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\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 10.150.13. Let $R$ be a ring. Let $\mathfrak p \subset R$ be a prime ideal. Let $\kappa (\mathfrak p) \subset \kappa ^{sep}$ be a separable algebraic closure. Consider the category of triples $(S, \mathfrak q, \phi )$ where $R \to S$ is étale, $\mathfrak q$ is a prime lying over $\mathfrak p$, and $\phi : \kappa (\mathfrak q) \to \kappa ^{sep}$ is a $\kappa (\mathfrak p)$-algebra map. This category is filtered and

\[ (R_{\mathfrak p})^{sh} = \mathop{\mathrm{colim}}\nolimits _{(S, \mathfrak q, \phi )} S = \mathop{\mathrm{colim}}\nolimits _{(S, \mathfrak q, \phi )} S_{\mathfrak q} \]

canonically.

Proof. A morphism of triples $(S, \mathfrak q, \phi ) \to (S', \mathfrak q', \phi ')$ is given by an $R$-algebra map $\varphi : S \to S'$ such that $\varphi ^{-1}(\mathfrak q') = \mathfrak q$ and such that $\phi ' \circ \varphi = \phi $. Let us show that the category of pairs is filtered, see Categories, Definition 4.19.1. The category contains the triple $(R, \mathfrak p, \kappa (\mathfrak p) \subset \kappa ^{sep})$ and hence is not empty, which proves part (1) of Categories, Definition 4.19.1. Suppose that $(S, \mathfrak q, \phi )$ and $(S', \mathfrak q', \phi ')$ are two triples. Note that $\mathfrak q$, resp. $\mathfrak q'$ correspond to primes of the fibre rings $S \otimes \kappa (\mathfrak p)$, resp. $S' \otimes \kappa (\mathfrak p)$ with residue fields finite separable over $\kappa (\mathfrak p)$ and $\phi $, resp. $\phi '$ correspond to maps into $\kappa ^{sep}$. Hence this data corresponds to $\kappa (\mathfrak p)$-algebra maps

\[ \phi : S \otimes _ R \kappa (\mathfrak p) \longrightarrow \kappa ^{sep}, \quad \phi ' : S' \otimes _ R \kappa (\mathfrak p) \longrightarrow \kappa ^{sep}. \]

Set $S'' = S \otimes _ R S'$. Combining the maps the above we get a unique $\kappa (\mathfrak p)$-algebra map

\[ \phi '' = \phi \otimes \phi ' : S'' \otimes _ R \kappa (\mathfrak p) \longrightarrow \kappa ^{sep} \]

whose kernel corresponds to a prime $\mathfrak q'' \subset S''$ lying over $\mathfrak q$ and over $\mathfrak q'$, and whose residue field maps via $\phi ''$ to the compositum of $\phi (\kappa (\mathfrak q))$ and $\phi '(\kappa (\mathfrak q'))$ in $\kappa ^{sep}$. The ring map $R \to S''$ is étale by Lemma 10.141.3. Hence $(S'', \mathfrak q'', \phi '')$ is a triple dominating both $(S, \mathfrak q, \phi )$ and $(S', \mathfrak q', \phi ')$. This proves part (2) of Categories, Definition 4.19.1. Next, suppose that $\varphi , \psi : (S, \mathfrak q, \phi ) \to (S', \mathfrak q', \phi ')$ are two morphisms of pairs. Then $\varphi $, $\psi $, and $S' \otimes _ R S' \to S'$ are étale ring maps by Lemma 10.141.8. Consider

\[ S'' = (S' \otimes _{\varphi , S, \psi } S') \otimes _{S' \otimes _ R S'} S' \]

Arguing as above (base change of étale maps is étale, composition of étale maps is étale) we see that $S''$ is étale over $R$. The fibre ring of $S''$ over $\mathfrak p$ is

\[ F'' = (F' \otimes _{\varphi , F, \psi } F') \otimes _{F' \otimes _{\kappa (\mathfrak p)} F'} F' \]

where $F', F$ are the fibre rings of $S'$ and $S$. Since $\varphi $ and $\psi $ are morphisms of triples the map $\phi ' : F' \to \kappa ^{sep}$ extends to a map $\phi '' : F'' \to \kappa ^{sep}$ which in turn corresponds to a prime ideal $\mathfrak q'' \subset S''$. The canonical map $S' \to S''$ (using the right most factor for example) is a morphism of triples $(S', \mathfrak q', \phi ') \to (S'', \mathfrak q'', \phi '')$ which equalizes $\varphi $ and $\psi $. This proves part (3) of Categories, Definition 4.19.1. Hence we conclude that the category is filtered.

We still have to show that the colimit $R_{colim}$ of the system is equal to the strict henselization of $R_{\mathfrak p}$ with respect to $\kappa ^{sep}$. To see this note that the system of triples $(S, \mathfrak q, \phi )$ contains as a subsystem the pairs $(S, \mathfrak q)$ of Lemma 10.150.7. Hence $R_{colim}$ contains $R_{\mathfrak p}^ h$ by the result of that lemma. Moreover, it is clear that $R_{\mathfrak p}^ h \subset R_{colim}$ is a directed colimit of étale ring extensions. It follows that $R_{colim}$ is henselian by Lemmas 10.148.4 and 10.149.7. Finally, by Lemma 10.141.15 we see that the residue field of $R_{colim}$ is equal to $\kappa ^{sep}$. Hence we conclude that $R_{colim}$ is strictly henselian and hence equals the strict henselization of $R_{\mathfrak p}$ as desired. Some details omitted. $\square$


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