Lemma 37.10.4. 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
$Y' \to S'$ is locally of finite type,
$X' \to S'$ is flat and locally of finite presentation,
$f$ is flat, and
$X = S \times _{S'} X'$ and $Y = S \times _{S'} Y'$.
Then $f'$ is flat and for all $y' \in Y'$ in the image of $f'$ the local ring $\mathcal{O}_{Y', y'}$ is flat and essentially of finite presentation over $\mathcal{O}_{S', s'}$.
Comments (0)