I'm reading Qing Liu's Algebraic geometry book, and here's a problem that I can't find an answer:
Find a ring $A$ and a $A$-module $M$ such that $M\otimes_{A} \hat{A} \to \hat{M}$ is not surjective, where $\hat{A} = \lim A/I^{n}$ and $\hat{M} =\lim M/I^{n}M$ are $I$-adic completions for some ideal $I\subset A$.
I know that the map is surjective if $M$ is finitely generated over $A$, so $M$ should be infinitely generated over $A$ to be a counterexample. I tried $A = \mathbb{Z}, M = \mathbb{Q}$ and $I =p\mathbb{Z}$, which gives an example of non-injective case, not surjective. Can anyone give a hint to find such example? Thanks in advance.