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*}

10.140 Overview of results on smooth ring maps

Here is a list of results on smooth ring maps that we proved in the preceding sections. For more precise statements and definitions please consult the references given.

  1. A ring map $R \to S$ is smooth if it is of finite presentation and the naive cotangent complex of $S/R$ is quasi-isomorphic to a finite projective $S$-module in degree $0$, see Definition 10.135.1.

  2. If $S$ is smooth over $R$, then $\Omega _{S/R}$ is a finite projective $S$-module, see discussion following Definition 10.135.1.

  3. The property of being smooth is local on $S$, see Lemma 10.135.13.

  4. The property of being smooth is stable under base change, see Lemma 10.135.4.

  5. The property of being smooth is stable under composition, see Lemma 10.135.14.

  6. A smooth ring map is syntomic, in particular flat, see Lemma 10.135.10.

  7. A finitely presented, flat ring map with smooth fibre rings is smooth, see Lemma 10.135.16.

  8. A finitely presented ring map $R \to S$ is smooth if and only if it is formally smooth, see Proposition 10.136.13.

  9. If $R \to S$ is a finite type ring map with $R$ Noetherian then to check that $R \to S$ is smooth it suffices to check the lifting property of formal smoothness along small extensions of Artinian local rings, see Lemma 10.139.2.

  10. A smooth ring map $R \to S$ is the base change of a smooth ring map $R_0 \to S_0$ with $R_0$ of finite type over $\mathbf{Z}$, see Lemma 10.136.14.

  11. Formation of the set of points where a ring map is smooth commutes with flat base change, see Lemma 10.135.17.

  12. If $S$ is of finite type over an algebraically closed field $k$, and $\mathfrak m \subset S$ a maximal ideal, then the following are equivalent

    1. $S$ is smooth over $k$ in a neighbourhood of $\mathfrak m$,

    2. $S_{\mathfrak m}$ is a regular local ring,

    3. $\dim (S_{\mathfrak m}) = \dim _{\kappa (m)} \Omega _{S/k} \otimes _ S \kappa (\mathfrak m)$.

    see Lemma 10.138.2.

  13. If $S$ is of finite type over a field $k$, and $\mathfrak q \subset S$ a prime ideal, then the following are equivalent

    1. $S$ is smooth over $k$ in a neighbourhood of $\mathfrak q$,

    2. $\dim _{\mathfrak q}(S/k) = \dim _{\kappa (\mathfrak q)} \Omega _{S/k} \otimes _ S \kappa (\mathfrak q)$.

    see Lemma 10.138.3.

  14. If $S$ is smooth over a field, then all its local rings are regular, see Lemma 10.138.3.

  15. If $S$ is of finite type over a field $k$, $\mathfrak q \subset S$ a prime ideal, the field extension $k \subset \kappa (\mathfrak q)$ is separable and $S_{\mathfrak q}$ is regular, then $S$ is smooth over $k$ at $\mathfrak q$, see Lemma 10.138.5.

  16. If $S$ is of finite type over a field $k$, if $k$ has characteristic $0$, if $\mathfrak q \subset S$ a prime ideal, and if $\Omega _{S/k, \mathfrak q}$ is free, then $S$ is smooth over $k$ at $\mathfrak q$, see Lemma 10.138.7.

Some of these results were proved using the notion of a standard smooth ring map, see Definition 10.135.6. This is the analogue of what a relative global complete intersection map is for the case of syntomic morphisms. It is also the easiest way to make examples.


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