The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 28.32.19. 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 28.24.12 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 28.32.12 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 28.32.16 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 28.27.2 we see that $\dim _ y(Y_ s) = b - a$. Hence we conclude that $Y \to S$ is smooth at $y$ by Lemma 28.32.14 part (2). $\square$

[1] In fact this is implied by (1) and (2), see Descent, Lemma 34.11.3. Moreover, it suffices to assume $f$ is surjective, flat and locally of finite presentation, see Descent, Lemma 34.11.5.

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