TITLE: Help to compute $\mathrm{Tor}_{n}^{\mathbb{Z}_{4}}(\mathbb{Z}_{2},\mathbb{Z}_{2})$?
QUESTION [5 upvotes]: Consider $\mathbb{Z}_{2}$ as a $\mathbb{Z}_{4}$-module. How to compute $\mathrm{Tor}_{n}^{\mathbb{Z}_{4}}(\mathbb{Z}_{2},\mathbb{Z}_{2})$?

REPLY [8 votes]: 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 $n$.