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
X' is flat over S',
f is flat,
S \subset S' is a finite order thickening, and
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'.
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