Lemma 59.104.5. Let $f : X \to Y$ be a morphism of schemes which is surjective, flat, locally of finite presentation. The functor
is an equivalence of categories.
Proof. Exactly as in the proof of Lemma 59.104.2 we claim a quasi-inverse is given by the functor sending $(\mathcal{F}, \varphi )$ to
and in order to prove this it suffices to show that $f^{-1}\mathcal{G} \to \mathcal{F}$ is an isomorphism. This we may check locally, hence we may and do assume $Y$ is affine. Then we can find a finite surjective morphism $Z \to Y$ such that there exists an open covering $Z = \bigcup W_ i$ such that $W_ i \to Y$ factors through $X$. See More on Morphisms, Lemma 37.48.6. Applying Lemma 59.104.4 we see that it suffices to prove the lemma after replacing $Y$ by $Z$ and $Z \times _ Y Z$ and $f$ by its base change. Thus we may assume $f$ has sections Zariski locally. Of course, using that the problem is local on $Y$ we reduce to the case where we have a section which is Lemma 59.104.1. $\square$
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