Orthogonal direct sum of cocycles across acting groups

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 $c_{i}$s, 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

• Orthogonal direct sum of cocycles is cocycle