So, recently I was looking at Lemma 03L7 because of a question asked a comment on Definition 022B. The condition that a single flat morphism f : X —> Y determines an fpqc covering {f : X —> Y} is the following topological condition on f: given any affine open V of Y there should exist a quasi-compact open U of X with f(U) = V.

My question is this: is it sufficient to ask for a quasi-compact open U with V ⊂ f(U)?

The lemma says “yes” if Y is quasi-separated (so in practice always).

Does anybody have a counterexample to the general case? Or is it sufficient?

I remember successfully avoiding thinking about this when I first wrote the material on fpqc coverings. But I guess no harm is done thinking about it a little bit during these hot summer days…

Update 5/7/2024: The answer is no by an example of Tony Scholl.