I am working on an "unimportant" exercise (c.f. Vakil, exercise 13.5.J) which goes as follows
Exercise 13.5.J: Find an example on a two-point space showing that $M:=A$ might not be a torsion-free $A$-module even though $\mathcal{O}_{SpecA}=\tilde{M}$ is torsion-free.
So what I gather from this question is to show an (affine?) scheme with two points over which $M$ might not be torsion-free even though localisation of $M$ at every point is.
If I were to construct a scheme with two points, that would probably be something like $\text{Spec}(\mathbb{C}\times\mathbb{C})$. But I run into trouble soon because of the following confusion:
If $A$ is torsion-free, it means for all $ab=0$, either $a$ is a zero divisor or $b=0$. So for $A$ to be not torsion-free means there exists a relation $ab=0$ such that both $a$ is not a zero divisor and $b\neq 0$, which to me is ridiculous...
How should I solve this question? Am I going in a wrong direction? (i.e. misinterpret the definition of torsion-free sheaves etc)
