˜R) = 0.<br />
We obtain an exact sequence in T :<br />
It follows that<br />
0 → R → H 1 T ( ¯X L , ˜S 1 ) → Pic 1 ( ¯X) D → 0.<br />
H 1 T ( ¯X L , ˜S 1 ) = Hom T (π 1 ( ¯X L ,p),y(S 1 )) = π 1 ( ¯X L ,p) D<br />
is representable by an abelian Hausdorff locally compact topological group. Indeed,<br />
H 1 T ( ¯X L , ˜S 1 ) is representable locally on Pic 1 ( ¯X) D .ButPic 1 ( ¯X) D is discrete (recall that<br />
Pic 1 ( ¯X) is compact) and the Yoneda embedding y : Top → T commutes with coproducts<br />
(see [4, Corollary 1]), hence the sheaf H 1 T ( ¯X L , ˜S 1 ) is representable by a topological space T .<br />
The functor y : Top → T is fully faithful and commutes with finite projective limits. Hence<br />
the space T is endowed with a structure of an abelian topological group since y(T) =<br />
H 1 T ( ¯X L , ˜S 1 ) is an abelian object of T . The connected component of the identity in T is<br />
isomorphic to R, sincePic 1 ( ¯X) D is discrete. Hence T is Hausdorff and locally compact.<br />
Therefore π 1 ( ¯X L ,p) DD = y(T D ) is representable by an abelian Hausdorff locally compact<br />
topological group as well.<br />
By Pontryagin duality, we obtain the exact sequence in T ,<br />
0 → Pic 1 ( ¯X) → π 1 ( ¯X L ,p) DD → R → 0. (10)