$\biggl( \prod_{i \in \mathbb{N}}\mathbb{Z}/p^i\mathbb{Z}\biggr)\otimes_{\mathbb{Z}} \mathbb{Q}\ne 0$

Question

I've found this claim $$\biggl( \prod_{i \in \mathbb{N}}\mathbb{Z}/p^i\mathbb{Z}\biggr)\otimes_{\mathbb{Z}} \mathbb{Q} \not\cong \prod_{i \in \mathbb{N}}\biggl( \mathbb{Z}/p^i\mathbb{Z}\otimes_{\mathbb{Z}} \mathbb{Q}\biggr)$$

Why is this true ? Any reference ?

Answer 1

Here's one way of viewing it.

First $\mathbb{Q}\otimes \mathbb{Z}/p^i\mathbb{Z}=0$ for all $i$, so the right hand side is zero.

There is an injection

$$0\to\mathbb{Z}_p\to\prod_i \mathbb{Z}/p^i\mathbb{Z}$$

where $\mathbb{Z}_p$ is the $p$-adic integers. Since $\mathbb{Q}$ is flat we obtain an injection

$$0\to\mathbb{Z}_p\otimes_\mathbb{Z}\mathbb{Q}\to\prod_i(\mathbb{Z}/p^i\mathbb{Z})\otimes_\mathbb{Z}\mathbb{Q}$$

So, it suffices to show that $\mathbb{Z}_p\otimes_\mathbb{Z}\mathbb{Q}$ is non-zero. But, the multiplication map produces a non-zero pairing $\mathbb{Z}_p\otimes_\mathbb{Z}\mathbb{Q}\to\mathbb{Q}_p$.