Lemma 37.10.2 (06AF)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.10: Infinitesimal deformations of schemes
Lemma 37.10.2 (cite)

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 $f'$.

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

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