$SL_2$ action on the cohomology of a rank two abelian group with arbitrary coefficient domain

Abstract:

A rank two abelian group $C_n\times C_n$ is in a natural way an $SL_2({\mathbb Z})$-module. This induces an action of $SL_2({\mathbb Z})$ on its group cohomology $H^m(C_n\times C_n,R)$ for any trivial coefficient domain $R$. In the present note we determine this module, including the question of when the universal coefficient theorem sequence splits.