@boarders@joshuagrochow@MartinEscardo
Indeed, it was already a surprise when Serre proved that the Zariski topology could recover the sheaf cohomology of coherent sheaves ,(FAC and GAGA papers), but that did not furnish the “correct” cohomology groups that would help proving the Weil conjectures (along the way suggested by Weil himself).
At some point, Serre observed that “algebraic fiber spaces” were not locally trivial for the Zariski topology, but we're made trivial after passing to an étale covering.
He made a talk at the Chevalley seminar about it
Grothendieck was there and, at the end of the talk, told Serre that they had (« on a ») the Weil cohomology.