Biprincipal spaces and their composition, following Serre's "Galois Cohomology"

Question

When reviewing my notes and Serre's book "Galois Cohomology" Chapter 5 dealing with non-abelian group cohomology, I realized that I don't fully understand the concept of biprincipal spaces such as a-(A,B)-space (defined in page 49).

I think that what Serre ment is that is a G-set P with two G-groups A and B such that:

$\forall p,q \in P : \exists ! a \in A : q=a*p$

$\forall p,q \in P : \exists !b \in B : q=p*b$

$\forall p \in P , a \in A , b \in B: (a*p)*b=a*p*b=a*(p*b)$ (A and B action commutes)

Then he talks about composition of such spaces: if P is a (A,A')-space and Q is a (A',A'')-space then

$P \circ Q = P \times ^{A'}Q $

which he claims has a (A',A'')-space structure.
If so, for what is this composition is good for? Q is already a (A',A'')-space.

Serre defines $ P \times ^AF$ as follows: let A be a G-group, let P be a principal homogeneous space on which A acts from the right and F a G-set on which A acts from the left. Then it is a quotient space of $P \times F$ induced by the following equivalence relation: $ (p,f) \sim (p*a, a^{-1}*f) $ for all $a \in A$.

He uses the construction of biprincipal spaces and their compositions in several instances (instead of using the cocycles language), such as in the proof of Proposition 35 and in the construction of fibers for normal subgroup in page 52.

I wonder if there are some examples which I hope will give me a feel of this construction and another good text about it. Moreover, translation of this biprincipal spaces to a cocycles language,

given that $P \times ^AF \cong _a F$ by using a cocycle $a_s$ to twist F: $^s \prime f = a_s \cdot ^s f$ which can be defined for $p \in P$ by $ s \mapsto a_s \mbox{ such that } ^s p = p * a_s$ and the mapping $ f \mapsto [(p,f)]$ induces $P \times ^AF \cong _a F$,

would be helpful.

Thanks, Zachi

Answer 1

This is all structured like a "bimodule" argument, and indeed like the construction of the tensor product in the general case. What is going on is that there is really only one group involved. But as it is acting on the left/right there is scope for talking about the group G and its "opposite", and just actions. (This is the attitude in category theory, naturally, where the old "contravariant functor" notion is better replaced by the idea of a functor from the opposite category.) That all being said by the way of Bourbakiste background, is this really so deep? You do have to read the notation properly, as has been said. The superscript in the definition of PoQ is the only place where anything happens. Each PHS is a single orbit of G, so it is really just at the level of orbit-stabiliser reasoning, I think.