Lemma 37.23.5 (0571)—The Stacks project

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Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.23: Slicing Cohen-Macaulay morphisms
Lemma 37.23.5 (cite)

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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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