Lemma 37.10.2. Consider a commutative diagram

$\xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (S \subset S') }$

of thickenings. Assume

1. $X'$ is flat over $S'$,

2. $f$ is flat,

3. $S \subset S'$ is a finite order thickening, and

4. $X = S \times _{S'} X'$ and $Y = S \times _{S'} Y'$.

Then $f'$ is flat and $Y'$ is flat over $S'$ at all points in the image of $f'$.

Proof. Immediate consequence of Algebra, Lemma 10.101.8. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).