Lemma 38.12.5. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $s \in S$. Assume that

1. $f$ is of finite presentation,

2. $\mathcal{F}$ is of finite presentation, and

3. $\mathcal{F}$ is flat over $S$ at every point of the fibre $X_ s$.

Then there exists an elementary étale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram of schemes

$\xymatrix{ X \ar[d] & X' \ar[l]^ g \ar[d] \\ S & S' \ar[l] }$

such that $g$ is étale, $X_ s \subset g(X')$, the schemes $X'$, $S'$ are affine, and such that $\Gamma (X', g^*\mathcal{F})$ is a projective $\Gamma (S', \mathcal{O}_{S'})$-module.

Proof. For every point $x \in X_ s$ we can use Proposition 38.12.4 to find a commutative diagram

$\xymatrix{ (X, x) \ar[d] & (Y_ x, y_ x) \ar[l]^{g_ x} \ar[d] \\ (S, s) & (S_ x, s_ x) \ar[l] }$

whose horizontal arrows are elementary étale neighbourhoods such that $Y_ x$, $S_ x$ are affine and such that $\Gamma (Y_ x, g_ x^*\mathcal{F})$ is a projective $\Gamma (S_ x, \mathcal{O}_{S_ x})$-module. In particular $g_ x(Y_ x) \cap X_ s$ is an open neighbourhood of $x$ in $X_ s$. Because $X_ s$ is quasi-compact we can find a finite number of points $x_1, \ldots , x_ n \in X_ s$ such that $X_ s$ is the union of the $g_{x_ i}(Y_{x_ i}) \cap X_ s$. Choose an elementary étale neighbourhood $(S' , s') \to (S, s)$ which dominates each of the neighbourhoods $(S_{x_ i}, s_{x_ i})$, see More on Morphisms, Lemma 37.35.4. We may also assume that $S'$ is affine. Set $X' = \coprod Y_{x_ i} \times _{S_{x_ i}} S'$ and endow it with the obvious morphism $g : X' \to X$. By construction $g(X')$ contains $X_ s$ and

$\Gamma (X', g^*\mathcal{F}) = \bigoplus \Gamma (Y_{x_ i}, g_{x_ i}^*\mathcal{F}) \otimes _{\Gamma (S_{x_ i}, \mathcal{O}_{S_{x_ i}})} \Gamma (S', \mathcal{O}_{S'}).$

This is a projective $\Gamma (S', \mathcal{O}_{S'})$-module, see Algebra, Lemma 10.94.1. $\square$

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