The Stacks project

Lemma 38.12.7. Let $f : X \to S$ be of finite presentation. Let $s \in S$. If $X$ is flat over $S$ at all points of $X_ s$, then there exists an elementary ├ętale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram of schemes

\[ \xymatrix{ X \ar[d] & X' \ar[l]^ g \ar[d] \\ S & S' \ar[l] } \]

with $g$ ├ętale, $X_ s \subset g(X')$, such that $X'$, $S'$ are affine, and such that $\Gamma (X', \mathcal{O}_{X'})$ is a projective $\Gamma (S', \mathcal{O}_{S'})$-module.

Proof. This is a special case of Lemma 38.12.5. $\square$


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