## § 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 $\mathbb R$.
Over this, we have a vector space of dimension $1$, $\mathbb R$. Now, if we consider $\mathbb R \otimes \mathbb R^*$,
this is isomorphic to $\mathbb R$ since we can replace $(r, f) \mapsto f(r)$. This
amounts to the fact that we can contract tensors.
So, $\mathbb R \otimes \mathbb R^* \simeq \mathbb R$. Generalize to bundles.
#### § References