\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*}

The Stacks project

Lemma 10.150.9. Let $R \to S$ be a ring map. Let $\mathfrak q$ be a prime of $S$ lying over $\mathfrak p$ in $R$. Assume $R \to S$ is quasi-finite at $\mathfrak q$. The commutative diagram

\[ \xymatrix{ R_{\mathfrak p}^ h \ar[r] & S_{\mathfrak q}^ h \\ R_{\mathfrak p} \ar[u] \ar[r] & S_{\mathfrak q} \ar[u] } \]

of Lemma 10.150.6 identifies $S_{\mathfrak q}^ h$ with the localization of $R_{\mathfrak p}^ h \otimes _{R_{\mathfrak p}} S_{\mathfrak q}$ at the prime generated by $\mathfrak q$.

Proof. Note that $R_{\mathfrak p}^ h \otimes _ R S$ is quasi-finite over $R_{\mathfrak p}^ h$ at the prime ideal corresponding to $\mathfrak q$, see Lemma 10.121.6. Hence the localization $S'$ of $R_{\mathfrak p}^ h \otimes _{R_{\mathfrak p}} S_{\mathfrak q}$ is henselian, see Lemma 10.148.4. As a localization $S'$ is a filtered colimit of ├ętale $R_{\mathfrak p}^ h \otimes _{R_{\mathfrak p}} S_{\mathfrak q}$-algebras. By Lemma 10.150.8 we see that $S_\mathfrak q^ h$ is the henselization of $R_{\mathfrak p}^ h \otimes _{R_{\mathfrak p}} S_{\mathfrak q}$. Thus $S' = S_\mathfrak q^ h$ by the uniqueness result of Lemma 10.149.6. $\square$


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