Lemma 38.12.8 (05KY)—The Stacks project

Lemma 38.12.8. 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 type, and
$\mathcal{F}$ is flat at $x$ over $S$.

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

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

such that $X' \to X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ is étale, $\kappa (x) = \kappa (x')$, 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. To prove the lemma we may replace $(S, s)$ by any elementary étale neighbourhood, and we may also replace $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$. Hence by Proposition 38.10.3 we may assume that $\mathcal{F}$ is finitely presented and flat over $S$ in a neighbourhood of $x$. In this case the result follows from Proposition 38.12.4 because Algebra, Theorem 10.85.4 assures us that projective $=$ free over a local ring. $\square$

The Stacks project

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