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§3. Principal Homogeneous G-Sets and the First Cohomology Set H (G, A) 149

(3.2) Definition. Let A be a G-group, and let X and X be two principal homoge-
neous G-sets over A. A morphism f : X → X of principal homogeneous sets is a

function which is both G-and A-equivariant.

Since for f (x) = x with x ∈ X, x ∈ X in (3.2), we can write any y ∈ X
as y = xa for a unique a ∈ A, the morphism f is given by the formula f (y) =

f (xa) = f (x)a = x a, and hence f is completely determined by one value f (x) =

x . Moreover, f is a bijection with inverse f −1 : X → X,and f −1 is a morphism
of principal homogeneous spaces. Since every morphism is an isomorphism which
is determined by its value at one point, it seems sensible to try to classify principal
homogeneous sets by looking at the G action on one point.
(3.3) Remarks. Let X be a principal homogeneous G-set over the G-group A.
s
Choose x ∈ X, and for each s ∈ G consider x ∈ X. There is a unique a s ∈ A
s
with x = xa s . Then s → a s defines a function G → A, but it is not in general
a group morphism. Instead it satisfies a “twisted” homomorphism condition. To see
s t st
what it is, we make explicit the associativity of the G action. The relation ( x) = x
shows that
s t s s s s
( x) = (xa t ) = x( a t ) = xa s (a t ) = xa st ,
and thus this function a s must satisfy
s
(CC) a st = a s (a t ) for all s, t ∈ G,
which is the cocycle formula or cocycle condition. Observe the (CC) implies that
a 1 = 1.

s
Next, choose a second point x ∈ X, and write x = x a . For a unique c ∈ A
s
we have x = xc, and we have the calculation

s
s s s −1 s
x = x c = xa s c = x c a s c .

Thus a s for x and a for x are related by the coboundary formula
s
s

(CB) a = c −1 a s ( c) for all s ∈ G.
s
Now we formally consider functions satisfying (CC) and relations (CB) between
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these functions, and this leads us to the pointed sets H (G, A).
(3.4) Definition. Let A be a G-group. An A-valued G-cocycle is a function a s from
G and A satisfying
s
(CC) a st = a s a t for all s, t ∈ G.
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Let Z (G, A) denote the pointed set of all A-valued G-cocycles with base point the
cocycle a s = 1 for all s ∈ G.
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Note that by putting s = t = 1 in (CC), we obtain a 1 = a 1 a 1 = a or a 1 = 1
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as with a homomorphism.