Let $G=Z/2Z=\{1, g\}$. Consider the ring of integer $Z$ with the alternating action of G, i.e., $g\cdot n=-n$ for $n\in Z$ and an abelian group $M$ on which $G$ acts. It is well-known that the group cohomology $H^{n}(G, M)$ can be obtained by the free resolution $$\cdots Z[G]\to Z[G]\to Z[G]\to Z\to 0$$ of $Z$ with the trivial action, where the the maps are given by $1+g$, $1-g$. Is it true that the group $H^{n}(G, Z\otimes M)$ is isomorphic to $H ^{n+1}(G, M)$? Could you explain to me how to show it?
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