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*}

A ring map is smooth if and only if it is smooth at all primes of the target

Lemma 10.135.13. Let $R \to S$ be a ring map. Then $R \to S$ is smooth if and only if $R \to S$ is smooth at every prime $\mathfrak q$ of $S$.

Proof. The direct implication is trivial. Suppose that $R \to S$ is smooth at every prime $\mathfrak q$ of $S$. Since $\mathop{\mathrm{Spec}}(S)$ is quasi-compact, see Lemma 10.16.10, there exists a finite covering $\mathop{\mathrm{Spec}}(S) = \bigcup D(g_ i)$ such that each $S_{g_ i}$ is smooth. By Lemma 10.22.3 this implies that $S$ is of finite presentation over $R$. According to Lemma 10.132.13 we see that $\mathop{N\! L}\nolimits _{S/R} \otimes _ S S_{g_ i}$ is quasi-isomorphic to a finite projective $S_{g_ i}$-module. By Lemma 10.77.2 this implies that $\mathop{N\! L}\nolimits _{S/R}$ is quasi-isomorphic to a finite projective $S$-module. $\square$


Comments (1)

Comment #3626 by Herman Rohrbach on

Suggested slogan: "A ring map is smooth if and only if it is smooth at all primes of the target."


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