In this post we find the Picard group of the circle using cohomology, and more generally discuss a cohomological description of .
This is going to be the first post in a series I’d like to call “Easy theorems, hard proofs”. The idea is simple. Take a classic theorem from some field, maybe whose proof is somewhat messy, and realize it as a corollary of a much more sophisticated theorem. Rarely will the proof be “better”, in the sense that it’s the proof one should initially learn, but hopefully the proof is neat enough, or elegant enough, to warrant attention.
I can think of no better way to start this series than with one of my favorite proofs of all time: the algebraic geometry proof of the Nullstellensatz.