Orthogonal direct sum of cocycles across acting groups
Then, consider . If all the -actions on commute with each other, then we get an induced -action on . We can define a -cocycle for this action in terms of s, as the orthogonal direct sum of cocycles as follows:
First, view as an internal direct product, so each is identified with a subgroup of . Now, define:
where the summation on the right happens in .
This does indeed give a cocycle.