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This question is kind of a follow up to the following question Compute $\mathrm{Tor}_{n}^{\mathbb{Z}_{8}}(\mathbb{Z}_{4},\mathbb{Z}_{4})$. I am working in a similar problem but I think understanding this first will help me with my problem. In this problem we end up having a complex of the form $$...\rightarrow \mathbb{Z}_8 \stackrel{r\circ p}{\rightarrow} \mathbb{Z}_8 \stackrel{q\circ s}{\rightarrow} \mathbb{Z}_8 \stackrel{r\circ p}{\rightarrow} \mathbb{Z}_8 \stackrel{q\circ s}{\rightarrow} \mathbb{Z}_8 \stackrel{p}{\rightarrow} \mathbb{Z}_4$$ where $p:\mathbb{Z}_8 \rightarrow \mathbb{Z}_4$ sending 1 to 1 and with kernel generated by $4$, $q:\mathbb{Z} \rightarrow \mathbb{Z}_8$ maps 1 to 4, $s:\mathbb{Z}_8 \rightarrow \mathbb{Z}_2$ maps 1 to 1 and $r:\mathbb{Z}_4 \rightarrow \mathbb{Z}_8$ maps 1 to 2.

The answer in that question reaches a point where they say "Apply the functor $\mathbb{Z}_4 \otimes_{\mathbb{Z}_8} (-) $ to the complex and compute its homology" but this is not quite clear to me. I have been trying to understand what exactly $\mathbb{Z}_4 \otimes_{\mathbb{Z}_8} \mathbb{Z}_8,\mathbb{Z}_4 \otimes_{\mathbb{Z}_8} \mathbb{Z}_4 $ are and how the differential maps $r\circ p$, $q\circ s$ and $p$ behave under the application of this functor. I would really appreciate any answer with a detail explanation of this! Thanks!

user26857
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user110320
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1 Answers1

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Suppose we have $A=\Bbb{Z}_m\otimes_{\Bbb{Z}_r}\Bbb Z_n$ with $m\mid r$ and $n\mid r$. As $\Bbb Z_m$ and $\Bbb Z_n$ are cyclic groups then $A$ is cyclic generated by $1\otimes 1$ under the relations $m\otimes1=1\otimes n=r(1\otimes 1)=0$. The last relation is redundant, so $A$ is generated by $1\otimes 1$ which has order $d=\gcd(m,n)$.

In the complex each map is induced from a map from $\Bbb Z_8$ induced by multiplying by a constant, $a$ say. Then the corresponding map from $\Bbb Z_8\otimes_{\Bbb Z_8}\Bbb Z_4$ is also multiplication by $a$.

Angina Seng
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