@stupid_question_bot I am using Qing Liu's definitions and it is not required to work on a perfect field.

@sriram Yes, there isn't the finitely presentation assumption in the defn of flat morphism. Indeed I think I solved my problem using just the "going down" for flat morphisms and working on affine opens.

@LaurentMoret-Bailly Thank you, I didn't know the "going down" for flat ring morphisms, but now I think the argument is clear since we can work on affine opens and $\eta$ is minimal (so it is in the image of $U$ for the g-down theorem and this is absurd).

@Kimball I know, you are right, but I am a student and I have to submit a relation about that theorem in about 2 weeks, so I need at least a partial answer as soon as possible.

@JorgeZuniga I sent him an email more then a week ago, but he hasn't replied yet.