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.

Comments (0)

There are also:

  • 2 comment(s) on Section 28.32: Smooth morphisms

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02K5. Beware of the difference between the letter 'O' and the digit '0'.