The Stacks project

Proof. (The only difference with Lemma 38.12.10 is that we do not assume $f$ is of finite presentation.) For every point $x \in X_ s$ we can use Lemma 38.12.9 to find an elementary étale neighbourhood $(S_ x , s_ x) \to (S, s)$ and a commutative diagram

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, 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

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

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