Lemma 37.64.7. Let $X \to Y \to Z$ be morphisms of schemes. Assume that $X \to Y$ is flat and surjective and that $X \to X \times _ Z X$ is flat. Then $Y \to Y \times _ Z Y$ is flat.
Proof. Consider the commutative diagram
\[ \xymatrix{ X \ar[r] \ar[d] & X \times _ Z X \ar[d] \\ Y \ar[r] & Y \times _ Z Y } \]
The top horizontal arrow is flat and the vertical arrows are flat. Hence $X$ is flat over $Y \times _ Z Y$. By Morphisms, Lemma 29.25.13 we see that $Y$ is flat over $Y \times _ Z Y$. $\square$
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