# The fppf topology is generated by Zariski coverings and surjective finite locally free morphisms

One last post this year. I came across the following statement in the Stacks project:

[...] describing the fppf topology as being equal to the topology "generated by" Zariski coverings and by coverings of the form $\{f\colon T\to S\}$ where $f$ is surjective finite locally free.

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.