How can I compute $\operatorname{Tor}(\mathbb Z_{p},\mathbb Z_{q})$?

Question

I am self-studying Vick's Homology Theory, and now it is on the topic of free resolutions. Since I am not familiar with it, I have little ideas about how to compute

$$\operatorname{Tor}(\mathbb Z_{p},\mathbb Z_{q}).$$

(An exercise in the book.)

Anyone help, thanks!

Answer 1

Hint:

There is a very easy free resolution of $\mathbb{Z}/p\mathbb{Z}$:

$$0\to\mathbb{Z}\xrightarrow{\times p}\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}\to 0$$

Combing this with the fact that

$$\mathbb{Z}/m\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z}\cong \mathbb{Z}/{(m,n)}\mathbb{Z}$$

Answer 2

An easy way to see that this group is 0 when $p\neq q$ is to see that, on this group, multiplication by $p$ is both zero and and isomorphism (by functoriality in either component). When $p=q$, I think you actually have to compute it (using Alex's answer) - I don't think there is a free lunch then.