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Consider $\mathbb{Z}_{2}$ as a $\mathbb{Z}_{4}$-module. How to compute $\mathrm{Tor}_{n}^{\mathbb{Z}_{4}}(\mathbb{Z}_{2},\mathbb{Z}_{2})$?

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    Indeed, this is one situation where "just apply the definition" works...2011-04-09

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You need a free (or projective) resolution of $\mathbb{Z}_2$. One is $\dots\to\mathbb{Z}_4\to\mathbb{Z}_4\to\mathbb{Z}_4\to\mathbb{Z}_4$ where every arrow is multiplication by $2$. Now you tensor this complex (over $\mathbb{Z}_4$) with $\mathbb{Z}_2$ and you get $\dots\to\mathbb{Z}_2\to\mathbb{Z}_2\to\mathbb{Z}_2\to\mathbb{Z}_2$ where the arrows are now zero. Your Tor's are the cohomology of this complex. As a result, they are $\mathbb{Z}_2$ for every $n$.

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    Hi my friend.Could you describe it more?I couldn't understand your answer.2013-07-25