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 10.43.3. Let $k$ be a field. Let $S$ be a $k$-algebra. The following are equivalent:

  1. $k' \otimes _ k S$ is reduced for every finite purely inseparable extension $k'$ of $k$,

  2. $k^{1/p} \otimes _ k S$ is reduced,

  3. $k^{perf} \otimes _ k S$ is reduced, where $k^{perf}$ is the perfect closure of $k$,

  4. $\overline{k} \otimes _ k S$ is reduced, where $\overline{k}$ is the algebraic closure of $k$, and

  5. $S$ is geometrically reduced over $k$.

Proof. Note that any finite purely inseparable extension $k \subset k'$ embeds in $k^{perf}$. Moreover, $k^{1/p}$ embeds into $k^{perf}$ which embeds into $\overline{k}$. Thus it is clear that (5) $\Rightarrow $ (4) $\Rightarrow $ (3) $\Rightarrow $ (2) and that (3) $\Rightarrow $ (1).

We prove that (1) $\Rightarrow $ (5). Assume $k' \otimes _ k S$ is reduced for every finite purely inseparable extension $k'$ of $k$. Let $k \subset K$ be an extension of fields. We have to show that $K \otimes _ k S$ is reduced. By Lemma 10.42.4 we reduce to the case where $k \subset K$ is a finitely generated field extension. Choose a diagram

\[ \xymatrix{ K \ar[r] & K' \\ k \ar[u] \ar[r] & k' \ar[u] } \]

as in Lemma 10.41.4. By assumption $k' \otimes _ k S$ is reduced. By Lemma 10.42.6 it follows that $K' \otimes _ k S$ is reduced. Hence we conclude that $K \otimes _ k S$ is reduced as desired.

Finally we prove that (2) $\Rightarrow $ (5). Assume $k^{1/p} \otimes _ k S$ is reduced. Then $S$ is reduced. Moreover, for each localization $S_{\mathfrak p}$ at a minimal prime $\mathfrak p$, the ring $k^{1/p}\otimes _ k S_{\mathfrak p}$ is a localization of $k^{1/p} \otimes _ k S$ hence is reduced. But $S_{\mathfrak p}$ is a field by Lemma 10.24.1, hence $S_{\mathfrak p}$ is geometrically reduced by Lemma 10.43.1. It follows from Lemma 10.42.7 that $S$ is geometrically reduced. $\square$


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