One last post this year. I came across the following statement in the Stacks project:
This fact is already discussed on the Stacks project blog. Recall that it should be interpreted as "the categories of sheaves for this topology are the same". A similar phenomenon happens for étale and smooth.
For posterity I would like to collect some facts about this statement:
- in the locally noetherian case finite locally free is the same as finite and flat;
- the fppf topology can also be taken as the "fppfqf topology", i.e. fppf coverings can be refined by fppf coverings where each morphism is quasi-finite (again an important reduction for the proof);
- the étale topology has a similar description, but instead of the four steps one takes for the fppf topology one only takes 3 steps (and this is crucial in the proof);
- it seems mysterious at first sight that Cohen-Macaulay morphisms pop up, but one first proves that for fppf morphisms Cohen-Macaulayness is an open condition which then allows us to slice morphisms locally of finite presentation into locally quasi-finite morphisms (these two steps seem to be the hard part);
With these interesting facts I will end the blogging year 2013. I hope to write more about Grothendieck topologies soon.