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150 7. Galois Cohomology and Isomorphism Classification of Elliptic Curves

(3.5) Definition. Two A-valued G-cocycles a s and a are cobounding (or cohomol-
s
ogous) provided there exists c ∈ A with
s

(CB) a = c −1 a s ( c) for all s ∈ G.
s
1
1
Let H (G, A) denote the quotient of Z (G, A) under this equivalence relation. This
is the first cohomology set of G with values in A.
s
s
If a = c −1 a s ( c) and a = d −1 s s −1 a s (cd)

a ( d) for all s ∈ G, then a = (cd)

s
s
s
s
and a s = ca (c −1 ) show that cobounding is an equivalence relation. Moreover, a
c
1
1
cocycle a s ∈ Z (G, A) represents the base point of H (G, A) if and only if there
exists c ∈ A with a s = c −1 s
( c) for all s ∈ G. Let [a s ] denote the cohomology class
1
of a s in H (G, A).
s
1
Observe that Z 1 (G, A) is a right A-set under (a s , c) → c −1 a s ( c),and H (G, A)
1
is just the quotient of Z (G, A) under this right A action.
(3.6) Remark. If A is an abelian G-group written additively, then cocycles, a s are
functions satisfying
a st = a s + sa t ,
1
1
and they form an abelian group Z (G, A) by adding their values. The group Z (G, A)
1
has a subgroup B (G, A) of all coboundaries b s = s(c) − c for c ∈ A. The quotient
group is the first cohomology group
1
Z (G, A)
1
H (G, A) = .
1
B (G, A)
1
Further, if G acts trivially on A, then B (G, A) = 0, and a cocycle is just a homo-
1
morphism so that H (G, A) = Hom(G, A).
1
Now we relate the pointed set H (G, A) to the pointed set Prin(G, A) of isomor-
phism classes of principal homogeneous G-sets over A.

(3.7) Proposition. Let X and X be two principal homogeneous G-sets with points
s

s

x ∈ X and x ∈ X where x = xa s and x = x a s . Then there exists an isomor-

phism f : X → X with f (x) = x if and only if a s = a for all s ∈ G. Further, X

s

and X are isomorphic if and only if a s and a are cohomologous.
s
s s

s
Proof. If f is an isomorphism, then it follows that f ( x) = f (x) = x = x a s
s

and f ( x) = f (xa s ) = f (x)a s = x a s . Thus a s = a follows whenever f exists.
s

Conversely, we can define f by the formula f (xc) = f (x)c from X to X as an

A-mosphism. Since a s = a ,the A-morphism f is also a G-morphism.
s
s

For the second assertion we compare f (x) and x where f (x) = f (x)a s and
s
x = x a as in (3.3). This proves the proposition.
s
(3.8) Corollary. A principal homogeneous space X over A is isomorphic to A over
A if and only if X G is nonempty, that is, if X has a G-fixed point. This is also equiv-
alent to the cocycle a s for X being a coboundary, namely a s = c −1 s
( c) for some
c ∈ A.