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Etale
cohomology of fields
by Johan M. Commelin, December 5, 2013
Etale
cohomology
The canonical topology on a Grothendieck topos
Let E be a Grothendieck topos. The canonical topology T on E is
given in terms of coverings by the collection of all families fXi ! X gi
of universal eective epimorphisms, i.e., for every X 0 ! X , the family
fXi0 ! X 0 gi = fXi zX X 0 ! X zX X 0 gi makes the diagram
Y
Y
!
Hom(Xi0 zX
Hom(X 0 ; Z) ! Hom(Xi0 ; Z) !
i;j
i
0
Xj0 ; Z )
exact for all Z 2 E .
Theorem. || The Yoneda embedding gives an equivalence of categories E æSh(E ; T ).
Profinite groups;
G-sets
A pronite group is by denition a group that is the limit of a diagram
of nite groups. Every pronite group is the limit of the diagram of its
nite quotients. A pronite group is a topological group in a natural
way: the nite index normal subgroups form a fundamental system of
neighbourhoods of the identity. In particular, open subgroups are of
nite index.
Let G be a pronite group. Let S be a set with a G-action. The
following are equivalent:
The G-action is continuous for the discrete topology on S.
For every point s 2 S, the stabiliser of s is an open subgroup of G.
The set S equals H SH , where H runs over the open normal subgroups of G, and SH denotes the H -invariants of S.
S
The category of G-sets satisfying the above equivalent condition is
a Grothendieck topos, denoted G-set.
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Sh(Spec(k)
The topos
et
)
. Write G for the absolute
Let k be a eld. Fix a separable closure k=k
Galois group, which is by denition the pronite group liml=k Gal(l=k),
where l=k runs over the nite Galois extensions contained in k.
Let X =k be an e tale k-scheme. Observe that G acts on X (k). For every
open normal
subgroup H G, we have X (k)H = X (kH ). Consequently
S
X (k) = H X (k)H , and therefore the G-action is continuous.
Theorem. || The functor
Spec(k)
et
! G-set
X 7! X (k)
is an equivalence of categories, and the e tale topology on the left corresponds to the canonical topology on the right.
Proof. We construct a quasi-inverse. Let S be a continuous G-set. Let
s 2 S be a point. Write H for the stabiliser of s. It is a open subgroup. By
elementary group theory, we have an isomorphism of G-sets G=H ! Gs.
This shows that S is the disjoint union of nite orbits. We map Gs to
Spec(kH ), and since Spec(k) has arbitrary disjoint unions, this dene
a functor G-set ! Spec(k) . It is left as an exercise to show that these
functors are actually quasi-inverse.
Since we are working with the small e tale site Spec(k) , every morphism in Spec(k) is e tale. It is then immediate that the e tale topology
coincides with the canonical topology.
qed
Corollaries. ||
et
et
et
et
1. Every sheaf on Spec(k) is representable.
et
2. Global sections on Spec(k) correspond with G-invariants on G-set.
et
3. Abelian e tale sheafs on Spec(k) correspond to e tale commutative
k-group schemes, and correspond to abelian G-modules.
et
F1f
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Group cohomology
Injective objects
Let G be a discrete group. The category G-mod consists of commutative group objects in G-set. For an object A we denote with AG
the G-invariant elements of A. Observe that G-mod is equivalent to
Z[G]-mod. Let H ! G be a group homomorphism. Then there is an
adjunction
HomG (A; IndGH (B)) HomH (A; B)
where IndGH (B) is the G-module Map(G; B)H .
In the category of abelian groups, the injective objects are the divisible groups (abelian groups such that multiplication by n is surjective
for all n 2 Z> ). It follows from the adjunction that, if A is a divisible group, then IndG (A) is an injective object in G-mod. In particular,
this gives a method to construct injective resolutions in G-mod from
injective resolutions in Ab.
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1
Computation with cochains
For r 2 Z , let Pr be the free abelian group generated by Gr . Naturally Pr comes with a G-action, by coordinatewise multiplication. Dene
boundary operators dr : Pr ! Pr x , by
+1
0
1
dr (g0 ; : : : ; gr ) =
Xr
i
(x1) (g0 ; : : : ; g^i ; : : : ; gr );
i=0
where ^ denotes the usual `ommision'. One may check that P is a complex. If we denote with " : P ! Z the map that sends every basis element
to 1, we can prove:
0
Lemma. || The complex P ! Z ! 0 is exact.
