The Stacks project

Proposition 38.12.4. 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

  1. $f$ is locally of finite presentation,

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

  3. $\mathcal{F}$ is flat at $x$ over $S$.

Then there exists a commutative diagram of pointed schemes

\[ \xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^ g \ar[d] \\ (S, s) & (S', s') \ar[l] } \]

whose horizontal arrows are elementary étale neighbourhoods such that $X'$, $S'$ are affine and such that $\Gamma (X', g^*\mathcal{F})$ is a projective $\Gamma (S', \mathcal{O}_{S'})$-module.

Proof. By openness of flatness, see More on Morphisms, Theorem 37.15.1 we may replace $X$ by an open neighbourhood of $x$ and assume that $\mathcal{F}$ is flat over $S$. Next, we apply Proposition 38.5.7 to find a diagram as in the statement of the proposition such that $g^*\mathcal{F}/X'/S'$ has a complete dévissage over $s'$. (In particular $S'$ and $X'$ are affine.) By Morphisms, Lemma 29.25.13 we see that $g^*\mathcal{F}$ is flat over $S$ and by Lemma 38.2.3 we see that it is flat over $S'$. Via Remark 38.6.5 we deduce that

\[ \Gamma (X', g^*\mathcal{F})/ \Gamma (X', \mathcal{O}_{X'})/ \Gamma (S', \mathcal{O}_{S'}) \]

has a complete dévissage over the prime of $\Gamma (S', \mathcal{O}_{S'})$ corresponding to $s'$. Thus Lemma 38.12.2 implies that the result of the proposition holds after replacing $S'$ by a standard open neighbourhood of $s'$. $\square$

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