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 10.45.8. Let $\varphi : R \to S$ be a ring map. Assume

  1. $\varphi $ induces an injective map of spectra,

  2. $\varphi $ induces purely inseparable residue field extensions.

Then for any ring map $R \to R'$ properties (1) and (2) are true for $R' \to R' \otimes _ R S$.

Proof. Set $S' = R' \otimes _ R S$ so that we have a commutative diagram of continuous maps of spectra of rings

\[ \xymatrix{ \mathop{\mathrm{Spec}}(S') \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(S) \ar[d] \\ \mathop{\mathrm{Spec}}(R') \ar[r] & \mathop{\mathrm{Spec}}(R) } \]

Let $\mathfrak p' \subset R'$ be a prime ideal lying over $\mathfrak p \subset R$. If there is no prime ideal of $S$ lying over $\mathfrak p$, then there is no prime ideal of $S'$ lying over $\mathfrak p'$. Otherwise, by Remark 10.16.8 there is a unique prime ideal $\mathfrak r$ of $F = S \otimes _ R \kappa (\mathfrak p)$ whose residue field is purely inseparable over $\kappa (\mathfrak p)$. Consider the ring maps

\[ \kappa (\mathfrak p) \to F \to \kappa (\mathfrak r) \]

By Lemma 10.24.1 the ideal $\mathfrak r \subset F$ is locally nilpotent, hence we may apply Lemma 10.45.1 to the ring map $F \to \kappa (\mathfrak r)$. We may apply Lemma 10.45.7 to the ring map $\kappa (\mathfrak p) \to \kappa (\mathfrak r)$. Hence the composition and the second arrow in the maps

\[ \kappa (\mathfrak p') \to \kappa (\mathfrak p') \otimes _{\kappa (\mathfrak p)} F \to \kappa (\mathfrak p') \otimes _{\kappa (\mathfrak p)} \kappa (\mathfrak r) \]

induces bijections on spectra and purely inseparable residue field extensions. This implies the same thing for the first map. Since

\[ \kappa (\mathfrak p') \otimes _{\kappa (\mathfrak p)} F = \kappa (\mathfrak p') \otimes _{\kappa (\mathfrak p)} \kappa (\mathfrak p) \otimes _ R S = \kappa (\mathfrak p') \otimes _ R S = \kappa (\mathfrak p') \otimes _{R'} R' \otimes _ R S \]

we conclude by the discussion in Remark 10.16.8. $\square$


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