The Stacks project

Lemma 29.32.19. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & S } \]

be a commutative diagram of morphisms of schemes. Assume that

  1. $f$ is surjective, and smooth,

  2. $p$ is smooth, and

  3. $q$ is locally of finite presentation1.

Then $q$ is smooth.

Proof. By Lemma 29.24.13 we see that $q$ is flat. Pick a point $y \in Y$. Pick a point $x \in X$ mapping to $y$. Suppose $f$ has relative dimension $a$ at $x$ and $p$ has relative dimension $b$ at $x$. By Lemma 29.32.12 this means that $\Omega _{X/S, x}$ is free of rank $b$ and $\Omega _{X/Y, x}$ is free of rank $a$. By the short exact sequence of Lemma 29.32.16 this means that $(f^*\Omega _{Y/S})_ x$ is free of rank $b - a$. By Nakayama's Lemma this implies that $\Omega _{Y/S, y}$ can be generated by $b - a$ elements. Also, by Lemma 29.27.2 we see that $\dim _ y(Y_ s) = b - a$. Hence we conclude that $Y \to S$ is smooth at $y$ by Lemma 29.32.14 part (2). $\square$

[1] In fact this is implied by (1) and (2), see Descent, Lemma 35.11.3. Moreover, it suffices to assume $f$ is surjective, flat and locally of finite presentation, see Descent, Lemma 35.11.5.

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