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*}

Geometric regularity descends through faithfully flat maps of algebras

Lemma 10.160.3. Let $k$ be a field. Let $A \to B$ be a faithfully flat $k$-algebra map. If $B$ is geometrically regular over $k$, so is $A$.

Proof. Assume $B$ is geometrically regular over $k$. Let $k \subset k'$ be a finite, purely inseparable extension. Then $A \otimes _ k k' \to B \otimes _ k k'$ is faithfully flat as a base change of $A \to B$ (by Lemmas 10.29.3 and 10.38.7) and $B \otimes _ k k'$ is regular by our assumption on $B$ over $k$. Then $A \otimes _ k k'$ is regular by Lemma 10.158.4. $\square$


Comments (1)

Comment #2110 by Matthew Emerton on

Suggested slogan: Geometric regularity descends through faithfully flat maps of algebras


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