The Stacks project

Lemma 38.12.10. 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 type, and

  3. $\mathcal{F}$ is flat over $S$ at all points of $X_ s$.

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

\[ \xymatrix{ X \ar[d] & X' \ar[l]^ g \ar[d] \\ S & \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \ar[l] } \]

such that $X' \to X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ is étale, $X_ s = g((X')_{s'})$, the scheme $X'$ is affine of finite presentation over $\mathcal{O}_{S', s'}$, the sheaf $g^*\mathcal{F}$ is of finite presentation over $\mathcal{O}_{X'}$, and such that $\Gamma (X', g^*\mathcal{F})$ is a free $\mathcal{O}_{S', s'}$-module.

Proof. For every point $x \in X_ s$ we can use Lemma 38.12.8 to find an elementary étale neighbourhood $(S_ x , s_ x) \to (S, s)$ and a commutative diagram

\[ \xymatrix{ (X, x) \ar[d] & (Y_ x, y_ x) \ar[l]^{g_ x} \ar[d] \\ (S, s) & (\mathop{\mathrm{Spec}}(\mathcal{O}_{S_ x, s_ x}), s_ x) \ar[l] } \]

such that $Y_ x \to X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S_ x, s_ x})$ is étale, $\kappa (x) = \kappa (y_ x)$, the scheme $Y_ x$ is affine of finite presentation over $\mathcal{O}_{S_ x, s_ x}$, the sheaf $g_ x^*\mathcal{F}$ is of finite presentation over $\mathcal{O}_{Y_ x}$, and such that $\Gamma (Y_ x, g_ x^*\mathcal{F})$ is a free $\mathcal{O}_{S_ x, s_ x}$-module. In particular $g_ x((Y_ x)_{s_ x})$ 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 $g_{x_ i}((Y_{x_ i})_{s_{x_ i}})$. 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. Set

\[ X' = \coprod Y_{x_ i} \times _{\mathop{\mathrm{Spec}}(\mathcal{O}_{S_{x_ i}, s_{x_ i}})} \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \]

and endow it with the obvious morphism $g : X' \to X$. By construction $X_ s = g(X'_{s'})$ and

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

This is a free $\mathcal{O}_{S', s'}$-module as a direct sum of base changes of free modules. Some minor details omitted. $\square$

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