The Stacks project

Lemma 37.23.5. Let $f : X \to S$ be a flat morphism of schemes which is locally of finite presentation. Let $s \in S$ be a point in the image of $f$. Then there exists a commutative diagram

\[ \xymatrix{ S' \ar[rr] \ar[rd]_ g & & X \ar[ld]^ f \\ & S } \]

where $g : S' \to S$ is flat, locally of finite presentation, locally quasi-finite, and $s \in g(S')$.

Proof. The fibre $X_ s$ is not empty by assumption. Hence there exists a closed point $x \in X_ s$ where $f$ is Cohen-Macaulay, see Lemma 37.22.7. Apply Lemma 37.23.4 and set $S' = S$. $\square$

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