The Stacks project

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

  1. $Y' \to S'$ is locally of finite type,

  2. $X' \to S'$ is flat and locally of finite presentation,

  3. $f$ is flat, and

  4. $X = S \times _{S'} X'$ 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 $\mathcal{O}_{S', s'}$.

Proof. Immediate consequence of Algebra, Lemma 10.128.10. $\square$

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