Let $R=\mathbb{Z}/ 8 \mathbb{Z}$ and let $M=\mathbb{Z} / 4 \mathbb{Z}$ be an $R$-module. How can I compute $\text{Tor}^R_n(M,M)$?
I was just introduced to the theory of Tor, and I am having difficulties to compute it. I know that $R$ is a principal ring and that if $0 \to N' \to N \to N'' \to 0$ is exact, then so is $$N' \otimes M \to N \otimes M \to N'' \otimes M \to 0$$ (where $N',N,N''$ are right $R$-modules and $M$ is a left $R$-module). So, $L_n(- \otimes_R M) = 0$ for all $n \geq 2$. But I dont know how to proceed with this really. Can someone explain how I can compute Tor$(M,M)$?
Careful, $R$ is a principal ideal ring, but it is not a domain! Can you find a resolution $$\cdots \longrightarrow R \longrightarrow R \longrightarrow M\longrightarrow 0?$$
The image and kernel of multipication by two in $R$ are $4R$ and $2R$ respectively, so you have a periodic resolution
$$\cdots \stackrel{\times 2}\longrightarrow R \stackrel{\times 4}\longrightarrow R \stackrel{\times 2}\longrightarrow R\stackrel{\times 4}\longrightarrow R \longrightarrow 0$$
of $M$.