TITLE: How to compute the group cohomology of $\mathbb{Z}/n\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z}$ with coefficient in a trivial module?
QUESTION [6 upvotes]: The group cohomology of cyclic groups can be computed easily due to the periodity. Now how can one compute the group cohomology $H^r(\mathbb{Z}/n\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z},M)$? As least in some special case, e.g. $M$ has trivial action, or even $M=\mathbb{Z}$? I would like to know the case that $M$ is not $\mathbb{Z}$.
Furthermore, can one compute $H^r(\mathbb{Z}/n\mathbb{Z}\times\cdots\times \mathbb{Z}/n\mathbb{Z},M)$?

REPLY [5 votes]: If $M$ is torsion-free (as a $\mathbb{Z}-$module), then the Kunneth formula still holds. If $M = \mathbb{Z}$ with trivial group action, then it holds by the corresponding result for the CW-complex category and abstract nonsense; more generally, it holds for chain complexes under some fairly mild restrictions (including having finite homology in each dimension, but that's clear here.)