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.
In this post we will review some of the basic properties of flat/faithuflly flat modules, define flat morphisms of schemes, and discuss some of the nice properties that these morphisms have.
In this series of posts, I will be documenting some aspects of étale cohomology, as I, myself, learn it. This will include a mixture of intuition, technical background, and examples. I will start with material that many readers may already be familiar with, through a course in algebraic geometry–topics such as flatness, smoothness, étaleness, etc.
There is no guarantee, and in fact it isn’t likely, that there will be anything here not present at some other place on the internet. There is also a high probability that some of what I say may be incorrect, either technically or intuitively. That said, I hope that some of my scribblings will be of use to some future learner of this brilliant and beautiful subject.
I will be following several sources, but most seriously will be Lei Fu’s Etale Cohomology, Milne’s Etale Cohomology, and SGA 4.5.