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§4. Long Exact Sequence in G-Cohomology 151

s
Proof. We apply (3.7) noting that 1 = 1 for all s ∈ G. For the second assertion
s
observe that a s = 1 when x = xa s for a fixed point x of X.
(3.9) Theorem. The function which assigns to a principal homogeneous space X
s
the cohomology class of a s where x = xa s for some x ∈ X defines a bijection
1
Prin(G, A) → H (G, A).
Proof. By (3.7) this is a well-defined injection. For surjectivity we have to prove that
1
a class of a cocycle a s in H (G, A) defines a principal homogeneous set X. For given
a s we require X = A as a right A-set, but we “twist” the action of G on X. Namely 1
s
s
in A = X has image 1 = a s by definition in X. Hence, in X the action u must equal
s s s s s
(1u) = 1 u = a s u, where the second u is calculated in A. Clearly, in X we have
s s s s s s
(ua) = a s (ua)(a s u) a = u a and X × A → X is G-equivariant. Since a s is a
st
s t
cocycle, the action satisfies u = ( u) in X. This completes the construction of X
giving the cocycle a s and proves the theorem.
(3.10) Definition. The principal homogeneous G-set X constructed from A and the
cocycle a s is called the twisted form of A by a s .

§4. Long Exact Sequence in G G G-Cohomology

Now we return to the exact sequence (2.6) at the end of §2 and show how to extend
it two or three terms. To do this, we need some definitions. In this section G denotes
a group.
0

(4.1) Definition. Let f : A → A be a morphism of G-groups. Then f ∗ = H ( f ) :
0
1
0

H (G, A) = A G → A G = H (G, A ) is the restriction of f and f ∗ = H ( f ) :
1
1
H (G, A) → H (G, A ) is given by f ∗ [a s ] = [ f (a s )]. These are called the induced

coefficient morphism in cohomology.
0
1
Now we will relate H and H in a short exact sequence.
(4.2) Definition. Let E be a G-group with G-subgroup A ⊂ E. The boundary func-
1
G
0
tion δ : H (G, E/A) = (E/A) → H (G, A) is defined by

cohomology class associated to the coset X viewed as a
δ(X) =
principal homogeneous G-set over A.
s
Hence for X = xA and x = xa s , it follows that δ(X) = [a s ].
(4.3) Theorem. Let E be a G-groupwith G-subgroup A ⊂ E and corresponding
short exact sequence
i
1 → A → E → E/A → 1.
Then the five-term sequence of pointed sets is exact