Line Bundles on Local Rings

Question

Let $A$ be a local ring and $L$ a module over $A$ which is projective and of rank one. Does it follow that $L$ is isomorphic to $A$?

Answer 1

Yes.

Every projective module over a local ring is free. See for example The Stacks Project.

Since your module is of rank 1, it is isomorphic to $A$ as an $A$-module.