Are the global sections of a flasque quasi-coherent sheaf on an affine scheme an injective module?

Question

I assume this is not true, because I would have seen it mentioned before, but I haven't been able to come up with a counterexample. If not this, do the global sections of such a sheaf have any other interesting algebraic properties?

Answer 1

I think the answer is no. Consider the ring of integers $\mathbf{Z}$, and consider $\mathbf{Z}/p\mathbf{Z}$ as a $\mathbf{Z}$-module. Then, the sheaf $(\mathbf{Z}/p\mathbf{Z})^\sim$ on $\operatorname{Spec}\mathbf{Z}$ is a skyscraper sheaf concentrated at $(p) \in \operatorname{Spec}\mathbf{Z}$, and is therefore flasque. On the other hand, $\mathbf{Z}/p\mathbf{Z} = \Gamma(\operatorname{Spec}\mathbf{Z},(\mathbf{Z}/p\mathbf{Z})^\sim)$ is not an injective module. One way to see this is that $p \cdot \mathbf{Z}/p\mathbf{Z} = 0 \ne\mathbf{Z}/p\mathbf{Z}$, hence $\mathbf{Z}/p\mathbf{Z}$ is not divisible.