§ Line bundles, a high level view as I understand them today
- What is a line bundle?
- What does it mean to tensor two line bundles?
- Why are line bundles invertible?
- Can we draw pictures?
§ Why are bundles invertible?
Because locally, they're locally modules. This leads us to
§ Why are modules invertible?
All modules are invertible when tensored with their dual.
To simplify further, let's move to linear algbera from ring theory; consider the field .
Over this, we have a vector space of dimension , . Now, if we consider ,
this is isomorphic to since we can replace . This
amounts to the fact that we can contract tensors.
So, . Generalize to bundles.