Lemma 76.18.2 (0CG5)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.18: Infinitesimal deformations of algebraic spaces
Lemma 76.18.2 (cite)

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Lemma 76.18.2. Let $S$ be a scheme. Consider a commutative diagram

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

of thickenings of algebraic spaces over $S$ . Assume

$X'$ is flat over $B'$ ,
$f$ is flat,
$B \subset B'$ is a finite order thickening, and
$X = B \times _{B'} X'$ and $Y = B \times _{B'} Y'$ .

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

Proof. Choose a scheme $U'$ and a surjective étale morphism $U' \to B'$ . Choose a scheme $V'$ and a surjective étale morphism $V' \to U' \times _{B'} Y'$ . Choose a scheme $W'$ and a surjective étale morphism $W' \to V' \times _{Y'} X'$ . Let $U, V, W$ be the base change of $U', V', W'$ by $B \to B'$ . Then flatness of $f'$ is equivalent to flatness of $W' \to V'$ and we are given that $W \to V$ is flat. Hence we may apply the lemma in the case of schemes to the diagram

$\xymatrix{ (W \subset W') \ar[rr] \ar[rd] & & (V \subset V') \ar[ld] \\ & (U \subset U') }$

of thickenings of schemes. See More on Morphisms, Lemma 37.10.2. The statement about flatness of $Y'/B'$ at points in the image of $f'$ follows in the same manner. $\square$

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