From 'Introduction to Algebraic Geometry' by Justin R. Smith:
What is the role of that intermediate sheaf $\mathcal{A}_{\mathbb{C}}$ in the chain in this example ?
Could we not have got from the constant sheaf $\underline{ 2 \pi i \mathbb{Z} }$ straight to $\mathcal{N}_{\mathbb{C^*}}$ via some sheaf morphism ?