Proof. Dene r : Pr ! Pr by
"
+1
r (g0 ; : : : ; gr ) = (e; g0 ; : : : ; gr ):
By construction dr r + r x dr = 0. If dr (x) = 0, this implies x =
dr (r (x)).
qed
Corollary. || For every G-module A, we have
+1
1
+1
Hn (G; A)æHn (HomG (P ; A)):
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Observe that HomG (Pr ; A) consists of maps : Gr
+1
!A
satisfying
(gg0 ; gg1 ; : : : ; ggr ) = g(g0 ; g1 ; : : : ; gr ):
Consequently, to know the value of (g ; g ; : : : ; gr ), it is enough to know
the value of (g g ; g x g ; : : : ; g x gr ). Therefore, it is enough to know the
value of on tuples of the form (1; g 0 ; g 0 g 0 ; : : : ; g 0 g 0 y ygr0 ). In this way, we
obtain a map
0
1
0
0
1
0
1
1
1
0
1
1
2
1
2
Gr +1 ! Gr
0
0
0
(g0 ; g1 ; : : : ; gr ) 7! (g1 ; g2 ; : : : ; gr )
gi0 = gixx11 gi ;
and this map induces an identication of HomG (Pr ; A) with Map(Gr ; A).
If we write Cr (G; A) for Map(Gr ; A), we nd that the induced boundary
maps are given by
dr : C r (G; A) ! C r +1 (G; A)
7! dr ;
where
dr (g1 ; g2 ; : : : ; gr +1 ) =
g1 (g2 ; : : : ; gr +1 )
+
Xr
i
(x1) (g1 ; : : : ; gi gi+1 ; : : : ; gr )
i=1
+ (x1)
r +1
(g1 ; : : : ; gr ):
Writing Zr (G; A) for ker(dr ), and Br (G; A) for im(dr x , we nd
1
Hr (G; A)æZr (G; A)=Br (G; A):
This allows for pretty explicit descriptions of Hn (G; A) for small n.
We give the descriptions for n = 1. The elements of Z (G; A) are called
crossed homomorphisms. They are maps : G ! A satisfying
1
g(h) x (gh) + (g ) = 0:
On the other hand, elements of B (G; A) are called principal homomorphisms. They are the maps a for each a 2 A, with
1
a (g ) = ga x a:
We will now put this description to use.
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Hilbert's theorem 90
If l=k is a nite Galois extension, and G its Galois group, then l{ is a
G-module. The following theorem by Hilbert goes by the name
Hilbert's theorem 90. || We have H (G; l{ ) = 0.
Proof. For this proof, we need the following fact, known as Dedekind's
theorem on the independence of characters : Let l be a eld, and G a group.
Every nite set of homomorphisms
i : G ! L{ is linearly independent
P
over l. In other words, if i ci i (g ) = 0 for all g 2 G, then ci = 0 for all i.
With this fact we can proceed with the proof of Hilbert's theorem 90.
Let : G ! l{ be a crossed homomorphism:
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(gh) = g(h) y (g ); for all g; h 2 G:
For all a 2 l{ , write
ba =
X
g 2G
(g ) y ga:
Apply the above fact to the nite set of homomorphisms
(g y) : l{ ! l{ .
P
Since (g ) 6= 0 for all g 2 G, the fact shows that g 2G (g )g is not the
zero-map. We conclude that there is an a such that ba is non-zero. Fix
such an a, and write b = ba . For all h 2 G we have
hb =
X
g 2G
h(g ) y hga =
But this means that
principal.
X
g 2G
(hg )(h)x1 hga = (h)x1 b:
(h) = b=hb = hbx1 =bx1 ,
which is to say that
is
qed
Galois cohomology
For pronite groups, all the above remains true, except that we need
to ask most of the maps to be continuous. In particular, we replace
C r (G; A), Z r (G; A), and Br (G; A) by their subsets of continuous maps;
denoted respectively Cr (G; A), Zr (G; A), and Br (G; A). In particular
cts
H
cts
r
cts
cts
r (G; A)
Zcts
æ Br
(G; A)
cts
(G; A)
:
Observe that there is a natural map
colimH Cr (G=H; AH ) ! Cr
cts
(G; A);
given by the compositions with G ! G=H , and AH ! A. (The colimit
runs over the open normal subgroups of G.) We claim this map is an
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isomorphism. Injectivity is clear. For surjectivity, take a continuous
map : Gr ! A. Then (Gr ) is discrete, and compact because Gr is compact. Hence (Gr ) is nite, and thus contained in M H0 for some normal
open subgroup H G. On the other hand, for every a 2 (Gr ), the
inverse image x (a) is open, and thus containsTa translate of Har , for
some normal open subgroup Ha G. Now H = a2 G Ha is an normal
open subgroup of G, and by construction factors via (G=H )r . Finally,
write H = H „ H , so that lifts to a map (G=H )r ! AH . This proves
surjectivity.
Because the system of normal open subgroups of G is ltered, the
colimits indexed by this system commute with kernels and cokernels,
and therefore with cohomology. We thus obtain:
0
1
1
(
r
1
0
1
Hr
cts
æcolimH Hr (G=H; AH ):
(G; A)
Corollary || We also have a statement of Hilbert's theorem 90 for
the absolute Galois group: If k is a eld, with absolute Galois group G,
then H (G; k{ ) = 0.
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