Lemma 38.12.10 (05L0)—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

$f$ is of finite presentation,
$\mathcal{F}$ is of finite type, and
$\mathcal{F}$ 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. $\square$

The Stacks project

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