Proposition 38.12.4. Let f : X \to S be a morphism of schemes. Let \mathcal{F} be a quasi-coherent sheaf on X. Let x \in X with image s \in S. Assume that
f is locally of finite presentation,
\mathcal{F} is of finite presentation, and
\mathcal{F} is flat at x over S.
Then there exists a commutative diagram of pointed schemes
\xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^ g \ar[d] \\ (S, s) & (S', s') \ar[l] }
whose horizontal arrows are elementary étale neighbourhoods such that X', S' are affine and such that \Gamma (X', g^*\mathcal{F}) is a projective \Gamma (S', \mathcal{O}_{S'})-module.
Proof.
By openness of flatness, see More on Morphisms, Theorem 37.15.1 we may replace X by an open neighbourhood of x and assume that \mathcal{F} is flat over S. Next, we apply Proposition 38.5.7 to find a diagram as in the statement of the proposition such that g^*\mathcal{F}/X'/S' has a complete dévissage over s'. (In particular S' and X' are affine.) By Morphisms, Lemma 29.25.13 we see that g^*\mathcal{F} is flat over S and by Lemma 38.2.3 we see that it is flat over S'. Via Remark 38.6.5 we deduce that
\Gamma (X', g^*\mathcal{F})/ \Gamma (X', \mathcal{O}_{X'})/ \Gamma (S', \mathcal{O}_{S'})
has a complete dévissage over the prime of \Gamma (S', \mathcal{O}_{S'}) corresponding to s'. Thus Lemma 38.12.2 implies that the result of the proposition holds after replacing S' by a standard open neighbourhood of s'.
\square
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