Orthogonal direct sum of cocycles across acting groups

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Definition

Suppose m and n are natural numbers. Consider a bunch of groups G_1, G_2, \dots, G_m, all equipped with actions on an abelian group A. Suppose, for each i, that we have a n-cocycle c_i: G_i^n \to A for the action of G_i on A.

Then, consider G := G_1 \times G_2 \times \dots \times G_m. If all the G_i-actions on A commute with each other, then we get an induced G-action on A. We can define a n-cocycle c for this action in terms of c_is, as the orthogonal direct sum of cocycles as follows:

First, view G as an internal direct product, so each G_i is identified with a subgroup of G. Now, define:

c((g_{11},g_{12},\dots, g_{1m}),(g_{21},g_{22},\dots,g_{2m}), \dots,(g_{n1},g_{n2},\dots,g_{nm}) = \sum_{i=1}^m c_i(g_{1i},g_{2i},\dots,g_{ni})

where the summation on the right happens in A.

This does indeed give a cocycle.

Proof that the definition works